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What does mathematical thinking look like? I don’t mean that metaphorically, I mean: when you walk into a mathematics classroom what do you actually see and hear that is ‘evidence’ of mathematical thinking?
When I was working on the Underground Mathematics project I was forced to engage with this question on a number of occasions. As part of the continued internal evaluation of the project, a number of small case studies were conducted in which members of the Evaluation team would visit schools, talk to teachers and students, and observe lessons in which Underground Mathematics resources were being used. However, if these case studies were to return useful information, the Evaluation team needed to have a clear understanding of what we, as resource designers, saw as intended outcomes of working with Underground Mathematics as a teacher and/or student. These needed to be shaped into ideals that could be measured or recorded in some way in order to be meaningfully shared with our various stakeholders.
We were often challenged, in the extreme, to discuss the following statement: “Using Underground Mathematics resources leads to better A Level results”. This was strongly opposed within the team as something that couldn’t possibly be measured meaningfully – exam results will depend on a huge number of factors, and even within the context of Underground Mathematics, we were clear that the resource itself was not enough to guarantee anything – it was how the teacher chose to use these resources and combinations thereof, and in turn, how the students interacted with them that would determine ‘value’.
So instead, we began to talk about our resources being designed to encourage ‘deeper mathematical thinking’. Now this sounds wonderful, but unfortunately our break-through was short-lived – what exactly does ‘deeper mathematical thinking’ mean, or look like for the observers of those lessons?
This was not a trivial discussion and the ‘beast’ of mathematical thinking made its presence felt in everything that we did from that point on. Key words and phrases began to emerge as somehow important: connections; talking mathematics; multiple representations; thinking ‘differently’; asking ‘what if?’ questions...
It goes without saying that we recognised that we were not describing something unique to Underground Mathematics, but actually a wider range of resources that, when partnered with a teacher or student, could help to develop desirable characteristics and mathematical behaviours that might act as proxies for mathematical thinking.
So, we’d translated the original problem into something more tangible: What mathematical behaviours were we looking for in students that could suggest that mathematical thinking is taking place?
It is safe to say that we did not come close to a definitive answer to this question but, over time, for us this became about identifying ways of working that could enhance students’ experiences of mathematics and designing our resources to promote these habits. The list below consists of some of the broader actions, behaviours, or characteristics that we chose to emphasize and explore with teachers when they engaged with us through professional development sessions:
• Talking about mathematics
• Asking questions of yourself, peers and teachers
• Drawing on previous mathematical experiences when looking at an unfamiliar problem
• Appreciating the importance of the journey through a problem, rather than the final solution
• Attempting to justify ideas and asking others to do the same
Now it has become ingrained in me as I work to question my own processes and the strategies that feel like ‘mathematical thinking’. I frequently find myself wondering “would I have noticed that thinking in someone else?” In the classroom that might translate into: “Would I have given a student credit for the mathematical thinking that may be taking place behind the scenes?”
This is also about my expectations of students and teachers: what I notice might depend on whether I go into that classroom wearing my ‘sceptical hat’, looking for errors and mistakes, or if I wear the ‘positive hat’, giving credit for the mathematical ideas and conjectures, however tentative. Finally, I am left with questions about what this means for my own teaching strategies, and the decisions that I make about when to intervene – or when to sit back and let the thinking proceed at its own pace.
Where is mathematical thinking found? Have a look at the two images below. What assumptions do you have to make about the teacher, the class, and the individual students in order to answer that question?