Utility or beauty: why teach mathematics

Something has been niggling for a long time and I think I will try to articulate it. Despite the widespread discussions of the value of introducing children to good (or even classic) literature, to art, to nature, and even to Latin, when homeschoolers discuss math, the thoughts turn away from beauty and focus on utility. Maybe I’m hanging out in the wrong circles, but I haven’t seen much discussion of the beauty of mathematics and how we teach our children to appreciate that. Even those who advocate the living math approach don’t talk in much depth about either the beauty or the fun of it. I sometimes get the impression they find this just a more palatable way of learning what they still consider a utilitarian subject. If you ever meet any mathematicians, you will find that utility is about as important to them as it is to philosophers or literary theorists. In fact, many would argue that mathematics is a humanities subject, not a science. The discussion raging over at Lissa’s provides an interesting example:

Lissa said : “School says that certain knowledge is more valuable than other knowledge. But you know what? Knowing how to fix a toilet would have saved me a lot of money, but school never presented *that* potentially valuable knowledge. I can draw really pretty circles with a compass, though. I’ve had more plumbing problems in my life than occasions requiring perfectly round circles.”

Willa replied: “But circles are more awesome and worthy of understanding in themselves than toilet plumbing, even though plumbing know-how is respectable and useful. Toilets come and go, but circles remain. It’s too bad that circles have to seem “schooly”, though, since really they aren’t.”

As with many subjects, the roots of the problem lie in the way that we were taught the subject ourselves. While there are many good math teachers out there, many of us get through school without ever encountering one. Or the ones we encounter are unable to engage with our preferred learning style. While self-confessed math-phobics try very hard not to pass that fear on to their children, they are not in an ideal position to bring a sense of beauty and fun to the study of the subject.

As you may know from my previous posts on math, I have “required” math for Tigger, mainly because I think that her dislike was due to what school did to her and I knew that she could love it and see the beauty in it. We went down some wrong tracks but we have now found materials that work for her and she is excited and enjoying it. We are using Challenge Math and doing the chapters out of order to follow her interests. We’ve done some algebra and trigonometry. Right now we are doing a chapter on “Distance = Speed x Time” problems. She enthused to someone the other day that she was really enjoying this because it was really algebra and she likes algebra. As we were looking at the other topics and talking about how long we might be working in this book she noticed that the last chapter is calculus. She is very excited about that possibility even though it might not be until next fall or winter.

I’m not entirely sure how I got to that place. One important step was to stop worrying about memorizing math facts and get on to using them in interesting ways. She still stumbles with her times tables and keeps a chart nearby to consult when needed. We’ve focused on problems rather than drill (a real advantage of Challenge Math) and talk about math as being puzzles. We’ve read some related history though not as much as you might think, given her interest in history. We visited that museum in Bonn and then played with arithmetic in other bases.

We haven’t done much geometry and I think I need to work more on the appreciation of circles. So far our treatment has been mostly as a vehicle for solving problems about perimeter and area (though those were pretty fun). Another local mom has organized a geometry and art class focusing on symmetry and optical illusions that starts in a week or so. That will be our required math for a while and should go some way to addressing this particular lacuna in our learning to date.

Because there isn’t that much detailed guidance out there for how to do this, I am realizing that I may need to require myself to do some math, too. This is a good thing. Tigger is very motivated to learn things together. In fact, when we were talking about that calculus chapter, she asked if her dad and I knew calculus. When I told her that her dad had never learned calculus, she suggested that perhaps he could learn it with her.

I have discovered Martin Gardner, who seems to be a promising figure for this way of engaging with mathematical ideas. I know very little about him but he wrote a column in Scientific American for many years including some puzzles. I found his Colossal Book of Short Puzzles and Problems in my public library and a quick skim suggested that I should investigate it further. In searching for that I discovered Gathering for Gardner a foundation “that works to honor the achievements of Martin Gardner by promoting the lucid exposition of new and accessible ideas in recreational mathematics, magic, puzzles, and philosophy.” A search on Amazon brings up a substantial list of both puzzle books and other writings.

Finding that Colassal Book has provided another term that seems useful in searching for more engaging mathematical texts: recreational mathematics. I’m hoping that sort of term might lead in directions that avoid having to plow through mounds of dry text books unconcerned with the beauty and joy of the subject. Certainly typing it into a search box at Amazon brings up a list with at least one book with “joy” in the title, which is a promising beginning. So I have more work to do investigating the possibilities. I’m not in a hurry. Challenge Math is going to last us until at least Christmas and if I make that foray into Simple Machines, there will be plenty of opportunity to include the mathematics of that as well.

So I am just setting out the general direction of my future travels. If any of you have been to some of these places and would like to contribute to our very own “Rough Guide to the Joy and Beauty of Mathematics” pipe up in the comments, or in your own blog. There are a few math-phobics who read this blog and the beauty is not necessarily obvious to them. A good travel guide can help us notice the stuff that we might have missed otherwise. I am amazed how many people ask us if we went to the Louvre when I mention Freya’s interest in Impressionist and Post-Impressionist art. The Louvre doesn’t have those paintings, but a good travel guide will tell you where to find them. Hopefully we can do the same for math.

Edited to add: After I posted this Saille posted a link to a rather good article decrying the traditional method. I rather like the title “Presidential Math Panel Vows to Increase Learning Difficulties”. If it read it right, he’s after something like what I’m trying to open up here.


7 thoughts on “Utility or beauty: why teach mathematics

  1. I only skimmed, becasue I got stuck on fixing toilets and geometry/math and how they are actually related.

    it’s not that school doesn’t teach useful real-world things, it’s that they don’t teach that the things you’re suposed to be learning can actually be applied in the real world.

    I find that’s how we’ve really only formally instructed anything.


  2. Great thoughts. We worked in the Primary Challenge math book a couple of years ago — we should get the next one too. I am with Willa in the above exchange — that is, that’s how my mind works.

    One of my gifted e-lists is run by a total math expert who recently won a national award as a math coach. He recommends the book “Knowing and Teaching Elementary Mathematics” by Liping Ma and the materials put out by The Art of Problem Solving (find them on line.)

    I have come to appreciate mathematics as a very elegant (in the scientific sense) language, in addition to a very elegant form of logic, and these things have given me a better attitude about math than I did as a student.

    BTW, Violet is still far from perfect on her times tables too, but it has been so much better for her to learn and memorize “math facts” as a function of studying concepts and solving problems. She may occasionally “skip count” on her fingers, but she gets the answer right and eventually it will stick.


  3. I am in the early stages of convincing a seven year old that math does not consist of arithmetic solely, and that learning those math facts will come, in time. He’s interested in geometry, so that’s where we spend our time. I am an advocate of Living Math to introduce concepts- because I am NOT skilled in math- generally.
    I have plans to use Primary Challenge Math soon.


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  5. How I wish that I had ever once had a teacher who could even articulate beauty in math to me, let alone teach it to me. This is why I have my kids take upper level math at our co-op, because once they were being taught by people who not only got it, but were excited about sharing it, they can move beyond me and my phobias!


  6. A great resource I use for fun, practical, beautiful math is How Math Works by Carol Vordeman. It has sections on algebra, geometry, measurement, statistics, and so on, with plenty of concept and plenty of activities.


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