What advanced math for high school student is actually competitive at highest levels?

My middle schooler is very good at math and loves it. Next year in 8th grade, he could finish algebra 2 (or even pre-calculus if he does it fast). While he takes geometry in school, he is taking the rest through Art of Problem Solving. He will have finished all the introductory Art of Problem Solving classes, as well as Intermediate Alg (Alg 2) and Pre-calculus by the end of 8th grade. He could take Calculus as a freshman. I say “could” because I don’t know if he needs to go that fast. When looking at the really competitive math camps for high schoolers, many say that a lot of students have taken calculus (and number theory and other things) - so my question is this - if he really wanted to pursue math at the highest levels (competitive math camps like MathCamp or Promys) and/or science at the highest levels (Summer Science Camp, etc.) and/or wanted to attend a prestigious university for math/science - what does he REALLY have to do? Is the reality that people attending these have all taken calculus in middle school? Physics in 9th or 10th grade? I know one answer is “of course not - enrichment before acceleration - you’ll get there and be fine” - but I really want to know the reality. It seems insanely competitive, and I don’t want to rush him that much if he’s already behind what would be expected for those things. Part of it is my own frustration because I am not a math person and I don’t understand much about different pathways. Would appreciate a blunt answer about what is actually required to be successful on some of the tests for these camps, in the camps themselves, and later at university.

As you are familiar with AoPS, I suggest reading Richard’s Calculus Trap paper.

D22 had Calc as a Sophomore and got much more out of advanced AoPS courses and local math team/clubs/competitions/etc., learning topics outside of the normal high school curriculum, in an environment of like-minded students.

No, Calculus in middle school is not “typical”, expected, or anything similar.

5 Likes

The biggest issue is what does your kid do when he/she runs out of all the math/science courses in his/her school? You might end up sending him/her to a private school that has more classes post Calc or the kid might end up taking classes online or at a local college. That’s harder than you think. There are maturity and access issues.

Here’s the rub. Kids need 4 years of math in high school. So you have to figure out transportation and costs as well. If your kid has taken that class you need 4 substitutes.

If your kid is many years ahead, they’ll probably end up doing math on their own time. It can be tough and frustrating.
One of my kids was skipped many times. It didn’t help. If they skip you two years but you really are four years ahead or if they skip you a year and you are two years ahead what happens? Your kid is still bored and now doesn’t have the support structure around them. They can even end up disliking the teacher or math in general.

I’d plan carefully. Do math/science as a deep hobby and recognize that there are may levels of knowledge. Plus there are tons of kids whose abilities are really deep, but they won’t get access to post Calc math until college. My kid dropped very high level math competitions which were dull for them and continued with other related activities.
Don’t worry about college in middle school. Your kid will find his own level.

And actually very few kids take calculus in middle school and those who do often will just take related classes often in science so they don’t repeat the same material.
You can look up the normal math paths in your district (public school) or ask about them in private school. Then you can adjust based on where your kid is and their interests.

4 Likes

Short answer: No. There is no benefit in terms of getting to Calculus anytime before 11th grade.

Our local high school system does not even allow anyone to take Calculus before 11th grade, and in fact only allows a handful each year to take it in 11th grade. And yet it does very well in terms of placing kids into those elite math and science camps, and is often followed by admission to the HYPSM colleges.

In terms of showing demonstrating talent, you are far better off having your son do well on the AMC-10/12 and AIME exams.

6 Likes

Thank you. That is reassuring. Diving too deep into the internet and message boards makes it seem like everyone in these camps was doing calculus in diapers. It’s hard to know what’s real and what’s marketing.

1 Like

Calculus will depend a great deal on the prerequisites, including algebra, trigonometry, and pre-calculus. In my opinion you want to make sure that a student has a very strong basis in each of these before jumping into Calculus.

Personally I did not take calculus until I was a freshman in university but that did not stop me from graduating with a bachelor’s degree in mathematics from MIT. The point is to learn the basis very well. Both algebra and calculus were then for me useful for quite a bit that I did later in life (including both graduate level classes and my first couple of jobs).

3 Likes

As the parent of a kid in the most accelerated math track in a nationally regarded STEM magnet, I’d say No. So many questions in this plan: Where is Trig? What is pre-calc actually covering? What is this student missing by trying to rush through Calc in addition to Alg 2, Trig, Pre-calc in a year? (Ans: a lot). What teacher would sanction this curriculum?

My kid was in the most advanced track for our district (Geom in 8th) before attending the magnet. In the magnet HS, she was admitted to a special track that condensed Alg 2 Trig and Pre calc into 1 year. It was grueling with 3+ hours of HW a night and trig problems that people w/ Math Masters couldn’t solve. But, the understanding was deep and solid. Next year (Analysis 2 or Calc BC+) – equally deep, though not quote as intense. BC and beyond. Fast forward, kid handled MultiVar and Linear Alg with ease and is now in Complex Analysis as a HS senior. Will a kid really do well in these advanced topics if rushing through Calc? Is there a need to go beyond this in HS? What is the point of college if kid has finished Calc in MS? If rushed like this, would there be any real sense of mastery when it comes to Calc? Again, so many questions.

People often think of math as tiers to climb as fast as possible to get to the top. But, all these tiers have depth, and sometimes the next tier builds on understanding that is enhanced by greater depth. Rushing impairs or even eliminates depth. Depth is good.

5 Likes

So your kid took Calculus in 10th grade? That’s good to hear, and was our initial plan before all those camps made it look like their participants were well beyond that. Our high school doesn’t offer classes after Calculus BC but has a partnership with the big public four-year college in town so kids can take those classes there each term. And there’s always AOPS, which our kid loves and seems more in depth than regular classes. I think the AOPS in conjunction with regular classes provides an amazing amount of depth (as well as work with a retired professor and a math coach). So I’m not worried about depth - and we would never push him on acceleration (I wouldn’t even know how to push him - again, not a math person). But I do want to give him realistic expectations about the future and participation in these programs - I believe in managing expectations not being unrealistic. Thank you for sharing your experiences - it is encouraging that he would be competitive if it continues to be something that he wants to pursue.

3 Likes

Remember, the most selective colleges do want to see a base of high academic performance all around, even if the applicant shows exceptional performance in one subject like math.

Also, a highly advanced math student who will major in math may be constrained in college choice in order not to run out of courses in math at college.

2 Likes

There’s no hurry. And there are many aspects of a young person that need developing.

3 Likes

My S22 took calc in 10th and then has been taking classes at the state flagship the past 2 years. He also got into and attended competitive summer math programs all three summers. My observation is that the main factor in getting into those programs was performance on the entrance exams. And then amc and aime performance seemed to help. When they got to calc seemed to be less important especially since those curriculums aren’t about calculus.

As others are saying, I don’t recommend self teaching for foundational high school curriculum courses. If they like competition math, they can practice that for fun and also can see results from that.

4 Likes

My son took Calc the summer before 8th grade at a T20 as part of a program for gifted STEM students. Now getting his PhD.

I think there was a benefit.

3 Likes

AOPS offers classes in counting and probability and number theory that is not in the usual math sequence but really good for kids whom live math. So you may want to check them out and see if they fit in your schedule

1 Like

A learning benefit? Sure.

But the OP explicitly asked if there was benefit in terms of admission to the most selective math and science programs, followed by admission to the most selective universities.

My son and some of his friends did both despite no formal instruction in Calculus until 11th grade. My son did learn Calculus earlier informally on his own, but then again that was the norm when it came to mathematics. Prior to college, for exceptional students, all of the real learning in math happens outside the classroom.

We are all fortunate that AoPS exists now, as their mathematics education can’t be beat. Because their courses admit by talent and not by age, there are often very young kids taking those courses and doing well in them. Want to learn number theory in fifth grade? No problem.

I know my son listed those AoPS courses on his applications to those math and science camps along with his scores on the AMC and AIME and the awards associated with those scores. I believe those certainly helped for admission into those math camps.

For the college apps, the AoPS courses were not listed, but his AMC and AIME scores and awards were listed, and the math and science camps were listed as well.

3 Likes

Yes. I think there is some misconception here that Calc as a Soph is somehow average. This is absolutely NOT the case. Calc (with any degree of depth) as a Soph is incredibly advanced, and even then could be rushed and/or shallow. I don’t know how any earlier would even be an advantage. Also, as others pointed out, competition math is a whole different thing. Your son may be interested in Discrete Math (encompasses a lot of competition math) as an area of interest. So, I would say augment w/ the competition stuff (AOPS is great for this), but don’t rush the core material. You want your kid to not only take Calc, but get a 5 on the AP test and be able to use it as a foundation for Multi-Var, Complex etc. These goals require increasing amounts of depth and comprehension. Good luck!

2 Likes

I agree with most here that your child is already well ahead. Don’t rush it too much unless your child is the one demanding more material. Let their excitements and passions fuel their direction. Perhaps this is the engineer in me writing, but encourage your child in ways to take this unusual skill for mathematics and translate it into something meaningful and helpful. For example, that Seattle area teen who wrote one of the better COVID tracking websites when the pandemic hit. Math can be an end of itself, but IMO when it is used as a basis to serve and improve society it can have the most impact.

Many of the math enrichment summer classes out there purposefully don’t follow the typical math curriculum and focus on things like set theory, numbers theory, etc.

1 Like

I believe there was a college admissions benefit as well. At least to some extent, AOs are looking at the student’s application to determine if the student has the ability to succeed at their institution. A student who demonstrates that they can excel at college level work at a young age (in this, I think actual graded courses at a top institutions is much superior to AP classes) is certainly helpful in admissions to top programs.

Math isn’t a race.

6 Likes

I have a Mathcamp kid, and according to him most (all?) of the campers are familiar with or have been exposed to at least a few of the undergraduate level math topics, especially when it comes to writing proofs. Does it mean that they all took undergraduate college classes? No, they might have been exposed to those topics via AoPS or math circles or self study … Is it expected that students took AP Calc BC in middle school? Definitely not, for many even this may simply not be possible regardless of their abilities. Are they all into competitive math? Again, the answer is no. Some kids love math but couldn’t care less about math competitions.

Your son isn’t behind. Let him explore different math topics. AoPS is perfect for that. Have him work on proof based problems. Have him join math circles to work with other students. Have him do competitive math if he enjoys it. See where it gets him. The Mathcamp problem set is very important. They of course look for a strong math background, and how that looks for each of the students may differ. I personally would give more credit to those who succeeded in AoPS Olympiad Geometry than those who took Calc BC as sophomores. Being involved in a math community - math circles, clubs also shows interest in being a part of a math community.

It’s also important to realize that there is a limited number of spots in each of the programs. For example, about half of the Mathcamp spots go to the past students. And for those who aren’t sure if to apply for such camps, please do. According to S22, Canada/USA Mathcamp is one of the best things ever.

3 Likes

I’m not a fan of math acceleration for the sake of math acceleration. What’s the purpose of the acceleration? If it’s to impress the adcom at the most selective schools in STEM, it isn’t going to work because they have no idea how impressively a student has done in the advanced math courses taken elsewhere (getting an A, say at a local flagship, doesn’t mean much to them). The student probably has done well enough in regular math classes in school to be deemed highly math proficient without those advanced math courses. The only exceptions are those who won the few recognized national/international competitions, and those who participated in the few extremely selective math/science programs and can get the professors in the programs to vouch for their exceptional abilities. Even excellent AIME/AMC scores can serve no more than tie-breakers when applicants who appear otherwise equally competent in math.

Colleges also know that a student will be able to study only a few limited areas in math, ever. If one includes areas in math-intensive sciences, a student’s knowledge (even with a PhD later) will be very limited, no matter how much s/he studies. The breadth of knowledge is helpful in connecting ideas within math and outside math, but the depth is what help a student stand out.

1 Like