The Economy of Teaching Mathematics Laurinda Brown, Jan Winter: Graduate School of Education, University of Bristol; Alf Coles, Tracy Wylie, Louise Ordman, Alistair Bissell: Kingsfield School; Barry Orr, James Pretty: Ashton Park School and Chew Valley School; Dave Hewitt: University of Birmingham Finding practical teaching strategies for teaching mathematics using Caleb Gattegno’s work What are the powers of children? Extraction, ‘ finding what is common among so large a range of variation ’ ; Making transformations, based on the early use of language ‘ This is my pen ’ to ‘ That is your pen ’ ; Handling abstractions, evidenced by learning the meanings attached to words; Stressing and ignoring, without which ‘ we cannot see anything ’. How can these functionings be used in the teaching of mathematics? - Students can notice differences and assimilate similarities - Students can use their power of imagery: ask students to shut their eyes and respond with mental images to verbal statements - Students can generalise given that ‘ algebra is an attribute, a fundamental power, of the mind ’. What are we doing? We are one year into a two-year project where a group of experienced and newly qualified teachers have formed a collaborative group meeting 6 times for day meetings to investigate Gattegno’s ideas and develop practical strategies for teaching and for continuing professional development. - Tracy Wylie: MEd dissertation, language for imagery - In the summer term of the first year members of the group visited each other’s classrooms to work on issues together - The four new teachers will each be developing their ideas through working on a small-scale research project accredited through a master’s module in the second year of the project - Alf Coles: web-site development with access to a programmer - 5 members of the group attended a conference on Visualisation, Easter 2007 and have written for a future issue of Mathematics Teaching - Every day meeting we co-observe one of the group teach or jointly watch a video of teaching to develop a common language for what we are seeing. What strategies, questions and issues are we working on relating to using imagery? 1) using dynamic images in Geometer’s Sketchpad 2) finding images for non-shape and space topics 3) what are canonical images? 4) using Cuisenaire rods in algebra 5) using more visualisations 6) starting with an image and adding dimensions (simplicity  complexity) 7) deciding what makes a ‘good’ image and finding some 8) ‘‘fading out’ of image 9) slowing down the pace when working with images 10) what does a teacher need to do when working with an image to become more comfortable with using it? On the tablet you can see Tracy Wylie teaching. We are looking at what we call metacomments. After encouraging students to say what they see, the students then use their powers of discrimination to notice patterns and, mathematically, to come up with conjectures. How do the students know what conjectures are? Tracy picks up on what the children say and comments about it. She also stresses actions that the children take that she wants to become part of the culture of the classroom, such as writing about things they’ve noticed. The following extracts of Tracy’s classroom talk are taken from the lesson she taught to her year 7 with all the members of the project watching! - Have your books open, be looking at your homework, because we are going to be talking about anything you've found out from your homework; be looking at that, be looking at the person's next to you, be having a little conversation about it while we just wait for the others. - While you're doing this, I'm saying this to individuals so I may as well say it to everybody; part of your homework was to think about Jacob's conjecture and Jodie's conjecture some of you have already commented about that in your homework. I was really impressed by those people who were writing about the things that they've noticed. - Now what I want you to try and do now is while you are doing these sums be thinking about the conjectures we talked about last lesson, be thinking about whether you think they are true, be thinking about whether you can explain why they are true and why you don't think they are true and also writing down all of these things. So make sure you are thinking about those things as well as working on the sums, because both those things are important; so maybe one or two more minutes just be reflecting on those conjectures and what you think about those, given what you've done here and what you've done in your homework, and then we're going to have a discussion about them. - This is what mathematicians do; they develop their conjectures so they begin with something they believe to be true and then they might change their minds having got some results, so it's interesting that I've got the same two people in the group, whose conjectures have actually changed. But what you need to be doing as mathematicians is thinking about starting testing these conjectures. And it might be that leads you to develop conjectures of your own. It might be that it leads you to disprove one of these. And what Tommy has given us an example of and what Jodie has given us an example of is where this doesn't hold to be true; called a counter-example, an example which doesn't fit the conjecture. The images on the second poster One of the key issues to emerge for us has been the use of imagery in classrooms. On the other poster are examples of some of the images we have used and discussed. Each of them links with at least one of the numbered questions in the box above. As an exercise try to make links and ask us about anything you’re not clear about.

The Economy of Teaching Mathematics – Using Imagery Laurinda Brown, Jan Winter: Graduate School of Education, University of Bristol; Alf Coles, Tracy Wylie, Louise Ordman, Alistair Bissell: Kingsfield School; Barry Orr, James Pretty: Ashton Park School and Chew Valley School; Dave Hewitt: University of Birmingham (and David Reid for the spiral photograph) I was struck by one student justifying his solution to me by saying "I know the 5p must be the big rod" - he then pointed to a drawing in his book: He had written next: 5p - 8 = 2y and then y = (5p - 8) / 2 What do you see? 5p 2y8 This arrangement of shapes was shown on the board and the pupils were asked to write down any similarities and any differences between any of the shapes. These could be similarities/differences between all four shapes, or between just two of the shapes. Pupils seemed to find it harder to think of differences in the shapes. If a pupil told me they couldn’t find any differences then I would choose two of the shapes at random and ask if the pupil “Are these two shapes the same?”. When the pupil replied “no”, I would then ask “What is it that makes these two shapes different?”. This always seemed to enable the pupil to see more, even though I hadn’t given any suggestions as to what the differences might be. A fixed circle. Two points moving in the same direction, one point moving twice the speed of the other are joined by a piece of elastic. If you trace the mid-point of this elastic, what does the path look like? The image, drawn in Geometer’s Sketchpad prompted some great discussions and mathematical comments. I was asked what would happen if the faster point was moving at 3 times the speed of the slower point. Trig image

Watch the stars go round……