# Fellow Profile: Matt Lafferty

**G-TEAMS Cohort:**2010-11-
**Graduate Program:**Mathematics -
**Teacher Partner:**Jennifer Gould -
**School:**St. Michael's Parish Day School -
**Grade level:**6-8 -
**Topics:**Pre-Algebra and Algebra I

"As one of my favorite quotes puts it, 'Computation is to mathematics as a spelling bee is to writing a novel'. Despite this, many students see math as no more than memorization of arbitrary formulas and definitions, and not the creative process it is. The G-TEAMS program gives us the opportunity to change this perception in the students we work with."

## Research Interests

Matt Lafferty is a graduate student in the Department of Mathematics at the University of Arizona. His research interests are in the area of algebraic number theory, which is the study of the algebraic structure of generalized integers. Prior to attending the University of Arizona, Matt received his master's in mathematics from Clemson University where his thesis was on the construction of square ideals in certain number fields used in the number field sieve, a modern integer factorization algorithm.

## Classroom Activities

Matt is working at St. Michael's Parish Day School with Jennifer Gould. Due to St. Michael's advanced curriculum and highly motivated student population, Jennifer and Matt focused their partnership on challenging the students mathematically in a way that exposes them to modern mathematical research and applications while emphasizing the inherent creativity of mathematics. Within the classroom Jennifer and Matt accomplish these goals by,

- Relating class material to modern mathematics. For example, during the unit on the coordinate plane the students played the "coding theory game" (see games below) where they used the coordinate plane to encode and decode secret messages in a way analogous to that used by algebraic coding theorists in modern day digital communications.
- Assigning a challenging problem of the week, which either expand on an idea from class or exposes the students to a new and interesting area of mathematical thinking. See below for a comprehensive collection of problems of the week given to Jennifer's math classes.
- Exposing students to the history of mathematics, in hopes that giving the material some character and personality will disabuse the students of their notion that mathematics is rote memorization of formulas.

Outside of the classroom Jennifer and Matt have created a non-competitive math club. This club, which meets every Monday after school is called "Monday Math Mania!" and gives the students a chance to tackle additional challenging problems and explore any mathematical ideas they might come up with in a fun collaborative environment. To facilitate the this math club Jennifer and Matt have

- Created a weekly newsletter that contains an article on some interesting aspect of modern day mathematics, profiles current and past mathematicians, and gives additional challenge problems for students to solve. Students that correctly solve these "star problems" have their name printed in the following weeks newsletter.
- Created a Monday Math Mania! Website containing fun and challenging math problems, an archive of the weekly newsletter, as well as profiles of famous mathematicians.
- Organized a St. Michael Family Math Night. Over 140 students and their parents participated in the six rotation stations that were based on the six mathematical strands defined by the National Council of Teachers of Mathematics.

## Lessons Learned

Working with students at St. Michael's and observing Mrs. Gould has greatly improved my understanding of how and ability to communicate mathematical ideas. While many of the lessons covered material I have long taken for granted, I found myself having difficulty explaining it to students in a way that they could easily understand.

For example, during a sixth grade unit on integers I was asked why subtracting a negative number is "adding the opposite". This is something that I had long taken for granted, and found myself having a hard time giving an answer the student truly understood. That night I thought about the intuition 6th graders revert to when they were considering problems of adding and subtracting positive and negative integers, namely the number line. Given a problem such as 2 + -7, they would imagine themselves on the number line standing on the 2. Since the second number is negative they would walk in the negative direction a total of 7 steps. Similarly, if the problem was 2 + 7 they would imagine themselves standing 2 and they would walk in the positive direction a total of 7 steps. Relating this intuition to the new situation of say, 2 - (-7), I told them to once again imagine themselves standing on the 2. The addition or subtraction sign would tell them which direction on the number line they should be facing. So in this case they would be standing on the 2 facing the negative direction. Next, because the second number is negative they would walk backwards a total of 7 steps. If the second number was 7, they would have walked forward seven steps. With this intuition in place, the student saw that subtracting a negative number is like facing the negative direction and walking backwards, which is the same as facing the positive direction and walking forward.

By relating new problems to old (and correct) intuitions, I found students were able to better understand the concept and more easily adapt their understanding to new concepts. Not only has this made me a better instructor, but it has also made me a better student and researcher as I find myself employing these same techniques to better understand the mathematics I'm currently working on.

## Teaching Materials

**Lessons and Talks:**- The Distributive Property (7th Grade)
- Mixing Problems (8th Grade)
- Math is Everywhere - Family Math Night presentation
- The Nature of Infinity

**Games:****Problems of the Week and Newsletters:**- Problems of the Week
- Newsletter #1 - The Mathematics of Invisibility
- Newsletter #2 - Serious Math, Not so Serious Place
- Newsletter #3 - The Monty Hall Controversy
- Newsletter #4 - The Holiday Issue
- Newsletter #5 - Fermat's Last Theorem