# Pulling it All Together

*Strengthening Strategic Competence through Modeling Mathematics Ideas*

## PRACTICE-BASED ACTIVITIES TO FOCUS ON

models and modeling within our standards

Through the research lessons from our lesson studies, we have learned a great deal about engaging teachers and students in meaningful and worthwhile problem-solving and mathematical modeling tasks. Working with teachers on understanding the different facets of modeling mathematical ideas revealed that teachers use models and modeling in important and diverse ways in teaching mathematics.

The important but distinct ways teachers used models and modeling were through

- (a) modeling math with tools—manipulatives, representations, and technology;
- (b) interpretative models of mathematics concepts (i.e., models for fractions, meanings for operations); (c) modeling math through problem-solving tasks; (d) problem posing; and (e) mathematical modeling through unstructured real-world problems that require problem formulation, real data, and building a model that makes sense in the real world.

Unpacking the different ways models and modeling is used in the curriculum that is important for teachers for it serves as means for developing strategic competence and for tailoring optimal teaching sequences that engage students in problem-solving and critical-thinking skills. Through designed professional development activities and lesson studies, we worked with teachers while developing students’ and teachers’ strategic competence in formulating, representing, and solving mathematical problems.

Research has shown that content-focused professional development leads to improvements in teacher content knowledge that is focused on student learning goals, highlighting the concepts being addressed, how they are developed over time, difficulties students may encounter, and how to monitor student understanding (Garet et al., 2001; Cohen & Hill, 2001; Desimone, Porter et al., 2002). Sztajn (2011) reports that mathematics professional development is an emerging research field that needs high-quality reports on description of the math professional development and a standard for reporting, including design decisions. As designers and researchers, we were intentional in our design decisions with the goals of developing teachers’ specialized knowledge for teaching mathematics.

In designing the professional development, we also considered cognitive demand in the tasks as one of the essential tools that the teachers were introduced to in the summer institute and lesson study. This involved helping teachers understand how to develop open-ended tasks and evaluate the level of cognitive demand the task offered while implementing standards-based mathematics instruction. In particular, the teachers were introduced to the work of Boston & Smith (2009) which helped them to identify the level of the cognitive demand in the tasks they created (low vs. high) and also learnt how to identify the factors related to the decline of the high-level cognitive demand tasks.

Our observations confirmed results presented by Arbaugh & Brown (2005) where teachers showed growth in the ways they considered various proportional reasoning tasks and how this influenced some of them to change their patterns of task choice. This helped us as a measure to evaluate teachers’ pedagogical content knowledge. Teachers get frustrated when they do not know what to do, an uncomfortable feeling for the “knowledge authority.” Several teachers remarked that “now, I know how my students feel.” This same remark was present in numerous reflections as well. Teachers who experience the frustration of struggling will be more acutely aware of it in their students.

Collaboration was not only a great stress reliever but also a rich learning environment. Several teachers remarked that they would use student groups in their classrooms more frequently because they saw the benefit of such strategies for their own work. We saw many instances of teachers asking others in their group to explain their reasoning. This helped the teachers to understand their colleagues’ reasoning and also helped the speaker clarify her/his own thoughts through explanation. Then, the groups at the table discussed whether or not the logic was valid and if they wanted to use that approach. Teachers benefitted from these collaborations in several distinct ways.

First, they began to see that real problems involving proportional reasoning are not simply plug-and-play exercises; they are multi-layered challenges, which require analysis, sound reasoning, and understanding of the relationships among quantities. Second, they recognized the profound importance of conceptual understanding as a baseline for strategizing approaches to problem solving. And, third, they gained an acute appreciation for the frustration of their students who apply incorrect procedures and cannot understand why their answers are incorrect. Several teachers mirrored that idea in their writings. Lastly, another teacher reflected, “I am also starting to think differently about analyzing student work. When problems have the opportunity of yielding a variety of correct answers, it is important to consider what the student is doing and what math they can do and understand.”

**Lessons Learned from Our Lesson Study and Professional Development Sessions**

Through our professional development activities, we learned that developing strategic competence through modeling mathematical ideas encompassed the five important ways we defined modeling mathematical ideas: (1) modeling math with tools, (2) conceptual and interpretative models of math ideas, (3) modeling math through rich problem solving, (4) problem posing, and (5) mathematical modeling of situations through unstructured real-world problems.

We looked at these ways of modeling by immersing teachers in problem-solving and mathematical modeling tasks that then led to the development of their research lesson during lesson study. Throughout these cycles, we examined how both students and teachers made use of models and modeling to understand mathematics.