# Fellow Profile: Kevin Powell

• G-TEAMS Cohort: 2010-11
• Teacher Partners: Karen Rakowitz and Lourdes Oros
• School: Nash Elementary School

"Mathematics is a science in its own right. It can be explored, handled, and tested. Understanding it leads to a deeper appreciation of the world around us."

## Research Interests

Kevin completed a Masters degree at Brigham Young University in Analytic Number Theory. There, he began enjoying approximating the solutions to counting problems by analyzing coefficients of series (or generating functions) for the counting problems. He coauthored a paper that addressed counting the number of cube-free, fourth pwer free and fifth power free integers. In his thesis, he studied the zeros of classes of series that are often used in the construction of generating functions for multiplicative counting problems.

At the university of Arizona, he recently passed all of his qualifying examinations and has been preparing to conduct further research. He just participated in a research tutorial in group theory. He chose group theory, because of its combinatorial aspect; that is, group theory is quite useful for counting the elements in a set by examining group actions and characters. In his tutorial, however, he realized how much he liked using generating functions and number theory when he saw its utility in determining certain properties of graphs he was analyzing in the tutorial. He plans to learn and further pursue number theory and find an advisor in that area.

## Classroom Activities

Kevin is working at Nash Elementary School with Karen Rakowitz and Lourdes Oros. At the beginning of the semester, Kevin worked with his teacher partners to create glyphs with the students. Glyphs are pictorial representations of data from which the students can practice collecting data, making graphs and drawing comparisons and conclusions. He and his partners spent time using kinesthetics and tables with tens to help the students visualize and practice making tens when adding. They also described the process of making tens graphically from which Kevin based a challenge story problem that involved each student in the class imagining that he or she recieved a certain number of cookies. Each student then had to identify where his or her quantity of cookies would be on the graph.

Some other classroom activities include the class collectively removing boards with numbers that do not round to a given multiple of ten, sorting shapes into categories and finding the intersection, lines of symmetry, stories for carrying, and more. One key activity was learning about fact families using triangles and little bear figures which the students could move around the triangles. More details of this activity are described in the next section. A very successful activity has been learning mulitplication from quiesenaire rods. The students have done suprisingly well.

## Lessons Learned

Kevin has learned much from his teacher partners about teaching. For young children, there are three basic teaching steps, which may be repeated as necessary: the first is to provide an example of or demonstrate what is to be taught; then, work together with the students through the demonstration; and last, let the students work on their own. Kevin's teacher partners have given him a new vision of lesson development. There is a time and a place for each part of teaching. An adequate amount of time should be spent learning together as a class so that homework becomes practice rather than assessment or even group assessment. He really practiced and became fluid with the three step process for younger children while teaching borrowing in subtraction.

One purpose of G-TEAMS is to bring more mathematical insight and thinking into grade school classrooms. Kevin's favorite and most successful contribution has been path identification on fact family triangles. That is, given a sum with two addends as vertices on a triangle, one can identify four "number sentence paths" that are each two edges long. There are simple rules that govern these paths and that can easily be seen on the triangle. After learning these rules, the second graders have become proficient at using these triangles and the paths on them to complete number sentences with missing addends or subtrahends; this is a terrific skill as they prepare for algebraic manipulations. This opportunity to develop frameworks for students to learn concepts in has inspired Kevin. In the future, he would like to create a little Java applet for his university teaching that will aid in teaching students about coordinate transformations. The students will be able to explore and practice by making transformations on points so that the process will become more tangible to them.