December 17, 2012

Is math beautiful?

7 Reasons to Love Math: A Q&A with Dr. Nathalie Sinclair

In a recent interview with CBC's Radio Canada, Dr. Nathalie Sinclair, SFU Education professor and Canada Research Chair in Tangible Mathematics Learning, reveals the beautiful relationship between math and everyday life. You can listen to the interview conducted entirely in French here.

1. Why do students (and often their parents) have a fear of math?

Actually, until they reach a certain grade level at school, most children do not fear mathematics--indeed, they quite like it. But there comes a point in their schooling where they become, let's say, un-aestheticised, either because mathematics turns into something that is overwhelmingly about memorization and performing procedures, or because they don't see anything interesting, surprising, wonderful or valuable in it. No one would take music lessons if it were all about learning and practicing scales; no one would want to play soccer if it was all about drills. The fear comes about through a vicious circle of getting a wrong answer or getting it too slowly, then losing confidence (and interest), which compromises thinking, which leads to being more slow or getting more answers wrong. Then, instead of saying "I made a mistake" students start to say "I'm bad at math," which is an identity statement that gets ratified in popular culture and becomes hard to shake. I think the fear comes from not feeling like one has any resources on which to call upon in order to solve a problem—it's about feeling that you always have a 50% chance of being wrong.

2. What are the consequences of emphasizing numbers and equations over geometry?

I see three major consequences.

  • One is related to the goals of education, which I think include developing the strengths that young learners already have before they start formal schooling. Research shows that young learners come to school with an enormous amout of implicit or informal understanding of shape and space, including, for example, a strong sense of symmetry. Studying geometry would help develop these understandings, which could be used in a wide range of everyday as well as school-based activities.
  • The second consequence is about the role of geometry in learning mathematics more broadly. Spatial reasoning has been shown to correlate with better mathematical understanding and better problem solving abilities. Doing geometry can be a powerful way of supporting students' overall mathematical understanding, including their understanding of numbers and algebra.
  • This leads to the third consequence—geometry doesn't have to be seen simply in such utilitarian terms. It is a domain in mathematics in its own right. If you aren't learning geometry, you aren't doing mathematics. Indeed, all new breakthroughs in mathematics have had an important geometric component—this is especially true for the mathematics that is studied in school. Further, for many people, geometry provides the most accessible and striking source of mathematical beauty and wonder.

3. How can we connect math closer to everyday life?

Connecting mathematics to everyday life can be very valuable, whether it's to help students see how mathematics does get used in a variety of activities with which they are familiar (and not just buying groceries or choosing between cell phone plans, but also in professional activities such as computer animation, medical imaging and financial modelling), or to support students in being able to identify situations in which mathematics can help solve a problem. That doesn't mean that school mathematics has to be all about connections to these kinds of utilitarian aspects of everyday life. Surprise, joy, desire, conviction and doubt are also aspects of everyday life, and they can be felt very strongly in mathematics as well. For example, being sure about things is a huge theme in life, at any age. But while you may never, in any career path, have to consider the sum of odd numbers, you can, in mathematics, convince yourself—and also your friends and family—that, for sure and without having to actually try every possible combination of odd numbers, the sum of any two odd numbers is an even number. That can be gratifying, comforting and empowering. It can satisfy a desire to know things for sure, especially in a world in which so few things seem certain.

One can also think of the doing mathematics for the same reason we read fiction, to escape everyday life in order to imagine new worlds. I think some students, and perhaps especially adolescents, aren't as motivated by "real life" problems as we think, and may actually enjoy playing with and exploring the virtual world of mathematics. Of course, this virtual world needs to be interesting.

4. Why does geometry give us a model to understand almost anything?

It provides spatial, and often kinaesthetic, ways of relating mathematical objects. This is important because mathematical objects don't seem to exist for us in the material world. Consider the number line, for example, which is a geometric model that was invented by mathematicians in order to be able to understand the evolving number system. By thinking of numbers as being on a line, we are able to also imagine how numbers could keep going on and on, just like a line does. We can imagine going off in one direction, for positive numbers, but also in the other, for negative ones. We can also imagine that in between any two spots on the line, there's a spot in between, and if you can't see it with the naked eye, you could imagine zooming in. All this imagining rests on spatial intuitions that we have that enable us to make sense of and operate on numbers.

5. What is beautiful about math?

Beauty is not an objective value judgement, as we know. Even mathematicians disagree about what particular theorems or proofs they find beautiful. What's important is that it is possible to make value judgements in mathematics—and not only is it possible, it's the very basis of the development of the discipline of mathematics. But even if people don't agree on what is beautiful in mathematics, they often describe beauty in similar ways by talking about a "sense of fit" they feel, both emotionally and intellectually. They also talk about surprise, especially when surprise makes you see a new relationship or engenders a sense of wonder. Grade 4 children, for example, can be surprised to learn that you need to use a circle to construct an equilateral triangle—the surprise comes in part because it seems strange that something round can make something straight, but also because triangles and circles are usually separate curriculum topics.

Efficiency and clarity are also values that can be related to mathematical beauty—for example, when you find a really quick way of solving a problem that used to be tedious, or when you find a really clear, convincing way of making an argument. But beauty can also be about mathematical objects or ideas themselves. For example, many children are very attracted to the idea of infinity. They may not call it beautiful, because that word is usually reserved for other things, but in their curiosity and wonder, there's a sense of beauty. Of course there is! Infinity is this thing that we never really ever see or touch or hear, but we can imagine it and talk about it—it's not there but we can bring it into being. If students could see the way that all of mathematics is actually shot through with infinity, it might change their experience of it.

6. You still hear people say, why do we need multiplication tables when we have calculators? What other things does math promote?

It's very hard to argue, at this point in time, that anyone needs the multiplication table or, to take another favourite example, long division. Calculators are now ubiquitous, making the ability to multiple big numbers or to do long division more of a curiosity than a need. Mathematics often falls prey to the myth that one needs to know the foundations before being able to move up in the scaffold of increasingly abstract ideas. But it turns out that there are many different ways of constructing the edifice of mathematics and that students can learn to make generalisations and explain relationships about very sophisticated and seemingly abstract ideas.

Those who insist on the necessity that everyone learn, through rote, such "basic" facts are being as short-sighted as Plato was being about the emergence of another technology, that of writing. He worried we would lose our oral culture, that is, our ability to remember—and recite—huge amounts of verbal information. And he was right, but that's because we use writing and paper now to store information, which frees us to be able to do more cognitively complex things. The computer not only stores information but it does the tedious work of operating on that information as well. Mathematicians love avoiding tedium—they always search for shortcuts.

The multiplication table can be a source of wonder, with all its patterns and relationships. Being able to recall some of the benchmark products (multiples of 5 and 10, which are easy) can be rewarding and also very generative in enabling you to work out all those products in between. But knowing it by heart is not a justifiable goal of learning in and of itself. Students deserve more interesting, challenging and mathematically respectable goals.

7. What are some of the new and innovative ways of teaching math?

Most of my research involves exploring the use of digital technologies in the mathematics classroom, from the kindergarten level all the way to the University geometry courses I teach for Masters students. There is plenty of research that shows that certain kinds of digital technologies improve student engagement in and understanding of mathematics.

I see two central affordances of these digital technologies. One is that they can offer much more palpable ways for students to explore and make sense of mathematical ideas because of their visual, auditory, tactile and kinaesthetic modalities. This means that an idea like a triangle is not a simple one static shape in the textbook or a written definition. It becomes the three-sided configuration that moves on the screen and can take on any size, any orientation and any shape. The triangle now has a personality and this is what might make it something that is interesting for students to want to know more about.

The second is that they turn learners into authors or inventors who can bring into being their own mathematical objects, be they numbers, shapes or functions, and can use these objects to create new ones, quickly and precisely. And, perhaps most importantly, they can test their creations and revise them based on the feedback of the computer, which means they no longer have to rely on a teacher or a textbook as the sole mathematical authority. Using these kinds of digital technologies in the classroom is very challenging though, even for expert users. There are many tensions that arise in relation to the existing resources, beliefs and practices. Many teachers are eager to try using them, but are being offered very few opportunities for relevant professional development.

View Dr. Nathalie Sinclair's faculty profile here.