The Global Math Department talk tonight was titled “Draw a Picture: Using Diagrams to Make Sense of Word Problems”, from Nicora Placa. Really impressed with her synthesis of mathematical research and experience in actual schools with actual kids. Big takeaways on word problems and pictures, mostly Nicora’s ideas and some things she has spurred me to think about.

1. Diagrams reduce what students hold in their head. In more technical jargon, it reduces the working memory load. Increases accuracy, the complexity of questions students can access, and quality of learning.

2. One of the best things some of my students do in complicated word problems is to draw a line to separate their workspace. It’s neat, it’s logical, and it keeps work organized in a way that reflects the thinking they are doing

3. We need concrete ways to help kids with hard problems. What I want from students is critical thinking, perseverance in the face of hard tasks, varied problem-solving skills, and divergent thinking. But telling kids to think about something, to persevere when they’re hopelessly confused, to try a different method when they never had a method to begin with, is enormously disheartening. It teaches, with a very rapid feedback loop, that they are incapable of doing hard math, and so there’s no point in trying. Chris Lusto has an excellent post on the topic here. Draw a picture is *concrete* and *specific*. It’s not “try a different method”, it’s “if you get stuck on a word problem, try drawing a picture”. It’s not, “show your work for a reason you don’t understand”, it’s “remember how writing some things down helped last time? Try it again”. This is a bit of a tangent, but I want to teach grit and perseverance and problem solving in my classroom. I don’t think I can do that by telling kids that grit is important, or telling them that if they just have enough grit they will be successful. I think part of teaching grit is giving kids specific tools that they can use to be gritty.

4. Pictures can be a great way to introduce very basic algebraic thinking very early on. This is a big step on the staircase of abstraction, and the earlier kids are exposed to algebraic ways of solving problems, the more skills they will have in algebraic representation when they really need them in the upper grades. Consider this question, from Nicora’s presentation:

“Nicora wants to buy herself a new bicycle that costs $240. She has already saved $32, but needs to make a plan so she can save the rest of the money she needs. She decides to save the same amount of money for the next four months. How much will she need to save each month in order to buy the bicycle?”

Kids can interpret this a number of ways, but they split based on whether they see it as “subtract, then divide” — which leads to the right answer, but cuts out the most complicated thinking — vs thinking about an unknown, and how many unknowns they are, and how they relate to $240 and $32. Students don’t need to use a variable, but drawing a picture *is* algebraic reasoning here, and if that connection starts happening in 5th grade rather than 8th, Algebra will be a much easier go for everyone involved.