Guest Post: Beauty, even in the teaching of mathematics
This is a guest post by Manya Raman-Sundström.
Mathematical Beauty
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.
Example
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.
mathbabe 
I think that synthetic division of polynomials is beautiful. I think the Heaviside cover-up method for finding partial fraction decompositions is beautiful. I think SOHCAHTOA is beautiful. I think derivative tables and integral tables are beautiful. I *do* teach what I think is beautiful. But I don’t expect my students to gasp in awe when I first show them these beautiful parts of mathematics. It takes, time and effort to appreciate the beauty. Many students experience this as tedium — so did I the first time I learned this. Later, I recognized the beauty, but not because somebody taught me the “right way”, rather because I had grown as a mathematician.
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Good morning Manya Raman-Sundström,
I have a somewhat funny and complicated relationship with math that I believe stems back to my experience in elementary school, and your post reminded me of one example. In the grade three my teacher (Mrs. Kay) gave us an in class assignment of adding up all the numbers between 1 and 100. Everyone sat down with a pencil and their work book and started going through the motions of performing the sums.
As I started doing the addition, I recognized that there were some interesting and useful patterns present, e.g. 1+9 = 10, 2+8 = 10, 3+7 = 10, 4+6= 10, which gave me 4 sums of 10, plus 5, which was 45, plus another 10, giving me 55. At that point I also realized that instead of summing up 1 through 100, I could instead sum up 0 through 100, which made that initial pattern recognition a little bit easier to work with (e.g 0+10 + 9 + 1 + 8 + 2 + 7 + 3 + 6 + 4 + 5). And then I realized I could apply the same pattern again at the scale of 0 to 100 and perform the entire sum very quickly because I had 50 sums that added up to 100, plus another 50, or 5050. So I wrote down 5050 in my notebook and took it up to the teacher. I finished the assignment before anyone else had finished the sums of the teens.
Then Mrs. Kay dressed me down in front of the entire classroom for ‘cheating’. I didn’t know why she thought I was cheating, the logic (at least as well as I was capable of describing when I was 7 years old) was right there in my notebook. In retrospect, it seems apparent to me that she didn’t understand the point of giving students that particular exercise, she had just heard it / seen it somewhere and decided to give it to our class as ‘busy work’. So when I finished the assignment in less than 5 minutes, her only explanation was that I must’ve cheated somehow. I was a particularly un-popular kid (as one might imagine), so I found this dressing down quite difficult and humiliating. I was already being teased nearly relentlessly by the other children. She just gave my classmates all sorts of cheap ammunition to continue humiliating me during recesses, and beyond that, I no longer felt that my teacher was a safe harbour.
What I learned that day was that my insights and intuition into math where ‘wrong’ and ‘bad’. This substantially contributed to my anxiety and dislike for math and I learned not to trust my intuition either (although she wasn’t the first or last teacher to do that, in grade 1 when we learned about subtraction I was scolded rather loudly for giving the answer -2 when asked to subtract 5 from 3. The teacher responded by saying ‘you can’t take 5 away from 3, if you have three apples, then you can’t give me five’, to which I responded by saying ‘of course I can, I just have to give them to you later’- that got me sent to the office for talking back to a teacher. I had similar experiences with fractions and remainders- remainders didn’t make any sense to me, but fractions did, but I digress). Through out the remainder of my elementary, middle school, and high school experiences in mathematics I learned that what I thought was the correct answer was probably wrong, which I found deeply confusing.
In retrospect, it is apparent to me that this happened because my teachers didn’t have the faintest understanding or appreciation for math. Appreciating the aesthetics of math, even if they hadn’t appreciated the cognitive aspects, would have made them better teachers.
I hope this post doesn’t make me come off as too bitter (although I am), I just think it’s a good example of the importance of teaching people to appreciate the aesthetics of math, even if they’re not interested in doing the work themselves.
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What I’ve found time and again is that students find mathematics most beautiful when they have spent a long time (30 minutes is enough, for most students) trying to solve a problem and fail, *and then* they see a beautiful proof presented.
That is, much of the beauty stems from how simple a problem is to state, how difficult a solution appears, and then the realization of how simply you can solve it in hindsight. It’s like a movie where the plot twist makes all the pieces fall together.
I’ve given guest lectures to high schoolers where I do this, and have received many “oohs” and “ahhs” (and even a standing ovation) for presenting even the simplest proofs! See my article here for a more detailed description of it: http://jeremykun.com/2011/06/26/teaching-mathematics-graph-theory/
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I agree completely with this. The problem for most of us (or at least me and the people I know) is that we’ve only got x amount of time to cover y topics. Yes, even in upper level undergrad courses. So while ‘finding the beauty in math’ is laudable, spending that extra thirty or so minutes on another proof of the same theorem is hard to justify.
On a similar note, I’ve found that students don’t like it when you pretend not to know the answer and step through the derivation while making an intentional mistake and then backtracking and fixing it as a demonstration of how this stuff is actually done.
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The time constraint is a huge issue. I think, though, that this is caused in part by bad standards/textbooks that invent useless content to memorize for the sake of filling pages. I think students would get a lot more out of the struggle of understanding a smaller number of problems.
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I think this is a very important issue lurking within math education.
Two people worth engaging are Gunther Ziegler (Freie Universitat) and Dierk Schleicher (Jacobs University). They can speak to this issue from the three perspectives of research mathematicians, people with a strong sense of aesthetic appreciation, and experience working with high school students.
I advise you to avoid spending too much time on defining mathematical beauty or identifying particular proofs or results as being beautiful. I’m not too worried as your post recognizes the ambiguity here. I also suggest you can sidestep the question of whether aesthetic appreciation is inherently valuable to teach (it is!) because the research suggests it is a strong tool for developing comprehension (so it will help test scores go up = yay! I guess).
What I suggest you take as a focal points are the questions of what tactics can be used to help students develop their own aesthetic sense, tactics that help math teachers deal with the fact that this sense could be sharply different from their own, and ways to address the fact that this whole discussion will be deeply counter to the culture of math education in most math classrooms on the planet.
My personal bias is that the answers are linked to thinking of math as games to play and puzzles to explore.
Last school year, I ran a math club at a high school. The students were enthusiastic, but not advanced. I had them play games with mathematical content, which is to say probability, strategic reasoning, inference, etc. You know: hearts, spades, bridge, poker, blackjack, some dice games, etc. Along the way, I would pose puzzles to them: if X happens in this game, what does it mean? Can Y ever happen in this other game?
How does that relate to aesthetics? I would ask them which games they preferred and why. Through the conversations, they realized that there were important structural differences between the games: the balance of strategic skill vs randomness (and associated preferences different kids had between analyzing probabilities vs thinking through game trees), differences in how much information was available to the players, and different levels of complexity depending on how many legal choices they had in how to play.
Another activity involved four “proofs” of the Pythagorean theorem:
(1) a collection of examples lengths for sides of a right triangle with the calculations showing a^2+b^2= c^2
(2) a proof that a triangle whose sides lengths satisfy the formula a^2+b^2= c^2, then it is a right triangle
(3) was one of the geometric proofs involving cutting and reshuffling a pattern of squares and triangles
(4) was Einstein’s “physics” proof using similar triangles and scaling areas.
Some of my questions to them:
– how convincing did they find these arguments? why?
– which helped them understand the theorem?
– which did they “like” the most? What about one appealed more than another?
– did they have any thoughts about the 4 taken as a group?
I got a lot out of their answers. It showed what they understood, what they didn’t and gave hints about how they were confused. I wasn’t sure how much they got out of the whole process, but they made it clear that this was very different from “learning math.”
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