# Fellow Profile: Ryan Smith

**G-TEAMS Cohort:**2011-12-
**Graduate Program:**Mathematics -
**Teacher Partner:**Alyssa Keri -
**School:**Catalina Foothills High School -
**Grade level:**9-12 -
**Topics:**Number, Algebraic Reasoning, Probability and Statistics, Geometry

"A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas." - G. H. Hardy

## Research Interests

Ryan will be completing his PhD in algebraic number theory in Spring 2012. Number theory is the study of "elementary" properties of numbers, such as the distribution of prime numbers. Many of the questions are simple to state, yet extremely difficult to prove. One of the oldest questions in number theory was Fermat's Last Theorem, which states that the equation x^n + y^n = z^n has no rational solutions when n > 2. Fermat conjectured that this was true in 1637, but despite the interest of many of the greatest mathematicians it remained an open problem for 358 years.

Serre's conjecture is a more recent problem in number theory which describes mod p representations of Galois groups and has far reaching implications. One measure of its significance is that Serre's conjecture directly implies Fermat's Last Theorem (which has been known to be true since 1995, but for different reasons). Ryan's dissertation work concerns generalizations of Serre's conjecture to totally real number fields.

## Classroom Activities

I have been working with multiple high school math classes at Catalina Foothills High School. I work with two AP Calculus classes, two Pre-calculus classes, and an algebra 1 class. My role in each class is slightly different, but in all of them I present some of the material, work with groups of students, and do brief presentations on topics beyond the course which might interest students.

## Lessons Learned

During my time at the high school I've become better at using group work in the classroom. Although I wanted to implement more group work when I taught at the University of Arizona, I found that it rarely worked as well as I hoped since most of my training was in mathematics and not teaching. In order to make group work an effective way to structure class time I've had to significantly change how I teach. It's not enough to plan out how long I expect my part of the lesson to take, I need to think about their role in the lesson as well. When I first started involving students more I would get derailed easily, so I learned that I have to try to anticipate where students will be confused so that I can have contingency plans.

Before working with high school students I didn't realize how many leaps I would make in my explanations since they seemed natural to me. I've found that I needed to slow down and show more steps, since students don't think like I do and find it confusing when I do several things at once. I've become more aware of the difference between how I think and how students think, which has led me to be more deliberate in how I communicate.

## Teaching Materials

- Black hole Worksheet. This was a worksheet I put together based on one of Jordan Schettler's worksheets. It describes the escape velocity of light near a black hole using rational functions, with a focus on the physical interpretation of the asymptotes of the function.
- Three stage production functions. This is a worksheet I put together on three stage production functions from economics. Students have lots of applications of quadratic polynomials, but this is a real world cubic application.
- What is i^i?. These are the notes for a talk I gave to the CFHS math club introducing complex exponentiation.