Developing Proportional Reasoning in Intermediate Mathematics

Proportional reasoning involves the deliberate use of multiplicative relationships to compare quantities and to predict the value of one quantity based on the values of another. Our project focused on how to provide varied intentional opportunities for students to use multiplicative relationships in intermediate grades. In the math curriculum, the development of proportional reasoning is a very challenging concept. In our experience teaching intermediate math, we have found that students seem to struggle more with the concept of proportional reasoning than they do with most other concepts. Although we have varied our instructional practices through the years, the struggle remains.

“All ability to reason using proportional relationships is a complex process that develops over an extended period of time. It takes many varied physical experiences to develop an understanding of what a proportional relationship is and then more time to gain the ability to deal with it abstractly.” (Cordel & Mason, 2000)

Our proposal involved four phases: Research, Planning, Co-teaching/Co-assessing, and Reflecting/Revising/Sharing. During the Research phase, we arranged to meet with curriculum consultants to discuss strategies to develop proportional reasoning. We also investigated technology and conducted a review of the literature in this area. Our goal was to deepen our understanding of the underlying skills and the developmental processes involved in developing proportional reasoning.

During the second phase, we used the information obtained through our research to work together to develop a landscape for proportional reasoning (Fosnot), devise good questions for Three-Part Math Lessons and develop strings for mental math fluency. In our planning, we focused on differentiating for the diverse needs and learning styles of our students as well as the inclusion of technology as a tool in modelling concepts (e.g., the use of virtual manipulatives). Through this project, we found ways for our students to understand proportional reasoning and enable them to make connections between the concepts and in the real world.

Once the planning phase was completed, we planned to co-teach and co-assess student work. We videotaped one lesson each and used our observations to enhance our questioning and sequencing. Important in our project was opportunity to co-assess formative and summative student work, and to co-reflect on student progress and next steps throughout this unit. The opportunities to collaborate with colleagues helped us to develop consistency in our assessment.

The final phase involved reflecting on our learning, revising our unit and sharing our learning with other teachers. Upon completion of the learning cycle, we reviewed our learning and developed next steps for our professional learning. During this reflection, we reviewed our questions, three-part lessons and strings, and revised our unit. We also sought opportunities to share our work with teachers in our respective schools and in York Region. We conducted a workshop at OAME Spring Conference at Humber College.