## Knowing --> Explaining

The central idea of this session was that mathematics is first empirical and then deductive. That is, we come to realize that something is true based on experimentation and observation. It is only after we've experienced this truth that we are driven to prove or explain it. For example, after calcualating the squares of the sides of x number of right triangles, you might know with all your heart that a^2+b^2=c^2. But you would still wonder why. This curiosity would then drive you to prove the pattern. However, we often teach in the exact opposite way. First a teacher explains an idea and then students use or apply it. In doing this, students don't get to experience the curiosity that comes with the expectation of structure and the drive to explain why it works.I find this to be an extremely convincing framework for the doing of mathematics. In the problems we've worked on this week, I have found that I am reasonably convinced of a conjecture (based on a combination of experimentation and intuition) and then want to figure out why and prove beyond a doubt that it is true. In thinking about the school year, I want to make sure that my students get the opportunity to have this type of experience in doing mathematics.

One of the other teachers posed follow-up question: If students can thoroughly observe and describe a pattern, do they then need to be able to explain

*why*it happens? I think this is a tough question because for me, the answer is "it depends." It depends on the topic and the student's own need to know why. Because it is at the heart of mathematics, I would like to cultivate in all of my students this compulsion to explain. However, sometimes I may need students to trust their own powers of observation without explanation of why a phenomenon exists. If I go this route, I need to have a good reason for why, as a class, we might not pursue an explanation.

In talking about doing mathematics, Professor Stevens also shared the following ideas:

- math is natural

- math exists independent of us

- experience precedes formality

- math is the science of structure

- math is the art of figuring things out

When asked about structure, Professor Stevens said "I can't define it, but I know it when I see it." This pushed me to think about how I define structure, particularly in the context of Math Practice 7: Look for and Make Use of Structure. I think that structure is any type of organizing principle. And that it is these organizing principles that allow us to be convinced of the truth of a conjecture before the proof, but also to create the proof itself.

## Problems

Finally, two awesome problems that I spent*a lot*of time thinking about.

1. In how many ways can one spell out "ABRACADABRA" by traversing the following diamond, always going from one letter to an adjacent one? (from Polya)

2. With Pascal's triangle, how many odd numbers are in the 100th row?

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