Want to Major in Math at an Elite University? Getting A’s in High School Math is Not Good Enough
If all the knowledge you show up with is high school math and AP Calculus, and you're not a genius, then you're going to get your ass handed to you.
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Prospective math majors often graduate high school with A’s in their math classes, leading up to AP Calculus BC in their senior year, thinking that they are prepared for their university’s math program.
But if you’re trying to major in math at an elite university that is known worldwide for its math program (MIT, Caltech, UChicago, etc.) –
and all the knowledge you show up with is high school math and AP Calculus, and you’re not a genius –
then you’re going to get your ass handed to you.
Here’s why this happens.
The problem is that high school math – even the “honors” track, even getting a 5 on the AP Calculus BC exam –
doesn’t accurately depict the level of background knowledge (especially proof-writing ability) that is assumed in serious math-major courses.
There are two overlapping groups of students who succeed in top math programs:
- mathematical geniuses, and
- students who have outsized background knowledge in mathematics –
for instance, it is not uncommon for elite university admits who are serious about majoring in math to graduate high school
- having already taken linear algebra & multivariable calculus and
- having already received plenty of exposure to proofs including inklings of real analysis (e.g., epsilon-delta limit proofs) and abstract algebra (e.g., structure of the additive & multiplicative groups of integers).
These students make up such a tiny slice of the overall student population that you’re unlikely to encounter them as classmates in high school.
But they exist, and they’re going to be concentrated in the math-major math classes at these elite universities.
When the professor is writing furiously at the chalkboard, assuming that their students are following along and absorbing the information in real time, these students actually are.
So much of the content – or, at least, the overall way of thinking about it – is familiar to them.
These students are able to keep up, and if you’re not able to do the same, then the class is not going to slow down just for you.
Not to mention, you’re going to feel dumb, which is going to severely impact your motivation even if you manage to find help outside of class.
Basically, it’s like you get accepted to some world-class martial arts training program on the basis of meeting some relatively low threshold of baseline fitness.
You show up thinking you’re prepared, but as soon as you get there and step onto the mat you get thrown clear across the room by people who were far more prepared.
Here’s a case study.
I recently had a conversation with someone who majored in physics at UChicago.
He initially started in math, and he thought he was prepared having taken AP Calculus BC…
but he got smacked in the face by the level of abstraction and proof-writing ability that was assumed, and he couldn’t keep up.
He ended up switching to physics where
- proofs were less of a thing and
- the playing field felt more level in terms of how much prior knowledge was assumed (as well as how much prior knowledge other students were coming in with).
He would have liked to study math if he had more time to catch up, or if he knew earlier about how far behind he was –
but he did great in all his high school math classes, and was recognized as one of the “smart kids” in the class, so he never suspected he was actually behind the curve.
Zooming out, this case study is representative of a general phenomenon that can sneak up on you when you’re at, say, the 99th percentile of a skill.
At first, you’re exceptional enough that you receive praise from virtually everyone, and you may never go head-to-head with someone who can beat you.
That is, until you join some specialized program where everyone is at the 99.9th percentile – where, suddenly, you’re the worst one there.
And here’s the real kicker: if it’s a time-sensitive program, you may be so far behind that it’s infeasible to catch up.
If you knew the caliber of these people earlier, you could have spent time working harder to join their ranks in the 99.9th percentile…
but that moment has passed, and now the door is closed on this opportunity.
So… what can be done about this?
If you’re an aspiring math major, you should really consider taking Math Academy’s Methods of Proof (MoP) course the summer before heading off to university.
MoP covers so many different types of proofs, in so many different contexts. You see the whole zoo –
and you develop tons of experience with elementary concrete examples that get built upon in all those branch-off subjects like real analysis & abstract algebra.
Epsilon-delta limit proofs, structure and operations in the additive & multiplicative groups of integers modulo N, you name it.
Even more niche stuff like number theory – for instance, MoP covers modular arithmetic & linear congruences really comprehensively, which is another one of those things that students typically don’t learn well enough before taking a number theory course.
Anyone who takes MoP before university should feel a seamless transition into serious math-major courses, as opposed to the massive shock & overwhelm that typically occurs.
Seriously, if every aspiring math major took MoP the summer before heading off to university I bet it would cut the math major drop rate by more than half.
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