More on the Russian math textbook

The other day I described the math textbook that was used in the Soviet Union when I attended school there.  My question was: how could this textbook teach so much in just 220 pages? My tentative hypothesis was that it eliminated needless “scaffolding” (yes, I recognize that the term is problematic) and went right to the matter. I realized later that this was only a partial explanation.

An acquaintance suggested to me the other day that Russians (and Soviets) had a particular way of presenting mathematics that made it clear to the reader and student. As I started to think about my textbook in this light, I saw that its material was presented not only lucidly, but also artfully. There is a well-considered and well-timed movement from specific problems to larger topics and back. 

The textbook’s editor and lead author is the eminent mathematician Andrei Kolmogorov (1903–1987). I suspect that he shaped a good deal of the book and didn’t simply attach his name to it. But since I don’t know that for certain, I’ll leave that aside and look at what’s actually there.

I’ll begin by describing the contents of the first four pages of the book. Then I’ll examine their structure.

The book begins with a discussion of induction. It presents a series of odd numbers (1, 3, 5, 7, … 2n – 1) and posits that for every natural number n, the sum of the first n elements of the series is n2. That is,

1 = 1 = 12;

1 + 3 = 4 = 22;

1 + 3 + 5 = 9 = 32;

1 + 3 + 5 + 7 = 16 = 42;

1 + 3 + 5 + 7 + 9 = 25 = 52;

and now the hypothesis: “For all natural numbers n, this equation holds true: 1 + 3 + 5 + … + (2n – 1) = n2.”

All that is on the first page (or half-page) of the text. The book leaves the reader with this concept: having observed a number of examples, you make a hypothesis about the general rule that unites them.

Then, on the next page, the book examines a few more problems in order to shed light on hypothesis-making. It shows, first, how a single counterexample is enough to disprove a hypothesis. For instance, if you have the formula P (x) = x2; + x + 41, and considered the natural numbers 1 through 5 as values of x, you might conclude that P (x) is always a prime number. However, an obvious counterexample is P (41).

After this, it gives two more problems where you can easily verify the hypothesis.

Now it goes on to explain the principle of mathematical induction. It returns to the original problem and restates it in this manner: If we know that the equation is true for a certain value of n, and if we can also demonstrate that where it is true for n, it is also true for n + 1, then we know that it is true for every natural number n

It then puts this in simple notation. The letter A represents the function. A (5)  is true, we know, because 1 + 3 + 5 + 7 + 9 = 52. We can also see that the truth of A (6) follows from that of A (5). (I wont’ go into that; it’s straightforward.) We represent this as follows: A (5) => A (6).

So our goal is to prove that A (k) => A (k + 1). That is, we want to prove that from the equation

1 + 3 + 5 + … + (2k – 1) = k2

this equation follows:

1 + 3 + 5 + … + (2+ 1) = (k + 1)2

In fact, this is easy to demonstrate, since

1 + 3 + 5 + … + (2k – 1) + (2k + 1) =

= (1 + 3 + 5 + … + (2k – 1)) + (2k + 1) =

= k2 + 2k + 1 = (k + 1)2.

(In the penultimate equation, just substitute k2 for everything contined within the outer parentheses. You can do that because we’ve already established that for a cerain value of k, the sum of the first k elements is k2)

From here, the textbook goes on to more complex examples.

How is this structured? First we are given a series with a hypothesis about the underlying pattern. The possibility of proving this pattern is dangled before us. We learn a little more about hypotheses themselves and how they might be proved or disproved. We see a few examples. We go on from there to the concept of induction. At this point we return to the original problem and prove, through induction, that it holds for all values of n, where n is a natural number. What in particular makes this presentation compelling, easy to understand, and at the same time challenging?

For one thing, there’s the lure of something to come. It’s a real lure. We know that a problem has been left hanging and that there’s more to it. Also, each concept is developed and at the same time reduced to the simplest possible notation. The student gets both the concrete examples and the abstractions. The careful reader understands the meaning of A (k) => A (k + 1). And then the proof of the original problem may jog the mind slightly at first, but then it’s obvious.

Now, some readers may be saying, “I have no idea what you’re talking about.” But that’s the thing. When I first read these pages, I could maybe make a glimmer of a tail’s feather out of them, but that was it. But then I sat down and puzzled over them. Then, when they came clear, they were straightforward and elegant to boot. This applies not only to the first four pages. I found that where this held true through page n, it also held true through page (n + 1).

Maybe that’s part of the “secret” here, too, if there is one: the understanding that the student must take time with the material, at home, without distraction.

Can math scaffolding hinder learning?

When I was fourteen, we spent a year in Moscow. I attended a Soviet school that “specialized” in French–that is, it taught French from the early grades. The other subjects (math, literature, history, technical drawing, geography, physics, chemistry, and biology) were in Russian. No one expected me to participate in class, but I insisted on being added to the class list and asked teachers to treat me like a regular student. I was eventually doing the work in all of my subjects except for chemistry and biology, where I lacked the necessary background knowledge and was usually a bit lost. (I barely got by in physics, but I did learn something.)

My favorite classes were math and French. Here is a picture of the math textbook. It took us through algebra, beginning calculus, and some trigonometry. Its 220 pages contained more substance than many a hefty textbook I’ve seen since. When I returned to the U.S., I was ready for calculus but had to take a year of precalculus first, along with my classmates. (It didn’t hurt, as I got to do more trigonometry.)

Recently I have been wondering how this textbook manages to convey so much in such short space, and how I learned so much without finding it particularly difficult. To answer this question well, I would have to work my way through the textbook again, this time with pedagogy in mind. That’s a project for another time. In the meantime, I’ll toss out a few hypotheses.

Well, one obvious reason we were able to learn so much is that there was a standard curriculum through the grades. All students came to this course with similar knowledge and practice. Some were better at math than others, but it wasn’t because they had better preparation. (Of course this isn’t entirely true, as some students had additional resources at home and elsewhere.)

It could also be that the curriculum included fewer topics than math courses in the U.S. do; thus there was more time to learn them thoroughly.

But what strikes me about this little textbook is that it plunges right in. The first chapter talks about inductive proofs. The second goes into combinatorics. There are no pictures except for graphs of functions (and a few circles and rectangles). There are word problems, but they are relatively few. There are no needless “scaffolds.”

Scaffolds in instruction are temporary supports intended to bring students to the point of self-sufficiency. All good instruction uses them to some degree.  But certain kinds of “scaffolds” can actually  become barriers, complicating the student’s entry into the subject matter. In mathematics, excessive reliance on “visuals,” “manipulatives,” and “real-life” applications can stand in the way of the math itself.

This textbook, by contrast, “scaffolds” the instruction in one way only: it builds from simpler problems to more complex ones. It  lacks the “scaffolding” that plagues many a math textbook that I have seen: those colored graphics, tips and strategies, needless word problems, and so on. It has a few word problems, but there are reasons for them to be word problems. The vast majority of the problems use mathematical notation. Thus, students become fluent in it and learn to think in it.

I was recently looking at AMSCO’s Geometry–better than many in terms of presentation. Very little clutter. But even AMSCO has word problems like this: “Amy said that if the radius of a circular cylinder were doubled and the height decreased by one-half, the volume of the cylinder would remain unchanged. Do you agree with Amy? Explain why or why not.” There is no reason to bring Amy into this; Amy’s presence does nothing for the problem. Also, turning this into a matter of opinion (“do you agree or disagree”) confuses the matter. Instead, the student should be asked whether the statement is correct or incorrect.

In bending over backwards to make math “accessible,” we may actually make it inaccessible. What do you think?