Thinking Like a Mathematician - Patterns and Growth
Over the year Arnalise de Robillard has improved her ability to plan, implement the ‘pre-lesson’ phase and anticipate how her students are likely to answer a question. All this preparation has provided some incredible wow moments along the journey.
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Wow, what a year! I have been contemplating how exactly to wrap everything up into a nice, neat, colour coordinated package. However, I’m certain that it won’t be possible, nor will I do this experience any justice. Therefore, I am going to revisit some of the key points that have really stuck with myself and my students this year. Planning, differentiation, implementation and responding on the fly (with consideration of course) will receive a mention. As well as reasoning practices and future pedagogy.
Since my last blog post, which you can read here, I have become much more skilled in implementing the ‘pre-lesson’ phase. In particular, targeting a key mathematical concept and anticipating the answers of my students. As a teacher who was a fan of consecutive and more encompassing learning intentions, I initially found it difficult to narrow the learning goal down enough. Honestly, I was also struggling with the curriculum pressure, as it was difficult for me to rationalise spending such a large number of allocated Maths minutes on a single idea. However, focusing my efforts on less restrictive questions and really narrowing down the learning focus allowed my students to be fully engaged in each problem-solving task in a whole new way. To my surprise it led to an increase in how they were transferring knowledge from other mathematical areas as well as noticing similarities and differences of questions we had worked on before.
The act of anticipating has been, at least for me, the cornerstone of this process. Anticipating how my students are likely to answer a question has added a different dimension to my teaching practise. In combination with the consideration of prior learning and analysis of diagnostic assessments, the anticipating action allows for more effective planning for success with the overall implementation. More importantly, it is an unquestionably practical element in this approach. Predominantly providing an essential component in terms of being prepared for and successfully differentiating within the context of a single task. Particularly in terms of allowing for targeted enabling and extending prompts. As well as the consideration of possible misconceptions and the plan for the rich mathematical discussion. Who doesn’t love a plan!
Over the year I have become more skilled with the planning for these types of problem-solving lessons. As I mentioned, anticipating possible responses with greater accuracy has had a major impact and it has in turn had a positive effect on the way in which I can direct students as I run a lesson. Additionally, during my most recent lessons, I felt that I gave a larger number of my students the opportunity to respond during solving and discussion points. Particularly with students whom may not be accessing the question at the preferred level. I have observed that they can still absolutely and positively be involved and hold an important role in contributing to the mathematical discussion. It is just up to me as the facilitator of the discussion to better understand at what point to bring them in, in order to achieve the overall learning goal simultaneously for individuals and the whole class.
Related closely to this is the actual implementation where all pre-lesson thought comes into play regardless if it is ‘spot on’ or if my students surprise me with something unexpected. How well I monitor what they are doing, talking about and focusing on as they work through the set problem in their small groups is a big part of how successful the post activity discussion is. Whether it is a task that they are working towards solving in the most efficient way, working to find a set answer or contemplating how to analyse, generalise and justify in a reasoning task, all their thinking comes together in the discussion, so I must select and sequence accordingly. But first, I monitor! I have to admit physically taking notes as I monitor my students during a problem-solving task doesn’t come naturally, as I find it a bit awkward. I much prefer not to carry anything as I move around the groups. However, I am improving and definitely feel less uncomfortable, especially as I notice that my students rarely take any notice of my notetaking. Regardless, this point of monitoring continues to prove crucial to appropriate selecting and sequencing to encourage a meaningful and relevant mathematical discussion.
Our workshop focus for the later part of the year was Mathematical Reasoning. This was something I absolutely loved because it felt like a sneak peek into the world of a real-life mathematician. I feel like my students thought the same because really who doesn’t appreciate a good debate? They absolutely love being able to analyse a problem and hunt for mathematical proof to support their conjecture, not to mention being given the floor during the discussion to convince others of their justification. This in particular has been one of those huge progress moments when compared with the beginning of the year. My students’ willingness to share, verbalise their thinking and search for logical, mathematical evidence for their justifications as they bring in a variety of mathematical concepts has been a bit of ‘wow’ moment.
This experience, although challenging, has had a positive impact on the way in which I view Mathematics and more importantly how I teach it, as I continue to put pieces of the puzzle together. By no means am I finished learning and to refer to Susie’s last blog post in relation to ‘accountability’, I am motivated to catch up with our Champions of Maths Team in the future to see what we have all been up to with our new-found knowledge. I am also looking forward to having time to reflect and process how this will continue to fit within my teaching practise. As term four is coming to an end I am thinking about how to incorporate these practices into my teaching in a bigger way; raising my students’ awareness of their Mathematical learning, enabling rich Mathematical discussions and above all promoting growth.