What should they teach in school?

Bill Mill links to this blog by Peter Gray arguing that we should stop teaching arithmetic in elementary schools. He cites a research study from 75 years ago!

L. P. Benezet (1935/1936). The teaching of Arithmetic: The Story of an Experiment. Originally published in Journal of the National Education Association in three parts. Vol. 24, #8, pp 241-244; Vol. 24, #9, p 301-303; & Vol. 25, #1, pp 7-8.

I imagine there’s been some research done on this since then?? Not my area of expertise, but I’m curious.

P.S. You gotta read this anonymous comment that appeared on Gray’s blog. I have no idea if the story is true, but based my recollection of teachers from elementary school, I can definitely believe it!

32 thoughts on “What should they teach in school?

  1. We can probably do fine with less math if you're talking about kids who grow up playing games involving counting, number lines, and so on. But there is reason to think that coming to school without those experiences, and so with a lack of basic intuitions about numbers, leads those kids to fall further and further behind. I'd say that at the very least early math education needs to focus on helping children to build a foundation based on understanding things like cardinality, the successor function, etc. See for example research by Robbie Case a while back, or Robert Siegler at CMU more recently.

  2. Educators are quite adverse to experimenting

    Do you have any evidence for that statement or is that just your way of signalling your moral superiority to educators?

  3. Unintentionally funny part of that comment: “She knew I was smarter than her cohorts…”

    One wonders why this false dichotomy of techie/fuzzy has such a tight grip on our cultural imagination? Is that an underlying cause for these problems? Nice to find a blogger who bucks the trend….

  4. That 22/7 tale is indeed entertaining (and appalling). I have one of my own…although, in fact, Andrew was there too. In high school, Andrew, I, and our friend Craig went out to a pizza joint. We ordered a large pizza, and asked the (high-school-aged) waitress how many pieces that would be cut into. She said sixteen. We asked if the kitchen could please cut it into 12 pieces instead, to make it easier to split between the three of us. She said No. I asked "why not," and she said "Well, it's a 16-inch pizza." "So?" Sigh; eye-roll; "we can't cut it into 12 pieces because 12 doesn't go into 16."

  5. Educators are quite adverse to experimenting…

    Whether they're adverse or not, I suspect the burden of getting such an experiment approved by the appropriate IRB and supported by the parents and school board would be a Sisyphean undertaking. Just thinking about it makes my head hurt.

  6. The answer to the question, "What, can we drop from the elementary school curriculum?" is easy: the teachers. Morris Kline wrote a book in 1973 called Why Johnny Can't Add showing just how incompetent public school math teachers were at that time– the good old days. He cites an example where the math teacher asks, "why is 3 x 4 = 4 x 3?" The teacher then proceeds to answer with, "because multiplication is communative." The readers here should be able to tell immediately why this answer is wrong. The teacher tried to illustrate the concept with reference to putting on your shoes and socks. As Kline pointed out, it's a big mistake to introduce these concepts to young children because you can only provide trivial examples, and the kids get to thinking all these fine points are about nothing. Better to wait until the students study something like matrix multiplication. The point being we don't need curriculum reform, we need better teachers.

    My daughter went to a highly selective elite private high school. I sat in on her senior calculus course. It was pretty bad. The instructor just filled the board with equations, which the students dutifully copied. No motivation. No insight. But at least the equations were right unlike chemistry where the teacher made errors of fact. I offered to provide motivational lectures to the school in physics and math, but they would have not of that. Leo Breiman told me he tried to help out in the Santa Monica school system, but they rejected him. He was actually pretty depressed about what happened and didn't want to talk about it. In the 1970s some of the staff at Bell Laboratories similarly offered free time to the New Jersey school system. The teachers fought back ferociously.

    It looks like things have just gotten worse today. Most math teachers did know that 22/7 was simply a crude approximation to pi thirty years ago, although most didn't quite get the concept of an irrational number, let alone a transcendental. I suspect that even today public school math teachers know pi is transcendental, but have no idea of what is being transcended.

    For years I wondered what was the matter with these teachers. Then one day I heard an educational official say, "we teach students, not subjects." Bingo. It all fell into place. They personalize the educational process to where the content becomes secondary. It's all about technique, not the subject material. They hardly care if the material is even correct. The whole idea is to create an experience for the students. They really don't care if pi is 22/7 or not. That's not what's important to them. You want better math education, fire the teachers. Fire the administrators too. Eventually people will home school in large numbers. Rich people will hire private tutors. It will be like it was in the day before public education.

    We could make an immediate improvement in teaching arithmetic by taking calculators and computers out of the classroom in K-6. Replace the calculator with an abacus. Bring back the slide rule for middle school, and then bring in computers in high school.

  7. William,

    As a teacher who believes in experimenting in the classroom to improve results, I think that I am in the minority. Most of the teachers around me think of experimentation as something negative. To most teachers I know, the idea experimenting and thus giving some children the equivalent of an educational placebo pill, or a potentially harmful one (innovative but unproven methods?), is unthinkable.

    I just think it is interesting that your comment presupposes that everyone's moral schema places scientific processes and experimentation at the top. I certainly hold scientific process up high, but a glance at the news reminds me that I am in the minority.


  8. Just to respond briefly to the comments above:

    Say what you want about individual teachers, but education researchers experiment all the time. I just don't know what to think about an experiment that was published in 1935. There's got to be some work following up on this; I'm just not familiar with the literature.

  9. In elementary school my brother had a teacher who taught the class how to divide by zero. (I think the answer was supposed to be "zero".)

  10. "why is 3 x 4 = 4 x 3?" The teacher then proceeds to answer with, "because multiplication is communative."

    my 12 cents worth:

    Make a rectangle of pennies and say, "Remember what multiplication actually is — it's adding up 3 four times, 3+3+3+3. Look you can either count up the 4 rows of 3 to get 12, or you can count up the 3 columns of 4 to get 12. Either way it's the same 12 pennies." Then lock it in with something like, "so if don't like memorizing the tables when the first number is bigger, just switch them around, and you only have to remember when the first number is smaller. That's half the work!" (or vise-versa) That way the kids have a reason to care about commutativity. Obviously don't mention the C word.

  11. To the commenter who doesn't like teachers arguing "multiplication is commutative": Would mathematicians say anything different? That is clearly the "right" answer, and while I am all for using examples involving integers ("if you have a rectangle with three columns of four spoons in each column, this is the same as a rectangle of four rows with three spoons in each row"…), what is the 8 year old supposed to think when he sees a fraction or a real number? There is no "why" to multiplication being commutative other than the fact that it is defined that way. It's asking a bit much to teach to kids, perhaps, but breaking the link between real world objects and mathematical objects is the most important thing the field has done in the last 200 years!

  12. In ordinary arithmetic, a x b = b x a follows from the way in which we define multiplication. Let's look at the first big generalization high school students get introduced to: complex numbers. They get taught how to multiply two complex numbers using what I would regard as a helper device, the imaginary number i. We tell the students its square equals -1, by some kind of magic. Then they use the algebraic manipulations they have already learned to do complex arithmetic. Then we should let them discover that a x b = b x a holds for a and b complex. We could have defined complex numbers as ordered pairs of real numbers and then defined multiplication purely in terms of operations with the ordered pairs without reference to that mysterious and magic i. The better students are often trouble by i. Unfortunately they don't usually get exposed to the next generalization, quaternions, where (as defined) multiplication is not communative. Then they would get to see a non-trivial example where commutativity fails.

    To say that a x b = b x a because multiplication is communative puts the cart before the horse. Commutativity is a consequence not a cause. The teacher in Morris Kline's book, did not understand the concepts she was parroting to the kids. As Kline wrote more 37 years ago, it's a mistake to introduce these advanced concepts to young school children without making them seem trivial. They have never seen, and probably will never see a non-communative division algebra. They don't need to know this. They need to learn how to do arithmetic quickly and accurately and how do estimation.

  13. There is no "why" to multiplication being commutative other than the fact that it is defined that way.

    Depends what you mean by "why." If it is a college or high school math class asking, that might be an answer to give I think.

    But that answer to a grammar school student may come across as "because I said so." A smart and or brave grammar schooler will ask, "Why can't I define it differently? What happens if I do? What goes wrong?" and you end up in more or less the same place.

    The 3×4 story above does not give the age of the students. Perhaps the lesson was about the C word and the question was posed by the teacher to start the discussion. Then it's about the syllabus, and not specifically about how/why of teaching commutativity of numbers.

  14. You wrote,

    "We all need to know a little arithmetic, but not that much – far less than is taught. We might need to multiply by 1/2 or 1/4; we are unlikely to need to multiply by 2/5 or 4/7 or what not. That's why there are calculators. I suppose it's of some utility to be able to calculate quickly and accurately, but it's not a key thing."

    You seem to be saying that calculators have made arithmetic ability obsolete. Assuming I understand you correctly, I have to disagree. I could be a wise guy and ask what happens if you battery runs down, but I won't because there are better reasons for not turning our brains over to calculators. I've seen too many people just push buttons and get absurd answers, and they don't even know the answers are absurd because they have no "feel" for numbers, and orders of magnitude. One should be able to get an approximation in one's head. Getting the exact number with a calculator is a check. Most of the time only bankers and accountants need the exact answer. The physicist John Wheeler once said that a physicist should never use a computer unless he first knows the answer. There's a lot of wisdom in Wheeler's dictum. I once asked a physicist who was utterly dependent on a computer code for the answers he was getting, the following. "Suppose I secretly changed the source code to multiply all your answers by a factor of 2 and recompiled your code. How you know I did that?" He had no answer. If you want to scare a physicist, suggest his answer is off by a factor of 2, which could mean too big or too small. Everybody knows factors-of-2 mistakes are very easy to make and sometimes very hard to find.

    In my opinion, even people who will never do physics or statistics or any kind of mathematics beyond arithmetic still need to have a feel for numbers. Tasks like carpentry, floor tiling, auto repair, budgeting often require "doing fractions" and getting the right answer.

  15. Commenting on the above comment by afinetheorem:
    "but breaking the link between real world objects and mathematical objects is the most important thing the field has done in the last 200 years!" — which might be true, but for small children one very important thing is to _make_ that link (it can't be broken before it is maid!), otherwise math will look just like some abracadabra.

    I would explain the 3*4=4*3 by explaining that both are area of rectangles which happen to have the same area — the last be geometric intuition, or mirror image, or …
    (that is, thatś the idea, I did't try to put in wording appropriate for 8-year olds. But before explaining it, they should be made to do a lot of exercises, and then the teacher discovers the result in the exercises.

  16. Richard Feynman wrote:

    "A zookeeper, instructing his assistant to take the sick lizards out of the cage, could say, 'Take that set of animals which is the intersection of the set of lizards with the set of sick animals out of the cage.' This language is correct, precise, set theoretical language, but it says no more than 'Take the sick lizards out of the cage'"

    The point about commutativity parallels Feynman's comment. Commutativity is how we axiomatize integer arithmetic (for example), but it is pedagogically for small children a silly thing to do, just as set theory is a silly thing to push on children at the wrong age. It adds nothing and just muddles the issue.

    More here:


  17. Must have been teaching future programmers. There are many application where you want to substitute a zero to avoid a divide by zero error.

  18. I have kids in elementary school now, and my wife just about finished with her teacher's certification before getting so disgusted she quit, so I have some experience with the current state of education. In my opinion, the problem with education is that the process of training new teachers has become too meta. That is the primary focus of what prospective new teachers learn is teaching/education theory. Making sure the teachers actually know and understand the material they are supposed to teach is secondary to making sure they know the latest theory and jargon. The implicit assumption seems to be that if you know "how to teach" you can teach anything because the text book has all of the answers. In my opinion this is almost exactly backwards, if you really know and understand the subject, you can probably teach it.

  19. "To the commenter who doesn't like teachers arguing "multiplication is commutative": Would mathematicians say anything different?"

    Mathematicians would say that multiplication in ordinary arithmetic is communative. But they wouldn't say that 3 x 4 = 4 x 3 because multiplication is communative. The commutativity is a consequence of the way arithmetic multiplication is defined. The statement is true independent of the notion of commutativity.

  20. Reminds me of the High School science teacher who taught us that the seasons occurred because the earth wobbled.

    I think a major question is: to what degree do teachers at various grade levels specialize in their topic? My feeling is that, especially at lower grade levels, teachers are generalists who are assigned particular topics based on seniority, "you're good at ", or other non-topic-related reasons.

    If you have no expertise in a topic, you'll teach it to the curriculum and textbook, not venturing far afield — both for preparation time's sake and also because you can't go any deeper into the topic yourself. Student questions then either become actual threats — threatening to reveal your ignorance — or are not properly comprehended and in either case, are answered by repeating what was just said make the questioneer feel like an idiot for asking.

    I remember learning the multiplication tables as a kid, with my dad, and I remember a horribly boring math class in elementary school where we were handed a piece of paper with dozens of (trivial) multiplication problems on it. After that, it's all a blur until geometry (didn't like it), and finally algebra-trig in something like 10th grade. Considering how math-oriented I am now, that's a pretty sad commentary.

  21. Hello A. Zarkov

    I think you slightly misunderstood what I meant.

    I think there are different skills – arithmetic and quantitative sense. Arithmetic is the ability to (say) multiply 134*472 and get the right answer. Quantitative reasoning is the ability to realize that if your calculator tells you "606" then you pushed the wrong buttons. I see little utility to the former, but great utility to the latter.


  22. I recall my college calculus teacher complaining that he ran into this nonsense at his child's school. I ran into it just this year — my child's math teacher told the class that pi = 22/7.

    Here is an Abbott and Costello room that reminds me of these teachers….Abbott proves that 13*7=28


  23. I agree that one can have quantitative skills without being a whiz at detailed arithmetic. As an extreme, we have lightning calculators, also called "mental calculators." These people are savants who can multiply and factor large integers. John Von Neumann had this ability in addition to all his other extraordinary talents. Wikipedia has a good writeup on mental calculators. It points out that some mental calculators qualify as autistic savants and have deficiencies in other areas including quantitative skills.

    All that being said, I still think students need to know their "times tables" as we used to say. In order to have quantitative ability, I think one needs at least some arithmetic facility, and this is best taught in the early grades. I could be wrong about the "early" part, and we need to do careful experiments to see if this is true. One would think this has already been done more recently, but you never know. China and Russia seem to have successfully used the abacus to teach school children arithmetic. When I was a young student I bought an abacus and learned how to use it. I found it sharpened my ability to do arithmetic. I also went out and bought a slide rule and learned that too. At one time, New York City schools did teach the slide rule, but they stopped doing that when I was a student. I know that because some of our classes had a giant (unused)over-sized slide rules hanging above the front blackboard. I suppose they stopped because of some change in fashion in educational technique.

    To sum up my position, I think we need more focus on content, then teaching technique for K-12 mathematics in public schools. It seems to me that to many people can't do even simple arithmetic, and fractions are something alien and frightening.

  24. Phil, I would have been happy to join y'all and thus help solve her (and your) problem.

    BTW (to anyone), does this site no longer accept OpenIDs? I can't seem to use mine.

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