Discussion point for 28th September- Visualisation part 1

I replied and managed to lose the post. I am going to try one of these (the 5-7 one) on one of my pupils this Saturday. It's making me look at visualisation in a different way. I get the children to visualise and then describe patterns etc. and can see how creating different visual images can provide a way into a problem. I seem to remember seeing something that is maybe like like this on one of the Singapore maths videos? . The teacher had the children go into pairs and gave them each the same three or four triangles, and the same task. It was really interesting to see how differently each pair approached it and discussed it, and then the range of 'designs' they produced at the end must mean that they all visualised the same 'problem' quite differently? I think it was meant to show that there are many ways to approach and work through a problem and that they can almost all be 'right', but perhaps in some ways it showed visualisation of the 'end game' too? Was the teacher encouraging visualisation here? This week's question has really made me think...hard.

I tried out a couple of these tasks with one of my 9 year old home schoolers. Over the last few weeks we have been looking at different problems, investigations and exploring different ways of looking at the problems. With the squares within squares task, my tutee recognised at once that the shapes were rotated squares and decided to see how many fit into the base of the square through measuring. When finding out they didn’t fit exactly, he then looked again and saw some triangles. He then copied a triangle and spent some time fitting them into the squares. He then did the same for the smaller triangles and found it very difficult to fit them in. However, he persevered - something he wouldn’t have done a few weeks ago. Once he had done this, he worked out that there were 18 squares in total and 9 had been shaded. It was interesting to watch as I approached the task almost in the opposite way and, it demonstrated that it is important to let the children work through their ideas to the end point, whatever that might be. I also was surprised at how hard he found it to fit in the triangles into the square - shows that we need to give lots of opportunities to make shapes and fit shapes into others. The interactive whiteboard and being able to rotate the shapes really helped with this. I then showed him the way I saw the problem and he commented that he knew I would see it a different way as I usually see things differently to him!

The odd squares task was really interesting. I tried it out on 2 different home schoolers both in one to one sessions. They both worked out very quickly that the totals would always be even and both quoted the same ‘rule’ to me (I have never done this with them) that when you add two odd numbers together, you get an even number, etc. Because the task mentioned odd and even numbers, this almost blinkered their ideas - just quoting the rule over and over. When I showed the diagram, because it didn’t fit with their rule, they almost dismissed it. One of the kids took the diagram and then created rectangles with the counters that were left, both having 4 columns. He then added the extra counters on the end, to demonstrate the rule that he knew.He also observed that both parts of the rectangle were the same but this was more a passing comment. It took a lot more questioning to make the connection between 2 equal parts and the link to even numbers. I wonder whether this is a link that children have really made or we just assume they have made. He also created his own two rules that he wanted to investigate - the first being taking an odd number, multiplying it by 2 and then subtracting the starting number. He was very surprised to find that all the answers were the starting number! We then used diagrams to explain why.

I think these types of problems give children much more ownership of their learning and provide so many opportunities to show what they know and be creative and to know that there isn’t just one right way to do something. I am thinking that for children suffering from anxiety (the section of the course that I am currently working through), these types of problems could be very beneficial especially if they were carried out in groups.