Posted Wednesday, 28 August 2019
What do you see? How many?
Scott and Ryan use unit chats as quick and engaging ways to warm-up a maths lesson.
Important update: Access to lifts at Scitech is currently unavailable until further notice.
Important update: Access to lifts at Scitech is currently unavailable until further notice.
Hilary gives a deeply personal account of her journey as a reflective and continually improving maths teacher.
I am Hilary Campbell, a Year 2 classroom teacher at Eaton Primary School, which is located on the outskirts of Bunbury. I am currently involved with the Alcoa Champions of Maths program. When I was asked to write this blog I responded with something along the lines of “Ooh, lovely. Can’t wait.” This was of course code for, “OMG! Why me?” I am not telling you this because I am trying to sabotage my chances of succeeding in the Alcoa Champions of Maths program, but because it mirrors my reaction to teaching mathematics early in my career. I was a good self-reflector in those days, which means I used to beat myself up most evenings as I was aware or worried that I was not doing it right. I felt ill-equipped to teach maths. I was unsure where to start. I was in a remote area in a two teacher school with another graduate who was pretending she knew what she was doing too. Let’s face it, at high school and uni I barely knew how to learn it – who was I trying to kid? So I made a decision to do something about it. I signed up for as much professional learning opportunities as was humanly possible, I joined WAMA (now MAWA), I joined maths communities, I went to seminars, read books, purchased resources and agreed to have important people from Murdoch University observe my mathematics lessons for research. One of the most important things I learned was that if a student is getting something wrong, it’s not because they want to, but because they are missing some information they need to solve the problem. That was in 2010 and I’m still doing something about it.
So as my career has progressed I have come to realise that the information that many students are missing is that mathematics is fun. I like it. My students learn to like it. I am exceptionally good at modelling mistakes and that isn’t a bad thing. Anyhoo, now I am in training to be an Alcoa Champion of Maths, which I would add ‘echo’ to if this was a podcast. I still feel like the dumbest girl in the room but now I don’t mind so much.
If I want my lessons to focus on mathematical problem solving and conversations, rather than power struggles for the whiteboard eraser, I realised I needed to make some of the issues disappear. I employed a few management strategies such as making sure there were plenty of new working markers, plenty of erasers and clear expectations of behaviour. Then I turned my attention to the problems to be solved.
Problem Number One
Even though we were all getting used to the process, I was aiming for success and so I made our first problem very simple. No matter what else happened I wanted my students to feel like they had achieved something and to feel good about themselves. Some of the groups worked ok. Some argued a lot. I actually modelled the lesson to introduce a similar problem, hoping my students would transfer their knowledge from one problem to another. I also had concrete materials at the ready. To cut a long story short, nobody used the concrete materials, I’m not sure whether the students transferred what they learnt from one problem to the next, but there were some interesting and inspiring moments. On a class level the lesson was a success because everyone enjoyed it. From a mathematical perspective, the lesson was a success in varying degrees.
Meet Jack. Jack and his group shared their solutions to the problem on the whiteboard, and although they spoke to each other, they actually solved the problem independently, while looking at each other’s work and building on it. Kind of relay problem solving – like on MasterChef! Jack drew an array and with questioning as recommended by my Alcoa Champion of Maths mentor, Shyam Drury, he was able to identify the bananas and the trees in the array. Someone else did the repeated addition. Someone else did the multiplication. Collaboration of sorts; Great learning.
Problem Number Two
The next problem required ‘5 practices planning’, which I would also use an ‘echo’ sound effect for. I like planning. I like planning documents especially. The ‘5 practices planning’ was nice, but to be honest I didn’t really ‘get’ what I was doing and I was trying to make my planning fit and it didn’t exactly. But I kept pushing that square peg into that round hole, because I am also a list checker and I needed to tick that box. Needless to say, the second problem didn’t go so well, mainly because I misinterpreted it! Nice one Campbell! Thank goodness I’m not a doctor – you know ‘first do no harm!’ I actually misinterpreted ‘four more dogs than cats’ as meaning there should be four times as many dogs (multiplication rather than addition). As I said early, I am very good at modelling mistakes, so although I should probably be embarrassed and ashamed I am more ‘Hhhph. Another learning opportunity.’
A good thing did come from problem number two. I was still riding shotgun over the children’s behaviour, so I was focusing very hard on trying to anticipate every little thing that possibly could cause an upset to the collaborative
process. I realised that in this cats and dogs question there was a possibility that some students would get hung up on being able to draw a cute cat or a sweet little dog. I have seen this before. I have seen students get up and leave a class because they felt they couldn’t draw well enough. The point of the lesson is the mathematics. It’s not an art class, so I decided to substitute the animals with symbols, triangles for cats and circles for dogs. Everyone can draw those. This was a good idea for the reasons I’ve just explained, but it was also a good idea, as Shyam pointed out, because mathematics uses symbols to represent things. It was a good way to get that idea started and it stopped people from feeling embarrassed and frustrated. You will also be pleased to know I have fixed my own misunderstanding and presented a new problem, let’s call it problem 2a, which was well received and well solved. Same dog; different leg. Better result and we can use the same symbols – genius!
Problem 2a was well received. Most of the students identified that it was a problem solving activity. As a group we discussed what type of problem it was and what operation was required to solve it. Most of the students recognised it was an addition problem due to the word ‘more’. Although some thought it was a subtraction problem, which we have been learning about, and everyone agreed that subtraction problems give us less. This question was about more, therefore it was addition. We defined the word ‘more’ as meaning extra and away we went. Most importantly, no one thought it was a multiplication problem.
The success of this problem was 50/50 from a mathematics perspective. On a small group level most of the students were discussing and working together, and explaining their thinking. Some students were in over their heads but were still participating, listening, drawing and even making suggestions. I could see the penny drop for some. They look at you through squinted eyes while you ask them questions like, ‘is four more, the same as four?’ ‘Is that four more, or just four?’ ‘If that’s four, what would four more look like?’ It’s funny how eyes open and brows go up when an ‘aha’ moment occurs. For one particular group I knew they had it early when they could solve the problem verbally. For example, they had drawn that 3 pizzas would mean 7 pies. When I asked how many pies you need if there were 10 pizzas, they answered 14 after a few seconds of obvious counting in their heads. Some groups sounded like they had the gist of the problem, yet when we came together to discuss our findings, they weren’t as confident as they had been. Is it bluff, intimidation or perhaps both? Something to work on and watch for.
Our work with the Alcoa Champion of Maths program is having obvious effects on my class and me. We are definitely still learning and embracing the problem solving process. It has had a big impact on collaborative working and behaviour. From the maths perspective the Alcoa Champion of Maths program is making my students more self-assured, engaged and interested in maths. They are learning, some faster than others, vocabulary, communication of ideas and a problem solving process that has equipped them to approach other areas of learning and maths with confidence. I know this program is having a positive impact on my students when I hear them telling their families about what we are doing. I have seen some of my most vulnerable students asking valuable questions during the recent problem-solving activities. Before an activity we did the other day I asked if there were any questions before we got started on the problem in the image below. Question from vulnerable student: What size are the tents? Nice!
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