When did Mathematical Thinking start?

To people whose experience of mathematics does not extend far, if at all, beyond the high school math class, I think it’s actually close to impossible for them to really grasp what mathematical thinking is.


If high school math covers algebra, geometry, analytic geometry, functions and calculus, then it covers parts of math from antiquity up to some in the 19th century.

If we consider pre-19th century algebra, analytic geometry and calculus as procedural and not conceptual, then we would exclude Euler, Newton, Fermat, Lagrange, etc. from thinking mathematically.

Certainly Euclid is excluded.High school math geometry texts are considered more rigorous than Euclid.

Even if students find the epsilon delta definition difficulty, they do read it, and Newton did not.

Set theory including ordered pairs, functions and relations are covered in a way that is 19th century even if only some of it is covered.   This was certainly not known to the 17th century.

So Euler, Newton, Fermat and Largrange did not only not do mathematical thinking, they could not even grasp what it is.

And a high school physics course that covers a little relativity?  That isn’t physics thinking either.


About New Math Done Right

Author of Pre-Algebra New Math Done Right Peano Axioms. A below college level self study book on the Peano Axioms and proofs of the associative and commutative laws of addition. President of Mathematical Finance Company. Provides economic scenario generators to financial institutions.
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5 Responses to When did Mathematical Thinking start?

  1. Michael Paul Goldenberg says:

    You asked me to take care in my choice of language, so I’ll do you that service. I wish you were a bit more careful in your choice of windmills at which to tilt. I believe you’re completely missing Devlin’s point. And so now you’re going to launch a fleet of posts in which you “prove” both that various experts (the ETS, Stanford, etc.) agree with you AND in which you show that if Devlin were right (about something I seriously doubt he is saying to begin with), then it follows that a bunch of top-flight mathematicians were obviously not doing mathematical reasoning. And hence, Devlin’s wrong, you’re right, and nyah, nyah, nyah, nyah.

    The problem is that Devlin’s not saying what you claim he’s saying. If you want to argue with people who actually believe some similarly insupportable ideas, DO join that math-teach@mathforum.org list. It’s got some remarkably hard-headed elitists and you can REALLY have fun screaming at them. I did so for way too long and gave it up.

    What Devlin is saying, on my view, is indisputable: very few people coming out of a typical US K-12 mathematics education will enter college with much idea of what mathematical reasoning entails. They will know little or nothing of higher mathematics. If you read the blog piece by Mathbabe that Devlin linked to today, you’ll see ample evidence that he’s not alone in seeing that lots of folks who do extremely well in K-12 math come to places like Harvard and Stanford with little or no clue what it means to do mathematics. They’ve studied a shadow of the work of the great mathematicians you mention. They certainly haven’t any idea how any of that work was developed. Thus, one of the common questions many people (myself included) wind up asking, if we ever bother to stop to reflect, is, “How did someone ever think of that to begin with?”

    Now, we might want to debate whether it’s possible to teach the K-12 curriculum in ways that entail mathematical reasoning. I believe it’s glaringly obvious that it can be and that it has been. Just not on any sort of consistent or wide-spread basis. Try looking at Robert B. Davis’ work with the Madison Project in the early 1960s (tons of info free online and you can download the book they used (DISCOVERY MATHEMATICS: A Text for Teachers) free. It has a stunningly beautiful overview by Davis and contains all the problems elementary students were given (as a supplement to their regular math work) and the teacher notes. Published in 1964. Probably not the first time anything like that was done, certainly not the last.

    I don’t know if Keith Devlin’s seen it. I only found it myself recently and got a hard copy before seeing that it could be downloaded as a pdf. But I already knew of people doing wonderful work with kids that entailed mathematical reasoning and I bet Keith Devlin does, too. It simply isn’t standard fare. And hence, for the vast majority of American kids, what he said is 100% correct. Therein lies one of the true challenges. Particularly given how opposed to such things some very influential mathematicians have been. None of whom are named Keith Devlin.

    My advice, given freely: give this one a rest. If you want to find bad guys to beat up, they’re out there. But you’re barking up the wrong tree.

    • You really think that the quote above is 1) accurate 2) inspiring 3) informative for students coming out of high school?

      I think the making of math as so mysterious is not accurate, is not inspiring and is not informative to students. I don’t see high school math as being so out of what mathematical reasoning is. Nor do I see procedural math as being so bad.

      The point of symbolic logic was to reduce logic to procedure. Recursive math is procedural by one argument, and recursive functions cover a large part of what most people think of as math.

      High school geometry does cover proofs. Those proofs and derivations of formulas like the quadratic formula are mathematical reasoning.

      If the quadratic formula seems hard in high school, going from the J Bessel function to N or Y Bessel function is a hard tricky derivation. Few people can do it without looking at a text to follow. There is always some hard but important derivation most people can’t do, even though they use Bessel functions in their work. You can publish a paper with a new math formula using a Bessel function and not know how to do most Bessel function derivations. This is typical in fact.

      People are said to sleep walk through their discoveries and this might be a more informative guide to students. The truth is that math understanding is a constant mixture of skills, memory, and uncertainty. It is not polished perfection for the most part. So I don’t think the dismissal of high school math as not thinking is accurate or a good guide.

    • I was unable to find the link from Devlin to mathbabe. I searched inside the source of his webpage that I link to above and that did not find it.

    • You raise a good point on development and age level. However, at what age people learn to not leave out steps of proofs or engage in critical thinking to understand why they are critical is different then Devlin’s categorical statements. Devlin’s statements were not directed at an age level, but all persons whose math stopped in high school.

      This also suggests that letting students place out of math in college is a mistake. It also suggests that courses like Harvard Math 55 are a mistake.

      Devlin’s two introductory sentences now reclassified as a joke following your suggestion also were not age directed. He didn’t write if you are 18 and can’t follow these two sentences without difficulty, he said anyone who can’t follow them. That is different than saying age level is linked to ability to spot missing statements in a proof.

      For reference his two statements (now officially a joke)
      =Devlin’s Joke

      What is mathematical thinking, is it the same as doing mathematics, if it is not, is it important, and if it is different from doing math and important, then why is it important? The answers are, in order, (1) I’ll tell you, (2) no, (3) yes, and (4) I’ll give you an example that concerns the safety of the nation.

      If you had any difficulty following that first paragraph (only two
      sentences, each of pretty average length), then you are not a good mathematical thinker.

      =End excerpt Devlin’s joke

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