If we’re to believe the press, mathematics education is in the midst of a war. Education saw the reading wars a couple of decades ago, and academia still seems to be in the middle of a science war. At least according to the press.

Before getting into the ideas, let’s just stop for a moment and say something about the metaphor. Real wars are violent, disgusting affairs that result in death, destruction and social instability. Nothing of the sort is going on. The math wars are blood sport only insofar as political careers can be destroyed by standing on the wrong side of an angry press.

In short the problem is simple. Elementary school children are currently being taught a number of pencil and paper methods that are clearly

- Not the same as the methods their parents were taught;
- Not mathematically efficient;
- Reportedly frustrating for students, parents and teachers.

So what’s going on?

Traditionally (for the last century or so, anyways) elementary school mathematics has largely been devoted to developing knowledge about numbers and to developing facility in computation. You begin by learning to count, learning the names of numbers, learning to add, subtract, multiply and divide. Students typically began with whole numbers and after developing knowledge and skill, they extended their work to fractions. Later on (typically Junior High, or Middle School), this arithmetic knowledge and skill expanded further to integers and real numbers. High School typically saw further extension to computation with radicals and polynomials. Depending on the location and political climate, complex numbers were added to the mix. Sure there were other aspects to the study of high school mathematics—a bit of geometry, theory of functions and so on—but school mathematics has traditionally been about taking arithmetic from whole numbers in the early grades to polynomials in high school. The central mathematical and educational idea was that this progression allowed for increasing mathematical generalization and increasing computational sophistication, leading to the development of some rudimentary mathematical understanding. (Before you get too excited about the “depth” of high school trigonometry and calculus, remind yourself that at the university level, first and second year mathematics textbooks invariably contain the words “introduction” or “elementary” in their titles. There’s a reason for this.)

As a general plan, I like the sketch I’ve given. Mathematics is largely about generalizing and abstracting the particular context away from particular problems, calculations, and whatnot. What’s the evidence that the “traditional” system gave students the payoff that was planned? To be sure, most English-speaking countries do not enjoy a population that by and large loves mathematics and has fond recollection of school math studies. Worse yet, the great majority of people do not need more than the most rudimentary mathematics to fulfill their life-plans, including education, employment and personal satisfaction. It would be nice to know if people were somehow better off because of their school mathematical experiences, but alas, we have nothing to indicate that this is so. Maybe we were on the wrong path. Hold that thought. I’ll return to it in a couple of days. There is some uncertainty whether the math education of the past few decades adequately prepared academically competent students for further study, and whether it provided much of value to those students whose futures were not dependent on mathematical achievement.

A second line of thought looks at the basic achievements of students past. Computation is a very valuable thing. You can’t do your taxes without it; you can’t take care of a significant amount of daily business without it. Accurate computation keeps the trains on time, and the books balanced. Nobody doubts this. But the days of accurate manual computation are gone, and there is no sign of them returning. A $5 calculator is better at all basic computation than anyone would ever want or need to be. Better, in fact, than any human ever could be (if you include the capacity to compute trigonometric ratios and logarithms). So why bother teaching children to compute? Why indeed?

There are two schools of thought on this question. One says that we should abandon manual computation because machines are all we need. Perhaps that’s too strong. Computation should be learned, says this camp, only insofar as it can be demonstrated to increase student understanding of mathematical concepts that are worth learning—whatever those turn out to be. Not so fast, says the other school. Computation is valuable in and of itself. It not only is a worthwhile thing to learn, it is the key to unlocking number sense in young minds, and it is the best way to increase student understanding of deeper mathematical concepts.

How to adjudicate? To date there is absolutely no compelling research evidence that is useful in addressing this matter. We have two ideological positions, each grounded in respectable intuitions, but neither having the weight of empirical evidence to support it. This saddens me deeply. With all the educational research published every year, why in the world is no one pursuing this question in a responsible way? I’m sorely tempted to give up my position in bureaucracy to get my feet back into a research institution just to answer this question.

In the meantime, what is happening in schools? Not surprisingly, two things. In some jurisdictions, you’d never notice that the times are changing and students are computing away like it’s 1969. In others, we have (yet another) “new” math. Students are no longer given extensive drill on basic number facts, and are learning a number of pencil and paper algorithms that look very strange to the students’ parents. These algorithms are not the tidy methods that were taught decades ago; they are often long-ago-discarded methods that involve a lot of extra writing and computing, but which are supposed to reveal the deeper structures that hide inside arithmetic. (E.g. Division by repeated subtraction.) Are these students gaining deeper understanding into arithmetic than their peers that learn the same algorithms as their parents? Nobody knows. To the best of my knowledge, there are no reliable evaluations of these new programs.

So here we are, stuck between two ideological positions, with a distressing lack of evidence. We’ll look at a bit more detail in a few days.

Excellent summary of the issues. The fact that there is so little research data available is quite alarming. One would think that education would lend itself quite readily to scientific testing of the various hypotheses. Much of the dialogue in education reminds me of the field of alternative health, whose proponents always makes grand claims about the efficacy of this or that method while expertly avoiding any objective testing.

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As a student, I think it’s better to do the rote drills first before understanding why they work. I didn’t know how two digit multiplication worked until I had done it for a few years. In retrospect, I don’t think that’s a problem, as long as a student eventually understands it. And even then, I think it’s overkill to require students to write all the steps out. If anything, it’ll turn them away from liking math at an early age. And there’s a lot more interesting stuff to learn in math than multiplication and division.

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Oh, so much to say about this. But I think basically it boils down to this: different children learn differently and at different rates.

Therefore, the competent teacher has a big bag of methods. And makes connections whenever possible. Eclecticism, I suppose, is my religion. All roads, and so on. But Rome, in my opinion, is the aesthetic realization about something profound about the structure of the world we live in, and the nature of our powers of thought.

We can begin almost anywhere. Say for elementary school students:

1

1+3=4

1+3+5=9

1+3+5+7=16

1+3+5+7+9=25

What patterns do you see? Does it continue? How do you know? What kind of numbers are involved in the pattern? Why should it be? What would happen if we took different colored counters and told this story? What shapes could you make with the counters from the numbers on the right hand side of the equations?

Would you rather understand things from a picture or an equation? Could you build a series of counters that made triangles? What patterns could you describe there? And so on.

Now, if a student does not know the perfect squares by heart, they may miss some of this. So the students who know their multiplication tables by rote at this point have an insight that others may not. OTOH, students can be led to this and actually understand that when they “square” a number, it has a geometric correlative.

Some students grasp things like this very quickly; others do not. Some respond to manipulatives; others do not. When I was in school I was bored silly with manipulatives representing the tens and units. OTOH, I have been successful using manipulatives in teaching basic addition, multiplication and subtraction in a very short time to students who were considered very behind in math.

For high school students, one of the most valuable things that can be done is to make the whole system of postulates and axioms as explicit as possible. I had an excellent Geometry teacher when I was in high school. The first day of school he asked the class, “What is the sum of the measures of the angles in a triangle?” My hand shot up, proud that I knew that the answer was 180 degrees. He called me up to his desk and opened up a thin geometry book, and pointed to an underlined passage: “The sum of the measures of the angles in a triangle is always greater than 180 degrees.” I protested that the book was wrong. Then he flipped some pages and pointed to another underlined passage: “”The sum of the measures of the angles in a triangle is always less than 180 degrees.”

Thoroughly confused and perplexed, I went back to my seat, and the teacher kindly explained about assumptions and conclusions. It was a very important lesson that day. Much more important than the weeks I spent learning how to factor trinomial.

Anyway, I’m rambling, but my point, I think, is somewhere here: I don’t really care that much what the research says. They’ll use that research no matter what it says as a cudgel to make teachers teach everything the same way. And that is one thing that is death to learning.

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[…] her daughter’s math homework. The issues are the basic “math wars” battles that I outlined in post in December, 2014. In short, students are often being taught a wide array of techniques—some efficient, some […]

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