This week, we began exploring proportionality as a our focus concept for the next several weeks. Much like most of what we have talked about this year, my students seem to have a procedural understanding of ratios — as in, they can define the word ratio and solve problems in the section of the textbook called, “Ratios.” However in the task that we looked at in class, I didn’t use the word ‘ratio’ or ‘proportionality’ once. All I said was the following:

- I made a sonobe cube out of 6 x 6 origami paper.
- Then, I made a sonobe cube out of a larger size paper.

Your task: Can you determine the dimensions of the paper I used to build the second cube using only what you know about the original size paper and whatever other information you’d like to gather about the two boxes.

I was surprised by the variety of different strategies that came up as they worked through this problem. As a whole, they approached the problem by gathering a variety of different dimensions from each of the boxes — the side length, the volume of each box, the size of the triangles, the area of the different faces of the two boxes. They then worked to try to related the dimensions to each other to try to figure out what how the two boxes were related to each other.

Even though almost every single student can tell me what a ratio, and almost every student can complete isolated ratio questions, they didn’t make the connection between their prior work with the procedures related to ratios. One of the things I wonder the most about is whether or not they would have immediately tried to use a ratio if I had given the problem immediately following a lesson ratios.

That’s one of the things I that have noticed about student learning. Students demonstrate understanding of a topic during a unit of study — the teacher teaches a lesson within that unit, students answers questions and do tasks related to the topic, because they know that’s the topic that they are learning about — the might be applying the concept to a ‘new’ question, but they know what topic they are supposed to be applying because they know they are in the ‘unit’ of ratios, or adding fractions, or algebra and so they draw on that knowledge. And then we move onto the next unit. And when we do that, it’s like all the things they have learned get filed away in the back of their minds, as they expect that since they are done with the fractions unit, they don’t need fractions any more until next year. This is the biggest argument I have come across for considering teaching math in concepts and not linear, isolated units. Students need to spiral through and around math concepts over and over in contexts that are meaningful and thought-provoking. Students need to explore the connections between ideas and concepts in ways that they are able to draw upon their prior foundational knowledge.

Of course, assessment of this type of learning is a disaster at times — At the end of the day, I’m told I have report on my student learning in isolated strands, and it’s hard sometimes to separate out all of this rich learning into separate strands. Until we change our reporting practices, I will untangle the web of ideas and learning that comes from this approach to mathematics. It’s something I’m willing to struggle through, though, for the sake of my student’s learning and understanding.