What Happens when Middle School is Put to Good Use
Typical honors students can learn all of high school math plus calculus *in middle school* if they are taught efficiently. They don't have to be geniuses, don't even have to spend more time on school. Just need to use time efficiently. Few people understand this, as well as the kinds of opportunities that get unlocked when a student learns advanced math ahead of time. The road doesn't end at calculus, that's just an early milestone, table stakes for the core university math that empowers students to do awesome projects.
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The worst segment of the K-12 mediocrity is 6th-8th grade math.
Typically, kids learn counting/arithmetic in elementary school (K-5) and then spend the next 3 full years spinning their wheels without learning much new math, only moving on to Algebra I in 9th grade.
If you take those 3 years and put them to good use, you can actually enable many kids to cover all of high school math during those years and pass the AP Calculus BC exam by the end of 8th grade, without spending any extra time doing math.
What? How is that possible? And what share of students do you estimate this to be? Is this just the top 10%? 1%? 0.1%? 0.01%?
The top 10% can learn all of high school math plus calculus *in middle school* if they are taught efficiently. Just typical honors kids. They don’t have to be geniuses, don’t even have to spend more time on school. They just need to use time efficiently.
I know that may sound shocking but it’s what we did in our original school program. It’s the most accelerated math program in the USA, there have been plenty of other news articles written about it, and there’s plenty of other information straight from the horse’s mouth including a summary of events 2014-20 (from Sandy and Jason’s perspective), a summary of events 2019-23 (from my perspective, with a focus on teaching in the school program and getting the algorithms in place to turn it into a fully automated system), and another summary of events 2014-20 that I gave on Anna Stokke’s Chalk and Talk Podcast #42 (I’ll paste the relevant snippet below):
- "So back to this eighth graders taking AP Calc BC story. We originally started as a nonprofit school program founded by Jason and Sandy Roberts. One of their kids, Colby, was on the fourth-grade math field day team, and his parents were coaching that team. Their kid and his friends were all really excited about learning math, so they did the standard fourth-grade field day stuff. But the kids were so excited that they didn’t want to just stop at fourth grade. Something they would often ask Jason and Sandy was, “What’s the highest level of math?”
Jason and Sandy would have to say, “Well, it goes really, really high, but for your purposes, let’s just say it’s calculus, because that’s what seniors in high school take if they are on the honors track.” And the next question was, of course, “When do we get to learn it? Can we learn it now? Can we learn calculus tomorrow?” They were just so excited about it.
Jason and Sandy were teaching a bunch of these kids advanced math, even through fifth grade. They got up through a bunch of high school math and to the point where they could start learning calculus. One thing led to another, and this turned into an official school program that was not just a pullout class but became a daily Math Academy class. There were other cohorts that came in following years.
What this turned into was that we would get students in sixth grade who were solid on their arithmetic. They might know what a variable is, but they didn’t really know how to solve equations or anything. They were kind of at an early pre-algebra level. We would scaffold them up, teach them all of high school math within the next two years—sixth and seventh grade. Pre-algebra, algebra one, geometry, algebra two, and pre-calculus. In eighth grade, they’d be ready to take calculus.
Then, they would take the AP Calculus BC exam. We got to the point where most of the students who took the AP Calc BC exam in eighth grade passed, and most who passed got a perfect five out of five on the exam.
A couple of things I should say, these are not national talent search students.
How the kids were selected was that they scored at or above the 90th percentile on a middle school math placement exam, which is typically taken by all fifth graders in the district around February or March. They were then invited to join the program. It's a seventh-grade math skills test, so it provides a somewhat high skill level, but it's not designed to identify math aptitude.
This is also in the Pasadena Unified School District, where about two-thirds of the student population qualifies for the federal free and reduced lunch program, and about 44 percent of all K-12 students are educated in private schools, compared to the California average of 11%.
This is not a particularly talented group of students. It's not a biased group of the top students in the nation. Just think of a standard school and kids in the standard honors class. They can be accelerated way, way, way higher than they currently are.
When Jason and Sandy were teaching, they were doing this all manually and achieving very good results. But these results got even better once students started working on the Math Academy system. Jason got tired of the kids saying, “I forgot to do my homework,” or “Oh, I forgot a pencil,” or all these excuses for not doing work. So, he just built a system where he could pick problems for them to do, and then all they had to do was log in at home and do the problems online.
It would automatically grade the problems and keep track of all the kids’ stats, keep track of the class accuracy, and various topics. Over time, this evolved into a system that did more and more of the teaching work.
In the summer of 2019, that’s when Jason pulled me in to make this system a fully automated platform that would actually select learning tasks for students. So, we built this automated task selection algorithm and continued refining it. By the time the pandemic hit in 2020, the big question was how to maintain this level of efficiency from manual instruction.
The answer was, “Well, we have this halfway baked task selection algorithm. Let’s just get it all in place over the summer and put the whole school program on it.” And that’s what we did. That’s how our AP Calc BC scores skyrocketed, from putting them on the system."
Okay, so even if many kids can learn calculus before high school, what's the point? What's left for them to do in high school?
I touched on this briefly in the podcast:
- "In 9th through 12th grade, what they do is learn a bunch of undergraduate math. We have PhD-level math instructors who teach the 9th through 12th graders, and they learn linear algebra, multivariable calculus, probability statistics, real analysis, abstract algebra, and algebra.
They go through all this content, and they are also often working on some independent math projects. In terms of full outcomes for the students, it's still pretty early, so the first cohort is still in their junior year of college, and they haven't really hit their careers yet.
We’ve been hearing a lot of really cool things from them. One kid is doing an accelerated master's degree in undergrad. Some other kids got into MIT and Caltech. Another kid is currently a senior in high school, and he did an internship at Caltech the summer of his sophomore year, then worked there on a research project for his junior year. He actually let me know a couple of weeks ago that he got a paper published as a high schooler, a solo-authored paper in a legit journal. It's interesting to see his author affiliations: Pasadena High School and California Institute of Technology."
But I’ll go into more depth here.
Few people understand the kinds of opportunities that get unlocked when a student learns advanced math ahead of time. The road doesn’t end at calculus. That’s just an early milestone, table stakes for the core university math that empowers students to do awesome projects.
For instance, a former student, Matteo, leveraged his outsized math/coding chops to – as a high schooler – conduct research that “revealed 1.5 million previously unknown objects in space, broadened the potential of a NASA mission” (not hyperbole, a direct quote from Caltech’s website). He also published his results solo-author in The Astronomical Journal and won 1st place ($250,000) in last year’s Regeneron Science Talent Search. And this is still just the beginning – I can’t wait to see where his interests, skills, creativity, and work ethic take him in college and beyond.
But here’s the thing that’s important to understand: this kind of story won’t play out for students who are marching through the standard curriculum. Not even to students whose skills are a year or two ahead.
If you want to do hardcore university-level quantitative research, you need hardcore university-level quantitative skills. You need it for the work itself, and you also need it to land a quality mentor who provides high-level guidance to keep you working in the right direction. If you want far outsized results ahead of time, you need far outsized skills ahead of time.
Matteo and other Math Academy students learned
- all of high school math (Prealgebra / Algebra I / Geometry / Algebra II / Precalculus) in 6th and 7th grade,
- AP Calculus BC in 8th grade, and
- plenty of multivariable calculus / linear algebra / differential equations in 9th grade.
With such solid and comprehensive math foundations, they came into 10th grade ready for some serious quantitative coding.
So in addition to continuing down the math-proper track (real analysis, abstract algebra, etc.), we were also able to offer these students a quantitative CS course sequence where we scaffolded them up to doing masters/PhD-level coursework by 12th grade (reproducing academic research papers in artificial intelligence, building everything from scratch in Python). We called this the “Eurisko” program.
Matteo joined Eurisko as a 10th grader, during the last year it was offered, and worked hard to complete almost all 2-3 years’ worth of assignments in a single year. (Eurisko ended when I relocated; nobody else in the district had the requisite knowledge to teach it.)
That summer, he participated in a research internship/mentorship program at Caltech, which was meant to be a brief 6-week taste of research, but he was skilled and driven enough to knock it out of the park, stay on afterwards, and achieve some serious results.
This is exactly the position that we were trying to put students in with the Eurisko program – get them to a point of skill that they can capitalize on some math/coding-related opportunity and turn it into a chain reaction of fortunate events. And it’s been so great to witness some of these chain reactions get underway.
But the best part is that we’re gradually able to do more and more of this at scale. We’re taking everything we’ve learned from doing math/coding talent development manually, and building it into our online system, to make it available to the whole world.
We’ve already built a pipeline from 4th grade through core undergrad math courses, and we’re working to extend that pipeline further in both directions, eventually spanning everything from the simplest arithmetic to the entirety of an undergrad math major, the Eurisko program, and more. I can’t wait to hear more of these amazing stories.
Follow-Up Questions
Q: Nice that he did all this, but really, did he have to do it so young? What’s wrong with him just being a kid for a few years?
For some kids, this IS their version of being a kid. Some kids are driven to pursue serious interests at a relatively young age. I was too. This level of drive is uncommon but not unheard of.
Q: While Pasadena has low income folks, it also has Caltech – smartest school in the country. Were the kids’ parents Caltech professors?
A: I taught many of these classes personally for 3 years and didn’t have a single Caltech prof’s kid – they typically sent their kids to private schools. Most of my students’ parents had more typical and non-quantitative jobs.
Sometimes the parents had so little math knowledge that I would get questions like “my friend’s kid is practicing arithmetic at school, but my kid’s math in your class doesn’t look like that, he seems to know his arithmetic when I ask him but is he getting enough practice?” – the parent didn’t understand that their kid’s algebra problems were building on and providing implicit practice on arithmetic, and I had to explain how math is really hierarchical. They thought arithmetic and algebra were completely separate, like different novels in English class or different countries in history class.
Q: You sound like you’ve never met a middle schooler.
A: I taught many of these students personally. As a year-round, full-time public school teacher. For 3 years. I was there, boots on the ground at ground zero managing the classrooms while simultaneously building the software.
Yeah, you gotta stay on top of them, strike a balance between keeping them focused/productive and letting them be silly, you have to hold the line on the quality of work you expect, you have to keep parents in the loop when their student is putting in a good effort or not.
But if you get students on the rails in an efficient practice environment, and nudge them back onto the rails whenever they start drifting off, they can accomplish much more than you might expect.
Q: Sure that all sounds great in theory. But in practice it would never work at least at the level you believe. Precal? Sure. Calc? Zero chance even your top 5 would be able to do it let own top 10%. The maturity isn’t there yet.
A: I’m not talking hypothetical. I’m saying this already happened. Top 10%, super-high-quality instruction (mastery learning, spaced repetition, frequent broad-coverage quizzes, etc., on the MA system) throughout middle school, started with prealgebra in 6th grade, covered all of HS math in 6th/7th grade, took AP Calc BC in 8th grade, most passed, and most who passed got a perfect 5 out of 5.
Q: It would be nice if parents didn’t have to fight for this in public school. My son took Calculus his sophmore year. But it took a lot of advocating at the district level.
A: Agree. I had the same experience myself in high school. Getting properly placed into a class just 1 year ahead of the highest honors path was like pulling teeth.
In my case, after enough pestering the math dept head, he lent me his notes/HW sets for the year above – likely trying to dissuade me, but at the time I thought he had just given me a workbook to do, and I carried it around with me everywhere in and out of school working through it like 10h/day, got through half of it by the time he tracked me down a few days later to scold & tell me “I need that back so I can make copies for the class’s HW next week!”
I had written my work/answers in the booklet, so he had to erase all that, which made him realize how driven I was and that I could handle the workload.
Q: Of course school should give all kids as much as they can handle, but that’s impossible.
A: That *used* to be impossible, back when when instruction was bottlenecked by the manual bandwidth of the teacher.
But today we have technology to automate the heavy lifting in delivering adaptive, individualized instruction.
I elaborated in the podcast, here’s a snippet:
- "Let me start by explaining how Bloom tried to solve it. He didn’t just frame the problem—he had an entire research program aimed at solving it.
Our approach is similar to his in some ways but different in critical ways.
He hoped the benefit of a human tutor could be captured by combining various evidence-based learning strategies. That sounds like a great idea, right? You take something effective, deliver it to students, then add another evidence-based strategy—whether it's pedagogy, the study environment, or something else. You keep layering these scientifically supported techniques, hoping that as you build them up, you reach the two-sigma effect—or at least get as close as possible. You try to improve learning outcomes as much as possible.
The thing about Bloom's approach is that he restricted his search space to strategies that could be implemented manually, not necessarily to the fullest extent. For instance, one of the strategies he examined was mastery learning, in which you ensure students know their prerequisites before moving them on to more advanced material.
In Bloom's solution to the two-sigma problem, mastery learning was not designed for an individual student but for the class as a whole. You can only approximate this for an entire class. That’s not bad—it does improve learning outcomes if a teacher attempts to apply mastery learning at a class level—but it’s not as effective as tailoring it to each individual student, who will have a unique knowledge profile.
This was the fatal flaw in his approach. I don’t mean flaw in the sense that it didn’t work at all, but rather that it didn’t fully achieve the effect of human tutoring. His search was ultimately unsuccessful because it was constrained by the limits of human teaching labor.
At the time, that was a reasonable constraint because computer technology was far less mature. Computers were very expensive. You were lucky if a school even had one. Things are totally different today.
For nearly a decade, Math Academy’s challenge—and in fact, our purpose, the whole reason we exist—has been to carry this torch forward and reattempt a solution. We overcome the limitation of human teaching labor by leveraging technology to implement individualized learning techniques to a much fuller extent.
We started out in a public school district, teaching manually and applying individualized learning techniques as much as possible. It was similar to Bloom’s approach—doing it manually to the greatest extent feasible.
The key difference is that we gradually built an online system to automate pieces of the work and apply them more effectively than we could manually. That freed us up. We would teach as well as we could in person, get a handle on the problem, and once we understood how to structure the solution, offload it to a computer program—whether for mastery learning on a knowledge graph, spaced repetition, or other techniques. Then we would return to manual instruction and ask, "What’s next? What else needs to be offloaded?"
By the end of this process, we created a teaching machine that is shockingly effective. For instance, students have passed the AP Calculus BC exam as early as eighth grade."
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