The other day I was listening to a class discussion about independent and dependent variables. The teacher provided students with two related variables and asked students to determine which one was dependent on the other. For example, price of a ticket and number of customers.
What I observed is that few students responded. More importantly, after the discussion was over, I still had no idea which students could identify and justify which variable would be dependent and which would be independent.
The teacher and I met afterwards and planned an activity that we thought would generate more student engagement and provide information about what students really understood. The plan was to ask students to generate variables they thought were related. Next, each student would share the variables and the rest of the students would move to one side of the room or the other as to which variable they thought was the dependent variable. Students would provide their reasoning for their choice. Repeat...
So what happened? Here are some samples of what students wrote:
Hmmmm... Not what we anticipated. It appears that students identified "things" that are related but without thought about the measurement or quantities. How can we discuss independent and dependent variables when students don't recognize the quantities that are represented by the variables?
The second Standard for Mathematical Practice is to "Reason abstractly and quantitatively." What does this mean, especially to reason quantitatively? What does this mean students can do? How do we, as teachers, design experiences that increase students' ability to reason quantitatively?
I recently read the article "6 Principles for Quantitative Reasoning and Modeling" by Eric Weber, Amy Ellis, Torrey Kulow, and Zekiye Ozgur (Mathematics Teacher August 2014 Vol. 108 pp. 24-30.)
The authors describe quantitative reasoning as a "specific way of thinking about mathematics" and focuses on its role in the modeling process. The first of six principles for integrating quantitative reasoning in instruction is "Rewrite a problem situation or prompt so that students must identify the quantities that they believe are relevant to solving the problem." The key idea that really stands out to me is the students identifying the quantities.
I am coaching the teacher to infuse this principle through every activity she can throughout the next module and then repeat the activity again. We agree that this must be an ongoing theme and not something a single lesson will address.
Through this experience I have become "hypersensitive" to whether students are reasoning quantitatively and if teachers are providing opportunities for students to develop this habit of thinking about mathematics. So, when a teacher asks me for advice on a lesson plan or an activity this is one of lenses I am using.
How do you address this mathematical practice? Share your experiences and strategies.