# Undergraduate Research

Math students at Houghton have a number of options to do undergraduate research, which usually is done in groups of one or two with a faculty member. Students have participated in the Summer Research Institute, took independent studies, and completed honors projects.

## What is undergraduate research in math?

Your first question might be what it means to do research in mathematics. As in all areas, research means learning or thinking about something new. In mathematics, our learning and thinking is usually focused around questions or problems about mathematical objects and concepts. The work to answer the questions frequently includes learning background information, trying examples, noticing patterns, finding out in which cases the patterns hold true and explaining why this is true through proof. For every question you answer, you frequently ask many more questions to keep investigating!

How can doing undergraduate research help me?

Undergraduate research helps you put together what you learn in your math classes, both concepts you learned and skills like problem solving, persistence and proving. While this is clearly important for those who wish to continue in graduate school in mathematics, it can also support many other career paths. Many of our students doing undergraduate research were education majors and felt this helped them better learn processes that they want to teach their students and not just content. The experience with problem solving supports almost any career, but would be especially helpful in a variety of applied mathematics and industry careers. In addition, students frequently present their research, both at Houghton, undergraduate and national conferences. This gives opportunity to carefully communicate new ideas and engage an audience, an important skill for every career!

## What are some examples of undergraduate research?

Below, you can find information about two different honors projects that students have completed. Because math research is focused on answering questions, the ideas are organized as answers to questions – some of which give background information and some of which tell you about the problems solved by Houghton students. Rob presented his honors research as a poster at the national Joint Mathematics Meetings in Washington, D.C. in January 2009.

Rob Zima ‘08: “Non-negativity of the γ-vector for 3-dimensional polytopes”

In this honors project, Rob Zima looked at patterns in the angles of polyhedra. In math, the search for patterns helps us to better understand the world around us. We look to extend patterns, asking questions about which circumstances guarantee a certain outcome.

- How do you measure the angles of a polyhedron?
- What sorts of patterns can you find in the angles of polyhedra?
- What other questions can you ask about angles in polyhedra?

April Bowers ’09 and Mindy Swancott ’09: “Tiling 4 by n Rectangles with L, T, and Z Tetrominoes”

In this honors project, April Bowers and Mindy Swancott extended some of their earlier work during the Summer Research Institute to count the number of ways you can fill a rectangle with tetrominoes. Mindy and April presented their research at the Penn State Undergraduate Research Conference in 2007 and the MAA Seaway section meetings in 2008.

- What is a tetrominoe?
- What does it mean to tile a 4 x n rectangle with tetrominoes?
- How can we compute the number of ways to tile a 4xn rectangle with tetrominoes?
- What other questions can we ask about tiling with polyominoes?