Explaining and Showing our Thinking in Maths

I want my students to think deeply, to identify patterns, to make connections and explore relationships, and to make their thinking visible – explaining and justifying as they do so.

With this in mind, over the past twelve months I’ve endeavoured to build a culture of thinking, most noticeably in number sense in mathematics in my senior primary classroom.

The learning journey has been challenging yet thoroughly invigorating. Regular check ins with Margarita Breed, who I chatted with on my Pushing The Edge podcast (see PushingTheEdge.org/53), reading and listening to Pam Harris, and devouring Peter Liljedahl’s, Building Thinking Classrooms in Mathematics.

Here’s what I’ve learnt:

The Task Matters

My students more deeply engaged in tasks, having richer conversations, full of challenges and disagreements when the task allowed for different strategies or different ways to get to the answer/s. Here low floor, high ceiling tasks were key.

Standing and Vertical Whiteboards Are Key

My students mathematised standing up around vertical whiteboards. With the upright boards, I could scan the room and determine commonalities and differences among the groups and tailor my teaching responses accordingly. I could ascertain which groups were working efficiently and which groups needed further support. Students could also, at a glance, see what was happening in other groups. Interestingly, I rarely saw students copy everything on another group’s board. More often they picked up an idea and then ran with it. 

Standing up was key to active engagement with my students. Positioning oneself away from the board or sitting down generally meant that a student had tuned out from the task.

Interestingly in other learning areas, when students were sitting down facing the whiteboards at the front of the room, they seemed to take it as a cue to disengage while I (hopefully) did the work for them. Years of facing the front whiteboard, listening to the teacher directing learning had obviously had an impact. 

Sitting at their desks, students’ books were hidden from me unless I swept around the room; a task that was much more time consuming than scanning vertical whiteboards. I’m aiming to extend what I do in maths – with vertical boards and standing up to encourage and support active engagement across the day – and lessen sit down tune out mode.

For further information: Peter Liljedahl’s research speaks to the value of standing and vertical whiteboards.

Random Groups Are Essential

For much of the year, my students worked in self-chosen groups with different levels of efficacy. Then in the final term, I switched to random groups (as suggested by Peter Liljedahl).

When I began, students weren’t told we were switching to random groups. I just displayed an online random group maker on the whiteboard and clicked it to form random groups of three. Students could then see that it was a fair process without teacher interference.

Often there was consternation; so many students unhappy with the groups they were allocated to. I just encouraged them to get started in their groups, supporting them to do so and the complaints generally dissipated. 

Where difficulties in groups arose, there were varying reasons for such. Some students for example, didn’t know how to work with students at different skill levels to them. There were also personality clashes and differences in willingness to complete tasks.

Rather than taking control, seeking to sort out the issues, I tuned in to see what strategies students employed, noticing what was working and what wasn’t working. I gave attention to, or amplified, the strategies that worked – unpacking the various components of the strategy with the class. I also engaged more deeply with those at the end of their tether with team-mates and supported them to come up with ways forward with their group. 

At the end of the year, referring to their random group tasks (for maths), I asked students what they had most improved in over the term. The vast majority of students said they hadn’t always liked working with others in the class but they were now used to it. They had also gotten better at working with students who weren’t their friends.

Some students noted that they often worked better in groups that didn’t have their friends in it. This was something I had increasingly become aware of. Students who complained the loudest about random groups participated far less when working in groups with their friends. Indeed there was often more conflict in their friendship groups. 

Having a Focus is Important

Over time I came to see the value of having a team-work focus and a show your thinking focus for each group maths task. 

For team-work, we focused on aspects such as commitment to task, active participation, persistence, supporting team-mates, and explaining our ideas. Peter Liljedahl suggests that students brainstorm what particular skills or qualities are and are not.

I tried Liljedahl’s suggestion focusing on commitment to a task, with students brainstorming, what is commitment? and what isn’t commitment? There was much commonality across groups, with students clearly highlighting the behaviours that indicated commitment or a lack of commitment. I summarised the key items and that commitment list became our focus over the next few sessions. It was what we reviewed ourselves against.

Students had an answer-centric approach at the start of the year, with getting the answer being the goal of any task, followed by quickly moving on. Less important to them was explaining their strategy (how they solved the problem), representing it through a model, and critically reflecting upon such.

Explaining, modelling and reflecting therefore became key components of our lessons. It was important that we verbalise strategies like:

• adding 10 (sometimes referred to as a friendly number)
• adding multiples of ten
  • 58+34=
  • =58+30=88
  • =88+4=92
• getting to a multiple of ten
  • 58+34=
  • 58+2=60
  • 60+32=92

This enabled students to develop a language to explain their strategies. I also encouraged them to annotate alongside their models – if they hadn’t written their strategies (or what they were thinking).

Also key was identifying and having extensive experience with a variety of  models to represent their strategies such as open number lines, equations or ratio tables.

I have learnt much about strategising and modelling from my maths mentor, Margarita Breed, and Pam Harris’ Maths is Figure-Out-Able podcast and her various maths publications.  

Noticing Must Be Part of the Routine

Key to student learning – and building of effective, efficient strategies was focused noticing by myself and the students. 

As I moved around the room, I circled, made annotations, and labelled sections of their whiteboard workings.  Crucial here was that students didn’t use the same colour as me. That meant that everyone knew that the red scribing was mine. 

I would highlight efficient strategies utilised, write effective team-work questions being asked, clear statements that explained their strategising approaches, clear representations (via models) of their strategies,  challenges that students were facing and many other things.

At times, my noticing related to our task and team focus for that day. Sometimes it was commonalities or differences amongst the group or differences that deserved attention. Also skills that needed to be developed.

Noticing sometimes happened during the lesson. I’d stop everyone, draw their attention to something, then get them to focus on that aspect as they resumed working in their groups. I’d also do a gallery walk at the end of lessons, using my noticings as the focus for the next day’s task.

Students also participated in gallery walks, noticing similarities and differences to their group’s work, along with things that interested them or they wanted to know more about. I would also encourage student to notice things we’d been focused on such as clarity in explaining their strategies, and clarity in their models. I asked questions like what makes this model clear and easy to follow? 

Passion and Interest Matters

I grew up in an era where you were or weren’t a maths person, finishing my maths education at secondary school in what was dispiritingly called, ‘vege maths’. It was like math skill or expertise was or wasn’t an intrinsic part of you. Memorisation, algorithms, and answer getting reigned supreme. 

When I returned to primary teaching a few years ago, I was determined to challenge the negative attitudes toward maths that so many of my senior students arrive with each year. I want them to know that, as Pam Harris says, maths is figure-out-able; that there’s no challenge that we can’t figure out or work our way through.

I want my students to know that we learn through verbalising and critically reflecting on our thinking and representations, and that mistakes can be starting points for our development in maths. To the extent that students come on this journey with me, I believe my passion and interest – shown through my words and actions – matters big time. My learning continues alongside my students.

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