The Structure of Thinking About Visual Patterns

I’m planning a lesson for 3rd Grade right now. We’ve been working on making representations of patterns using graphs and models, and as part of that work I’ve been asking them to find “jump-ahead” steps, e.g. the 43rd or 21st step or whatever.

Their work on Friday was great to see. In particular, there was a controversy over the number of total squares in the 21st step. Some students drew attempts at accurate diagrams that they miscounted or misdrew, or that somehow didn’t capture the structure of the pattern. Other students reasoned without trying to draw the 21st step, and they often made misgeneralizations.

The most accurate students tended to do a particularly sophisticated thing: they drew schematic diagrams of the highest step that accurately captured the structure of these patterns.

Coming in to today, I’m thinking about making explicit two important features that distinguish the most accurate and efficient work on this problem in my 3rd Grade class from the least.

• Seeing The Structure: The strongest work articulated a productive way of seeing the pattern and then applied that to a higher step. That productive way of seeing it involves thinking about how the number of WHATEVER is related to the step, e.g. “There are 2 columns in the first step, then there are 3, then 4” or “There are two rows on top of each other and at first the grey are just 1 long, then 2 long, etc.”
• Drawing a Schematic, Rather than Accurate, Picture: The strongest work also used a schematic representation of the jump-ahead step rather than trying to draw an accurate picture and count it.

If I knew what productive thinking looked like for “jump-ahead” work with visual patterns over a wide-span of grades, I could have a bank of mathematical goals, feedback, and hints to give students. Further, I could tightly integrate goals, feedback and hints and really focus my instruction in productive ways.

I’m about to head into class. I wonder what other productive mathematical moves I can find in their work today. I also think that explicitly articulating these important features of the strongest work will help close the gap in the student work. I’m OK modeling it, though I think I could also draw their attention to these features by analyzing some excellent student work.

Maybe articulating the structure of student thinking about visual patterns would be a helpful first mini-project to take on for me?