Digging Deep into Proportional Reasoning

This past fall, we continued a series of cross-panel co-teaching days with some of our grade 7/8/9 math teachers. In addition to co-designing and co-teaching a lesson in a grade 9 classroom, we also spent part of the day dedicated to digging a little deeper into a continuum of concept development for proportional reasoning.

Ahead of time, we asked each teacher to give the following task to their students. Calculators and manipulatives were allowed, and the question could be read to the students, but no instruction or guidance was allocated.

The point of the task was two-fold. First, we wanted to introduce teachers to the idea of how students develop proportional thinking. In upper elementary grades and in secondary math courses, we often jump right into the more advanced concepts, without looking back to see how students learned the basics (or even if they have learned the basics).

Second, we wanted to give teachers a chance to see how their students would fare on a question with no coaching, and in some cases, no teaching (some classes had not reached their proportional reasoning units when this was assigned).

The Proportional Reasoning Continuum

Teachers brought their students' work to the co-teaching day, and in cross-panel teams, each student's solution was placed on a continuum of proportional reasoning concept development.

We created the continuum based heavily on Dr. Lawson's work on the multiplication and division continuum, and a continuum of concept development for proportional reasoning presented at a Renewed Math Strategy symposium last year. It outlines how students first learn about patterns and begin to shift their thinking away from absolute thinking (where caterpillars and leaves are considered in isolation) and toward relative thinking (focusing on the relationship between the caterpillars and leaves). 

The further down the continuum, typically the more efficient the strategy for large quantities. Our qualifiers (along with descriptions) for the continuum can be found here.  

The Numbers

The original 2:5 caterpillar:leaves problem was introduced to us at a Renewed Math Strategy symposium, however we changed the questions asked of the students. The number of caterpillars we chose in each part of the problem (10, 60, 11, 23) was done so with great intentionality.

• We chose 10 because we wanted to see how students would make the jump from 2 caterpillars to 10 caterpillars. It was accessible enough that it could be drawn or modelled, or not too tedious for skip counting. Perhaps students would easily recognize a multiplier of 5 and use that (or go right to the unit rate of 1:2.5).

• We chose 60 to try and nudge students toward a more efficient method, if they had used a basic method like drawing, or repeated addition in the first question. We chose a number that we hoped they would recognize as one of their ten facts (6 x 10), and use that either as a regrouping tool for repeated addition (going up by 10s instead of by 2s) or as a multiplier (previous answer x6).

• We chose 11 to try and nudge students toward the idea of a unit rate, if they hadn't already found it. It was close enough to 10 that students might recognize that they need the same answer as in the first question, plus half as many more leaves. Or they might continue the pattern in the first question to 12, and then find the "middle" between 10 and 12 to determine the number of leaves for 11 caterpillars.

• We chose 23 to try and nudge students toward implementing different tactic - be it a unit rate, or doubling their answer to the third question and adding a half-step again. Of course, some students used the same method for each question.

What We Learned (and what we are still questioning!)

1) This was the first time many of the teachers had seen a concept development continuum like this, so there was a large learning curve in terms of looking at student work and identifying where on the continuum the students' thinking lay. This was a valuable first step in "noticing and naming" how our students were thinking through a problem.

As we looked at student work, we classified the thinking as the highest strategy used successfully in the task. If a student skip counted in the first part, but then demonstrated proper use of a scale factor in the last part, we coded the solution as using scale factors.

2) Surprisingly to us, most students - regardless of grade and/or exposure to proportional reasoning in class - were solving at least part of the problem using strategies at the high end of the continuum:

Students' work coded according to the most efficient strategy used successfully on the caterpillar task

Was this due to the way we had scaffolded the problem? Or are we underestimating what our students can do? Would the students have immediately jumped to higher-efficiency methods if we had only asked for the number of leaves needed to feed 23 caterpillars?

3) Some of our high-achieving students, who typically do not struggle with math, were demonstrating solutions lower on the continuum, such as repeated addition. Do students interpret "show your thinking" as "break the math down into basics?" Were they giving us what they thought we wanted to see? Would they have found unit rates if we had just asked them to solve the question without showing their thinking?

4) Some of our students would determine the number of leaves for one caterpillar (2.5 leaves) - a unit rate - however would not use it as a scale factor. Instead, they would count leaves for 10 caterpillars, and then add 2.5 leaves to solve for 11 caterpillars. Even though they found a unit rate, they weren't applying it proportionally. Despite finding a unit rate (high on the continuum), they used an additive strategy (low on the continuum).

5) What is the next step? Now that we are learning to notice and name, how can we nudge students who are still engaging in additive strategies to start thinking multiplicatively? How can we encourage students working on any given level of the continuum to progress through higher-level strategies? And how can we use the idea of intentionally choosing numbers in proportional reasoning problems to allow students to show multiple ways of thinking through a problem?