November 2023 Practical Practices

Connecting Representations: The Power of Why

By Joe Bolz, George Washington High School, Denver Public Schools

“You’ll NEVER know what your students are thinking unless you ASK your students what they are thinking”

One of the proudest moments of a teacher is to see a student apply what they have learned from that teacher in their everyday lives. No, I don’t mean seeing the student do great on a test, although that does make us proud. Seeing them really use the material - apply it - be it in their careers or everyday life. After attending the first session of the CCTM Book Study (studying “Teaching for Thinking”) in October, I hope I made my teachers proud as I took what I learned to help my team of teachers at the high school I work at effectively implement the reasoning routine, Connecting Representations.

Our Integrated Math 1 team had made it their goal to utilize this strategy to get students talking to each other. They wanted an activity that had a low floor and a high ceiling, so they settled on Connecting Representations. Connecting Representations starts with the task of students being asked to connect two representations of a mathematical concept and explain why. This can vary from an equation and a graph to an algebraic expression and algebra tiles to an expression for area and its corresponding diagram. The low floor exists here as there are many ways students can make these connections. The high ceiling lies in those many connections that can be made. 

While our team was well on our way to creating tasks that push thinking, we were lacking a solid structure. Applying what I learned in the CCTM Book Study, I worked with my team to come up with the following example:

Scatterplots - Connecting Representations (DESMOS): Student Version   Teacher Version

While I won’t bore you with the details of the task (I trust you to look through the task and see for yourself firsthand the evolution of this Connecting Representations), I will highlight what we learned and why it was so powerful.

The task asked students to decide which line of best fit went with which scatterplot and why. While I expected a lot of justifications including “the scatter plot is going up and to the right so the line of best fit should too”, the reasoning I heard was different than what I expected. Most of the students’ reasonings were tied to the starting values (or y-intercepts). Why was this important?

For me, as a teacher, this told me the students had a strong grasp and understanding of a starting point. While they relied on this integral piece of information about a linear function and had an intrinsic understanding of what starting values and starting points were, it told me students lacked the vocabulary to connect to the concept of y-intercept (or at least that vocabulary). 

Yet, sometimes what we don’t see is just as important as what we do. While students were comfortable with talking about starting values, they felt less so talking about concepts around slopes. We, the students and the teachers in the room, could discuss the notion of the scatterplot increasing and decreasing (although that was prompted by the teacher), yet students were remiss to discuss the concepts of slope on their own. They also struggled to discuss the steepness of the scatterplots. The word slope was hardly, if ever mentioned. This, again, told us so much about where our students were at with their understanding. We knew we had to reinforce concepts of slopes and how they applied to linear functions, particularly scatterplots.

None of this would have come to light had we not asked our students what they thought, had we not asked why. Simply put, you’ll never know what your students are thinking unless you ask your students what they are thinking. Connecting Representations is a powerful protocol to elicit the “whys” from our students. And when you ask your students why, the lessons you learn - both in what is seen and unseen - will empower your teaching, giving you the capacity to effectively address the conceptions your students hold. This is where true learning can occur. This is where we can create students who see the joy, the beauty, and the usefulness of mathematics in their everyday lives. And hopefully, this is where we can create learners who give us that sense of pride that forms when we see our students using the skills they learned in our classrooms, applying those skills in their everyday lives, as we did when we implemented this protocol of Connecting Representations taught to us via the CCTM Book Study of “Teaching for Thinking”.


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