I have been feeling a bit odd about that last post. Although some good ideas came out in it, I feel a bit like it got away from the idea of math as a subject that could be appreciated in the same way as literature or art or music. It seems that we are more comfortable discussing mathematics utility than its beauty. Or perhaps, in our quest to imagine a different way of teaching mathematics, we find it easier to imagine a method based on utility. Appreciating beauty is somehow more abstract. It is much harder to imagine how to nurture that appreciation. Instruction in useful techniques and concepts seems more straightforward.

Today, I was brought back to my original musings on the importance of beauty in the teaching of mathematics. Angela found a wonderful article by Paul Lockhart (a mathematician). I suggest (as she does) that you read it. To entice you I offer a couple of quotations. On page 5, Lockhart addresses this question of beauty and utility directly:

I’m merely suggesting that just because something happens to have practical consequences, doesn’t mean that’s what it is about.

His discussions of pedagogy resonate with discussions amongst homeschoolers about unschooling and other forms of natural learning.

So how do we teach our students to do mathematics? By choosing engaging and natural problems suitable to their tastes, personalities, and level of experience. By giving them time to make discoveries and formulate conjectures. By helping them to refine their arguments and creating an atmosphere of healthy and vibrant mathematical criticism. By being flexible and open to sudden changes in direction to which their curiosity may lead. In short, by having an honest intellectual relationship with our students and our subject.

Of course what I’m suggesting is impossible for a number of reasons. Even putting aside the fact that statewide curricula and standardized tests virtually eliminate teacher autonomy, I doubt that most teachers even want to have such an intense relationship with their students. It requires too much vulnerability and too much responsibility— in short, it’s too much work! (p. 10)

and

Teaching is not about information. It’s about having an honest intellectual relationship with your students. It requires no method, no tools, and no training. Just the ability to be real. And if you can’t be real, then you have no right to inflict yourself upon innocent children. (p. 11.)

I love this point about relationship. Lissa’s post on patience was, in many ways, a reflection on unschooling as a better way of nurturing this relationship with her children. Willa has also commented on the importance of relationship and Julie has directly stated that her whole method of teaching writing is based in the importance of the relationship between parent and child in the learning process. And as I settle in and find my rhythm in homeschooling, it is the quality of the relationship with Tigger that takes priority.

Lockhart also says,

Teaching means openness and honesty, an ability to share excitement, and a love of learning. Without these, all the education degrees in the world won’t help you, and with them they are completely unnecessary. (p. 11)

That made me think of numerous discussions that Becky and I have had about schools and our decisions to homeschool. Not to mention something a high school headteacher said in a meeting I went to once about how what he thinks is important and absolutely necessary in a teacher is a love of children. If they love kids, they can learn the technical bits necessary to teach them. But all the subject knowledge and technical skills in the world won’t make a good teacher out of someone who doesn’t love kids.

And when I read this…

If we honestly believe that creative reasoning is too “high” for our students, and that they can’t handle it, why do we allow them to write history papers or essays about Shakespeare? (p. 17)

… well that just resonated with those thoughts about why did we all want to teach our kids Shakespeare, and poetry, and even Latin, but reduced mathematics to utility.

By page 23, I was no longer surprised to be nodding and smiling at:

Efficiency and economy simply do not make good pedagogy.

I have thought the same thing myself several times. I have at least considered blogging about it (though perhaps not specifically in relation to math) and may have done so.

What follows this statement is a beautiful argument for the value of unschooling, in my opinion, though he does not call it that. And I’m not even sure if he is aware of all those debates. At the very least it suggests that those of us not constrained by an imposed curriculum could imagine a whole different way of approaching mathematics.

Which is, of course, the really scary part. Because, as a friend pointed out the other day, no matter how problematic you find a curriculum, it at least gives you a framework to start from. Most of us don’t want to invent this thing from scratch, and most of us don’t.

I’m not sure what the practical consequences of this article are. I think it means that all that playing with arithmetic in other bases was definitely a good idea. I think it means that the series of sessions on Geometry and Art that I’ve signed Tigger up for is a good idea. I’m not sure what other good ideas there are but I have some vague inklings of how I might begin to look at the world differently. And I’m committed to figuring out what those good ideas might be in my own unsystematic and probably drawn out way. I’ll share what I find here and hope you do, too.

I do not plan to give up the integration of math with the teaching of subjects that use math. That is also a good idea. And Tigger thinks that learning about physics, starting with mechanics and kinematics, might be fun. What I want to guard against is sheltering in that relatively safe harbour of utility because I am scared of that big unknown world of the beauty of mathematics.

*Edited to add:* surfing around I found this article by Kevin Devlin which begins to make that link to beauty

Very interesting. This debate between beauty and utility reminds me of the difference between science and technology as subjects. Though they may externally concern the same subject matter (the workings of nature), they approach it in such different ways. Science appreciates knowledge for it’s own sake, while technology is all about how we use that knowledge. I imagine theoretical physics and mechanics have a similar relationship, as do art and design.

Regardless of the subject, I think it is important to have an appreciation for both sides of the coin, so to speak. But how do those of us who were only taught to memorize equations, much less understand their utility, help our children to discover the beauty of mathematics? I mean, I know it exists, I get glimpses of it now and then,but it just is not a clear enough picture for me to share it with my children. At least not like it is with science, you know?

I have a kid who hated math in public school and my highest priority for him when I pulled him out was to re-nurture a love of math. I was successful and he loved math for the next 3 years. But this year I see it fading again and I am not sure how to stop that from happening. Needless to say, I will be interested to know what you come up with on this one.

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I have been thinking about this topic non-stop since reading Lockhart’s lament. I am seriously working toward an alternative to pre-algebra, and have gathered a few ideas already. I want to instead present the concepts of algebraic thinking in concrete ways, and do the same for geometry. The joy is I have months to really wrap my brain around this. I am so thankful that you too, will be investigating ideas. This is exciting!

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I was one of those who simply liked the little brain puzzles of math. This transferred over to chemistry and physics and algebra and I simply never bothered to inquire what these little formulas were FOR. It didn’t matter, any more than it mattered what crossword puzzles were for.

But that doesn’t seem to be enough for my kids who want to know WHY — especially, what is the USE of all this. My oldest simply loves math and science for their beauty. He delved into the “why” of the subject for himself.

But I find myself unprepared to instill this sense of wonder in the younger children. I am really enjoying this discussion for that reason. I loved the quotes and have printed out the pdf to read.

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Math + beauty = QUILTS. 🙂

Or how about the swhirl of a sunflower and its seeds? That’s pattern-based.

I really need to formulate some thoughts on this and post an entry. 😀 I missed doing one in time for the Canadian homeschooler’s carnival.

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