I’m working with a school in Delaware (one of my all-time favorite states) and the teachers there are reflecting that in order to really teach great math, they need to teach reasoning. In helping them, I’ve been reflecting on what good math reasoners and learners do…

This is off the top of my head… to me it feels like taking what I know from the Mathematical Practices, the Process Standards, others’ work on Habits of Mind, and trying to put it into a short, kid- and teacher-friendly list.

What do you think?

The next step is to connect these with some activities to get the kids doing and reflecting on these reasoning moves…

Students who are ready to learn from problem solving, make and critique mathematical arguments, and apply their understanding to new problems do the following things:

Working on problems and applying their learning:

  • They read problem statements/scenarios multiple times and ask themselves: do I understand the story?
  • They represent math scenarios in multiple ways, and understand other representations they don’t choose to use
  • When faced with a problem that seems hard, they have ideas to try, like making a guess and seeing what happens, drawing a picture, using manipulatives, or estimating.
  • If they aren’t sure, their go-to is NOT an algorithm or a process
  • When they use an algorithm or a process, they ask themselves:
    • Why did I do that?
    • Does this answer make sense in the story?
  • They check their work in different ways:
    • Checking the story
    • Solving the problem another way
    • Using another representation
    • Making sure their work is accurate
    • Asking a friend to compare

Reflecting, getting better, and learning from others:

  • After they have an answer, they are interested in other ways to solve the same problem
  • When someone else shares a different approach, they pay attention to:
    • Things that are similar
    • Things that are different
    • Things that don’t make sense, yet
  • They ask questions to understand similarities, differences, and work through confusion
  • They take notes on other people’s thinking, and mark up their notes to help them make sense and remember
  • When they learn new ways to solve problems, they quickly try the new ideas out
  • They ask themselves, “does this make sense?” and if it doesn’t, they ask a question
  • They pay attention to processes and repetition in processes, in order to look for generalizations and shortcuts, and can explain why those generalizations or shortcuts make sense

Being a good math community member:

  • They seek out collaboration and offer collaboration when asked
  • They listen to ideas, and when they don’t understand or disagree, they ask a question
  • They try to find common ground (I think we agree up to…)
  • When there is a disagreement, they use definitions and assumptions to find the source of the disagreement (when you say this isn’t a rectangle, are you thinking that’s because it doesn’t have 2 long sides and 2 short sides?)