The goal of this project was to better understand how students who are learning English as a second language (i.e., English Learners, or ELs) engage in the practice of reasoning abstractly and quantitatively while using dynamic representational technology (DRT). Reasoning abstractly and quantitatively entails relating real-world situations to mathematical representations to solve problems. DRT is software that allows students to manipulate multiple, linked representations. For example, in our DRT, students can manipulate a graph showing a runner?s distance over time, which modifies a linked animation of the runner (See Image 1). Prior research involving small numbers of ELs suggests that using DRTs, together with carefully designed instructional activities, can have a positive impact on ELs' mathematics learning. Our study used larger numbers of students to compare ELs using DRTs with ELs engaged with similar tasks and using only paper and pencil representations. 

Reasoning abstractly and quantitatively is one of the eight mathematical practices given as targets for instruction across grades K-12 in the Common Core State Standards for Mathematics. Most prior research that defines this practice has been based on case studies, and these cases often analyze the reasoning of students who are easy for researchers to find and to understand. As a result, there is little existing research that focuses specifically on how ELs reason abstractly and quantitatively. This is a major gap because mathematics educators have noted that students' abstract and quantitative reasoning is greatly influenced by their life experiences, so we would expect students with varying life experiences to have varying ways of reasoning abstractly and quantitatively. Understanding this variation can broaden our understanding of this mathematical practice. Additionally, ELs are an increasing number of students in many schools around the country, so understanding how ELs engage in the Common Core State Standards Mathematical Practices is essential for equipping teachers with the knowledge and tools they need to teach their EL students. 

The outcomes of our research include middle school mathematics activities focused on reasoning abstractly and quantitatively and designed for use by ELs and non-ELs, along with guidelines for facilitators of these activities. One version of these activities was designed to be used in a DRT environment and a parallel version was designed to be used in a paper and pencil environment. Along with these activities, we developed design principles for the development and use of DRTs with ELs. For example, one specific design principle that came out of this work was the notion that designers of DRT should make important quantitative information visible and manipulable within the representations. In our DRT, we found it was useful to create a visible "limit line" on the graph that corresponded to the finish line of the race depicted in the animation in the DRT (See Image 2). Either the limit line or the finish line could then be manipulated to change the length of the race. This small change allowed students to provide more accurate answers and facilitated communication. 

After creating and pilot testing the DRT and paper-and-pencil activities, we recruited middle school EL and non-EL students to be video-recorded while completing these activities, and we analyzed the examples of reasoning abstractly and quantitatively that we saw during these activities. Based on this analysis, we developed a coding scheme that identifies forms of reasoning abstractly and quantitatively that students use as they engage with the activities. We also identified cycles of reasoning abstractly and quantitatively, in which students move from the real-world context to a mathematical representation and back to the context again to check their work. This coding scheme can be used by other educators interested in how EL and non-EL students engage in this important mathematical practice. 

Finally, our coding scheme allowed us to answer the main questions of our project about how ELs who use DRT vs. paper and pencil reason abstractly and quantitatively. We coded 46 mathematics activity sessions (22 involving DRT and 24 involving paper and pencil) with two ELs in each session. Our main findings include the following:

• Students used their first language (Spanish), second language (English), gestures, and mathematical representations on paper and on the computer screen as resources for communicating about and doing mathematics. Coding students' reasoning required attending to all of these resources.
• All groups engaged in reasoning abstractly and quantitatively, even if they did not achieve a satisfactory answer to the main question in the activity.
• DRT groups engaged in more cycles of reasoning abstractly and quantitatively than paper and pencil groups.
• DRT groups were more likely than non-DRT groups to achieve a satisfactory answer to the main question. 

Overall, we conclude that the middle school ELs in our study engaged in reasoning abstractly and quantitatively and were supported in doing so by their use of two languages, gestures, and, in particular, DRT.

Last Modified: 12/20/2019
Modified by: Philip Vahey