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The work of the Center for the Scholarship of School Mathematics is based on three key principles:
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The mathematics people study feels personally relevant to the people engaged in its study.
Different people have different needs for and interests in mathematics. On the surface, this suggests that people with different professional
interests might not want to work on mathematics together. In fact, just the opposite is true. The superintendent who cares about students’ performance on area and perimeter problems on high-stakes exams, the teacher who is teaching area and perimeter, the mathematics educator
who is studying how middle school students learn geometry, and a mathematician in the field of differential geometry might all find the following
question interesting and "relevant":
Are two triangles with the same area and the same perimeter necessarily congruent?
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Participants encounter the mathematics as mathematicians.
In his article, “On Proof and Progress in Mathematics,” William Thurston poses the question, “How do mathematicians advance
human understanding of mathematics?” He responds to his own question: “This question brings to the fore something that is fundamental and pervasive: that what we are
doing is finding ways for people to understand and think about
mathematics.” He also writes, “what people want is usually not some collection of ‘answers’ – what they
want is understanding.”
Part of generating understanding is thinking about questions. We will ask participants to pose a question of their
own choosing. One participant describes her experience like this:
… at times of exasperation, some of us wondered if we had stumbled upon another great question of mathematics
(like trisecting an angle) that might require years of work by the greatest mathematicians to solve or prove unsolvable. In fact, one of the
students said that while he was working on his problem, to him, his problem was one of the great problems in mathematics. Our problems
may not have been research problems in the sense of being unsolved in the mathematical field, but they were very real to us because we had posed
them and did not know if or how anyone else had answered them. This provided an avenue to experience the joy of posing and solving problems that
matter to us.
Another participant reflects:
What seems so central here is that the mathematics we studied mattered to us.
Changing roles from teacher to student and from question answerer to question poser was sometimes a scary transition,
but this transformation had the potential to awaken the mathematical mind and to allow mathematics to be intrinsically
meaningful. There are so many questions that have not been answered yet, and so many that have not even been asked.
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The focus of the work is on mathematical-mindedness, rather than on developing particular content.
We’ll study mathematics that has a low threshold and a high ceiling – in the sense that people with
diverse backgrounds and prior experiences in mathematics can engage in thinking about the mathematics, because it’s accessible as well as
interesting from an advanced standpoint.
For example, one of the most important – and difficult to teach – skills in mathematics is how to make connections between mathematical topics. A student could memorize all of the techniques in his Algebra 1 textbook for factoring quadratics, but still might not have any idea how to connect those techniques to any other mathematical ideas. And he would have difficulty memorizing “all” of the potential connections, because there are simply too many to list. Mathematical-mindedness leads to being able to recognize and generate connections among mathematical topics.
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