]]>The point of the Bucky Badger activity is not calculating the number of push-ups Bucky performed, rather it’s devising experiments to test our hypotheses for both of those two questions above, drafting and re-drafting our understanding of the relationship between points and push-ups.

Maths includes problem solving, but not all problem solving is mathematical.

The only important part of this task is to discover the variation in how the scores accumulate. That’s about football. The arithmetic part is trivial and hugely repetitive.

At the end of the task, what mathematical concepts have they deepened? My bet is most learned more about football.

Some would argue that “problem solving” is itself a useful task. I have two counters.

Firstly, it’s probably not true. Research shows that generic problem solving skills are not transferable. Problem solving is hugely context dependent.

Secondly, there are problems which involve a useful mathematical concept, which practice both problem solving and some technique or concept. We should be using those ones.

So while you may have learned a lot about how students think, I believe the students would have been better off doing a genuine mathematical task. One that links to other mathematical skills and concepts, rather than being a stand alone task.

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Regarding productive teacher beliefs, I’ve thought a lot about what they are, but I am less sure about how to help teachers develop them. Maybe they’re developed through just listening to kids and experiencing how much we learn from their thinking?

]]>But, yes, beliefs that support giving students opportunities to engage with doing mathematics (such as in ways you describe here) are likely to be those that you suggest in this post:

Doing mathematics is more than answer-getting.

Everyone’s mathematical thinking can constantly evolve and shift. Continually. There is no end to this.

Everyone’s current mathematical thinking has value and can be built upon.

An important role of teachers is to interpret students’ thinking before evaluating it. Holding off on evaluating and instead engaging in negotiating meaning with students supports their learning. And teacher’s learning.

Everyone learns in the classroom. Teachers are learning about students’ thinking and their thinking about mathematics evolves as they make sense of kids’ thinking.

The list goes on, but I’m reflecting on some of the beliefs that are underlying the ideas in this post.

(I’d go so far as to say that misinterpretations are not a helpful way to think about students’ thinking (or anyone’s thinking), but instead I’d recommend “current” interpretations. [My advisor for my PhD was Jack Smith, who wrote this significant article that shapes my thinking on this matter.])

In your post, I love love love how the everyone’s thinking about the problem itself evolved. That sounds like how mathematicians think and work!

Also, this made me laugh: “That’s unfortunate because push-ups are the worst and we should hope to do fewer of them rather than more.”

Thanks so much for sharing the work you did with fifth graders in this post!

– Yes, I agree with Dan that interpretive questions over evaluative questions matter. Also, having an interpretive stance rather than an evaluative one when listening to students is HUGE.

– I noticed that at the end of the lesson, Dan had students reflect on what they had learned. Giving students opportunity to become aware of shifts in their thinking and giving them agency to name and label those shifts seems like a powerful teacher move, too.