I have thoroughly enjoyed the TAM course and, travelling to Birmingham early on Tuesday morning for the penultimate day, was filled with excitement at another couple of fantastic days and sadness that these were the last two days of the course.
Upon arrival we were asked to write a definition of what we perceived modelling to be. I had always interpreted modelling to include concrete resources or diagrams of some sort, so I listed resources such as multilink cubes, lives demonstrations and the bar model.
We looked at problem-solving tasks as the first main task, but with a twist. Rather than just working on a problem in a group, we had to nominate a Silent Scribe. The scribe wasn’t allowed to comment on what was being done or contribute in any way. They just had to write down any observations that were made of how the rest of the group solved the problem. I thought I would volunteer to challenge myself… it was one of the hardest things I have ever done! The question we had was about minimising the surface area of a cylinder with a volume of 500 cm^2, and involved rearranging formulae, substitution and differentiation. After the groups had had time to work on the questions, the scribes met up to discuss what we had observed. It was really interesting to see that all groups had followed pretty much the same pattern:
Interpretation – read and discuss the question, read the question again emphasising key words and comparing it to similar questions that have been seen previously;
Useful formulae – for example volume and surface area of cylinders in this case;
Start solving – setting up an equation of sorts;
Speculation – discussing what could be done and what needs to be done, linking in key facts with key words; and
Solving and checking.
The speculation phase was most interesting for me – some members of the group needed more convincing that others, but all were willing to give everything a try. They tried rearranging the formulae, differentiation and graphing what the equation might look like. We discussed the importance of problem-solving afterwards, and how it can help us teach valuable transferable skills.
Following this, we went back to look at modelling in more detail: we looked at two scenarios and thought about what the graphs might look like for these. This was a great discussion, and it was interesting to see what others around the room had drawn based on how they had interpreted the scenario. Some had shown the opening of the parachute to instantly change the speed of the parachutist’s fall and some had shown it a gradual change. The balloon caused great discussion and it was highly entertaining to see how many of us modelled this with our hands!
We were then handed 6 more scenarios and 7 graphs to match up from this resource on Underground Maths. Often when we do these matching tasks one of the cards is held back to ensure we think about all of the graphs, rather than guessing the last two. This is something I do in any lesson including a card sort to attempt to replicate these thinking processes in my students. We also had to think about the units when matching the scenarios with their graphs and some were far more obvious than others. When discussing assumptions made and queries that we had about the graphs it was interesting to see how some accepted these mathematical models and how some wanted to completely pull them apart. Matching up the equations to the graphs was relatively easy for us as experienced teachers – we knew to look at whether the gradient was positive or negative, what the y-intercept was and whether the graph was linear, quadratic or trigonometric. I think we may have taken this for granted when we were working through it, but on reflection this would be a fantastic task for my students to try.
The assumptions and queries, when discussed as a group, were all very similar. Lots of us were unsure about the tennis ball scenario and the equation for its graph. We discussed the need for different equations for different domains, and then which would be needed for different questions. Some of us were definitely easier to convince than others. We (mostly) agreed that as long as the mathematical model was good enough to answer questions such as maximum height, distance travelled and speed. There is a need for compromise with mathematical modelling between how accurate the model is required to be and how doable the maths needs to be.
Dan Meyer has plenty of videos of scenarios that are readily available to support modelling discussions with students. We were also shown the Vernier Video Physics app that enables us to create our videos to model scenarios and trace a specific object to draw a variety of graphs, including distance-time and velocity-time. I’m looking forward to trying out this app, and hopefully I’ll have time in the next few weeks for my students to explore it too.
Filled with ideas of how to model scenarios for mechanics, we looked at a real life statistics issue using binomial distribution – the over-selling of seats by airlines to maximise profit. The emphasis this time was placed on assumptions and the need to sometimes create assumptions to simplify the model. There is also a need sometimes to find a way to add value when the simplified model isn’t necessarily realistic. A good introduction to a mathematical model would be to pick it apart first – if there are any changes that can be made, then great, but if not it will have to do. This type of mathematical modelling is another way to show students that maths doesn’t always have to be right or wrong.
Day 7 was to be based on mathematical proof so we started to look at KS3 and 4 proof at the end of day 6 with another card sort. 13 of the 15 cards we were given had a proof question written on them and we had to sort them into three groups: proofs all our students would engage with, proofs our most able students would engage with and proofs none of our students would engage with. This was a complicated task straight away as teachers on the course come from a variety of different schools. It led to interesting conversations. Once we were happy with how we’d sorted them, we then had to think of a couple more proofs to write on the blank cards.
Our homework had been to complete a proof and think about prompts we could give students if they got stuck, so our first task was to get into groups of 4 and take turns to prompt the ‘students’ within our group so they could complete the proof. I really enjoyed this task, despite getting completely stuck when proving there are an infinite number of prime numbers.
We had to write down our thoughts and feelings for each of these proofs, and a 'starting point' was a key theme in what all of us had written down. Where prompts were well-thought through, we had a way in without the whole proof being given away. One member of the group had written hints on strips of paper for us to look at one at a time if we got stuck - I loved this idea.
We looked at visual proofs of Pythagoras' Theorem briefly. I really liked this one that's very easy to make and easy for students to see the areas are equal.
Moving on from a visual proof of Pythagoras' Theorem, we had a go at some of our own. This visual triangular number proof was drawn by George and is ace. I'm more than a little envious that I didn't think of it first! We also looked at "every prime number greater than 3 is 6n+/-1" using a number grid across 6 columns. It was easy to see why this was the case once we had crossed off multiples of 2 and 3.
We had several conversations over the two days about what makes a good proof. We decided that as long as the audience was convinced then that was enough. But then we talked about the angles in a triangle summing to 180, and whether ripping the corners counted... I'm still not sure I would count this as a proof, but instead a demonstration that it is the case.
At the end of the two days, we were asked to reflect and think forward about what we were going to do in the future. This is my long list...! If you read this blog post, please Tweet me or message me through the website to remind me of one of these, or just to ask me how I'm addressing them.
I know I've said it before, but I'll say it again, this really is a fantastic course. If your school will let you go, take them up on it! It's some of the best CPD I've been lucky enough to receive.