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ROUND 12

Students should not meet real standards for the first time at 18

Some students don’t even meet a real math class until college. And I’m not talking about the subject matter or how hard the math is. What I’m talking about is Before that, everything had cushions. There were corrections, Extra credit, the review sheet looks just like the test You get loads of partial credit for almost every single line of work you write down Even if you have no idea what you’re doing. You can pass a class if you just put your name on the assignment and upload it to Canvas with some work written down for each problem, even if it’s not right at all. And then they go to college and they get into a course where the professor is a little more, you know, some might say a hard ass, but I would just say more honest, actually assigning grades on the basis of whether you have learned the skills or not. There’s no extra credit. There’s no review sheets that tell you exactly what’s gonna be on the test. You don’t get partial credit just for showing up and trying You get it for results, for making real successful progress. And now the student thinks that, you know, because of all that, that college math is just impossible Can’t do it. Expectations are too high. And maybe it’s challenging. I mean, I get it. The subject matter can be challenging, but often the overwhelm that they’re feeling is just that all this rigor and accountability arrived all at once. And maybe it’s just me But I don’t think students should be meeting real standards for the first time at age 18.

Biology is math

Let’s say you’re an aspiring biologist, and you finished calculus in high school. Do you stop taking math in college? No! No, no, no Why not? Because biology is now a quantitative field. And the people who can do the coolest things in biology and have the biggest impact? Well, they have both domain expertise and serious math and coding chops. Almost every aspect of biology now involves reading DNA The genetic code and analyzing millions, if not billions of letters. And if you wanna do this, you’re gonna need probability and statistics Linear algebra, coding… even differential equations become necessary if you want to model things like population growth, disease spread, how body structures form during development. See, that’s, that’s the thing is that modern biology is not just about microscopes and test tubes. It’s also math and data and computation.

Just tell them the tool

I don’t need a seventh grader to rediscover the distributive property from scratch. It’s just, it’s not a good use of time. You know, just, just give them the idea. Show them why it works, have them use it, practice it until it’s reliable. I mean, the whole point of instruction The whole reason why teaching exists is because humans have already figured out a lot of knowledge, right? We already know a lot of math from centuries of geniuses grinding away at it. So the whole point is kind of like, hey, let me show you how this works So you don’t have to waste a whole bunch of time rediscovering it So you don’t have to make every mistake, go down every dead-end road just to finally get to the techniques that actually work. There’s a lot of knowledge to learn, and the edge of the field, in any field really, is a lot further than it used to be. So if you wanna get there and push it further, you’re gonna have to get there faster than your predecessors did so that you actually have enough time to push the edge over the course of your career.

You cannot be creative at a high level unless you are robotic at a low level

People love to talk about creativity as if it floats above skill. But the reality is creativity needs raw material to work with. It needs patterns, it needs facts, it needs examples, rules, exceptions, and procedures and all this has to be loaded into your long-term memory in order for you to think creatively with it. Cause you can’t think with knowledge that you don’t have. You can’t cook with ingredients that you don’t have.

Repetition is not the enemy of creativity

When basic skills are not automatic, They just hijack every thought in your brain. You might think you’re trying to be strategic, but really you’re still thinking about how all the pieces move. It’s like trying to run basketball plays when you’re still looking down all the time at your dribbling, ‘cause if you don’t, you’ll lose the ball. You gotta automate those basics, get really solid so that they don’t consume all of your thinking so that you have the mental bandwidth to actually move beyond them ‘cause they’re so solid that you can start layering more advanced skills and strategies and modes of thought on top of it. And that’s why the grunt work matters. Repetition is not the enemy of creativity. It is a compression process that frees up all your mental bandwidth for creativity. That’s how you get the creativity. If you master the basics until they become unconscious, then your brain finally has room to start operating at those higher levels.

Weak math ruins your day

When a student struggles during a physics problem, Half the time it’s not even the physics that is the source of the struggle. You know, half the time it’s the algebra or the math, the calculus, whatever, inside of the problem. Like they understand that the cart is speeding up They can describe it in words. They can even draw a little motion diagram. That’s not the problem. The problem is moving mathematical symbols around. That’s the thing that comes back to haunt them over and over Cause it’s no longer just, you know, imagine a scenario, what happens It’s been abstracted into these mathematical symbols. They’re less concrete. And it’s like well, what, what do you do? You gotta know your algebra. You gotta know your math. Gotta know your calculus. Whatever level of math that you’re building upon, you gotta know it! Cause weak math does not just stay in math class. It also walks into physics class, it walks into chemistry class, into economics class and just ruins your day. Follows you around to every class, ruining your day, Haunting you, sabotaging your every move into eternity.

You can't escape math

Weak math does not just stay in math class. It also walks into physics class, it walks into chemistry class, into economics class and just ruins your day. And if you decide, like, “Hey, I don’t wanna go into science, engineering, I wanna go into business,” guess what business is run on? Money. Guess what money runs on? Math Guess what happens if you mess up your money ‘cause you can’t do your math? You screw up your life! So yeah, math is this tool that you gotta learn because pretty much everywhere you go, if you get serious enough about it, it’s gonna involve math at some point.

Math is everywhere

Math gets used in so many different subjects from physics to chemistry, economics, finance, engineering… and if you’re trying to learn one of those subjects and you don’t know your prerequisite math skills, well, guess what? You not only have to learn that subject, but you also have to learn your math. You’re gonna be struggling with two difficult things at the same time And it’s gonna be overwhelming, and that’s what makes it so hard. That is generally why a lot of these quantitative subjects are seen as really hard in college it’s the math And it’s often not so much that the math itself is super hard, but it’s rather there’s a bunch of math that is required that you don’t know. You have foundational skill gaps stretching back years of material that you never really learned well coming in And you have to fill in all these knowledge gaps just to grasp what it is that you’re supposed to be learning in this math-heavy course. But if you know your math, If your prerequisite skills are solid, it’s like You can go into any field you want. You don’t have to worry about like, “Oh, this class is really hard cause it takes a whole bunch of math that I never really learned that I’m gonna have to spin up on. It’s gonna move at a fast pace. I don’t know if I can keep up with it. No, you don’t have any of those thoughts if you actually know your math skills. All these constraints removed. You can go into whatever kind of fields you want. It’s up to your taste, what you find interesting And that pays off big in the long run if you get to choose a career field that you actually find interesting.

A lot of homework isn't even graded for accuracy

A lot of teachers just don’t even grade homework anymore. I mean, I remember lots of students that I tutor basically, the story was They’d get their assignment, they’d start working on it And there’d be parts where they’re not really sure what to do, but they’d just write something down, just YOLO something Because they know it doesn’t actually matter. They just gotta get some words on the page, even if it’s wrong. And I’d be like, “But hold up, that’s, that’s not quite right. There’s some issues here that we probably should go over.” And then I’d realize in the process of going over these issues with them Their problem is not just that they’re struggling with this material that was assigned. They have knowledge gaps dating back weeks, months, years. They don’t know how to do the component skills cause those things were never graded for accuracy. They just get in the habit of writing something down, however wrong it is. Uh, it, it doesn’t matter. You just upload something that looks like you made an attempt that you tried So that’s the game, is just can you write something on the paper and upload it to Canvas? That is the bar. So low. Not whether it’s correct or not, just did you upload it? So now we kind of got this shit show where a lot of students are lacking foundational skills in large part because their assignments were never checked. If you’re not checking whether those foundational skills are there, don’t be surprised when it turns out they’re not.

ROUND 11

How to turn math into a superpower

Being really good at math, it’s kinda weird. Cause by itself, math is kinda underpowered. But when you combine it with some other area of expertise, it’s so overpowered Just ridiculously overpowered See, here’s the thing Most of the world does not care about pure math. But if you bring it as a power up to something else like coding, science, engineering, finance… or really any other deep domain obsession. Those serious math chops become a gigantic force multiplier. And that’s why learning math early is so valuable. The earlier those tools become automatic The more time you have to explore and find a field that you really care about and go deep. And that’s the thing, the goal is not to collect intellectual trophies. It is to become so technically skilled That you can solve real problems in ways that other people can’t even see.

Math trauma isn't really about math

Math trauma it really tends to be less about math itself and more about somebody asking you to perform advanced maneuvers before you’ve even mastered the basics. Imagine this. Imagine going to skating lessons ice skating lessons and your coach tells you to try skating backwards on one leg, and you can barely even skate forwards on two legs. But that’s what coach is telling you to do. So you try it and you fall, you get up, you try again, you fall again And I mean, deep down you know that this is hopeless ‘cause you can’t, you can’t even skate forwards. Like, how are you gonna skate backwards on one leg?! You can’t even skate forwards on two legs But your coach just keeps on telling you like, “Try harder, try harder.” And you’re just like, “I c- can’t do it.” And you’re just like crashing out ‘cause there’s this, you just know that there’s this huge prerequisite gap Between what you’re actually able to do and where your coach expects you to be. That’s the frustrating part when the coach is not recognizing that there’s this massive gap and helping you bridge that gap. They’re expecting you to just jump to the highest rung of the ladder in one go. And if you can’t do it, then it’s just ‘cause you’re lazy or you gotta try harder. Just try again, even though it’s just hopeless. That’s really what Math trauma is about. I mean, whether it’s math, skating, swimming, whatever That kind of experience would be traumatic for anybody.

The technical bar is skyrocketing

If you are a student who’s interested in tech, You need to be aware that you gotta graduate with serious technical skills. See, you used to be able to just learn a few basics, Make a few toy projects, And then get serious and skill up on the job. But those days are over The bar, the standard is just so much higher nowadays. So you gotta be able to hit the ground running. You have to be a net positive very early. You have to be more capable than an LLM. And that requires years of serious upskilling that most schools, even most top colleges, are not delivering on See, a degree is no longer proof of skill. An A in a course is no longer proof of skill. The bar for joining that technical workforce, it is skyrocketing And the bar for graduating school is cratering. That gap needs to close, and as of now, you are the only one who can close it for yourself.

The status quo isn't good enough

We can’t help people who don’t think that they need help. And the reality is that most students aren’t even at grade level in math. And a lot of schools have decided that they can just ignore the problem, that it’s okay if students are not prepared for the next grade Or that it’s an unsolvable problem Or that everybody has the same problem, so we don’t need to fix it. I mean, hey, if you’re fine with the results, then sure, keep doing what you’re doing. But If most kids reach high school unable to do basic algebra, then maybe the status quo isn’t good enough.

The limits of classroom teaching

Even a great teacher can’t easily implement mastery learning in a classroom. You know, where students only move on to a next topic after mastering its prerequisites. See, here’s the problem You’re a public school teacher. You got 4 classes, each with 20 to 30 kids That’s like 100 students that you’re having to manage all in parallel. And one single teacher can’t keep track of every single topic every kid knows or they don’t know Or just how well they learn it and what should they learn next It’s, it’s, it’s just too much work. And unless the teacher has some kind of technology to help them do that work, to manage the knowledge profiles and learning schedules of all these different students Unless they have that, the teacher just has to move the whole class at the same pace. And that means some kids just sit there bored waiting for everybody else to catch up while other kids are rushed through and they gotta move on before they’re actually ready. And because math is so brutally hierarchical, They just fall further and further behind with every new topic.

When homework isn't a good signal

A lot of homework in schools today is graded for completion, not accuracy. Did you turn something in? Did you upload it to Canvas? Did you check a box? But that’s, that’s useless as far as actual learning. It doesn’t tell you, like, does the student actually understand the concept? Did they actually do the problems correctly? Did they even do it themselves? Did they just outsource it to AI or Their older sibling, their smart friend…? Who knows? See, homework is supposed to be practice. And practice only helps if somebody is actually checking whether you’re practicing correctly and giving you feedback on the things that you gotta tweak so that you can improve on them. Otherwise, you can spend a whole hour doing homework and reinforce the exact mistakes that you were supposed to fix.

Learn faster, don't aim lower

Middle schoolers can learn all of high school math Yeah, algebra 1 all the way through calculus in just 3 years if they have the right setup. Now, I know that might seem kind of crazy, but here’s how to think about it. Think about driving 100 miles. In back roads, that might take like five hours. But on a highway, that only takes 2, maybe 1 and a half. See, the distance didn’t change, but the conditions did. You can go fast when the road is paved and there aren’t tons of bumps and potholes. And it’s the same thing with learning. The whole idea is you don’t have to bring the destination closer. You don’t have to decrease the distance from algebra to calculus. If you just improve the learning conditions: Make sure students know the prerequisites for each new thing you’re trying to get them to learn, scaffold each new thing so that they can learn it smoothly instead of constantly staring off into space, racking their brains about what the heck is going on. That is how you get students there faster.

The potholes of learning

The potholes of learning are when you find out that you have a knowledge gap that’s blocking you from learning the thing that you’re trying to learn. You try to learn this new thing and you’re like, “Oh, wait, this doesn’t make any sense. Why does it make sense?” And you have to kind of backtrack and figure out what’s the bottom of your foundation where things first are not really making sense? Then you gotta build all that up again. And if you are continually having to do that on every single topic that you learn, you waste a lot of time just figuring out what the heck is it that is confusing me that I have to learn before I learn this new thing? And often it’s really hard to figure that stuff out Cause you don’t have a great mental model of the subject. I mean, if you did, you’d already be an expert who knows a lot about it. And so that’s why it takes an expert to kinda help you chart a course that avoids all these potholes in your learning journey, so you can just step on the gas and go fast.

Learning debt compounds

Learning debt. It starts when adults stop correcting all these workarounds, these hacks to get by. Like, like you always use your forehand instead of your backhand, so You never develop your backhand. That’s a learning debt, and so is when you let a student keep on recalculating their multiplication tables Instead of actually memorizing them. Or you let them use a calculator instead of doing some simple mental arithmetic. I mean, I get it. Adults don’t want to nag, right? They don’t want conflict So they let these workarounds slide. But that skill gap, it, it doesn’t disappear, it just hides. And it compounds. And eventually it slows down progress. Students hit these ceilings that never even needed to exist. But the skill gap grows so big and so ingrained that everyone just assumes that it reflects innate ability or lack thereof. And if you scale that across millions of students, Then eventually you get a system That leaves millions of students without the skills to succeed.

Stop grading attendance

Attendance should not be a part of a student’s grade. I mean, your grade should answer one question, right? How well do you know the material? That’s it so if a student skips every lecture, teaches themselves from the textbook, and then aces the exam, they know the material, so give them an A. And if another student has perfect attendance, takes beautiful notes, and still can’t solve the problems, then they don’t know the material. So they shouldn’t get a grade that suggests that they do. So stop grading proxies for learning And don’t pretend that sitting in a chair is evidence of mastery.

When a workaround becomes a trap

Some alternative learning methods feel easier because they avoid the actual skill. I mean, take lattice multiplication, for example Kids often like it because it’s really pretty and visual and it’s fun to draw But it’s also slow And if they get attached to it, They just stop practicing the more efficient standard method. And if you let this continue, then eventually the alternative method becomes a workaround and the workaround becomes a trap, a crutch. The homework takes longer, More advanced problems get harder, And the student has to go back and relearn that method that they should have been practicing all along. I mean, it’s inefficient and leads to all sorts of frustration.

Projects can help or harm

Projects can make math really fun, but only if the students actually have the prerequisites in place to do them. Cause if they don’t, if they don’t know the basics and you’re asking them to do this advanced project, then it is an awful experience. They’re sitting there confused like, “What am I even supposed to be doing?” How do I even start? So they just look over and copy what the smart kid does, right? And even if the project is something that they find intrinsically interesting, If they can’t actually do it, then they’re just being set up to fail. And that sends the exact wrong message. It sends the message, “I’m not cut out for this. There’s this thing that I think is really cool, intrinsically interesting, and I’m not cut out for it.”

Division of labor is division of learning

In a lot of classroom group projects, The strongest student does everything. They carry their entire group, and the weaker students just don’t really learn anything. And that’s not just some minor classroom inconvenience. That, uh, this is a structural failure mode of learning, of education. From the outside, it can look pretty collaborative, right? Makes for a good photo op, everyone’s discussing, engaged… or so it looks. But the thing is engagement does not imply learning. Learning requires direct contact with the work, pushing it forward yourself in a productive way. You gotta attempt to retrieve things that you’ve learned, do so successfully, Make a few mistakes, correct them, repeat again All right, I did it right this time. Now you’ve learned And somebody else cannot do that process for you. You don’t learn by watching somebody else do that process. You gotta do it yourself. So division of labor in a group project is, it’s ultimately division of learning. And concentrating that labor on one student, the smart kid who’s carrying their whole group, doing all the work for ‘em, That is a concentration of learning on that one student.

The anti-memorization movement

The anti- memorization movement has left millions of students unable to think because every little operation consumes their working memory. For instance, solving equations. It’s really smooth when basic arithmetic is automatic. It’s like moving puzzle pieces around. You just gotta figure out how they fit together. But if you are not automatic on your basic arithmetic, if you don’t know your times tables, if you’re always having to recalculate all that from scratch, then every puzzle piece that you try to move around when solving these equations, it’s like a heavy weight. It’s really hard to lift at all. So you can’t really move it around much, much less figure out where it should go. And it’s the same way all the way up that ladder of math from arithmetic to algebra to calculus to all sorts of university level math and beyond.

Taking notes isn't learning

You think that taking detailed notes in class means you’re actually learning? Think again See, learning doesn’t happen when you listen to a teacher and just copy down what they’re saying or just nod along and look studious. No, learning happens when you pull information from memory. That is what strengthens your knowledge: Retrieving information from your brain and bringing it out to the outside. And I mean, yeah, it’s gotta get inside your brain first, but if you just let it sit there without trying to exercise retrieving it from your brain to, to bring it out, then it’s just gonna kinda decay and dissipate and you’re just gonna forget it. It’s use it or lose it, right? Everybody knows that, but few people want to actually put it into practice. It’s the solving problems. That is where the learning happens.

Learning is memory

Everyone tells you not to memorize, but that’s backwards. At the end of the day, all learning, it comes down to memory. That’s what it is. That is the substrate of learning. Even deep understanding is memory. It’s deep ingrained memory. See, the difference between memorizing versus understanding is, it’s not about whether you use memory. It’s about how much memory you have stored and how well it’s connected. Facts connect onto concepts, procedures become skills, And all sorts of new ideas come from combining things that you already know. You can’t connect the dots if you got no dots. You can’t cook with ingredients that you don’t have. So you see, memory is not the enemy of understanding it’s the foundation of it.

Learn math 4x faster

We increase math learning 4 times over a traditional classroom without even increasing the amount of time that students spend doing math. I know that sounds crazy, but the way you do it is you increase the learning efficiency. Have students work on exactly what they are ready to learn at every second. And they practice it until they master it. None of these false starts where a student kinda gets ahead of their skis cause they made a bunch of mistakes in the homework cause they don’t really know what’s going on, and then a week later they’re just totally lost and you have to rebuild their foundation. No, just, just make them master it to begin with, and then you don’t have to backtrack all the time. It’s just smooth forwards progress. And even after they establish some, some baseline mastery, you continually have them review it right before they’re about to forget it. That way they maintain their knowledge and you don’t look up a month later and ask them to build on foundations that have just decayed. That is how you do it. No wasted time on material that’s, that’s too easy or too hard. You meet the student where they’re at, and you fit their knowledge profile like a glove. And then you build upwards from there. Same amount of time, 4 times learning.

Silly mistakes are still mistakes

No, you did not make a silly mistake. You made a mistake. You dropped the negative sign. You forgot to carry. You didn’t flip the fraction before. Now, those mistakes might seem silly because you know how to avoid them But knowing how is not the same thing as actually doing it reliably. And if you keep making silly mistakes, then it means the skill is not actually automatic yet. And the solution isn’t to just shrug them off Nah, it’s more practice until getting it right becomes the default.

This will haunt you in math

This is the ghost of math that will come back to haunt you forever. It’s math facts, particularly times tables, multiplication tables. If you don’t memorize those early, then everything afterward, it gets much, much harder than it really needs to be. I mean, imagine solving a whole algebra problem while you’re spending a bunch of time trying to figure out, like, what is 9 times 6? That’s only one little part of the problem, and you got a whole bunch of other multiplications to figure out. So now every algebra problem is really an arithmetic problem, a handful of arithmetic problems, and you just can’t think about the algebra that’s going on ‘cause you’re so focused on these arithmetic problems ‘cause it’s a full brain problem ‘cause you never memorized your math facts! Every time one of those low-level things comes up, every time a multiplication fact comes up, you gotta stop, turn off your attention to everything that you were doing, and then focus on this one multiplication fact, work it out, and then pick your head up and be like, “What was I doing? Where am I? Oh, right, we’re doing this problem. This is where we left off. What are we trying to do?” It’s just totally disorienting And you can’t build fluency that way.

The wrong way to study

Here’s why you can study for hours and remember almost nothing: Thing is, a lot of students study like this. They study by reread the chapter, Highlight your notes, watch the lecture again… I mean, everything feels familiar, right? So you feel like you know it. But then you close the book, you’re trying to explain it back to yourself, And you’re just at a loss for words. Like nothing’s coming out. It’s like, it, it’s almost as like your, your mind got wiped And if that’s your study technique, then that’s a totally normal experience because that’s what happens when you’re just consuming information. You feel like you got a lot of it stored up there. You’re filling your brain with it, and then pff, just gets wiped You barely remember anything And the solution to this problem: If you want something to really stick in your brain, you gotta practice retrieving it. You gotta solve problems from memory. And if you really can’t, you can look back, and then close the book again and then try again, Just look back for a little hint, then close the book, go back to the problem, keep pushing it as far along as you can. Don’t just keep passing information over your brain. You gotta practice pulling it back out.

So you wanna be an ML engineer

So you wanna be a machine learning engineer Well, here’s just some of the math you’re gonna need: you’re gonna need linear algebra to represent data and model transformations, You need probability and statistics to reason about uncertainty And learn things from the data, like get your machine to learn things. And the way that the machine learns things, well, you’re gonna need multi-variable calculus to understand gradients and the way that models are trained. See, what you have to remember about these technical careers is that the really exciting stuff, the stuff that makes you feel alive Like, holy crap, it did this? I wrote a program to do that? I built a model that can do this? All this exciting stuff sits on top of this huge stack of math. I mean, if you wanna really build stuff yourself and extend it, really understand what’s going on well enough to extend it in super interesting and novel ways, you gotta know the math. It’s more than just importing some- somebody else’s API. Ultimately, if you wanna do novel things and, and provide it to everybody, you gotta know what you’re doing at a fundamental level Cause all that math is very hierarchical And if you wanna get to that top, you gotta build the foundation first.

Want to improve your creativity?

Everybody wants to think creatively, but nobody really wants to build their knowledge base. See, the thing is creativity It doesn’t come from nothing. Almost never can you discover a clever proof if you don’t know any theorems. You know that saying, A writer is the sum of their experiences. Well, that applies in math too. And math is really, really hierarchical So you can’t really even collect experiences at the top if you’re just sitting there at the bottom. You need to climb your way up into the level of skill and domain expertise That all those symbols and theorems And clever tricks at the top actually mean something to you beyond just generating a pretty picture on a YouTube video. Ultimately, creativity comes from taking things that you actually know and combining them in new ways. It’s a, it’s a combinatorial explosion. So if you wanna be more creative and really expand the possibility space of all the ideas, perspectives, all the insights that your mind is capable of coming up with, Then build a bigger knowledge base. Give your brain more to work with.

What can you learn after calculus?

What can you learn after calculus? Well, there is a whole world of math afterwards. If you go down the applied math path, You might find yourself taking differential equations where you model really complicated real-world systems. And you’ll use a lot of probability and statistics Where you study randomness and uncertainty and distributions. And by the way, that’s calculus-based probability and statistics I’m talking about. It gets really, really complicated And when you go far enough, you actually end up using tools from calculus, not just algebra, but calculus integrals, derivatives to model all that complexity. And you also need bunches of tools from linear algebra: vectors, matrices, eigenvalues, eigenvectors… see, the thing is calculus is not the end of math at all. It’s really the starting point where you get to choose your own adventure.

Unserious edtech is rotting your brain

Your brain gets really soft when you avoid hard thinking. And that’s so common now, right? I mean, a lot of mainstream ed tech tries to make math feel really fun and effortless. All you gotta do is, is watch a little video and answer 1 or 2 conceptual questions. And if you get them wrong, ah, that’s okay. As long as you’re having fun, kid It’s just so much fluff and so little actual thinking, actual working out problems, actual retaining what you learned And being made to demonstrate it again the next day, the next week, the next month… actually being held accountable for all that.

You gotta build mental strength

See, the thing is math is cognitive athletics. That’s basically what it is. You wanna make progress? Well, you gotta build up your sub-skills, practice them really well So they’re just automatic And you can pull them together and practice close to the edge of what you know, where things are tricky, Where you make a mistake every now and then, and you get feedback on it, And you have to try again And again And you gotta really demonstrate the skill Prove that you can actually do it before you keep moving on. You need to retrieve it from your head. You can’t just consume stuff And then do projects where you outsource every difficult step to a calculator or to AI. If the tool is doing the hard part, then you aren’t really getting the rep. And without reps, you you don’t get stronger.

Why math hits people differently

The bigger factor controlling how difficult math feels for most students in classrooms is that they’re not all starting from the same place. The kids who seem the brightest Sometimes they are, but a lot of the time It’s really coming from the fact that they have mastered the prerequisites. So they’re really prepared for what it is that they’re being asked to learn. When they’re learning algebra, their arithmetic is automatic, rock solid. They know their times tables. They can factor numbers like nobody’s business. They don’t need to finger count. They got no baggage slowing them down from learning algebra. Same thing in calculus: They’re automatic on not just their arithmetic, but also their algebra. They have no issues solving equations, Manipulating expressions, they can do it in their sleep. And that’s why, you know, natural talent, it is a thing, but what looks like natural talent is often largely the result of being better prepared.

It's not talent, it's better preparation

Here is why some students seem naturally gifted at math And they learn new concepts instantly while everybody else is really struggling. The kids who seem the brightest Sometimes they are, but a lot of the time It’s really coming from the fact that they have mastered the prerequisites. So they’re really prepared for what it is that they’re being asked to learn. And when you’re in that situation, whatever new concept you’re being asked to learn, well, that’s the only thing that you really have to think about. It’s just the new thing. You don’t have to worry about that huge swath, the mountain of prerequisite skills that this is sitting upon. If you got those skills on lockdown, then it’s just, it’s no big deal. But if you don’t know your foundational skills, then oof. Being asked to learn one new concept can bring along so much baggage that you might as well be asked to learn an entire semester of math in a day. And that’s why, you know, natural talent, it is a thing, but what looks like natural talent is often largely the result of being better prepared.

The ghost of learning gaps past

This is the ghost of math that will come back to haunt you forever. It’s math facts, particularly times tables, multiplication tables. If you don’t memorize those early, then everything afterward, it gets much, much harder than it really needs to be. So now every algebra problem is really an arithmetic problem, a handful of arithmetic problems, and you just can’t think about the algebra that’s going on ‘cause you’re so focused on these arithmetic problems ‘cause it’s a full brain problem ‘cause you never memorized your math facts! And you can’t build fluency that way. See, that’s why gaps in your fundamentals are so costly. They don’t stay behind you. They just keep on following you forever and ever and ever Like a stalker Just trying to sabotage your every move as you climb up this ladder of math slowing down your progress, weighing you down, and the higher you climb, the heavier they become.

This sets students up to fail in math

Putting a student in the wrong math class doesn’t make them more advanced. It doesn’t make them smarter. It just makes them crash and burn. They might wanna be an engineer, and then they get thrown into a math class that’s just way above where they are currently. Maybe just ‘cause they were never held accountable for learning the prerequisite material. And then they crash and burn and come out thinking “I’m not cut out for engineering” when really They very well may be, And a placement exam would have identified like, Listen, kid We need you here, and here’s where you are, and here’s all the skills that you gotta fill in Which might look daunting at first But if you start the student in the right place, then they can start making progress, and progress produces motivation.

Why placement matters

Student signs up for calculus and the transcript says that they passed pre-calc. They got an A in the class, right? So they should be prepared. And then the first quiz has a problem with fractions in it, and everything falls apart. Not the calculus, the fractions. See, this situation is why placement exams matter. You can’t really guess math readiness by vibes or a padded inflated grade from last year. At some point, somebody’s gotta ask the question: Can you actually do the prerequisite work? Now, my question might feel rude. It might feel like an interrogation, an audit, and Well, it is But it’s necessary because putting a student in the wrong math class doesn’t make them more advanced. It doesn’t make them smarter. It just makes them crash and burn while simultaneously making it harder to diagnose what even happened.

Why automaticity matters

People sometimes talk about multiplication facts like they’re some kind of old-fashioned tiny little detail that doesn’t matter But understand That they do matter a lot And if they’re not automatic, then every algebra problem gets way heavier. See, when a student isn’t automatic on arithmetic and they’re trying to do algebra, They’re not just thinking about variables, equations, or structure. They’re also spending a lot of mental energy on that basic arithmetic. And that mental energy, that working memory, all the stuff they can hold in their brain at once, it’s pretty limited. So now the algebra problem is competing with the multiplication problem inside the student’s head, Each vying for attention. That’s why automaticity matters cause it frees up the student’s brain to actually think and pay attention to the harder thing. Just like a basketball player thinking strategy while simultaneously dribbling the ball and passing to teammates, cause that’s all so automatic that they don’t even have to think about it. They can just think about the game.

Sometimes a push is what opens the door

The most mathematically gifted student I ever worked with, he was still kind of resistant to learning calculus. He liked math that felt interesting, which to him often meant familiar. Kind of like a gifted basketball player who learned a bunch of cool moves and now loves playing at the park, showing off sometimes, having a good time But kind of starting to stagnate, not really developing their game. And the same thing happened with this kid. He was having a good time, but he was just kind of stuck in limbo at his current skill level. And if you really wanna use math throughout the rest of your life, like really have a good time with it, work on mathematically interesting problems, well guess what? Algebra skills, arithmetic skills, not enough. You gotta keep climbing higher. That’s how you unlock more of this thing that you love But kids, you know, they don’t always see the long game. Anybody who’s worked with kids knows how kind of just in the moment short-sighted they can be But ultimately, as an adult, it’s kinda your responsibility to help kids see that long game and at least take actions that are progressing them along in that long game. So that kid Once his parents and I pushed him to learn calculus, He was a little resistant at first, but he ended up loving it! And now he’s doing some serious math and engineering using ideas from this area that he originally resisted. You know, sometimes a push is what opens the door.

Nothing succeeds like success

The best kind of practice is challenging, but it still lets you succeed. It’s like an exercise, right? If you, if you can’t do a pushup, Then don’t just keep trying harder and failing to push your body off the ground, getting frustrated, demotivated. Just, just accept the situation, Put your knees down, make it a little easier Do some reps of that to build up some strength, and then, and then reattempt a full pushup after a few days of that. And learning anything should work the same way. You’re learning to solve a new equation? Well, start out with the simplest case So that you get comfortable with the process and then gradually ramp up the difficulty. Master the full skill at an easier level. You know, the, the full range of motion, whether it be physical or cognitive. Do the real skill, but on a simpler case, and then build up to more advanced cases, more advanced settings.

Full ass everything

Feeling drained doesn’t automatically mean that you’re doing too much. It can also happen if you’re not really doing anything fully. You work while thinking about resting You rest while you’re feeling guilty about not working You’re always just halfway somewhere else. And halfway is exhausting. Now, that’s the problem Then the solution is really simple: it’s just when it’s time to work, work. Immerse yourself and give it everything. And then when it’s time to rest, actually rest. Like recover When you go to bed, actually focus on sleeping. Don’t turn your rest into a half-conscious state of worry or guilt or whatever. Don’t half-ass your rest Cause if you’re not full-assing your rest, then you’re not full-assing everything.

Much practice, little progress

A lot of people say that they practice a lot But then they end up making very little progress towards the things that they’re trying to do. And often the reason why is that they’re not actually practicing anything. They’re just spending time near the skill. They’re doing the cognitive equivalent of watching sports on TV and thinking that it’s turning them into a better player. That’s what it’s like when you’re just watching a bunch of math videos and not working out problems, just looking at pretty images generated by somebody else who actually did the work. See, good practice makes progress visible. And The work has to be hard enough to force improvement. You gotta be breaking a sweat, whether that means physically, literally, or, or cognitively. Cause if you just keep doing things you already know how to do and can do reliably, consistently, You can practice for hours, days, months, years without actually getting any better. And ultimately, that inefficient practice kills your motivation Cause you don’t see yourself getting any better! Whereas efficient practice that actually builds skill It creates visible progress and it really makes you want to keep going.

Still counting on your fingers?

What happens when you don’t memorize your addition facts? Well, for one, you become painfully slow at subtraction. And trust me, you will encounter this in your daily life. It’s unavoidable. Let’s say you’re setting the table for nine people. Four plates are already out, and how many more do you need to grab? I mean, if you don’t instantly know that nine minus four is five, You gonna pull out your phone calculator in front of all your guests? Seriously?! Or stand there counting backwards on your fingers: 9, 8, 7, 6… no, that’s, it, it’s ridiculous. You should not need technology or a five-step finger counting exercise to set a table. Just, just memorize your addition facts. You’re gonna need them.

Educational malpractice

One of my friends tutored Calculus 1 students at WashU last year, And during one of their review sessions, They realized that instead of doing the regularly scheduled calculus work, they had to do a remedial math session. Not just like you’re a college student, you need some review on your high school math. These knowledge gaps traced back to not only middle school math, but elementary school math. Calculus students unable to add fractions Much less work with exponent rules or solve equations with trig or square roots. How do you expect students to take derivatives of these functions when they can’t even work with them algebraically, when they can’t even add fractions?! And so many of these students, they think that they have a calculus problem. They don’t. They don’t have a calculus problem. They have a years of missing prerequisites problem. Years of knowledge gaps, of holes in their foundations. But instead of actually filling these holes, Much of the education system has decided just to keep on shoving them into more advanced classes and calling it rigor. It isn’t rigor. It’s educational malpractice.

Calculus students who can't subtract negative numbers

One of my friends tutored Calculus 1 students at WashU last year, and there’s so many crazy stories of educational dysfunction. Like one day, they were helping a student who was stuck on subtracting negative numbers. Yeah, a calculus student stuck on subtracting negatives. They were doing a problem that asked them to calculate a derivative, And the student couldn’t do -27 minus 14. I mean, no wonder they had no idea what was going on in the class or how to choose what derivative rule to apply to each function or how to work with functions. Like, for crying out loud, they don’t know how to do arithmetic! And just because you graduated doesn’t mean you get to leave all that math you’re supposed to learn behind cause everything builds on each other. Algebra builds on arithmetic. Calculus builds on algebra. You’re never done with anything. It always comes back, And you gotta keep building on top of it. If you wanna solve calculus problems, it’s not just the new stuff in calculus that you need to know. It’s also all the foundational material from algebra and arithmetic, trigonometry, everything that is gonna come back and get exercised. And if you can’t do it, you’re gonna struggle until you refresh that knowledge. So when we’re putting students into calculus who don’t know middle school math, It’s no surprise that they’re completely underwater.

What can you learn after calculus?

What can you learn after calculus? Well, let me tell you, there’s this whole world of math afterwards where you learn how to construct proofs And then you might take real analysis, where you attack calculus with those proofs rather than computations. You kind of go up another level of abstraction. And then there’s number theory Again, more proofs, and you focus on the behavior and properties of numbers. There’s discrete math, where you study objects like graphs and networks. See, calculus is not the end of math. It’s really just the point where you start getting to choose your own adventure.

How to see the big picture in math

You ask a 10th grader to factor something… nothing crazy, just like x squared plus 11x plus 28 And before they even get to the algebra, they’re stuck on the multiplication! What’s 7 times 4? Seven times what? What is that? Let me recompute it. You do four plus four plus… oh, wait, I should just add seven. Seven plus seven plus seven… geez! Y’all already got one job, which is to do the algebra And now you’re spinning out on computing basic facts? And even if you’re using some kind of strategy like seven times five, then minus a seven, or seven times two, you know, double it and double it again, I mean, Your whole brain is thinking about this. It’s pulling away your attention from the algebra that you really need to be focused on If you’re gonna see the forest for the trees and really understand and get intuition about what you’re doing. You need to be able to look up Like a basketball player who’s dribbling the ball and is able to look around them, see what’s happening you can’t get good at basketball if you’re constantly having to like look at the ground and you’re dribbling the ball. Like, it doesn’t work that way. Anybody who plays sports knows the importance of getting automatic on your basic skills so that you can build on top of them and they don’t continually hijack your mental bandwidth. And it’s the same way with math.

Cognitive ankle weights

If you’re having to do algebra and think about basic arithmetic, If every problem makes you recompute multiplication from scratch, Doing strategies or repeated multiplication or God forbid, finger counting You’re just carrying these cognitive ankle weights around that nobody can see but they stay with you, they get heavier over time, they drag you down. And you’re going as fast as you can, putting as much effort as you can and it’s all getting wasted on lower level stuff that you haven’t automated yet. And everybody else who has automated those skills, they’re just passing you up. You’re huffing and puffing, going slow, and they’re just joking around and having a good time and passing you up.

How math can level up your career

A lot of people realize later who go into software engineering maybe at first they don’t really care about math in school But eventually they build up their foundations of coding, and then they realize that if they just had taken it seriously and learned all their math, They could be doing so much more! Just imagine a kid graduating high school Not only do they know precalculus with coding applications But also calculus, linear algebra, multi-variable calc, differential equations, basically all your core engineering math, right? And they come to college And they’re just blowing the socks off of anybody who gives them an opportunity to do some research, an internship They already got the basics of everything, and they’re actually ready to make a serious impact Do you know how rare that is? How rare that is to see in an undergraduate researcher or an intern? Usually you can’t count on them, so you just throw them a toy problem for them to play around with. But if you can actually make a serious impact at a young age Cause you have the skills to do so, You can just compound that into a massive compression of time.

Universities are having to teach middle school math

One in 12 students entering UCSD in 2025, they don’t know middle school math. We’re talking below Algebra 1: Fractions, order of operations, what’s an exponent? I mean, it’s so ridiculous. You know, UCSD, they had a remedial math course for students coming in who are missing their high school foundations. And that remedial course was too advanced for a lot of these students coming in So they had to make a remedial remedial course. Remediation for the remedial course for the calculus course. I mean that’s crazy!

Standardized tests are a better signal of knowledge than grades

You know, UCSD, they had a remedial math course for students coming in who are missing their high school foundations. And that remedial course was too advanced for a lot of these students coming in So they had to make a remedial remedial course. And that’s not even the craziest part though. The craziest part is that you don’t actually get a good signal of who’s gonna be in remedial, remedial math based on their high school grades. Cause the grades are fake Instead, the best predictor is really the SAT and ACT scores. And it’s been that way for decades, and that’s what the UCSD math department found: that you can’t trust high school grades. You have to look at the standardized tests if you really wanna know what math skills the students have. And guess what’s happened in recent years? Well, all the UCs, all the University of Californias, They didn’t require those standardized test scores to be sent. They’re just YOLOing the admissions on the basis of fraudulent high school grades. So now you’re flying blind in admissions, you’re admitting tons of people who don’t have their foundational skills I mean, the, the math department has been pleading about this for years Like please bring back the standardized tests, but You know, not everybody is interested in ascertaining the truth of the situation. So now we have this kind of shit show that’s happening.

Project-based learning can be taken way too far

The whole project-based learning can be taken way too far. See, projects are built on a foundation of skills. You can’t do projects without the skills. And if you’re trying to fill in all these skills on the fly at the same time as trying to do the project, it’s just it’s not efficient at all. It makes the project take forever, You don’t really learn the underlying skills that well. You just learn it just enough to get over the hump ahead of you. You don’t really master it. The teacher kind of carries you Or the smartest kid in your group. And you don’t really learn the things that the project was supposed to be a milestone certifying that you’ve learned.

Stop making kids reinvent math

The idea that students should discover everything for themselves is one of the biggest wastes of human potential that I’ve ever seen. Thing is, we have thousands of years of accumulated knowledge Entire civilizations of thought compressed into things that can be taught and learned efficiently, quickly I mean, that’s the whole point of instruction, right? is to get you to inherit all sorts of insights and knowledge and discoveries that took other people lifetimes to figure out. And not just like other random people. I mean like literal geniuses The brightest minds of each century grinding away for lifetimes Through thousands of years, Handing the baton and extending human knowledge. Just take the baton and run with it! You don’t have to rediscover all of that on your own from scratch. It’s gonna take you forever, and you’re gonna be spinning out doing years of research on things that are already known. So just suck it up, learn your foundations, and then build new discoveries on top of them where it’s actually novel, Where your inventions are actually pressing forward human knowledge.

Difficulty vs. Rigor

People confuse difficulty with rigor. If you take undergrads and you give them a graduate level textbook, You skip the concrete examples, And you just make them fight their way through it, Yeah, the course is hard, but hard does not automatically mean well-designed or that they’re learning more. Sometimes hard just means inefficient. It takes you a lot more work to ultimately get to the same place you could have gotten with a lot less work, just concentrated on the areas that are gonna move the needle the most. Sometimes a hard math course is just painful for no reason. And that’s the thing about rigor vs. difficulty. Real rigor is not about making the path as brutal as possible. Real rigor is about getting students to understand and master the material without wasting their energy on avoidable confusion.

Don't hold students hostage to the calendar

The rule should not be, “Hey, you’re in eighth grade, so you’re gonna do eighth grade math.” No. No, no, no. The rule, it should be What have you mastered? What new things do you have your prerequisites in place to learn? If you’ve mastered the prerequisites, then move the student forward. If they haven’t mastered the prerequisites, then fix it. That’s it. It’s that simple. The calendar should not be the thing controlling the learning. The mastery profile across all the skills for each individual student, that is what should be controlling their learning. Cause when students are ready and you make them wait, then you just waste their time. And when students are not ready and you push them forward anyway, then you create these knowledge gaps, these holes that follow them for years, sabotaging their attempts at every new thing that they try to learn and build on top and making them think they’re just inherently not cut out for math.

The most hard-hitting two sentences in all of talent development research

The most hard-hitting two sentences in all of talent development research is this: deliberate practice requires effort and is not inherently enjoyable. Individuals are motivated to practice because practice improves performance. That’s according to K. Anders Ericsson, one of the most influential researchers in the field of human development and expertise. In other words: Maximal learning does not happen naturally as a result of maximizing other things like enjoyment, comfort, convenience, ease of practice. In fact, maximal learning is at odds with a lot of these things.

This one idea would fix a lot in math education

Picture a kid who finishes long division in fifth grade Then sixth grade happens, and seventh and eighth, and somehow the big achievement Is finally starting algebra 1 in ninth grade?! That’s a lot of time. That’s a lot of calendar and not much distance. Just spending all of middle school doing laps around the same math. Nah, you know you know a kid who can handle arithmetic they should be doing algebra. They should be building on top of what they learned. And a kid who can already handle basic equations they should be doing functions. Kid who can handle functions, they should be moving towards trig. Readiness should matter a lot more than the calendar and this one idea would fix a lot.

Most math major math is taught backwards

When you’re a math major in undergrad and you get up to those junior and senior level courses you know, real analysis, Abstract algebra, topology You start to realize that a lot of the stuff is really taught backwards: theorem proof, theorem proof But you don’t even have a concrete grasp of what’s going on. You’re just pushing all these symbols around. Maybe you can follow the rules, but you don’t really understand the subject. That’s because you need the concrete examples first. You gotta work through the computations with actual numbers, see how it plays out, get a feeling for the process, and then you do the abstraction. Skipping that concrete stage With actual numbers and test cases that can help you build some experience and intuition Skipping that doesn’t make the course rigorous. It just makes it inefficient.

Abstract math needs concrete roots

Abstract math is not the problem. Proofs are not the problem. Those are not really the primary reason why math gets hard. The biggest reason why math gets hard is that students are thrown into abstraction before they even understand anything concretely. When that happens, the math just turns into symbol pushing. Definitions, theorems, proofs, More theorems, more proofs, But no intuition, no examples No real grip on the objects There’s nothing to hold onto, nothing to grasp And it’s backwards Build the concrete understanding first, Make the calculations instinctive, And then the abstraction actually means something.

There are countably infinite courses after calculus

People think that calculus is this big finish line, But it’s really just the beginning. Cause even after calculus, You still got linear algebra, vector calc, Differential equations, probability and statistics, And all these other specialized courses that branch out. I mean, there are so many University math courses above calculus that a serious student could not even fit them into a standard four-year academic schedule, even if they overloaded every single year. So getting to calculus early is not about being done early. It’s about having enough runway to actually reach all that math that comes after.

ROUND 10

Training wheels syndrome

If you can’t solve a problem without looking at an example, you might have training wheels syndrome. Training wheels are designed to do one thing. They help beginners get started. But they’re really, they’re not designed to stay on forever. And if you refuse to take them off, then eventually, They become the very thing that’s limiting how fast you can ride. And worked examples are really the training wheels of math. I mean, don’t get me wrong. Yes, you wanna use them when you’re first learning the motion. They can really help you get started, work efficiently, wrap your head around what it is that you’re trying to do. But, once you’re going through the motion successfully, that’s when you want to take ‘em off. Don’t let this good initial scaffolding hang around forever and turn into a crutch that you become dependent on and that you can’t move beyond.

Take the hints away

The easiest way to tell whether you’ve actually learned something is to take the hints away. ‘ Cause if everything falls apart once you take the hints away, then you’ve got training wheels syndrome. Training wheels, they don’t prove that you can ride a bike. They prove that you can ride a bike with training wheels. That is a good intermediate step towards riding a bike, but it’s not the same thing as actually riding the bike unassisted on your own. And in math, worked examples are the equivalent of training wheels. They’re supposed to teach you the motion when you’re first learning it, not perform the motion for you into eternity. Again, don’t get me wrong. Yes, use them at the beginning when you’re first learning something new. But understand that the only way to learn how to solve problems on your own is to practice solving problems on your own without assistance.

Borrowed brains

If your first instinct is to ask AI before actually thinking for yourself, then you might have borrowed brain syndrome. Now, it’s not an official term, but it just describes so well this phenomenon that’s becoming more and more common. See, the the thing is, AI should amplify your thinking, not just replace it. Like imagine hiring somebody to go to the gym for you. The workout gets done, right? But you do not actually get any stronger ‘cause you didn’t do the workout. Well, your mental muscles are exactly the same. And if you wanna actually grow them, you can’t just let AI do all the workouts for you. Now, I get it, this technology is amazing, so you wanna use it in some way, right? So here’s the trick. Use the AI after you’ve thought, not before. Use the AI as the feedback on your own thinking. That’s how you leverage technology while simultaneously continuing to grow your own brain and improving your own capabilities.

Cognitive crutches

Crutches are meant to help you walk again. Not to become the only way that you can walk. And learning tools should work exactly the same way. Whether it’s hints or worked examples or notes that you take or AI. These things are all meant to help you become independent. That’s their purpose. The goal of every learning tool should be to eventually make itself unnecessary, not to make you permanently dependent on it. ‘ Cause if you never become able to solve problems without them, They’re no longer helping you build up your strength. They’ve just become cognitive crutches.

Skill inflation

Watching 10 videos on how to perform a skill, any skill, like how to solve an equation, how to play an instrument, anything. It can make you really feel, really convince you that you’ve understood the process. But that feeling is often artificial. And this illusion is shattered when you actually try to do it yourself And realize you can’t. The thing is, watching creates familiarity. Doing creates skill. And these are not the same thing. The watching can kind of show you what you need to practice. And that’s good ‘cause you need to be practicing the right things. But it’s the actual practice itself of you practicing Not you just watching somebody else practicing you practicing that imbues the skill into you. That is how you turn the feeling of understanding into real understanding.

Phantom mastery

Recognizing how to do something is a whole lot easier than actually doing it yourself. And that’s why phantom mastery feels so convincing. You read through a solution, everything makes sense. You’re nodding along. Yeah, of course, that’s how it works. That makes sense. That’s obvious. yeah, I get it, right? But no, you don’t actually get it yet. You just feel like you do. Recognition can create the illusion of mastery. And you can be convinced that you’ve mastered something until you actually have to do a problem yourself, not just watch somebody else do it, not just read a solution, but actually do it yourself. And then you realize, “I don’t actually know where to start.” And that’s why it’s so important to actually go beyond recognition and practice retrieving it from your own brain. That is what reveals whether it’s actually there. That’s the real test. It’s not whether an answer looks familiar, a solution technique makes sense. It’s whether you can produce it from memory.

Fake practice hours

You can watch a thousand hours of somebody playing an instrument, or shooting a basketball, Solving math problems. But none of those are your practice hours. Thing is, watching teaches you what good performance looks like, but it’s not the performance itself. Doing it yourself is what teaches you to perform. The hard part isn’t seeing what to do, it’s actually doing it yourself. Making the mistakes, correcting them. Everything is obvious in broad strokes. It’s the micro level where things get more tricky. And if somebody else is doing the work, they are getting the practice, not you.

Mental outsourcing

AI can answer your questions, but it can’t build your brain for you. It can help direct you towards productive practice, but it can’t do that practice for you. You have to do it. See, your own thinking is the part that actually changes your brain. That’s the part that strengthens your mental connections that allow you to do tasks at higher quality or faster or better in some way. And if AI does all the thinking, then it gets all the practice. You get the answer, but not the skill and not the knowledge to know if the answer actually makes any sense. I mean, don’t get me wrong, like use AI to check your thinking. It is incredible tool. Use it to challenge your thinking, to expand your thinking. But don’t let it replace your thinking.

The comfort ceiling

If every practice session feels comfortable, then you’re probably just practicing things you already know, and you’re probably not getting a whole lot better. Think about lifting weights, right? If every set feels easy, you don’t really get stronger. The growth happens at the edge of what you can handle. And learning works the same way. See, practice should not always feel easy. If it does, then you’re mostly just reinforcing things you already know. But the real progress happens when you start working on things that are not already in your comfort zone.

Learning debt

Some shortcuts save time, but others just move the work to a day when the stakes are higher. That’s learning debt. See, skipping a skill or a concept, it doesn’t make it disappear. It just hides it until A harder topic depends on it. Which is inevitable in any hierarchical subject. And eventually you have to learn both at once. Not just the harder topic, but also the thing that it depends on that you skipped. Which makes the task way harder, way more stressful. And if you keep putting stuff off, The work just piles up in the background, you put yourself in an impossible situation where there’s just not enough time to fill in all the gaps, and then you have to resort to things like cheating. It’s a slippery slope. So just understand that the work will never be eliminated, just postponed Until you climb high enough to need it. And at that time, you will have to pay back the debt and interest.

Borrowed time

Every corner you cut on your learning, even if you manage to pass the class despite it. Every gap in your foundations If it’s in an area of life that you’re gonna be pursuing further, you will eventually meet it later inside a harder problem. Except now you’re trying to learn the missing prerequisite and the new material at the same time. The shortcut doesn’t save you from the work, it saves the work for later When it costs more, when it’s less convenient, And the stakes are higher. The longer you put off learning it, the more expensive it becomes.

Unconscious skills

Fast typing isn’t really about moving your fingers faster. What it’s really about is just not thinking with your fingers anymore. See, if you have to hunt and peck to find each key, then your brain’s attention is divided between the search and your ideas. And the same thing happens when climbing any hierarchical skill tree, whether it’s typing or math, playing an instrument, a sport, you name it. If you are still thinking about the basics, then your brain just doesn’t have the room to focus on the higher order tasks. That’s why you gotta practice the basics, really hammer into them and layer on top of them until they become so automatic, so well ingrained that it’s just like breathing. Like, you don’t even have to really think about it. That’s how you get to the complex maneuvers. They become possible once the simple maneuvers become unconscious.

Competence mirage

Can you still do it without the notes, without Google, without AI? The real test of learning is what remains after the supports disappear. It’s easy to feel confident when the answer’s just one search away. But a real competence isn’t measured by what you can do with help. It’s measured by what you can do on your own.

Brain calluses

If remembering something feels difficult, then good. Your brain is doing the work that it needs to do. It’s developing the calluses. See, calluses let you keep doing hard physical work by protecting your skin. Your skin adapts, so next time the work hurts less. Mental work, it’s the same way. That discomfort you experience while recalling information, you know, when it’s like it hurts your brain, but you, you get over it and you, you recall it successfully. Or maybe it’s just on the tip of your tongue, And you get almost there and you just need a little hint, and that little hint then gets you over the edge, and then you remember it. That discomfort is what makes it easier to retrieve next time. It’s what callouses your brain. And the more brain calluses you can build, the more difficult tasks you can handle.

Foundation termites

When a student hits a wall in math class, it’s usually not as sudden as it seems. I know it might seem like they were fine and then just one day hit a wall. Some- something that they were asked to learn just made no sense. They couldn’t move beyond it. That’s what it looks like on the surface, but really typically it’s more like they had termites in their foundation. Termites don’t destroy a house overnight. It takes years of them eating away at it underneath the surface before you really notice anything. And math gaps, they work the same way. A student never really masters fractions or even multiplication, But they just keep on moving forward. They don’t really know what they need to know. And then they reach calculus, and they just, they can’t simplify algebraic expressions much less think about integration techniques. The skill gap is just so wide that they can’t really game or hack their way around it anymore. Now, on the outside, everyone thinks that calculus is the problem, right? But it isn’t. Calculus just exposed the damage that happened years earlier. And just like termites, you can’t solve the problem by covering up the visible damage. You have to eliminate the problem at its source. The kids gotta learn their foundational skills. That’s the only way to solve this problem.

Mental tollbooths

Imagine having to stop at a toll booth every few hundred feet. You can still reach your destination, technically, But every stop breaks your momentum, slows you down, and forces you to pay before you can keep moving. Terrible, right? Well, that’s what algebra feels like When multiplication facts are not automatic. The student’s supposed to be solving equations, transforming both sides, simplifying expressions. And instead they get stuck on factoring. What are all factors of 48? Is it a perfect square? So the student isn’t just doing algebra. They’re doing algebra while constantly stopping To multiply numbers. And each stop takes another payment of the student’s limited attention. And it increases the chance that they’ll make an arithmetic mistake. And you have to kind of reorient yourself afterwards. What was I even doing? What am I trying to do? Where am I? What am I doing? That’s why multiplication facts need to be automatic. That’s the only way that the student is gonna be able to focus their limited attention on the algebra ahead of them.

The drag of weak foundations

Two students can be working on the same algebra problem, but one of them is being dragged down by extra weight. They’re recomputing their multiplication facts. They’re wrestling with fractions. They’re second guessing their negative sign every step of the way cause they didn’t really master any of this stuff. They’re not fully automatic on it. It’s like swimming with jeans on. I mean, yeah, they can still move forward, I guess But the extra drag, it makes them much slower, much more error prone, And just quicker to fatigue and lose focus, forget what they’re doing because they had to devote all their attention to some sub-skill that should have been automatic. Now, the solution to this problem is not to make the algebra easier or to ask them to swim a shorter distance. It’s to remove the extra weight by having them train their foundational skills until they’re really automatic.

Civilization mode

Human progress works because each generation starts where the previous one left off. We don’t reinvent the wheel. We don’t rediscover electricity. We accept guidance from the previous generation. We learn about these discoveries, we train ourselves to use them and understand how they work, And then we extend them and then we pass them on to the next generation. And that’s how students should be taught too That’s what efficient learning looks like. Civilization mode. Teach them the tools that people already discovered. Give them enough practice to use them reliably. Show them why they work. And understand that the goal of education is not to make every student retrace thousands of years of dead ends that took geniuses to overcome and figure out what’s what. The goal is to help the students inherit these genius level insights in a way that does not depend on the students themselves being geniuses.

The classroom conveyor belt

In most schools, the classroom conveyor belt moves forward continually at the same pace for every single student. Ready or not, the students just get carried forward. Even if fractions are only halfway mastered, They barely understand algebra. Well, they’re just gonna keep moving because the calendar says it’s time. Thing you gotta understand is the conveyor belt does not just move the students forward. It also moves their knowledge gaps forward. Those don’t just get left behind. Those are on the conveyor belt too. And eventually those gaps arrive at a class that depends on them, and everything jams. That’s why progression has to depend on mastery, not the date of the calendar. If the foundation isn’t ready for a student, then just stop the belt and fix it.

Academic life raft

Some students don’t really learn the math in their course. They just learn how to survive the course. They get their smart friend to help carry them through the homework. They hunt for partial credit, lobby for retakes. They cobbled together just enough hacks to pass the class. And it keeps them afloat for one semester, but it never really teaches them how to swim. And then the next course assumes that they can, that they know how to swim. Now the raft is gone, the water’s deeper, And they’re in trouble. The goal should not be just helping students survive math class with some kind of makeshift academic life raft. The goal should be actually teaching them how to swim.

Bubble wrapped students

Some students go through school just wrapped in bubble wrap. Every time they hit something hard, the system cushions the impact with extra credit, completion grades, Review sheets that look just like the test. Then they get to college, and the bubble wrap is gone. Suddenly the mistakes count, the deadlines matter. Partial understanding isn’t enough, And the student concludes, “I’m not cut out for math.” But that’s not really it. It’s not because of their abilities. It’s because They were never held to proper standards. It’s not that they were never cut out for math, it’s that they were never trained up to math. The goal of school is not to protect students from failure forever. It’s to teach them how to recover and correct while the stakes are still low.

Why understanding multiplication isn't enough

Students have to understand conceptually what multiplication is, and they gotta know how to compute. But even if they have these two things, the understanding and the computation, that doesn’t mean that the practice is done. They also gotta have it memorized. ‘ Cause even if they know that 7 times 8 means add together 7 equal groups of 8 Even if they know the procedure to get to the answer and they can carry it out If they have to do that every time to get to 56, they have to add 8 plus 8 plus 8 plus 8 plus 8 plus 8 Or even the doubling trick, you know, 7 times 2, then double it again, then double it again Even if they can do this reasonably quickly, it still takes a lot of mental effort. And if you put that 7 times 8 inside of a long division problem. Which is hard enough as it is with all these other computations going on. You know, they’re trying to learn long division While multiplying numbers over and over again in their head. They can’t focus on the long division ‘cause all their focus is on the multiplication cause they haven’t memorized it, ‘cause they have to go through these procedures every time. It’s mentally exhausting So no wonder they get tired they get overwhelmed they wanna quit. In order to keep learning these new levels of math, Kids should not have to just stop and solve 7 times 8 by recomputing it every single time it shows up. At some point, 56 should just be there So their attention can focus on the new material.

Spaced repetition = wait-lifting

Your memory improves When you actually have to work to remember something Not when it’s just right at the front of your brain. See, it’s, it’s like weightlifting. To get stronger, you have to continually increase that weight. And to increase your memory, the idea is you have to wait longer before the next time you retrieve the knowledge. Waiting longer is like putting weight on the bar. You wanna increase that weight. But the thing you gotta remember: You don’t get stronger just by putting more weight on the bar. You actually have to lift that weight successfully. Cause if you add so much weight that you can’t even move it Or you wait so long between reviewing information that you have to just kinda relearn it from scratch Like you’ve just forgotten it completely, not even on the tip of your tongue and you just need a hint, but you’re just like, “I have no idea. I need to relearn this.” That’s too much weight So this general principle behind both building muscle and building memory is that you want enough resistance to challenge you But not so much that you can’t do it.

Practice more often, not longer

If you have just two hours a week to do math, don’t cram it all into one single Saturday session. Four half-hour sessions spread out throughout the week, that’s gonna get you a lot further, even though it’s the same amount of time. And here’s why: Number 1 It builds a habit. One of the most important elements of your habit structure is that you can keep doing it consistently. And it’s a lot easier to fall off the wagon with something you only do once a week as opposed to every other day. Number 2, you can stay focused way better during a short session than an hours long marathon. Like the quality of your practice It’s gonna be way better if you’re full-assing it for half an hour every other day instead of dragging yourself through this multi-hour marathon every weekend. And number 3 You forget less between the sessions. If you wait several days, a week, you know, you’re probably gonna spend A significant chunk of that next session just getting back up to where you were. I mean, just ask any teacher how much their students forget over a weekend or over winter break and how many class periods they have to spend just spinning back up to where they left off? If that’s what you’re doing every session, you’re basically just running in place. So the takeaway is that short and frequent practice, that is what moves you forward.

A good grade is not equal to mastery

Getting an A on the test doesn’t necessarily mean that you’ve mastered the material. You know, maybe you studied fractions all week And you crammed a week’s worth of material into just a few days And then your teacher gave you a review sheet so you knew exactly what kinds of problems to expect And everything was fresh in your mind? You were kind of primed and ready for this performance. But then, later on, everything kind of falls apart when you have to subtract two mixed numbers with different denominators and you have to remember all these different fraction manipulations that you didn’t just study the night before. Can you still recognize what to do? Like, without reviewing it first? If not, then you didn’t master it.

Silly mistakes mean you need more practice

If you understand the concept but you keep on making these silly mistakes You know, say you’re solving this really long algebra problem And you know that you need to isolate the variable And you know that to do that, you gotta carry out The order of operations in reverse order But you keep on making all these sign errors and you keep making these multiplication mistakes, then I’m sorry to tell you, but you gotta shore up your basics. Cause if you can only do a skill some of the time, you’re not ready to stack harder skills on top of it. Those little mistakes, they compound The more skill you stack on top, the more robust the foundations need to be to support this structure of skills that you’re trying to do. And eventually, if your foundation cannot support you, you just won’t be able to keep on moving up. You’ll hit a plateau.

Concrete skills build abstract understanding

People often say that kids should learn the concept before they do the procedure And sometimes that’s true, but Other times it’s concrete procedural practice that actually builds the abstract understanding. It goes both ways. The concepts and the procedures, they’re mutually reinforcing. Alternating rungs up this ladder. For instance, a kid first learns to count with physical objects Like beads or marbles, blocks. And after enough of these examples of counting these concrete items, The number seven stops meaning seven specific beads right in front of em And it starts to mean an abstract quantity. Carrying out this counting procedure has enabled them to climb up to this abstract concept. And that matters when they get to arithmetic, right? I mean to do 340 times seven, You can’t count out 2,380 beads. You can’t try to visualize it like that every time. You need to work with numbers as abstract symbols. But before you make the jump up to algebra, that next level of abstraction, You gotta to do plenty of concrete procedural arithmetic to lay this concrete foundation For the even more abstract idea of a variable, a placeholder in this arithmetic. And this pattern keeps on going throughout math. Every layer of concrete skills gives your brain something to build the next level of abstraction on.

Mental traffic lights

Imagine trying to drive across town and you hit every single red light. And you’re just like, “God!” Well, that is what algebra feels like when multiplication facts are not automatic. I mean, you will eventually reach your destination, technically But every stop slows you down and makes you wait. And it is so frustrating. You know, the student is supposed to be solving equations, right? Simplifying expressions, isolating the variable And instead they get stuck on factoring. What are all the factors of 48? And they’re sitting there waiting at this red light Before they can get back to driving through the problem again. See, that’s the problem. The student is not just doing algebra. They’re doing algebra while constantly having to stop to multiply numbers. And every time they do that, it takes up extra time, it divides their attention. They kind of forget what they’re doing, where they’re even going. And it increases the chance that they make an arithmetic mistake I mean, just like the red lights, it’s incredibly frustrating. That’s why multiplication needs to be automatic first So that the student can then focus their attention on the algebra road ahead.

You're probably taking too many notes

You are probably taking too many notes in math class Thing is, if you are copying down literally everything that your teacher says, then you may be transcribing more than you’re actually thinking. And when you get to the homework, the real problems, You may be tempted to refer back to those pristine, detailed notes all the time instead of actually pulling the information from memory. I mean, that’s what increases your retention when you retrieve information from memory, not when you transcribe it from some reference. It, it, it’s not just that the information passes through your brain, it has to get retrieved from your brain to the outside. That is the number one biggest lever in increasing your retention. Now, don’t get me wrong. Writing can help during class. It’s not always a waste of time But the key is that you should be writing down sparse key bits of knowledge just to, you know, maintain focus or draw connections to other bits of knowledge. That’s not transcribing. Think of it more like listening on paper where you are actually thinking about the information just as much, if not more than if you weren’t taking notes. See, notes can help you pay attention But they can’t replace retrieving information from memory.

The best way to take notes

You are probably taking too many notes in math class. Thing is, if you are copying down literally everything that your teacher says, then you may be transcribing more than you’re actually thinking. Writing can help during class. It’s not always a waste of time. But the key is that you should be writing down sparse key bits of knowledge just to, you know, maintain focus or draw connections to other bits of knowledge. That’s not transcribing. Think of it more like listening on paper where you are actually thinking about the information just as much, if not more than if you weren’t taking notes.

College math classes are often a waste of time

If you walked into a college math class, you might see 100 students, bent over their notebooks, furiously copying down whatever the professor is writing on the board. And the professor is just copying that from their own notes or from the textbook. They’re just facing the board the entire time Maybe turn around to say like, “Hey, you all understand this? Okay, sounds good. Keep moving on.” Meanwhile, nobody even knows what the heck is going on enough to even ask a question, to like…? And the professor mistakes this for everybody knows what’s going on. And then nobody actually learns anything until the problem set that they do for homework. I mean, what a waste of class time. Students are just copying information that they could easily get somewhere else. They go to class with the hope that all this information will just magically diffuse into their brains. But that’s, that’s not how learning works. You learn math by actually solving problems. I mean, if that’s the situation, Then learning-wise, you’d probably be better off just skipping class and using that time to work on your problem set.

Radical math acceleration isn't the problem

Some parents worry, If my kid gets really accelerated, really ahead in math Will they have knowledge gaps? Will it hurt them socially? What do they do when they run out of math? Do they just sit there bored? But thing is, rapid acceleration in math is not bad for kids. What’s bad is making them wait when they are ready to move on. And trust me, this road of math goes as far as you want it to go. You know, the real question, the only question here is have they mastered the prerequisites? If a kid knows algebra and geometry, then making them wait just ‘cause they’re “too young” for calculus doesn’t somehow protect their development. All it does is it delays them from reaching their potential. ‘ Cause if they’re ready, then moving quickly lets them spend more time doing something that they’re kinda good at And that can open up so many incredible opportunities for them. And you know, we all love doing things that we’re kinda good at and that can earn us some attention, respect, fulfillment. It’s like a fundamental law of human nature. So there’s no evidence that radical math acceleration is psychologically bad for students who are ready. The problem is just when you move on before a student has mastered the prerequisites. If they’ve got their prerequisites in place, then they’re ready.

Stop holding back kids who are ready

Some parents worry, If my kid gets really accelerated, really ahead in math Will they have knowledge gaps? Will it hurt them socially? What do they do when they run out of math? Do they just sit there bored? But thing is, there’s no evidence that radical math acceleration is psychologically bad for students who are ready. The problem is just when you move on before a student has mastered the prerequisites. if they’re ready, then moving quickly lets them spend more time doing something that they’re kinda good at And that can open up so many incredible opportunities for them. And trust me, this road of math goes as far as you want it to go. If they’ve got their prerequisites in place, then they’re ready.

How to build problem-solving skills

A lot of people ask, “Don’t kids need to struggle for a while to train their general problem-solving ability?” But thing is, in math, problem solving mostly comes from knowing more math. a student who knows fractions and ratios and equations has way more tools to attack a hard problem than a student who doesn’t. So instead of leaving students stuck for a long time and hoping that some magic will happen and they’ll, they’ll somehow acquire tools for solving hard problems? Just teach them the tools! Give them examples, give them guidance, give them problems that they can practice out their new tools on. Now, I know that being able to sit with a difficult problem and not give up too quickly, it’s an important life skill. I know But that’s not actually what solves the problem. The tools solve the problem And to a large extent, the willingness to sit with that difficult problem, stick with it, not give up too quickly comes from being confident that you have the skills, the tools required to solve the problem.

Learning is bridge building

Learning math is like trying to cross a wide river. Some people can naturally jump further than others. But eventually, everybody, Even the smartest kid in the class, they reach a gap where they can’t just jump straight across it on natural ability. That’s why we build bridges. And in math, those bridges are prerequisite skills. The more prerequisites you’ve mastered, the smaller the mental leap to the next concept. Arithmetic gets you across to algebra. Algebra gets you across to calculus. And each new skill gives you a foundation to build that next bridge. What makes math education effective and efficient is when it builds enough bridges to allow everybody to make the crossing.

The math habit that actually works

If you have just two hours a week to do math, don’t cram it all into one single Saturday session. Four half-hour sessions spread out throughout the week, that’s gonna get you a lot further even though it’s the same amount of time. One of the most important elements of your habit structure is that you can keep doing it consistently. And it’s a lot easier to fall off the wagon with something you only do once a week as opposed to every other day. The quality of your practice is gonna be way better if you’re full-assing it for half an hour every other day instead of dragging yourself through this multi-hour marathon every weekend.

How concrete math becomes abstract

A kid first learns to count with physical objects Like beads or marbles, blocks. And after enough of these examples, of counting these concrete items, The number 7 stops meaning 7 specific beads right in front of them And it starts to mean an abstract quantity. Carrying out this counting procedure has enabled them to climb up to this abstract concept. And that matters when they get to arithmetic, right? I mean To do 340 times 7, You can’t count out 2,380 beads. And you can’t try to visualize it like that every time. You need to work with numbers as abstract symbols. But before you make the jump up to algebra, that next level of abstraction, You gotta to do plenty of concrete procedural arithmetic to lay this concrete foundation For the even more abstract idea of a variable, a placeholder in this arithmetic.

Remedial math at Harvard

Harvard having to add a bunch of remedial support to its calculus classes ‘ Cause they’re getting the same situation of people coming in and they’re nowhere near able to pass a calculus course ‘ Cause they got these gaping holes of missing foundations in high school and probably even middle school math. I mean, that’s what the University of California San Diego found, right? That 1 in 12 incoming freshmen college freshmen were not proficient in middle school math. But like at Harvard too. All this remedial math, it’s just ridiculous.

Middle school math is the biggest waste

The worst part of K - 12 math is sixth through eighth grade. ‘ Cause kids learn arithmetic in elementary school and then spend three whole years spinning their wheels until they actually learn Algebra 1 in ninth grade. So they finish arithmetic in elementary school, And then move on to algebra in high school. What do they even do for those three years of middle school? It’s just, it, it is such, such a waste To have them running in place for three years. If you actually use those three years well, you teach efficiently, You require mastery, and then you move kids forward when they’re ready, honors students can be finishing all of high school math by the end of eighth grade. And not even by doing extra math, just by not wasting three years.

The rot starts with multiplication tables

Mathematical rot It starts early. It starts with things like, we don’t really need kids to memorize their multiplication tables. But then, multiplication never becomes an effortless skill. It always takes up a bunch of space in their brain, and they can’t easily reverse it when it comes time to find factors and other stuff in algebra. So algebra becomes way harder than it really needs to be. And because it’s so hard, because students are struggling so much more than they really need to, It gets watered down. And then we let them lean on calculators and Desmos ‘cause they can’t actually graph things. They can’t solve things. They can’t manipulate expressions by hand. And now there’s a weakness a big weakness a widespread weakness in their foundational skills a rot That’s gonna collapse whatever type of knowledge structure they try to build on top of it. And this weakness just climbs up the ladder with them. You don’t fix that in calculus. You gotta get it at the beginning.

Calculus is not the finish line

A lot of people think that calculus is the end of math. The final boss. You beat calculus and then you basically beat the game. And it’s not, it’s not even close! After calculus, there’s linear algebra, There’s multi-variable calc, differential equations, probability, statistics… and that’s just a tiny subset of applied math But it branches off in all different directions. There are so many specialized courses. In fact, there are even more university math classes above calculus Then there are high school math classes below calculus. So getting to calculus early That’s awesome. That’s great, but it does not mean that you’re done. It means you finally get to start.

Differential equations is one of the worst taught math courses

Differential equations. It’s one of the worst taught math courses. Thing is, there’s a lot of breadth to it. It has a lot of branch off points. And what I’ve experienced in my own education and many students that I’ve tutored Was that typically, the instructor will go through like integrating factors and linear differential equations, Then do characteristic polynomial, second order. I mean, that’s barely anything. But then they say, “All right, we’ve covered the basics.” Now let’s just go into my favorite area of research. So it’s like, depending on who your instructor is, You can get a completely different view on what even is the subject?! What does it cover?!

Why you need to interleave a practice

Say you’re learning how to calculate the area of shapes. You got some rectangles Some triangles, some circles. If you do 20 triangle problems in a row, then you already know which formula to use. That step has basically already been done for you. And so by problem 20, you feel great at triangles. And you think you got it down pat. But problem is that feeling is completely artificial. You never really had to choose the formula. You never really had to match the formula to the situation. You just blindly, robotically applied the triangle area formula to triangles. Now, if you do this instead, you mix all the shapes together You’re not just doing all the rectangles, then all the triangles, then all the circles. You’re mixing them up, you’re interleaving between them. Do a triangle, rectangle, triangle, circle, circle, rectangle, triangle… now you have to look at each problem and actually decide which formula applies and then correctly apply it. And yeah, those problems are gonna feel much harder But the only way to actually know something is to be able to retrieve it without context clues.

A math concept doesn't need different explanations

If a student needs five different explanations before one finally “clicks,” The problem may not be the explanations. The problem is probably that the student has lots of missing prerequisites that are preventing them from understanding the concept. Here’s an example. Say you’re teaching slope, but your student is really shaky on fractions. Well, you can keep finding clever ways around fractions So that they don’t actually have to know fractions to do the problems, but Now they only understand the easy cases when the slope is an integer. They don’t really know slope yet. And if you pretend that they do and just say, “All right, you got it. Let’s keep on moving on,” well It’s gonna cause a lot of problems later. So in this situation, what you gotta do is you gotta go back, you gotta find and fix those missing skills, And then keep building from there.

Why college proofs seem impossible

Why do so many students get crushed by proofs in college? The problem is that they’re learning two really hard things at once. I mean imagine taking matrix algebra And you’re learning all this brand new math While also having to write proofs about it? I mean, that’s like learning Shakespeare while you’re still learning how to even write an essay. It’s ridiculous, and this jump is simply too big for most students to make. See, students need to master a topic at the concrete level first. So for instance, they should master calculating eigenvalues and eigenvectors and performing operations on matrices before proving any properties about them. Then when the proofs are introduced, They’re not trying to understand the math and work out the proofs at the same time. They already know what these objects do. And now they can focus on proving the why.

What makes an expert

What separates experts from enthusiasts? The answer is having done the reps. See, a physics enthusiast might watch lots of documentaries Quantum mechanics, relativity… I mean, yeah, they can explain the big ideas at a really surface level And that might be enough to impress people at dinner parties But a physics expert has actually solved problems, applied their knowledge, tested and refined hypotheses. I mean, yes, they were probably guided down this learning path so that they could work efficiently. But the point is They learned what works and what doesn’t by doing concrete reps. They didn’t just watch a bunch of videos. They actually did the problems. And without those reps, you might be familiar with physics, but you can’t actually do physics. Expertise comes from actually doing the work and producing something, not just consuming information about it.

The secret to building confidence

The secret to building confidence is to first build skills. Every time you get stuck and you successfully work through it, you build evidence that you can do that again. And in math, that means having a bunch of tools to try. Break the problem into smaller steps, rearrange the algebra, Use a formula you know… the more tools you have, the less likely you are to get completely stuck. So when a harder problem shows up, You don’t panic. You think, “Hey I’ve been stuck before. That’s okay, ‘cause I know how to get unstuck.

You need to measure learning

A lot of people think that measuring kids’ learning in school is bad for them. And it stresses them out, that it gives them test anxiety. But really, all this does is it leaves kids unprepared. And being unprepared, that is stressful. That induces anxiety. Maybe not in the moment when there’s no pressure on you, but in the future, When you’re in situations that need those skills or you wanna go do things, grasp opportunities and you can’t ‘cause you don’t have the foundational skill base. That’s a million times worse. And if you don’t measure learning, You don’t know what kids know and don’t know. You can’t tell if your teaching methods even work! And if nobody knows what works, then there’s no pressure to improve what doesn’t. You can’t improve a system that refuses to measure its results.

The case for mastery learning

Typical letter grades They basically measure how much a student learned within a “standard” amount of class time. So of course, over the course of the year, students are gonna end up at different levels. I mean, some students only need 5 problems to learn how to add fractions. Others need 20. And if the homework only gives them 10, then the second student has to do practice on their own to master it. And if they don’t, they end up with a lower but passing grade. And then they move on with all these knowledge gaps that haunt them indefinitely into the future of their math learning. Every single time they try to learn something new It’s like, well, this depends on that, which you don’t know, which depends on this other thing that you didn’t really master. So everything becomes a lot harder than it really has to be. I mean, how about this? How about instead of giving everybody the same amount of time and just accepting all these different levels of mastery, Why not require a baseline level of mastery where the student knows it well enough to move on And then give different students different amount of practice so that they can hit that bar?

Calculator addiction

Calculators can become an intellectual drug. You get stuck on 7 times 8, so you punch it in. Instant relief! And then the next time you get stuck, you reach for another hit without even trying to use your brain at all. Soon reaching for that calculator becomes automatic. You can’t do the arithmetic yourself. The calculator owns you! The more you use it for that simple stuff that you should be able to do in your head, that you really should have memorized, the harder it is to kick the habit and the more your basic foundational skills decay. They just rot away. So what started as a time saver ends up becoming this crutch that you just can’t function without.

The secret to success in life is consistent effort

The secret to success in life it’s consistent effort. That’s what Jason Roberts said, and he’s totally right. Whether it’s exercise or learning math, Learning a language… whatever the heck it is that you’re trying to do. You don’t have to do the superhuman effort thing. Just get started and then make a consistent push every day. Even if it’s only 15, 20 minutes.

What happens when students don't know their math facts

We hear from teachers and chief academic officers and tutors who are so frustrated. They’re getting kids in 6th, 7th, 8th, 10th grade who are still finger counting! High schoolers, finger counting. They don’t know their multiplication tables. How the heck are you gonna factor a quadratic when you don’t know your multiplication tables? You can’t do it, not really. And then guess what? You can’t factor a quadratic, you can’t do algebra. And even if you do manage to grind through just finger counting into oblivion, It’s gonna take you way too long. It’s gonna take you like 10 minutes to factor Which some kids can just do in their head in like 10 seconds. So you’re not gonna wanna do any problems. Like, “Teacher gave us 4 factoring problems and it took me all night. It’s so unfair.” Like, come on And when you get to calculus, if you’re taking 10 minutes to factor a quadratic And that’s only one component of a calculus optimization problem, You’re gonna be spending half an hour on this problem that should take you a couple minutes.

Don't look back at the reference too often

One mistake that I have seen students, even adult students, make all the time is looking back at the reference too often. Thinking that it’s free to look back at the reference When in reality, if you just reflexively look back at the reference Instead of trying your best to recall from memory, You’re basically just letting your spotter lift the weight for you. You’re supposed to be doing mental weightlifting, And you’re like, “I can’t get it up,” and then the guy behind you just lifts it for you. And it’s, it’s ridiculous, right? I mean, it’s so ridiculous that this kind of situation never really happens in the gym. But it happens all the time in learning that people ask for the spotter’s help before they even get to the point that they’re trying really hard. The moment that it stops becoming super easy, they’re like, “Ah, let’s just look back at the example.” When in reality, that is the moment when you are getting the most bang for buck out of recalling the information. The challenge during the active recall process, that is when you’re strengthening the memory. So don’t look back at the reference right away. Try to pull it from memory. Don’t look back unless you absolutely need to. And definitely do not, under no circumstances, never do this. Don’t pull up the worked example and look at it side by side when solving a new problem. Cause if you do, you just shoot yourself in the foot. You just shortchange the learning process. You might get through today’s homework, but you’re not training yourself to retain it, much less do it unassisted.

ROUND 9

The grade isn't the skill

An A in math, you know, sometimes it’s just a receipt that you turned something in, not proof of actual learning of math. I mean, yeah, a parent can sit at a kitchen table, open the grade portal. Hey, kid’s got an A in math. Everything looks fine, right? Right? Then you ask your kid, like, “Can you turn 0.6 into a fraction? Can you really do that?” And no, they can’t. It’s at grade level, and they have supposedly learned this, right? That’s what their report card says, But the kid doesn’t know how to do it. And it’s not ‘cause they’re dumb, it’s just ‘cause the system has been giving them a real-looking signal, a receipt for fake learning. So that’s the part that parents really have to watch. It’s that a grade can sometimes hide a missing skill. And a real math program that practices real learning will make that missing skill show up early while it’s small enough to fix. Instead of letting the kid just carry it endlessly down the road so that it handicaps them in all their future math courses.

Take the calculator away for one minute

Just take the calculator away for one minute And just ask the student to sketch the line by hand. You know, the y = -2x+7, whatever. Something easy, something that should be a layup for ‘em. And the graph, it, it doesn’t have to be perfect. Just, you know, where does it cross the X-axis? How about the Y-axis? Is it going up or down? ‘ Cause you know, if they can’t do these things manually, that tells you a lot. It tells you that they have been “doing algebra” for months by just pressing buttons around the part that they misunderstood. I mean, listen, tools are fine when you know how to do the underlying skill, when you’ve done this thing manually enough times that it’s just in your bones. But if the tool is carrying all that, the tool is carrying the skill, you, you can’t do it manually ‘cause you’re dependent on the tool to do it for you, and you, you just know how to press a button on the tool, It’s not math mastery.

Calculus is just the doorway

Getting an A in calculus, having that on your transcript as a high schooler, that might feel like the finish line. Like, “Hey, I’m done with math. I did calculus.” That’s what your average person thinks is advanced math. And that’s kinda true, but here’s the thing. That’s not the finish line of advanced math. That is the doorway. That is the entrance. You learned calculus. You actually learned it well, got an A, did a solid job. Well, congratulations. You now have a key to the building And now you can really go off into, you know, physics, economics, data science, engineering, where there’s a bunch of vectors, matrices, partial derivatives, probability models. There is still so much to explore, so much to learn, so much to use after calculus. And that’s why getting to calculus earlier, getting it out of the way, just getting it locked down, it matters a lot. Not so you can brag about being done. I mean, you’re not done yet. There is a whole stack of math after that. But the point is that students who lock down their fundamentals early, They have more runway, And they can actually use college to explore all the super advanced math and/or applying it to particular domains that they find interesting, Working on research problems, building stuff Instead of, you know, just spending the first two years catching up with the fundamentals of the things that they wanna do.

Lost is not the same as rigorous

I don’t trust a math class just because everybody seems miserable. I’ve seen that version: A professor writes a definition, then a theorem, then a proof. And everybody just nods along thinking that they’re the only one who doesn’t understand it. And then you get to the homework and the first problem is like, “Hey, prove this theorem about this thing that you have no concrete experience with. And nobody knows where to start, Except for maybe the really smart kid who goes off and learns about this stuff on his own just for fun in his free time. And he’s already seen this theorem play out with plenty of concrete examples. For the rest of the class, that’s not depth. Being serious, being rigorous with proofs doesn’t mean that you just skip the concrete examples and go right into abstractions. You can work with the objects first, and you should. compute something. Put numbers through some process, see what happens, Then be like, you know, “Oh, that’s interesting. Let’s see if that works in general. Let’s try running it through symbolically for all numbers.” And then that’s when you work with the abstractions. When you do it like that, you get your reps with the concrete examples, then the proof actually has somewhere to land.

The easy math can become a hiding place

A strong student can look advanced and still be stuck. I know that sounds weird, but it happens. They’ll do puzzle books, mental math tricks, and contest problems that they already know how to approach, they know how to attack it. It’s familiar. And then you put legitimately new material in front of them, say trig functions, And suddenly they’re like, “Ah, I’m, I’m not interested.” I mean, maybe, but that’s unlikely. What is vastly more likely is that they just like the math that they’re familiar with because it keeps them in their comfort zone. And guess what? Growth doesn’t happen in your comfort zone See, adults have to notice that. Kids are not always the best at noticing that. But you can’t just camp out at your current level. You gotta keep climbing every step of the way. Go up, go up. You’re comfortable? Well take another step. That’s how you build real mastery. That’s how you get really good at something.

Students need to retrieve

A math class can look smooth because nobody’s really thinking very hard. The class starts and then before students even get a problem to work on, teacher says like, “Hey, you know, remember yesterday when we divided by a fraction? Well, we’re gonna do that again here, and this is how you do it,” Walk through the whole thing again, get a nice refresher, and now, yeah, the class is warmed up. They got it. They’re, they’re doing it on this next problem, but the problem is They didn’t actually have to retrieve that information from memory. You robbed them of the opportunity to retrieve the fuzzy information from memory. That is really where the retention gain comes from. When you think like, “Huh, how did I do that again? That was… I know we did this yesterday. What did we do?” And then you have to think about it. You’re like, “Oh yeah, that’s right. This is what we…” And then you, you know, you do it. That’s what really extends your memory duration. You don’t get your memory duration extended by somebody reminding you right then and there. That’s like going to the gym and then having the spotter lift the weight for you all the time. You have to lift the weight, not your spotter. It’s the same thing in math or any other subject where you have to remember information. So if a student learned it yesterday, they should have to reach for it again in their head today. And you know, if you, you ask them and they just blank stare initially, That’s not automatically failure. Sometimes that’s just the moment when the learning is actually being pulled back into place.

When projects camouflage skill defects

Projects can hide weak math. I’m sorry, but they can. See, like a little group makes a design your bedroom, your dream bedroom poster, and that’s their math project. There’s a bed, there’s a desk, there’s a rug, And it’s all colored in, looks really nice. I mean, th-this can be a fun project, right? And if there’s math baked into it and the students are actually doing the math properly, then yeah, it can be a good learning experience. It can pull a lot of skills together. But if you’re checking over the project, you, you check the scale and the kid has the, the bed is 20 feet wide. And the budget has tax added twice, or they, they just added a Percent instead of multiplying, like 6% sales tax. So you just add six cents to the end, something like that. Or the, the area calculation is just like, ugh, it’s just totally wrong, then the project’s really not doing a whole lot for the math learning. And these problems are, are not things that you solve on the fly in a project. They are foundational skill problems. If a student can’t calculate, they can’t measure stuff, they can’t estimate or check whether an answer is ridiculous, The project is, is not a deep project. It’s just camouflaging all these foundational skill gaps.

When algebra gets harder than it should

If six times seven still makes a student stop and think, maybe recompute what it is, then, you know, the, the algebra that they’re supposed to be learning next, it gets a lot harder than it really needs to be. Here’s an example: 6 times ( 2 x – 7 ) + 5 = 23. That’s the equation you wanna solve. You’re in algebra class, that’s what you’re doing. So you’re supposed to be thinking about distribution, inverse operations, keeping both sides balanced. And on top of all that, Part of your brain is going like, “Wait, what is six times seven?” And now your attention is being pulled to that. Now sure, you might say like, “Well, no big deal, right? Just recompute it.” But Now you have to context switch, take some time to recompute this thing, Then pick your head back up and be like, “Wait, what was I even doing before this? Where, where was I?” And this happens like five times in a problem. I mean, just wait till calculus. It’s gonna happen like 20 times in a problem. Like The basics, they need to be easy ‘ Cause hard math is already hard enough that it needs all your attention. And if you’re spending all your attention on the basic maneuvers, you’re just not gonna have any attention to devote to the higher level stuff and really see the forest for the trees, keep track of what you’re doing. That’s when you get lost, and that’s when math gets hard.

Same course, different universe

Two people can take the same college math course and come out thinking that it’s just wildly different things. And I think that should bother people more. You know, one differential equations course spends a lot of time on phase portraits. Another one, no phase portraits, but tons of coverage on numerical methods. And another one, none of that, but tons, tons of coverage on Fourier series. I mean The course title is identical, so what is the course? What is it? Especially if it’s a core class. I mean, come on. A core class should not be just about whatever a professor feels like explaining or digging into this semester. It shouldn’t be a highlights reel of their research interests. The, the students should come out knowing comprehensively What is all the stuff that the subject is built around?

The same wrong step 15 times

One unchecked mistake can quickly become a habit, and that’s the thing a lot of people miss, especially when it comes to math learning. Here’s an example. Say a student sees negative and then open parentheses, then X + five, close parentheses. And on the next line, they write negative X + five, and they had to do it again on the next worksheet. They’re just practicing this thing wrong over and over again. See, if nobody checks this for accuracy and Gives them feedback, you know, “Hey, here’s what you got wrong.” You didn’t get full points on this. If you wanna get full points, You gotta do it correctly. If nobody does that. The kid just thinks like, “Hey, I did the homework. But really what they did is they just rehearsed the wrong move over and over again. So that’s why math practice needs feedback and incentives to respond to that feedback, to take it on board and integrate it into your practice going forward. ‘ Cause otherwise students are not practicing accuracy. They were just practicing whatever their hand already wanted to write.

Here's the missing floor

A student signs up for calculus, and in the first week you got limits. You have to be simplifying algebraic expressions in order to figure out what the limit is. And the student’s really struggling first week. You know, the real problem is earlier. They’re struggling in the first week of calculus because they’re shaky on fractions. They’re just guessing how exponents work. They can’t even clean up an expression without making it worse. That student doesn’t need a pep talk about believing in themselves. It’s not gonna make a difference. What they need is somebody to say accurately, “Hey, here’s your missing floor. Here’s why everything is so hard. Here are your missing foundational skills. And let’s get you to fix them first. Fill them in so that you can actually learn calculus.”

Stop feeding them the next step

I can make a student look way better than they really are. I mean, any tutor can. They get stuck solving an equation: 3 x + 8 = twenty six. And I say like, “Hey, what should we subtract from both sides?” And they pause and I’m like, “What do you think?” And they’re like, “Subtract eight?” Then you’re like, “Yeah, good job, you remembered it.” But The thing is they didn’t actually remember it. You just tipped them off. They have not exercised the skill in full. They haven’t retrieved the process. They’ve just kind of fit to the way I asked them, ‘ cause there was information there that tipped them off. So really what I should be doing is I should just shut up. I should let them sit in silence for longer than is comfortable Because that’s the only way that I’m gonna give them the opportunity to retrieve their knowledge and extend its retention. And that’s the only way that I’m gonna actually find out what they actually know. What are they ready to build upon and what do they need more practice on? You’re the spotter here. You’re, you’re at the gym, you’re doing reps, that you’re the spotter. They are lifting the weight. And being the spotter doesn’t mean that you help them lift the weight on every rep. You catch them when they really can’t lift it at all And you help them get over the edge. But you should really be doing as little as possible. ‘ Cause whatever you’re doing for them, they’re not doing themself and they’re not getting stronger for it.

The email is easier than the skill

I understand why parents send the email. I do. I get it. You know, your kid gets a rough test back, They’re all stressed out. You’re worried about their grade. So you’re like, “Can this one not count? Can there be some kind of extra credit?” You know, there’s a saying in life, everything’s negotiable. Well, the situation here is a little different. It’s the math has to be learned, And higher level math does not negotiate with its prerequisites. You know, if a student doesn’t understand fractions on Friday, they don’t magically become ready for rational equations on Monday. The teacher can’t just say Yeah, don’t worry. This one doesn’t count. I know you struggled on this one, so let’s just forget about it and move on. Well, you never move on in math. You keep climbing the ladder. That ladder keeps building upon things that you’ve previously learned. So if you can’t do those things, you can’t do the new things. And you know you’re just gonna be in the same situation. “Well, we tried this new thing and it was really hard. I didn’t really get it. Can we negotiate the grade, get some extra credit, just drop this test? Once you get into this mentality, this cycle, it just never ends. And eventually you’re so far out of your depth that you, you can’t resolve it anymore. So that’s why it’s so important to just not let this thing kick off, not let it compound into an unresolvable problem. That’s why the standard has to hold.

Three years of boredom

Three years. That’s a long time to be bored But that’s what a lot of middle school math feels like for students everywhere. A student already understands fractions, decimals, negative numbers. They got their arithmetic down pretty well, and they know some basic equations, some basic pre-algebra. They are ready for something real. They’re ready. Let’s go. Let’s go. Let’s learn. But instead it’s just, you know, review packet after review packet. Some more word problems, exercising skills they already got down. It’s just like three years of review, three years of boredom. And what they learn is basically just how to finish their work early and doodle in the margin. Just spinning their wheels and waiting. Waiting has a cost. You know, a student who has their prerequisites in place They should just keep on moving up into higher and higher level math. They shouldn’t just sit around there doing endless review and word problems and different scenarios that are basically the same thing. There’s a lot of talent that can be developed in three years if you seriously try and push it. But if you just do the same stuff over and over and over, it’s a lot of potential lost.

The math decides what you're able to make

it’s kind of crazy how quickly a lot of math will come into play as soon as you try to code up something really cool. You know, s- say a kid builds a little video game. They got a character, and they’re trying to make the character jump. It looks so cool in your mind, and then you code it up, and it sorta resembles it, but There’s something just off, and it’s because you have not baked real physics simulation into the dynamics of your game. You don’t have really, like, gravity acceleration when you’re jumping. You’re not slowing down towards the top and speeding up as you hit the bottom. And the solution to that problem is really just a whole bunch of math. You know, there’s a lot of vectors, a lot of rates of change, and it’s, it’s really deep. You know, I-if you have not learned this math up until this point, if you’re just trying to backfill it, that’s a lot of math they’re gonna have to backfill. We’re not just talking like one day, two days, a week, a month. This is years of math. And that’s why it really matters to get it in place early. Not for decoration, not to collect course names, but to really produce the cool thing that you were imagining in your head. So these math fundamentals, they, they end up deciding what you’re even able to create. You don’t have them? Well, that’s, that’s a pretty low ceiling on what you can do.

The easy A has a cost

An easy A in a math class, that’s not always a gift. Sometimes that’s really just a bill that’s delayed and is racking up interest, just waiting to pounce on you and at the most inconvenient time possible. See, a student could get through with perfect homework scores, but the homework didn’t really ask much of them. You know, maybe it wasn’t even graded. Or maybe it was graded for completion, so you know, just check that they scribbled some symbols on the page. Who knows if they actually learned any math. And the quizzes, the tests, they weren’t great, but they made it up with some extra credit, And now they got an A. And now they’ve graduated up to a course that’s actually asking them to do things? To perform skills, to perform them correctly. And all those skills pull together these sub-skills that they learned last year, right? Except they didn’t actually learn them last year. They’re missing. They got the grade but not the skills. And they hate it because every day they show up and they’re asked to do something that they can’t do. Not because they couldn’t learn it with instruction But because they have been moved forward so far that they are not getting that instruction on their missing prerequisites That pretty much sums up many students’ experience in math.

One problem set a week is dumb

One giant problem set every week is a dumb way to learn math. Yeah, I said it. that’s what happens in universities, colleges, everywhere, and it’s dumb. You sit through lectures all week. You don’t learn much in the lectures ‘cause you’re just passively watching as the instructor does the work. The problem sets are where you actually learn the math But that’s outside of class and you don’t have any guidance. And the problems are pretty challenging ‘cause they’re meant to kinda stretch the ideas that you learned in lecture, except you didn’t actually learn them ‘ cause you weren’t actually practicing them yourself. And so this experience ends up with you and three of your buddies sitting in a library on Thursday night looking at each other like problem six just dropped down from space, and you’re just waiting for an alien to come down and just help you through it. And you know, maybe you figure it out. Probably you don’t. Maybe one person does and nobody else really gets it, but it’s like, “Hey, just, just write it down, Just copy off of them. Or nobody really figures it out, so you send a representative over to some other study group to figure out if they figured it out. And regardless, maybe the feedback from the instructor just comes like 10 days later or more, maybe you wait weeks, months for this stuff to be graded, if it’s graded at all. That’s not a learning rhythm. Math has got to be trained in these smaller, tighter loops. Try, get feedback, fix it, try again. That’s how you get good at something. That’s how the coach really helps you. The coach doesn’t just put on a little show for you and then you do the thing on your own.

Just tell them the tool

I don’t need a seventh grader to rediscover the distributive property from scratch. It’s, it’s just not a great use of time. You know? Just, just give them the idea, Show them why it works, and have them use it, practice it until it’s reliable. I mean, instruction exists Because humans already figured out a lot of math. And the whole point is kind of like, “Hey, let me show you how this works so you don’t have to spend so much time rediscovering it.” So you don’t have to make every mistake, go down every dead-end road just to get to the techniques that work. The whole point is to kind of boost you, speed run you towards the edge of the field So you can get there faster than your predecessors did, and you have enough time to push it further.

Weak algebra ruins your day

During a physics problem, you know, half the time it’s not even the physics where the student breaks, where they have just no idea what to do. You know, half the time it’s just the algebra inside of it. I mean, they understand that the cart is speeding up. They can describe it in words. They can even draw a little motion diagram. They understand the situation, but it’s moving the mathematical symbols around that’s the problem. ‘ Cause it’s no longer just like a imagine a scenario. It’s been abstracted into mathematical symbols, and now it’s less concrete, and it’s like, well, what do you do? Well, you gotta know your algebra. You know, weak algebra does not stay in algebra class. It walks into physics, Ruins your day. Walks into chemistry, ruins your day. Walks into economics, ruins your day. Everywhere it will follow you around, ruin your day. Because it is this tool that you’re supposed to carry and use, and it gets used in so many different subjects. And if you’re trying to learn one of those subjects and you don’t know your algebra, well, guess what? You not only have to learn the subject, you also have to learn the algebra. You’re gonna be struggling with these two difficult things at the same time. You’re gonna get overwhelmed, And that’s what makes it so hard.

Sitting there is not learning

A kid can sit through every math lesson and still not learn the math. And yeah, maybe that sounds obvious, but people act like attendance has these magical powers. You know, they, they watched the lesson on solving inequalities. They, copied the notes, They highlighted the little flip the sign warning. Fine. But if you assume that means that they’ve learned it, then you’re wrong. Two days later, just ask them to solve negative 3x is less than 12. They’ll divide by the negative 3, but they’re probably gonna keep the sign the same ‘ cause they never actually practiced this manipulation. I mean, yeah, they were present for the full lesson, sure, but Math is not absorbed by just being near it. It’s not an osmosis thing. The student actually has to practice solving problems. That’s how it’s absorbed.

The AP student trap

A student is signed up for AP Calculus. Great! But they’re missing a lot of algebra. You know, they don’t know their function notation. That’s the language that calculus is written in. And you can’t fix that just by being ambitious on a course request form. Ambition is good, but math is also stacked. It’s hierarchical. And if the layer underneath is missing, The advanced course doesn’t really make the student advanced. It just makes the weakness harder to go back and admit and remediate. It locks that weakness in. And once that happens, the game changes from how do I learn this new math to how can I fake my way through the class and convince the teacher that I know what I’m doing? How can I cheat? How can I get by without actually learning the skill?

Learning math is like training a sport

Learning math is really similar to training a sport. Take basketball, for instance. If a basketball player only took wide open layups, and they just skipped every drill where they missed, they wouldn’t get any better, right? Nobody would call that serious practice. Everybody knows this. And you know, math works the similar way with the problems that you solve. The useful reps are often the ones that feel a little annoying, a little uncomfortable, a little unfamiliar ‘cause you can’t do it quite right and you, you screw it up sometimes, and that’s why you need to practice it. It’s not impossible. It’s not humiliating. It’s just, it’s just annoying, just tricky enough that your brain has to change to really do it correctly, consistently. That is where the progress is.

ROUND 8

You can't design standards around the rare kid

There are some kids who work really hard by default. They’ll do the assignment, they’ll study, they’ll push through even when nobody’s really forcing them. But those kids are really rare. Most kids are not like that. Most kids respond to the environment. They respond to incentives: whether their work is checked, whether a grade is assigned, whether the standard is real and adults really mean it. So if the whole system is built around this idea of like, “ Well, the motivated kids, they’ll figure it out.” Then you’re designing around the exception and just abandoning everybody else.

The pain signal got misdirected

In school, the pain signal should be this student has not learned the material. That should be the problem that everybody feels. But in a lot of school, the signal just becomes, the parent is upset. So now the teacher feels pressure, The administrator feels pressure, the school feels pressure to lower the standards, make everything easy, just make everything really enjoyable in the moment and kick the can down the road when it comes to actual learning.

Working memory is the bottleneck

The average person only has a few working memory slots, like four to seven. And that means that when a student is solving a problem, They can only hold so many pieces in their head at once. So if basic arithmetic is not automatic, If algebra is not automatic, if notation is confusing, Those things start taking up the whole whiteboard in the student’s mind. And then people say, “Well, why can’t they problem solve?” And the answer is, well, they can’t even see the problem anymore. That’s how full their working memory is.

A grade can become a customer service outcome

When parents are paying a huge tuition bill, The grade can start to feel like part of the service. Not officially. Nobody says it out loud, but the pressure is there. “My kid worked hard.” “My kid’s stressed.” “My kid needs this GPA. We’re paying all this money.” And then the school has a choice: Hold the academic line or reduce the conflict. And over time, if enough decisions go towards just reducing conflict, Then the grade stops being a clean measure of learning, and it starts becoming a customer service outcome.

Opportunity only matters if you're ready

People talk a lot about giving kids opportunities: Research opportunities, internships, advanced classes, special programs. But The thing that a lot of people don’t understand is the opportunity only matters if the student is ready for it If they don’t have the math, they don’t have the coding, they don’t have all the foundations Then the opportunity just turns into a toy version of itself Just something to, you know, keep them busy, ‘cause you can’t hand them the real problem yet where they can actually make progress and contribute So the real goal is not just give the kid an opportunity. You know, the real goal is to build the student who can actually do something and contribute in a real way once the opportunity shows up.

Projects are not magic

Projects are not magic. A project does not automatically make learning deep. A project does not automatically make students independent. A project does not automatically create understanding. If the student doesn’t have the underlying skills, then the project just becomes a mess. They don’t know what they’re doing, and then every single thing takes 10 times longer than it really should. Now, don’t get me wrong, projects can be great. They can be amazing. But only when they sit on top of real skills. Otherwise, you’re not doing project-based learning, you’re doing confusion-based learning.

The standard has to be visible

A standard only works if students can actually see it. , Then the real standard becomes just turn something in. Not understand the material, Not get the answer right, not fix the mistake. Students are smart. They figure out what the system actually rewards. So if the system rewards uploading nonsense, Then you should not be shocked When nonsense is what you get.

A course should not depend on instructor taste

In a core math course, the student’s understanding shouldn’t really depend so much on the instructor’s taste. One professor likes this branch, another one likes that branch, Another professor rushes through the basics so they can get to their favorite advanced topic. But that’s not fair to the student. A core course needs a core structure. The professor can add personality, examples, style, depth. But the spine of the subject should not be optional. Students should not finish the same course With totally different ideas of what they learned.

Mastery is what lowers stress later

A lot of people try to lower student stress By lowering the demand, by lowering the standard. Less work, easier grading, fewer mastery checks. More ways to get by. But That only lowers stress in the moment. In the long term, Mastery is what really lowers stress. Because the student can actually do the next thing. Their arithmetic is automatic, so they’re not panicking in algebra. Their algebra is solid, so they’re not drowning in calculus. The compassionate thing is not always making today easier. I mean, don’t make it hard for no reason, but don’t lower the standard just to make it easier. Because the real compassionate thing is making sure that tomorrow is not impossible.

The transcript can become a costume

The transcript is supposed to represent what the student knows Right? But when the grades get inflated And the homework’s only checked for completion, The tests get removed or you got retakes on retakes on retakes, the transcript can become a costume. It makes the student look academically ready But underneath The actual skills may not be there. And math is brutal about that Because eventually the costume comes off. The student hits a class where every missing foundation matters and they can’t hack it. And they think it’s ‘cause, well, they’re just not smart enough, like they don’t have the potential learn that level of math, but really it’s just they weren’t prepared enough. And there’s years and years and years of missing preparation that’s all coming to light.

Stop making kids reinvent math

The idea that students should discover everything for themselves is one of the biggest wastes of human potential I’ve ever seen. We have thousands of years of accumulated knowledge, Entire civilizations of thought compressed into things that can be taught. I mean, the whole point of instruction is to get you to inherit what took other people lifetimes to figure out. Not just other average people, like literal geniuses through centuries, through thousands of years to figure out all this stuff. So trying to rediscover all of that yourself, that’s like trying to reinvent the microchip with a rock.

You don't learn from introspection, you learn from action

Self-knowledge is not part of your base install. You don’t spawn with it. You just gotta figure it out. And a lot of people think that they’re just supposed to sit around and wait for passion to strike. But that’s, that’s not really how it works. You gotta generate signal. You gotta try things. You gotta build skills. You gotta push through the part where you’re bad at it. And then you pay attention. What did you get obsessed with or at least interested in? Or just what kind of difficulty makes you feel more alive and, and less dead? See, passion is often downstream of competence. You don’t always find it first and then get good. More often, you get baseline good first and then you realize you care about getting better at that thing. So bottom line, don’t wait for clarity before taking action. Movement, action, that’s what produces clarity. It produces information, and information helps you understand who you are and what you want.

Your study method is suboptimal

A lot of students who say and think that they’re studying, they’re not actually studying. They just do things that feel like studying. Rereading stuff, highlighting, Rewriting your notes over and over and over again. All of it feels really productive, but It’s mostly just building familiarity, not learning. The information passes through your brain, so it feels easy, it feels comfortable. And your brain might mistake that comfort for actual retention. When the context is gone, you’re just retrieving from scratch from your own head, That’s the real work. It’s the retrieval of the information, not the consumption, The production. So close the book, cover the solution, solve the problem without looking, only peek back when you’re truly stuck. Because learning isn’t “I recognize it,” it’s “I can produce it.”

Students aren't bad at math, they just have invisible gaps

 Most students who think that they’re bad at math, they’re not actually bad at math. No, see, they’re just carrying these invisible gaps. Thing is, math is not just a list of topics that you can get a little familiar with and then pass and then forget. No, no, no. It is brutally hierarchical. Everything stacks on top of each other. Nothing really goes away. If you miss fractions, you struggle with algebra and if algebra is shaky, then calculus, it’s impossible. But often the system just moves you forward anyway into the next class even if you haven’t really mastered the material in the previous one. And all these missing prerequisites, they keep stacking up over time. They make future classes harder and harder. And from the outside, it just looks like, oh, they’re not good at math. But that’s not the problem. The problem so often is nobody’s checking for mastery before moving on.

Recognition is not recall

One of the biggest misconceptions in math education is this: If something feels familiar, you think that you know it. But recognition is not the same as recall. You can look at a solution and think, “Yeah, I’ve seen this before.” But that doesn’t mean that you can produce it yourself from your head, from scratch, without any clues. And that gap between familiarity versus actual recall, it’s a huge gap. Cause if you can’t recall something, you can’t apply it to new situations. And this is where a lot of students get misled. They think that they can understand something ‘cause they can follow it in a textbook, in a lecture When everything’s already set up for them. But real use of math, of information in general, doesn’t work like that.

Age-based classrooms are structurally wrong for math

We organize math education by age. But math itself, it has nothing to do with age. It’s hierarchical. Every concept, every skill depends on previous ones. So what happens when you group students by age instead of mastery of the previous skills? Well, you’re forcing a system where some students are missing prerequisites, and other students are already past them. In the same classroom, you can have two students, same age. One of them is struggling with fractions, and the other one is solid all the way up through algebra, and they’re ready for calculus. And the teacher is forced to teach one speed, which is nobody’s real speed. The slower students feel lost, the faster students get bored. And everybody is pretending that this is somehow good for everyone’s learning. But math doesn’t care how old you are. All it cares about is what can you do right now. And if we ignore that, we don’t get fairness, we just get an inefficient learning environment for everybody.

The danger of waiting for motivation

One of the most damaging habits in learning is this: waiting to feel motivated before you start. Motivation is just, it’s not a reliable input. Really, it’s more of a result than an input. ‘ Cause so much of motivation comes from success. So when you wait for motivation, You delay practice, which delays competence, which delays motivation even further. It’s a feedback loop in the wrong direction. But when you start anyway and you push through a little bit of discomfort, you eventually start to see progress. A problem you couldn’t do yesterday, but you can do today. Yeah, that, that progress feels good. And that feeling makes you wanna continue. So the loop flips: action to progress to motivation to more action. And everything compounds in this virtuous cycle.

Learning styles are one of the most persistent education myths

A common belief in education is that people have different “learning styles.” Students might have a preference for one style: visual, auditory, or tactile. But the thing is, there’s no strong evidence that matching instruction to a student’s preferred learning style actually improves learning outcomes. What actually matters is a lot simpler, just practice and feedback. A student doesn’t fail algebra because it wasn’t visual enough. They fail because they haven’t mastered the prerequisite skills. And really no reformatting of the explanation fixes that. I mean, yeah, some skills and concepts are best communicated visually, like graphing functions. But that’s a property inherent to the content. It’s not just the visual learners who learn how to graph functions by drawing them. It’s everybody. And in the same way, there’s also lots of skills that don’t lend themselves to visualization so much. It’s more like a procedural muscle memory. Like manipulating expressions and using those manipulations to solve equations. And these are just cognitive movements that you gotta practice. You do the work, you get corrected, and you get better at it. And that loop is not something that you can replace with a picture.

You failed. Congratulations, you're promoted.

I overheard a student on the train say that they failed Italian 1 this year. And then got placed into Italian 2 for next year. I mean, yeah, she was rightfully confused. She’s like, “What do you mean Italian 2? I don’t even know Italian 1!” And that’s insane! But this is what schools do all the time. And I mean, there are some subjects that are more forgiving to that, but in hierarchical subjects like language and math, A student fails algebra and guess what they’re taking next year? Geometry?! , And it, It happens because Making a student repeat a class is inconvenient. It looks bad for the school, it upsets parents and students. So instead, the grades are fudged, Kids go to summer school to simply check off the box that they did it, and they’re promoted to the next class even if they’re missing a ton of prerequisites. And the problem’s just punted for the next teacher to deal with.

Students can learn so much more with efficient teaching

People massively underestimate how much math students can learn When the learning process actually makes sense. A lot of students quit because they think they’re bad at math, but imagine being thrown into pages of definitions, theorems, proofs Before you even have an intuition for what any of it means? I mean, of course it’s frustrating. It’s a bad experience. But if you start with concrete examples, You build intuition and make the basics feel natural, All those abstract ideas, they stop feeling impossible. They start feeling like logical extensions of existing knowledge. And when that’s what it feels like, when that is a student’s experience with math, The student can go way, way further than they ever thought that they would.

Getting to calculus early changes what's possible

People think that high school calculus is the final destination. Like you made it to the top of the mountain and now you’re done. But that couldn’t be further from the truth. See, calculus is the first step to unlocking Huge areas of science, engineering, economics, any quantitative field. So when students learn calculus earlier, it’s not that they finish earlier. It’s that they get more time to learn what’s actually beyond it. More time for probability and statistics, linear algebra, differential equations, And all the ways that math gets applied to real problems. That’s where the most interesting opportunities are… when you combine math and the knowledge of a domain that you’re interested in. So calculus isn’t the finish line. It’s really the starting point where a whole bunch of doors start opening.

When a student's algebra is fake

I can tell when a student’s algebra is fake. I don’t even need a test score. One line. They’re in pre-calc. We’re doing rational functions. And I say divide both sides by X – 4. Half the room freezes. And that’s a problem because the course is supposed to be about functions: graphs, asymptotes, behavior. But suddenly we’re back in algebra land trying to remember what’s even allowed. How do you divide by x – 4? Do you divide by x then subtract 4? No you put one term over x and the other one over 4, and then you subtract them. Math does not forgive skipped steps. It stores them, and then it sends you the bill later at a really inconvenient time when you got a whole bunch of other stuff that you’re trying to figure out. And you just come to the conclusion, “Oh, I’m not a math person.” No, that’s, that’s often not what it is. So often you’re just paying for this old gap that you’re supposed to learn previously and nobody made you learn it.

A lot of calendar for not much distance

Picture a kid who finishes long division in fifth grade. Then sixth grade happens, then seventh and eighth. And somehow the big academic achievement is finally starting algebra 1 in ninth grade? That’s a lot of calendar for not much distance. All of middle school just doing laps around the same math. No, you know, a kid who can handle arithmetic, they should be doing algebra. And a kid who can already handle basic equations, they should be doing functions. And a kid who can handle functions, they should be moving towards trig. Readiness should matter a lot more than the calendar. That one idea would fix a lot.

Math should be carrying the project

A project can look amazing and teach almost no math. Nice title, colored arrows, maybe a little cardboard bridge. Then you ask one question, “Where did this number come from?” And everybody’s quiet. That’s the part people skip. If a student is designing a mini golf ramp, They need slope, they need measurement, they need proportional reasoning. Maybe a little quadratic thinking if the ball leaves the ramp. Otherwise, it’s not really a math project. It’s arts and crafts with some numbers sprinkled on top. I mean, projects are fine, they’re great, but the skills have to be real first. The math should be carrying the project, not hiding behind it.

The road after calculus

A student says, “I’m taking calculus. I’m almost done with math.” And I get why they think that. That’s how school makes it feel. But open a course catalog. After calculus, the road gets even longer. Linear algebra, multivariable calculus, probability, statistics, differential equations. Then you get to courses with names that don’t even sound like math to a high schooler, like game theory and topology. So getting to calculus early, It’s not really some trophy at the end. It’s more like you get the key card to actually enter the building. And if a student wants physics, machine learning, economics, engineering, You don’t want them arriving at that door at the last possible second. You want them with some room to walk around and explore.

Why placement matters

Student signs up for calculus. Transcript says they passed pre-calc. GPA looks fine. And then the first quiz has a problem with fractions with it, and everything falls apart. Not the calculus, the fractions. See, that’s why math placement exams matter. You can’t really guess math readiness from vibes or a padded inflated grade from last year. At some point, somebody has to ask the rude little question, can you actually do the prerequisite work? Because putting a student in the wrong math class doesn’t make them more advanced. It just makes the failure more confusing.

Proofs after numbers

A student reads a proof about matrices. They nod, looks serious, looks mathematical. And then you ask them to row reduce a three by three matrix, and they’re lost by the second line. It’s backwards. The proof is not supposed to be the first time the objects feel real. You should have handled the thing first. You know, moved the rows, seen the pivots, made some mistakes with some actual numbers. If you do that, then the theorems, the proofs, they actually have somewhere to land.

The smart kid who camps

Some really strong students get stuck ‘ cause they only want math that already feels good. They’ll do puzzle after puzzle after puzzle. But ask them to learn trig identities, and then suddenly the math is boring. They don’t wanna do it. They don’t wanna try. And that’s the moment that adults have to start being adults. Not mean, not dramatic, just clear. You don’t get to camp forever in that part of math where you already feel impressive. There are so many doors that only open after a less fun stretch. A student who avoids trig is also avoiding physics. A student who avoids calculus is avoiding half the serious quantitative world. Kids don’t always see that, so somebody older has to.

That's what the calculator showed

A student types something into the calculator and a graph appears and they say, “I get it. I understand it.” Well, maybe. Let’s check. Why does the graph cross there? Why is it flat at the top? What happens if I change this coefficient? If their answer is just, “Well, that’s what the calculator showed,” then they didn’t really learn it. And in math, that difference matters a lot.

Code asks you math questions

A kid builds a little game, the character moves left and right. The score works, the menu works, and now they want the ball to bounce realistically. Well, now it’s math. Now it’s velocity, it’s angles. A little trig, maybe some vectors. And this is the moment a lot of people realize that coding was never really separate from math. They just hadn’t quite reached the point where the math showed up. See, basic coding makes you make buttons do things. But stronger math lets you make systems behave. That is a very different ceiling. So when people say, “Do kids really need all that math?” For some jobs, maybe not. But for the interesting technical work, oh yeah. Eventually, the code asks you math questions.

The fast kid who lacks persistence

Some students look amazing for 10 minutes. They see the trick, they get the first problem right. They’re fast. They’re very fast And then the assignment gets a little longer. Problem 6, a little annoying. Problem seven’s got this weird fraction. Problem 8, It uses something that they screwed up yesterday and had to correct. And then they’re just gone. I mean, not physically, but just mentally they’re no longer here. They’re no longer engaging with the process at the first sign of discomfort. And that’s a very important part of math, staying with the process even after your first attempt fails. Checking stuff, fixing stuff, coming back tomorrow to keep iterating. And then the next day and the next day. See, a slower student who actually does that can blow right by a faster student who just keeps on bailing at the first sign of discomfort.

Lowering standards equals future pain

Nobody wants to be the adult who has to say, “This isn’t good enough.” I mean, the teacher doesn’t want an angry email from the parents. Administrator doesn’t want a meeting. The parents don’t want tears at the kitchen table. I get it. You know, you just accept the work, soften the grade. And just whatever missing skill it was, just rename it as stress or a different learning style, or just a rough week. And everybody gets one peaceful afternoon. But the thing is, the math doesn’t disappear. It waits. And it shows up later. As a student who just can’t handle the first college class they take that doesn’t bend, doesn’t let them off easy, actually holds a standard. See, that’s the part that people miss. Lowering the standard may feel compassionate in the moment. It’s just more pain in the future. Nobody fixes the skill, nobody removes the pain. You just schedule it in the future, lots of interest. A big learning debt bill at the most inopportune time.

Cognitive ankle weights

You ask a tenth grader to factor something. I mean, nothing crazy, just like x squared plus 11x plus 28. And before they even get to the algebra, they’re stuck on the multiplication! What’s 7 times 4? 7 times what? Huh, let me think. See, the problem is that hesitation is not tiny ‘ Cause now the algebra problem has two jobs. Job one is do the algebra, and that’s already a full-time job. Job two is recompute the basic facts. Another full-time job. And you’re having to do both of these at the same time. And it just slows you down. It scatters your mind. And it makes it so much harder to see the bigger picture of what you’re doing. You see, math facts are the stuff that you’re supposed to stop spending attention on. You gotta know them cold. Because if every problem makes you compute multiplication from scratch, then you’re just carrying these cognitive ankle weights around that nobody can see. And these weights just get heavier and heavier over time.

Check what they can do from memory

Give a student a take-home math test and you may learn something. Just maybe not about the student’s knowledge. You may learn who knows what site to paste it into. You may learn who works together with who. Or who has an older sibling nearby to do their work for them. That’s not the same as knowing math. Put that same student at a desk with a blank paper, then it’s honest. But if you never check what a student can do alone from memory, You don’t know what they can do.

The first week matters

The first week of class is not when you prove that you’re nice. It’s when you prove that the work is real. That you have standards that need to be met. ‘ Cause if students learn that late work slides and halfway done work counts for most of the credit And just saying, “I don’t get it” all the time gets them off the hook for everything, then you have taught them the course already. Not the math course, the hidden course. The one where effort is optional. Just tire out the teacher and then you get a good grade. Now, don’t get me wrong, you can be warm. You should explain things clearly. You can laugh with your students. But the standard has to show up early. A student should know, If I turn in nonsense, somebody’s gonna notice. If I skip the practice, it will matter. If I wanna build my grade, I have to build my skill. The stakes have to become serious enough for learning to matter.

We covered it

We covered exponents last month. Okay, but did they actually learn exponents last month? ‘ Cause those are not the same sentence. A teacher can cover negative exponents on Tuesday, the worksheet gets collected on Wednesday, class moves on on Thursday. And then a month later, a student sees 2 to the -3, and writes - 8. That topic was not learned. It was just visited temporarily. You see, math has a very annoying feature. It keeps asking whether things that the student learned last month actually became a part of the student and whether it was actually reviewed enough to stay a part of the student. Not whether it appeared on the calendar, not whether the slide deck had it. Not whether everybody nodded in class. See, covered is easy, but learned and retained, that’s the standard.

A course-wrecking problem

The worst time to find out that a student can’t handle algebra is the first week of calculus. Now the teacher has two choices. A, they can slow the whole course down Or B, they can just keep going and watch the student slow down, fall off the rails. Neither one is good. And it’s just a bad situation. See, the fix should have happened years earlier when the gap was still small. I mean, yeah, a shaky fraction skill in fifth grade is annoying, But it’s also relatively quick to fix. It’s not such a big deal compared to a shaky fraction skill in calculus, that, that’s a course-wrecking problem. Same gap, different price. And that’s why early standards matter.

Familiarity versus learning

Students do this all the time. They look at a problem and they say like, “Oh, yeah, I know this one. I know how to do this. This is easy.” And then they start actually working it out. They start writing. And by line two, it’s already wrong. See, the thing is, “I’ve seen this before.” That’s not the same as “ I can do it.” That little feeling of recognition, of familiarity, of having seen it before. It’s comfortable, it’s fast, And it feels like learning. But math is not just some vocab quiz where pure recognition gets you over the hump. It’s more like a mental sport. You gotta be able to carry out the techniques. It’s not enough to have seen somebody else carry out the technique. You have to be able to do it. A student can watch 10 videos on factoring and still have no idea how to factor. Only honest test is working the problems yourself.

The puzzle kid ceiling

There is a type of student who can absolutely destroy math puzzles. They got their number tricks, they got their clever shortcuts, competition style little gems. Yeah, they’re sharp, no question. But then you start giving them material from pre-calculus. It’s not flashy. It’s just functions, Trig, identities, graphs, inverse notation. And they hate it! Because math, it’s not just a clever one-off, it’s an actual language that they have to build. They don’t already have it all loaded into their head. They have to continue learning the new vocabulary, The new objects, the new moves, the new techniques. And unfortunately, that’s the ceiling for some strong kids. They get really good at that isolated cleverness with things that already feel comfortable, and they just avoid the boring connective tissue. The stuff that’s actually leading them deeper into math, not just expanding sideways. You can get by for a while with clever tricks, but you can’t puzzle solve your way around every missing prerequisite forever. At some point, talent has to become new coursework. Otherwise, the student just hits a plateau and stops growing mathematically.

Talking to individual customers is exhausting but extremely useful

When we, this whole thing started, Sandy was like, “Look, you know, we don’t have any way for people to pay or sign up, so I’m just gonna schedule Zoom calls. And then if they wanna purchase it, a subscription, then you can just send them the Stripe link. And I would do like, 3, 4 hour-long Zoom calls a day. I’m talking like a demo with a new customer. They’re not always giving you a lot to work with. They’re kind of just sitting there listening, so you have to keep all the energy up and you’re trying to get, um, some understanding of what it is they need and their points of confusion or whatever. And even if it goes well, it’s still tiring. And I think I did over 200 in the first six months, maybe six to nine. I mean, it was exhausting. But it was e- it was extremely useful because, you know, when you talk to hundreds of customers, you really get an understanding of what their pain point is. As an entrepreneur, you, you have some idea of the problem you’re solving, right? You know what problem you wanna solve, but that may not be exactly the right problem, or at least the way you’re solving it or the way you’re communicating it or what the priority of the aspects of the problem that’s being solved. You know, all these things are, are kind of unclear. You have this sort of probability distribution and ranking of all these things, and those things kinda get resorted over time the more people you talk to, and it’s like, “Oh, geez, there are more adults,” or, or there’s more of this situation or more of that situation. So it’s extremely useful, but it’s time-consuming and it’s exhausting.

Customer feedback gives you the confidence to move faster

That’s why it’s so helpful to talk to customers. ‘Cause, you know, before you talk to customers, you could kind of delude yourself into thinking that you really know, and you might have a good instinct. It might be right. But when it really, when the rubber meets the road, it’s not 100% accurate, and you, deep down you know that, and so you’re always kind of second-guessing yourself and it slow- slows down. So when you get confirmation and your confidence builds that you have the right signal and the right idea, then you can just move. You really move.

Why we don't have a sales team

We don’t have a sales team, we don’t go sell to schools, it’s like I’d rather them come to us and say, “We have a problem. We are way underperforming. We have a ton of kids who are below standards, and I am willing to do what it takes, you know, within reason, but I’m willing to put in a solid effort to get there. Can you help me?” And I’m like, “Yes, we can, 100%.” we can’t help people who don’t realize that they need to be helped or willing to put in some effort and, and engage with a solution.

Successful learning makes you independent

The student gets stuck, so they tap the button, you know, “Give me a hint.” Eh, still stuck. Another hint. Now the first step is basically written for them, and they just continue doing this through the problem, just hint after hint after hint. When they get to the end of the problem And they think like, “Okay, I got it. I, I figured it out. I’ve learned it,” right? I mean not, not really. That’s not how it works. The platform just carried the heavy part for them. The real question is, can you do this problem without hints? I mean, don’t get me wrong, help is useful when it teaches you how to think, scaffolds you through. You gotta see how it’s done right, and then you emulate it. But that constant step-by-step every single time you do a problem, you know, and it never really forces you to do it on your own. It’s just always there to help you out. If you don’t strip it away, it becomes a crutch. You can’t think without it, and you’re really not thinking that much in the first place. That’s why the math practice has to gradually strip away that scaffolding, strip away those hints until the student is just alone with the problem and they’re able to do it independently. That’s the desired outcome of successful learning.

Strong math prep opens doors

A kid shows up to college and they’re thinking, “You know, I might wanna do a quantitative major, like something technical, you know, like physics or electrical engineering, economics, computer science. But then the thing that turns them off is they find out that they’re just so woefully unprepared for the kinds of mathematical skills that the major is gonna ask them to exercise. They’re just missing so many prerequisites, and there’s just this huge swath of knowledge to fill in in order to be prepared to take really even the first class for that major. And you know, it’s, it’s not even always that they were lazy. It’s not that they’re incapable. It’s just that for whatever reason, they didn’t build that math early enough. All these cool majors, every option requires them to scramble. And they got enough going on in their life as it is. You know, just let’s just do an easier major. There’s some lost potential there. You know, you don’t really build up your foundation, and doors just end up closing. And that’s really the importance of a strong foundation. It’s not about just forcing like one career path. You don’t have to be a mathematician. But there’s a lot of other stuff that you might wanna be that requires serious math. And you know, you wanna build up the foundation that leaves those doors open when the student finally knows what they want.

Give me three examples first

A professor goes up to the board, writes a definition, then a theorem, then another theorem, then another theorem. And all the students are copying furiously, just transcribing that information from the board into their notes. And I’m just sitting there thinking like, “You know, can we actually just see the thing first?” I mean, and not, not the formal version, just the thing. Give me three actual concrete calculations with, with numbers that we can run through, like real concrete examples of this thing that we’re talking about. If you do that, then the definition actually has something to grab onto. And see, that’s why so much advanced math feels much harder than it really has to be. The reason why it’s hard is that people so often just skip the step of concrete examples. You arrive at the abstractions with no actual examples in hand. And then you wonder why it feels like just pushing symbols around.

Notes are not the goal

One of the most misleading things in math is a perfect notebook. Yeah, you know what I mean. The color-coded headings, all these nice boxed formulas, Every example copied so neatly from the board. And you flip through and it’s just like eye candy. It looks so, so pretty and so beautiful But then you ask the student to, you know, close their notebook and just do this basic problem that they should know how to do by now. And they can’t even start it! You see, why that happens is the student didn’t actually learn the lesson. They just recorded it. They transcribed it, and that’s not the same thing. See, math notes are not really an indication of learning. They’re barely even scaffolding. Now, before you jump down my throat, I know, yes, they are a way to keep you paying attention when you’re listening or you’re reading, and that can be helpful for some students. But that’s just the consuming information part, and that’s not really where the learning happens. See, the learning isn’t really happening until you’re pulling information out of your own head. It’s the retrieval of information from your own memory. That’s what produces the learning: Retrieving information from memory and using it to solve problems. Not just transcribing information from one place to the next.

You can lower the bar, or you can teach the missing math

Sometimes a school doesn’t really fix math skill deficiencies and instead just changes what counts as okay. What’s the bar for, you know, being able to do something versus not being able to? Well, people just push it down low and just make up stuff. You know, like instead of can you solve the problem correctly, it just turns into did you try to solve it? Or maybe you get a B, but then you do a bunch of extra credit, and now you bump up to an A. I mean, yeah, the class average looks better, But the students didn’t really get stronger. They’re not more capable. It’s just a, a metrics, you know, cooking the books kinda thing. And that’s a problem because everybody can point to these new numbers and be like, “Oh, you know, look, it’s fine. They got, they got an A in the class, right? They must know their skills, right?” But the problem is math itself does not really work like that. It’s a brutal hierarchy. You can trick ‘em. You can trick their parents into thinking that they’re actually doing really well. But, you know, eventually the truth comes out when they try to do the more advanced maneuvers that require these more basic components. Now they’re confused, their parents are confused. And we come up with all these fake explanations like, “Oh, the advanced teacher is, is unfair.” Or like, oh, the kid just hit a wall ‘cause they’re bad at math. But the truth is simple. They just never learned the prerequisite skills because they were never forced to do so because somebody just lowered the bar last year and the year before and so on. So, I mean, you can either lower the bar or you can teach the missing math. Only one of those actually helps the student.

Behavior is not the same as accuracy

There’s a student that teachers really like. They’re really polite, they engage in class, they submit everything. I mean, they’re on top of it, and they take really pretty notes. They don’t cause trouble. They’re kind to their classmates. I mean, what more could you want? Well, there’s actually one really important thing. Accuracy. Sometimes this kind of perfect student can actually be wrong a lot. But they continue moving through math for years Because the adults, the, the teachers, they reward the behavior around the work. And the behavior matters, but it is not the same as accuracy. See, etiquette and effort are necessary, but they’re not sufficient. They cannot replace the skill.

What skill actually failed here?

Student asks for help in a geometry problem There’s a triangle, there’s a missing side. I mean, yeah, it looks like geometry. But the real problem is with the fractions! I mean, they set up the proportion correctly, but they kind of screwed it up when they cross-multiplied. And if you only focus on the triangle, you might miss the diagnosis of what’s actually preventing them from solving this problem. See, this happens constantly in math. The new topic that they’re trying to learn is not always the bottleneck. You know, the, the lesson’s about similarity, but the gap is in the division. Or maybe they’re trying to learn trig, but the gap is factoring. Or the worksheet says calculus, but the gap is actually algebra from two years ago. And that’s why good help in math starts by asking, “What skills do you actually know?” What gaps are gonna fail when you try to do higher level material?” Not just, you know, “what are you supposed to be learning?” “What chapter are you in?”

The unit test trap

So the class marches through the units, right? Unit 1: exponents, unit 2: radicals, Unit 3: quadratics. Unit 4: rational expressions. It’s nice and tidy, right? One unit at a time, just get through it, mark it done, go on to the next one. Well, it’s nice until the final exam mixes them all up, and then students act betrayed. They really struggle because each of these skills that they learned was in this tiny little fenced off area. When the board said radicals and the worksheet said radicals, and you know, this whole week we’re doing radicals, well, it was pretty clear what to do. Everything’s radicals, so just think about your radical rules, and don’t even think about anything else. Well, the problem is when that context disappears, The skill disappears with it. And that’s not real readiness. Real readiness doesn’t have that kind of context dependence. You know, all the math that you encounter, it’s not gonna keep walking around with a label on its forehead. Maybe we do that initially just briefly as a kind of scaffolding to really just help you focus down on the new thing that you’re trying to learn. But pretty soon after that, your practice has to start mixing things up. Not just being able to apply a skill when it’s all teed up for you, but actually knowing when to apply that skill as well.

The back of the book

A student says, “I understand it now.” But what they mean is they just looked at the answer in the back of the book. And it showed a few intermediate steps, and then they just followed along with it. I mean, don’t get me wrong, That’s useful for checking your work, But it’s not the same thing as actually solving the problem. The real moment is before the answer arrives. That’s where the skill lives. And if the student can only understand the solution after seeing it, they’re not independent yet. And solving problems, being able to carry it out yourself, not just watching along as somebody else carries it out and you just kind of nod, no, the, the independence is the whole point. And that’s the bar for what it means to actually understand the material.

Explanation is not transfer

The teacher explains it perfectly. You know, the board is so clean. There’s a really nice example. Everything makes perfect sense when you’re watching it. And then you open the homework. You see the first problem and you’re like, “ wait, how do you do this again?” Yeah, y- you know what I’m talking about ‘ cause this happens to everybody. And it doesn’t mean that the explanation was bad or that you’re dumb or anything. No, all it means is that explanation is not transfer. You don’t really learn something by watching somebody else do it. You learn it by doing it yourself. By having guidance beforehand, that speeds up the learning process. But that’s a catalyst for the learning. The learning itself happens when you carry the idea yourself. you can think that you understand somebody else’s solution, but it might still take some practice to actually produce your own. The practice is where the teaching either becomes real and you absorb it and you retain it, or it just evaporates. See, a, a great explanation, it starts the process, it guides the process, but it does not finish the process. The only thing that does is actually working problems yourself.

When enrichment becomes avoidance

Sometimes a kid who’s strong at math gets enrichment, You know, like some logic puzzles, some math games, some challenge problems. That’s fine. Some of that can be fun and some can be helpful for their learning. But if a student is ready for algebra, and we keep handing them these cute little puzzles or arithmetic challenges, And we never move on to algebra? They just stay in arithmetic land? That’s not enrichment anymore. That’s avoidance. See, the main sequence of math matters: equations, functions, geometry, trig, calculus, matrices, differential equations, and onwards. I mean, it matters not because those names are fancy, but because they unlock the next serious thing. A student can spend years feeling clever on side quests while the road ahead just sits there. They kind of stay put. I mean, I, I like puzzles. But puzzles should not become a holding pen for a student who’s ready to move.

Can they do more than they could before?

Your kid stops complaining about math and that feels great, right? No arguments at the table, no tears, no panic before homework. But check one thing before you start celebrating. Did the math actually get easier because they got stronger? Or ‘cause the demands got softer? And that difference matters a lot. maybe math class became peaceful because your kid finally understands fractions. That’d be awesome. Hopefully that’s the case. But it can also become peaceful because nobody’s asking them to deal with fractions anymore. And that’s not great. So what I’m getting at is that low stress is not automatically a healthy signal. Sometimes it means the student is really well prepared and the work is well calibrated and they’re learning a lot, making a lot of progress and it’s going really smoothly. But other times, it can mean that the weak spot is just being avoided. So don’t just ask, you know, are they less upset? Ask, can they actually do more than they could before?

A low score can be useful if you listen to it

Class average was a 54, so tests got curved. Now everyone feels a little better, right? I mean, hold on. Did the curve actually teach anybody how to solve the systems of equations that they messed up on the exam? Did it fix their sign mistakes to make their fractions less shaky? No. It just changed the story that the number told. But a low score can actually be really useful if you listen to it. It can tell you like, “Hey, you’re missing a lot of prerequisite skills and we gotta get those in place.” You didn’t really learn the material that well, so we’re gonna address that now and fill in those skill gaps, get you back on track with your learning process. But if you just soften the number and just move on, then that doesn’t really fix anything. And students just go on and carry that missing math forwards with them.

The best kind of help can sound a bit annoying

A student turns in work that’s half right. Teacher knows it, parent might know it, student might know it too And everyone feels kind of bad, so the teacher’s just like, “You know, it’s okay. You tried.” And that can be a really kind thing to say, but it can also become kind of a trap if that’s the end of the conversation and it earns them credit. I mean, don’t get me wrong, trying is good, should be encouraged. That is such a critical component of learning. But trying is also the beginning. It’s not the same as actually learning the skill. A better kind of help, it might sound more annoying, but it’s like, Good effort. You’re getting there. And you’re gonna get there, I believe in you. But take a look at line three. There is a mistake in there, so see if you can identify it. And if not, then go take a look at that example, and then go and try again. That’s not being harsh. That is really the kindest kind of help that you can give the student. Because otherwise it’s just false comfort now That just compounds into a whole bunch more confusion and pain later.

Students should not meet real standards for the first time at 18

Some students don’t meet a real math class until college. And I’m not talking about the subject matter or how hard the math is. What I’m talking about is before that, everything had cushions. There’s corrections, there’s extra credit, the review sheet looks just like the test, partial credit for almost every single line you write down, even if you have no idea what you’re doing. And then they hit a course where the professor is a little bit more, some might say a hard ass, but I’d say honest. There’s no extra credit. There’s no review sheets that tell you exactly what’s gonna be on the test. You don’t get partial credit just for trying. You get it for results, for making real successful partial progress. There’s no rescue built into the grading policy. And now the student thinks that college math is just, you know, impossible. And yeah, maybe it’s challenging. But really what they’re feeling is more that the rigor and the accountability just arrived all at once. Maybe it’s just me, but I don’t think students should meet real standards for the first time at 18.

Did they actually do the work?

That little submitted tag can fool you. The kid turns in 20 algebra problems. Grade book says complete, everybody relaxes. But then, you actually look at it and it’s full of nonsense, even gibberish. Mathematical symbol soup. The whole assignment is like this. See, that’s the thing. The submission portal can say it’s done, but the math is still broken, unlearned. They didn’t actually produce the work. They didn’t learn how to do the work.

ROUND 7

Puzzles can become a comfortable plateau

Puzzles can be really fun. They’re interesting, they’re challenging, they let the kid feel mathematically alive. But eventually puzzles can become a comfortable plateau, where a student is still ahead of grade level doing impressive things but not really climbing up into that next layer of math. At some point, they have to make the leap into calculus, linear algebra, differential equations, you know, the serious tools. Because the tools can unlock a much bigger world than puzzles alone can reach.

A holistic story cannot replace prerequisites

Holistic admissions might tell you a lot about a student’s story: Their activities, recommendations, and grades. But the math placement has one very specific question. Does the student know the prerequisites? Do they know algebra? Do they know calculus? Because if they don’t, then the next course is gonna expose that immediately. See, a beautiful story does not make the prerequisites disappear. At some point, the math asks, what can the student actually do? And the answer has to be real.

Concrete examples are not baby math

Concrete examples, that’s how the real understanding gets built. And a lot of undergrad math skips that step because examples feel less prestigious than theorem, proof, theorem, proof. But that is backwards. The computations are what make the objects feel real, and once the objects feel real, the proof is not this mysterious ritual anymore. It’s just explaining something that you already understand. So skipping those examples doesn’t make the course deeper. It often just makes the confusion harder to diagnose and remediate.

The real prize is compressing time

The real prize of learning advanced math early is not just bragging rights, it’s time compression. If a student shows up to college already knowing the core engineering math and all the coding applications, then they don’t have to spend the first two years just skilling up enough to become barely useful. They can start trying things right away: research, internships, harder projects, real problems. And that matters because young people are still figuring out what they’re interested in. And the earlier they become capable, the earlier they can test serious opportunities and the faster they can find the thing that they actually wanna pursue and lean into that.

Math facts are not a tiny skill

 People sometimes talk about multiplication facts like they’re some kind of small old-fashioned detail. But the thing is they’re not, because if those facts are not automatic, then every algebra problem gets heavier. See, the student is not just thinking about variables or equations or structure, they’re also spending mental energy on basic arithmetic. And working memory is limited. So now the algebra problem is competing with the multiplication inside the student’s head. That is why automaticity matters because it frees up the student’s head space to think about the harder thing.

Don't hold students hostage to the calendar

The rule should not be, “ Hey, you’re in eighth grade, so you do eighth grade math.” No, no, no. The rule should be what have you mastered? If the prerequisites are mastered, move the student forward. If the prerequisites are not mastered, fix them. That’s it! The calendar should not be the thing that’s controlling the learning. Mastery should control the learning. Because when students are ready and you make them wait, you’ll waste their time. And when students are not ready and you push them forward anyway, you create holes that follow them for years.

The worst time to remove measurement

The pandemic created a ton of learning loss. And then right around that same time, a lot of schools decided to remove the test scores, which were the only metrics that were giving a useful signal of actual math readiness. That is the worst possible time to remove the measurement because now you’ve got students with inflated grades, weak foundations, less objective placement data, and everybody’s pretending that the transcript tells the full story until the student gets to college level math and the truth shows up all at once. Not because the problem suddenly appeared. It didn’t come out of nowhere. It’s just that the system stopped measuring it.

Being ahead of grade level is not the same as progressing

A kid can be way ahead of grade level and still not progressing much anymore. That’s what adults have to notice because on the outside it looks impressive, right? The kid is doing puzzles. The kid is doing math above their peers. The kid is clearly talented. But talent development is not just about being ahead. It’s about continuing to climb, and at some point, you have to move into the next level of the domain. And sometimes that means learning the thing that the kid is avoiding because that thing unlocks the next mountain.

The fastest looking path can waste the most time

Sometimes the fastest looking learning path is actually the slowest. You rush ahead, you stack topic after topic, you avoid mastery checks. You turn a blind eye to weak skills, you don’t do the problems. This may feel like progress because the student is moving. They’re going through the material, they’re consuming, they’re nodding their head. It may feel like information is passing through their brain, but none of this is actually being learned or retained. And so then later, everything has to be relearned. So the shortcut was not actually a shortcut. It was a debt. And eventually the student has to pay it back.

A transcript can look better than the student's actual skills

A student’s transcript can look pretty good while their actual math skills are just full of holes. And that’s the scary part because from the outside, you see the grades, you see the course names, the recommendations, the whole polished package. But then you put the student in front of fractions, exponents, basic algebra, and suddenly the signal collapses. And you see that the paperwork was not actually measuring what people thought it was measuring. It was measuring the system’s ability to make the student look ready, not whether the student actually was ready.

Weaknesses don't stay in one grade level

A missing skill doesn’t stay where it started. If multiplication is weak, algebra gets weak. If algebra gets weak, calculus gets weak. If calculus is weak, it’s hard to build more college-level math on top of it. The weakness just travels along with you on your educational journey. That’s how it goes in hierarchical subjects like math. And that’s why early standards matter so much. See, you’re not just deciding what a student does this year. You’re deciding what they carry along with them into every course after this year.

Difficulty vs. Rigor

People confuse difficulty with rigor. If you take undergrads and give them a graduate level textbook, and you bury the intuition, you skip the concrete examples, and you make them fight their way through it. Yeah, the course is hard, but hard does not automatically mean well-designed. Sometimes it just means inefficient. Sometimes it means painful for no reason. See, the thing is, real rigor is not about making the path as brutal as possible. Real rigor is about getting students to understand the material without wasting their energy on avoidable confusion.

The thing kids resist may unlock the thing they love

Even a gifted kid can resist the exact subject that later becomes one of their favorite tools. I’ve seen it happen over and over and over again. They need adults to help them see beyond the mood of the moment. I mean, the kid’s just thinking, “I don’t feel like learning this right now.” The adult has to see that this unlocks the next level And try to communicate to the kid or otherwise just get them through it. Because so often at first it’s resistance, and then it becomes kinda useful, and then it becomes enjoyable, and eventually, it shows up inside that advanced work that the student really cares about. I mean, sometimes the door to the interesting stuff is the topic the kid just doesn’t wanna learn yet, and they need an adult in their life to keep them making progress.

College admissions needs real academic signals

The problem with removing test scores is that you still need some real academic signal. If grades are inflated and every recommendation is glowing, then how do you know what the student actually knows? Especially in math, because math has prerequisites. You can’t just vibe your way into calculus If you’re missing algebra or fractions, exponents. See, a placement signal is not an insult. It’s just a way to avoid pretending that a student is ready When the foundations are not there yet.

Improvement is the motivation

Deliberate practice is not motivated by the practice feeling amazing. That’s not the point. The motivation is that practice improves performance. You do the hard rep because it makes you better. You face the weakness ‘cause that’s where the growth is. And this is where a lot of education gets confused. If every activity has to be enjoyable in the moment, You lose the thing that actually develops skill. The reward is not always in the rep. Usually the reward is realizing I can do something now that I couldn’t do before.

The steady student wins

People sometimes over-focus on raw aptitude. I mean, it’s important, But there’s another variable that matters a lot. And it’s called conscientiousness. You can have a student who’s not the fastest, not the flashiest, They’re not getting everything right on the first try, but they align with the process. They do the work, they correct their mistakes, They keep moving, they’re reliable. Meanwhile, the smart student who only works when it feels easy, they can fall apart the moment the work starts pushing back. Talent is great, but you need also the reliability because that reliability is what compounds.

Soft standards become hard consequences

When standards get lowered, It can feel really compassionate in the moment. Just let them move on. Give them the higher grade. Don’t make the homework too painful. Don’t test directly. But these skill deficits don’t disappear. They travel with the student And eventually the student hits a serious course, And the consequence arrives all at once. These soft standards early can become really catastrophic consequences later.

Too much depth in one day becomes fake progress

There’s this version of learning that looks really aggressive, But it’s actually just fake progress. You take one topic and then stack another on top, and then another after that, and then another on top. You build this tower of hierarchical knowledge all in one day, and you think it’s sturdy, But it’s not. ‘ Cause none of those ideas have consolidated yet. Every new layer is sitting upon A foundation your brain has not really stored yet in long-term memory. That’s why going too depth-wise in a really short amount of time can overload working memory. It feels like speed, But really you’re just trying to build a tall tower on wet concrete.

The parent-teacher boundary disappeared

When I was in high school, teachers did not meet with parents all the time. That boundary was just different. But now, Parents send emails to the teacher, then the principal, And just make this whole situation painful enough that the school starts bending. And that changes the system Because now the teacher is not just teaching the student. The teacher is also managing the parent, And administration is managing the parent, And the standard gets negotiated and bent by all that pressure. So you get your never-ending supply of extra credit, retakes, retakes on the retakes, And parents learn That if you just apply enough pressure, Your kid can come out with a grade that’s way higher than they deserve.

A check mark is not feedback

A check mark on homework? That, that’s not feedback. If a student writes something wrong, and it still gets marked as complete, full credit, then the student learns the wrong lesson. They learn that all that matters is you turned it in, and it doesn’t really matter whether you got it correct, did it properly, actually learned anything. So they can half-ass it, just turn in slop, gibberish, not really try, not really learn. But the errors are accumulating, the lack of knowledge, lack of skill, it’s accumulating. And by the time a student hits a serious class, It looks like they suddenly fell apart. But they didn’t suddenly fall apart, it’s just they were never getting corrected along the way.

ROUND 6

A toy problem is what you get when you're not ready

Justin Skycak (00:00) When undergraduates get research opportunities,

a lot of the time they get a toy problem.

Not because the professor is being mean,

just because the student isn’t ready for anything bigger yet.

They’re still missing too many basics, need too much supervision.

They can’t exactly be counted on to move the real work forward.

But,

if a student

shows up already knowing the core engineering math,

and they can code,

and they understand the applications,

that changes the entire situation.

Now they’re not just some puppy dog to manage.

They’re someone who can actually contribute.

You can't tighten the reins later

Justin Skycak (00:00) A lot of teachers make the same mistake.

They start too loose, too friendly, too casual,

and then when their class gets completely out of control,

they try to tighten the reins later.

But by that point, the kids have already decided

how seriously or unseriously they need to take you.

And once that tone is set,

it’s very hard to undo.

So that’s why you start firm,

you make the standard obvious immediately,

and you hold students accountable for meeting it.

And then you can loosen up later

because the respect is already there.

The parent has to be on board

Justin Skycak (00:00) With gifted kids, the parent has to be on board

because the kid may be brilliant,

but the kid doesn’t have the long perspective yet.

They don’t know what happens three years from now.

They’re probably not even thinking about three months from now.

So if the parent treats resistance as the final answer,

the kid can just stall out.

But if the understands the long game,

then the kid is gonna move forward one way or another.

Not because you’re crushing their curiosity,

but because you’re helping them get to the place

where their curiosity actually has more room to run.

Advanced math makes domain experts more valuable

Justin Skycak (00:00) I guess you don’t need advanced math for every job

in computer science or medicine, economics.

But the people in those fields who do know

their advanced math

are extra

because they can combine two things.

They understand the domain,

and they have the mathematical tools to go deeper.

That’s where the leverage is.

Not just knowing code, not just knowing biology, not just knowing business,

but being able

to bring serious math into the field

and see things that other people can’t.

Calculators can hide weak algebra

Justin Skycak (00:00) Calculators and Desmos, they’re not problems by themselves.

The problem is when they become a smoke screen.

A student can produce a graph how to graph.

get an answer without knowing how to solve.

They can move through the course without the algebra ever becoming solid.

And then everyone says, well, they passed.

But passing is not the same thing as mastering.

If a tool is hiding the weakness, the weakness is still there,

just waiting

to come back and smack you down in a harder class.

Placement tests are not the enemy

Justin Skycak (00:00) You gotta understand, a good placement test

is not there to insult students.

It’s there just to tell the truth

before the course tells it more painfully.

Because if a student is missing the basics,

and you place them into a course

that assumes all of those basics are automatic,

then you are not being kind, you’re setting them up to fail.

When tuition turns into grade pressure

Justin Skycak (00:00) At some private schools, parents are paying 30, 40, 50, 60

dollars a year?

When that much money is involved,

school completely changes.

Because now if the kid is getting a B or a C,

the parent is not just thinking, my kid needs to learn more.

What they’re is, much money am I paying for this?

So the emails start.

The meetings start, the pressure moves up to the administration.

And over time,

the school’s incentive

becomes just keep the parents at bay instead of the standard.

Proofs become easier after computations

Justin Skycak (00:00) Writing proofs does not have to be this terrifying leap,

but it becomes terrifying

when students never really even learned the computations first.

See if row echelon form feels natural

and eigenvectors feel natural,

eigenvalues feel natural,

then the proofs are not coming out of nowhere.

At that point proving something about linear algebra

is not really harder than proving something about or parity

because the objects make sense to you.

See, the proof is not the first step.

The proof comes after the intuition has something to stand on.

There are countably infinite courses after calculus

Justin Skycak (00:00) People think calculus is the big finish line,

but even after single variable calculus,

you still got linear algebra, multivariable calc,

differential equations, probability and statistics,

and all these other specialized courses that branch out after that.

There are so many

university math courses above calculus

that a serious student could not fit all of them into a standard four-year undergrad schedule,

even if they overloaded every single year.

So getting to calculus

early

is not about being done early.

It’s about having enough runway to reach the math that comes after.

High school grades stopped being a clean signal

Justin Skycak (00:00) The craziest part about the UCSD situation

is not just that students were missing middle school math.

No, it’s that the high school grades

that these students came in with

were not a good signal as to what was happening.

You had students coming in with strong looking grades,

but they still didn’t know fractions,

order of operations, exponents.

And there’s no test scores like SAT, ACT.

The UCs removed those, they went test blind.

Even though those test scores

were actually giving a cleaner signal of math placement.

So now you have inflated grades,

pandemic learning loss,

and fewer objective measurements.

And you end up surprised by this problem

that you simply chose not to measure.

Students inherit the system adults created

Justin Skycak (00:00) When students show up with holes in their math foundation,

it’s really easy to blame the student.

But the reality is adults created this system.

Parents, teachers,

administrators, school boards, politicians.

Everybody contributed something:

inflated grades, weak standards,

missing tests,

watered down courses,

smokescreens everywhere.

And the student eventually reaches a class where weakness

cannot be hidden anymore.

At some point, you have to say,

The system is broken, and we have to dig out this rot.

Most professors are not trying to teach well

Justin Skycak (00:00) There are some professors that take pedagogy really seriously, but it’s rare.

Most are just not building the course

around how students actually learn.

You know, concrete examples, review,

making sure prerequisites mastered before moving on.

They’re just presenting the material,

and they’re hoping that the strongest students survive it.

The rot starts with multiplication tables

Justin Skycak (00:00) Mathematical rot.

It starts really early.

It starts with things like,

we don’t really need kids to memorize multiplication tables.

But then,

multiplication never becomes an effortless skill.

It always takes up a bunch of space in their brain,

and they can’t easily reverse

when it comes time to find factor pairs or stuff in algebra.

So algebra becomes way harder,

and because it’s so hard, it gets watered down.

And then we let them lean on calculators and Desmos

because they can’t actually graph things or solve things

or manipulate expressions by hand.

Now there’s a weakness, it just travels up the ladder with them.

You don’t fix that in calculus. You gotta get it at the beginning.

Smart but unreliable students

Justin Skycak (00:00) There

things that people confuse all the time.

There’s aptitude and there’s conscientiousness.

Some students are not the fastest.

They’re careful.

They follow the process.

They do what they’re supposed to do,

and they end up moving really steadily through a ton of material.

And then you have the smart knucklehead type.

High accuracy when it’s easy.

Fast on the stuff that comes naturally.

But the moment they struggle,

they just rage quit.

and completely fall off the rails.

See, here’s the thing, talent matters.

But if a talented student won’t align with the learning process,

then the talent just leaks out everywhere.

UCSD needed remedial remedial math

Justin Skycak (00:00) One in 12 freshmen entering UCSD

didn’t know middle school math.

We’re not talking shaky on calculus,

they forgot their trig identities. No, no,

We’re talking like fractions, order of operations.

What’s an exponent?

Like, they already had a remedial math course,

and that was too advanced for some students.

So they had to create a remedial, remedial math course.

Two layers back.

And the craziest part of all this is that high school grades didn’t give a good signal

about this problem.

The best signal was the SAT or ACT math score,

which a lot of schools decided they just didn’t want to accept anymore.

Gifted kids still need to be pushed

Justin Skycak (00:00) The most mathematically gifted student I ever worked with,

he still had to be pushed to learn calculus.

He was way ahead for his age, but he didn’t see the long game yet.

I mean, kids usually don’t, right?

Adults can see it.

just thinking about what’s interesting right now.

What do I want to do right now? It’s gotta be right now.

Anyway, this kid, eventually,

learned the calculus.

And then calculus became one of those things he actually enjoyed.

And now he’s doing advanced math research.

He’s throwing derivatives and integrals all over the place,

and he’s having a great time. He loves it.

Not just for the long game, but also for the right now.

But if nobody had pushed him through that resistant phase,

he would not be doing what he’s doing right now.

Teachers have to set the tone early

Justin Skycak (00:00) Listen, as a teacher,

you have to set the tone early.

You can’t just walk in there like, hey,

I’m your buddy, I’m your study pal.

We’re all friends here.

Yeah, that doesn’t work.

Because a lot of kids, they’ll hear that and will think

What a joke.

You can just blow this class off.

And once they stop taking you seriously like that,

it is very, very hard to get that back.

So that’s why you

You make it clear that the work actually matters.

And then once the standards are established,

that’s when you can lighten up.

Grades stopped measuring learning

Justin Skycak (00:00) A lot of homework now, it isn’t even graded for accuracy.

It’s just, did you upload something to Canvas?

Not whether it’s correct, not whether you actually learned it,

just whether something got turned in.

And then people are shocked

when students get to algebra or to calculus or to college,

and they have these massive holes in their foundation.

This is how the smoke screens get built:

inflated grades, weak homework checks.

There’s no real signal.

And then everybody finally acts all surprised

a student finally hits a course where they can’t fake the math anymore.

Harvard having remedial calculus support is insane

Justin Skycak (00:00) Harvard having to add a ton of remedial support to its calculus courses, it’s just insane.

Harvard.

These students are coming in who are supposed to be ready for calculus,

but a lot of them have missing foundations from high school.

Maybe even middle school?

And this is the part that people don’t want to say out loud:

If the foundations are rotten,

then calculus isn’t the problem.

Calculus is just where the rot finally becomes impossible to hide.

Concrete examples first, proofs second

Justin Skycak (00:00) One of the biggest mistakes

in undergrad math

is making students push symbols around before they even understand what these symbols even mean.

You do the theorem proof, theorem proof.

Everybody just pretends that this is deep learning.

But without concrete examples,

this is just an inefficient game of symbol manipulation.

You gotta, gotta, gotta get your reps with the calculations.

Make it all feel instinctive in your bones.

And then, when you write proofs about it,

you’re not proving something alien.

You’re proving something that you actually understand.

Make honest work the easiest option

Justin Skycak (00:00) Homework is a tricky topic now, especially given chat GPT.

It’s really not that tricky if you just set up the right incentive structure

that if you cheat you lose.

You can detect cheating

really, really easily.

If you have the kid in person,

and they turned in a homework full of

perfect solutions to problems, and you don’t think that’s real?

Just get the kid in front of you, ask them to do the have to solve every single problem in front of you, just pick out one. You don’t have to even do it every day for every homework assignment. do it frequently

and have the stakes high

that the kids know they don’t want to be caught out.

And I’m not saying blow up their life, I’m just saying,

make them understand that that is actually the lowest effort way to get through your class is just do what you’re supposed to do.

Why STEM dreams die

Justin Skycak (00:00) Eventually, the truth comes out when the kids graduate and they go to college.

Whatever kind of cool thing that they said they wanted to do, like, I wanted to

build rockets or whatever. Guess what? You actually have to know your math.

And if you’re coming out of years and years of just being graded on completion and not being held to account for actually learning the skills, you don’t know your math.

And this is not a problem that you can just fix in couple months of tutoring.

it’s gonna take of time and effort to resolve. So what’s gonna happen is you’re just gonna say, know, math is hard, I was never cut out for it. I’m just gonna switch my major to something else.

If the people who are responsible

for making sure that you develop these skills earlier in

you’re going to miss out on a lot.

You don't grade it, they don't do it

Justin Skycak (00:00) You don’t grade it, they don’t do it.

That’s how it goes. If you don’t grade the work, vast majority of students are not going to do it.

If you don’t grade it for correctness, the vast majority of students are not going to do it correctly.

They’re just going to submit some slop.

Make it just look like work so if the skims over like they made an attempt. Check. Completed. Guess what? They didn’t actually do anything. didn’t actually learn anything. They didn’t even try.

You grade the homework,

but you weight it small enough compared to test performance

that a reasonable amount of mistakes in the homework

is not going to pull down the student’s grade.

When teachers are forced to dumb down classes

Justin Skycak (00:00) If you just give completion grades on homework,

that makes

the next year suck for the next who has to teach them. It’s terrible.

What happens down the line is the teacher, they have no choice

but to kind of make the class a more A little more fluffy.

Everybody just collectively agrees that this isn’t really a problem,

that we can do a fluffy watered down course just graded on completion. I mean, Eventually, the truth comes out when the kids graduate and they go to college. and they get smacked their first course in math or physics and realize they can’t do anything they didn’t learn shit

in high school.

But by that time they’re out of high school, so high school’s like, it’s not our problem.

Kid got into college.

We did our job here.

So everyone’s just kind of passing around this problem like a hot potato.

The pull of mediocrity in schools

Justin Skycak (00:00) If you just give completion grades on homework,

that makes

the next year suck for the next who has to teach them. It’s terrible.

Cause you come in as like a calculus teacher and you think your kids have learned their basic algebra? Guess what? They haven’t!

It’s like this pull of mediocrity. Once a

school program starts

especially in these hierarchical domains like math and physics,

you get caught in this trap of fluffy curriculum.

And you’ve gone so far

that you’re just stuck in it unless you are willing to go back to step one, which is like learn your basic

learn your basic algebra.

There’s no way to learn anything that builds on top of that unless you’re willing to go back down to your basics and rebuild

Teaching physics to students who don't know algebra

Justin Skycak (00:00) If you just give completion grades on homework,

that makes

the next year suck for the next who has to teach them. It’s terrible.

Cause you come in as like a calculus teacher and you think your kids have learned their basic algebra? Guess what? They haven’t!

I mean, I was in that position myself as a teacher.

I had to somehow teach physics kids who couldn’t even solve a linear equation,

but supposedly they had taken algebra.

Their algebra class was just such a waste

because they just had their class discussions and they’re plotting on Desmos.

Nobody learned how to solve the equations,

so they just don’t have any of this knowledge.

And I’m supposed

teach them

long it’ll take for an object to fall, and you gotta calculate of the angles and stuff.

So What happens down the line is the teacher, they have

but to kind of make the class a more

because nobody knows anything.

Completion grades destroy math education

Justin Skycak (00:00) If you just give completion grades on homework,

that makes

the next year suck for the next who has to teach them. It’s terrible.

Cause you come in as like a calculus teacher and you think your kids have learned their basic algebra? Guess what? They haven’t!

They don’t know how to do the most basic things and somehow you still have to them calculus?

And you get in that state because somebody the line

decided it’s fine to just give everybody completion grades.

If you’re a teacher and you want to completely wreck your school’s math program, then just a foundational course and just give everybody completion grades. Pass them on. Fail them upwards.

Because after a year of that passes by,

the problem becomes so big that nobody wants to accept that it’s even a problem anymore.

The rookie mistake that kills careers

Justin Skycak (00:00) Here’s a rookie mistake and it’s so common, it’s so painful to watch it play out over and over and over again:

Do not show up and immediately start trying to become important.

That’s not how you become important.

You gotta first focus on just becoming useful.

And if you become useful enough,

then you start earning leverage.

And if you earn enough leverage, then you become important. That’s how it works.

A surprising amount of wisdom

is just doing the next obvious thing

correctly repeatedly

without letting impatience

trick you into skipping the level that you’re actually on.

The real world rewards execution, not potential

Justin Skycak (00:00) You gotta understand this:

A natural edge that you don’t cash in, it eventually expires.

I mean, students often don’t realize this, but

after you graduate,

the game changes.

It eventually becomes 100% about what you’ve actually done,

not what you just have the potential to do.

And this transition, it happens faster than you think.

And you know, this can be a hard truth.

Students get used to hearing, oh you’re gonna do great things, and then they never take action.

But it’s really a welcome truth

for those students who are willing

to put in serious work and get serious results.

Abstract math needs concrete roots

Justin Skycak (00:00) Abstract math is not the problem. Proofs are not the problem.

The problem is when students are thrown into abstraction they even understand anything concrete.

The math just becomes symbol pushing.

Definitions, theorems, proofs, more theorems, more proofs.

But no intuition, no examples,

no real grip on the objects.

That’s backwards.

Build the concrete understanding first,

make the calculations instinctive,

and then abstraction actually means something.

The grade inflation trap

Justin Skycak (00:00) High school grades are not giving us a clean signal anymore.

A student can have a really strong transcript, but still be missing math foundations.

And that’s the trap. Everybody sees the grades, and they assume that learning has happened.

But really the grades can be inflated, the homework can be completion based.

The tests in class can be diluted on retakes on retakes,

participation points,

extra credit.

You just keep beating the grade

with this onslaught of let’s pretend you learned this activities.

So that by the time the student reaches college, transcript says they’re ready,

but the placement test says they’re not.

Bad feedback creates fake confidence

Justin Skycak (00:00) If homework is only checked for completion, students can be wrong for weeks and not know it.

They submit the assignment, they get the check.

The grade looks fine.

The parent thinks everything’s fine, the school thinks everything’s fine.

But guess what? It’s not fine, because the math was never actually learned.

And nobody knows that because the homework wasn’t checked.

Then later, weeks later, maybe months later,

on the test, where all the missing pieces show up.

But by that time, the situation is too painful to correct because it’s compounded

into this big skill gap.

And at this point it’s easier to just kind of brush it under the rug,

kinda fudge the grades, give retakes on retakes on retakes.

How about some extra credit for something else?

And so it continues being the case that the math was never actually learned.

And that’s why feedback matters very soon after the student completes the problem.

Not just encouragement, not completion points, actual correction. Did you get it right or not?

Undergraduate math teaches you to survive, not to understand

Justin Skycak (00:00) A lot of undergrad math is taught like this:

Professor shows lectures, assigns a problem set.

Good luck!

And that’s basically the course.

I mean, yeah, there’s some professors that take the teaching more seriously,

but that’s a small minority.

The reality is most courses are not structured how students actually learn.

They skip examples, they skip intuition, jump straight to abstraction, just pushing symbols around

before they even know what the computations mean or how to run real numbers through them.

And then everybody just pretends that the students are learning math.

And really, they’re surviving the course.

Projects are not an efficient way to learn math

Justin Skycak (00:00) Projects sound great, and they can be great, but they’re not magic.

They only work when students have enough skills to actually do something with them.

Otherwise, project just becomes this giant mess.

Students don’t know what they’re doing. The teacher has to carry everything.

And the learning is inefficient, but really it’s probably not happening at all.

Thing is, you can’t skip the foundation

and jump straight to the fun part.

The fun part works because of the foundation.

ROUND 5

Kids are capable of so much more than standard school allows for

Justin Skycak (00:00) What is effective math training? What can students be doing? When I joined this radically accelerated math program to teach in it, these kids are AP Calc BC, passing the exam, often getting five on the exam in eighth grade. You get them in sixth grade, and these kids are nowhere near your conception of math genius, they’re like the typical honors level sixth grade. Using effective pedagogy and cognitive science-learning strategies, you can just transform them into having the skills that you would normally associate with somebody who has a much different cognitive profile. These things are so rare because standard education does so little. What you’re looking at is mostly natural talent and motivation. If school were actually leveraging all of the research that we know about what effective training is, you should get these kids who look almost like normal kids, and then you put them through this skills training program, and they just come out the other side, just complete monsters.

Good training programs transform ordinary people

Justin Skycak (00:00) If you have a really good gym training program, it’s not just the people who are absolute tanks walking in there who are going to walk out being absolute tanks. If this thing works effectively, you’re going to see a lot more people who just look like normal people, and then they go through this training regimen, and they follow it for months, a year, couple of years. And like they come out the other side and everyone just assumes that they had always been like this Greek God looking strong man. And that’s not at all what they looked like when they started. A hallmark of a really strong training program is the amount of transformation that you see in the students.

You build a life around what you can't do

Justin Skycak (00:00) It’s kind of like a self-fulfilling prophecy. A lot of people only need basic math for everyday life because they never got the math skills, the coding skills and everything to make something other than the default standard life. And then life solidifies around their current capabilities. You don’t learn a bunch of math, you don’t learn a bunch of coding. Well, guess what? You’re very limited in what you’re able to do skill-wise. And so you build up a life that doesn’t leverage the skill. And then you question why would anybody need the skill?

Calculators are a crutch when learning math

Justin Skycak (00:00) If you don’t learn the actual stuff, the calculator just becomes a crutch for you. You’re limited to the mathematical ceiling that is a calculator, which is a very low mathematical ceiling. If you don’t learn any of your math, all you learn is how to punch buttons on the calculator, you don’t actually have the skills that you need to support the higher level math that lets you go above the level of a calculator.

Lectures are a waste of time

Justin Skycak (00:00) Lectures are terrible. They’re just awful. You just talk at the students for like an hour. Whose eyes are not going to glaze over after an hour of anybody talking anything? The only people who are going to enjoy this are the students who love the subject, but they’re already operating at this kind of high level. Us who think about math all day, we are not the ones that you can design an education system around. It’s not a basketball practice if the coach is the only one bouncing the basketball and just doing slam dunks. As entertaining as it might be, it’s not going to get them better at the sport. And it’s the same way in math.

The myth of the "aha" moment

Justin Skycak (00:00) People think that there’s some phase transition that happens where at one point you went from like being unskilled to being skilled. That’s not how it works. Once in a while, when you get an aha moment, that’s really this last missing prerequisite clicks into place, and it’s like this big unlock for you. But there’s not anything particularly about that missing prerequisite. That just happens to be the last keystone that you needed to support this next level. And so it feels special because it’s the last one. There’s that satisfaction when it clicks into place.

Consuming content doesn't lead to mastery

Justin Skycak (00:00) You can’t even talk about a subject the same way as somebody who actually does it. Talking to somebody who has watched a lot of Three Blue One Brown, talking to somebody who has watched a lot of AI hype channels online versus somebody who is actually building the stuff, who is actually working out the math problems. Talking to somebody who watches a bunch of history on physics versus somebody who actually solves the physics problems and can tell you about it, do research into physics, figure this out, compute that. Like it’s just completely different.

Miscalibrated coursework leads to students hating math

Justin Skycak (00:00) Math class is moving everybody forward at the pace of the median student of the class. Almost nobody is the median student. For the bottom half of the students, they’re just like utterly confused. They have no idea what’s going on. They’re being asked to do things that they just have no idea how to do. And they don’t even know how to get started. And they’re building all this emotional hatred up around the subject. If you put them in that situation long enough where they’re just like, this subject is not for me, they’re not going to participate in the process. Even if you give them material that’s perfectly calibrated, the situation has gotten so bad that not only do you have a pedagogical problem, not even a motivational problem, not even just like a build relationship from scratch, but like an actual completely broken relationship.

The active learning loop

Justin Skycak (00:01) You gotta have students solving problems. The thing that’s gonna improve students’ performance is really doing the action themselves. This is not to say that no instruction should be provided. You actually have to show them what they’re supposed to be doing, but it’s just the minimum they need to start doing it on their own. Again, minimum effective doses. Now they know what they’re supposed to, now they practice it. And then at that point, you say, OK, good. Now let me show you the next thing. And you keep dangling the next thing in front of them. It’s a cycle. That’s the active loop.

Upskilling is the result of consistency

Justin Skycak (00:00) Just like when you’re doing a sport, you’re developing muscle tissue, your muscle memory, in math, it’s the same way. You’re physically changing your brain. And this requires the same type of reps. It’s a constant cycle of strain and adaptation. You microstrain, micro adaptation, rinse and repeat over and over again. And you’ll look up months later, years later, and you’re at a totally different level of skill. People think that there’s some phase transition that happens where at one point you went from like being unskilled to being skilled. That’s not how it works. It’s this day in, day out, these little wins happening.

How to master anything

Justin Skycak (00:00) You need these tight cycles of instruction to activity to feedback, correction. That’s the loop to mastering anything, whether it’s math or it’s sports or it’s music. Any skill that you can learn is most effectively learned in this way. The more of these loops that you can cram in to the learning session, the further you’re going to go, the more you get done. Each loop is a small incremental improvement, but you compound a lot of them, and you get really solid skills.

The right way to learn math

Justin Skycak (00:00) Repetition is necessary. But, you’re not just taking one problem repping on into the zone of diminishing returns. You wanna focus those reps on kind of the minimum effective dose and then switch over to doing on other reps. Now of course, you have to make sure your prerequisites are in place for all of these. So it’s to arrange this sort of practice setting because depending have, like what are they ready for? What are they not ready for? How many problems do they before moving on? This all depends student to that’s what it looks when you do it right.

If you build on shaky ground, you're doomed to struggle

Justin Skycak (00:00) When students struggle with math, usually the reason they’re struggling is because they don’t have prerequisites in place. And you’re assuming that they have prerequisites in place and then there’s this new combo punch. You’re going to struggle with the combo if you can’t even do the things individually. You ask them to factor cubic before they can even factor a quadratic. You ask them to solve a linear equation with a variable on both sides before they can solve an equation with a variable on only one side. You build on shaky ground and that’s what makes it hard.

Bottom-up learning is the most efficient

Justin Skycak (00:00) It’s a difficult problem to identify exactly which prerequisites they’re missing. The solution is really just start at your level with things that you able to do and just work through a well-sequenced curriculum up from there. If you build it from the bottom up, you can just spend all your time building, building, building, instead of searching for missing pieces.

Different students need different amounts of practice

Justin Skycak (00:00) Bottom up learning: you start at a student’s level and you don’t want to be too aggressive and put them ahead of their skis because things are just going to topple down. The way to build up to more challenging stuff pretty quickly is just you need to adapt the dosage of the practice to how well the student is doing. Don’t give them too many problems if they’re just knocking it out of the park. If they’re struggling, then you just ask more questions. You basically just adapt the amount of practice to how well the student catching on to the skill.

Most of class time needs to be individual learning

Justin Skycak (00:00) If you have a class, it has personalized to each student. You have to make the material appropriate for them at their pace. And there’s no way to do that if you are teaching the same thing to every student at the same time. Now, you can do group activities in class that depend on knowledge that all the students have learned, but that’s not going to be the majority of class time. That’s going to be kind of like the icing on the cake. Most of the time it’s just going to have to be occupied by skills practice and that’s going to be different for every student.

Accelerated learning unlocks opportunities you can't even see yet

Justin Skycak (00:01) Accelerated math education and coding always unlocks these new things that you probably would not have even thought of. You don’t even see the road ahead of you until you have the foundations in place. The more skills that you have the more surface area you have those skills further, combine them in various sorts of opportunities. These things are not years away. These are like, could sit down and start working on this tomorrow. And that makes a huge motivational impact.

Stop rotting, start compounding

Justin Skycak (00:00) Spending one hour per day on brain rot, scrolling, scrolling, scrolling is just insane. Spending one hour a that doesn’t some kind your life is just insane because that’s like 6 % of your waking day. That may not seem like a lot, but when you just think about how much of that time are you spending accumulated throughout your life, 6 % of your day, that’s 5 years of your waking life. That’s half a decade, just gone. Zero return, zero fulfillment, zero meaning, zero contribution to any other parts of your life.

Prerequisite knowledge is intellectual capital

Justin Skycak (00:00) It is so easy to think that you’re untalented, maybe even dumb, when really you’re just unpracticed on some prerequisite skills. Talented students can struggle if they’re missing prerequisites. The flip side of that it’s so easy to think that you’re just hyper talented, that you’re a genius when really you’re just better practiced on prerequisite skills than everybody around you. If you conclude that, I’m a genius, and I don’t need to practice, well, your genius is going to be short lived. So you have to recognize that the amount of practice you put in got you into this situation. And if you take your foot off the gas and you stop practicing, you’re gonna get behind. So the moral is that prerequisite knowledge is intellectual capital. And it can take you from academic rags to riches or from riches to rags, if you squander it.

You can't develop intuition without lived experience

Justin Skycak (00:00) Everybody wants intuition as a way to bypass this hard work of getting experience, of grinding through the computations, the life experiences. You cannot skip the experience and still come out with the intuition. Somebody can tell you all the kind of intuition that they have built over the years. But what you don’t realize is that this is just a compression of all their experience. It’s a lossy compression. You don’t know the edge cases where this is generally true, but watch out for this, watch out for that. Intuition emerges when you’ve done the grunt work so many times that your subconscious has automated the pattern recognition. It’s a very bottom-up kind of thing. It’s not a top-down thing.

You gotta get your reps

Justin Skycak (00:00) Repetition is necessary. You have to do the reps if you want learn and retain the material, just like you strength engaging in resistance training repeatedly. Say you go to the gym, you’re doing a 45 minute workout. You’re not doing bicep curls for 45 minutes. You’re not doing 45 minutes of deadlifts. That’s the same way that it needs to go in math education is you need reps, but you’re not just taking one problem and repping on into the zone of diminishing returns. You wanna focus those reps on kind of the minimum effective dose and then switch over to doing reps on other things.

Stop repeating, start interleaving

Justin Skycak (00:00) Repetition is necessary. You have to do the reps if you want to really learn and retain the material. But, you’re not just taking one problem and repping on into the zone of diminishing returns. You wanna focus those reps on kind of the minimum effective dose and then switch over to doing reps on other things. So you’re not doing this worksheet of like 30 problems that are like the same type of problem. Imagine dividing this worksheet into six sections and then five problems from a completely different area of math in each section.

The illusion of prior knowledge in fast learners

Justin Skycak (00:00) Maybe you have some student who actually doesn’t know the skill, but they just catch on so fast because they’re incredibly bright, have high generalization ability. The prerequisites are very strong. And they can grasp it so quickly they almost look like somebody who has learned it previously and is just remembering it really quick. That’s kind of what it feels teach some of these incredibly bright just like, wait, do you know this already? Did you learn this already? No? Okay, wow, that’s crazy.

You can't teach every student the same material

Justin Skycak (00:00) If you have a math class, and you’re doing a one size fits all, you’re trying to teach the same thing to everybody, you’re not going to be able to challenge high flyers. Likewise, there is no way to reach the struggling students without filling in their missing prerequisites. It has to be personalized to each student. You have to make the material appropriate for them at their pace. And there is no way to do that if you are teaching same thing to every student at the same time.

What if students mastered college math by 10th grade?

Justin Skycak (00:00) In this super advanced math and computer science program that we taught in the Pasadena district, the idea is we would build up students coding skills and also have them like really super powered in math, so basically your core undergrad engineering math. The students would get this in place by 10th grade. That leaves a couple of years a lot of that depend on these foundations. Like, have students build their own machine learning libraries from scratch and research papers in 90s artificial intelligence. In high school!

You're not dumb, you just lack the prerequisites

Justin Skycak (00:00) It is so easy to think that you’re untalented, maybe even dumb, when really you’re just unpracticed on some prerequisite skills. This reminds me of a time that I tutored a real analysis student who hadn’t gotten much practice with proof writing beforehand. She thought she was gonna fail the class. She thought she might not be cut out for it. But we just shored up some of those missing proof writing foundations. And she came out of class with a well-deserved A. And then she took Fourier analysis the following year and just absolutely crushed it. Didn’t even need my help at all. So she was a talented student who’s capable of doing highly advanced math but was just missing some prerequisites.

People don't hate math, they hate missing prerequisites

Justin Skycak (00:00) When you just hit a wall in something that you’re trying to learn, this is typically just a massive debt of unlearned prerequisites that are being called due. People don’t really hate math. They hate the feeling of being lost, of not knowing what the hell is going on. They hate this compounded debt of prerequisites that they haven’t learned and being asked to continually learn things that depend on them.

It's never too late to do the work

Justin Skycak (00:00) If you are willing to put in the work, it’s the same amount of work, whether you start in your teens, 20s, 30s, 40s. What changes is not the amount of work. It’s the amount of free time that you have to do the work. People often say like, it’s too late for me. But usually what that really means is just that I’ve got less time now. I’ve got more responsibilities now and it’s harder to carve out the hours. And that’s real, but the work itself is still doable if you can make time for it.

Silent educational failure

Justin Skycak (00:00) What do you mean homework was never meant to be graded? If you’re not going to grade the material at all, everybody’s pretending that this learning is happening. Nobody’s measuring it. Everybody’s pretending that it’s happening. You can go on like this for a long time. Maybe a student doesn’t learn in high school all the math that they’re supposed to have learned, and they get to college, and can’t do can’t even do the remedial precalculus. And what this is chalked up to is typically, know, this stuff’s hard and I’m not cut out for it. Things just get so bad that people just accept because it’s so far the original circumstances that were causing this situation. responsible are no longer really in your life. You can’t really hold anyone accountable anymore. You just got to deal with the situation. So it’s easier to just pretend this was an inevitable thing. But that’s really what’s happening is, Nobody’s measuring the learning and making sure it’s happening.

Why math education has gone to crap

Justin Skycak (00:00) The dumbest of feedback I ever got from administrator who sat in and watched one of my classes. At the beginning of class, I asked students to recall information that I’d taught them the previous day. And of course, they just learned this information, so they’re gonna be a little fuzzy on it. You try to make them experience that effort of retrieval and get them retrieving successfully because that’s what strengthens the long-term retention. This admin saw this friction, and she was like, Well, you should really remind the students of what you did yesterday. That way it goes smoother. And it’s like, we’re doing strength training, right? And I’m the coach, and I’m telling the students, hey, go lift that weight, because I want you to get stronger. And admin walks in, she’s like, You should lift the weight for the students. That way the weight will get lifted faster. Like, seriously? This was a math coach for the district. So you wonder why math education has just gone to crap. It’s because we got so many people who are in charge of math education who just don’t know the most basic stuff about how memory works, how the brain works, learning works. They just run on vibes like, what kind of feels like it should be good for learning? You have to know about how learning works. And retrieval practice is one of the most important techniques long-term retention.

Problem solving ability comes from technical skills

Justin Skycak (00:00) Here’s the thing. Everybody wants to talk about like, we should teach students problem solving skills. They should be solving all these problems, these hard problems. That’s where the learning really happens. It’s like, you realize that the problem solving skills, they depend on not just abstract, like meta skills. They depend on actually being able to do things: to solve equations, to manipulate mathematical expressions, to code stuff up, to write for loops, if statements, functions, modular systems, like all this stuff. You can’t solve technical problems without technical skills. There is no such thing as technical problem solving in the absence of technical skill building. You can’t solve problems with skills you don’t have.

Good challenges convert into progress

Justin Skycak (00:00) Sometimes I’ll talk about inefficiency and about how student had such a poor educational experience. This is what should have been done to skill them up more efficiently, and some people will just comment things like, you know, adversity isn’t a bad thing. It builds character. Do you really where if you take that to its logical completion? It’s like, why even have a teacher guiding students, trying to teach them things in ways that they will understand? The kind of challenges you want students to be facing in the classroom challenges most efficiently into progress You should help them overcome these challenges and make it like a mental workout the students. If you’re just letting them flail around making things harder than they need to be. A lot of that, it’s just like heat loss. This is like a thermodynamic engine. And so much of what could be converted into skill just gets lost. It’s a waste.

What's the point of homework if nobody checks it?

Justin Skycak (00:00) What do you mean homework was never meant to be graded? So you’re gonna make a student do half an hour to an hour or more of every night and you’re not gonna grade the material? Like what are they even doing it for then? You know that saying like practice makes permanent? I mean, just doing an hour of practice, waiting for the next day or next week to get feedback on that, that’s bad enough. But if you’re not going to grade the material at wall, hat is even the point of this whole thing? It’s just a performative exercise where you’re just asking the student, scribble on this paper, and then turn it in, and we will pretend that that indicates that you have learned the material even though I’m not going to check it. We’re just going to assume that it’s all correct and that you’re understanding it.

School "works" because nobody is held accountable

Justin Skycak (00:00) What do you mean homework was never meant to be graded? So you’re gonna make a student do half an hour to an hour or more of every night and you’re not gonna grade the material? You know, what’s really happening is it’s like, it’s less work for you. It’s less work for me. The student doesn’t have to really make sure it’s like all correct. The teacher doesn’t have to grade it, doesn’t have to deal with the consequences when a student doesn’t know the material. Principal’s happy, parents happy, because everybody’s pretending that this learning is happening. Nobody’s measuring it. Everybody’s pretending that it’s happening.

Teacher training programs ignore the science of learning

Justin Skycak (00:00) So many of these admins don’t even know what spaced repetition is. They don’t know about blocked versus interleaved How are you gonna run any sort skills training program if you don’t know these basics? It’s just ridiculous. I guess to be fair to them, this is not taught as part of their credentialing. This is not taught as part of their programs and education schools programs are just so unserious. It’s been taken over by these academics who run on vibes too. And so they set the bar of what it means to pass teacher credentialing for moving up in the education administration world. And these bars are also just set on vibes. It’s not based on science. It’s not based on actual effective learning results.

Unnecessary difficulty in learning is not helpful

Justin Skycak (00:00) Sometimes I’ll talk about educational inefficiency and about how this student has such a poor educational experience. This is what should have been done to scale them up more efficiently, stacking little wins instead of just continually asking them to do things they don’t have the prerequisites in place to do. and some people will just comment things like, you know, adversity isn’t a bad thing. It builds character. You’re going to get enough adversity in life outside the classroom. You don’t need additional adversity inside of your skill building.

Kids won't do work that's not graded

Justin Skycak (00:00) All these comments about you’re not supposed to grade homework. “It’s independent practice of a new skill, not an assessment.” If you don’t assign a grade, do you really think that the kid’s gonna take it seriously? Like this is ungraded. This is just for completion. What kind of world are you living in? That does not work at all. Kids are effort minimizers for the most part. Like they’re gonna do the minimum amount of get the grade in the class. There are exceptions to this rule, but generally speaking, that’s how kids work. That’s how students in general work.

Breaking down the excuses against grading homework

Justin Skycak (00:00) Now there’s a bunch of really dumb excuses being brought up like, “oh, I have 160 students. Should I look over every problem of homework during my 50 minutes of prep time? Do the math on how long that would take.” Well, you don’t have to look over every single problem. You can choose a subset of problems to grade. Just don’t tell the students what they are beforehand. That still puts in the incentive that the students have to try their best on all the problems because any of them could be sampled for the grade. It’s just an excuse to say things like, I can’t grade every problem because you don’t need to do that. There are ways to work smarter than that.

Kids are effort minimizers

Justin Skycak (00:00) Kids are effort minimizers for the most part. There are exceptions to this rule, but generally speaking, that’s how kids work. That’s how students in general work. That’s just how people work. They’re going to minimize the amount of effort that’s needed to attain the goal they want. And for most students, the goal that they want is not really mastery of the material. It’s just the grade, getting the teacher off their back and the parents off their back because they generally don’t really understand the importance of having a good foundational base of knowledge.

Kids don't naturally think long-term

Justin Skycak (00:00) Kids are effort minimizers for the most part. Like they’re gonna do the minimum amount of needed get the grade in the class because they generally don’t really understand the importance of having a good foundational base of knowledge. Even if the teacher or their parents explain it to them, “you this in order to opportunities in the future,” kids, they don’t have really the whole experience to see this play out. It takes a certain amount of foresight that kids just generally don’t have.

Why kids need immediate incentives and consequences

Justin Skycak (00:00) So the role of the teacher is to make sure that the kids are actually learning the material. And one way to ensure that kids don’t master the material is to not set up an incentive for mastering the material. Not just a couple of weeks we’ll have an exam and you need to be prepared for it incentive. Kids aren’t going to take it seriously until it’s imminent. And kid imminent is a lot shorter than adult imminent. Adult imminent is like a few weeks in the future. Kid imminent is like, If tomorrow can go by and my not doing something today doesn’t really matter the outcomes of tomorrow, then guess what? I’m not going to do it.

Homework has to be graded

Justin Skycak (00:00) Homework has to be graded. And not just checked for accuracy, but also factored into the grade to some amount. Now, obviously not as much as tests and quizzes. But there has to be some incentive for the kids to actually do what they’re supposed to do every step of the way. If you don’t collect it, you don’t check it, it’s not going to get done. And if you don’t check it for accuracy, it’s not going to get done correctly.

Students fall off the rails without immediate accountability

Justin Skycak (00:00) Homework has to be graded. If you don’t collect it, you don’t check it, it’s not going to get done. And if you don’t check it for it’s not going to get done correctly. You ever hear that saying practice makes permanent? If students are just doing the same misconceptions over and over because they don’t care about whether they’re actually getting it right or not, they’re just, how can I just write something in the fastest way possible so I can get back to video games or hanging out with my friends or whatever else I’d rather be doing. The incentive has to be there and it has to be tight, where if they’re not doing what they’re supposed to there’s going to be a penalty right away. if they’re doing what they’re supposed to do, there’s got to be a reward right away. That’s how you keep things on the rails.

Calculations before proofs

Justin Skycak (00:00) When I took linear college, it was embedded in abstract and it was mostly proofs. And it was way harder than it needed to be. You can’t skip intuition and go straight to abstraction. Students need to compute first. Row echelon form, eigenvalues, eigenvectors. This stuff should be instinctive, and then the proofs aren’t scary anymore. Once you understand the objects concretely, the proof is just the next step.

Grades don't mean they know math

Justin Skycak (00:00) A lot of homework now. It’s not graded for correctness. It’s just graded for whether it was even uploaded. A student can write down complete nonsense and then submit it to Canvas still get full credit. And everyone thinks things are fine because the grade looks fine. And then they get to college, and it is revealed that they’re missing fractions, exponents, order of operations, middle school math, sometimes even elementary school math. Because that’s what happens grades stop measuring learning.

Coding makes math click

Justin Skycak (00:00) A lot of people don’t really care about learning math in school. But then they get into coding, and they realize. Wait a second, if I knew more math I could build so much more. And that connection should happen way earlier. Imagine graduating high school knowing not just calculus, but also multivariable algebra, differential equations, all with coding applications. Now you’re not just doing toy projects. You can walk into college ready for research, internships, and serious technical work. Math becomes capability.

Projects don't replace skills

Justin Skycak (00:00) Project-based learning can go way too far. Projects are built on skills. And if students don’t have the skills, the project doesn’t magically teach them efficiently. It just becomes slow and confusing. You need the foundation first. The same in math, coding, research, anything really. Meaningful, fulfilling, and exciting work comes after a lot of reps, and you don’t get to skip the skill building stage. Because that stage is what makes the interesting projects even possible.

Calculus is not the finish line

Justin Skycak (00:00) A lot of people think calculus is the end of math. And it’s not, it’s not even close. After calculus, there’s linear algebra, there’s multivariable differential equations, probability, statistics. Then all the specialized courses after that. There are more university math classes above calculus than there are high school math classes below calculus. So getting to calculus early, that doesn’t mean you’re done. It just means you finally to start.

Differential equations is taught so badly

Justin Skycak (00:00) Differential equations. It’s one of the worst taught math courses because the subject has all these branch points. I mean there’s a trunk at the beginning where you do linear differential equations, then second order equations with characteristic polynomials. But then, suddenly…The instructor just goes to their favorite research area. So depending on who teaches the course, students get a totally different idea about what the subject even is. A core math course should not feel like a random tour of the professor’s interests.

Parents only want three things and it’s impossible

Justin Skycak (00:00) A lot of parents want three things at once: They want their kid to get amazing grades, they want their kid to learn a lot, and they want their kid to not have to do too much work. And that doesn’t work. Kids don’t learn unless they put in a real amount of work. But when the grade drops, parents often just start lobbying. Email the teacher, the principal, push the school, and over time that lowers standards. School’s not supposed to be grade negotiation. It’s supposed to be skill acquisition.

The most gifted kids still need pushing

Justin Skycak (00:00) The most mathematically gifted student ever worked actually resistant to learning calculus. He was already way ahead. He liked puzzles. He liked math that felt interesting, which really often just meant familiar. But eventually he just wasn’t really progressing. You see, there are levels of math that you have to climb in order to unlock the next thing. And kids don’t always see the long game, so adults have to see it for them. And so this kid, once his parents and I pushed him to learn calculus, he ended up loving it. And now he’s doing serious math using ideas from this area that he originally resisted. You know, sometimes a push is what opens the door.

Most math major math is taught backwards

Justin Skycak (00:00) When you’re a math major in undergrad and you get up into the junior and senior level courses - real analysis, abstract algebra, topology - you start to realize that a lot of this stuff is really taught backwards. Theorem, proof, theorem, proof. But you don’t really have a concrete grasp of what’s even going on. You’re just pushing symbols around. Maybe you can follow the rules, but you don’t really understand the subject. That’s because you need the concrete examples first. You gotta work through computations and then do the abstraction. Skipping the concrete stage doesn’t make the course rigorous. It just makes it inefficient.

Homework should be checked

Justin Skycak (00:00) A lot of students aren’t actually getting feedback. They submit the homework and the grade is basically, did you upload it? Not, is it correct? And that’s a huge problem because students can be over and over again and still think they’re doing just fine. And math doesn’t work that way. You need reps, you need correction, you need to know what you got wrong. Otherwise, you’re just accumulating knowledge gaps.

The SAT was measuring something real

Justin Skycak (00:00) For more than two decades, UCSD’s math department found that the best predictor for math placement was a student’s SAT math score. And ACT was just as good. But not high school grades. And then a whole bunch of colleges and universities removed those signals - the test scores - right after massive learning loss, grades getting inflated. And now students show up with strong transcripts, but missing basic math: fractions, exponents, order of operations. That’s not a mystery. Everyone just stopped measuring the thing that they needed to measure.

ROUND 2

Middle school math is the biggest waste

Justin Skycak (00:00) The worst part of K-12 math is sixth through eighth grade. Kids learn arithmetic in elementary school and then spend three years spinning their wheels before Algebra 1 in 9th grade. That is such a waste. If you actually use those years well, you teach efficiently, you require mastery, and you move kids forward when they’re ready, honors students can be finishing all of high school math by the end of eighth grade. Not by doing extra math, just by not wasting three years.

The most hard-hitting two sentences in all of talent development research

Justin Skycak (00:00) The most hard-hitting two sentences in all

talent development research: Deliberate practice requires effort and is not inherently enjoyable. Individuals are motivated to practice because practice improves performance.

That’s according to K. Anders Ericsson, one of the most influential researchers in the field of human expertise and performance.

In other words, maximal learning does not happen naturally as a result of maximizing other things like enjoyment, comfort, convenience, and ease of practice. In fact, maximal learning is at odds with some of these things.

Must read: Developing Talent in Young People by Benjamin Bloom

Justin Skycak (00:00) Every young person and every older person who has young people in their life needs to read this book: Developing Talent in Young People by Benjamin Bloom.

That’s the same Bloom as Bloom’s 2 sigma problem, Bloom’s taxonomy,

This guy is one of the most influential scientists in the history of education.

Everyone seems to have their own opinion about how talented people become talented and what roles working hard, working smart, and getting lucky play in the process. But Bloom being a scientist, he didn’t want to settle for some subjective opinions. He wanted to formulate conclusions based on real data. So Bloom studied the training backgrounds of 120 world-class talented individuals across 6 talent domains: piano, sculpting, swimming, tennis, math, and neurology. And what he discovered

was that talent development occurs through a similar general process, no matter what domain. In other words, loosely speaking, there is a formula for developing talent, though, of course, executing it is a lot harder than simply understanding it.

What’s great about this book is the level of detail presented. This is not one of those corny pamphlets where somebody tells you about their three keys to success framework, and all the information is obvious.

and too abstracted to be useful. Each chapter covers a different talent domain and goes into extreme depth,

making heavy use of direct quotes from the interviews with world-class individuals and their parents.

Instead of handing you some kind of abstract framework and expecting you to accept it at face value, the book walks you through the process of starting with first principles, detailed interviews and backstories, and zooming out to identify the general trends. You go from microstructure to macrostructure.

The big conclusion is that talent development occurs through a similar general process, no matter what the domain. First, fun and exciting playtime, then intense and strenuous skill development, and then finally, developing one’s individual style

while pushing the boundaries of the field. It’s an older book, so you can get a used copy relatively cheap.

The worst segment of the K-12 mediocrity is 6th-8th grade math

Justin Skycak (00:00) The worst segment of the K-12 mediocrity is sixth through eighth grade math. Typically kids learn counting and arithmetic in elementary school - grades K through five - and spend the next

three full years spinning their wheels without learning much new math, only moving on to Algebra 1 in 9th grade. If you just take those three years and you put them to good use,

just avoid wasting kids’ time,

teach them efficiently,

make them master the material,

and move them on to new material once they’ve mastered the prerequisites, you can actually enable gifted kids to cover all of high school math during those years and pass the AP Calc BC exam by the end of 8th grade

without spending any extra time doing math.

Just do the frickin' work

Justin Skycak (00:00) Just do the fricking work. If you want to develop serious skills, you have to engage in intense taxing workouts.

Amateurs sometimes make up all sorts of excuses for why this rule doesn’t apply to them, but real pros don’t try to weasel their way out of the hard work.

You think you’re too good for the grunt work?

Too smart to listen to your coaches feedback? Then what are you waiting for? Go on, succeed all by yourself in your current state. Either prove your inherent greatness or fall and get your ass handed to you enough times to knock some humility into your head.

At the end of the day, can either waste your time debating your coach on the training regimen or you can actually use that time to put your head down and do some freaking work.

Do you want to turn into a pro or do you want to stay tethered to amateur level the rest of your life? It’s your choice. If you want outsized results, then you’re going to have to put in an outsized amount of work. Achievement, expertise, greatness, whatever the hell you want to call it. It

doesn’t happen naturally.

It’s about transforming yourself from normal to abnormal in ways that confer a competitive advantage. There’s nothing natural about it.

Your goal is not to prove you're smart, it's to make problems go away

Justin Skycak (00:00) Your goal is not to prove that you’re smart. It’s to make problems go away.

Yeah, the more knowledgeable you are, the better you are equipped to solve problems. But if your primary focus is peacocking your intellect, then

you’re gonna create problems instead of making them go away. You don’t get points for creating an unnecessary problem on which to demonstrate your smarts.

You don’t get points for creating an overcomplicated solution to a simple problem. You lose points for these things. What you get points for is taking a problem and making it go poof.

completely solved, easy to maintain, nobody has to think about it anymore.

Your mathematical potential has a limit, but it's likely higher than you think

Justin Skycak (00:00) Your mathematical potential has a limit, likely higher than you Most people can learn basic math like arithmetic and some algebra, but beyond that, higher levels of math become increasingly abstract and technical, and fewer people have the cognitive resources to learn it quickly enough to make a career out of it, much less get to that point relatively early in their lives. But at the same time, few people actually reach their full mathematical potential because they get knocked off course early by factors like

missing foundations, ineffective practice habits, inability or unwillingness to engage in additional practice when needed, or lack of motivation.

Effective people maintain a gigantic locus of control and bias towards internalizing blame

Justin Skycak (00:00) The most effective people I’ve met, they all maintain a gigantic locus of control and bias towards internalizing blame.

Whenever something doesn’t go their way, they deeply analyze the circumstances to identify what they should have done differently. Or even if they acted appropriately based on all the information available, how could they have gotten more information or taken intermediate actions to extract more signal to inform later bigger actions?

They don’t dwell. don’t make excuses. They just learn and apply that learning to the future so they don’t make the same mistake twice.

You need to practice retrieving information from long-term memory

Justin Skycak (00:00) You ever have that experience of being unable to remember something despite repeated exposures because you kept automatically looking it up from a reference instead of trying to retrieve it from memory? happened to me an embarrassing number of times with addresses, phone numbers, directions, you name it.

And any books you read, any movies you watch, the only ones you remember in proper detail are the ones that you periodically think about and replay in your head. If you just consume and you don’t reproduce the information, then you just forget it almost entirely.

Retrieval is the act of pulling information from long-term memory into working memory.

That’s what increases your ability to remember and use the information. Each time you successfully recall a fuzzy memory, it stays intact longer before getting fuzzy again. Each time you successfully recall a memory with less priming, its recall becomes less dependent on priming in the future. But if you don’t practice retrieval, then this doesn’t happen. The information just quickly dissipates. It stays with you briefly, just long enough to trick you into thinking that it’ll stick with you

when it’s really on the way out the door.

But of course, you don’t notice that it’s gone if you’re not actually testing whether it’s there.

The most generalizable skill is climbing hierarchical skill trees

Justin Skycak (00:00) The most generalizable skill out there is probably learning how to climb skill trees, and in particular ones that are relentlessly hierarchical.

Of course, you can’t acquire that general skill without practicing it on particulars. You gotta lean into specific domains and acquire measurable demonstrable skills. But once you see it, once you get a feeling for the process, you see it everywhere.

Individual learning styles are a myth

Justin Skycak (00:00) There is a common myth that goes like this: Everybody has the same cognitive horsepower and learns at the same rate, but different people learn differently depending on their preferred learning style.

In reality, the exact opposite is true. Different people generally have different working

and they learn at different rates. And while people may have preferred learning styles, like visual or verbal whatever, they don’t actually learn better when given information in their preferred style.

The style needs to be matched to the content, not the person.

The myth is that different people need the same amount of practice, but in different forms. But the reality is that different people need the same form of practice, but in different amounts.

Always play the long game

Justin Skycak (00:00) Always, always, always play the long game. Nothing really matters unless it matters in the long term.

The short game is just a trap,

and no matter how big you win it, it will leave you feeling empty at the end.

Lots of people in education disagree with the premise of maximizing learning

Justin Skycak (00:00) Took me too long to realize that many educational debates

aren’t really about the effectiveness of various learning strategies. They’re about whether education should even begin with the premise of maximizing learning.

A shocking number of people would prefer education to maximize other things like fun and entertainment while as a secondary concern

meeting some low bar for shallowly learning some surface level basic skills.

You can make serious progress climbing any skill tree if you just put in 30 minutes of focused effort every day

Justin Skycak (00:00) It’s just amazing how many people will rationalize avoiding skill development on the grounds of something like, nah, bro, I’d rather have a life. Here’s the thing. You can make serious progress climbing pretty much any skill tree if you just put in 30 minutes of focused effort every day.

But it has to be fully focused, a full-assed effort, and you have to be continually upping that level of challenge as your capabilities increase.

Effective learning feels harder than ineffective learning

Justin Skycak (00:00) A well-known finding in education research is that the perception of learning is often at odds with actual, measurable learning.

When using effective learning perform better on assessments, but they may feel like they’ve learned less. Why is that? It’s because effective strategies increase cognitive activation, enhancing learning despite students feeling that it’s harder. It’s like weightlifting. The strongest people lift weights heavy enough to make them feel weak.

It’s true in the gym, it’s true in the classroom, it’s true in every skill domain.

Bloom's 3 Stages of Talent Development

Justin Skycak (00:00) The journey to developing a high level of talent can be divided into three stages. That’s what Benjamin Bloom discovered while studying the training backgrounds of 120 individuals across six talent domains: piano, sculpting, swimming, tennis, math, and neurology.

Talent development occurs through a similar general process no matter what talent domain. There’s stage 1: fun and exciting playtime, where students are just starting to develop awareness and interest in the talent domain.

The teacher provides copious positive feedback and encourages students to explore whatever aspects of the talent domain they might find most exciting.

Students are rewarded for effort rather than for achievement, and criticism is rare.

Then there’s stage 2: intense and strenuous skill development. Students are fully committed to increasing their performance. The teacher becomes or is replaced by a coach, focuses on training exercises where the sole purpose is to improve performance. These exercises are demanding and the coach provides constructive criticism to help the student perform the exercises properly.

Positive feedback is provided in response to achievement and effort is assumed.

Finally, there’s stage 3: developing one’s individual style while pushing the boundaries of the field.

At this point, students are proficient in all the foundational skills in the talent domain. And they are so committed that they center their entire lives around the talent domain, no matter the sacrifice. And they typically work with a world-class expert in the talent domain.

The expert helps the student identify and lean into their individual strengths so that they can excel beyond perceived human capabilities.

You need to practice retrieving information to transfer it to long-term memory

Justin Skycak (00:00) What’s the thing that transfers information into long-term memory? Retrieving from memory. When you take notes, you know what you’re not doing? Retrieving from memory. When you take great notes and you constantly refer back to them, you know what you’re still not doing? Retrieving from memory. Retrieval is the specific action of pulling information from one part of your brain, long-term memory, to another part of your brain, working memory. It’s like your brain is lifting a weight.

off the ground of long-term memory and raising it up into working memory.

If you load information into working memory by looking at a reference material, instead of pulling information from long-term memory, then you’re not strengthening your retention. It’s like you’re going to the gym to lift weights, but you’re just going through the motions and you’re letting the spotter lift the weight for you. No strength is being developed.

This leads to a vicious cycle of forgetting. You take notes because you can’t remember things. You can’t remember things because you’re not transferring them into long-term memory. You’re not transferring them into long-term memory because you’re not practicing retrieving them from memory.

and you’re not retrieving them from memory because you’re always looking back at your damn notes.

If you fall into this vicious cycle of forgetting, it will completely rip apart your whole learning process.

The reference material becomes a crutch and you’re lost without it. You miss out on making connections and understanding things deeply. You can’t connect the dots because you don’t actually have the dots in your head.

Sure, they’re in your notes, but not your knowledge base. And you can’t cook with ingredients you don’t have.

You learn slower, you forget faster, and eventually you just grind to a halt.

And you might think it’s because you’re not doing enough review really it’s because you’re not doing those review problems properly, pulling information from memory.

Ability is built, not unlocked

Justin Skycak (00:00) Too many people waste their time searching for a magical motivational unlock that’s going to bring about a transition in their ability.

Don’t bother, doesn’t exist. Ability is built, not unlocked. You do not have a latent superpower that can be unlocked in a day. You gotta build up a skillset. You build it up high enough and then other people think it’s a superpower you always had.

If you're struggling with discipline, try getting some work done first thing in the morning

Justin Skycak (00:00) If you’re struggling with discipline, try getting some work done first thing in the morning, before you eat, before you shower, before anything.

Getting some work done is your meal and shower ticket. The longer you go in your day being lazy, the harder it is to get yourself to break out of this laziness and put forth some effort. But if you start your day off with some momentum, then that momentum can help carry you through the rest of a productive day.

Passing a class does not mean you've mastered the material

Justin Skycak (00:00) It is absolutely shocking how many people think that passing a class at school implies having learned all of the material in proper depth.

That’s just so false.

I mean, just think about it.

Are students made to a bar for mastery on every single one of the hundreds of granular topics covered in a course?

Nuh-uh.

A student can typically pass a course despite not having mastered a ton of material.

even a student who gets an A may have only mastered 90 % of the material in the course.

Well, less actually, if some of that grade comes from participation.

And to top it all off, the course might not even be comprehensive. It might not even cover all the topics that ought to be included in a proper treatment of the subject.

Even in the topics that it does cover, learning tasks might be cherry-picked or watered down to be artificially easy or otherwise not truly representative of what needs to be learned.

Kicking the can down the road in education

Justin Skycak (00:00) This is how kicking the can down the road happens in education.

It starts with students passing a course without demonstrating sufficient mastery of the material.

In the next course, instructor discovers that these students are struggling to learn the material due to missing foundational knowledge.

The only true remedy in this situation is to hold the students accountable for learning all the material in the course, including the prerequisite material that they’re missing.

But this requires the instructor to work overtime and support the students with remedial assignments and help sessions.

And it’s not just the instructor. I mean, the students also have to work overtime to pay off their learning debt,

which leads to tension when the instructor has to hold the line on what it means to pass the course.

So what usually ends up happening instead is that everyone turns a blind eye and kicks the can further down the road.

Instructor just gives the usual lectures and assignments, then curves or otherwise inflates the grades, and students go along with it, and the problem is left for the next instructor to deal with or not deal with.

The hard truth about educational products

Justin Skycak (00:00) It’s a hard truth that if you want to build a serious educational product, you can’t be afraid to charge money for it.

You can’t back yourself into a corner where you depend on a massive user base.

Why?

The reason why not is that most people are just not serious about learning.

And if you depend on a massive user base of unserious learners,

then you have to employ ineffective learning strategies that don’t repel unserious students.

Which makes your product suck.

ROUND 1

Prelearning math sets you on a virtuous cycle of opportunities

Justin Skycak (00:00) The greatest educational life hack is learning math ahead of time. Why learn math ahead of time? Because it guards you against numerous academic risks and it opens up all kinds of doors to career opportunities.

When you pre-learn the material in a math course before taking it at school or college, You guard yourself against all sorts of risks, such as the course moving too quickly, explaining things poorly, assuming knowledge of

prerequisite material,

goes on and on.

That’s especially true at university when lectures are often unsuitable for first introduction to a topic. But if you pre-learn the material, then you’re not depending on the teacher to teach it to you, which means you’re immune to even the worst teaching.

Of course, the natural objection is, won’t you be bored in class? But here’s the thing, if you do super well in advanced classes, especially at university, then that opens all kinds of doors to recommendations for internships, projects with professors, and so on.

Basically, you can use pre-learning to kick off a virtuous cycle. Even if you aren’t a genius, you appear to be one in everyone else’s eyes, and consequently, you get a ticket to those opportunities reserved for top students.

Students who receive and capitalize on these opportunities can launch themselves into some of the most interesting, meaningful, and lucrative careers that are notoriously difficult to break into.

There is so much more math after calculus to learn

Justin Skycak (00:00) Many people think that calculus is the end of the road for math and that it doesn’t matter if you get there many years ahead of schedule. But that is so far from the truth.

There are even more university math courses above calculus than there are high school courses below calculus. is not even halfway.

After a single variable calculus course like AP Calc BC, most serious students who study quantitative majors like math, physics, engineering, and economics have to take core engineering math courses, including linear algebra, multivariable calculus, differential equations, and probability and statistics.

Different majors include plenty of specialized courses that branch off in different ways.

There are so many courses that a student could not fit them all into the standard four-year undergrad course load, even if they overloaded their schedule every year.

However, the more of these courses a student is able to take, the more academic opportunities and career opportunities are open to them in the future.

And while it’s true that students don’t need to know much beyond algebra to get a job in fields like computer science or medicine,

The people in those fields who do also know advanced math are extra valuable and in demand because they can combine domain expertise and math.

Competition math has little ROI for most mathematically gifted students

Justin Skycak (00:00) When a middle or high school teacher has a bright math student, and the teacher directs them towards competition math, it’s usually not because that’s the best option for the student. Rather, it’s the best option for the teacher. It gives the student something to do while creating minimal additional work for the teacher.

Competition math problems generally don’t require students to learn new fields of math. Rather, the difficulty comes from students needing to find clever tricks and insights to arrive at solutions using the mathematical tools that they’ve already learned.

But if you look at the kinds of math that most quantitative professionals like rocket scientists and machine learning researchers use on a daily basis, those competition math tricks show up rarely, if ever. What does show up everywhere is university math level subjects like linear algebra, multivariable calculus, differential equations, and calculus-based probability and statistics.

So given that most students who enjoy math end up applying math in some other field as opposed to becoming pure mathematicians,

it would be a lot more productive for them to get a broad view of math as early as possible

so that they can sooner apply it to projects in their fields of interest.

Mathematical acceleration is developmentally appropriate for students who have learned the prerequisites

Justin Skycak (00:00) Many people think that learning math early is not appropriate for students’ social emotional, academic development.

But the reality is that educational acceleration does not lead to adverse psychological consequences in capable students. According to a study titled “Academic Acceleration in Gifted Youth and Fruitless Concerns Regarding Psychological Well-Being:

a 35-year longitudinal study,”

“there is little evidence that academic acceleration has negative consequences on the psychological well-being of intellectually talented youth.

Those who were accelerated had few regrets for doing so. Indeed, if anything, tended to wish that they had accelerated more.”

The takeaway is that whether a student is ready for advanced math depends solely on whether they have mastered the prerequisites. If a student has mastered the prerequisites, then it is appropriate for them to continue learning advanced math early and not appropriate to stunt their development by holding them back.

Educational acceleration allows you to win the race against time

Justin Skycak (00:00) Here’s the thing that many people don’t understand about educational acceleration. At the core, it’s not really a race against your peers. It’s a race against time.

When somebody gives up on their dream or gives up on figuring out what that dream is, it’s typically the result of time closing in on them. No matter how many times you claim you’ll never settle,

it doesn’t prevent the time from passing, and it doesn’t prevent you from increasingly desiring things that only a stable life can provide.

The further time gets ahead of you, the more likely you are to settle into a life that is fine, or even good. But deep down, you just can’t shake the feeling that it’s less than something more you could have found if you had more time.

Educational acceleration is about winning that race against time.

It’s about opening doors early and running down avenues that you might be interested in exploring.

That’s why learning advanced math ahead of time is the greatest educational life hack for any student interested in a science, tech, or engineering career. It helps you find your place in the world before time closes in on you and forces you to settle for something else.

Accelerating mathematically talented students is inconvenient for schools

Justin Skycak (00:00) Why do so many people think that academic acceleration is developmentally inappropriate, given all the research to the contrary? Well, it becomes pretty clear if you look at the incentives.

Acceleration requires extra work, but people typically don’t like to do extra work, so they will gladly rationalize that the extra work wouldn’t have really helped them anyway, even if that rationale is incorrect.

Now you might ask, What about

If accelerating their capable students will lead to beneficial outcomes, wouldn’t they be pushing it or at least not discouraging it? The problem is that acceleration is also very inconvenient to schools.

In schools, each grade typically progresses through the math curriculum in lockstep, which means that accelerated students would need to be placed in above-grade courses, and this can lead to major logistical challenges.

When your ego makes you uncoachable

Justin Skycak (00:00) A common thing that keeps mathy people from actually learning math is having an ego that’s too damn big.

It’s like a basketball hobbyist who shows off at pickup games at the park, thinks they’re the star of the show, signs up for private coaching with an ex-NBA player, and thinks they know better than the coach. And consequently, they’re uncoachable.

Learning ennui

Justin Skycak (00:00) You know how if you spend the whole day sitting on the couch watching TV, you get kind of restless, but somehow also too tired to get off your butt?

Like you’re tired of doing nothing, yet you’re also tired from doing nothing. You know what I’m talking about, that state of blah. Well, that also happens with learning. If you binge a bunch of lecture videos, documentaries, textbooks, whatever, without actively working exercises, solving problems, building stuff, then you’re gonna fall into that same state of blah.

You are consuming too much and producing too little.

Satisfaction comes from pulling through and achieving something that you know took real work, something that you really earned. If you don’t go for something that takes real work, you just waste away in a state of blah. The blah is a

The only way out of the blah is to start actively doing things that require you to put in work.

your problem isn’t about being tired. It’s about being unsatisfied.

And I know, I know, it’s tough because the satisfaction only starts appearing after you put in some work. But if you muscle through and accomplish something that took real work, even if it’s the tiniest thing, you will start to feel the blah and you’ll find some motivation

to accomplish incrementally bigger things.

You keep doing this and before you know it, you’re out of the blah. You’re doing stuff, you’re earning satisfaction. And by the end of the day, you feel tired, but in a good way, where you can actually feel yourself absorbing rest and recharging your battery for more action and satisfaction the next day.

Channel your anger into productivity

Justin Skycak (00:00) One of the greatest meta skills in life is learning to channel anger. Anger can be an incredible source of motivation. When it consumes your focus, you can channel that into positive action. But if you don’t channel it into something productive,

It can consume your entire life to no end at all. You can stew an anger for hours, days, even weeks at a time. And we all know people who do.

You can lose years of your life that way.

Some mathematicians wear the inaccessibility of their work as a badge of honor

Justin Skycak (00:00) In college, I did some work under a math professor who told me, with a sigh, that some of his peers, other mathematicians, that is,

considered it cool to publish papers that were hard to follow. Like, the more time and scratch work and mental energy it takes to understand your work, the cooler you are.

I’m not suggesting that research papers be held to the same pedagogical standard as courses, but it’s something that stuck with me as a mentality to avoid.

Personalized learning dramatically accelerates student progress

Justin Skycak (00:00) When you let students move at their own pace, some will take off flying faster than you could ever imagine.

many students can learn calculus in middle school

if they are given individualized learning tasks and allowed to progress as soon as they’ve mastered the prerequisites.

All you have to do is let them sprint independently on a paved road instead of chaining them up to their classmates and making everybody trudge through mud in lockstep.

High expectations can only be met with a rigorous learning experience

Justin Skycak (00:00) When students aren’t expected to climb a sky-high skill tree, you can get away with putting fun front and center

while checking the box on a few basic skills as a byproduct.

If every student in gym class were expected to do a backflip by the end of the year, the instructional approach would have to change. But nobody expects that. Playing kickball works fine. In math, however, students are expected to climb a sky-high skill tree.

Every student takes many, many years of math courses, increasing in difficulty, continually stacking more advanced skills on top of previous ones.

So when students do the mathematical equivalent of playing kickball during class and then are expected to do the mathematical equivalent of a backflip at the end of the year, it’s pretty easy to see how struggle and general negative feelings can arise.

How to do a ton of intense work

Justin Skycak (00:00) It’s a lot easier to put forth a massive volume of intense work when you 1) love what you do, 2) are heavily incentivized to do it, 3) you know that you’re learning a that you’re continuing to skill up at a rapid pace, and 4) you have

so much and such a high diversity of work to do that you can take a break from one thing

by working on another task that is very different in nature.

Max-efficiency learning in a nutshell

Justin Skycak (00:00) In a nutshell, this is how max efficiency learning works: 1) you solve progressively more challenging problems that are bite size, given your level of prerequisite knowledge. 2)

Implicit spaced repetition credit trickles down to component skills that you’re practicing implicitly.

and 3)

You choose new learning tasks along your knowledge frontier that knock out all of your due reviews

while minimizing the amount of explicit review. All of the required review you’re trying to knock out implicitly.

Standard milestones are based on an unserious approach to work

Justin Skycak (00:00) A grade level’s worth of learning can be compressed much, much shorter than a year if you just avoid wasting time. And once you see this in academics, you see this everywhere in life. Standard milestones are based on what anyone can accomplish with a high volume of unserious, inefficient work. Why? Because that’s the standard approach to work. Show up, mess around, waste time, do the bare minimum, run out the clock, rinse and repeat every weekday.

But if you just take things seriously, work efficiently, and put in the same volume, you can take off flying.

It’s the biggest edge: actually giving a shit.

But in order to capitalize on this edge, you have to get into a line of work where outsized results reap outsized reward. Not every line of work is like this.

Most people don't train seriously

Justin Skycak (00:00) If you are willing to train seriously to achieve a goal, don’t let yourself get faked out and discouraged by how long it takes people who don’t take their training seriously.

This is especially true in academics where standard grade level learning paces are based on unserious, inefficient training.

A grade level’s worth of learning can be compressed much, much shorter than a year if you avoid wasting time.

One group work problem is not a brain workout

Justin Skycak (00:00) Sometimes people think it’s enough to have a class as a whole discuss and solve a single problem over the whole duration of the period.

That’s not enough. That’s like going to the gym with your friends, having everybody crowd around a single standard weight dumbbell, lift it together for one rep, and then put it down, leave, and call it a workout.

Satisfaction is earned by doing real work

Justin Skycak (00:00) Satisfaction comes from pulling through and achieving something that you know took real work, something that you really earned. If you don’t go for something that takes real work, you just waste away in a state of blah. The blah is a

The only way out of the blah is to start actively doing things that require you to put in work.

your problem isn’t about being tired. It’s about being unsatisfied.

And I know, I know, it’s tough because a brief period of time when you switch from doing nothing to doing something when it feels like you’re just getting more tired. and more blah. But if you muscle through and accomplish something that took real work, even if it’s the tiniest thing, you will start to feel the blah and you’ll find some motivation

to accomplish incrementally bigger things.

You keep doing this and before you know it, you’re out of the blah. You’re doing stuff, you’re earning satisfaction. And by the end of the day, you feel tired, but in a good way, where you can actually feel yourself absorbing rest and recharging your battery for more action and satisfaction the next day.

Learning requires you to produce, not just consume

Justin Skycak (00:00) You know how if you spend the whole day sitting on the couch watching TV, you get kind of restless, but somehow also too tired to get off your butt?

Like you’re tired of doing nothing, yet you’re also tired from doing nothing. You know what I’m talking about, that state of blah. Well, that also happens with learning. If you binge a bunch of lecture videos, documentaries, textbooks, whatever, without actively working exercises, solving problems, building stuff, then you’re gonna fall into that same state of blah.

You are consuming too much and producing too little.

Producing is where learning happens. Consuming is only helpful insofar as it enables you to produce.

Your problem isn't about being tired; it's about being unsatisfied

Justin Skycak (00:00) your problem isn’t about being tired. It’s about being unsatisfied.

And I know, I know, it’s tough because the satisfaction only starts appearing after you put in some work. There’s a brief period of time when you switch from doing nothing to doing something it feels like you’re just getting more tired. But if you muscle through and accomplish something that took real work, even if it’s the tiniest thing, you’ll find some motivation

to accomplish incrementally bigger things.

You keep doing this and before you know it, you’re earning satisfaction. And by the end of the day, you feel tired, but in a good way, where you can actually feel yourself absorbing rest and recharging your battery for more action and satisfaction the next day.

Interleaving topics is counterintuitive but highly effective for learning

Justin Skycak (00:00) Effective learning strategies sometimes go against our human instincts about conversation.

During a conversation, people generally want to focus on a single thought and explore it to the fullest extent and completely finish the thought before moving on to other things. We converse depth first.

But effective learning strategies like interleaving and spaced repetition are all about stopping the flow of thought, doing other things for a while, and then coming back to remember the flow of thought just before you’ve forgotten it.

breadth first, not depth first.

Now granted, when you’re creating content for a course, it’s easiest to proceed in the form of a story and close the loop each time before moving on and opening another loop.

That’s how to create good content, but it’s not how the content should be presented.

Now, to be clear, I’m not suggesting that learners should be switching to a different topic every other minute while first learning the material. That would be excessive.

What I’m saying is, imagine taking the usual week-long or month-long units that a traditional classroom marches through and chopping those up into mini-segments that are cohesive, yet scoped down to 10 or 20 minutes and interleaving through those segments. A 60-minute class period would cover about 15 minutes across each of those four different units.

That’s been our approach at Math Academy. Students work through one full topic at a time. The topics are bite-size, atomic, and consist of about three to four knowledge points of increasing difficulty, each knowledge point consisting of about two to five practice questions, depending on a student’s performance. Students go through these individual topics one at a time, but the individual learning paths through future topics are interleaved.

Richard Feynman admitted that his classes failed to teach 90% of students

Justin Skycak (00:00) According to Richard Feynman himself, his classes were a failure

for 90 % of his students.

In his lectures, he did a phenomenal job of appealing to intuition and conceptual thinking and making complex physics feel simple and accessible without getting too deep into the math. But on the flip side, however, when it came time to solve actual problems on exams, Feynman’s students failed.

He wrote, I quote,

When I look at the way the majority of the students handled the problems on their examinations, I think that the system is a failure.

These are Feynman’s own words.

some may view the Feynman-style pedagogy as supporting inclusive learning for all students across varying levels of ability, Feynman himself acknowledged that his methods only worked for the top 10 % of his students.

Now, it’s worth noting Feynman taught at Caltech, which is one of the most selective universities in the world, possibly the most STEM-focused university in the world,

The top 10 % of Feynman’s students were well above the top 1 % of students in general.

So, for the other 99 % of students, what method of learning does work? Well, what works is starting out with a minimal dose of explanation

in which intuition and conceptual thinking do have a place, but then immediately switching over to active problem

During the active problem solving, students should begin with simple cases, but then climb the ladder of difficulty to cover

the cases that the student could reasonably be expected to demonstrate their knowledge of on an assessment.

Teacher credentialing programs focus on political ideology over the science of learning

Justin Skycak (00:00) Speaking as somebody who had to suffer through a teacher credentialing program, it’s actually an anti-signal when somebody references their teaching as a qualification to speak about how learning happens. It’s typically centered around political ideology rather than the science of learning.

You see, there exist learning strategies that have been scientifically shown to improve student learning, such as mastery learning, spaced repetition, retrieval practice, and interleaving.

these learning strategies have been researched extensively since the early to mid 1900s, with key findings being successfully reproduced over and over again

Yet when I completed my teaching credential from 2020 to 21 and attended numerous professional developments from 2019 to 23,

Not once did I hear any mention of these learning strategies. Instead, the focus was 100 % on diversity, equity, and inclusion.

anti-racism training, sharing circle training, and even a presentation on the gender unicorn complete with an extraordinary complex gender classification flow chart. I mean, forget the science of learning, even the most obvious practical skills that a teacher would need to exercise on a daily basis, such as managing a rowdy classroom communicating with parents holding students accountable for their dealing with academic dishonesty, none of these were covered at all.

there was no shortage of time to cover them.

It’s just that that time was devoted towards pointless activities.

I also wish that my experience with teacher credentialing was an edge case, some kind of dysfunction that wasn’t widespread. But if you look at the curricula of standard teacher credentialing programs within well-reputed universities, find the same phenomenon.

The curricula are focused entirely on making education engaging, diverse, and unbiased, and there’s little to nothing about the science of learning.

Like workouts, the best study sessions are short and frequent

Justin Skycak (00:00) The best study sessions are short and frequent as opposed to long and sparse.

For instance, suppose you’re budgeting three hours per week to learn math. If circumstances allow, it would be much better to study 30 minutes, six days a week

as opposed to 90 minutes twice a week. And here are several reasons why.

First, you want to form a habit. The more consistently you study math, the more it will become a habit that you naturally do each day without thinking.

The habit is what will carry you through the long game once the initial adrenaline wears off.

Second, you want to operate at peak productivity during your session. It’s easy to maintain high levels of focus and intensity throughout a short 30-minute session, but during a long 90-minute session, you become significantly less productive as fatigue sets in.

And third, you want to minimize the amount that you forget between sessions.

However, there are some caveats to consider. First of all, whenever you switch to a new activity, it can take a few minutes for your brain to spin up on the new context. And if you make your sessions too short,

then the proportion of study time that’s wasted on context switching will outweigh all the other benefits of daily practice. So while it’s good to spread out your practice, each session should be long enough that context switching cost is proportionally negligible.

Additionally, if you have a hectic schedule and six days per week in theory ends up being just three days per week, then you need longer sessions just to achieve the same volume of practice. think of it like working out. 30 minutes, six days per week. That’s no problem. Easy.

45 minutes, four days per week, 60 minutes, three times a week. I mean, it takes more discipline, but it’s doable. 90 minutes, twice a week

of high intensity workout? damn, it’s gonna feel like a grind with slow progress, and you’re gonna be constantly skip workouts.

PEMDAS follows what you would do naturally

Justin Skycak (00:00) Why is the order of operations the way that it is,

where multiplication takes precedence over addition?

the following explanation feels so intuitive to me that I’d be shocked if there were a different reason.

When you use multiplication and addition

For simple real life applications,

you’re typically trying to find the total of some number of this and some number of that.

For instance, say you buy two 3-kg bags of rice and four 5-kg bags of wheat.

And you want to count how many kilograms of grain you have altogether.

Well, you do the 2 times 3.

get that then add it to the four times 5.

You do the multiplications first and then you add them afterwards.

Cognitive activation vs. cognitive load

Justin Skycak (00:00) Common confusion in education.

is cognitive activation versus cognitive load.

Everyone talks about maximizing cognitive activation because that’s good for learning. But you also talk about minimizing cognitive load

Because when students have too much load, they get overwhelmed

and can’t learn effectively.

So here’s the way I think about it: The purpose of cognitive activation to get students to think deeply about what they’re learning

so that they incorporate it into their mental tree of knowledge as opposed to just going through the motions and memorizing facts and procedures.

Whereas the purpose of minimizing cognitive load is to make lessons accessible to all students avoiding undue stress on working memory.

since students have different working memory capacities and different amounts of background knowledge.

Serious learning is just like training at sports practice

Justin Skycak (00:00) the purpose of minimizing cognitive load is to make lessons accessible to all students avoiding undue stress on working memory.

And I think there’s a strong analogy to sports practice where athletes do drills and scrimmage against each other.

During the drills, athletes are generally focused on improving a particular skill. If you want to improve your three-pointer accuracy, first thing that you should do is drill shooting from the three-point Once you’re good at that, you can run progressively more advanced drills,

And that drill progression is like cognitive load minimization. If you take somebody who can’t even shoot a three pointer,

and you immediately have them practice shooting against a hardcore defender, then it’s going to be too overwhelming and they’re not going to improve.

That’s what happens when a learning task exceeds a student’s working memory.

But what happens if you do 100 % drills and you never scrimmage?

Well, you end up getting pretty good at the skills practiced in the drills, you have no idea how or when to apply them during an actual game.

Should I try to go in for the layup or should I hang back and try to do a fadeaway shot or could I pass to one of my open teammates?

Well, in order to answer these questions and perform well during an actual game, you need a good meta level understanding of the game.

So,

you also scrimmage during practice and critique tape recordings of prior games.

Learning is like climbing a staircase

Justin Skycak (00:00) Learning is like climbing a staircase, and students get stuck at any individual stair that is too tall for them to climb. So the smaller you make the individual stairs, the more students can climb all the way to the top.

The size of a stair that a student can climb is like their working memory. By minimizing cognitive load, you’re making the individual stairs smaller so that more students can climb them.

Cognitive activation

is the meta reasoning behind the why and the where. Why are you climbing the staircase? Where are you trying to go?

The purpose of cognitive activation to get students to think deeply about what they’re learning

so that they incorporate it into their mental tree of knowledge

Students need to know the component skills before they do projects

Justin Skycak (00:00) Can you teach a math class on the basis of projects only, as opposed to more traditional exercises? not really. Sure, projects can be useful for pulling a lot of skills together to do something cool and exciting, and that’s valuable. But the thing is, students need to acquire the component skills first. Most students can’t learn component skills on the fly in the context of the bigger project.

They need plenty of practice on each particular skill, where lots of scaffolding is provided at the beginning and then gradually removed.

For example, let’s say you want to do a project where students sketch an image by graphing a bunch of straight lines in Desmos.

Obviously, going into the project, students are going to need to know how to represent points in the coordinate plane and find the equation of a line between two points. And they should also know how to manually create the graph of a given linear equation.

Each of these component skills has to be built up beforehand,

and it takes plenty of time and practice to build them up. Most students are going to need a worked example and a handful of practice problems with feedback on each separate case of each component skill.

Because if they don’t get this practice, then they’re just going to be totally overwhelmed by the project. And they’re going to spend a ton of time working unproductively. And ultimately, they’ll learn little to nothing from it.

Like in fitness, making progress in learning math requires sticking to a plan

Justin Skycak (00:00) When somebody tries to learn math by poking around Wikipedia, checking out whatever math articles pique their interest,

They often end up marveling at equations and diagrams,

But for the most part, they’re unable to understand what they’re Unless it’s something that they already learned in more structured environment.

The end result is that a wide variety of mathematical objects and fields of math sound familiar to them, they have lot of difficulty describing what those objects and fields actually are, and they are completely incapable of using them in any sort of problem-solving context.

To understand why this strategy is so ineffective, imagine somebody who frequently goes to the gym but doesn’t have a solid workout plan in place. They sporadically walk around to whatever piece of equipment piques their interest, and they perform an arbitrary number of reps with some arbitrary weight.

And then after a month of this, they’re disappointed to realize that they look no different in the mirror.

To make any sort of fitness progress, this gym goer will need to develop and stick to a proper workout plan, will likely require the assistance of a personal trainer. The personal trainer will help them choose a sequence of exercises will help them make progress towards their fitness goals. And in each exercise, the amount of weight will be calibrated so that the weight can be successfully lifted, but there is sufficient struggle to stimulate muscle growth.

Nonstandard mathematical algorithms distract students and slow down learning

Justin Skycak (00:00) Should you teach alternative multiplication algorithms like the lattice method?

Well, personally, I would be very cautious when introducing any algorithms other than the standard methods. The standard algorithms are standard for a reason. They’re easy to set up and they’re hard to mess up.

The lattice method, for instance, it’s hard to set up. It takes a lot of time and effort to draw the entire grid with the diagonals, And it’s easy to mess up.

I mean, I’ve seen this happen so many times where a student draws a sloppy grid with misaligned diagonals and then screws up the calculation as a result.

But the biggest issue is probably that kids will often latch on to whatever method they like best, And their incentives are often misaligned. For instance, I’ve tutored students who straight up told me that they preferred the lattice they liked being able to take a break from the math to draw the And believe me, they took their sweet time drawing the grid and making it perfect.

Of course, it took these students forever to complete their problems because they were working with incredibly low efficiency, And that frustrated them.

But another factor leading them to resist switching to the more efficient standard method was that they had completely forgotten it.

Why? Because they were using the lattice method for so long not practicing the standard method. And so they got into a situation where relearning the standard require some additional upfront time and effort on top of what they already an overwhelming workload.

I mean, listen,

If the alternative method is just as efficient and just as general, mean, sure, introduce it.

But if not, then I wouldn’t introduce it because students who latch onto it and resist letting go are going to be in for a world of hurt.

Even if you try to introduce that alternative method as a fun, temporary vacation away from standard techniques, some students will try to stay on that vacation forever.

Struggle does not imply inability

Justin Skycak (00:00) Struggle does not imply inability. If you do poorly in a math class, it doesn’t necessarily mean that you are incapable of learning that level of math. There’s a number of reasons that could be the root cause of your struggle.

While it’s true that everybody’s mathematical potential has a limit, in practice, the ceilings we hit rarely represent our true abstraction ceilings. All sorts of factors can artificially lower our ceilings, such as missing foundations, ineffective practice habits,

inability or unwillingness to engage in additional practice, or lack of motivation.

Mathematical struggle is caused by gaps of missing foundational skills

Justin Skycak (00:00) Struggle does not imply inability.

Struggle can be caused by missing foundations.

It’s kind of like how when people age, they accumulate biological damage that eventually reaches a tipping point and leads to a cascade of catastrophic health issues. The same thing happens to students learning mathematics.

Students accumulate weaknesses and knowledge gaps as they progress through math.

Once they accumulate a critical number of gaps, then the student is doomed to struggle unless proper remediation is enacted to fill in those gaps.

Remediation is extremely difficult to accomplish outside the context of an adaptive automated learning system. It rarely happens in the classrooms. just don’t have

Enough time

with each student to figure out exactly which pieces of foundational knowledge are missing.

Students will almost certainly accumulate these deficits in traditional classrooms. It’s only the most gifted and most motivated students who are able and willing to identify and self-repair their gaps on their own.

Effective learning requires deliberate practice

Justin Skycak (00:00) Effective learning is active. It’s not It’s not effective to attempt to learn by passively watching videos, attending lectures, reading books, or rereading notes.

Effective learning should center around deliberate practice.

Deliberate practice requires repeatedly practicing skills that are beyond one’s repertoire.

However, this tends to be more effortful and less enjoyable, which can mislead non-experts to practice within their level of comfort.

Classroom activities that are enjoyable, collaborative, and non-repetitive, like group discussions and free-form or project-based or discovery learning, these are only supplements, not substitutes, for deliberate practice.

Lastly, deliberate practice must be part of a consistent routine. The power of deliberate practice comes from compounding

incremental improvements over a longer period of time.

It’s not a quick fix like cramming before an exam.

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