#16. Representations can influence classroom talk and mathematical connections

Posted on January 8th, 2015

From the Institute for Learning, this is Laurie Speranzo.

In another podcast, we talked about what makes for a high level task and why the use of high-level tasks matter in math classrooms. High-level tasks, by definition, offer multiple entry points. High level tasks require that students discover mathematical connections or relationships. Yet this is often difficult. In this podcast I want to suggest that we choose tasks that situate the learning in a context and require students to use representations so that their learning is scaffolded and then they are more likely to discover the relationships.

Consider this example of a 7th grade classroom of students working on a thermometer task. The teacher hopes they will understand absolute value. When she was in school she was given the rule that absolute value symbols around a number mean that the number is positive, but she would prefer that her students actually understand the meaning of absolute value.

The tasks reads: On Tuesday, the temperature is 20 degrees. On Wednesday, the temperature goes below zero and it now registers at -7 degrees. Students are asked to identify the difference between the temperature from one day to the next. Some of the students see the word "difference" and immediately identify the problem as a subtraction one: subtract -7 from 20. Some write 20 - 7. Some students can be heard trying to recall a rule about how to work with integers.

Everyone in the room stops working when a student points to the thermometer and says, "Oh I get it. On the first day in the problem it's 20 degrees so the temperature drops the 20 degrees to zero and then 7 more degrees below zero on the second day. So this is a 27 degree difference in two days' time." Students in the room agree with the student. The teacher asks another student to come up and use the thermometer to show the difference between the two days' temperatures. The student indicates verbally and on the thermometer that you can add 20 plus 7 because 20 is the difference from 20 to 0 plus 7 is the difference from 0 to -7. The student draws brackets on the thermometer showing the distance from 0 to 20 and the distance from 0 to -7 and how these distances together are the difference between 20 and -7. The teacher asks the class if anyone knows how to define the distance from 0 to a number and the class launches into a discussion about the term absolute value, and what it means in terms of representation on the thermometer.

The teacher challenges students to write an equation that makes use of the -7 in which the answer is 27. Because some students see the difference between 0 and 20 as 20 units and the difference between 0 and -7 as 7 units, the students write addition equations using the absolute value of 20 and of -7.

The students write |20| + |-7| = 27. Some students write 20 - (-7) = 27.

The teacher follows up with another thermometer application: the temperature yesterday was -6 degrees but the temperature rose to 14 degrees today. Model the difference between yesterday's and today's temperatures on the thermometer. Write both addition and subtraction equations to represent the difference between the two temperatures.

By bringing the mathematics back to the context and by asking students to continuously touch and refer to the actual thermometer, what was gained? Not only did the teacher see the areas of student struggle, but students also gained a very deep conceptual understanding of absolute value. By using the thermometer, students did not just memorize a definition of absolute value, they visualized the concept and in future lessons can better operate with the absolute value of negative integers.

Now that you've heard this example, you might want to think about what other use of contexts with representations would also help students add value to their understanding.

Laurie Speranzo

Laurie Speranzo is a Fellow of the Institute for Learning's mathematics Disciplinary Literacy Team.

More information on Laurie Speranzo.