In this post, I am going to highlight some bits that resonated with me from the article 'In conversation with Brent Davis' from MT261. I have been interested in Brent Davis' work since reading his book Teaching Mathematics: Toward a Sound Alternative, which I cannot recommend highly enough.

He starts by describing his experience of mathematics at high school, and one teacher in particular who changed his view of mathematics:

When I was studying maths at university, every now and then whilst browsing through recreational maths magazines in the student lounge, I would notice that one problem or another had been solved by John Heuver of my high school in Northern Alberta. It was sort of mind blowing that my high school teacher was solving maths problems that were being published in world-distributed journals. Weirdly, it was only then that I appreciated the way he invited his students into thinking about mathematics problems differently... There are incidents in your life where "That's different now." It was one of those. I walked out of [his] room thinking, "Maths is different now."

Reading this brought to mind a Combinatorics lecturer I had at university called Victor Bryant. Many years later, I was exploring the subject again, and came across his wonderful Aspects of Combinatorics. I also found out that he used to contribute to (edit) Mathematical Gazette. I remembered the joy he brought to maths lectures, the way he used to lunge across the boards... This recollection also brings to mind John Mason's suggestion (MT248):

What matters most, perhaps, is the Mathematical Being of the teacher: their bearing, their mathematicalness, as exhibited in how they use and display their mathematical powers, how they prompt learners to make use of their own mathematical powers...

In his interview, Brent Davis goes on to describe teaching an "alternate class", akin to a bottom set, but more so. He describes how they taught him how to teach:

The mode of teaching the alternates was gentler. I was listening more. I had to. Whereas with the other streams I could take much for granted, I had to be fully present every moment with the alternates... One of the things they really underscored for me is that humans are not logical beings and so teaching wasn't the same as explaining...

I have found that working with groups that I have found challenging has taught me the most about teaching. This is particularly true of my attempts to teach primary children this year. I have just about managed to hold my own with a group of 8-11 year olds, but was at a loss with the 6-8 year olds - I didn't feel I was able to offer them what was required, and asked to stop teaching them.

Brent Davis goes on to describe the different modes of listening (listening-for, listening-to, listening-with) that he explored in his dissertation (and book), which have become relatively well-known in maths education. I found his insights on listening invaluable as a maths teacher, but also with regards to my training as a counsellor this year. Listening requires energy; it involves more than the ears. One must try to still one's own reactions and responses long enough to hear what is being communicated.

Davis then describes some of his recent research. One project, working with a group of teachers and children over several years, has resulted in his discovery of something he calls

*"pedagogical impasses"*:

Pedagogical impasses... are moments in teaching when you're stuck and don't know what to do next... The first time we tried [this mechanism], every single one of the 25 maths teachers reported an event that, to my hearing, had to do with shortcomings or breakdowns in their understanding of number... The experience was a trigger for a year-long concept study of number. What images and metaphors for number are necessary at what levels? What new insights do they make available? What experiences are necessary to make them available for learners?

I think that these questions are great prompts for a teacher or department who want to think about what they are offering learners.

I too have become aware of pedagogical impasses, or what I call 'pedagogical ruts'. I describe one in my recent diary extract in MT, when I was unable to see how the learner (L) was viewing the examples. Davis describes how difficult it is to see the shortcomings that are at the root of such impasses:

It almost seemed to be taken for granted that learners would shift to new meanings as the need arose... It became clear that teachers had all developed rich, coherent, integrated understandings of number that rendered these issues almost invisible to them. We had to struggle together to develop appreciations of the difficulties that might be encountered by learners.

It will be interesting to reflect on where the shortcoming in my understanding is, the next time I reach a pedagogical impasse. I wonder whether recognising one's shortcomings is even possible, without the help with others.