 Hello, my name is Tim Rowland. In this session, time together, I'd like us to think about those times when we're teaching mathematics and things don't quite go according to plan. So it's not just what happens in unexpected moments, but what we might learn from those. We'll go straight to the classroom. So here we are in the classroom with Jason. He's working with a year three class. And let me say now that any episodes classroom episodes that I speak about or introduce actually happened. We didn't make them up. They were observed and in most cases, videotaped. So he's reviewing some work on fractions that they've done earlier, and each of the people's had a small oblong whiteboard and a dry white pin. And Jason asked them to split their whiteboards into four equal parts. Well, most of the class did what Jason had expected. They drew the two lines parallel to the sides is Rebecca's version. But Jason was surprised to find that Elliot drew the two diagonals. So Jason asked the class, what has Elliot done that is different from what Rebecca has done. So he's working because he's done the lines diagonally. So Jason asked Sam, has Elliot split his board into quarters. And as we see Sam wasn't at all sure. So what's Jason going to do next. Well, the question as to whether Elliot's board been partitioned into quarters was not mentioned again. So the issue if you like was not resolved. What I'd like to happen now is that imagine either that you're Jason and that you're with the class and Elliot had said what he did with his diagonals. How would you respond to that idea. Or you might refer to imagine that you're Jason's colleague or his mentor, you're observing the lesson. And you've agreed that your talk to Jason have a discussion with him afterwards. What might you want to say to Jason about this episode. And at this point I'd like to suggest that you pause the video to give you time to think about and hopefully discuss your response to these questions before moving on. Well, Jason was surprised. He hadn't anticipated Elliot's response. And this brings to the surface and demands on Jason's teacher knowledge resources of three kinds. Which have called fundamental mathematical and I want to just emphasize the pedagogical one for now. The way that Jason is going to pick up this suggestion of Elliot needs to somehow tie in with what the class have already experienced in terms of area. So what kind of explanation if you like will be meaningful and helpful to them. I've asked many friends, teachers, colleagues and so on, how they might respond to that and what they might do. And most of them turn to the nation of the area of the triangle as half the base times the height. In the pink triangle, the area we see there is half of the base, which is a and the height, which is a half B. And likewise for the blue triangle, it would be a half of B times a half A. And just manipulating those expressions, we can say that both of them are equal to a quarter A times B. But this particular approach to explanation isn't going to be very helpful to these seven to eight year old pupils. Jason can tell them that the areas are equal, but how can he justify the claim to them in a way that they would relate to and find helpful. If he thinks about it later, or discusses it with a colleague, he might come to know how he could respond another time. Well, this is just one possible approach. And there are others, of course, suppose we divide those two non congruent triangles into half, then we can now see that half of that rectangle is separated into four triangles of the same shape and size and congruent. And we could say, well, each of those two triangles, the blue one and the pink one, are now made up of two smaller triangles of the same shape and size. So we're not referring there to any formula or anything simply it's a matter of perception. This topic of Equal Area Triangles is a very interesting one. And I was in Mexico a few years ago, and the colleague Olympia Figueras told me about something that she'd seen in a primary school there. I've included a link to an article that I wrote for the journal Tangention. And if you skip to page 43, you'll be able to read it. It shouldn't look like that. But we'll move on for the moment. My research in the last 20 years or so has caused me to develop an interest in events like this, the kind of surprises that happen during lessons, unexpected events. We've come to call them contingent events. Contingency is one of the four dimensions of a theoretical framework about mathematics teachers knowledge and how it plays out in the classroom. You might already be familiar with this theoretical framework and I've given again a link to our website at the bottom there. The contingency dimension is about these unexpected and unplanned moments when the teacher has to sink on their feet. And that's the book that we wrote about the Norwich Quartet. Let's now go to Heidi's classroom. She was working with a year eight class. She was revising fractions and percentages and the students had calculators. I'll use two examples with them and I'll just home in on the second one. My mean uncle wants to decrease my allowance by 7% each month. If I used to get 450. Sounds like very generous. How much will I get after three months. So these were the things that the students came up with. One suggested funding 7% by calculating 450 multiplied by 0.07 and then multiplying by three. So Heidi asked what would need to be done before multiplying by three. So the same students said well take it away from 450. She recorded these two steps 7% and then subtraction and came up with 418 pounds and 50 pounds. She said this gave them out after the first month. But they would need to do the same twice more to find out what she would be receiving after three months. A separate student said, could you do 450 times 0.93. She said that was absolutely right and by taking 7% away you end up with 93% of the previous month's allowance. And so that brought you to your amount after one month in one step. The class checklists are not calculators and got the same answer. She then said they could use the same calculation twice more in order to answer the question about her allowance after three months. A third student Oliver came up with something different. He said, instead of timesing by 0.93 loads of times, you could just times by 2.79. I didn't quite know what this was about. So she said, where did the 2.79 come from? And Oliver explained it was 0.93 plus 0.93 at 0.93. Well this took Heidi by surprise and she didn't quite know how to respond to it. So once again I put it to you. Imagine either that you're the Heidi in that situation, how would you have responded to Oliver's suggestion? Or again suppose you're observing Heidi's lesson as a colleague and you're going to discuss it with her afterwards. What might you want to say to Heidi? I've included here a link again from a page from our Knowledge Quartet website where you'll find our discussion of that situation. So there we are. These contingent moments in the mathematics classroom, they make demands on teachers knowledge resources. They bring about the possibility of reflection in or on action. They can be powerful opportunities for learning, both the subject content and the pedagogy. And I like this quote from James Cass. To be prepared against surprise is to be trained. To be prepared for surprise is to be educated. Well I'll finish there. I wish you well and hope you'll find it useful to think about these contingent events in your own classroom or those of others in the future.