The Missing Middle in Test Prep
There's a large gap between the standard math curriculum that students learn at school, and the additional skills that show up on standardized exams like the SAT, ACT, etc. We're working to fill it.
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This “inscribing” trick is a perfect example of the kind of skill that shows up on the SAT, but most students won’t learn it even if they ace all of their math classes at school.
There is a gigantic “missing middle” on these exams, a chasm between the standard curriculum and what’s on the test, the purpose of which is to raise the ceiling of the test’s ability to measure
- cognitive advantage (IQ, generalization ability, etc.) and
- willingness to put in the work to train up their skills outside the standard curriculum.
But unfortunately, most SAT prep resources either address little to none of this “missing middle,” and whatever is addressed is typically presented with so little pedagogical effort that even highly capable and motivated students find it difficult to process.
As a result, the “missing middle” primarily serves to measure (1) and not (2).
Our goal is to change that. We are putting a full-assed, full-nerd effort into identifying this gigantic missing middle and adding it to our finely scaffolded knowledge graph, so that we can put hardworking students in the best possible position to succeed on these exams.
By the way, we’ll be doing this for the ACT too, and all the other common exams. And it’s the same idea for competition math, where the missing middle is even bigger.
(H/t to Jason for describing the chasm between the standard curriculum and exam tricks as “missing middle.” The phrase really drives home the idea that you can visualize the standard curriculum as a smaller subset of knowledge within the space of knowledge tested on these exams.)
(Update 2025-09-06) Why Students Need To Be Solid On The Missing Middle
We took a deep dive into past SAT exams this week and excavated a ton of additional topics that students really need to know for the exam, but won’t explicitly encounter in school or any standard textbook/course.
Many common exam question types are intentionally set up to increase cognitive load and confuse/trick students, and if a student is encountering these questions for the first time while taking the exam, then they’re cooked. (Here’s an example.)
It doesn’t matter if you could have figured it out eventually on your homework, but ran out of time on the test. (You only get about 90 seconds per question on average.)
It doesn’t matter if you got the core of the solution right but just fell for a “silly mistake” trap while executing it.
It doesn’t matter if the question is long-winded and all you missed was some little detail that you would have realized if the question had just been stated more clearly and concisely.
The thing a lot of people don’t realize about the SAT is that it is a well-trained opponent who has made a career out of studying your game and figuring out what’s most likely to mess you up.
Coming up against these types of questions is like entering a fight with a professional boxer who has made a career out of exploiting any weakness their opponent might have. If you have solid fundamentals but are untrained on what to expect from this particular opponent in the ring, you are absolutely going to get your ass handed to you.
It doesn’t matter if you could eventually figure out how to beat your opponent in a longer fight. It doesn’t matter if you could knock them out with a wind-up punch but they keep moving around and throwing off your balance. It doesn’t matter if you can block a freaking bulldozer but they keep faking you out and sliding punches around your blocks.
When you are going up against an opponent who is pulling out every trick in the book, you absolutely have to train on specific situations that might show up in the game. Baseline athleticism is necessary but it is not sufficient to win.
You need to know in advance what kind of tricks your opponent might pull, what kind of weird attacks to expect, how to defend against them – and even further, you have to build reflexes that will enable flawless execution come game day.
That’s the kind of training experience we are working to deliver with our SAT course.
(Update 2025-08-29) Watching the Manifold Hypothesis Play Out
I’m currently reverse-engineering the distribution of problems on the SAT so that we can automatically generate high-fidelity practice exams.
It’s looking like the manifold hypothesis holds true here.
Yes, you need to know lots of skills for the SAT, and yes, any given SAT problem can combine several of those skills. If you run the computation for how big that space of possibilities is, you get a combinatorial explosion.
But the types of problems that actually show up is a proportionally tiny subspace (i.e., a manifold).
Don’t get me wrong, the subspace is still pretty big in an absolute sense, pedagogically speaking. But it’s not astronomically big.
And here’s the kicker: From what I can tell so far, it’s not too big to enumerate exhaustively via explicit instruction.
Typically, people throw their hands up and say “The complete space of possible problem types is too big to teach explicitly. There are hundreds of skills you need to cover and any problem can combine several of them. Millions of combinations are possible. You can’t explicitly teach that many problem types. All you can do is cover a scattered sample and hope that your students have enough generalization ability to fill in the gaps.”
To that I say: you just don’t know what the manifold is.
But in order to find the manifold, you have to go on a serious archaeological expedition. You can’t just glance at a couple exams and throw your hands up and shout “it’s incompressible” after seeing a bunch of different problem types.
For the past couple days, I have spent most of my time functioning as a manifold archaeologist, constructing a taxonomy that is conserved across the exams.
I still have a ways to go, but I’ve gotten far enough to see that it’s working tremendously well. You have to dig a lot to find pieces of the skeleton, but once you see it, it is mind-blowing how well it compresses what you originally thought might be a combinatorial explosion.
It makes me wonder how compressible competition math is.
Follow-Up Questions
Q: Are you guys thinking about tackling the math relevant to the MCAT, GMAT, etc.?
A: Yep, we’ll be coming after all the common exams. But SAT/ACT is the first step. There’s quite a bit more for us to do to fully get our arms around exam prep – there’s a large missing middle (much larger than anyone would expect), and then on top of that fundamental skill-building arena there also needs to be a full-blown test prep practice arena, and in that test prep practice arena we’ll need to estimate your current score and project how much practice you’ll need to nudge your score another rung upwards.
Q: Is test prep ethical?
A: This question is so silly that in hindsight it probably doesn’t even need a response, but I’ve heard it a couple times, so I wrote up something to reference any time it comes up in the future.
Personally, I think it’s ethical to present students with opportunities based on the end result of their work ethic, and provide avenues by which hard work can more effectively compensate for things like
- cognitive advantage (high generalization ability / WMC / IQ / etc.) and
- social/financial advantage (e.g., families know the game being played and can locate and pay an arm and a leg for a top-of-the-line test prep tutor).
To me, bringing down the price of elite-quality test prep and making it accessible to many, many more students seems like an unequivocally good thing.
I can’t tell you how many students work their asses off to do all they can to get into the colleges and careers of their dreams, and when they finally get to where they want to be, they realize that most of the people around them had knowledge of and access to all sorts of additional efficiency gains enabling them to work smarter.
(I was one of them. I didn’t come from a particularly disadvantaged background, but nobody in my family knew anything about how to get into the colleges/careers I was interested in, and it’s only after encountering more advantaged students/families that I realized how many times I shot myself in the foot and could have capitalized much more on my hard work if I just had a better sense of the game being played.)
Most people who push back against test prep also have a misguided conception on what test prep even is. Hence the next question:
Q: Why focus on the test? Why not equip students with general skills to do well on any assessment of their mathematical knowledge?
A: That’s a false dichotomy. Test prep isn’t about gaming the test. It’s about
- producing students with a high degree of skill in the subject matter, like well-trained athletes,
- and ALSO making sure they are particularly well-practiced defending against the particular plays/tricks that their opponent is most likely to run in the upcoming game.
Q: Can’t the test-makers just change the exam?
A: That’s completely theoretical. Standardized exams don’t change rapidly, just like an opposing sports team isn’t going to rewrite their whole playbook from scratch every time they play a game.
Sure, exams periodically get updated and there may be some “distribution drift” over time in the types of problems that show up, but the drift happens at a glacial pace (think: continental drift), and it’s nothing that test prep providers can’t adapt to.
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