I love this description of the difference between understanding a problem and being able to propound the theorem that explains how it applies in all circumstances. As a lawyer, I see a strong parallel here with the task of formulating a legal principle. Lawyers sometimes list examples of the application of a principle as if that was the principle, without explaining the higher guiding rationale.
I appreciate the connections you are making here between how dialogue in a math community fosters students developing their expertise and "what it feels like to be a mathematician." Fostering environments where 'lightbulb' moments like these happen and students really see the deeper connections to the structure under the examples is a driving force in education.
I've been reading Heal and Berlin's new book Mental Models and Max's thinking reminds me of one of the early chapters that digs into experts and novices. The authors highlight the differences between how novices and experts approach problems, noting that "an expert doesn't see information as a series of isolated facts but rather as a network of interrelated concepts" (p14) and that "experts don't just know more than novices. They observe and organize information differently" (p15). Max's exclamation reflects his shift from novice-level thinking about subtraction problems in isolation toward generalizing something true about subtraction. This makes me appreciative of the power of working in community to develop expertise and wonder about the teacher moves that happened in that classroom to foster such a community.
Thanks for restarting your blogging, Bill! I look forward for more musings!
That's a super interesting connection, and captures something about the difference between what Max said and what the others said that I hadn't quite formulated. The others are observing facts about the situation, he is explicitly connecting the adding 2 with the "subtracting less leaves you with more: and presents them as a unified bundle (i.e. network).
I love this story and how well it illustrates what the experience of doing mathematics is really like. I have been in classrooms where this kind of mathematical work was unfolding, and it is one of the most wonderful things to see.
I love this description of the difference between understanding a problem and being able to propound the theorem that explains how it applies in all circumstances. As a lawyer, I see a strong parallel here with the task of formulating a legal principle. Lawyers sometimes list examples of the application of a principle as if that was the principle, without explaining the higher guiding rationale.
That's a very interesting connect, Lucy! It reminds me that I did consider studying law instead of mathematics, in my youth.
I appreciate the connections you are making here between how dialogue in a math community fosters students developing their expertise and "what it feels like to be a mathematician." Fostering environments where 'lightbulb' moments like these happen and students really see the deeper connections to the structure under the examples is a driving force in education.
I've been reading Heal and Berlin's new book Mental Models and Max's thinking reminds me of one of the early chapters that digs into experts and novices. The authors highlight the differences between how novices and experts approach problems, noting that "an expert doesn't see information as a series of isolated facts but rather as a network of interrelated concepts" (p14) and that "experts don't just know more than novices. They observe and organize information differently" (p15). Max's exclamation reflects his shift from novice-level thinking about subtraction problems in isolation toward generalizing something true about subtraction. This makes me appreciative of the power of working in community to develop expertise and wonder about the teacher moves that happened in that classroom to foster such a community.
Thanks for restarting your blogging, Bill! I look forward for more musings!
That's a super interesting connection, and captures something about the difference between what Max said and what the others said that I hadn't quite formulated. The others are observing facts about the situation, he is explicitly connecting the adding 2 with the "subtracting less leaves you with more: and presents them as a unified bundle (i.e. network).
I love this story and how well it illustrates what the experience of doing mathematics is really like. I have been in classrooms where this kind of mathematical work was unfolding, and it is one of the most wonderful things to see.
Thanks Kristin, would love to hear those stories!