# Wonder as motivation

Like all mathematicians, I have a deep appreciation of the beauty of mathematics. Many theorems I find amazing even after I fully understand their proofs. (Example: Euler's formula, $V-E+F=2-2g$. That the genus $g$ can be determined by counting $V,E,F$ remains astounding to me.)

(Image from David Eppstein's webpage on Euler's theorem.)

In my teaching, I try to convey that wonder, and hope that my students start to see mathematics in the same light. But I wonder :-) if this approach can backfire, with students turned off by what they see as an alien aesthetic, shared only by math nerds.

Q. Have you encountered this reaction? If so, do you have strategies to mitigate it?

Certainly I do not consider it an option to hide my enthusiasm.

• We shouldn't have to mitigate who we are in teaching. I have no answer. I think sharing this feeling of awe is crucial to properly passing the torch to the next generation. – James S. Cook Mar 9 '15 at 14:19
• I think you risk alienating your students if you expect them to be in awe of mathematics. I suspect, however, that expressing your wonder or excitement in class can make a real positive impact on student-learning – David Steinberg Mar 10 '15 at 18:06
• Somewhat related: Check (or contrast with) the quotation from Fixx in MESE 1553 on puzzles. – Benjamin Dickman Mar 10 '15 at 18:18
• @BenjaminDickman: "Remember: you are an intellectual at play, not a missionary."---Thanks for the reference! – Joseph O'Rourke Mar 10 '15 at 18:20
• Perhaps you are doing your job if you turn some off by revealing the motivations of a mathematician. Part of an education is learning what doesn't appeal to you. – Tom Copeland Mar 11 '15 at 12:28

I have given this issue a lot of thought over the years -- in fact a large portion of my dissertation is devoted to related issues. I think of this challenge in terms of a "mathematical sensibility" -- a way (more precisely, a cluster of related ways) of appreciating and participating in mathematics -- and frame the question as "What is the role of school mathematics [and I would extend this to University level as well] in cultivating a mathematical sensibility?"

In order to get a handle on this question, I identified a number of distinct dispositions or 'categories of appreciation' -- ways that mathematicians use to label or attach value to individual mathematical results. These dispositions seem to appear in dialectical pairs; some of the ones I have described are:

• surprise / confirmation
• simplicity / complexity
• utility / abstraction
• theory-building / problem-solving
• formalism / realism

It's not at all unusual for mathematicians to refer to any and all of the above with the catch-all label "beauty". This is not really surprising; we are not trained as aestheticians, and consequently we aren't particularly skilled at teasing apart the different strands that evoke that aesthetic response. In the case of Euler's Theorem, I see in the OP mainly the disposition of surprise ("amazing... That the genus g can be determined by counting $V,E,F$ remains astounding to me"), with perhaps some implied appreciation also for the simplicity of the formula.

It is worth pointing out that sometimes we have much the same appreciation for mathematics that has exactly the opposite characteristics. The construction of a faithful complex representation of the monster group, for example, is astounding precisely for its complexity, and the proof of a long-standing open conjecture is appreciated for confirming what we already were pretty sure was true.

It is also worth pointing out that K-12 mathematics focuses almost exclusively on just one of these dispositions, namely, utility. In nearly all of students' mathematics education, the value of a piece of mathematics lies (if anywhere) solely in what it can be used for. This is why students ask "When are we ever going to need to know this?" If the only way you have of appreciating the value of something is in terms of its instrumentality, than something that lacks an obvious application is quite literally valueless.

So, why is it so hard to cultivate an appreciation for these values? Certainly part of it is personal proclivity; some people are more naturally inclined to appreciate these values than others. But I think a larger problem is that it is extraordinarily difficult to assess whether students are learning a mathematical sensibility. Without a way of assessing something, we can't hold students accountable for it; and instruction naturally defaults toward those things for which accountability can be maintained.

If that hypothesis is true, then another way of framing the question is: Is there a way to assess whether students are learning to appreciate mathematics? One of the things that I found in my dissertation was that standard forms of assessment (exams, quizzes, daily or weekly homework) seem to be ill-suited for that purpose, but other forms (specifically, open-ended research projects conducted over a longer timeframe) may provide an opportunity for students to get interested in things.

Besides my dissertation, I've written about these matters in a couple of places: an article for classroom teachers on 'Thinking Like a Mathematician' and a new article (just out this week!) on the theory-building disposition.

• Thanks for the links to your papers. I'll definitely read them. "Is there a way to assess whether students are learning to appreciate mathematics?" Good question! – Joseph O'Rourke Mar 11 '15 at 18:03
• The recent article reminds of a quotation from Paul Cohen about whether he sees himself as a problem solver: "Yes, I would say that. I’m not particularly proud of it though. I don’t think it’s a good thing to be, but I don’t think I’ve had much choice... I mean, it’s a somewhat, well, egotistical way of being. You know—you want to do one problem. There are other people who have a larger view of mathematics. I would regard it as a higher activity for someone to have a wider perspective from which many new ideas and interactions emerge..." – Benjamin Dickman Mar 11 '15 at 19:37
• There is, I should mention, a small misprint in an equation in the new ESM paper. (I mention this only because I wouldn't want people to think I don't know what Ceva's Theorem says.) :) – mweiss Mar 11 '15 at 23:52
• My favorite would be simplicity, abstraction, theory-building, formalism. I especially love various combinations of these. – PyRulez Apr 11 '15 at 1:59

This answer is from my experience running a Maths Learning Centre. I help students learn and use maths, mostly when they are struggling, and I also hear their opinions of their lecturers and other teachers.

I am very interested in all sorts of maths, and I will take almost any opportunity to be excited about a mathematical idea. For example, I made t-shirts with mathematical things on them which I wear every day in the Centre, and only need students to show the slightest interest in what the t-shirt is about to wax lyrical about it.

Do the students find my sense of fun and excitement with maths alien? Yes, most of them do. A lot of them shake their heads and simply can't understand how I can love it so much.

Do they find my sense of fun and excitement with maths alienating? No. Let me tell you why.

Firstly, students love it when their teachers show enthusiasm. They often praise teachers who are enthusiastic about the subject matter, even if they dislike every other aspect of their teaching. They want to be confident that the person who is teaching them knows what they are doing, and cares about what they are doing, and excitement with the subject matter is a good sign.

Secondly, I am not excited by only the "high and mighty" maths -- I am just as excited about the maths they are doing right now. I have a favourite fraction (3/8), I love that division has multiple interpretations, the classification of quadrilaterals is fascinating to me, and the fact that a set of points in the plane has an equation at all is the coolest thing ever. If I was only ever excited about things beyond what they were doing -- things they won't learn for years, if ever -- then they might get the impression that I think their problems are beneath me. Students have cited this very idea to me when they say they don't like some of their lecturers. They say they feel like the lecturer is great with their fancy cutting-edge research and is only teaching their class because they have to. Moreover, it's tricky to be inspired to wonder if you have to wait for a couple of years before you're allowed to feel it properly.

Thirdly, I sympathise with their experience. When they find it difficult to learn something, I tell them about how much I hated differential equations when I first learned it. When they find the lecturer's style of presentation boring, I can relate to all the times when I wished the lecturer gave more actual examples with actual numbers. And when they just want to get their assignment done, I remember what it was like with those pressures and work with them to get that bit further. Students prefer teachers who care about the students' struggles and consider what it is like to be a student. They hate it when the lecturer loves their subject but is "out of touch". A particularly relevant and hated brand of "out of touch" is when a lecturer has such a sense of wonder that they set assignment questions based on their mathematical coolness, without regard for the difficulty students have knowing what they hell they are supposed to achieve.

So in short, displaying your sense of wonder is a good thing, but you need to find wonder in the things the students are doing now, and you need to balance it with a healthy dose of understanding what it is like to be a student.

• I don't think it is off-topic here to ask -- why is your favorite fraction 3/8? – Chris Cunningham Mar 12 '15 at 14:49
• It is the number of litres of milk in every block of Cadbury chocolate, or the number of litres in a can of Coke. The fact that those common amounts have such a simple fraction, which isn't quite a half, appeals to me. But really, as with most favourites, I'm not entirely sure. It just feels nicer than other fractions. – DavidButlerUofA Mar 12 '15 at 18:39
• @DavidButlerUofA What would you say your least favorite number is? – PyRulez Apr 11 '15 at 2:07
• A very late response to this @ChrisCunningham, but I've just discovered something new about 3/8: Draw a regular hexagon and join the midpoint of every second side to make an equilateral triangle. Then the triangle is 3/8 of the area of the hexagon! – DavidButlerUofA Oct 21 '15 at 19:27

This spoke to me, enough that I decided to answer a related question.

I see in your question the drive to do better continually and reach everyone. Don't give up the struggle, or even relax much. Be sure to celebrate your accomplishments, know that you are reaching some, and use that as fuel to find, share, and improve upon ever-inspiring pictures. (It isn't an alien aesthetic. It is your aesthetic, the sharing of which will help people find and develop their own aesthetic.)

Gerhard "One Of Those Saluting You" Paseman, 2015.03.10

A student asked me to explain how to graph a hyperbola.

$\frac{\left(x+2\right)^2}{3^2}-\frac{\left(y-6\right)^2}{2^2}=1$

My first question was if she understood ellipses. And specifically the ellipse,

$\frac{\left(x+2\right)^2}{3^2}+\frac{\left(y-6\right)^2}{2^2}=1$

Drawing that ellipse will give us the rectangle needed to form the two asymptotes as well as easily see the hyperbola vertices.

For sake of completeness, the conjugate equation

$-\frac{\left(x+2\right)^2}{3^2}+\frac{\left(y-6\right)^2}{2^2}=1$

is also shown above.

The punchline is this student responded with, "that's so cool." She went on to ask why the teacher didn't present it this way. She (the teacher) hadn't make any connection between the ellipse and the hyperbola. For me, the reward comes from aiming to have as many "that's so cool" moments as I can. It won't be every topic I address, nor will every student react this way. But I keep my enthusiasm and cherish those moments.

On a lighter note - When asked to explain the nature of asymptotes, I explained that they think they are MC Hammer, always humming "can't touch this." I'm in this to have fun and need to inject humor like that when I see the opening.

• This is beautiful mathematics, @JoeTaxpayer! Thanks for sharing the story. – Joseph O'Rourke Mar 15 '15 at 22:20

Even if the students are turned off by your enthusiasm, they will be even more turned off by a bored teacher. So go with your enthusiasm, realise that some kids will react badly to it, but it is probably better than the alternatives.

Below I discuss some particular factors involved in the negative reaction towards enthusiasm for mathematics.

### Maths curricula are too crammed to develop aesthetics

I share your enthusiasm for Euler's formula. I managed to get some kids really interested in it, and they now share our wonder of it. This is how I did it:

• I taught kids who already loved maths
• I had unlimited time to spend on the topic, so that they could adequately explore the problem domain before we discovered the formula. Understanding the complexity of the problem is key to appreciating the simple relationship.
• After we had the formula we used it on a big project, finding all the possible Pythagorean and Archimedian solids and creating nets for most of them on Geogebra
• We spent time extending the relation in more general topological contexts (spherical/hyperbolic geometry)

I have also taught class room maths, and I have no idea how I would be able to do it in a way which communicates the aesthetic within the time constraints of a standard curriculum. Developing aesthetics takes time. Like art classes, we would need to give kids room to experiment and play around and develop their own responses to the aesthetic. We probably developed our own aesthetics in our own time rather than class time.

## Maths is uncool

The bigger problem is that enthusiasm for maths often seems to alienate kids. I think there are strong cultural factors at play here. Kids have been programmed into thinking that people who enjoy the mathematics aesthetic are uncool, and therefore listening to them is uncool. By loving/valuing the beauty of maths, you are demonstrating that your values are at odds with their values, and you lose their interest.

If have personally seen teachers over come this in a number of ways:

• subtly make fun of the class nerd. This shows that you have the same values as they do and you respond appropriately to those who transgress those values. You might even poke fun at other teachers who students may see as uncool, and point out how their subject is even more uncool than yours.
• cover the walls with pictures teens like, ie. half-naked young adults (this was a female art teacher - I don't know if a maths teacher could get away with this. And yes, it was super-creepy.)
• let them know you agree that it is unreasonable that they have to learn this uncool stuff, but here are some easy tricks to pass the test without really needing to understand it. This is a popular tactic for some tutors.
• turn a small topic into a big project that has very little maths content but which gives each student a sense of belonging which turns their lives around ... I admit, I've only ever seen this one in movies.

Those of us with a basic sense of ethics and professional responsibility don't have the flexibility to be really cool, so I guess we need to rely on our own relationships with the students. Hopefully some of them will forgive our geeky love of what we teach, and actually learn something or maybe even come to appreciate our perspective.

• I wouldn't necessarily think of it being 'uncool', just that not all students are interested which impacts their priorities in studying. Instead of spending time to study math, they try to maximize what their interested in. They don't want to spend the extra time developing their mathematical thinking. – Chris C Mar 11 '15 at 0:28
• @ChrisC I would like to agree with you. But I have seen too many kids movies where the maths kids are ostracised, or the maths kid gets his validation from being a sidekick to the cool kid. I have heard teachers make comments about the maths geek which they never made about the English nerd. I have seen parents specify that they want a non-maths specialist for maths tutoring for their child, but never heard someone ask for a maths teacher to tutor English. As many people think they are not good at maths, it becomes easier if they also believe that it is not worth being good at. – Richard Mar 11 '15 at 1:11
• I really appreciated the shocking conclusion of this. Well done. :) – Chris Cunningham Mar 12 '15 at 14:54
• A way to be hip without any of the negativity matheducators.stackexchange.com/questions/7619/… – Tom Copeland Mar 16 '15 at 23:42