I had another interesting experience while listening to a student thinking about math today. This time I had the opportunity to listen to a grade 4 student’s thinking on the following question:

“I am reading a book with 207 pages in it. I have read 188 pages so far. How many more pages do I need to read to finish the book?”

This question is a part of our board-created interview assessment that provides teachers with an opportunity to dig into student thinking with questions that address the Quantity and Operational Sense Big ideas in our Number Sense and Numeration curriculum. Missing addend questions, like the one above, seem to pose a particular challenge for our students, according to our EQAO data (and confirmed by our teachers). So, these questions are a mainstay across the grades in our assessment.

One particular grade 4 boy was working on this problem and he asked to use the hundreds math rack. I was curious about his choice of tool and so rather than giving him one, I asked him to explain to me his plan for solving the problem. This is what he very clearly articulated to me:

“First I would make 188 on the math rack” (which I found interesting, since he had asked for only ** one** 100 math rack). “Then I would add ten to get 198 and then I would add 10 more to get 208” (with no hesitation on 208). “But I only want 207, so I would take 1 away. Then I could see what the answer is.” WOW!

I was very excited and troubled at the same time in hearing this. What a fabulous strategy, but why did he feel he needed the math rack to figure out what he had already figured out? I recorded his words on his paper (afterwards I reflected that I could have perhaps modelled his thinking on the number line, but considering this was an assessment of his thinking and he hadn’t gone to the number line himself, perhaps this could go down as a next step for him). I had him read back his words and I asked him if he could figure out his answer now without the math rack. He thought out loud “10, 20, 19 – 19 more pages!” I celebrated with him that he had used such an efficient strategy – and used a mental model to do it!

I’ve thought a lot about what might have happened if this student had immediate access to the math rack. Would he have found himself getting lost in trying to model 188 on this tool, possibly forgetting the purpose of doing it in the first place? Would there have been so many steps that he might have made an error. The solution he articulated to me was so efficient that there practically wasn’t room for error. It also highlighted his understanding of counting on by 10’s from any number (such an important understanding/skill for number line use). To me, this student already has created a useful mental model that he can use for addition and subtraction, yet he is still closely attached to his concrete math rack.

I think some next steps for this student would include more experience with similar contextual situations, where his thinking could be modelled (by a teacher at first) to an open number line. In this question, the numbers could be held onto mentally, but what would happen with larger numbers or numbers with a greater difference? Encouraging a more flexible, mental tool may empower him to work with more challenging numbers and various contexts.

This experience is causing me to wonder again about the intentional use of concrete tools and the transition to graphic and mental models. I do believe that concrete models/manipulatives are very important in building conceptual understanding and exploring strategies and big ideas. However, we need to know how to identify when these concrete tools become crutches and may actually prevent students from further building flexible, efficient and accurate ways of solving mathematical problems. We need to be prepared with a deep understanding of a variety of models, including flexible graphic models (like open number lines, arrays, rectangles for partitioning, etc.). We need to know how to support students to see their thinking on different models and make connections between them, with the intention of creating powerful mental tools for thinking over time and with varied experiences.

Using concrete models for computational thinking is not the end goal, but is an important step in the journey. Becoming fluent thinkers with a variety of strategies and tools based on conceptual understanding is the end goal. I am looking forward to continuing to learn more about how to support students in making this transition towards flexible mental models.