by Jeroen Bouterse
“Am I ever going to use this later?” As a math teacher, I seem to be getting this question about once a month (which is actually less frequently than I would have predicted). It is asked with varying degrees of openness to the idea that a satisfying reply is even conceivable, but almost invariably by students who are probably justified in believing that their tertiary education or future career is going to involve preciously few linear equations indeed.
Rather than trying to conjure up some practical situation in which one might need to solve a fractional equation, I usually suggest that math may be worthwhile even if it turns out you can safely forget the techniques you learned in school. Isn’t cracking this puzzle fun, doesn’t it make for good mental exercise, don’t you feel yourself getting a little bit smarter? Implicitly, I am banking on the idea that math improves your thinking. But what does that mean?
Recently, I stumbled upon a pamphlet from the 1920s that turned out to be both a feast of recognition and a source of further questions. Its author was Tatiana Ehrenfest-Afanassjewa. A Russian mathematician and physicist, Ehrenfest-Afanassjewa spent much of her time teaching mathematics and developing a program of mathematical pedagogy. Having worked in St Petersburg, she moved to the Netherlands in the 1910s when her husband, Paul Ehrenfest, became a professor of physics at Leiden University.
Ehrenfest-Afanassjewa’s Argument: Intuition and Logic
Her 1924 essay is titled “What can and should education in geometry bring to the non-mathematician?” That this question is relevant, is clear: it is not just struggling students and their parents who doubt the use of mathematics, Ehrenfest-Afanassjewa explains, but teachers as well. Few people will deny that spatial visualization and imagination are valuable skills, she says; but are those skills improved by a scholastic approach to them? – Is the painter or the hunter helped or hindered by the Pythagorean theorem?
However, she continues, we teach geometry not just to improve our students’ conceptions of spatial relations, but also to encourage them to think logically. And indeed, Ehrenfest-Afanassjewa says, geometry can do that – it can stimulate logical thinking, but only when it is taught in a certain way. “Education in geometry will only bear full fruit […] when we assign to intuition its proper place in thinking.” (5)
Ehrenfest-Afanassjewa goes on to develop a subtle but very interesting argument on the relation between intuition, geometry, and logic. Fully aware that ‘intuition’ is a rather vague term, she takes care to explain her intentions: she discerns a distinction between ‘seeing’ a feature of a geometric figure, and our ‘becoming conscious’ of it. Her use of the terminology of consciousness is somewhat confusing, but the thrust of her argument becomes abundantly clear: psychologically, when we are ready to formalize the results our thinking, this is only because we already understand ‘intuitively’ what we are going to do. We can prove a theorem because we have grasped it, not the other way round. Or, in Ehrenfest-Afanassjewa’s own words: “a proof phrased in syllogisms is a sign that one has sufficiently thought through the matter. It is false, however, that syllogisms are the very instrument of thought.” (8)
Now, Euclidean geometry, Ehrenfest-Afanassjewa notes, is at the same time a matter of spatial reasoning and a matter of formalization. Our knowledge of space gets enriched and clarified by treating it in an abstract and logical manner, but intuitive understanding remains the end-goal. Too many teachers err on the side of either logic or intuition: they assume that proving theorems is itself a way of increasing understanding, or they assume that their students have a fully developed spatial intuition from the outset. In fact, this intuition needs to be developed, which is not going to happen by forcing upon students a formal system of proofs. Euclidean proofs, after all, are the results of someone else’s thinking, and they do not as a rule replicate the process of thinking that leads to these results.
Very sensibly, given her argument so far, Ehrenfest-Afanassjewa proposes that a propaedeutic course in which students develop their spatial intuitions should precede a brief systematic course, which involves some proofs but only of those theorems that are not by now self-evident to all students in the group. Her propaedeutic course involves classroom activities such as estimating and then measuring the lengths of different objects, cutting and folding cardboard and paper models, drawing polyhedra, mentally extending lines going through certain points in the room, and a whole range of other physical and mental exercises.
Almost a century later, many math educators will recognize the merits of this approach: the teaching of abstract concepts should be grounded in more or less concrete, visual or even tactile experience. At the time, however, Ehrenfest-Afanassjewa’s ideas met with serious criticism from the Dutch historian of science and mathematics teacher Eduard Jan Dijksterhuis.
Dijksterhuis’ Criticism: Abstraction Over Intuition
Dijksterhuis’ point of contention is Ehrenfest-Afanassjewa’s core idea that intuition precedes proof: “she seems to assume silently that all axiomatics (taken to mean a science that delivers logical proofs based on foundational theorems) is a posteriori” (8). According to Dijksterhuis, axioms, theorems and deductive reasoning are not just the mathematicians’ equivalent of dotting the i’s and crossing the t’s; they are stepping stones towards actual understanding. Instruments that have nothing to do with imagination, such as effective notation or use of symbols, don’t merely help us write down our solutions; they help us find them. That is in itself a useful experience: “the conscious employment of these tools can help making a mathematical education fruitful for those who will never have something to do with mathematics later.” (22)
Ehrenfest-Afanassjewa’s distinction between spatial reasoning and axiomatization in Euclidean geometry, Dijksterhuis continues, seems to be her own invention; the tradition of Euclidean geometry does not in fact know such a distinction. “Now”, he admits, in a bout of rhetorical pessimism, “an appeal to tradition does not convince everyone. In our time, in which a lack of historical sense leads to a tendency to underrate those most classical instruments of intellectual education, ancient languages and math, it even often effects the opposite of its intentions.” (9)
Let’s pause here for a moment to see what the stakes are, according to Dijksterhuis. In his eyes, the value of an education in mathematics lies precisely in how it teaches us to think abstractly in a disciplined way – to ask what precisely we claim and why. This is a matter of thought more than a matter of imagination. “How, as a matter of fact, do the limitations of our imagination concern us?”, Dijksterhuis’s biographer cites from an (unrelated) letter he wrote around the same time (150). This penchant for abstract thought is what Dijksterhuis values in classical Greek mathematics. He would keep defending the value of a mathematical education on the classical gymnasium as well as the value of classical languages for students of mathematics and physics. Dijksterhuis’ opinions on the formative role of both mathematics and classics went against the spirit of the time, which was more inclined to distinguish between humanistic or literary education on the one hand and scientific education on the other, and to render unto each what was theirs.
For Dijksterhuis, mathematical thought has an eminently historical dimension, which is why it is so at home at the classical gymnasium. Together, mathematics and the classics open our eyes to the processes of scientific and mathematical thought in history. And again, thought is abstract: knowledge and insight are not synonymous with a capacity to imagine or visualize. “I […] confess”, Dijksterhuis dryly notes as to Ehrenfest-Afanassjewa’s proposed classroom practices, “that I do not consider myself capable in the slightest to draw a perspectival sketch of a regular icosahedron, but I am not yet convinced that I am therefore unsuited for my office.” (17)
As a matter of classroom practice, Dijksterhuis takes exception especially to Ehrenfest-Afanassjewa’s proposal that only those theorems require proof that aren’t already evident to everyone. Can’t our intuitions lead us astray, after all? What does the teacher do when the intuitions of her students conflict? What does she have to fall back on, if not previously proven theorems? Dijksterhuis is uncompromising in his demand that no stone in the abstract edifice of mathematics remain unsupported; that nothing is to be intuitively taken for granted, and everything proven.
If that sounds like an unreasonable or impracticable demand, then, well, that is sort of the point. The purity, honesty, order and discipline that mathematics demands simply goes further than the demands that we put upon each other in our daily life. That is precisely how mathematics education “bears fruit”. It trains its students, even the least mathematically inclined, to purify and discipline their thinking (12-13).
So Why Teach Math?
Dijksterhuis was not opposed to developing an intuitive understanding of spatial relations, nor was Ehrenfest-Afanassjewa aiming at intuitive understanding exclusively. Still, this was not merely a clash over classroom practices, but over educational goals and ideals – over why we teach math at all. While Ehrenfest-Afanassjewa’s pedagogical ideals may sound abstract and idealistic enough to us, to Dijksterhuis it was all too sensible. She ceded too much ground to the imagination at the cost of logical thinking, of the integrity of Euclidean geometry, and thereby of the proper place of mathematics in education.
On those matters where they differed, Ehrenfest-Afanassjewa looks remarkably ahead of her time. Dijksterhuis, however, deserves credit for fighting the times: his view on math in secondary school was informed by an ideal of a unified scientific and humanistic education. That is an ideal that deserves its champions.
What both have to offer, is a beginning of a clarification of what we mean when we say that math helps you think. Both are concerned about the ‘use’ of math for all students, but neither believes that the rationale for learning about mathematics lies in applying its concepts and techniques later in life. For both, mathematics is a formative subject. “Practice in thinking and in representing one’s insights would, on occasion, certainly provide a valuable service to everyone”, Ehrenfest-Afanassjewa says (6). Now I just need to convince my students she was right.
Klaas van Berkel, Dijksterhuis: Een biografie (Amsterdam: 1996).
E.J. Dijksterhuis, ‘Moet het meetkunde-onderwijs gewijzigd worden?’, Bijvoegsel op het Nieuw Tijdschrift voor Wiskunde I (1924-25) 1-26.
E.J. Dijksterhuis, ‘Problemen van het wiskunde-onderwijs’, Euclides 14 (1937-38) 99-118.
T. Ehrenfest-Afanassjewa, ‘Wat kan en moet het meetkunde-onderwijs aan een niet-wiskundige geven’, Paedagogische Voordrachten (1924) 3-27.