The mathematics we teach our kids…

Math education has made progress compared to my school days in specific areas. For example, I am really happy that statistics and probabilities are fully integrated with how we teach mathematics today. I cannot remember learning any notion of probabilities in class until I reached college. I do remember computing the probability that I would get a 7 if I were to throw the dice as a teenager, but this was never part of my schooling.

However, it is not clear that we are moving forward as fast as we need to.

  • As I have criticized before, we seem to stress too much the importance of rote memorization, in particular by pressuring student to memorize the addition and multiplication tables. I have never memorized these tables. To this day, to compute 6 times 9, I do… 6 times 10 minus 6. Not only do I feel no shame, I am convinced that rote memorization for its own sake can be detrimental. For one thing, it annoys students, making them hate the topic. For another, it encourages rote memorization as a way of doing mathematics.

    If you think it is odd for someone trained in mathematics like myself not to know his multiplication tables, note that Benoit Mandelbrot, arguably one of the greatest mathematicians of the XXth century, never memorized his multiplication tables.

  • There is a strong emphasis on naming the concepts and memorizing the names. My oldest son was given the following test: if 5 times 10 is 50, what is 5? The correct answer was “a factor”. I am quite sure I would not have known what was demanded. The way geometry is taught is almost entirely about memorizing the names of various figures. Who cares about what a scalene triangle is? Math is not about learning definitions.
  • Generally, we stress trivial but complicated algorithms. As a professional scientist, I rarely have to add fractions, or to do long divisions. To be honest, I had to pause for a few seconds before I could explain how one goes about computing 12.53 times 13.21. Really, you never compute such a thing the long way in the real world. Though I read that it has been de-emphasized, and this makes me happy… I still do not know why you would drill anyone by asking them to divide 115.5 by 24.21 by hand. It is useless and boring.
  • Computational thinking is not encouraged. The students have to be able to divide 115.5 by 24.21 by hand, but they are not be made aware that they are using an algorithm. In fact, computational thinking appears to be almost entirely absent while rote memorization and mindless drilling rule.
  • The students are asked to do “real-world” problems, that are designed to be encouraging and relevant. So instead of an openly arbitrary 4-line word problem, we get a 3-page story about Any who is trying to organize a party and she has 12 friends who each need to drink 120mL of juice, but she can only buy juice in units of 1.2L. Except that she can go across town where she can buy cheaper juice in units of 1.5L, but then she has to pay for her cab. Except that she can ask her sister to drive her, but then she also needs to pick up blue curtains for her mother and they are across town anyhow. Or she could go to the local store and buy red drapes, but it is out of her way… You think I am joking, but I barely exaggerate… I guess that they are also trying to train kids in the art of filling out complicated tax forms or do some corporate job, so they need to read through long instructions with many details and plan ahead… Except that if I am turned off by these problems as a trained mathematician, I can only guess how people who proclaim to hate math must feel. No, I do not think that they are fooled into thinking that this is relevant because of how it is phrased.

How would I train kids in mathematics? Good old word problems combined with an introduction to programming.

19 thoughts on “The mathematics we teach our kids…”

  1. There is a reason why you would make someone go through dividing 115.5 by 24.21. It is an un-natural process.

    It requires you to actually make an effort to solve it. Of course you will never be doing those calculation by hands if you are working on a paper or doing research. But like they say, you need to know the rules first before you break it.

    This turn off most students from math, but the most amazing thing is when it attracts some people. It attracts the mathematicians. Math is not for everyone.

    Those who go through solving it by hands will have no problems working with or creating algorithms. The others will use the calculators. And that makes up the world.

  2. I think learning multiplication table is important, it makes life easier. Absolutely agree with the rest of the article.

  3. Funny enough, I was reading opinions on math education I really agree with from Bill Thurston in 1990 here last night: http://arxiv.org/pdf/math/0503081.pdf

    His introductory remarks here are also relevant: https://www.youtube.com/watch?v=o6SucT2Zzys

    The short version: There are more important parts to mathematics than getting your times tables correct. I don’t think that is a point of contention. But I think what is at contention in these comments is the idea that “you have to slog through your times tables before you can do math. And if you don’t do that, you can’t expect to do math.”

    I think that is utterly wrong. Not only that, I think it filters out people who can think more abstractly but early on think they are not good at math because they can’t do their times table as fast as others, or perform division algorithms error-free.

    Thurston agrees. As does Terry Tao. Can’t find Terry’s blog post, but he does have one where he talks about the fact that math just takes work, and the people who become the best mathematicians are not always, or perhaps even usually, the people who are the fastest or the best at math olympiad competitions. In our arrogance to put up an acid test like pure computation deprives us of really good people in the field.

  4. If we look at the great mathematicians of the past we find a considerable number that had extraordinary mental powers in the field of memory and mental arithmetic. (Gauss, e.g., just to cite a single.) I think that in past times (no calculating or indexing machines) this was a prerequisite for excelling in any scientific field: by darwinian selection mathematicians were all, more or less, versed in arithmetic and memorisation.

    Your blog raises the question if this type of selection should, or should not, be imposed on 21st century kids. The answer should be based on experimental evidence, but the idea of a “computer aided mind” is compelling and could foster a new type of scientist and technician for which memory and mental arithmetic play no role at all. (And new and different skills will be fundamental.)

    May be I’m to old, but I think that memory (and arithmetic) is a typical human trait, and every one should train his memory, as every one should train his body. The mere existence of machines and cars is not a valid argument for muscle atrophy.

  5. My 13-year-old wholeheartedly agrees with you. I find, however, that I cannot encourage him to skip the pointless rote memorization as I still think it’s important that he get good grades. Oh well.

  6. The ability to multiply small numbers in your head needs to be learnt, and the multiplication table is a key part of that, but it’s to show the patterns that make many of the entries easy to learn/calculate on the fly.

    Children spot ten-times adds a 0 on their own, but many never progress further, e.g. nine-times is adding ten and taking off one at each step, and x9 is x(10-1), as you said, and the digits in the result also add to nine.

    There’s only a few outside these patterns that can be easily mopped up by rote.

  7. @Ben

    The grades he gets when he is 13 are important? Might it keep him from getting into college later?

    Even assuming that they are important, on the long run, the best thing you can do for your son is to make sure he likes math.

  8. I would teach kids math by basing lessons on Martin Gardner’s Mathematical Games column from old issues of Scientific American. That’s how I learned to love math, in spite of the tremendous effort that my teachers invested in getting to make me hate math.

  9. Benoit was never taught the alphabet and never learned multiplication tables past fives. Even today he claims not to know the alphabet, so that it is difficult for him to use a telephone book. Still, he had a special genius, and after the war Benoit enrolled in elite Paris universities and started to follow in his Uncle’s mathematical footsteps. He had a tremendous gift in math, but it proved to be quite different from his uncle’s, in fact quite different from anything seen before in academia. He had a visual mind, a geometric mind, in a school setting where this was discouraged. He solved problems with great leaps of geometric intuition, rather than the “proper” established techniques of strict logical analysis. For instance, in the crucial entrance exams he could not do algebra very well, but still managed to receive the highest grade by, as he puts it, translating the questions mentally into pictures. Benoit was clever and hid his gifts until he had obtained his doctoral degree in math. Then he fled academia and his uncle’s “bourbaki” math and began to pursue his own way.

    http://www.fractalwisdom.com/science-of-chaos/benoit-b-mandelbrot/

  10. “To this day, to compute 6 times 9, I do… 6 times 10 minus 6.”

    What do you do for 6 times 8 (resp. 6 times 7) … subtract 12 (resp. 18) from 60?

    —–

    “Who cares about what a scalene triangle is? Math is not about learning definitions.”

    Math isn’t about definitions, but they do make language concise. If you are working regularly with some mathematical object, it makes sense for that object to have a name (i.e. definition) and expect the student to know what that definition means.

    Instead of being just rote memorization, definitions can also serve to enhance a student’s understanding. If memory serves me correctly, I’ve asked the following on a homework assignment :

    “A student has defined a scalene triangle to be one in which the interior angles are unequal. Explain why this is or isn’t a valid alternative definition for a scalene triangle”.

    —–

    “I still do not know why you would drill anyone by asking them to divide 115.5 by 24.21 by hand.”

    If a student doesn’t have some fundamental understanding of long division, then what happens when the subject comes up again with polynomials?

    —–

  11. @Marvin

    What do you do for 6 times 8 … subtract 12 from 60?

    In this instance, my brain wants to use the fact that 5 times 8 is 40 (since 10 times 8 is 80), and then just add 8, to get 48.

    Math isn’t about definitions, but they do make language concise. If you are working regularly with some mathematical object, it makes sense for that object to have a name (i.e. definition) and expect the student to know what that definition means.

    I must have taken 8 or 10 college-level geometry classes (e.g., two classes of projective geometry, several classes of differential geometry…), I even met Coxeter in person when he was still alive.. and when my kids had to learn about scalene triangles, I had to look up what there were.

    I have no idea who uses the term scalene triangle outside schools…

    If a student doesn’t have some fundamental understanding of long division, then what happens when the subject comes up again with polynomials?

    I do not recall dividing polynomials by hand… though I probably did at some point in my education… yet part of my current research involves Galois fields generated from irreducible polynomials… still, I never manually divide polynomials…

    But if you must divide polynomials (what for?), you can just do it online these days…

    http://www.wolframalpha.com/input/?i=%28x%5E2%2B4*x%2B4%29%2F%28x%2B2%29

  12. @Peter

    I teach math to my boys using puzzle books for kids. I have not yet used Garner’s, but the idea is the same.

    I choose neat problems and I ask them to work it out… when they get stuck, I help. The process is a game. No grade, no pressure. All I ask from them is that they try their best.

  13. @Daniel

    “What do you do for 6 times 8 … subtract 12 from 60?

    In this instance, my brain wants to use the fact that 5 times 8 is 40 (since 10 times 8 is 80), and then just add 8, to get 48.”

    So to derive 6 times 8 you first have to decide which of the two numbers is closest to another number which is a factor of 10. In this case, 5 is closer to 6 than 10 is to 8. Then since you don’t know 5 times 8, you calculate 10 times 8 and divide that by 2. Finally you add 8 to that result to arrive at 48 as the final answer. Perfectly valid algorithm, but horribly inefficient.

    How about 7 times 7?

    —–

    “I even met Coxeter in person”

    Nice appeal to authority, but you missed my point about definitions which had *nothing* to do with scalene triangles.

    —–

    “But if you must divide polynomials (what for?), you can just do it online these days…)”

    A typical AP Calculus test problem involves integrating P(x)/Q(x) (where deg(P) > deg (Q)) and access to Mathematica (or similar) is not permitted.

    —–

  14. @Marvin,

    How do you discount an appeal to authority in one line, and in the very next one, justify polynomial division by appeal to authority (“A typical AP Calculus test problem involves integrating”… appeal to authority of a test rather than a person).

    There is no justification for why polynomial division is important other than “it is on the test.” Why is it on the test to begin with? Because people who set the test say it should be on there?

  15. @Marvin

    Perfectly valid algorithm, but horribly inefficient.

    Using this “horribly inefficient algorithm”, I can compute quickly 6 times x in my head for a wide range of x values.

    Granted, I could do it faster with rote memorization, but how many times do I need to multiply a number by 6? And how often do I need to do it in a time critical context? Never.

    And my “horribly inefficient algorithms” allow me to compute 15% (the tip) of most amounts in my head… Sadly, most people do not memorize the table of 15… what are they going to do?

    I mean, how are you going to compute 15 times 82 quickly? Should people memorize this as well? Or are people expected to pull out a pen and some paper and do it the primary-school way? That is going to be awfully elegant on a first date.

    It is a lot more powerful to learn to reason out these problems and design algorithms to solve them.

    Nice appeal to authority, but you missed my point about definitions which had *nothing* to do with scalene triangles.

    Fair enough regarding the appeal to authority, but I say that people who insist that knowing what a scalene triangle should be compulsory knowledge have the burden of the proof.

    Speaking for myself, I am convinced that it is pointless school trivia.

    We can easily test this theory: let us go ask 100 educated middle-age adults who do not work in education, when they used their knowledge of what a scalene triangle is for the last time.

    Do you want to bet on the percentage of adults who will report relying frequently on the concept of scalene triangle in their daily lives?

    My point is that as a college students in mathematics, I never needed to know what a scalene triangle is… even though I took a lot of geometry. So how can you pretend that most adults badly need to know what a scalene triangle is?

    and access to Mathematica (or similar) is not permitted

    So, we should teach long divisions because later students will be asked to divide polynomials without access to a computer?

    This has the flavour of a circular argument: force students to do contrived work because later we will force them to do contrived work.

    This being said, I taught calculus and advanced calculus, and I cannot recall using non-trivial polynomial division.

    Who on Earth really needs to divide polynomials without a computer in 2015?

    Much more interestingly, you could ask calculus students to derive a polynomial division algorithm… some might even be tempted to implement it. That would be fun! Doing it by hand? Not so much.

  16. @Matt

    Stating that Daniel never ran into scalene triangles in his various geometry courses is a valid comment. Saying he met Coxeter is not and Daniel acknowledges that.

    You may not like what’s tested on an AP Calculus exam, but until it is changed, knowing how to divide polynomials by hand is a reality to many high school students.

  17. @Daniel

    What is your experience teaching at the K-12 level?

    Also, what markup is allowed in the comments?

    —–

    Your fixation on scalene triangles has caused you again to miss my point. Let me quote myself :

    “Math isn’t about definitions, but they do make language concise. If you are working regularly with some mathematical object, it makes sense for that object to have a name (i.e. definition) and expect the student to know what that definition means.

    Instead of being just rote memorization, definitions can also serve to enhance a student’s understanding.”

    I then go on to give an example of that, using an alternative definition of a scalene triangle. Said example was not meant to be an endorsement of the value of defining a scalene triangle.

    —–

    “… let us go ask 100 educated middle-age adults who do not work in education, when they used their knowledge of what a scalene triangle is for the last time.”

    Very few, if any would have used it outside of high school. However substitute just about any concept from geometry in place of “scalene triangle” and the responses will be pretty much the same. Does that mean their exposure to geometry was mostly pointless? Since when does “ask 100 educated adults” determine the value of something?

    —–

    “Perfectly valid algorithm, but horribly inefficient.

    Using this “horribly inefficient algorithm”, I can compute quickly 6 times x in my head for a wide range of x values.”

    I can do that as well. Because I bothered to learn my 6 times table, I can do a problem like 56 times 7 more efficiently than you because I don’t need to stop to use an algorithm for 6 times 7.

    How about 7 times 7 … did you commit this to memory or do you have an algorithm for this?

    “Granted, I could do it faster with rote memorization, but how many times do I need to multiply a number by 6? And how often do I need to do it in a time critical context? Never.”

    If time is not as issue, why bother with any algorithm at all? Just let your calculator do the work for you. Since calculators and computers are so widely available, let’s skip learning how to add, subtract, multiply or divide. Once a kid knows how to count, hand him/her a calculator.

    —–

    “And my “horribly inefficient algorithms” allow me to compute 15% (the tip) of most amounts in my head… Sadly, most people do not memorize the table of 15… what are they going to do?

    I mean, how are you going to compute 15 times 82 quickly? Should people memorize this as well? Or are people expected to pull out a pen and some paper and do it the primary-school way? That is going to be awfully elegant on a first date.”

    We don’t bother learning our times table of 15 (or 14, 13, … 11) because single digit times table is all you need to know in order to do multiple digit multiplication.

    Before calculators were available, people leaving a 15% tip calculated it in their head or on the back of the receipt or on a napkin. No clue if this impressed their dates.

    —–

  18. @Marvin

    I don’t understand then what you are driving at. The argument here is “memorization of multiplication facts and memorizing the division algorithm is not a prerequisite to mathematical understanding.” You can memorize the algorithims and not understand why they work. Most do, I’d argue. Memorizing the division algorithm helps you access the form of how you do polynomial division, but why it works, what it is doing, and what the parts mean still makes no sense to most. And understanding of why it works is how you advance mathematical understanding — not putting another cargo-cult algorithm in your holster. Understanding that you can only group like terms (unknowns of the multiplicative form, digits representing the same place value/groups, fraction denominators) is the deeper “math” part of what is going on. The algorithm leverages that, but you can understand it without remembering the accounting bits by rote.

    Your argument of “Well, that is the way the tests are, like it or not” seems to not get at this. Yes, the tests are this way, whether I like it or not. But that is no argument for why memorization is prerequisite to mathematical understanding. Which is why I call it an appeal to authority. Perhaps I am misnaming it, though, but I don’t have a better name for it.

  19. @Matt

    “Your argument of “Well, that is the way the tests are, like it or not” seems to not get at this. Yes, the tests are this way, whether I like it or not. But that is no argument for why memorization is prerequisite to mathematical understanding. Which is why I call it an appeal to authority. Perhaps I am misnaming it, though, but I don’t have a better name for it.”

    It’s called an appeal to reality. Daniel stated “But if you must divide polynomials (what for?), you can just do it online these days…” and I gave an example of where that is not true.

    Where did I state I advocate rote memorization in reference to the division of polynomials?

    —–

    “…“memorization of multiplication facts and memorizing the division algorithm is not a prerequisite to mathematical understanding.””

    I never argued to the contrary. However, lack of knowledge of simple number facts can hamper student learning. I’ve seen it many times with younger students.

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