*This is the first in a planned series of posts about my course ‘Developing Mathematical Thinking’, a maths content elective for pre-service teachers training in primary and middle maths. All posts in the series are here.*

This week marks the start of Semester 2 classes at universities in South Australia and, with it, my ‘Developing Mathematical Thinking’ course. The university year for most students is early March to early July (Semester 1), then late July until mid-November (Semester 2). This elective runs once a year, so the first class is a release of eight months of excitement and anticipation.

During the last eight months, I’ve been mulling over what worked and what didn’t in the previous two years. I’ve also been fortunate to have rich discussions at conferences, workshops and on Twitter, and to read thought-provoking blog posts, books and articles. In short, I’m ready to make improvements! To do this in a purposeful way, I’ve identified three goals.

#### 1. Slow maths

*‘The Slow philosophy is about doing everything at the right speed.’*

One of the Ignite talks at NCTM 2016 that really resonated with me was Jennifer Wilson‘s ‘The Slow Math Movement’. (You can watch the Ignite talk, and read more about Slow Math, here.)

One reason that I initiated and developed this course was to give students time to experience mathematics (rather than just being told mathematics) — the same principles as the Slow Math Movement. But I’m sometimes too eager to move to the next experience, rather than savour the current one. In her talk, Jennifer (quoting from Carl Honoré’s book, ‘In Praise of Slowness’) said ‘The Slow philosophy is about doing everything at the right speed.’ My goal this semester is to pay attention to and respect the natural tempo of the mathematical experiences happening in my class.

In a similar vein, someone in the MTBoS recently tweeted that ‘Too much information is the same as no information’. If I want students to meaningfully reflect on their mathematical thinking (an intention of the course), then I can’t give them too much to chew on at once. So, I’ve slowed down the syllabus by spreading the big ideas throughout the course. To be a confident and competent problem solver, there are many important components: writing yourself notes, being attuned to emotional responses, developing resilience, questioning, mathematical reasoning, reflecting, and more. But, in the same way that you don’t need to read the whole manual before you start driving a new car, you don’t need to know all the aspects of effective mathematical thinking before you start tackling problems.

#### 2. Remember the close

There are two ‘closes’ that I’m talking about here: closing the circle, and closing the (mini)-lesson.

**Closing the circle:** Earlier this month I was part of a micro-conference on the Gold Coast that aimed to help participants develop their ways of thinking. I was listed as a presenter, but I took far more from the workshop than I contributed. Led by Anthony Harradine, this was a masterclass in facilitating how people work and think mathematically. Anthony emphasised ‘closing the circle’ — bringing resolution to participants’ work on a carefully-chosen problem. This sounds obvious, but again, I’m often too quick to move on, rather than bring the mathematical experience to a satisfactory close.

**Closing the lesson: **Tracy Zager, in her TwitterMathCamp keynote a fortnight ago, talked about the importance of ‘closing the lesson’ — pulling out the learning to create a lasting impression. A common theme in my past students’ responses to ‘What did not go well for your experience in this course?’ was that they couldn’t see the point of some of the activities we did. In her QAMTAC workshop, Gemma Mann noted that adult learners need to understand why they are doing something. I don’t always want to reveal the ending at the start of an activity, but I do need to remember to close each lesson (or mini-lesson), that is, to explicitly articulate and summarise how a particular activity was intended to help students develop particular aspects of their mathematical thinking or habits, and to link that activity to the overarching course framework.

#### 3. Guide students in using different lenses

To help students further understand the purpose of some of the learning experiences, and to clear up the confusion about whether this is a content course (‘but you don’t seem to teach us any maths’) or a pedagogy course (‘but you don’t seem to teach us how to teach’), this year I decided to explicitly start my very first class with a discussion of the different types of lenses through which students can view each activity.

The discussion progressively revealed these four lenses. These are my naive descriptors, thought up in a moment of need. I know that it is not as straightforward as delineating content and pedagogy in this way, but it suffices for my class. (By the way, I couldn’t find much elsewhere about this, so I mustn’t be searching with the right terms. Please — point me towards other, more informed, descriptions on different ways to view classroom learning tasks.)

- ‘Teacher’ lens: ‘How would I teach this activity?’
- ‘Student’ lens: ‘How would my students approach this activity?’
- ‘Learner’ lens: ‘How am I, as a learner, engaging in this activity with respect to mathematical concepts?’
- ‘Mathematical thinker’ lens: ‘How am I, as a learner, thinking mathematically? What mathematical habits and processes am I using/developing?’

We then followed up with an activity that I’ve written about before, Fold-and-cut, in which I first invited students to use the ‘learner’ lens as they engaged with the task. Later, we reflected on it through the other lenses, drawing out the connections with mathematical topics, articulating the thinking processes we used, and analysing the choices that I made as a teacher in presenting the task.

My goal is to make reference to these different lenses often through the semester. This will help make explicit the intended purpose of each task in relation to this course, but also encourage these future teachers to more thoughtfully choose the activities they use in their own teaching.

#### Summary

I know that there are so many more aspects of my teaching that I need to work on — asking better questions, classroom management (still necessary with adult learners, unfortunately), being succinct in my directions — but these are the ones that I choose to focus on now because I think they’ll have the most impact.

I still haven’t really described what we did in the first class, but 1100 words is already too many. Look out for that post soon. By the way, the Friday Five is making a return, but I mucked up my posting schedule this week. Look out for it next Friday — with luck!

Dr NicHi Amie

That was interesting. I totally get what you are saying about different lenses. When I take PD, sometimes the teachers are being the students, and then they have to reflect on what they did as students. And sometimes they are just learning stuff! I use little graphics on my Powerpoints to try to indicate the role they are playing at each point, but even then it is murky,

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