Learning mathematics

This is mostly reminiscences about high school. I also give some opinions about how mathematics ought to be learned. The post originally formed one piece with my last article, “Limits.”

I learned calculus, and the epsilon-delta definition of limit, in Washington D.C., in my last two years at St Albans School, in a course taught by a peculiar fellow named Donald J. Brown. The first of these two years was officially called Precalculus Honors, but some time in that year, we started in on calculus proper.

We sat facing the windows. Blackboards filled the walls on either side of us. The text of our course was the notes that Mr Brown wrote out on these boards. He planned ultimately to type up the notes himself; but the priority in doing such typing went to the regular, non-honors precalculus course, and he was not finished there. Meanwhile, in the honors class, he wrote on the boards in his neat hand, and we copied the words and symbols onto the lined sheets of paper in our three-ring binders. We were required to reinforce the holes in these sheets with the little adhesive annuli that were marketed for this purpose. Indeed, Mr Brown inspected our binders to make sure that this reinforcement was done.

The course did begin with some typewritten notes from Mr Brown on logic. I have kept those notes. Consulting them now, I cannot quite corroborate a memory that I have, although the memory is plausible. I recall an early assignment, requiring us to negate the formula “p ⇒ q”. I was not alone in writing “q ⇒ p” as an answer. Those of us who did this were made to stand in front of the class, as an example of whom not to be like.

What makes the memory plausible is that negation is not introduced till the third page of the notes. The first page defines the kind of statement called an implication (“If p, then q”, or “p implies q”), along with its converse, inverse, and contrapositive. The next page discusses quantification: “For all x, P(x)” and “For some x, P(x)”. The third page begins:

IV. Negations. It is important to understand how correctly to deny a statement.

I. Implications. Since p ⇒ q means that q follows once p is given, to deny this is to assert that p is given, but q doesn’t follow. That is to say, p and not q. We write ∼ for “not”. Thus

∼(p ⇒ q) ≡ p and ∼q .

Mr Brown does not point out that denying and negating are the same thing. Also, this passage represents the first use in his notes of the symbol ≡. He does not explain the symbol. He introduces the notion of equivalence on the first page, where he says, “A statement and its contrapositive are equivalent; that is, if one is true, so is the other; if one is false, so is the other.” But the passage on negation above does not mention equivalence. So it does seem possible that a student might be mystified by the request to negate p ⇒ q.

Should Mr Brown’s notes have been written differently, to avoid this mystification? I myself would present the material differently now. In fact I have come to disapprove of the practice of presenting the logic of mathematics before any actual mathematics. The thirteen books of Euclid’s Elements are not prefaced by a Book Zero, discussing the logical forms of the ensuing propositions. Logic as a topic of study is older than Euclid; but the logical symbolism of today is a recent invention.

In what turns out to be an all-too-common practice, Mr Brown abused the arrow ⇒, using it not only for the verb “implies,” but also for the phrase “which implies” or “this implies.” For example, in one of the first weeks of precalculus, Mr Brown gave the following proof that a2ab + b2 ≥ 0 for all a and b.

By [earlier results], a2 + b2 ≥ 0. Now use trichotomy.

  1. If ab = 0, then a2ab + b2 = a2 + b2 ≥ 0
    a2ab + b2 ≥ 0.

  2. If ab < 0, then –ab > 0
    a2ab + b2 ≥ 0.

  3. If ab > 0, then 2ab > ab.
    By [a previous result] a2 + b2 ≥ 2ab > ab
    a2ab + b2 ≥ 0.

Hence in all cases, a2ab + b2 ≥ 0.

What did Mr Brown say aloud as he drew these three arrows on the board? It could have been “which implies”; it could have been “and so,” or “as a result,” or “consequently,” or simply “thus.” The bare verb “implies” would have been illogical. Therefore, in my view, an appropriate English expression ought to have been written out.

Thus I have some criticisms of the practice of this old teacher of mine. And yet it is not a bad thing if students can recognize the imperfections of their teachers. Also, by producing his own text for his course, Mr Brown showed that mathematics is not something handed down by God, or a publishing house, as holy scripture; mathematics is rather created by each of us as we come to understand it.

We students of Mr Brown got the hang of his logic. We reached limits a few months later. We memorized the epsilon-delta definition. Knowing what it meant if the limit of the function f at the point a was L, we learned how to derive mechanically what it meant if the limit of f at a was not L.

Meanwhile, we had to use the definition to prove some limits. The task did not make sense to me. I remember trying to mimic Mr Brown’s own examples; but I had no feeling for what we had to do. Looking back at his notes now, I see nothing missing. I see the sort of account that I might give now; and yet I recall being mystified when I first saw this subject. I actually wondered if I had reached my own limit in mathematics. I thought my classmates probably did understand what was going on. They seemed to understand so many other things better than I.

I was surely wrong here, given the evidence that developed later. In each of my last two years at school, I ended up being the sole recipient of the school’s mathematics prize. But some of my classmates were quite articulate in other matters, and this must have impressed me. When we graduated, it was remembered in our yearbook that, back in fifth grade, after memorizing fifty digits of π, I had been elected class president. My classmates had thus confused a facility in mathematics with general merit. Likewise, it was my mistake to confuse outspokenness with true authority.

That is a theme developed by Susan Cain in her recent book Quiet. When people act as if they know what they are talking about, too often we assume that they really do know.

In the eleventh grade (or what was called Form V at my Anglophile school), when we studied limits, I wondered if I had what it took to continue in mathematics. I got over this feeling. I would later credit Mr Brown with introducing me to real mathematics: mathematics in which no assertion was accepted without justification.

Our tenth-grade honors geometry course had been rigorous too. We wrote proofs. But this was in the two-column format, one column for “Statements,” and the other for “Reasons.” I cannot argue against this format as a pedagogical tool. But in retrospect I find something unsatisfying about not writing out a proof as English prose. This is what we would do with Mr Brown, albeit with the too-liberal use of the double-shafted arrow “⇒” that I mentioned above. While taking the geometry course, I discovered Euclid. I obtained my own copy of the Dover edition of Heath’s translation of the Elements. I wished we read Euclid’s prose in class, rather than our condescending textbook.

Mr Brown’s practice of writing his own textbooks is one I tend to adopt now. But Mr Brown did have us buy supplementary texts, for their exercises if nothing else. In precalculus, there were two Soviet texts:

  • G. Dorofeev, M. Potapov, and N. Rozov, Elementary Mathematics: Selected Topics and Problem Solving (Moscow: Mir, 1973),

  • A. Kutepov and A. Rubanov, Problem Book: Algebra and Elementary Functions (Moscow: Mir, 1978).

Nothing American met Mr Brown’s high standards here. In calculus, there were

  • S. L. Salas and Einar Hille, Calculus: One and Several Variables, Part I, third edition (New York: Wiley, 1987), for routine computations;

  • Michael Spivak, Calculus, second edition (Berkeley: Publish or Perish, 1980), for theory.

I appreciated Spivak for treating his readers as grown-ups.

Strunk and White’s Elements of Style contains chapter a called “An Approach to Style,” added by E. B. White to William Strunk’s original work. White says of his addition,

The chapter is addressed particularly to those who feel that English prose composition is not only a necessary skill but a sensible pursuit as well—a way to spend one’s days.

Michael Spivak presents mathematics as a way to spend one’s days. I credit him, and therefore Donald Brown, for showing me that it can be so.

And yet Mr Brown’s personal manner helped tell me that I did not want to spend my days doing only mathematics. Perhaps the whole school told me that I did not want to spend my college days in the same style of academic pursuit that was encouraged there. My most accomplished classmates went off to Ivy League schools. I went to St John’s College, where everybody read Euclid.

I have written about St John’s elsewhere, in an article [in pdf format] in the De Morgan Journal (vol. 2, no 2). I shall not say more about the College here. I also do not want to say too much more about Mr Brown, as if the details of his life with us students could really explain how I ended up where I am now. There is a sense in which he drove me away from mathematics; but mathematics itself drew me back.

In my last year of high school, somebody from the class before me returned to the school for a visit. He told me that Mr Brown’s course had discouraged him from pursuing mathematics; but now, at university, he was rediscovering his love for it.

Mr Brown encouraged us to gloat over our SAT scores and elite university acceptances. Unfortunately he is no longer here to explain himself: he died young. He surely would have admitted that mathematics was something to be pursued for its own sake. And yet he did get us involved in mathematics competitions with other schools. The sense I got of the subject resembled that of G. H. Hardy, who wrote in A Mathematician’s Apology:

I cannot remember ever having wanted to be anything but a mathematician. I suppose that it was always clear that my specific abilities lay that way, and it never occurred to me to question the verdict of my elders. I do not remember having felt, as a boy, any passion for mathematics, and such notions as I may have had of the career of a mathematician were far from noble. I thought of mathematics in terms of examinations and scholarships: I wanted to beat other boys, and this seemed to be the way in which I could do so most decisively.

I read Hardy’s Apology in high school, having seen it in the “Suggested Reading” chapter of Spivak’s book. I understood the pleasure of beating other boys and girls at mathematics. I like to think that I recognized implicitly that the pleasure was indeed ignoble; plus, it was inevitable that somebody would eventually beat me.

Even before high school, I had vague misgivings about the educational practice in which students sat at separate desks, all facing the same direction. As a high school senior, I did have a chemistry teacher who broke with this practice. She was called Mary Beth Key, and she decided to run our class as a seminar. Everybody was supposed come prepared for discussion, having done the necessary reading. Unfortunately the seminar was not entirely successful, because not all students did come prepared. They were too used to letting teachers do all of the talking. Meanwhile, I had learned about St John’s College, where all classes were run in the cooperative style, and this was the reason why students attended.

Now I have able to run some of my own classes in this style. In particular, in my university department now, all first-year students have a class in which they take turns going to the board and presenting the propositions of the first book of Euclid’s Elements. This would seem to represent the complete opposite of a class in which the textbook is written by the teacher. And yet, in this class, it is really the students who write the text, in the form of their own presentations, with the help of Euclid (and sometimes a vigilant teacher). The students have Euclid in the original Greek; but in almost every case, students will use the parallel translation—which is in Turkish, since we are in Istanbul. The translation is highly literal, and so it may not be very good as Turkish; but then the students will find their own words when they make their presentations.

If students are to get a sense of mathematical rigor, they do well to see the original model of this, a model that served to inspire the greatest mathematicians for over two thousand years. In the preface to the Disquisitiones Arithmeticae, Gauss remarks,

Included under the heading “Higher Arithmetic” are those topics which Euclid treated in Book VII ff. with the elegance and rigor customary among the ancients …

About 140 years later, Hardy observes (in §8 of his Apology),

The Greeks were the first mathematicians who are still “real” to us to-day. Oriental mathematics may be an interesting curiosity, but Greek mathematics is the real thing. The Greeks first spoke a language which modern mathematicians can understand; as Littlewood said to me once, they are not clever schoolboys or “scholarship candidates”, but “Fellows of another college”. So Greek mathematics is “permanent”, more permanent even than Greek literature. Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not.

Hardy here makes the Greeks sound like the Russians, as described by Palmer and Colton in the text used for the European History course of our last year of high school:

In the closing decades of the nineteenth century Russia became more than ever a part of European civilization. Almost overnight it presented Europe with great works of literature and music that Europeans could understand. The Russian novel became known throughout the Western world. All could read the novels of Tolstoy (1828–1910) without a feeling of strangeness; and if the characters of Turgenev (1818–1883) and of Dostoevski (1821–1881) behaved more queerly, the authors themselves were obviously within the great European cultural family.

Ancient Greek mathematics did not just happen to be akin to modern mathematics. The Russians and the Western Europeans had common cultural ancestors. Of course they shared a common humanity. But the ancient Greek mathematicians were the ancestors of the moderns, who learned mathematics from them. In his extensive note on Euclid’s definition of proportion, Heath remarks,

Certain it is that there is an exact correspondence, almost coincidence, between Euclid’s definition of equal ratios and the modern theory of irrationals due to Dedekind.

But Heath does not seem to observe that this coincidence in the logical sense is not a coincidence in the figurative sense of an accident. If Dedekind’s definition of irrational numbers, and real numbers in general, resembles Euclid’s definition of a ratio, it is presumably because Dedekind was inspired by reading Euclid. I am not aware that this is clearly documented, but Lucio Russo suggests it in The Forgotten Revolution:

Let’s not forget that the Elements were the textbook at the foundations of Weierstrass’s and Dedekind’s early mathematical education.

[Note added 2016.06.20:] Dedekind gives his theory of irrational numbers in “Continuity and Irrational Numbers,” but documents his relation to Euclid only in the preface to “The Nature and Meaning of Numbers.”

Regardless of what they read, I suggest that students ought to have a sense for what it is like to be a teacher, standing up in front of a room full of watchers and listeners. On a couple of occasions, people have told me that they had never thought much of teachers, until they themselves were required to instruct a group of people. Then they learned what a job it was.

In my own experience, it is certainly discouraging when students at the board do not get the point of the Euclidean proof that they are supposed to present. It is more discouraging when they pretend to understand: when they copy out the steps of Euclid in a polished manner, without understanding them. Moreover, a class can certainly cover a lot more material if the official teachers do all of the lecturing. But this coverage may well be illusory. If the students themselves cannot explain the material, then they have not really learned it. To learn is to learn to be a teacher.


  1. Susan Cain, Quiet: The Power of Introverts in a World That Can’t Stop Talking, 2012.

  2. G. Dorofeev, M. Potapov, & N. Rozov, Elementary Mathematics: Selected Topics and Problem Solving. Moscow: Mir Publishers, 1973. Translated from the Russian by George Yankovsky.

  3. Euclid, The Thirteen Books of Euclid’s Elements, Translated with introduction and commentary by Sir Thomas L. Heath, second edition, New York: Dover, 1956.

  4. G. H. Hardy, A Mathematician’s Apology, Cambridge University Press, 1967.

  5. A. Kutapov & A. Rubanov, Problem Book: Algebra and Elementary Functions. Moscow: Mir Publishers, 1978. Translated from the Russian by Leonid Levant.

  6. R. R. Palmer & Joel Colton, A History of the Modern World, fifth edition, New York: Alfred A. Knopf, 1978.

  7. Lucio Russo, The Forgotten Revolution: How Science Was Born in 300 BC and Why It Had to Be Reborn, with the collaboration of the translator, Silvio Levy; Berlin: Springer, 2004.

  8. S.L. Salas and Einar Hille, Calculus: One and Several Variables, with Analytic Geometry, Part One, third edition, New York: John Wiley & Sons, 1978.

  9. Michael Spivak, Calculus, second edition, Berkeley: Publish or Perish, 1980.

  10. William Strunk, Jr, & E. B. White, The Elements of Style, New York: Macmillan Paperbacks Edition, 1962; sixteenth printing, 1968.


  1. Posted May 14, 2013 at 6:42 pm | Permalink | Reply

    One of my previous lecturers on a module in mathematical logic told us that \Rightarrow had an informal meaning, just as you mentioned; he used \to for the strict “implies” used in logical formulae. Such a distinction was quite helpful for my understanding, at least.

    • Posted May 15, 2013 at 7:17 am | Permalink | Reply

      Hmm. Did the lecturer actually use the double-shafted arrow ⇒? In print, I have been using this arrow in formulas, so that it will not be confused with the single-shafted arrow →, which may be used in writing functions. I got this idea from a teaching colleague, who thought students might be confused by the use of the same arrow with two different meanings. However, if one is writing out propositional logic with a pen or chalk, the single-shafted arrow is easier to write!

      • Posted May 15, 2013 at 9:27 am | Permalink

        I think it’s really impossible to avoid having to use the same symbol in different contexts, and when it comes to functions and formulae, I’d argue that the context is sufficiently different, not making the use of \to confusing (in particular because colons aren’t used in logic anymore ever since they got replaced with parentheses, as far as I know).
        He used \Rightarrow aswell yes, whenever he e.g. talked about logical arguments in the English language or basically any other place than inside a formula. He also made sure the different meaning of the two arrows from the get-go.

2 Trackbacks

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