How do we Integrate Math and Language Goals? Lessons from a Design Research Effort

April 20, 2020

How do we design integrated content and language learning objectives? What might it look like to have a progression of objectives for content, practices, and language throughout a curriculum unit?

The ELSF guidelines include a call for student materials to contain mathematics and language learning objectives [Specification 2b], and for teacher materials to articulate a pathway or progression of objectives for content, practices and language [Specification 2c]. In this post I share a few ideas based on my research engaging in design work.  

Over the past four years, I have been working with a team of secondary mathematics teachers and student researchers to design and study mathematics lessons that include an integrated focus on mathematics and language. We are focusing on a unit on linear functions as an exemplar, and the lessons we are designing are specifically tailored to meet the needs of bilingual/multilingual students who are classified as English learners (ELs). We are working in a linguistically diverse setting where about 25% of ninth graders are currently classified as ELs, and an additional 40-50% were formerly classified as ELs. To do our work, we are adopting a design research approach, which means the work proceeds in an iterative fashion.

The steps of our design cycle are the following:

1)   Identify student resources and challenges related to both the mathematical content and the language demands of the curriculum materials.

2)   Plan and design a unit of study that builds on student resources and responds to the identified challenges.

3)   Teach the lessons and collect data including video recordings and student work to document the effectiveness of the lessons.

4)   Examine the data to identify successes in the design and potential improvements in future iterations. 

To integrate a focus on mathematics and language, we have followed the steps outlined above.

  1. Identify resources and challenges related to language

For Step 1, we have conducted both student interviews and systematic analysis of the written curriculum materials used at the school where we are working. The ELSF guidelines proved valuable for the curriculum analysis process. We complemented the text analysis with student interviews using tasks from the school’s adopted curriculum. The student interviews revealed both resources we did not know about, as well as new design challenges we had not anticipated.

For example, one of the math problems about using slope to solve a problem about the steepness of a roof started with this sentence: “You need to cut four 2x6s for the vertical roof supports shown in the diagram.” We suspected that many students would struggle to interpret the symbol “2x6” as meaning “two by six.” This was born out—many students interpreted the 2x6 as an invitation to multiply. However, what we found surprising was that many students were able to give correct responses to the task by ignoring the unfamiliar words and symbols and instead focusing on a pattern in the numeric table that went with the text. This pointed to a need to include prompts for student explanation to understand the thinking behind student responses. 

  1. Develop mathematical and language development goals

During Step 2 of the design cycle, we created lessons with integrated math and language goals. The image below shows an example of the cover page for one of our lessons, including both mathematical goals, mathematical-language goals, and student-facing goals. During this stage of the work, we engaged in in-depth discussions about how we expected the students to use language in their high school mathematics classes. We also wrestled with how to integrate the unit’s mathematical goals with specific language goals.

Our overarching mathematical goal for our unit was for students to understand that the slope of a linear function measures the rate of change connecting the independent and dependent variables. We chose this focus because the rate of change meaning of slope incorporates other meanings of slope (e.g., the rate of change can be reinterpreted as a measure of steepness in a graph or to solve real-life problems involving steepness of hills, roofs, etc.). Our primary language goals were for students to explain relationships among representations and to justify claims using evidence. For example, for explaining the connection among representations, we wanted students to be able to explain why a steep line in a distance time graph indicates moving fast.

In order to integrate our mathematical and language goals, we created a trajectory of the mathematical ideas we wanted students to develop during the unit and we made a trajectory of the language proficiencies we wanted students to develop. Then, we developed lesson-level mathematical goals and mathematical language goals that integrated these foci. Across lessons, we developed  a progression of the goals. For example, early in the unit students were asked to identify or describe relationships.Later in the unit, students began to generalize and justify relationships.

  1. Incorporate supports to meet the goals in a linguistically diverse class

Note that we incorporated explicit linguistic support in each lesson to help students achieve our lesson-level and unit-level goals . For example, in the lesson above, the lesson plan included specific prompts to use the mathematical language routines (Zwires et al., 2017) Collect and Display and Compare and Connect. We also incorporated the use of dynamic representations on Desmos so students could experiment with ideas and reason with and reason about a shared representation. In a separate post I can say more about how we used these routines and technology resources to support our goals.

Conclusion

Looking across the whole design effort, we note that this work required careful consideration of student resources, curriculum language demands. Once we understood student resources and the curriculum demands, we were able to design instruction that incorporated the integrated mathematical and language development goals. Of course, this design work is iterative, and at each stage we have gained new insights into student reasoning, and developed new ideas about how to refine this unit in future iterations, and how to design better lessons moving forward.

You can watch a video describing this work at the link below:

Zwiers, J., Dieckmann, J., Rutherford-Quach, S., Daro, V.,Skarin, R., Weiss, S., & Malamut, J. (2017). Principles for the design of mathematics curricula: Promoting language and content development. Stanford University UL/Scale. http://ell.stanford.edu/content/mathematics-resources-additional-resources

Find more activities and related resources at http://meld.sdsu.edu

The research described here was supported by a grant from the National Science Foundation (#1553708) to William Zahner. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of The National ScienceFoundation.

Bill Zahner is a member of faculty in the mathematics department at San Diego State University. His research, teaching, and service activities all focus on how we can improve mathematics learning for all students, especially English Learners and students from groups underrepresented in STEM fields. He enjoys teaching and working with students from across grade levels, from middle school all the way through graduate school. He earned his PhD in Education from UC Santa Cruz in 2011, where he was a fellow with the NSF-funded Center for the Mathematics Education of Latinos/as. Bill’s most recent research aims to design learning environments to foster productive and powerful discussions among linguistically diverse students in secondary mathematics. Prior to his life as a mathematics education researcher, he taught high school mathematics for six years. For three of those years he taught in Chuuk, Federated States of Micronesia.

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