Our last *sniff, sniff* book club meeting was held this week. We are so appreciative of the teachers who signed up to participate with us. We are so grateful for the rich and thoughtful conversations and collaborative trouble-shooting that went along with our study of Making Number Talks Matter. We really believe that professional learning is so much better with colleagues and that setting aside time for professional learning is great for our students, but also helps us (as teachers) stay excited about our work on a day-to-day basis.

This week, we looked at Chapter 9 – Investigations.

We started our meeting with our usual check-in about how Number Talks have been going in the classroom. It sounds like most people have established a good routine with Number Talks. Some folks are taking a short break and shifting focus but planning to come back to Number Talks in the New Year. We did some trouble-shooting discussion about how to deal with students who offer silly answers or make up strange answers that don’t relate to the question posed. We talked about using phrases that help the student connect their answer with the question (*can you explain to me where in the question you got the numbers that you are using in your strategy?*). We also talked about moving on from a student who is having trouble explaining his/her thinking clearly with a statement like: *I’m having a hard time understanding your explanation and I would like us both to think about it some more – can I check in with you about your strategy after the number talk is over?*

Next, we talked about our BIG IDEAS for the day:

And outlined how to do an investigation:

We then delved into exploring the multiplication strategy of halving and doubling using the general procedure for an investigation. We started with the question 8 x 26 to try to elicit the strategy of doubling and halving. Once we looked at all the suggested strategies, we focused in on doubling and halving and talked about the big question of “Does it always work?” The group split up into partnerships to explore this question – we provided graph paper, colour tiles, rulers, paper and scissors and then circulated to try to see how the investigation went.

It was interesting to note that it is really difficult to be a skeptic in Math – the strategy might make sense to us, but actually thinking about what it takes to PROVE that it works requires much more depth to our thinking. Many groups got started by discussing WHEN it would be good to use this strategy (ie. what circumstances/numbers make it an efficient strategy). Some groups explored odd vs. even numbers, some explored big and small numbers, some tried to delve into fractions to see if it worked there. Some groups worked with the colour tiles to make arrays and others used the graph paper to show the strategy visually.

Then, we wrapped up with a discussion – different groups shared their approaches and it was interesting to note how varied the ideas were. We looked briefly at the questions offered in the book to guide small group work for this investigation:

- Will it only work with even numbers?
- What would happen if, instead of halving, you took a third of one factor?
- Can you represent this strategy visually/geometrically?
- What generalizations can you make?
- Would this work for division?

We had hoped to have time to also have teachers choose another investigation: either the same difference strategy for subtraction or the halving-halving strategy for division, but we ran out of time. I think participating in an investigation was really valuable. I enjoyed seeing how much mathematical thinking (curricular competencies) is involved in this type of activity. Of note – mathematical investigations take time, and it is worth setting aside some time to do activities like this in class.

Once again, a HUGE thank-you to our teachers who participated! We will be running some more after school PD for intermediate math teachers in the New Year – likely a Mathematical Mindsets book study at some point – so keep an eye on your email in January for information on how to join in.