Teaching for Understanding

Teaching for Understanding

One of the things that has always bothered me about the Common Core Curriculum is the idea of teaching for "understanding." On the surface, this seems like a term that is so vague that it is almost useless. How do you know when a student understands? Then I started to read Adding It Up: Helping Children Learn Mathematics—the CCSS borrows freely from this report.

As per Adding It Up, conceptual understanding is about making connections and organizing ideas. Students need to learn how to represent situations in different ways and know how different representations can be useful for different purposes. In particular, students need to explore similarities and differences among various representations. Students need to learn why ideas are important.

Let's look at some examples of varying difficulty.

• As per standard A.SSE.4, students are supposed to understand that the polynomials form a system similar to the integers. You can teach for understanding by having students explore the similarities and differences among addition, subtraction, and multiplication with integers and polynomials. In particular, students can see whether or not the usual properties of numbers hold for polynomials—commutativity, associativity, etc.
• As per standard 6.RP.2, students need to understand the concept of unit rate. You can teach for understanding by emphasizing unit rates in different real-world contexts. For example, speed as a rate, cost per item as a rate, etc.
• As per standard 8.F.1, students are supposed to understand the concept of a function. Again, you can teach for understanding by emphasizing the different ways that functions can be represented—symbolic, numerical (table), and graphical. For example, you can give a problem that asks students to compare two functions—one represented symbolically and the other by a graph. You can also teach for understanding by talking about the importance of functions in modern society—namely, how functions are a fundamental part of computers.