# Fellow Profile: Israel Vaughn

**G-TEAMS Cohort:**2013-14-
**Graduate Program:**Optical Sciences -
**Teacher Partner:**Brenda Ugalde -
**School:**Tucson Magnet High School -
**Grade level:**10-12 -
**Topics:**Honors Pre-Calculus II, Calculus AB

"It is the story that matters not just the ending"

- Paul Lockhart

## Research Interests

My interests involve the theory and application of imaging science which is a specialization of stochastic estimation theory. I enjoy this field because I can have have one foot in a math heavy discipline, and simultaneously have another foot in an experimental science. In imaging science we typically have to derive a mathematical equation which represents a physical instrument in a statistically robust way (the forward equation), and then we attempt to find the inverse problem to this equation. The purpose of the inverse equation is to find the actual physical quantities which were measured by the instrument. This usually ends up being quite difficult, but sometimes a closed form solution can be found. Finally we have to experimentally verify the forward and inverse imaging equations. The requirements of instrumentation, physics, and mathematics knowledge in imaging science fits me well.

The specific instrument that I am working on in my lab is called a polarimeter. This instrument measures the polarization property that is inherent to all electromagnetic radiation, specifically in the optical wavelength region. Polarization occurs as a result of electromagnetic radiation being composed of transverse waves in a three dimensional space. I am working on extending results of prior students to better reconstruct the measurement of this property, and I have designed and am currently building a very fast portable polarimeter.

## Classroom Activities

I work with two separate classes with Brenda Ugalde at Tucson High. I help Brenda in many different ways in the classroom. I typically help students with individual questions. I attempt to guide them to find the answer through their own abilities and thoughts about the problems. I have also developed a matrix/vector curriculum module (relying on their textbook) with Brenda's help. This module is important, because in engineering and the sciences most mathematics is done in multidimensional spaces. I have helped write interesting problems for the problem of the week (POW) assignment, and occasionally have taught an entire class.

I have also started a Math Club focused on the national American Mathematics Competitions (AMC). With help from UA Math, we have an undergraduate mentor coming once a week. After the AMC, we will transition to more general topics including calculus based problems and engineering applications.

## Lessons Learned

As previous fellows have noted, I was surprised by the difference in computational ability vs. conceptual understanding in the High School students. The material that I teach is at the college level, yet the understanding seems deeper among college students who are just a few years older. I am curious if this is a result of teaching methods or of fundamental biology. Most students are very interested in obtaining an algorithm with which to compute an answer, and few are interested in why we have a specific algorithm, or the implication of a specific algorithm. I believe that this focus on computation must be due to teaching style and methods.

Something else that was surprising was how easy it is to make mistakes at the board, and how difficult it can be to recover. I know most of this material very well, but the board can create nervousness and tunnel vision which leads to mistakes. I'm finding myself to be more comfortable in front of the class as my experience grows, which makes recovering from mistakes less difficult. These mistakes at the board have taught me to be lenient with students when they are working at the board or in front of the classroom.

## Teaching Materials

- Introductory Presentation (PDF)
- Calculus Problems
- Box Optimization (PDF)

- Precalculus and Trigonometry problems
- Vector Multiplication Exercise (PDF)
- Matrix Exercise (PDF)

- Math Club
- Club Flier (PDF)
- Polygon Area Proof (PDF)

Note that all LaTeX materials for generating the above PDFs are available by request.