Mathematically Correct; Pedagogically Questionable

One important aspect of the ongoing struggle over K-12 math education is over how much rigor is desirable. I’m among those who believe the typical U.S. high school goes way overboard in emphasizing rigor for the vast majority of students. The (college) sophomore calculus textbook I used as a student long ago has something to say about this that I think is well worth bearing in mind. This is from the Preface for the Teacher to A. W. Goodman’s 1969 Analytic Geometry and the Calculus, 2nd ed.  (I corrected a typo in Theorem R: the righthand side was printed as “f (u0) u’(x0)”.)

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…The student who is well prepared and who is interested in pure mathematics for its own sake may be able to understand and appreciate a rigorous course in the calculus. But the majority of students are still a little insecure in their algebra and trigonometry, and are far more interested in learning what the calculus can do and where it is going than in following a purely logical argument… I contend that one should not try to state all the hypotheses in a theorem, because the statement can become so long as to be incomprehensible to the average student… As an illustration, consider the following:

Theorem S.  If  y = f(u)  and  u = g(x) , then the derivative of the composite function y = f (g(x))  is given by

dy/dx = dy/du • du/dx

Here is a statement that is brief and simple, and the average student has a reasonable chance of understanding it. Now let is look at the same theorem when stated in a rigorous fashion.

Theorem R. Let f and g be two real-valued functions of a real variable and suppose that the range of g is a subset of the domain of f. Let  h = fg  be the composite function defined over the domain of g by setting  h(x) = f(g(x)) for each x in the domain of  g. If x0 is an interior point of the domain of g, and g is a differentiable function at x0, and if f is a differentiable function at u0 = g(x0), where u0 is an interior point of the domain of f, then h is a differentiable function at x0, and further the derivative is given by the formula

h’(x0) = f(u0) g’(x0)

There is no doubt that R is the correct statement and S is full of gaps. However, the average student can learn and use S, but when R is presented he will either fall asleep or totally ignore it. It is just too complicated for him to master at this stage of his mathematics study. The presentation of R rather than S does real harm because it serves to repel many students who are originally attracted to mathematics and who might turn out to be capable technicians or teachers (perhaps even creative mathematicians) if they are given a reasonable chance to develop.

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It’s interesting to compare the above with what appears in the 4th edition of Applied Calculus by Hughes-Hallett et al, a popular textbook these days for “brief surveys” of calculus for non-science students. Applied Calculus simply states the above theorem as the “Chain Rule”: it does not call it a theorem, nor does it provide any more than a vague intuitive argument for it. In fact, I don’t think it includes anything beyond handwaving arguments for anything. In other words, Applied Calculus is even less rigorous than Goodman’s Theorem S approach. Yet, based on my own experience, it’s all most college business majors (for example) can handle—and yes, an intuition for what calculus can do really is useful to them! The other extreme, as represented by Theorem R, is undoubtedly perfect for some students, but I’m sure Goodman is correct: it won’t do anything for “the average student” except convince them calculus isn’t worth the trouble.


There Are No Lazy Students (WIL #8)

How many lazy students have you known? How many stupid students? Maybe none. I’ve never known anyone, student or not, that I was convinced was lazy, and very few I was confident were stupid. Laziness sounds like a fundamental aspect of a person’s character. But how can you know that, especially about someone you’ve known only in the context of school? A much better way to think about someone that doesn’t want to work on whatever they’re supposed to be doing is just that they’re unmotivated. One very experienced teacher I made this argument to commented that “even the laziest people I have known were willing to work hard on things that interested them, and I suspect more than a few were depressed or discouraged.” It may not be possible to motivate students to do what you want them to do, but once you decide they’re lazy or dumb, you’re already most of the way to giving up on them. That’s an easy out for the teacher—too easy.

I think this is a really important point. As just one example, how about an exceptionally talented student who doesn’t work in class simply because the material is too easy and they’re bored stiff? This is important both because it may well be more common than educators realize (how can anyone know?) and because it’s particularly unfortunate to lose talented students for no good reason. Yes, little Ina Albertstein might truly be lazy or stupid, but probably not. As her teacher, it’s not likely you’ll ever know for sure, and assuming she is has far more potential to hurt than to help.

Baby’s First Infinite Series


Okay, okay, I admit this isn’t suitable material for babies, nor for any normal kid whose age can be expressed with one digit! The “baby” phrase just popped into my head and I found it too cute to resist. A more accurate title might be “Joanie & Johnny’s First Infinite Series”, or “Kids’ First Infinite Series”.

I’m always thinking about ways to get those apathetic middle- and high-school students interested in math again by showing them something wild and crazy, yet simple and concrete enough (for them, of course) that it shouldn’t be too difficult to understand. My latest idea is an infinite series with a surprising and very simple visual “proof”.  Can you guess what it is? Hint:  the “proof” is in each of the figures above (surprise, surprise). What it proves is…


I’m sure each of these informal proofs — really variations on a single proof  — has appeared in many places, but I ran across the one on the left first, in Roger Nelsen’s wonderful Proofs Without Words. (The one on the right is from the Wikipedia article “Geometric series”.) The two volumes of Proofs Without Words, published by MAA, contain dozens of marvelous and miraculous visual “proofs” on a wide variety of mathematical topics; I highly recommend browsing through them. Full disclosure, though: in my opinion, not many are as elegant or as easy to see as this one. Still, I believe that several of the proofs Nelsen has collected, certainly including this one, could be presented successfully on the middle-school level — though probably not with the above notation. Regardless of notation, of course, appreciating this proof requires some ability to do arithmetic with fractions.

I think most people with experience teaching high-school math would agree that (1) not many “apathetic” secondary-school students have much of an idea what a proof is (notwithstanding the emphasis on them in a typical geometry course), and (2) hardly any of them know why they or anyone should care! That may be a problem with my idea, but it strikes me more as an opportunity. I suspect that going through something like this proof would help with both problems; it would also expose students to the important idea of adding up infinitely many numbers and getting a finite sum.

Do you agree? Do you this could work (with a reasonable amount of scaffolding, naturally) as a discovery lesson for, say, 10th graders? If you think it’s worth trying, here’s an idea for “reasonable scaffolding”: Discuss the fact that 1/3 = 0.33333333…, with infinitely many 3’s, and ask the class what 0.33333333… with infinitely many 3’s actually means. Of course it’s a very simple infinite series, one with the same sum as the series I’ve been discussing:

InfiniteSumFormula2Since both series have the same sum, an obvious question might be whether it’s possible to draw a picture to illustrate the above equation, and if so, what it would look like. I leave those questions as an exercise for the reader.

Gamow, Bread Rationing, and the Normal Distribution

"angelic" baker & suspicious professorYou might be familiar with George Gamow, the mid-20th-century physicist who, with his student Ralph Alpher, came up with the Big Bang Theory long before there was any experimental evidence. Gamow is also the best writer for the layperson on science, and one of the best on math, I’ve ever read. In particular, his book One, Two, Three… Infinity is a masterpiece, jammed with fascinating ideas presented with absolute clarity. (Though much of it — mostly the non-math stuff — is kind of out-of-date now; the revised edition came out in 1961.)

A few years back, my friend Doug Hofstadter sent me a short article of Gamow’s about how the normal distribution was once used to expose a dishonest baker — apparently a true story. But true or not, it’s a fascinating story of mathematical probability in real life, one that I think would interest even a lot of apathetic middle-school and high-school students! A PDF of the story is available at


Terms, Notations, and (Mostly Needless) Confusions

I wanted to call this “Logician General’s Warning: Confusion about Terminology is Hazardous to Your Understanding”, but it takes too much space…


Needless Confusion Over Terminology and Notation

A friend of mine who has a degree in statistics commented a few years ago that he couldn’t understand why people were confused about the terms random variable, probabilistic variable, and stochastic variable; after all, they all mean the same thing. I instantly realized that I myself had been confused because I didn’t know that. Or, quite likely, I once knew but had totally forgotten! I’ve seen confusion — usually needless confusion — over terminology cause serious problems many times, both inside and outside the classroom.

And while I’m talking about probabilistic things, how about Bernoulli “processes”, Markov “chains”, and Hidden Markov “models”? In my experience, those are the usual terms for the three phenomena; but they’re all “processes”!

The same thing happens with notation. I was guest-teaching a lesson on Zeno’s paradox of Achilles and the Tortoise to a high-school math “exploration” class (see my post about it, As an example of a convergent infinite series, I wrote on the board


A lot of students had trouble with the 1/(2^n) part until their regular teacher pointed out that it means the same thing as (1/2)^n — a more familiar notation to them. And I probably would have used the latter form, if it had even occurred to me it might make a difference 😦 .

Hard-to-Avoid Confusion Over Terminology and Notation

How many students confuse quadratic expressions, quadratic equations, and quadratic functions? Many of my own students certainly did, but at least the terms are as consistent as possible. I’d say the situation with the two common notations for derivatives — dy/dx and y’ — is somewhere between “Needless” and “Hard-to-Avoid”. There’s some justification for both notations, but I wonder if it’s worth it.

It’s vitally important that students understand and remember the terms and notation we throw at them. If they’re mechanically following rules but they confuse widgets and wodgets, they’re dead; even if they’re really going for understanding, confusion about terms and notation can waste a lot of their time, and ours.

Early Number Sense and Success in Math

For the last few months, I’ve been teaching Algebra II to a class of one, a high-school senior who’s struggled with math for years. After spending two or three hours a week together for 11 weeks, I’m happy to report that Ellie (not her real name) has unlearned a lot of wrong ideas and learned enough right ones to get an A- on her first test (factoring quadratics, solving quadratic equations by factoring, etc.). Ellie told me the last time she felt she understood math in school was 6th grade and she hasn’t gotten better than a C or D (she’s not sure which) on a test since then. So she was happy with the A-, too!

I’ve always wondered how strong the connection is between number sense and learning math. The other day I mentioned to Ellie the recent news item (widely reprinted on the Web, e.g., about University of Missouri research showing kids with poor number sense in primary grades have trouble with math years later. She responded that she remembers struggling with concepts like adding 4 and 4 in 1st grade — unless she used her fingers so she could see what the numbers meant. Yep.

Be realistic about your strengths and weaknesses and adapt! (WIL #7)

It’s important to be realistic about your strengths and weaknesses and adapt accordingly. An important special case: Grading homework and exams can easily overwhelm the teacher. My brief tenure as a regular, full-time classroom teacher was as a maternity-leave replacement, and by the time I started, I knew I was a slow grader. I had four (count ’em) preps, and getting four lessons ready every day took so much time that I quickly realized that I’d have very little time to grade homework. From the beginning I didn’t try to grade as much homework as the very experienced teacher I was filling in for did. But I couldn’t even do what I thought (and had told my students) I could. It wasn’t until I graded the final exam that I realized how slow I was! I can see several reasons for this: wanting to give students really useful feedback, wanting to grade as consistently as possible, my inexperience, etc. If I teach much more, I’m sure I’ll get more efficient, but not by so much that it won’t always be an issue. But — as Ofer Levy pointed out to me later — the teacher usually has quite a bit of flexibility in how much grading they have students generate. One reason is that there are often ways to substantially reduce the load of grading without harming learning, so the teacher can avoid being a victim of a system they created themselves. Some ways that seem appropriate for teaching math, along with many other subjects: assign group instead of individual work; let students correct each other’s work; go over homework and quizzes in class but don’t collect or grade them. I did some of this, but could have done much more… and if I had, not only would I have been better off, my students would have been too because I would have had more time for their more serious problems.