After reading Bryan Meyer’s thesis (which I highly recommend you take the time to look at), I decided to do something I’m ashamed to admit I rarely do – actually that might be a lie – I don’t think I’ve ever done what I just did. Before I get into it, here’s a quick bit of context…
Bryan Meyer mentions an interesting lesson/sequence of lessons in which he asks the students to explore consecutive numbers in anyway they feel like exploring them. As you can probably imagine, some of the students were completely stumped – they had no idea what he was asking them to do. The students were simply too accustomed to being told exactly what to do and when to do it and had no sense of how to create “mathematical artwork” and play around with mathematics. Essentially, they didn’t know how to create their own mathematics – only how to learn some of the vast array of content which other people have created and possibly apply some of that knowledge to problems given to them by the teacher.
In a Mathemtician’s Lament, Paul Lockhart thinks this is a travesty. He asks the important question, “Why don’t we want our students to learn to do mathematics?”
I could get into a discussion here about external assessment issues, pressure on teachers or teacher perceptions of learning mathematics. Instead, I decided to ask myself the following question:
When have I ever sat down and explored an area of mathematics simply for fun?
I tell my students that mathematics is fun, creative and playful but then I rarely practice what I preach. Instead, I spend time learning about mathematics which other people have created and occasionally add to that by filling in some of the gaps myself – I rarely just mess around with a random topic for fun. Please don’t take this post as confirmation that I’m all about students solely creating their own mathematics and never learning anything that other people have created. I believe in a very balanced approach to a curriculum in which a particular teaching and learning approach during one activity isn’t the best way to do it for another activity. Direct instruction may work for one thing but then exploring a concept using an applet may be better for another. It just depends on the topic and the students you’re working with.
Anyhow, I felt is was highly hypocritical of me to preach the creativity of mathematics and rarely explore things for myself. Hence, I sat down at the table this morning and purposefully explored consecutive numbers for no other reason than to see what I could come up with and what direction it took me. I didn’t stop when I’d come up with something, I just asked another question which took me in a different direction. After 15 minutes, I was quite amazed at where I’d got to and what I was thinking about.
I stopped at the 15 minute mark and immediately wondered where other people would be at this point – what wonderful mathematics would come out of a discussion between a room of teachers who sat by themselves and explored consecutive numbers for 15 minutes? What would we learn from each other?
It’d be bloody brilliant if there are any teachers out there prepared to spend 15 minutes thinking about consecutive numbers to see what they’ve come up with and where they’ve got to. If anyone does this, I’d be very interested to discuss it. I’m giving a talk next week to our Excellence Challenge students titled, “Mathematics: Why some people fall in love with it.” It’d be great to discuss the playful nature of mathematics and use this activity as a springboard to expose the students to the true nature of mathematics.
Update: Here’s my 15 minutes of play. It’s a bit tidier than my original sheets but had to re-write with bigger handwriting so that the photos came out okay. I was pleased to re-discover the integral test for divergence of the harmonic series. I remember learning this about 10 years ago and not having the foggiest where it came from – great to actually discover it for yourself. Infact, when I checked whether my reasoning was correct – I found the exact same reasoning on wikipedia! Outcome: Must do 15 minutes of mathematical play each week.