Becoming mathematicians – together

Those who follow me on twitter might have noticed over the last few weeks that I’m engaged in some kind of project. A project to get my students thinking mathematically, working like mathematicians and thinking of themselves as mathematicians. We aren’t to the end of that, but I think it’s time for a bit of self-reflection.

Why am I doing this, where does it come from, and where might it go?

My students are all newly arrived English as an Additional Language (EAL) speakers enrolled in an English Language School for two to four terms to get their language up to speed for a mainstream school.

About two thirds of them are refugees or from refugee-like contexts, and many have significant gaps in their prior learning of mathematics. Some have had no schooling at all before they arrive with us. So shouldn’t I concentrate entirely on filling those gaps in their procedural maths? I don’t think so. My students are working in their second, third, fourth, even fifth language; a language they have only just begun to get to grips with, at least in an academic way. This means working fast is out of the question. I only have them for a couple of terms, so there is no possibility that we can fill all of those gaps in that time. And the one thing we absolutely must work on is advancing their mathematical academic English as far as possible so that they can survive and thrive when they transition to a mainstream school. I hope and pray that they will get fantastic maths teachers when they get there, but I may be the only maths and EAL teacher they ever meet. That means lessons with a lot of them talking (and some writing) about mathematics, and listening to each other doing the same.

But if they have very little English, and they come from backgrounds with very traditional prior schooling, are they going to be able to succeed at this? What I am finding so far is “yes – far more so than I imagined”. As we build the routines, and students begin to understand that different things are being valued they respond amazingly. They are giving better and better explanations, writing more interesting questions, beginning to justify and finding their own mathematical avenues to follow, seeking patterns and understanding rather than valuing only correct calculations. Engagement has increased as students see purpose in what they do and the teacher valuing more than the fast right answer to a closed question. And all that wrestling with the context or problem together, explaining and justifying, allows for the language production and listening we need.

And you know what? I think we are covering just as much content, but more depth and less breadth.

Jo Boaler’s Mathematical Mindsets and youcubed, has been a big influence here. So has the idea of MathsCraft (which I am excited about attending in November) and twitter conversations with a whole bunch of people such as @nomad_penguin and @DavidKButlerUoA, along with a dose of Dan Mayer’s “be less helpful” (dy/dan).

And the title of this post? I started with title in my head along the lines of “teaching the students to be mathematicians”, but that was too focused on me. That evolved into “[my students] becoming mathematicians”. But how can I facilitate that if I I’m not living it it. So “Becoming mathematicians – together”. One thing I don’t think we talk about enough as maths teachers is the need for us to do mathematics ourselves. Are we regularly and actively thinking mathematically? For me that’s whole range of different things: writing R code to process school data, timetabling, engaging in mathematical puzzles, writing Desmos activities that push the boundaries of what I can do in the Desmos graphing calculator and computational layer, …

The next step for me? Learning to think aloud publically as @DavidKButlerUoA models so brilliantly on twitter.

The next step for my students? I’m not quite sure. They are already taking me places I didn’t expect.

Number talks

“How do I get students talking about numbers and ‘sums’?”

I have been asked that a number of times, and it’s a good question. Students are not going to acquire the basic language of maths unless they are using it.

Having played around with a few ideas for a while, I have come to the conclusion that one of the best ways is to spend 15 minutes of each lesson on a number talk.

The essential idea of a number talk is simple. The teacher writes a single problem on the board (28 × 12, say). Students are not allowed to write anything and are asked to silently calculate an answer in their head. The teacher collects the various answers, and then asks a student to explain how they got their answer. In the EAL classroom I insist that they do so from their seat, and that they are not allowed to write anything. I will write what (I think) they said on the whiteboard. Other students are then invited to ask questions, and other strategies collected.

Building up effective mental strategies (and representing them) builds number sense and is far more practically useful in the age of the iPhone than written algorithms, but in the EAL maths classroom we have the added bonus of rich language production, and giving students a genuine need for that language. Great maths pedagogy and EAL teaching come together.

Executing number talks is a little less straightforward than it sounds, but fortunately there are great resources out there to get you started. The best I’ve found for secondary teachers is Cathy Humphreys and Ruth Parker:

Their book gives a teachers a clear guide on how to select questions and manage the routine for best effect, as well as what mental strategies we would like students to develop and how to represent them:

Making meaning without understanding a word

At the MAV Convention on Thursday, the team from Sunshine College, Thao Huynh and Alex Mills, gave an excellent presentation on how they use reciprocal teaching to scaffold their students into worded problems. To get us into thinking about the language demands of a maths problem for their EAL students, they gave us this problem to solve:

Without being able to speak the language, but assisted by their tool, all of us were able to make a start on understanding the question, and some of us were able to solve it.

But what got me rethinking about it this morning was reflecting on exactly what other resources we were able to bring to the problem. Without being to recognise any of the words in the question except Markt, Frau and Schweine, what enabled me to decipher the meaning?

  • A familiarity with the pattern of mathematics worded problems (a genre in their own right)
  • Multiple previous encounters with questions of this type
  • That the punctuation is close to English, so that I can tell what is a piece of information and what is the question
  • Some educated guesses about those small words like und
  • And, of course, the ability to read those numbers and recognise what is a count and what is a price.

What else have I missed?

Could I have solved the problem if it had been in Persian or Chinese?

Could I have solved it if there had been a superfluous piece of information in there? I very much doubt it, and yet we wonder why students struggle with questions where there is more data than they need and want to use every number in sight.

I won’t give the solution away, try it for yourself. But also give thought to what we, as maths teachers, mean when we ask “is this reasonable”. A solution with four-fifths of a chicken, for example, wouldn’t be. But would the actual solution, or even aspects of the question like the relative prices, seem reasonable to a student from an agrarian society where buying and selling animals at market is part of everyday life? How much of what we call “reasonable” is actually real-life reasonable, and how much is maths-classroom cultural construct?

It’s interesting to notice not only what I can manage without, but also how much else I need to use to replace it. We bring so many skills to bear to make meaning from a text, unaware that we are doing so. And then we don’t understand why our students, who don’t yet have those resources, are having difficulty.