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, … 2*n – *1) and posits that for every natural number *n*, the sum of the first *n* elements of the series is *n*^{2}. That is,

1 = 1 = 1^{2};

1 + 3 = 4 = 2^{2};

1 + 3 + 5 = 9 = 3^{2};

1 + 3 + 5 + 7 = 16 = 4^{2};

1 + 3 + 5 + 7 + 9 = 25 = 5^{2};

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

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*) = *x*^{2}; +* 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 = 5^{2}. 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 + … + (2*k* – 1) = *k*^{2}

this equation follows:

1 + 3 + 5 + … + (2*k *+ 1) = (*k* + 1)^{2}

In fact, this is easy to demonstrate, since

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

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

= *k*^{2} + 2*k* + 1 = (*k* + 1)^{2}.

(In the penultimate equation, just substitute *k*^{2} 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 *k*^{2})

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.

You have to have the intellect to puzzle over it. You have to care about it.

Your example, again, shows how completely clueless you are as to what challenges face low ability kids. Induction? Are you kidding?

It’s not that I think you’re an idiot, or anything. I’m not trying to be mean. No, really.

It’s that you are so profoundly and deeply ignorant about what it means to teach math to kids who can’t handle abstractions–not because they weren’t taught properly, not because the math wasn’t “scaffolded”, but because their brains don’t work that way.

If it were just you, it’d be one thing. But your idiocy is exactly what is behind the “college for all” push.

Really, really high IQ people (say, over 140) understand what low intelligence is. Simply smart people (say, over 115) apparently need to have their noses rubbed in it constantly through experience and until that point, they apparently think it’s fine to be feckless and shout their stupidity to the world.

If my post were about teaching low-ability kids (which, in this context, I take to mean kids who have great difficulty with mathematical abstractions and reasoning), then I might consider your point, despite your insulting tone and your implications about my IQ.

But my post was not about low-ability kids to begin with. Nor am I a fan of college for all.

Unfortunately, curricula such as Everyday Math treat ALL kids as though they were low-ability kids. The advanced ones survive those years and get into honors courses in high school. But the honors courses are still not as advanced or thorough as they could be, and there was all that time wasted along the way.

Kids with medium ability can understand induction. They can understand the textbook’s abstractions (which are presented very clearly). Why shouldn’t they get a chance to do so?

And then the ones who just don’t get it when it’s taught this way can be taught in other ways. Nothing wrong with that. That’s why I support flexible tracking in mathematics.

But there’s no real issue about mid-ability or high ability kids. You have no evidence they are doing poorly, a great deal of evidence that they are doing better, and absolutely nothing beyond an ill-formed notion of what scaffolding is to base your belief that scaffolding is to blame.

So if you’re not talking about low ability kids, then this is a non-issue. You’d have to start by proving that mid-ability kids were doing worse, and not only don’t you have that data, you have tons of data showing that they are taking far more math now than they were back in the day this book was published.

Since traditional math fell out of favor with the ed world, I think there’s been lots of tutoring (parents, tutors, Kumon etc) which has enabled math success for advantaged kids, who tend to be on the right side of the ability curve. When spiral curricula like Everyday Math are questioned, schools point to the successful kids without acknowledging the outside help. I know that was happening in Scarsdale, NY until parent pressure was applied to force a recent switch to Singapore Math. The director of the local Kumon program admitted that decision was likely to impact his business and local teachers were also being paid for private tutoring.

Right. There has been a huge cultural change in how middle class and upper-middle class parents approach homework over the past 30 years. When I was a kid, my had almost no involvement in my school work, unless I asked them. These days, middle class and upper-middle class parents are much more involved, there’s much more tutoring and enrichment activities, and the cultural consensus of materially comfortable parents is that they simply aren’t willing to sit back and let their children fail.

If anything, this change has partly been prompted by a lack of faith in the schools. For the schools to take credit for these kids’ achievements is ridiculous.

If we’re interested in maximizing achievement for

all students, then we can’t just focus on those who are doing poorly.We need to be maximizing, not thresholding.

Wow! Still no calibration of IQ with content and skills. Apparently, if kids do poorly, then it must be because of low IQ. Cal knows how to separate the variables, because, well, Cal just knows. These poor kids should consider having their IQs tested just to protect them from people like that.

Great post, Diana!

I, too, have math and Russian in my background, and found Kolmogorov’s real analysis textbook (written with Fomin) to be very useful during my undergrad analysis courses. In my Russian major, I focused on literary translation, so when I started raving about Kolmogorov & Fomin to my classmates, they begged me to translate it for them to use! Most of my second semester of analysis, I was learning primarily by translating the textbook for my study group. Lots of fun in the end. 🙂

Thank you, Chris! Great to hear about your experience.

I see what you did there 🙂

After changing schools this last year, I was assigned to teach Geometry. The Geometry textbook was crap. It had some useful stuff, but it was buried in lots and lots of fluff and it didn’t cover all of my state’s standards. I told the students to take it home and use it for reference. I pulled my lessons from just about everywhere but that book.

For the students it would have been much better to have a book that the teacher somewhat follows — to have that book to the point, and not so heavy that the student avoids carrying it around.

Too cool to find out that Kolmogorov wrote a math textbook! Can one get it in English anywhere?

Not that I know of, but it would be a worthy translation project.

Doesn’t matter how good a coach you are…if you don’t have the horses, the load won’t be carried. Your team will fail.

Low ability, low motivation: too dull to recognize the need for knowledge; or, just don’t care…you can’t make them drink.

Textbooks have very little to do with education.

The way I took this post (and the previous post) – was that students would have had the background knowledge to understand this book in the first place. Their previous courses would have led up to this book (with similar style instruction I assume). It wouldn’t be like dropping a class full of Everyday Math educated students in this!

While I haven’t done this kind of math in like… er… 20 years… my impression is one of a “Charlotte Mason” style textbook or “living book” if you will. The book was written by one person who really knew and loved his subject matter. And that would show in his writing.

This is very cool stuff. Thank you for sharing it with those of us who don’t speak/read Russian.