Today as my principal and I were having a meeting looking at our EQAO results we came across a great example of a prompt where the importance of being a multiplicative thinker really becomes evident. As we chatted we discussed how it tied in with our focus at the two schools which is really digging in deep and knowing our learners in their operational development from counting to **multiplicative thinking**. (Note: There is a stage higher than multiplicative thinking and that is exponential thinking) We choose this prompt to look at because it was the question this group of grade 6’s scored the lowest on. Only 18 percent of the students answered it correctly. Here is pic of the prompt:

The prompt is actually coded as a thinking prompt under the strand measurement. Take a minute and break it down into what the students need to know and do in order to solve this task. Then look at the list I made in the meeting with my principal below. Maybe I have missed some and if I have leave me comment. I would love to see other people’s thoughts.

**(Note: I only looked at the concepts needed not the problem solving strategies etc.)**

The fact that this measurement prompt sits on top of multiple number sense concepts is worthy of another blog post entirely!

Even if I missed some other pieces in my break down of the prompt I wanted to focus on what I have highlighted in yellow. The idea that students need to move to multiplicative thinking in order understand scaling, unit rates and proportions becomes fairly clear in this prompt. If they can’t compare the quantities in a multiplicative way they are going to struggle with this task. I like Kathy Richardson’s explanations, here are some pics from one of her books:

She clearly states that **“Children need more than how to get answers to multiplication and division problems. They also have to develop multiplicative thinking, because multiplicative relationships underpin many number-related concepts, such as fractions, percentages, ratio and proportion, similarity, functions and graphs, rates of change and algebra.’** As you can see by the list of concepts in the excerpt from Kathy’s book, multiplicative thinking becomes one of the most important milestones in students math learning. It can stall much of a student’s learning from junior level on if they don’t successfully make the transition.

This task is great example to break down for teachers during PD to help highlight how students will struggle if they can’t think multiplicatively. It also can be used for an example of how just knowing your facts by auto-recall or using an algorithm to multiple is not enough. The students were allowed to use calculator on this prompt yet still struggled. It could be that the students didn’t retain the measurement concept of volume or the application piece of using the volume formula, but our suspicion based on diagnostic interviews of the grade 7’s at that school this year is that many are not yet using multiplicative thinking. There are still many counters and early additive thinkers in the class. In the diagnostic interview most of the class were not able to view relationships as “times as many” or proportionally. Its work that we are setting out to address but I wanted to share this prompt and the break down to highlight why it is so important to know our students developmentally in operations and to start closing gaps with students who are not multiplicative thinkers. I would love to read your thoughts on this so please leave a comment if you like! We are going to give this prompt again to the grade 7’s so we can then look at the student work that we can’t see on the EQAO results. I will let you know how it goes!