One goal I have in my teaching is to support students in developing productive beliefs about what mathematics is and how they perceive their relationship with mathematics.
One phrase I’ve found myself using more and more with respect to student beliefs about mathematics is sense-making, and it seems like a useful exercise to try to define sense-making and articulate what I do to promote it in my students.
sense-making (n): a belief that mathematics is a logical system where new knowledge is consistent with and connected to previous knowledge; a disposition to search out that logic when learning something new
I don’t have any easy solutions for making this happen for my students. I do have a few things I try to do, and most important for me is spacing these activities over a unit to give students multiple at-bats with this type of thinking.
One useful method is what Ben Blum-Smith calls jamming — “posing a mathematical task in which the underlying concepts are essential, but the procedure cannot be used”. This Illustrative Mathematics task is a nice example. Kids tend to see function notation as “plug that number in for x and see what pops out”.
Here they have to connect that knowledge to what they know about functions and equality, and do so in a way that is often unfamiliar for students.
My second go to is just asking students to explain why something is true. For a function y=(x+a)(x+b), why are the x-intercepts at (-a,0) and (-b,0). In my experience, even kids who have discovered that principle on their own are likely to forget it and have trouble explaining it in the future. Spacing practice with this type of thinking helps to mitigate that forgetting.
Finally, I explicitly ask students to draw connections between problems. Figure out the odds of rolling three dice where the product of the three numbers is odd. Now figure out the odds of rolling three dice where the product of the three numbers is even. How can you use the first problem to help you solve the second problem?
These approaches often feel mundane to me. They don’t result in my most spectacularly engaging teaching. But they are short tasks, they lead to fruitful discussion, and they very often reveal to me that my students understand less than I think they understand. Then I can go back, dig into their understandings, and try to support their sense-making. No shortcuts, just hard work worth doing.