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?

Math teachers at work and play

In blog carnival news, Math Teachers at Play is up at mathrecreation.

The Carnival of Math is making a come back, thanks to Walking Randomly and other math bloggers.

A delusion of rigor in math

U.S. high school students are taking more advanced math classes and earning higher grades, but math achievement hasn’t improved, writes Mark Schneider, a visiting scholar at AEI and a vice president at American Institutes for Research.

More of our high school students are getting through Algebra II and calculus, while fewer and fewer of them are stopping at general math and Algebra I. And transcript data show that even as they take more difficult courses, they are earning higher grades.

. . . while the math skills of elementary and middle school students entering high schools have improved, what American high school students know and what they can do in math have barely changed over the course of thirty years and not at all over the last fifteen. And when we step outside the United States to compare our high school students to students in other advanced industrial countries that are our peers and our competitors, the picture is also grim.

The new NAEP scores for math achievement will be out next week.

In the second year of the American Diploma Project‘s multi-state end-of-course exams for Algebra II, 80 percent of students were judged not prepared for entry-level college math.

"Standards are not curriculum"

The revised Common Core standards are ready for review. The NGA clarifies in its release that “standards are not curriculum.” Robert Pondiscio comments that “it’s good to see a measure of clarity” about the distinction between the two.

These standards do look clearer than the previous version, although, as before, the math is more specific than the English. Chester E. Finn, Jr., at Flypaper comments:

We’re still reviewing the latest version but at first glance it appears that the math standards, while not perfect, have a lot going for them. The “English” standards are harder to appraise. They’re not actually English standards, but, rather, standards for reading, writing, speaking and listening. The drafters acknowledge that they would need to be accompanied by solid curriculum content, and they’ve provided a handful of examples—good ones, mostly—of such content. But they’ve also left most of the heavy lifting to states, districts, schools and educators. That’s not necessarily a bad thing. But it also means that the “common core” standards, at least in this version, are more a vessel waiting to be filled with curriculum than an actual framework for what teachers should teach and students should learn.

Yet even with the relative vagueness of the English standards, they have more substance than some of the state ELA standards I have seen. Here’s what the standards say about the quality of reading material:

The literary and informational texts chosen or study should be rich in content and in a variety of disciplines. All students should have access to and grapple with works of exceptional craft and thought both for the insights those works offer and as models for students’ own thinking and writing. These texts should include classic works that have broad resonance and are alluded to and quoted often, such as influential political documents, foundational literary works, and seminal historical and scientific texts. Texts should also be selected from among the best contemporary fiction and nonfiction and from a diverse range of authors and perspectives.

I looked at the illustrative texts and the commentary. I have some minor quibbles, but all in all they look fine. My main concern is that English class would turn into “a little bit of everything.” There should be literature class, and then there should be extensive reading and writing in the other subjects.

The math standards look promising, though the illustrative examples seem a bit on the easy side. Also, I am not sure why they avoided organizing the material around areas of mathematics such as geometry, algebra, linear algebra, calculus. Only probability and statistics get their own categories. Otherwise the material is organized around general skills and concepts. Why?

Innumerate, unemployable

A British haberdashery hires only applicants who can do mental math, reports the Daily Mail,  Most fail.

Colin Bamberger, 82, whose parents founded the Remnant Shop in 1944, said that less than one in ten applicants are now able to solve basic maths problems without turning to a calculator or till.

In the past, around eight in ten made the grade.

. . . He said: ‘Most of the youngsters who come to us for jobs are unemployable because they are not numerate.

. . . ‘It is all very well using calculators but if you have not got some idea what the answer is, how do you know if you have pushed the right button? It’s so easy to make a mistake.

Bamberger blames poor teaching and over-reliance on calculators.

As easy as 1, 2, 3

Preschoolers should learn their 1-2-3’s as well as their ABC’s, concludes the National Research Council. From NBC:

The National Research Council finds kids ages 3 to 6 are already learning numbers and geometry through everyday experiences.

“When we’re going outside we’re lining up and then we’re all gonna count. Count how many friends we have,“ teacher Anuschka Boekhoudt said.

“They’re learning addition and subtraction but they don’t really realize it you know. It’s just, it’s fun for them,“ Helling said.

Kids are ready to learn the report says. It’s preschool teachers who need more math training.

There are fun ways to introduce math before children decide it’s scary or hard, researchers say.

Calculators? Don’t answer

At a training session for teachers in Bellevue, Washington, an elementary teacher asks a reform math consultant what to tell parents who ask whether use of calculators will hinder children’s computational skills. Here’s a video of Phil Daro, co-director of Berkeley’s Tools for Change, telling teachers to dodge the question.  (He’s preceded by Uri Treisman of the Dana Center at the University of Texas.)  Cal State-LA Math Professor Wayne Bishop, writing on Math Forum, provides a transcript of Daro’s answer:

. . .  it’s part of the math wars. The best advice is, Don’t answer that question. You are being asked to fight a battle on a hill that has been custom made to turn you into a fool. And there’s no way to win. So basically, the general advice I give in the math wars is Advice 1. You have to realize that their strategy is to attack you, not your ideas and
they’re going to fool you by making you think they are attacking your ideas.

The first thing you do is you stand up and identify yourself to this
audience of worried and frightened parents. Tell them who you are and say I believe that all students should be able to add, subtract,
multiply, and divide without calculators. That’s the first thing you
say when the calculator issue comes up. And everything after that
when they say “calculators”, you say “technology”. If they ask about
calculators, you say, “Well, technology is important but it’s no
substitute for mathematics.”

In my experience, many parents and teachers believe introducing calculators in early elementary school hinders students’ mastery of computation, weakens their “math sense” and makes them permanently dependent.  Others think there are ways to introduce calculators without letting them become a crutch. This really is about ideas, not personalities. The question deserves an honest and complete answer.

Learning to love skills-free math

On Kitchen Table Math, Barry Garelick quotes from a 2006 report on federally funded training in “standards-based” math teaching, which Garelick defines as “how to teach the crap programs that NSF’s Education and Human Resource Division funded (like Everyday Math, Investigations, IMP, CMP, Core Plus, etc).”

The report lauds “changes in teachers’ beliefs” about the need for ability grouping.

“Before IMP, I felt that there were mathematically unreachable students. I felt that students could not go on to more challenging ideas like algebra, statistics, probability, or trig without basic skills. Fortunately, with my IMP training, I have a different feeling about students. I strongly believe in access to mathematics for all. (Teacher, 6–12 mathematics)”

Garelick writes:

Before this teacher started using IMP, he/she felt that basic skills were necessary in order to proceed in mathematics. After IMP, which essentially avoids content whenever possible, he/she saw the light. Yes, wonderful things happen when you pretend that content doesn’t matter, and that higher order thinking skills occur just by giving students “authentic” problems without the bother of all those and boring drills and instruction. They are able to reach for the stars. Unfortunately they do so by standing on a two legged stool.

After many years working in science, Garelick is preparing for a second career as a math teacher.