The Math Stack

by Justin Skycak on October 05, 2026

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Learning happens through memory and active practice (13)

Active practice, not passive exposure, builds skill. (5)

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.

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.

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.

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.

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.

Learning depends on memory, and retrieval strengthens it. (5)

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.

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.

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.

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.

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.

Working memory limits how much unconsolidated material can be handled at once. (2)

We know from cognitive science is long-term memory you’re sleeping. Which means there’s only so much you can do in a certain day. Why? you have a certain amount of working memory. The average person has four to seven slots of working memories. Think of it like the size of your desk and each part of the problem is paper and you can only have so many papers on the desk before you just can’t look at them all at the same If there’s a lot of things that have to do with this problem taking slots up in working memory, you can’t see everything in the problem, can’t problem solve, can’t think your way. If you just learned one to two things at most a day, you let this stuff consolidate to long-term memory, you’re operating with your full with your working memory.

There’s a portion of people who putting hundreds of XP per day. They’re learning per day. Instead of taking those 10, 15 topics and spreading them out wise along knowledge frontier, if you allow them to go wise, now they are going to overload their working memory. when you go breadth wise, you’ve got the prerequisite knowledge consolidated for all these you want to Cause any one of those, only material But when you start like, let’s go 10, 15 topics stacked on top of each other, well, every single one of those topics you’ve learned that day has not had time to consolidate yet. So it’s really your working putting pressure on your working memory much more than if you had spread that

Aptitude and conscientiousness are different dimensions. (1)

are students have astronomically high accuracy. Maybe their accuracy is 80%, that’s solid. But you can have students like that who pass % of their tasks. what that indicates is that average an aptitude level, but in terms of diligence, conscientiousness, doing what they’re supposed to, aligning themselves with the learning process. They’re really good. you can also have students % accuracy, but their pass rate tasks is maybe like % or 90%. These are kind of like of the knucklehead smart really fast on the things that come easy to them and a lot comes easy to them. But then once they struggle, rage quit or just like fall off the rails, It’s like one of those quadrant memes, like aptitude and like conscientiousness.

Instruction should make the next step learnable (17)

Give students explicit guidance and scaffolding instead of making them rediscover known tools. (5)

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.

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.

The whole project-based learning can be taken way too far. Projects are built on a foundation of skills. You can’t do projects without skills and have it be efficient at all. It’s just going to be incredibly inefficient because the students don’t really know what they’re doing.

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.

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.

Concrete examples and procedures should precede abstraction and proof. (7)

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.

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.

At the University of Chicago, if you’re a math major, the linear algebra was embedded in abstract algebra, and it was mostly proofs. And it was terrible. I remember our textbook was Hungerford, which was an abstract, it was a graduate school textbook. It was just so much harder than it needed to be. I guess if you’re dealing with some absolutely brilliant students, and they just work like crazy, and they work in groups, you can kind of get them to fight their way through it, but it’s so inefficient, and it’s so painful, and it’s just unnecessary. That is not the way, and so what we do is take a concrete, straightforward, let’s get your skills, your intuition up. And then once really understand how to do these calculations, you really understand what eigenvectors and eigenvalues and row echelon form, and all this stuff is just like instinct to you, then we can make proofs about it. And writing proof about that is not harder than writing proofs about divisibility and parity because you just understand it intrinsically, and then it’s just a matter of having the insight through the proof.

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.

Most undergraduate math majors, We’re playing this little game of pushing symbols around, but nobody really has a great intuitive grasp of the subject itself because it’s not really taught. not really structured in that way. Pushing symbols around without concrete examples is just a really inefficient way to go about things. They just skip the concrete examples, and so you just do the theorem, proof, theorem and then so, so poorly done at the undergraduate level, almost without exception. There are probably a few, probably like 2 % of math professors in the undergraduate university level who take the pedagogy seriously and are really trying to teach them the rest just show up lecture, problem set, whatever. Good luck.

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.

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.

A core course should have a coherent trunk rather than becoming a tour of the instructor’s interests. (2)

Differential equations is one of the worst taught courses because there is this breadth to it, right? Where there’s all these different branch off points. What I’ve experienced in my own education and in many students that I’ve tutored was typically, the instructor will get through your integrating factors and linear equations, then go do characteristic polynomial second order. And then it’s like, okay, we covered the basics, let’s just go into my favorite area of research. But it’s like, just depending on who your instructor is that you can get a completely different view on what is even the subject? What does it cover?

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.

Class time should center useful practice rather than passive lecture and crammed homework. (3)

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.

At the university level, you go, you sit in a lecture hall, up there and it’s typically three days a week, and then you have like a weekly problem set. The problem sets, depending on the kind of university, they can be really, really challenging when you got to kind of work with two or three other people. maybe you show up at TA session and everybody’s like, ⁓ what the hell was problem three? The guy’s like, ⁓ you know, you might want to think about this. And that sort may or may not be very you’re kind of anything. I just found it extremely inefficient. First of professor is typically just writing theorems and proofs and things. They’re all in the book anyway. So why am I even coming here? I might as well just go to the library and work on my problem set. the problem is that you have one problem set a week. So you’re all that into a short period of time where you should be doing it every you should be getting everything you’re doing. And then when you turn the problem set especially if they were wouldn’t get it back for like two Maybe a week if you’re

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.

Mastery, not age or proxies, should control progression (20)

Ready students should advance based on mastery, not age or grade level. (5)

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.

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.

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.

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 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.

Even highly capable students sometimes need adults to push them through the next level. (4)

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.

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.

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.

The most gifted kid that I’ve ever worked with, he was actually resistant to learning calculus around seventh or eighth grade. He had kind of learned a lot of arithmetic on his own and I had found some puzzles online that he wanted to work on. And that was a good use of time at the time. Eventually, you get to a point you’re not really progressing a whole lot in your mathematical development. I mean, you’re way ahead compared to grade level, but like at some point you gotta make the leap. Like there’s levels of math that you have to climb in order to just get further along in the talent domain in order to unlock new things for you to do. Adults think like that. Kids don’t think like that cause kids don’t have the longer perspective. They don’t know what the long game is. They haven’t seen the long game play out. And so what ended up pushing this kid over into okay, fine, I’ll learn calculus was that he wanted to go take college level math courses, like in ninth grade. And so I was talking to him and his parents like, yeah, he can totally do that. Problem is though, if he doesn’t know calculus, then not only is he going to struggle in these courses, they’re not even going to let him into these courses if he doesn’t have the five on the AP Calc BC exam. The big thing was getting his parents on board with it because if the parents on kid’s going to be on board one way or another. The interesting part is once he learned all the calculus stuff, calculus became one of the things that he really enjoyed. And now today, I still work with him every other week. He’s gotten through a lot of undergrad math, and so he’s actually sinking his teeth into research, working university mathematician. And there’s like bunch of derivatives being tossed around. And it’s in this area that he was resisting back in seventh or eighth grade, and he’s having the time of his life right now. And if we had not pushed him through this segment of the journey that he was resistant to, he would not be doing what he’s doing right now.

Measure actual mastery rather than proxy signals. (7)

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.

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.

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.

Jason: If you go down to the high school level, they don’t even homework anymore. Justin Skycak: I remember back when I was living in your daughter had assignment. Jason: She was just writing down in some of these cases, kind of nonsense. And you were looking at graded stuff, they were just giving a check. She would just write down stuff that was totally wrong and they were just checking off the homework. Justin Skycak: Yeah, it was just whether or not the assignment into Canvas. Not whether it’s correct or not, just whether it was uploaded. So it kind of got to the point where a lot she learned because the stuff wasn’t being

If you go down to the high school level, they don’t even homework anymore. I remember your daughter had an assignment. She was just writing down in some of these cases, kind of nonsense. And you were looking at graded stuff, they were just giving a check. She would just write down stuff that was totally wrong and they were just checking off the homework. Yeah, it was just whether or not the assignment into Canvas. Not whether it’s correct or not, just whether it was uploaded. So it kind of got to the point where there’s a lot she hadn’t properly learned because the stuff wasn’t being checked.

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.

I don’t ever believe in take home exams. Definitely with LLMs coming, you can’t write papers. When I went to school in college, University Chicago, the final exam were with those blue books, you those little paper blue books, and you would write those in class, and it was a 2 or 3 hour final exam period and it was just like you’d fill up, you know, 6, 7, 8, 10 pages of a blue book for each essay question, you just write on the fly. And that’s what you gotta do for all of these more paper writing classes, history and English and things like that because it’s too easy to fake.

Adults and institutions have to maintain real standards instead of continually softening them. (4)

You kind of have to do that as a teacher. You kind of have to scare the kids, you don’t mess with this teacher. You do a good job and otherwise it’s gonna be a horrible experience. If you come in, you’re like, we’re all going to be friends. And I just think of me as your study buddy, and I’m going to help you. Kids are like, oh, this is a joke. And they just have no respect for you. And then you never get it back. And then if you try and tighten the reins later they just can’t cause kids are like, And so you set the tone early. then you lighten up. Unfortunately, any teacher thinks you can come and say, well, I don’t do that. I just I’m just happy all the time. I like, guarantee you that kids are not doing anything close to what they’re supposed to be doing.

When I was in high school, teachers didn’t meet with parents ever. But now because parents lobby through a barrage of emails to the teacher to the principal to the whatever and then just make everything really painful. The parents are acting like an agent. It’s like a sports agent. what they want, a lot of them want is their have like great grades so they can go to whatever college they want to go to, But they also want their kids to learn a lot, but they don’t want their kids to do a lot of work. And kids aren’t going to learn much unless they put in a fair amount of work, unfortunately, that’s how the world works. So, but parents, they wanna do less work. want my life, I want there to be low stress at home, but I want my kid, yeah, I my kid to be really educated, but I also want them to get great grades. Okay, well, the kids, there are some kids who like work really, really hard by default. Most of them don’t. there’s a whole spectrum. But anyway, the parents at the end of the day, once they realize that, my God, my kid’s gonna get a C, or is he gonna get some Bs when I tell them they’re not gonna get A’s and they’re like, my God, this is really, they’re not gonna get to go to Harvard or whatever dream they have. And so then they serve as this, they wanna lobby the school. And so they do, and so that has helped lower the standards, I think, especially these private schools, right? paying 30, 40, 50, $60,000 a year and, and, and, and, know, and my kid works really hard in this. these, these costs and the, and the superintendents and the head of school and stuff, they’re like trying to keep the parents at bay. Cause that’s what pain for the pain for the schools, for the administration and the teachers are parents. Parents P for pain.

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.

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.

Foundations compound upward (19)

Core low-level skills have to become fluent and automatic. (9)

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.

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.

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.

heard from a lot of families tutors and teachers that kids have not mastered their math facts. They don’t know the multiplication tables. They’re finger counting. Not finger counting in fourth or fifth grade. They’re finger counting in 10th grade. And if you can’t do, you know, even multiplication, if you don’t know multiplication tables, you’re gonna really struggle even with basic algebra.

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.

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.

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.

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.

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.

Prerequisites compound upward; missing ones eventually surface as failure in more advanced work. (10)

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 rot can start early and start in elementary middle school. It’s like, we don’t teach multiplication tables. And because they don’t know multiplication, the kids are struggling with algebra. So we have very watered down algebra. And we let them use Desmos and calculators and stuff because they can’t actually graph things. They can’t actually solve things. So that stuff’s weak. And so then the rot goes all the way And so you just got to nip this stuff in the bud. And you got to hold standards early. You got to fix core skills early. You can’t allow students to progress up the ladder and carry along all of these weaknesses and then come up with all kind of ways to put up smoke screens and hide it by not giving standardized tests and giving inflated grades and glowing recommendations and whatever nonsense you’re doing. These are adults created this problem. You know, it’s no one person’s fault. It’s not the teacher’s fault. It’s not the school administrators. It’s not the school board. It’s not the parents. It’s not the politicians. It’s everybody. We all created this together. sometimes You know, you just have to look at a system and say, OK, this thing is broken. There’s rot and we got to dig out the rot and we got to start fixing these things and so that we have a healthy educational system so students able to be successful and learn that they want to learn and be able to do things they want to do and become the kind of people live the lives they want to lead. That starts early and you got to stay on it their whole career.

Harvard having to add a ton of remedial support to its calculus courses. Cause they’re getting the same kind of situation of people coming in, and they are nowhere close being able to pass a calculus course. They got tons and tons of missing foundations in high school and probably even middle school, Like at Harvard, really? It’s just incredible.

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.

1 in 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. It’s so ridiculous. They had a remedial math course for students who were missing high school math foundations. And that was too advanced for a lot of the students coming in. So they had to make a remedial, remedial math course, two layers back. So the crazy part is you don’t actually get a good signal of who’s going to be in remedial math based on their high school grades. “In fact, for more than two decades, the mathematics department has found that out of all available student data, the single best predictor for math placement has been the SAT math section score, with the ACT score being an equally good predictor.” And guess what happened in recent years? They didn’t require those scores to be set. Pandemic happens, there’s a ton of learning loss. And now we’re just going to say like, okay, we’re going to do like this holistic admissions without test scores. We’re going to remove things that actually measure actual learning well. And we’re going to rely on high school grades, which are known to be inflated now, especially after the whole pandemic thing. So now we have this kind of shit show that’s happening.

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.

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.

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.

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.

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.

Advanced math expands capability and leverage (11)

Calculus is a doorway into a much larger world of mathematics. (6)

What comes after calculus. (2)

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.

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.

Calculus is not the finish line; early arrival buys runway. (4)

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.

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.

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.

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.

Advanced technical understanding expands what you can build and understand. (3)

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.

A lot of people realize later, who go into software engineering, like they don’t really care about math in school. And then eventually they build up the foundations of coding and then they realize that if they just knew all their math, then they could be doing so much more. The earlier you make this happen, somebody gets interested in coding, sees how important math is to doing non-trivial coding. Just imagine a kid graduating high school. Not only do they know pre-calculus with like coding applications but also calculus, linear algebra, multivariable calc, and differential equations, basically your core engineering math all the coding applications and they come into college and they’re just blowing the socks off of anybody who gives them an opportunity to do some research, an internship, they already get the basics of everything. They’re ready to actually make serious impact. It’s so rare to see in an undergraduate researcher or an intern, right? you can’t count on them You just like throw them a toy problem usually, but like if you can actually make a serious impact at a young age, because you have the skills to do so, then you can just compound that into a massive compression of time and figuring out what you’re interested in and everything.

In software, just importing a solution from the library and saying I imported this model and I ran it. “now I’m a machine learning researcher engineer.” Like, no, no, no, you can’t just use the off the shelf. Like you can’t just use the, the theorem and wield the theorem and say, “now I am all powerful.” You actually have to go code from scratch, re-derive the result from the bottom up to really understand the mechanics of what is it. It’s not enough to just take it off the shelf and use it. You need to know what went into building this thing.

Real technical skill creates career leverage. (2)

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.

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.

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