Promoting English Learners’ Geometric Reasoning

Jill Neumayer DePiper began her career as a teacher and has extensive expertise in working to improve mathematics instruction, support students at risk, and enhance teacher preparation and professional development. For over seven years, Jill served as a professional developer, taught graduate-level mathematics education courses, and conducted research targeted to advance the quality and equity of mathematics education. She is a member of an EDC research team that has conducted several studies to identify effective strategies to support teachers in enhancing instruction for English language learners, including EDC's new Visual Access to Mathematics study. In this post, Jill shares some of her team’s findings from these studies.

Students who are learning English or designated as English Language Learners (ELLs) by their school or district may face multiple challenges in the mathematics classroom. ELLs, like all students, need rich opportunities to learn mathematics. Yet they also need specific supports and opportunities to develop academic language or communicate mathematically. To meet ELLs’ needs, it is necessary to “expand what we know about good teaching” and develop practices that “specifically address the language demands of students who are developing skill in listening, speaking, reading, and writing in a second language while learning mathematics” (Celedon-Pattichis & Ramirez, 2012, p.1).

The work of my team—including Mark Driscoll, Johannah Nikula, Rachel Wing DiMatteo, David Bamat, and myself—sits at this intersection of mathematics teaching and learning and supporting ELLs. 

For example, in the Fostering Mathematical Success of English Language Learners project, a collaboration with Horizon Research, Inc., we sought to better understand how we could prepare teachers to support ELLs engaging in rich geometry tasks. In the project, we examined the effects of EDC’s Fostering Geometric Thinking Toolkit professional development materials on middle grades teachers who have ELLs in their mathematics classrooms. 

A Closer Look at Geometric Thinking
The Fostering Geometric Thinking Toolkit emphasizes four specific geometric habits of mind. The attention to geometric habits of mind extends EDC’s long-standing work focused on mathematical habits of mind to how mathematical habits of mind appear in students’ work with geometry in classroom instruction. Specifying geometric habits of mind captures distinctions, for example, between geometric thinking and algebraic thinking in terms of its attention to visual relationships and reasoning.

The four geometric habits of mind featured in the Fostering Geometric Thinking Toolkit are:

  • Reasoning with Relationships: Attending to geometric relationships within and between geometric figures and using such relationships to understand or solve a problem
  • Generalizing Geometric Ideas: Striving to understand the “always” and the “every” as related to geometric phenomena
  • Investigating Invariants: Examining what is invariant—that is, what remains the same—and what changes under a transformation such as dilation, reflection, or dissection
  • Balancing Exploration and Reflection: Exploring a geometric situation, taking stock of the exploration, and deciding where to take the exploration next

To examine how the Fostering Geometric Thinking Toolkit supported teachers and ELLs, we conducted classroom observations, administered teacher assessments, and analyzed videos of students completing a problem-solving task. In the student problem-solving task, a facilitator engaged pairs of students in completing a task called the Parallelogram Problem. The task presented students with three points that did not lie on a single line and asked them to identify one more point that would allow them to use all four points as vertices to form a parallelogram.

Before they started the task, students reviewed a handout with the facilitator to go over key concepts in the task. The handout defined a parallelogram, provided examples and non-examples, and prompted students to create both examples and non-examples of parallelograms. The facilitator prompted students to share their thinking about these examples and non-examples before they started the task. Students began working independently on the task, switched to working together at the request of the facilitator, and then described their reasoning to each other and the facilitator. From analyzing the videos, we hoped to understand how the task and its design elicited the thinking and reasoning of ELLs, as well as how students communicated and reasoned together.  

Our Findings
Guided by our geometric habits of mind framework, we analyzed the videos of the student work and interactions of many pairs of students. Our analysis suggests that the following four features of task design and implementation promoted ELLs’ geometric thinking and learning: 

  • Activities to set up work on the task. The review provided students with an opportunity to gain feedback from the facilitator and to become familiar with the representations and definitions of parallelograms that they later referred to during their work.
  • Prompts and tools to support student work and perseverance. Written prompts, such as, “Do you think you found all the possible points where the fourth vertex would be? How do you know you found them all?” elicited students’ geometric reasoning. Facilitator prompting also encouraged students to persevere in their problem solving and communicate with increasing detail and precision. The facilitator encouraged students to explain their ideas in multiple ways and specifically encouraged drawing and gestures.
  • Designated time to collaborate and share ideas. Specific time for students to work together prompted collaboration and further reasoning. Sharing solution strategies, and being pushed to generalize, supported students in justifying their reasoning. As students worked together, they reasoned about parallelograms and made generalizations, which furthered their problem solving and construction of arguments to support their results.
  • Structured tasks to encourage exploration and testing conjectures. The ways in which the steps of the task were related and embedded in each other gave students multiple opportunities to explore parallelograms and their properties in different ways. The options to try a strategy and test the ways in which is it generalizable encouraged them to explore and reflect on their strategies and the utility of the strategies in different contexts.    

The task features we identified as fostering ELLs’ use of geometric habits of mind are not new to the mathematics education research literature on task design and implementation. However, our findings add to the literature base by making specific connections to students who are ELLs and illustrating task design features for geometry content. Understanding how to engage ELLs in these types of thinking to support their problem solving and communicate their mathematical thinking can help teachers and researchers design and implement tasks that provide ELLs with rich opportunities to learn.

These same task design ideas lie at the core of a series of supplemental activities designed to support ELLs and created to accompany mathematics tasks from the Fostering Geometric Thinking Toolkit. You can find these activities on our Mathematical Thinking: Supports for English Language Learners website. Designed for middle grades mathematics teachers who work with ELLs, the website includes 14 mathematics tasks from the Fostering Geometric Thinking Toolkit, as well as a set of support tools for each task. The support tools, which map onto the findings from our video analysis, include: warmups, sentence starters and frames, word charts, support notes, and a Spanish translation of the task. The website also provides links to other useful resources that support the teaching and learning of mathematics for ELLs. 

Our work identifying effective instructional strategies to support English learners is timely and important considering the large number of students who are from diverse language backgrounds. Our team has also enjoyed working at the intersection of two research areas, mathematics education and the education for ELLs. Bringing these two fields together can further build our understanding of how to meet teacher and student needs in mathematics teaching and learning.

This material is based upon work supported by the National Science Foundation under the Grant No. DRL-0821950. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.


Tuesday, January 26, 2016 - 1:00pm