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?)

Thanks Max for this insightful article. I live in the UK and the government has just introduced new Maths exams for GCSE (age 16) and A-level (age 18), which have a much larger problem-solving component than the previous qualifications. Teachers here are really struggling to instil problem solving skills in students, as students are used to the procedural or algorithmic style of thinking that you mention above. In particular, STEP exams (for Cambridge university entrance) require resilience and many of the skills you mention above, but as they aren’t given prior training in problem solving, they really struggle. What would you recommend to teachers who are trying to develop problem solving skills within their students? Michael

Michael, thanks for your reply. I’m always glad to hear when assessment are focusing on problem-solving and resilience because I think that’s what we as a society really need from students, so the more we can assess it and focus instruction on it, the better.

So how do we make the shift? The UK certainly has some great resources in the Shell Centre and NRICH. I also wrote a book,

Powerful Problem Solving, published at Heinemann in the US (I think they’re distributed by Pearson in the UK, or available on Amazon.co.uk). I’m hoping to expand this post about what we’d like students to be able to do, into a series of posts about classroom activities to help students become aware of, reflect on, and get better at each of these skills, using activities fromPowerful Problem Solvingand other sources. So… stay tuned?Thanks,

Max