(1) Elegance with Substance. Amsterdam: Dutch University Press, 2009. 112 p. ISBN 978-90-3610-138-7. PDF at http://mpra.ub.uni-muenchen.de/15676
(2) Conquest of the Plane. Using The Economics Pack, Applications of Mathematica, for a didactic primer on Analytic Geometry and Calculus. Scheveningen: Thomas Cool Consultancy & Econometrics. 238 p. ISBN 978-90-804774-6-9. PDF at http://www.dataweb.nl/~cool/Papers/COTP/Index.html
Elegance with Substance (2009) and Conquest of the Plane (2011) are two books by econometrician and mathematics teacher Thomas Colignatus, science name of Thomas Cool. In this review I shall refer to the books as EwS and CotP. Both are concerned with mathematics education (at high school level or first year of university for non-maths majors). EwS says that there is a whole lot wrong with how this is presently done, with itemization of 15 notational disasters, 5 disasters in nomenclature, 4 in the standard line of development, … The author has proposals for fixing most of these problems but all in all, this book will come across as somewhat destructive. CotP is then a good antidote: here the author actually gives us a high school mathematics text book in which he attempts to rise to the challenges detailed in EwS. You could call it a “proof of concept”.
Let me first report the author’s views about his works, and then proceed to discuss my findings on a fairly quick reading of the two books. First of all though I should tell you my conclusion: yes, these works are thought provoking. You will probably not agree with a lot of what the author says. However, teaching mathematics ought to be fun, and it ought to be thought provoking, both for students and teacher. When someone who is something of an outsider to the pure mathematics community comes up with a carefully and thoughtfully crafted alternative to all the old books and all the old methods, then whether or not the reader agrees with the author, I think he or she can expect to be both stimulated and entertained. I certainly was.
The author claims in EwS that “failure in mathematics and math education can be traced to a deep rooted tradition and culture in mathematics itself. Mathematicians are trained for abstract theory but when they teach then they meet with real life pupils and students. Didactics requires a mindset that is sensitive to empirical observation which is not what mathematicians are basically trained for.” And: “While school math should be clear and didactically effective, a closer look shows that it is cumbersome and illogical … What is called mathematics thus is not really mathematics. Pupils and students are psychologically tortured and withheld from proper mathematical insight and competence”.
The reader of NAW may find these unfair charicatures. My own opinion is that mathematicians are quite properly trained in what one might disparagingly call “abstract theory” but can equally well call “fundamental structures and patterns”. I like very much philosopher of science Imre Lakatos’ picture of mathematics as driven by empirical research into the structures which we see in our minds, and which reflect what we see in the real world. There have to be some people around in advanced societies who understand logical structures and who are able to operate at a high level of abstraction. Didactics is indeed a different matter. I am fond of the old British saying “those who can, do; those who can’t, teach; those who can do neither, teach teachers how to teach”. Thus I do not lay failures in mathematics teachers with the training of mathematicians. In fact, in my personal experience, the school teachers who fired my enthusiasm so much that I later became a mathematician were also teachers whose teaching was highly respected by those of my fellow pupils who later went on to become artists or bankers.
Anyway, this review should not be about my own prejudices. I just make these quotes so that the reader is fore-warned: come to EwS and CotP with a sense of humour.
Let me turn to the more constructive, and in my opinion quite fascinating CotP. The author here works out in detail his ideas for reforming the teaching of elementary calculus, trigonometry, and algebra. The book contains a large number of innovations. For this review I will concentrate on just one, which particularly caught my imagination, but there are plenty of others which also provide food for thought. Here is again a quote, now from the introduction to CotP. “Calculus can be developed with algebra and without the use of limits and infinitesimals. Define y / x as the outcome of division and y // x as the procedure of division. Using y // x with x possibly becoming zero will not be paradoxical when the paradoxical part has first been eliminated by algebraic simplification. The Weierstraß epsilon and delta and its Cauchy shorthand with limits are paradoxical since those exclude the zero values that are precisely the values of interest at the point where the limit is taken. Much of calculus might well do without the limit idea and it could be advantageous to see calculus as part of algebra rather than a separate subject. This is not just a didactic observation but an essential refoundation of calculus.”
The idea is to define differentiation through algebraic manipulations. We start with a “known function” f. “Known” means, a function which we have seen before. At this stage of CotP we have seen polynomials and trigonometric functions, multiplication and division and composition, absolute value and sign function, but no exponentials or logarithms. Trigonometrical functions, by the way, are defined through plane geometry; we are supposed to have a prior intuitive understanding of circles, length, …
We are also at this stage of the book accustomed to various processes of simplification (reduction) of complicated functions. The author’s proposal is to define differentiation by reduction of differential quotients (f(x+Dx)–f(x))/Dx [note to the typesetter: the D should be replaced throughout this passage by the Greek capital delta] followed by substitution of Dx=0. These steps include the parallel manipulation of the domain of Dx to warrant soundness: first exclusion of 0, then extension with 0, finally focussing on 0. He insists that this is different both from the “official” epsilon-delta definition and from an approach using infinitesimals. More importantly, he argues that this is a definition which you can give at high school level, thus giving a glimpse of the wholeness and beauty and logical rigour of mathematics: contrast with the hand-waving stories of approximating tangent lines to curves!
This could indeed be a valid definition on the presently stated domain. It certainly reproduces the “usual” derivative there, though already Colignatus needs more than just reduction and substitution in order to differentiate the trigonometric functions. In fact, he invokes a new principle to deal with them: if a given function is bounded from above and from below by two other functions, all three passing through the same point, and the two bounding functions have the same derivative at that point, then the given function has the same derivative at that point as the two bounding functions.
To pull off this trick on the trigonometrical functions he needs some more geometric insight as well as algebra.
Most high-school pupils are not going to ask the two questions: (1) Will I have to go on inventing new tricks every time I meet some new class of functions? And (2), Is there any guarantee that the result is going to be self-consistent? Most pupils trust their teachers and cannot conceive of functions different in nature from what they have seen so far: explicit recipes for piecewise smooth functions. They are living in the mid 19th century, even if their teachers have paid lip-service to late 19th and early 20th century set-theoretic concepts of function.
But readers of his book who are already professional mathematicians might well ask themselves those questions. On a little thought we will realise that we do know that the answers to questions (1) and (2) are essentially No, and Yes, respectively. But that is because we already know that so far Colignatus’s derivative coincides with the usual epsilon-delta derivative, where the latter exists. His sandwich trick now deals with all new functions, as long as they are twice differentiable at the point in question, using second order Taylor series to sandwich them locally, in the way which worked for differentiating the sine function. The same sandwich trick also shows that the result will always coincide with the usual derivative. I already mentioned that I suspect that as far as high school pupils are concerned, all functions (or all functions of interest) are piecewise smooth.
Later Colignatus defines the exponential function as being the unique function equal to its derivative (up to a constant factor), calling informally on Brouwer’s fixed-point theorem to justify its existence. However his presently available function space is already infinite dimensional, so the existence remains an intuitive matter only. The definition is amusing, but it makes me wonder if is possible to reproduce all of standard high-school calculus in a completely rigourous way without making use at all of epsilon-delta mathematics, just using algebra instead (perhaps together with some basic intuitive geometry). Has this been done before? If not, is it in principle possible?
Before set theory, which led to a “modern” notion of function as being an abstract set of ordered pairs (the graph of the function), mathematicians thought of a function as being some kind of formula: a procedure or recipe. Just as the derivative might, in a parallel universe, have come to be officially defined through infinitesimals, if only our conceptualization of the number line had taken an alternative route, could not differentiation, in yet another parallel universe, have been defined through applying algebraic operations to a convenient class of “known functions”, and only extended to larger classes of functions later, if and when required? After all, we can add rational numbers before we learn to add real numbers... So, I do not have a principled objection to Colignatus’s approach. Whether or not it would work in the class-room is not for me to judge, but it seems to me that there is a lot of sense, in imitating the historical development of mathematics as one passes through subsequent phases of teaching mathematics to children and young adults. I am reminded of philosopher and historian of mathematics Imre Lakatos' studies of the history of the calculus, in which he argues that it is largely by historical coincidence that our present-day “official” calculus took on the form it did. Some famous elementary mistakes by Cauchy and others were not mistakes if seen in the light of the then insufficiently formalized notions of number and function; they would have been correct theorems (according to Lakatos) if history had embraced non-standard calculus before standard, in other words, had formalized those notions differently. I have been brain-washed by my mathematics education to think that mathematics is about sets, but younger mathematicians tell me that this is limiting my imagination. It's about categories. I wonder if present day barriers in mathematical physics to establishing some clearly fruitful structural categories (speaking loosely) are merely an indication that wanting to see everything as a set is starting to be a hindrance, not a help? We need a formalism which legitimizes the formal manipulations which give the right answers (in physics, we have nature as ultimate authority). We need freedom from contradictions and inclusion of existing formalisms.
Could history have managed - for quite a while longer - with restricting calculus to what can be got with algebra and geometry alone? That might only be a fantasy, but still I find it an interesting fantasy. One thing I do object to in standard high school mathematics education is its tendency to adhere slavishly to the dogma of the day, which leads to a conservatism and dogmatism concerning how mathematics has to be done on the part of cautious souls. Die Wissenschaft von Heute is das Irrtum von Morgen. Mathematicians tend to think their discipline is immune to this phenomenon. That, if anything, is the culture of mathematics which I think is wrong in high school teaching. In my experience we learn by mistakes, we need to learn to discard ideas as we grow out of them, in mathematics as elsewhere. And mathematics is wonderfully alive and growing dynamically today … it is not frozen and engraved on tablets of stone. This is what is wrong in present day school mathematics education: no sense of wonder, no amazement, no notion that mathematics is a living part of living science. Just a tool to calculate how many rolls of wall-paper to buy when furnishing a new apartment. A necessary nuisance for future entrepreneurs and managers in the kennis, kunde, kassa society. (This is the present Dutch government’s sickening slogan on science policy: knowledge, skills, cash. No wonder the country is going through a crisis of identity). Thomas’ book could give school children a bit of that wonder back again, through his own reinvention of elementary calculus.
Finally I turn to my only big complaint with CotP, and it is quite petty, namely the author’s choice of a Mathematica package as companion to his books. Surely, the Sage project (http://www.sagemath.org/) has developed far enough to be able to support everything which Colignatus does. His ideas will only be taken on board when interested school teachers are able to try things out for themselves and their pupils. I doubt that most schools have a budget to allow all staff and students access to Mathematica. And what about the Third World? Now, Colignatus has worked a long time on developing Mathematica packages for teaching economics and clearly Mathematica has strongly influenced his thoughts on mathematics and teaching mathematics. And one can see this again as a “proof of concept”, namely to integrate teaching of mathematics with computer algebra, which again seems to me to be a sensible aim. In my own university teaching of probability and statistics (to mathematicians) I do my best to integrate the mathematics with the practice through use of professional but free computer tools (R, http://R-project.org).
So while talking about free software (“free”, both as in free lunch and in free speach) wouldn’t it be beautiful if someone would also create a free web-based version of CotP so that other authors can more easily build on his attempts? Let us see half a dozen alternative CotP’s. Take the writing and distribution of mathematics texts for schools back from the publishing houses, and give them back to the teachers and students themselves.
Richard Gill, Leiden, January 2012