Guest Post: Beauty, even in the teaching of mathematics
This is a guest post by Manya Raman-Sundström.
If you talk to a mathematician about what she or he does, pretty soon it will surface that one reason for working those long hours on those difficult problems has to do with beauty.
Whatever we mean by that term, whether it is the way things hang together, or the sheer simplicity of a result found in a jungle of complexity, beauty – or aesthetics more generally—is often cited as one of the main rewards for the work, and in some cases the main motivating factor for doing this work. Indeed, the fact that a proof of known theorem can be published just because it is more elegant is one evidence of this fact.
Mathematics is beautiful. Any mathematician will tell you that. Then why is it that when we teach mathematics we tend not to bring out the beauty? We would consider it odd to teach music via scales and theory without ever giving children a chance to listen to a symphony. So why do we teach mathematics in bits and pieces without exposing students to the real thing, the full aesthetic experience?
Of course there are marvelous teachers out there who do manage to bring out the beauty and excitement and maybe even the depth of mathematics, but aesthetics is not something we tend to value at a curricular level. The new Common Core Standards that most US states have adopted as their curricular blueprint do not mention beauty as a goal. Neither do the curriculum guidelines of most countries, western or eastern (one exception is Korea).
Mathematics teaching is about achievement, not about aesthetic appreciation, a fact that test-makers are probably grateful for – can you imagine the makeover needed for the SAT if we started to try to measure aesthetic appreciation?
Why Does Beauty Matter?
First, it should be a bit troubling that our mathematics classrooms do not mirror practice. How can young people make wise decisions about whether they should continue to study mathematics if they have never really seen mathematics?
Second, to overlook the aesthetic components of mathematical thought might be to preventing our children from developing their intellectual capacities.
In the 1970s Seymour Papert , a well-known mathematician and educator, claimed that scientific thought consisted of three components: cognitive, affective, and aesthetic (for some discussion on aesthetics, see here).
At the time, research in education was almost entirely cognitive. In the last couple decades, the role of affect in thinking has become better understood, and now appears visibly in national curriculum documents. Enjoying mathematics, it turns out, is important for learning it. However, aesthetics is still largely overlooked.
Recently Nathalie Sinclair, of Simon Frasier University, has shown that children can develop aesthetic appreciation, even at a young age, somewhat analogously to mathematicians. But this kind of research is very far, currently, from making an impact on teaching on a broad scale.
Once one starts to take seriously the aesthetic nature of mathematics, one quickly meets some very tough (but quite interesting!) questions. What do we mean by beauty? How do we characterise it? Is beauty subjective, or objective (or neither? or both?) Is beauty something that can be taught, or does it just come to be experienced over time?
These questions, despite their allure, have not been fully explored. Several mathematicians (Hardy, Poincare, Rota) have speculated, but there is no definite answer even on the question of what characterizes beauty.
To see why these questions might be of interest to anyone but hard-core philosophers, let’s look at an example. Consider the famous question, answered supposedly by Gauss, of the sum of the first n integers. Think about your favorite proof of this. Probably the proof that did NOT come to your mind first was a proof by induction:
Prove that S(n) = 1 + 2 + 3 … + n = n (n+1) /2
S(k + 1) = S(k) + (k + 1)
= k(k + 1)/2 + 2(k + 1)/2
= k(k + 1)/2 + 2(k + 1)/2
= (k + 1)(k + 2)/2.
Now compare this proof to another well known one. I will give the picture and leave the details to you:
Does one of these strike you as nicer, or more explanatory, or perhaps even more beautiful than the other? My guess is that you will find the second one more appealing once you see that it is two sequences put together, giving an area of n (n+1), so S(n) = n (n+1)/2.
Note: another nice proof of this theorem, of course, is the one where S(n) is written both forwards and backwards and added. That proof also involves a visual component, as well as an algebraic one. See here for this and a few other proofs.
Beauty vs. Explanation
How often do we, as teachers, stop and think about the aesthetic merits of a proof? What is it, exactly, that makes the explanatory proof more attractive? In what way does the presentation of the proof make the key ideas accessible, and does this accessibility affect our sense of understanding, and what underpins the feeling that one has found exactly the right proof or exactly the right picture or exactly the right argument?
Beauty and explanation, while not obvious related (see here), might at least be bed-fellows. It may be the case that what lies at the bottom of explanation — a feeling of understanding, or a sense that one can ”see” what is going on — is also related to the aesthetic rewards we get when we find a particularly good solution.
Perhaps our minds are drawn to what is easiest to grasp, which brings us back to central questions of teaching and learning: how do we best present mathematics in a way that makes it understandable, clear, and perhaps even beautiful? These questions might all be related.
Workshop on Math Beauty
This March 10-12, 2014 in Umeå, Sweden, a group will gather to discuss this topic. Specifically, we will look at the question of whether mathematical beauty has anything to do with mathematical explanation. And if so, whether the two might have anything to do with visualization.
If this discussion peaks your interest at all, you are welcome to check out my blog on math beauty. There you will find a link to the workshop, with a fantastic lineup of philosophers, mathematicians, and mathematics educators who will come together to try to make some progress on these hard questions.
Thanks to Cathy, the always fabulous mathbabe, for letting me take up her space to share the news of this workshop (and perhaps get someone out there excited about this research area). Perhaps she, or you if you have read this far, would be willing to share your own favorite examples of beautiful mathematics. Some examples have already been collected here, please add yours.