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2011 May 24

Why Discrete Math Is Important and The Calculus Trap

Filed under: Uncategorized — gasstationwithoutpumps @ 20:19
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My son is nearing the end of his Art of Problem Solving precalculus class, and it is still going well, as I reported earlier.  We are now looking at what to do next.  Should he take Calculus BC at his high school in the Fall? Should he take the AoPS Calculus class this fall?  Or should he detour into a different branch of math?

The Art of Problem Solving people have written a couple of essays about pre-college math preparation:

  • The Calculus Trap. The basic premise of this piece is that the lock-step march through arithmetic, algebra, geometry, algebra, trigonometry, precalculus, and calculus is not the only or best way to study math.  Elementary and secondary math education is not a race to see who gets to the “finish line” (calculus) soonest.  Indeed, from a mathematician’s viewpoint, calculus is just one of many starting points for interesting math.  A lot of what gets tossed aside in a race to calculus is more interesting.
    Even more important: “the standard curriculum is not designed for the top students.”  They point out that being the top student in your class is not the way to make progress in your learning—you are better off getting to a level of challenge that really makes you exercise your problem-solving skills.  Racing through courses full of drill problems is not going to do that the way working on harder problems will.  There are plenty of hard problems that do not need a lot of mathematical machinery to solve, so students can start on them before having learned all the machinery.
  • Why Discrete Math Is Important. As a computer scientist and computer engineering who taught applied discrete math a few times, and as a bioinformatician who has to teach grad students all over again how to count (that is, how to do simple combinatorics) and how to do simple Bayesian probability, I am certainly in agreement that discrete math is important.  In fact, a big part of the reason I ended up in computer science rather than pure math was that I liked discrete math (graph theory and combinatorics) better than real or complex analysis, and I had made the mistake of starting my graduate math education in a department that had no discrete math. Luckily there were four or five great people doing combinatorics, graph theory, and graph algorithms in the computer science department, and I was able to switch departments.  That turned out very well for me, so perhaps it was a good thing that I’d chosen the wrong math department.

Most likely we’ll continue the standard progression, doing either Calculus BC or AoPS calculus next year.  After that, there will be time for applied discrete math and for probability and statistics before he goes off to college.


  1. My son attended the most recent “Math Jam” on the AoPS website, which was a discussion of how the classes work. (He’s taking their Python class this summer.) One thing Richard R. mentioned was that AoPS has started the process of getting accreditation, but that it is a long process. He did not know whether classes would be retroactively accredited. Don’t know if this matters to your son’s school.

    Comment by Yves — 2011 May 24 @ 20:35 | Reply

    • Accreditation would be a very big deal—he could get credit directly for the course, rather than having to find an equivalent at the high school and take a final exam for that course. For the Precalc class, he’s already intending to do the Trig and Analytic Geometry final exam (and the SAT2 math level 2, though that only counts for college, not high school).

      But it would be good for him to be able to get high school credit for, say, a combinatorics class, which the school certainly doesn’t offer.

      Comment by gasstationwithoutpumps — 2011 May 24 @ 22:59 | Reply

  2. After teaching Calc and Discrete Math/Stats at both the high school and college level I am convinced that calc is good for about 1% of the students and discrete/stats is for the other 99%. This is from looking at what mathematics students use and need in their majors. I would claim that the 1% that actually use calculus eventually have to take a discrete/stat course.

    Comment by Garth — 2011 May 25 @ 07:34 | Reply

    • Actually, I’d put it more like 5–10% need calculus, 5–10% need discrete math, and 100% need probability and statistics.

      Comment by gasstationwithoutpumps — 2011 May 25 @ 08:14 | Reply

  3. Our oldest son is finishing up the same class. He will take BC Calculus next fall if only to open up the door to more of the OHS university level courses. That won’t stop him from taking more discrete math courses in parallel. Another option that he enjoyed quite a bit was the mathematics / logic course offered by eIMACS. which our younger son will begin this year.

    Comment by Metroplex Math Circle — 2011 May 25 @ 07:53 | Reply

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  5. One of my daughters had the choice of taking AP Statistics or AP Calculus in her senior year of high school (she graduated June 2011). I recommended she take AP Statistics. She is now majoring in college in an interdisciplinary program in the social sciences. Calculus would have been useless. I’m sure she’ll have many opportunities to use the statistics she learned.

    I myself majored in both math and computer science as an undergraduate. While the math I learned gave me “mathematical sophistication,” practically the only math I used later (in my Ph.D. studies in computer science) was an elective course in abstract topology.

    I think one needs calculus to go into (physical) engineering and some hard sciences (although more fields unnecessarily require it). I think more students would do well with combinatorics and discrete math (anyone who uses computers in a “deep” way, and there are more and more of us). I think a working knowledge of probability and statistics is necessary to be considered numerate.

    In using the term “numerate,” I mean the analogue to “literate.” No one is proud to be illiterate. No one should be proud to be “innumerate.” It should be objectionable when someone says, “I don’t do math.” On the other hand, there are increasing elements in society that is anti-science.

    For those who don’t consider probability and statistics to be important, consider this: The lottery is a tax on the innumerate.

    Comment by Arthur Keller — 2012 April 17 @ 00:16 | Reply

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