The View From Mathematics

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I have a lot of respect for Professor Sylvia Serfaty. Not only is she a brilliant and esteemed mathematician, but she recently brought two of my favourite things together when she said this:

“You enjoy solving a problem if you have difficulty solving it. The fun is in the struggle with a problem that resists. It’s the same kind of pleasure as with hiking: You hike uphill and it’s tough and you sweat, and at the end of the day the reward is the beautiful view.”

There is a certain exhilaration you feel when – after carrying a heavy pack on your back for kilometres on end through mud, up hills, feeling that gross sweat trickle down your back, and running out of things to say to your hiking partners – you arrive at your destination. There you are in the middle of dense bushland, with not a roof, road or electricity wire in sight. Instead, stretched out around is unending greenery and the vastness of the sky above. You are in a patch of the world that very, very few people will ever get to see. And yes, you can be proud in knowing that you worked hard to get there.

And this natural beauty can be compared to maths?!

Just like hiking, there is much in the journey of problem solving that is hard work and will challenge you. Mathematicians – and I use that term broadly, to describe educators, academics, students and those who are in some way engaged in the field – take joy in getting to the destination. Problem solving is not like relying on your GPS to get somewhere, where each step you are told what to do next. “At the roundabout take the third exit. In five hundred metres, use the second from the left lane to turn left. You have arrived at your destination.” Nope. Why would we bother with mathematics if it was that mundanely easy? It’s hard and mathematicians knowingly struggle. Serfaty took nearly 18 years to solve one problem. She’s also not the first to show such extreme mathematical persistence (e.g. Andrew Wiles‘ momentous  journey with Fermat’s Last Theorem).

On solving a problem, mathematicians reach a point of (sometimes momentary) finality. There is perspective on the method used to get there- what was effective, what held them back, how they failed, but then learned from it. And, just like the hiker’s view, there is immense satisfaction that comes with overcoming your own personal limitations to arrive somewhere new.

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(As inspired by Serfaty and Ben Orlin)

Encouraging higher-order thinking in students

Went to a fantastic professional development seminar today. It was given by Glen Pearsall, a teacher at a public high school and an expert in classroom dynamics.  In amongst the many things he taught us, was the importance of encouraging higher order thinking and drawing this out in students.

Much to my own annoyance, frequently in class I find myself asking basic questions so that I can elicit some response from students. If I can get students to answer a basic question, doesn’t that mean they are on the right track and understand what I am teaching them? Maybe. Questions that elicit front-of-mind responses or only require simple recall or recognition are not going to embed the learning for students.  The type of thinking that is required to answer these questions, will not take learning to a higher level.

Instead, students need to be working at a higher level of Bloom’s taxonomy: analysing, ranking, deducing, convincing, assessing, generating, etc. Sure, the basic level of response is useful initially, but not for creating deeper learning. And this is something I struggle to encourage in my more difficult classes.

Glen gave us a couple of examples of activities that can be applied across subjects to enable higher-order learning. One, which I will be implementing with my politics class tomorrow as a tie-in with Harmony Day, makes use of the website wordle.com. Prior to giving students a reading task, put the assigned reading into Wordle and click “Go”. Out pops a word cloud, sizing words according to their frequency in the article. You can play around with it a bit (taking out some words, changing font, colour, shape).

Have students think about the word cloud, individually and in pairs, discussing what they predict the article is about. Aside from Humanities or English, this task could be done with Science or Maths, looking at new concepts or procedures.

The purpose is to have students think about key words, and provide evidence for their predictions – guaranteed to engage students and get the limbic system firing!

Here is the word cloud I will be using, based on Chris Bowen’s recent speech on multiculturalism: Harmony Day word cloud