I have been teaching mathematics now for the last 14 years, and have had several roles over that time including classroom teaching, curriculum leadership and mathematics coaching on a school and regional level. I feel I have grown a lot in those 14 years, to the point that I do not recognise that teacher that I was all those years ago at the start of my teaching career. But I also like to think that I do not even recognise the teacher I was 4 years ago. There have been two groups that I have learnt an incredible amount from over the last four years, one is is the MTBOS, an online tribe of mathematics teachers who I have been following for over a year now. The other is The Flinders Centre for Science Education in the 21st Century, who are responsible for the success of the of the Empowering Local Learners Project in the region in which I work.
I have been writing this blog now for a while, but only a little more seriously in the last year or two. I don't think I ever intended anyone to read this. In fact I probably hoped they wouldn't. Writing this stuff down has helped me to make sense of my teaching, my professional growth and how the thinking of my student's develops over time. I could have done that offline in a notebook or document, but in many respects putting it online in my own mind meant that I was making myself accountable to the work I was doing in my class as anybody who stumbled upon it could read it. However I haven't taken the plunge into trying to share it more widely as I still don't think it is as good as I would like it to be. However with some of the talk on MTBOS at the moment it seems that many are in the same boat as me. They have been silently watching the MTBOS for a long time, and are learning a lot from it, but have never posted anything. I know it is an incredibly supportive group who want are keen to tap into as many ideas from Maths teachers as possible, but the fear has been around how much I respect this group and in hoping that I am honestly applying their ideas to my own practice within my context.
So this post is about me being brave and jumping in. The aim of this post is to hopefully share how much I have learnt through the lens of just one task on algebra, in fact it was my first task that I did on algebra with my year 8's. I am not sharing this task because I think it is a fantastic feat of mathematical teaching brilliance, I am sharing this task because how I approached it has become a normal part of my practice, it is how i try to approach all lessons. Therefore it represents how I have fundamentally changed my views on a lot of things from just a few years ago, growth I definitley wouldn't have seen without the aforementioned networks. I want to first give credit to where some of the ideas for this lesson have come from and then describe the lesson and the student thinking
I know I will forget to mention people in this section, you won't know it, but I have taken so many ideas and lessons form so many people that I find it hard to keep track of what I have taken from who. What is represented below is not all the things I have learnt from these people, but rather just some of the things that are pertinent to this lesson
Angela, Kristin, and Deb
I really appreciate, more than you will ever realise, the frequent conversations we have. They constantly challenging my thinking, keep me honest, and they consistently keep me reflecting on what it is exactly that I am trying to achieve. They have helped me to bring clarity to my teaching and to iron out many of the bugs. Your expertise in task design and the effective use of questioning in class discussions, through the lens of executive function has help me to ensure that it is the kids thinking that is most strongly represented and valued, it is their voice, not mine that is heard most often. I have also appreciated the strong evidence base to our work. Thinking about the research that sits behind the work has helped me to think about my practice in some very deep ways, on my own I probably wouldn't have had to the time to come across this research myself.
Jo Boaler (Stanford University)
Your online course for teachers has changed my practice in a lot of ways, but probably for algebra more than any other. The clear importance of visual representations for building algebraic understanding through the examination of patterns is not something that I had given enough thought to until I did the course. This one change has done more for improving kids understanding of algebra than anything I had done in the past.
Dan Meyer (MTBOS)
Your work has influenced me for a long time but in recent times your talk on "beyond relevance and real world" has had the most profound impact. The idea of a maths dial that you start turned low approaching the content from a very intuitive base has allowed many more or my students to have an entry point and feel comfortable sharing their mathematical thinking, it has given a voice to those who didn't feel they had a voice in a maths class previously
Fawn Nguyen (MTBOS)
Your visual patterns website has been a great resource and an effective framework for my algebra work and hopefully this is represented here. But much more than that your passion for the students you teach and your drive to authentically and honestly represent the learning that goes on always keeps me accountable to trying to do the same for my students.
Andrew Stadel (MTBOS)
Your talk on the classroom clock has helped me to bring much greater clarity for how I choose to spend my time in my lessons and how I can further prioritise more time towards those more effective practices. It has helped me to regain the time I needed to give my students more time to think.
Robert Kaplinsky (MTBOS)
The #observeme process has really helped me to open up my class to others and to get the feedback that has helped me to grow as a teacher. Although I have not had many visitors, I am inviting many more people in and simply the process of writing the sign has helped me to articulate a vision for what I want my class to look like and what I am striving for.
The kids arrived at my lesson today on the first day of term, I stood outside to class in the common area to greet them. They asked if they can go into class, I told the "no we are staying out here", they asked what we are doing and when I told them algebra the groans went up almost in unison. It always happens, most kids seem to hate and fear algebra, so my first step is to attempt to take that fear of it away by showing them that it is not really as daunting as they think it will be. I can understand why they fear it, in my experience algebra has always been a topic that teachers feel the most need to teach from a proceedural base, they find it harder to develop conceptually. Algebra becomes about moving numbers and symbols around a page rather than really getting into what it is about, patterns and relationships.
I started by isolating one table in the common area and asking them how many chairs they feel would fit comfortably around it to which they replied 4, I then put two tables together and asked again and then three, creating the pattern below. When I moved these tables I intentionally moved the chair on the right out of the way, joined the next table on and then moved that same chair back to the right to help develop the idea of a constant (left and right chairs) I then brought two additional chairs in to put on the top and the bottom to help develop the idea of the rate of change and also how the pattern is growing.
As a starting point they were then asked to predict how many chairs would be around 4 tables and the quick answers provided were 10 and 11. When there was not agreement, rather than telling them what was correct, giving them a few extra minutes to discuss which was correct was enough to bring them all to the same thinking. Just giving them the time to stop and think, and to refine their original quick thinking has been valuable, they often answer quickly looking for me to give them the answer rather than taking the time to think more carefully. After settling on 10 as the answer for the 4th in the sequence their work for the task was to answer two questions.
The questions are very closed, but intentionally so, this was their first lesson on algebra and they had not seen questions like this before to the aim was to build my starting point, to see how they approached it, to see how I could build their intuitive understanding of the pattern into a deeper understanding of the concepts that sit behind the pattern. This is a task I could hook all of the learning on for the rest of the topic.
With limited or no input from me they were away and the approaches varied, but the three that were most prevalent were the ones that I have shown below
Some had attempted to draw the 54 tables and tried to count the chairs, and that is fine it worked for them. They hadn't noticed a pattern from the first three, but what drawing them all out allowed them to do is to recognise that they could then simplify their picture to one of the ones below it. The processing of drawing them all was the light bulb moment they needed. The thinking needed to be captured and shared, but it also needed to be refined, I wanted to take their thinking, which they were happy with but to them did not represent the algebra they were used to seeing and to transform their thinking to that more pure algebraic form by helping them to strip aspects away. The image below shows the process it took to do this
It started with their visual models and their calculations. As they talked about how they did it we recorded their words. We started a process of removing some of the words that they felt were not necessary in describing the pattern. By also substituting some words for symbols and then substituting other words for pronumerals they were able to refine their inital thinking down to an equation . The equation made sense to them as they could see with much more clarity how their thinking was represented in that equation, that equation was no longer a daunting and unfamiliar thing.
This process was repeated with the second of the two questions, but unlike the first question, there was a great deal of disagreement about the answer. In order to arrange 238 tables we had a roughly even split between the answers of 117, 118 and 119. We have had this at many stages over the course of this year and my response is always the same, first is to ask them whether this is a question that can have multiple answers, if it is then those three answers might be fine, but if not we need to agree on just one. They talked about it and came to the view that it was a question with only one answer, so the second part of this process is to have each of the groups try to convince the other that their process is correct. Whilst there was two or more answers on the table then as a group, we hadn't developed the understanding that we needed to in order to move forward. They were able to lead that conversation and the 118 group was able to successfully convince the other two groups that they had got it correct, there is a lot more power in them owning that process. Those so at the end of this we had four equations that looked very different. They were able to articulate that the operations use the same numbers but were the opposite operation (- instead of +) when you are trying to find tables rather than the chairs. This insight will serve us well as we move into solving linear equations at a later stage in the topic.
However in looking at those equations I was not yet convinced that they were in fact different representations of the same thing so I asked them "if I graphed each of these" what would I see, all different graphs, 2 different graphs, 1 repeated graph?" Some thought they would all be different because all the equations are different and others thought that the ones on the left would look the same and the ones on the right would look the same. plugging these into Desmos it became clear to them that they were all equivalent statement and they were able to articulate that they had to be the same graph because they were all representing the same problem, the same pattern.
I was happy with how this went, they did really well, there was a lot of thinking here and a lot I can tap into as we build this understanding as this topic progresses.
Senior Leader of Pedagogical Innovation and Mathematics Coordinator in Regional South Australia.
Opinions in this blog are my own and do not necessarily represent the views of my employer.