# 2015

# Investigating the Interplay of Argumentation and Mathematics in Classroom Tasks

# Group Members

## Supervisors

Steven Lemay

## Overview

This project was motivated by the current emphasis on argumentation in the Common Core State Standards for Mathematics. In particular, we focused on tasks that address the third standard for Mathematical Practice that highlights argumentation as an expertise all students should develop. Our data consists of 157 tasks from 40 elementary and high school teachers participating in a professional development program focused on mathematical argumentation. We analyzed how argumentation affects the cognitive demand levels of the tasks, as well as how argumentation detracts or contributes to the mathematical content in the tasks. Our findings show a change in cognitive demand levels when tasks were analyzed with and without the argumentation component. In addition, five different themes emerged from our analysis with regards to the interplay of argumentation and mathematical concepts, some of which were common across elementary and high school tasks.

## Presentation

# Mathematics and Intercultural Competence in the Middle School

# Mathematics and Intercultural Competence in the Middle School

# Group Members

## Supervisors

Kyle Evans

## Overview

As today’s world becomes increasingly globalized, there exists a greater need to develop *intercultural competence* (ICC) in children through education. Three interdisciplinary units were designed by graduate students at UConn in the past year to develop ICC in sixth grade students, but lack assessment tools that track this development over the course of a school year. With a focus on Michael Byram’s model of ICC, we created two assessment tools – a survey and a rubric. The survey contains Likert scale items measuring four dimensions – attitudes towards cultures, knowledge of cultures, mathematics learning, and interdisciplinary learning – along with open-ended questions to add a qualitative component for each dimension. The rubric, designed for teachers, provides a resource for evaluating ICC in students’ interactions and written reflections. The tools have the potential to form a new baseline for assessment of ICC in children and can be refined and adapted for use across school districts and grade levels. In addition, we created three examples of lesson plans – one for third grade, one for sixth grade, and one for ninth grade – that demonstrate how intercultural competence and global issues can be incorporated across different levels of student development while also adhering to the Common Core State Standards for Mathematics.

## Presentation

# Wave Propagation through a Fractal Medium

# Wave Propagation through a Fractal Medium

# Group Members

## Supervisors

Alexander Teplyaev, Luke Rogers, Ulysses Andrews

## Overview

We consider the wave equation on the unit interval with fractal measure, and use two numerical models to study wave speed and propagation distance. The first approach uses a Fourier series of eigenfunctions of the fractal Laplacian, while the second uses a Markov chain to model the transmission and reflection of classical waves on an approximation of the fractal. These models have complementary advantages and limitations, and we conjecture that they approximate the same fractal wave.

## Presentation

# Minimal Length Maximal Green Sequences and Triangulations of Polygons

# Minimal Length Maximal Green Sequences and Triangulations of Polygons

# Minimal Length Maximal Green Sequences and Triangulations of Polygons

## Group Members

Emily Cormier, Peter Dillery, Jill Resh, John Whelan

## Supervisors

Khrystyna Serhiyenko

## Overview

Maximal green sequences (MGS’s) are combinatorial objects that involve local transformations of directed graphs, also called quivers. We studied minimal length MGS’s for quivers of type A. It is know that such quivers are in bijection with triangulations of polygons. Moreover, these local transformations of quivers behave well under this correspondence, and there is a related notion of mutation of triangulations. This enabled us to study MGS’s both in terms of quivers and in terms of triangulations. We showed that any minimal length MGS has length n+t, where n is the number of vertices in the quiver and t is the number of 3-cycles. We also developed an algorithm that yields such sequences of minimal length.