The AMC vs. the usual tests and grades
Understand how to take advantage of this unusual opportunity
This blog, also hosted on Substack, is where I share what I’ve been discussing, writing, and reading in the realms of academic and cognitive development for gifted high schoolers in specific, and education in general. My tutoring and coaching practice is centered around the idea that gifted students usually get the short end of the educational stick… but that it doesn’t have to be that way. If you know somebody who could benefit from my perspective, please forward this to them or let them know they can set up a free intro chat with me.
I get lots of calls asking about the AMC 8, AMC 10, and AMC 12 tests. Most commonly: Are they useful for college admissions? and How can I tell whether my son or daughter would be a good candidate to participate?
I’d like to give you some important but not-so-obvious info, and also address some common misconceptions.
The big takeaway is that success at the AMC 10 and 12 requires a different approach. It’s not only different from regular school; it’s also different from the ACT and SAT.
When families expect AMC preparation to look like ACT/SAT preparation, it leads to disappointment.
But when they know what to expect, they get what they came for, and so much more.
First, what are the AMC 8/10/12?
The American Math Competition (formerly the American High School Math Examination) is a once-a-year opportunity for STEM-minded middle- and high-school students to demonstrate their excellent problem-solving skills and extra-curricular math knowledge. There are three versions; you take the one appropriate to your grade level and math knowledge. You get about an hour (more or less depending on your level) to answer 25 questions. The median score on all three tests is a one-digit number (!), and doing very well puts you in the running to train to represent the USA in the International Math Olympiad, which is a very big deal, just as its name implies.
Relatively few schools know about it, and fewer still offer it, but most schools are willing to get on board if they have even just one interested student. Despite its low profile, it’s a big competition: tens of thousands of students participate each year. Something like two thousand advance beyond the first round.
Note: the AMC 8 (the middle-school version) works a bit differently from the AMC 10 and the AMC 12. From here on, I’ll use “AMC” to refer primarily to the AMC 10 and AMC 12 exams.
The AMC is harder than you realize.
Before starting work together, I ask clients to identify their minimum goals and their stretch goals. This helps me stretch professionally, and it also helps me understand how realistic clients’ goals are.
Joseph does great in pretty much everything related to school. He’s just an unusual kid. This should be easier and quicker for him than it is for most students, right?
One thing that’s come up is that many believe that the AMC, like the SAT/ACT, can be gamed through high-end tutoring. That idea is seriously misleading. Instead, I’m going to lay out a better framework for thinking about the AMC and what successful preparation really looks like.
AMC prep is not like ACT/SAT prep.
People often expect prep for AMC to be analogous to ACT/SAT prep, but it’s not. AMC prep is different. And it’s quickly becoming more different, but it’s been hard to get straight talk about what’s changing. Let me help:
What ACT and SAT prep have become
For context, here’s how ACT/SAT prep has evolved over the last few decades: while the SAT/ACT people have been claiming with straight faces that preparation doesn’twork, those who participate in preparation have been consistently and systematically proving them wrong. It works so well, in fact, that many argue that it’s been a driver of inequity. (And so well that it sometimes gets students into schools that are too challenging for them to thrive there!)
Service providers have responded to market conditions, in other words, by getting very good at their jobs, particularly at the high end. There now exist SAT prep providers who can reliably assure success, given sufficient fees. (I am, by most measures, one of them.)
So against this backdrop, it’s no surprise that some parents see the SAT as a barrier to one’s collegiate admissions success that can be bought off. Because for many families, that’s what ACT/SAT prep has become: little more than a “cost of doing business,” with a relatively assured outcome, given a high enough price tag.
And that, in a nutshell, is why it’s no longer as valuable to colleges as it used to be. We preparers have figured out how to raise students’ scores reliably, so it’s now a better indicator of family wealth than of student potential.
Here we are in 2023. The SAT and ACT are less valuable to colleges (and they also feel PR pressure to downplay them), while applicants have become ever-more-qualified in spite of broadly weak COVID school years.
For many students, these tests are a necessary but not sufficient part of the admissions process. In other words, you have to do it, but it’s not enough by itself.
Easily distinguishing exemplary applicants is more valuable to admissions committees than ever before, and the SAT/ACT doesn’t cut it.
I’ll say it more plainly: neither grades nor the SAT/ACT are good tests of high-to-extreme ability, and college admissions officers know it.
- Grades are so inflated that there’s no way to signal exceptional performance. (What’s better than A+?) So what is a teacher supposed to do with a student who is literally the best in years? Sure, there might be a chance to write a stellar recommendation… but what do you think are the chances that “the best student I’ve had in years” isn’t part of that teacher’s standard recommendation template?
- ACT/SAT questions are simply not difficult enough to distinguish between very good students and great ones. This may seem “obviously” wrong to you, since clearly plenty of students put a lot of effort into doing better on those tests every year. But what you need to understand is that learning SAT material only gets you from a middle-of-the-curve score to a few hundred points above, i.e. from a non-starter score like 1000 to a decent but ultimately admissions-unhelpful score of 1350 or so. Beyond that point, the game is increasingly about meticulousness, patience, self-awareness, and self-control, not about knowing more facts and techniques: it really is about learning to perform like an error-free robot. It is emphatically not about becoming a genius. (Those who truly are exceptional can generally “see through” the questions well enough that a 1450 or so is easy for them to achieve. But this is not the skillset being taught to those who work their way from 1250 to the 1500s.) To sum up: scoring 1550 requires you to be gifted or prepared (or both). But telling these categories apart is impossible from the test alone.
The AMC is a different test, in a different part of its lifecycle
The AMC checks all the boxes for this moment in high-end university admissions. It’s been around for about 80 years. It’s an incredibly good test of general problem-solving skill. But because it’s been viewed as a “test for math geniuses,” which wasn’t a serious category until recently, almost no one has been paying much attention to it outside specialized math programs.
The test itself is quite different from the ACT/SAT:
- Preparation is necessary. (You simply cannot do well if you take it cold.)
- Preparation is significantly more involved.
- Quality of preparation help is not yet nearly as mature as for the ACT/SAT market.
- It is more all-or-nothing, in that there is an officially recognized cutoff score above which you get an admissions-ready “gold star.”
- It’s more niche: not everyone loves math puzzles. (Shocking, I know.)
- It’s underestimated: everyone goes in thinking they’ll crush it, and most don’t.
- It’s still under-the-radar, though that is changing fast.
- It’s incredibly hard. Out of 25 questions, fewer than half the participants answered 9 correctly. That’s not a typo. And, remember, these stats are about students who all typically get the highest score in their class on math tests.
So. The AMC is very different.
These differences make the AMC very attractive for helping excellent students get the admissions attention they want and deserve.
As families are beginning to notice this, the way people are preparing for this exam is starting to evolve in two main ways:
- Students are beginning to treat the AMC as a varsity sport, giving it enormous quantities of preparation time.
- Coaches are becoming more valuable in proportion to their experience and success rates, because increasing the effectiveness of those training hours (thus decreasing total hours spent!) is starting to matter a lot.
These last two items are why the test (and prep for it) have been evolving so quickly in the last few years.
In short, some families have started to recognize that the AMC represents a high-leverage admissions opportunity, just as the AMC has opened up to students outside the US education system. Many of those families commit literally hundreds of hoursof preparation to this test. Most US-based families don’t have this option. Thankfully, neither do they need it, if they prepare in an efficient way.
Meanwhile, the test itself is evolving to account for this influx, as are the various online tools available to help those who prepare on their own.
In a nutshell, the AMC world now is in a situation that’s a bit analogous to the SAT in the 80s and maybe early 90s: prep had been around for a while, but it wasn’t particularly widespread, and plenty of people still just took the SAT either cold or with nothing more than some self-study (as I myself did, oh those many years ago).
The prep professionals are gearing up, and the students are starting to take it seriously. And in response, the test has been getting more challenging. I expect the quality of prep and the difficulty of the test to both continue ratcheting up for the next few years.
And for serious students with interest in solving difficult puzzles (both now and perhaps as part of their coming careers), that’s good news. Finally there’s a way to demonstrate your abilities on a playing field that isn’t so systematically biased towards those with only moderately high ability.
So… the AMC is a no-brainer?
Hang on, not so fast.
Yes, it’s an individual math competition with almost a full century of history, and the endorsement of many highly regarded mathematicians and professors. It has a long statistical record showing its effectiveness at identifying talent. And it’s familiar at the top echelons: many of the people in charge of admissions at selective schools did it themselves.
And yes, it’s an obvious choice on paper for helping to distinguish your STEM-interested son or daughter.
And of course a natural parent response is to expect to be able to pay someone to help their kid do well on this exam.
He got a 1550 on his SAT and a 35 on his ACT. Our college counselor is telling us we should consider doing this AMC thing too. We were thinking of maybe doing that in March. Do you have availability in March?
Here’s the thing, though: getting good at this takes a serious commitment. And not in the sense of “my kid can do in a weekend what takes most kids half a year to do.” No, I mean an actual serious commitment even for extremely strong students. You see, the AMC is actually an accurate test of well-prepared students with high-to-extreme ability, unlike grades and unlike the SAT/ACT.
There are no tricks here. To get good at this test you need to actually get extremely good at math and at problem-solving under pressure. And I don’t mean just best-in-your-school good. I mean nationally-ranked good.
So at least for now, you can’t buy a good score nearly as easily here as you can with the ACT/SAT.
How does the nature of the AMC make preparation different?
Preparation is both much harder, and much more valuable. That’s because of how the test itself is different:
- It’s hard enough. The questions past the first few are nearly impossible for the merely well-educated, moderately talented, and merely-competently coached.
- It’s harder to game, in two different ways that reinforce each other: The questions aren’t similar enough to each other to allow for the creation of meaningful guessing strategies (unlike the SAT/ACT), and the scoring formula sharply penalizes not only random guesses, but also “hunch-based” guesses. (Some guesswork is profitable here, but the decision of whether to guess requires expertise and nuance.)
- It’s harder to “brute-force.” If you want to just memorize all the necessary formulas and tricks, that’s an option. But it’s a really costly option, because there’s so much to memorize, most of which doesn’t get covered in school programs, even excellent ones. And even if you know exactly which things need to be memorized, and which can be worked around, and even if you have access to a tool designed to help you memorize exactly those things (and yes, I built and sell this exact tool), it will boost your score, but it can’t bring you all the way to the top echelons.
- It requires college-level skills. Doing very well on this exam requires not only broad and deep math knowledge, but also superior critical reasoning skills, active self-awareness and self-control, and superior self-analysis and self-improvement skills.
So, it’s a lot harder to prepare for: it takes more student dedication, and more coaching skill and experience. But once you’ve prepared successfully for the AMC, you’re much better prepared to succeed at the most challenging universities, because these skills transfer very easily to other areas of study.
That makes it a much more valuable indicator to college admissions committees, andit makes it more worthwhile just for the personal growth it helps students undergo.
That’s too abstract. Show me the difference.
Sure. Here are two fictionalized case studies woven together from real client experiences. They both demonstrate very common patterns.
What ACT/SAT prep actually looks like:
Jenny is a sophomore preparing for the ACT. She got a composite score of 28 out of 36 on a practice test: English 28, Reading 31, Math 26, and Science 27.
The path forward for Jenny is as clear as can be:
- Work with her on reading charts and graphs efficiently and confidently. Her Science score then jumps to the same level as her 31 Reading. That puts her at 29 Composite.
- Examine her Math work to figure out how much of her problem is lack of knowledge and how much is carelessness. Assuming 50/50 for the sake of argument, work with her directly to get her out of the habit of doing so much math in her head (which also means getting her out of the “I have to rush on this test” mindset), and assign homework around flash card drills to refresh and lock in the content that she’s forgotten. What little math she hasn’t yet gotten to in her class won’t hold her back much; the basics are far and away the most important part. Math comes up to about 32, bringing the Composite score to 30.5, which rounds to 31.
- There are a bunch of rules of English grammar that she just never learned correctly. Use the flash-card system that we built up for the math section, and have her drill those rules; then work alongside her once or twice to make sure she applies them correctly in context. That brings her to English 30, which boosts her morale enough to allow us to push hard to 32, where her Composite score bumps to 32.
Either she’s now done, or—if she needs a still-higher score—she now understands the routine here well enough that we can dive into a more nuanced collection of content and techniques that will raise her score to 34 or 35 (at the cost of a lot of practice time so that she can execute quickly and reliably).
For the student, this is somewhat-challenging internal work. For the seasoned and expert tutor, this is as straightforward as falling off a log.
What AMC prep actually looks like:
Kim is a sophomore preparing for the AMC. She expects a prep experience like Jenny had with the ACT (above), except Kim believes she should be open to the possibility that she’ll do so well on the practice test that she’ll be able to skip prep and just take the test cold.
But that isn’t what happens.
Instead, she gets 7 out of 25 questions right on a practice test, with 8 wrong guesses and 10 left blank. She is demoralized and dubious; Kim simply “never gets under 94%” on tests, and 38% is inconceivable. She wants to burn her paper and forget that this ever happened. She is mad at the test, mad at the teacher who suggested she try it, mad at her parents and herself for having wasted an hour on this.
This is the moment at which we lose most potential AMC competitors.
That’s because Kim is unaware that she has performed better than 3/4 of the first-time participants, all of whom are (like her) at the top of their classes. Remember, this test is hard in a way that very few tests really are these days.
If we lose her now — if we lose the chance to help her train now — we are losing a chance to give her a huge “leg up.” It’s not just a missed opportunity for Kim, though. It’s also a missed opportunity for all of us, because students like Kim are the ones most likely to make huge systemic differences (e.g. to develop vaccines, to cure cancer, to address large energy and climate issues) later in their careers. The lessons students can learn here, right now, put them far ahead of their equally-gifted peers. But most will miss this opportunity. They will eventually have to learn these lessons anyway, but it will be years before it comes up again. By then, they will have lost a lot of precious time.
Back to Kim’s practice test, though. A deep dive there reveals a few salient details:
- Her sixth- or possibly seventh-grade math experience was lacking. Bad teacher, maybe, or she had just changed schools, or maybe it was her first year in a demanding sport. Whatever the reason, she is missing some key fundamental concepts that force her to do certain more advanced problems the hard way. But since she’s so good at math in general, no one ever noticed this; after all, Kim can work a problem the hard way better than most students can work it the easy way. So that’s been holding her back, but up to now, it’s been completely invisible.
- Because she is accustomed to getting every question right, she doesn’t check her work in any more than a cursory way. As a result, she missed three more questions: she didn’t catch that she had missed a word in one question, therefore getting the whole thing wrong; she messed up a complicated bit of mental math on another and never noticed the error; and she mistook one unfamiliar question for a similar type of question she was more familiar with. On that last one, she answered “her version” of the question correctly, but of course her answer was not the answer to the question that was actually asked. These are all common mistakes among very strong students facing unusual challenges, but she has no way to know that. She thinks she “just failed.”
- In math class, everything is pretty easy for her, so all her techniques are optimized for getting to easy answers quickly. But that same in-class work feels hard for her classmates. So they are used to dealing with work that is hard for them. Unlike her classmates, Kim hasn’t ever built up any techniques for dealing with hard work. Where classmates might apply some some determination, slow down, draw a picture, look something up, think back to a similar problem, or plug in numbers, Kim just stares harder at a tough problem, waiting for something to click. In class, that almost always works. That does not on hard problems, but Kim doesn’t realize that. (How could she, when she almost never deals with problems that are hard for her?)
- Because she is also so accustomed to answering every question, she paced herself in a way that would allow her to get through the entire test in the allotted time. She did not realize that no one finishes all 25 questions, and that her pace is therefore far, far too fast for this test. Around question #6, she started falling behind her pace, so she started moving faster, racking up her first careless mistake on that very question. She skipped #7 after spending 90 seconds thinking about it, then read question #8, which is more difficult still, and started to panic and doubt herself, a very unusual emotional situation for her, bringing her even farther outside her comfort zone. This led to a few unproductive minutes of jumping around to some of the later questions, for some of which she couldn’t even understand what they were asking. Rather than slowing down and regrouping, she continued running off the rails: she started to guess more wildly, hoping for some additional points. She continued on this path until she had answered the first fifteen questions (most incorrectly, it turns out), at which point she put her pencil down, barely satisfied that she’d answered 15 of the 25 with six minutes to spare. She is completely bewildered by this experience. Her best guess is that it was a disaster.
This might be the first time she’s seen what happens when things get really hard. She might even realize for the first time that a career in (or even just adjacent to) math and/or science will eventually get this hard and more. Yet she’s just seen hard evidence that she actually has no idea what to do when she hits that wall. For a student whose core identity involves being smart and academically capable, this is a serious (and scary) threat to her identity.
But Kim is actually at an important crossroads: this is the experience that can help her better manage her abilities. To properly leverage them. To open doors for herself that are not an option for most.
The path forward for Kim involves a reconstruction of her self-image as a great math student. While she is indeed exceptionally good at math, she is not behaving like a great student. (And why would she? Brains alone have gotten her this far, and she doesn’t yet truly understand that brainy people get given harder and harder challenges until braininess just isn’t enough.) The skills her classmates have been building will get them through this class and more besides, but Kim is sailing through on talent alone. Her classmates will run out of math classes they want or need to take, but Kim will keep advancing into harder and harder material. Her talent won’t always be enough.
The path forward for Kim is long, but worth it.
So our first priority is helping her recalibrate: with great talent comes great challenge, and success requires not just skill but effort and technique. And since that’s new to her, she’s not going to be good at it at first. And that’s completely normal for gifted students. It feels extremely uncomfortable to Kim, but it’s not because she’s doing it wrong. It’s because she’s 16, and doesn’t yet have the experience to realize that even the brightest functional adults rely on skills, habits, and technique at least as much as they rely on smarts. It’s a completely different mindset, and even the best learners have trouble adopting it at first.
Once we start to get some traction there, it’ll be time to have her learn (or re-learn) a wide swath of fundamentals. The goal here isn’t so much to fill her few knowledge gaps, but rather to allow her to get into the mindset that if I don’t know it, I don’t need it. When it comes to hard problems, the mind wants very badly to jump to “I can’t do this one, I’d better skip it for now” and the test authors lean on this psychological glitch very heavily: many of the questions just “feel like they are out of my league” even when they are not. The defense is to know your material well enough that when you inevitably become stuck, you can resist the urge to disengage, and instead dig deeper in order to find and decipher whatever makes it (incorrectly) seem so impossible. In other words, I help Kim build the habit of acting with confidence even when she doesn’t feel confident. (In time, the feeling will also come, but for now, the action is more important.)
- Growth mindset
Now Kim is (finally!) ready to re-approach this experience with a “beginner’s mind” or “growth mindset.” She’ll start to see how some of her initial uninformed reactions to certain questions stopped her from deploying considered techniques. She’ll begin to reform her own bad habits, replacing them with more adaptive choices. The lessons affect other parts of her academic life: for example, math class will still be easy, but she may start new ways to engage. She may find she has more interest in helping her classmates, since for the first time, she’ll be able to understand what about the class seems hard for them. For the first time, she is analyzing her own mental processes, and working at improving them. The history class she hates becomes an opportunity to experiment with different learning techniques, and as a result, she comes to find it interesting in a new way. She is laying the groundwork for a much richer intellectual and functional future for herself and for those lucky enough to work with her in the future.
Now armed with the actual tools required for an initial level of AMC success, she and I can begin to dive into the interrelations among the various rules of math she has fully internalized, the habits of thought she has examined, and the fascinating patterns captured by so many of these problems. After all, extremely talented math educators and mathematicians spend a full year preparing just a handful of problems for this competition. Where a novice sees “impossible,” the experienced participant sees delight and fascination. You simply wouldn’t believe some of the remarkable patterns and observations that get packed into some of these problems. But Kim is now at a level where she can start to appreciate these fascinating properties and patterns that are hidden in the math all around us.
This is the point at which she “catches the bug” and starts really enjoying engaging with this material. Yes, it’s hard to share with others at first, but it’s interesting enough to make it worth thinking about nonetheless. She begins to have opinions about the quality and elegance of certain problems: this one is clever, but that one requires more grindwork than I prefer. I should have seen thattrick coming, but I don’t see how anyone could solve this one in under ten or twelve minutes. The work takes on the quality of play. Enriching, exhausting, exhilarating, yes, but play nonetheless. This is a part of learning she’s been inadvertently denied for a long time.
- Wide & deep exploration
This opens up explorations into areas of math that school curriculums just don’t make time for: combinatorics, probability, number theory, game theory, complex analysis, and on and on. Each new topic is a playground, an opportunity to find new gems of insight and clever ways to think. There’s no grade here, no pressure, because for the first time in a long time, we’re doing something academic mostly just for the enjoyment of it. Sure, we are aware that this is leading to a game-changing bullet point on the college applications, but the competition is still half a year away. We can afford to relax a little and just enjoy the learning process for a while.
And when the competition does finally come around, Kim will be ready. More ready than she ever realized someone could even be.
And so she starts wondering: what would I need to learn, or how would I need to grow, in order to…
…and that question, asked by our strongest students, is how students begin teaching themselves. It’s how they start really pulling away from the pack. That question is the very point of the AMC, and of the preparation process that leads to champions.
But most people don’t get it yet.
I get a sad chuckle when I hear from a parent who earnestly believes that their child or teenager can go from novice to AIME-qualifier (i.e. the next round after the AMC) “in a few weeks or months.” Usually they believe that because the student achieved their goal score on the ACT or SAT in a similar timeframe.
Wait, you’re saying this will take more than a month? Even for a very bright student?
Here’s the hard truth: being bright is just table stakes for this test. Doing well here involves learning things you’ve never seen before, and building habits you’ve never tried before. It’s hard. But successful mathematicians know these techniques, and successful adults have these habits. So the hard work pays off not only in admissions, but in future success as a college student and as an adult.
Students who learn these skills and habits now are positioning themselves to be outstanding students for the rest of their academic careers: not only bright, but also resilient.
Other students have something of a ticking bomb on the horizon: it’s just a matter of time before not having this resilience causes serious difficulty. Worse: the later one grapples with this lack of resilience, the harder it is to work around it.
The bottom line
Who does great in school, and also on the ACT or SAT? Students who are meticulous and compliant, that’s who. Unfortunately, that’s the wrong aspiration for the world they’ll be adults in.
To do great at the AMC takes more. Successful students need to have the habits both of top students and of high-performing adults. They need to be able to put those superior habits into play smoothly when things get truly difficult. These are skills we all want our kids to carry with them for the rest of their lives.
In other words, this is about helping a student become the kind of person that everyone—their parents, the colleges, future employers, but most importantly they themselves—all want them to be able to become.
AMC prep can be a complete game-changer. Your decision to help your child or teenager explore this now could be one of the singular great decisions you look back on later, and say “that’s when we started to see our kid take off.” If you decide to take a first step into this realm, I’m right here with you both.
So if they’ve been exceptional so far, and you’d like them to have the option to stay that way, even as they enter more and more competitive circles, get in touch.
It is my honor to help every one of my students develop, excel, and even transform.
Thanks to Chris Avrich, Seppo Helava, Eric Nehrlich, and Audrey Khuner for their invaluable help and insights in editing this article.