I am reading Inclusive Mathematics 11-18 by Mike Ollerton* and Anne Watson (my two biggest maths education idols?). I’m only on page 7 and my ideas are being challenged and shaped. There are a lot of fthings they say that I believe in principle but find hard to apply in the classroom.
Maths (as a school subject) should be something to do, they argue. I agree completely. Often maths is viewed as a body of knowledge that others have created that we need to learn (and memorise). In this way maths skills can be seen like a checklist to master. Some of my lessons have this theme: the aim of the lesson is to master the skill of finding one amount as a percentage of another. But Ollerton and Watson contend that thinking mathematically and doing mathematics are what maths is all about. Memorising (“learning”) techniques is somewhat useful–because of future maths study or work-related skills or everyday numeracy. But the goal of thinking and doing mathematics, I believe, is to have a more mathematical mind, to be more logical, and to be a beautiful thinker.
In this way studying maths at school could be similar to studying art or music. Students take music, art, or drama in order to appreciate these realms of life. They broaden their minds through artisitic expression. All students are painters or actors who have something to learn in school performing arts classes because they are developing their creative talents. Only a small proportion of the students go further with art or music–and the same is true of maths (though perhaps the percentage is higher?). But the arts and maths are still valuable to those students since their minds and creativity are improved.
So how can I teach maths to be more like an arts subject? I can do this by focusing on maths as a process to be explored rather than memorised. I can also give problems which allow students to investigate and apply their mathematical ideas. Ollerton and Watson say that students “sometimes learn more when plunged into a complex mathematical situation”–so I can avoid giving five simple examples then asking for skill practice. I can develop activities that make students think about why things work.
Would you say you teach maths like a list of skills to be mastered? Or do you focus on “doing” mathematics?
*Ollerton seems to have no website. Take me on as your apprentice, Mike, and I will create your website in return!
Well, you can think about your question the other way around. For example, if we are to teach art like we’ve been teaching maths…Let’s tell the students to “master” the skills of say how to mix two colours or three together; how to apply certain strokes; how to use a B before using a 2B etc… And only after we think they have mastered the skills then we let them draw a picture. Does this not sound ridiculous?
We often say to students that “yes, you got the skills, but you were not able to apply those in problem solving”. I really should blame myself for that as I haven’t provided them enough opportunities to apply their skills. If students are presented with problems that urge them to explore the skills required to solve them, then learning the skills and problem solving come quite naturally. That’s why I love what Mike Ollerton shown us and that’s what I been practicing in my classroom.
And that makes you and me the followers of MO!
Clare x
Thanks, Clare, for the art analogy. It reminds me of The Mathematician’s Lamanet by Lockhart, where he dreams music education is similar: students must learn to draw perfect black dots, construct the staves, read time signatures, etc. They are told that *one day* they may be able to use it when playing or listening to a piece of music. It’s shocking. Why do I do that to my maths students? I feel bad for them. Here’s to more MO in the classroom!
I’m going to be giving my first (ever!) math lecture in a few hours and I’ve been thinking extensively about your notion of “how do I teach math as an art”, especially how it ties into my perennial problem as a marker of “how do I convince students to write more than just symbols” and my best idea so far is using the metaphor of *stories*.
There are many, many stories, and the industries/media of film, books, graphic novels, (most or all of) music, poetry, etc., are all predicated on stories. But bad movies (for instance) are just as full of stories as good movies, so what makes a movie a *good* movie? Well, the story must be *interesting*, or suspenseful, or romantic, or unexpected, or told by beautiful people, or any number of other considerations, according to the taste of the person who makes the judgement.
So then, if we think of mathematical fragments (textbooks, published papers, assignment answers, lectures, etc.) as “stories” in this sense, we can ask “What makes a math story a *good* math story?” and the answers are different — first, and foremost, it must be *convincing*, and this is not so strange. After all, this is what people are demanding when they complain that “the characters weren’t believable” about films. Another important element is that mathematical stories must not contain digressions–they are only about one thing at a time.
As I’ve expounded this idea to various people, sometimes they complain that “this makes good math seem like a matter of taste, and I have a really hard time making my students realize that it is possible (even necessary) to make RIGHT and WRONG designations in math, you will merely cloud this issue”. While it is true that there are esthetic judgments to be made — for instance, I loathe any use of suspense in math, while some of my friends think it is a mark of beauty—even saying that math which has no suspense doesn’t deserve to be published. However, suppose we concede the point totally—suppose that you, as a teacher, find yourself constantly engaged in debates on the relative merits of a piece of mathematical exposition generated by your students—already you are miles, miles, miles ahead of where you were.
Micah, thanks so much for your comment. I am very interested to hear how your first lecture went. (Teaching is so hard–and not just at first. But just the fact that you ruminate about how to do it best is miles, miles, miles ahead of a lot of teachers.)
Your analogy about stories is very insightful. I think I will run this by some of my upper school students and see if it helps them construct better answers. One of my problems is that I often teach in a way that a story is not required, or even requested, from my students. So I need to provide more meaningful problems. I guess a movie equivalent would be just watching a character open the door–push or pull? This is not worth telling a story about and there is only one right answer. Boring!
Please visit again soon. 🙂