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Gifted Math Education: acceleration, enrichment, and the Calculus Trap

September 18, 2007

Quite often the question arises as to what is better for gifted kids — acceleration or enrichment. And it seems that many, many parents of gifted kids are adamant that only acceleration could possibly be appropriate for their children. That if they are are ready to take the next step, they should not be “held back” from taking it, and if they don’t learn something “new” they’re being denied the education they have a right to.I’d like to counter that enrichment is not only an appropriate means of meeting at least some of the needs of gifted math students, but in many cases may be a better choice than pure acceleration.

The problem is that, too often, schools say “enrichment” when they mean “do nothing, or at least very little”. And so parents have learned to be highly suspicious of the term.

When I advocate for enrichment for gifted math students, I mean two things:

  1. A student who is gifted in mathematics should IMO be given more and more difficult problems to solve with the “tricks” they already know. Because the problem solving skills are far more important than the particular bags of tricks.
  2. These students should be exposed to topics not usually covered (either at all, or at least not very deeply) in a standard curriculum, such as logic, number theory, combinatorics, set theory, recursion, etc…

In many cases, these activities are more beneficial to students, than acceleration. Rather than just completing the standard, rather-bare-bones math curriculum up through high school in record time, students instead get to explore interesting topics, stretch their minds with challenging puzzles, and learn more than the minimum curriculum (which was certainly not designed with the needs and abilities of the gifted learner in mind).

Most standard curricula spent little or not time on “non-routine” problem solving — working on the kinds of problems you might find in math competitions, problem of the week websites, etc. The argument is that there isn’t enough time. Well, I’d argue that it’s well worth making time, because, IMO, that kind of thinking and problem solving is a very important part of mathematics. But for gifted kids who have extra time because they zip through the standard curriculum so quickly, it’s even more of a no-brainer to give these kids this kind of experience that will deepen their mathematical thinking skills and better prepare them for higher-level mathematics, or indeed many other disciplines that require excellent thinking and problem-solving skills.

IMO, math is not a race. I say this as a parent, a mathematician, and a volunteer math educator. I’d encourage you to read this article: The Calculus Trap. It addresses the issue from the point of view of high school curriculum, but many of the same principles apply earlier on. It helped me think more clearly about the kinds of experiences that could be beneficial to gifted math students, and helped shape many of my views as articulated above.

Those who know me well probably won’t be surprised to hear that we’ve worked with our kids’ school to take a hybrid approach with our children. They work with kids a couple of years older than themselves, but also spend time on enrichment activities to deepen their mathematical learning and experiences. We’ve elected to have them join the “standard” honors stream here in high school, which starts with honors geometry in 9th grade. This means the have all of middle school to get through “Algebra I” which gives them lots of time to work on extra topics, problem solving, etc.

I have more to say on related topics, such as gifted kids being used as tutors and mentors for other students, possible topics for enrichment starting even in the primary grades, etc. But since this post is getting quite long already, I’ll save those for another day.

21 Comments leave one →
  1. September 19, 2007 9:36 pm

    I think we are more or less on the same page. Did you see the recent, related post by Pissed Off Teeacher? (spelling intentional). I have the greatest respect for what she does, and how well she seems to do it, but I disagree with her on this one.

  2. September 19, 2007 9:58 pm

    I have been following Pissed Off Teeacher (never noticed the spelling until now!) recently, but I don’t know which post you’re talking about.

  3. September 20, 2007 9:25 am

    Oh, yeah, thanks. I had read that, but then forgot about it.

  4. September 22, 2007 9:36 pm

    I just read that article on the calculus trap. I agree with it completely. In my calculus class, i try to do just what the article suggests–we do lots of problem solving and I try to test with problems (not concepts) the kids have never seen. I present problems that are very different from what we have done, but the skills we’ve used solve them.

    The reason I put so much value on AP calculus is because it is the only class I have come across that teaches these things. Even in the community college I teach in, the math is too watered down. Learning calculus is not nearly as important as learning to think.

  5. September 22, 2007 10:58 pm

    Pissed Off — sounds like you teach a great AP calc class. My oldest is only a freshman, so we don’t have personal experience with our local AP calc but I hear good things about the teacher. It’s co-taught with AP physics which I think is a cool idea.

  6. September 23, 2007 7:58 pm

    Mathmom – I teach middle school gifted math students and you are absolutely correct about providing quality enrichment. After I give my students a heavy dose of number theory, set theory, and advanced problem solving, they tell me that their high school math courses, at least up to calculus are a waste.

  7. October 1, 2007 4:40 pm

    The “race to calculus”

    I’ve noticed that too and blogged on it. I think the problem is due to the fact that most people don’t know what happens after Calculus in college. They imagine that it’s just more complicated computations, since that’s all they ever really learned. Why not race there? Makes sense. The other issue that a lot of “enrichment” programs that parents are familiar with are very low quality. An example is a friend of mine with a child who qualifies for the G&T program, the school’s idea of enrichment was to place him with a teacher who was extra critical about his spelling and punctuation in English. Students that I went to high school with reported that their enrichment classes amounted to busy work.

    I am pretty sure that my ten year old could have finished an Algebra I course before school started this fall, but we chose instead to take the time to teach him algebra axiomatically this year (that ought to slow him down a bit). I see no need to race to Calculus.

    Finally, I’d like to comment on, “If what’s in enrichment is so important, then why isn’t it in the regular curriculum?” I think that many people do recognize particular content areas as important to mathematics. Try to suggest that average kids learn this by including it in the curriculum and the retort is, “Well, not every kid is going to be a math major.”

    You can’t win.

    Calculus is the upper-middle class shibboleth.

  8. October 1, 2007 5:02 pm

    Hi Myrtle, thanks for the comment. You are right that poor enrichment programs turn parents and students off to “enrichment” as a good use of a gifted math student’s time.

    You’re also right on with your discussion of “why isn’t it in the regular curriculum”. I’ve seen it taken one step further, where even parents of mathematically gifted kids will say that their kid “shouldn’t have to” learn it if it’s not important enough for everyone to learn, and not even every gifted math student is going to be a math major. sigh. There plenty of things left out of th regular math curriculum for lack of time/priority, that are important and valuable to learn. It’s frustrating when parents refuse to embrace the opportunity that their kids have to learn something extra that will benefit them later on!

    Thanks for dropping by! Your blog looks great too. I’ve bookmarked it to browse through later.

  9. Rachel permalink
    October 6, 2007 10:20 pm

    I can see lots of reasons to go as far as AP calculus in high school — but since Algebra I in 8th grade seems to be the coming standard, calculus senior year is likely to end up being standard for many college-bound kids. But if you’re going to accelerate kids faster than that, I’d like it to be clearer what comes after AP calculus in 10th or 11th grade.

  10. October 7, 2007 7:56 pm

    Most kids who do accelerate end up taking community college classes after they finish the math sequence at their high school. I’m not at all convinced that this is worth “racing” toward, which is part of my point here. In some high schools where lots of kids finish AP calc early, they have a college professor come in and teach a higher level math course right on the HS campus. But these arrangements are rare.

  11. June 11, 2010 8:46 am

    I wrote a similar blog on this subject but you nailed it here.

  12. April 24, 2012 9:55 pm

    What you say makes much sense–start with depth and breadth, add related concepts, practice a variety of applications and only then consider acceleration. Once the concepts are understood to the extent that they are comfortably used in a variety of problem-solving situations, and the patterns and connections with other related concepts are well understood, then it is time to consider acceleration. I think the whole “race to calculus” mentality is strongly related to the “Baby Einstein” syndrome and has more to do with bragging rights than interest in deep mathematical understanding. Since calculus is synonymous with “higher level mathematics” in the minds of many, it becomes a sort of bench mark or status symbol.
    Thank you for sharing this!

  13. Yakimama permalink
    April 24, 2012 11:41 pm

    I agree with the depth vs. breadth argument for enrichment, but haven’t seen it done really well. Any suggestions for enrichment curriculum for Pre-algebra/ algebra level?

  14. April 24, 2012 11:59 pm

    Yakimama, take a look at the Art of Problem Solving textbooks at — they offer textbooks that offer enriched versions of the basic curricula, plus some extra topics (combinatorics, and number theory). I don’t have any connection to them but I really enjoy their textbooks. For middle school, and up through algebra I, the problems offered by the MATHCOUNTS folks suggest a number of great enrichment topics as well.

  15. April 25, 2012 12:13 am

    p.s. was this post linked from somewhere interesting yesterday?

  16. Peter L. Griffiths permalink
    January 19, 2013 12:30 pm

    I think the binomial formula and factorials should be taught before calculus. For an interesting calculus exercise, try the following. In the orbits of planets at the Sun focus end of the elliptical orbit, deceleration is v^2=(1/r), and velocity is its square root v=(1/r)^(1/2). Likewise at the empty focus end of the elliptical orbit, acceleration is v^2=d, and velocity is its square root v=d^(1/2). This is not quite the expected calculus relationship, in that (1/2) becomes a power not a multiplier. d+r equals the major axis of the elliptical orbit.

  17. April 14, 2014 6:24 am

    This is a fantastic post and sums own opinions on the matter of gifted mathematicians well. Thanks for sharing.


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