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.
Understand how to take advantage of this unusual opportunity
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 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.
What is 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’t work, 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-19 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 worked 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 skills. 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.
The 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 ofpreparation 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 hours of 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 an 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, and it 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 flashcard 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, and mad at her parents and herself for having wasted an hour on this.
This is the momentat 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 willmiss 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 of knowing that. She thinks she “just failed.”
In math class, everything is pretty easy for her, so all her techniques are optimized for getting 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 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 finished all 25 questions and that her pace was 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 isnot 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, 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
Recalibration 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.
Fundamentals 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 ifI 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.
Vision 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.
Engagement 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 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.
…and how some ambitious students can reap huge benefits
There have been some major shifts in the college admissions landscape over the last few years. These shifts have changed the value of the AMC for certain students, and they have also changed how a student should prepare.
Brief recap: what’s the AMC?
It’s an annual exam run by the MAA for high-school math fans and “mathletes” featuring much more difficult questions than the SAT/ACT. Therefore it does a farbetter job of identifying early talent. It is also the “feeder” exam for the US Math Olympiad team.
Why do I care?
Because the way students get into top colleges has changed so much that our intuitions about what “ought to be enough” no longer apply.
How has the AMC itself changed?
It’s now given in November instead of January
It’s now given online as well as at schools
It’s become consistently more difficult over the past five to ten years
These changes don’t make a huge difference yet, but what they imply about the future of the AMC does.
How has its importance changed?
I should probably write a longer essay about this, but in a nutshell: The ACT/SAT can easily distinguish between a mediocre student and a good one. But they can’t distinguish between good and great. Instead, these tests inadvertently distinguish between hard-working and very-hard-working, with little focus on talent beyond a certain moderately high level.
Talent—i.e. potential future success—is what’s most important to colleges, but the ACT/SAT isn’t giving colleges the insight they need into this dimension. (This underlies some of the recent shifts in the testing industry.) That task is increasingly falling to other sources, and the AMC is quickly becoming one of the main ones.
How has preparing for it changed?
The same way that preparing for the ACT/SAT changed about 20 or 30 years ago: it’s becoming professionalized. What used to be an exam that you could just decide to take cold on test day has become something you prepare for for an extended period.
Also, the way you prepare has changed, because there are really two routes to success on the AMC:
Pretend it’s the LSAT, and just work dozens and dozens of past exams until you have pretty much seen it all
Pretend it’s a “thinking competition,” and learn new problem-solving skills that work in a wide range of situations
Option #1 is more popular, especially outside the US, because it’s easier to plan out how to do it, and for many international students, simply spending hundreds of hours coming to master the test is a viable option.
Option #2 is a better choice for many US students because they have less time to devote, have more resources to devote to developing true expertise, and need more of the general problem solving skills this training develops.
I’ll say that last part again in a different way: the US has been lagging most developed countries in critical thinking and general problem-solving for quite a while now. The AMC helps our best and brightest learn the critical skills that are necessary for high-level competitiveness in future markets, and which most of the student population increasingly lacks.
What should I be asking in response?
If you haven’t considered the AMC, you should ask these questions:
Do I enjoy math (or at least, do I enjoy it when it’s interesting)?
Is it important to me to attend a top school and/or to have a challenging and rewarding career in technology, science, and/or research?
Could I see an academic study becoming for me what varsity sports are for student athletes?
If you’re already planning to prepare for the AMC, or have started preparation, you should ask these questions:
Do I have a concrete, realistic plan for success?
Historically, have I gained at least ten points per month of preparation on my practice tests?
Am I continuing to improve?
Is what I’m learning still both helpful and interesting?
If you’re heading to college, you’re almost certainly going to take the ACT or SAT at least once. I’d like to set the record straight on a few things so that we can all minimize unproductive time and effort, especially in light of the fact that most students take these tests during school years that are already very busy.
ACT/SAT preparation is a big industry encompassing a wide range of options. I don’t think that “low quality” providers are terribly common, but I do think that it’s important to match a provider to your needs, because there is a wide range, and fit matters. There are three issues in particular that come up around ACT/SAT prep that I think are worth calling out:
On variable depth: Lots of ACT/SAT prep is done by those who did well on the tests themselves but whose teaching experience (and, sometimes, incentive to ensure student success) can vary widely. If you are unaware, this can cause problems. If you are savvy about it, though, this can be an opportunity to get what you need for less money overall.
On score bands: What you need to improve your score is not only affected by what kind of person you are. It’s also affected by where you are on the score curve. For example, tell me where you are scoring, and I’ll tell you whether vocabulary, grammar rules, reading style, or attention management are probably costing you the most points on the verbal sections of these tests. Some teachers are aware of this and some aren’t; some specialize in a narrow score band and some don’t. But whenever I talk with a parent for the first time, I note whether they have considered this important part of the decision process.
On the grapevine: It’s great to be well-connected! Just be aware that Suzy’s great tutor may or may not have actually been a great tutor. Maybe she was just great for Suzie. (This is also why I encourage tutors to be selective about whom they take on. “Taking all comers” is good in the beginning, but it isn’t a good long-term plan for most.)
Basic pro facts
Here is some “inside baseball” about these tests. Most of the pro tutors I know are delighted when clients have already considered these ideas:
No school cares which test you pick. Really. (If you have a reputable college counselor telling you otherwise, then please trust that you’re in a very specific and unusual situation.) So just pick the one you do best on (see below), or pick the one whose test date happens when you need it to, and move on.
There is no “penalty for guessing.” Neither test currently assigns negative fractional points for incorrect answers. Which is unfortunate, because you’d be better off if they did. This video helps to explain why that’s true (even though it may seem “obviously” false). The punch line is: without “penalty” points, you’re forced to spend the last minute of your testing time bubbling in random answers to whichever questions you haven’t yet answered.
Most students will get comparable scores on the two tests. Of those who do better on one or the other, it’s a 50/50 chance which one you’d do better on (see also “subjective preference” below). The best way to determine whether you are a student who does better on one or the other is to take one of each in comparable circumstances and compare the two scores using a concordance table from either organization.
That said, if you believe that taking both tests is not in your best interests (e.g. because you have very limited time), then here are some rules of thumb:
If you read especially slowly (e.g. have a tested learning difference, esp. low processing speed) then the ACT is going to penalize you extra hard for it. Prefer the SAT.
If you are considered a linear thinker and/or your primary school strategy is to “grind it out,” then the ACT is going to be more in your wheelhouse. The ACT lends itself better to simply learning all the material and techniques, then delivering on test day. Prefer the ACT.
Conversely, if you are considered a good test-taker and/or your primary school strategy is to wing it, this suggests a preference for the SAT, which remains slightly more gameable in this way. Prefer the SAT.
Crucially, most students express a subjective preference for the ACT. This is not to be confused with “most students do better on the ACT,” which is simply not true; scores are normed to reflect percentile rankings.
This last point might be worth expanding a bit.
In other words, schools aren’t looking at your raw scores per se; they’re looking to see whether you did better than most others. The all-too-popular canard that “the ACT is better for most students” is not only wrong, it betrays a deep misunderstanding of the whole situation. The score is normed across all test-takers. That preference, therefore, gets normed out of the score. In other words, the ACT “feels better” to everyone else too, and so thanks to the norming that happens across this particular zero-sum game, it’ll all come out in the wash.
And let me emphasize: the best way to select is to take one of each test in comparable circumstances, and then to compare the two scores using a concordance table from either organization.
As I say to students who are ready to hear it: the hard truth here is that you’re not here to have a good time; you’re here to get a job done, and that means a good score. So don’t let your preferences guide your decision unless they are likely to affect your outcome. Your score is the thing to focus on. Structure your preparation (and your test selection) in the way that will lead to the outcome you want.
This framing helps put a more human face on the underlying mathematical truth about the way these tests are scored, and on the need in life to be able to jump through the occasional tricky hoop.
Some thoughts for top students
If you’re in the top tenth or so of students (i.e. if you test at the 90th percentile or higher), then the game is a bit different for you, since it’s no longer about being able to solve problems that few others can. Instead, for you, it’s about not making avoidable errors.
Let me unpack that a bit. The people who score in the middle of the pack — say, an SAT 1200 or an ACT 21 — are playing winners’ games, where doing better relies on getting more problems right. By contrast, the people who score at the top of the pack — say 1400+ on the SAT or 28+ on the ACT — are playing losers’ games, where doing better relies on just performing consistently while others make mistakes.
This means that beyond a certain point, the winning strategy shifts from trying to get more right, to trying to make fewer unforced errors.
This is where preparation passes outside the realm of what most of us think of as fact- and tactics-based “test prep” and into softer skills adjacent to psychology, habit formation, and self-awareness.
I created a short (ten minutes maybe?) free mini-course on this called What Am I Missing?that you are welcome to give to anyone you’d like, and to take yourself. It gets into more detail than comfortably fits here.
Consider the long game
Parents, please take an uncomfortable moment to consider the unlikely scenario where the student wins admission to a school that’s hard enough that they wind up dropping out. Now consider the less dramatic but much more common version: the school is hard enough that they wind up needing continued help throughout their undergraduate years and even beyond. (These do happen; I’ve seen both first-hand.)
My point is that we’re all working together here to execute a successful launch. This means both success andactuallaunch.
One of the ways this can positively affect SAT/ACT prep for active parents and their kids is the explicit recognition that a bad run at these tests can in many cases be a very positive milestone in the long run. There’s nothing like an unexpectedly difficult run at an important test to challenge expectations and catalyze a bit more self-awareness. After all, if the SAT/ACT is hard enough to shake you up, what does that say about every test you are going to face over the next four years?
Now is the time to either downshift or step up. It is absolutely possible to gain new knowledge, to gain new skills, to try on new perspectives, to embody new strategies, to experiment in the service of self-improvement.
You’re about to do that for four years. Why not start right now?
But again, you have to see this as a long game in order to get these long-term benefits from these immediate disappointments.
A historical side-note
Originally, the SAT was an aptitude test and the ACT was an achievement test.
The SAT has its origins in trying to assess which enlisted soldiers were officer material (and which didn’t belong in the army in the first place). I.e. assessing undemonstrated potential.
The ACT, by contrast, has always aimed to assess what material a student actually knows, as a fraction of the material that the student is supposed to know.
In other words, the SAT tried to determine how capable you were in general, and the ACT tried to determine whether you’d already learned what you were supposed to have learned by then. The tests have evolved over time (mostly with the SAT drifting towards the ACT), but there are still some shadows of their origins in their modern incarnations.
I know a lot of specialists who can help in various ways, and I see connecting people as an important (and enjoyable!) duty. If you need specific help, let me know what you’re looking for and I’ll do my best. And if you’re a fellow professional, please get in touch to let me know what you do best, so that I can help connect you with families who need it.
”I missed this one because I (stupidly) assumed that…”
Don’t be so hard on yourself, sister. Assumptions are just shortcuts that didn’t ace their job. I would say that you jumped to a conclusion, didn’t notice that you’d jumped, and so you didn’t think to question it. Happens to the best of us. Let me help you up, dust yourself off, and let’s try again, hey? Right on.
Improvement means being wrong a lot. That’s not stupidity. It’s just the “cost of doing business” when you’re trying to get better.
The key bit is that you examine why that unhelpful assumption was poised for you to grab onto in the first place. It was probably there for (what used to be) a good reason. Re-assessing the reasoning, and possibly re-visiting your conclusion, is a very helpful way to keep no-longer-helpful habits and assumptions from continuing to re-assert themselves.
Hey, thanks for showing up here. I hope you found it helpful, or at least interesting.
I’m [email protected], I read all feedback personally, and I really do care whether this was helpful, and how I can make it better next time around.
They say that the best time to plant a tree was 20 years ago, but today would be the second best. That’s how I feel about writing about the American Math Competition (AMC): I wish I had spent more time writing about it in the past, but I’ll do the next best thing and write right now.
What is the AMC 10/12?
The AMC 10 and 12 are both 25-question, 75-minute, multiple-choice tests crafted to not just test but also nurture your problem-solving prowess.
The AMC 10, tailored for our younger math enthusiasts in 10th grade or below, encapsulates the high school curriculum up to that grade. So, if you’re under 17.5 years old and in grade 10 or below, this is your arena.
On the other hand, the AMC 12 is a bit more advanced, covering the full spectrum of high school math, from trigonometry to advanced geometry. But don’t expect any calculus here! It’s designed for those in grade 12 or below and under 19.5 years.
Who (Should) Care About the AMC?
You should — If you have (or are, or know) a student in high school who is interested in STEM and good at school.
That student (plus their support structure) is usually focused on these three goals:
Get good at core skills needed through adult life
Build out a resume that opens doors later
Get through this craziness without excessive friction or stress
For these students, the AMC remains a little-known but very powerful tool for helping to accomplish these goals.
Should I Wait Before Taking the AMC?
Not if you want to participate this school year. The AMC 10 and AMC 12 exams, usually given in February, are being given in November this year. This means much less time to prepare, but crucially, it also means that participating seniors will get to use their good scores in the college admissions process.
Another benefit: Those who participate this year will have much more time to prepare for the second level of the competition (the American Invitational Math Exam, or AIME) in March, if they qualify in November. In past years, students would discover they qualified for AIME with a week or less to spare before competing. This year, competitors will have months to prepare.
This matters for a couple of reasons, but the one most people focus on is that top schools want to know when an applicant made the cut for AIME. It’s one of those distinguishers that can really matter, especially for schools with STEM-focused reputations and/or “elite standing.”
Recent Testing Trends
Over the last 20 years, high school has generally become a more systematized “accomplishment treadmill,” with the advantages and disadvantages that this implies
Over the last several years, the ACT and SAT have taken significant credibility hits. These include equity issues, doubts about correlation to college success, rejection by major universities, and most recently logistical difficulties simply being able to sign up for and take the test
Side note: One problem with these tests that has always been lurking under the surface is that whatever it is that they test, they test it best for the people in the middle of its bell curve, but their results are most scrutinized for the people at the top of its bell curve. I.e. a harder test would be better suited for the people who need it most
Nonetheless, competition for college admissions has remained very high
College admissions offices, understanding all these trends, are continuing to use the AMC as a distinguishing characteristic among otherwise hard-to-distinguish strong students.
I expect that the AMC will continue to increase in prominence in the coming years.
What I Have Observed
Students who prepare for the ACT or SAT tend to prepare by filling knowledge gaps, which is all to the good. For most students, this remains the focus throughout their preparation.
Students at the high end of the curve who prepare seriously, though, only start their preparation the same way. Before too long, they run out of facts to learn and techniques to leverage, because the test simply doesn’t cover that much material that they haven’t already learned. (This is a big part of what I meant when I said that the test is geared towards students in the middle of its curve, but the students who care about it most are the ones at the top of its curve.)
This is the point at which things start going sideways:
Preparation becomes an exercise in meticulousness, sometimes to the exclusion of engaged critical thinking. For these students, these tests are about simply not letting their attention drift as they answer easy question after easy question. (In my view, this selects for the wrong skills and habits among “top” students and sets them up for trouble later. If you see it differently, I recommend the movie Brazil. It’s really good, and I hope you like it as much as I do, even though it makes me cringe to watch it.)
Also, preparation often becomes an exercise in anxiety management. I feel better about this consequence since anxiety management is a real-life skill for sure. But I think we can all agree that that’s not the thing we all want to be selecting for in this context either.
I’ve also observed that students preparing for tests that really challenge them develop skills that in my view hold much higher value overall: critical thinking, self-awareness and self-management, self-evaluation and metacognition, and creative problem-solving.
What I Sometimes Advise About the AMC
If the ACT or SAT isn’t hard enough to allow you to show off your real skills, and you like STEM, then the AMC is probably worth looking at:
You get to work on problems that are harder than anything your teacher would subject your whole class to
You get to explore interconnections in the math you already know without having to learn a ton of new material
You get to practice the critical skill of making progress even on incomplete information
Top universities actively solicit your AMC 12 score. This tells you everything you need to know about its importance to them
Recent Trends in the AMC Exams
At the top levels, the AMC can be seen as a feeder program for the International Math Olympiad. The US, Russia, and China have been vying for top honors at the IMO for the last few years now. I genuinely don’t know how much of this should be attributed to good coaching; I’m impressed by head coach Po-Shen Loh — he’s clearly focused on teaching well instead of merely on being smart, and you’d be shocked at how many miss that — but that’s all I know because I’m not anywhere near the inner circle at that level.
But what I do know is that Richard Rusczyk’s project Art of Problem Solving has become the dominant force in this area. What started as a pair of (excellent) books and a thriving community has become the 900-pound gorilla in the space; practically every serious student competitor has some relationship with AoPS. There’s a ton of good stuff to say about AoPS, but I’ll also mention that two false impressions often come out of students’ initial foray there:
“There’s a ton of math I need to learn in order to be successful here.”
“Reading through other people’s solutions is the best way to learn.”
Regarding #1, it would be more correct to say that AoPS will help you learn a ton of math, and a relatively small slice of it is needed for AMC success. Figuring out what’s important remains more art than science, data analysis notwithstanding, because — at least at the AMC level — doing a good job of using what you know is generally more valuable than knowing more.
And regarding #2, AoPS rightly discourages this perspective, and actively, but you can only do so much when such a large proportion of your site is dedicated to the thing you’re telling everyone to use only judiciously. I mean, it’s been said plenty before: math is not a spectator sport.
Doing math — and analyzing your steps and missteps after the fact — is the way to get good at it. Deliberate practice isn’t easy, often isn’t fun, and it certainly isn’t your default behavior when you’re reading a website. But it’s the best way. Unfortunately, almost no students get this, at least not at the early levels.
My online AMC prep course is open for business. If you need to learn the fundamentals of AMC math, this is the place for you.
I know a lot of specialists who can help in various ways, and I see connecting people as an important (and enjoyable!) duty. If you need specific help, let me know what you’re looking for and I’ll do my best. And if you’re a fellow professional, please get in touch to let me know what you do best, so that I can help connect you with families who need it.
Other quick links
My buddy Amie Dorsey and I just recorded the 13th episode of our (now-monthly) podcast WATEC: The Excellent Conversation. You can just search for it on your podcast app, or watch it on YouTube. They’re about an hour each, and you’ll find links to snippets of the conversation in the comments so that you can browse for what’s most interesting to you.
Wes Carroll’s Puzzler introduces parents and students alike to AMC problems and related brain-teasers. (No scratch paper required!) It’s fun for families to challenge each other around the kitchen table, and it’s also good for listening on your walk or run.
As I’m sure you know, subscribing to either or both of them lets us notify you when new episodes are ready, and also helps others to find them.
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I’m [email protected], I read all feedback personally, and I really do care whether this was helpful, and how I can make it better next time around.
In this episode, we discuss current events and multiple intelligences, with me touching on how believing in multiple intelligences could lead to a fixed mindset. During this conversation, I talked about why I coach people who take the ACT tests:
Answer from that bit of the podcast:
Amie: We were talking, and you mentioned how the way we think and how our minds operate is like a series of hacks, right? So, I’m pondering this: some minds might be more adept at adopting these hacks quickly, finding the most efficient ones. If learning these hacks is central to becoming smarter, then how significant is raw intelligence or “horsepower”? Given enough time, can someone learn as many hacks as needed to be considered a genius?
Wes: Your point brings to mind the distinction often made in business development and entrepreneurial circles. Having a brilliant idea is commendable, but execution is paramount. Ideally, you’d want both a solid idea and stellar execution. Similarly, if you aspire to be exceptionally successful or what some might call a “genius”, you need both potential and the means to harness it. However, most often, our emphasis is on tapping into the potential we already possess since that’s more tangible.
Regarding aptitude, I believe there’s a certain inherent ability everyone has. It’s not that a child can’t become exceptionally skilled in an area they currently lack proficiency in; it’s entirely possible. But there are time constraints. For instance, if a student aims for a perfect ACT score within nine months, that’s quite ambitious.
I’ve often said to people, ‘I can get you a perfect score on the SAT or the ACT. However, you’re going to have to take a year off from school to achieve it. We’re going to cover a lot, and you might feel that you could have spent that year doing other things. It’s not that it’s impossible; it’s just that there’s a price to pay. In a way, there’s a toll it takes on your soul. Some of the things you have to learn for the test might have limited value outside of that context.
And yet, I’m going to contradict myself to some degree with what I’m going to say next. Granted, there are some kinds of improvement that you can’t achieve in the time frame you’re dealing with here. However, what really saddens me about that parent comment you mentioned is that there’s a ton of stuff you can change in that time frame. The beautiful part is, some of it is useful only in the context of the ACT, but not nearly as much as people think.
It is true that the ACT is testing, you know, how much of the math vocabulary you’ve internalized or “encyclopediazed” in your mind. Thank you for pointing that out. So yes, that’s true. To some degree, it’s about filling in the gaps in knowledge.
While some might argue that’s only useful for the ACT, the reason it’s part of a math curriculum is that it’s going to be useful for other things later on. So there’s intrinsic value there.
Why I coach students for ACT Tests — Wes Carroll
But the overarching or “meta” value overshadows all of it. We’re going to do an A to Z review of what you know in this broad field, identify where you’re lacking, and then explore ways for you to not only acquire new knowledge but also develop new ways of thinking. And all of this in a manner that remains robust under test pressure. There’s a lot to consider.
Pretty much everyone I talk to these days has a higher-than-usual level of difficulty and frustration, but we’re all working hard to keep that stuff from leaking into our conversations unwittingly. I really appreciate that emotional effort. I’m also grateful that being on the receiving end of that effort helps me do my part on the sending end.
My AMC students are hitting that wall where they realize that what they’ve always done before isn’t going to keep working, and they have to pick up entirely new skills. Scary …and exciting! Seeing that realization dawn really makes this work feel worthwhile for me.
I’m grateful to have a job that I like enough to keeping working hard at, even when things are rough.
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College admissions, STEM schools, puzzles, the future of learning, and everything in between. Monthly.