**Teaching for Understanding**

In their previous blogs, Allen and Dera both talked about "aha" moments, or moments of understanding, that can drive and inspire our teaching. Indeed, "teaching for understanding" is a buzz phrase in education. But just what does that mean?

How do we move students from feeling like this

to feeling like this?

According to Buehl, we as teachers can help students move in that direction by teaching them how to apply comprehension processes. So in this blog, I am going to talk about a lesson I've seen in which a teacher helped her students to apply those processes.

I observed a sixth grade mathematics lesson in which the teacher was introducing students to the concept of slope. She began by graphing y=x on a calculator that was projected on the overhead projector.

Then, she put a fraction in front of the x. So, she graphed y=(1/2)x.

Then she graphed y=(1/4)x.

Students then had to suggest other fractions they could enter in front of the X. Before entering these fractions, students predicted what the line would look like by drawing it on the board first. Then, Grace (the teacher) graphed the actual line and students compared their predicted line to the actual line.

Grace again returned to y=x and asked what other numbers students could enter in front of the x.

Students suggested different whole numbers they could enter in front of the x. So Grace first graphed y=2x.

Then she graphed y=4x.

Students then predicted what y=10x would look like by drawing a line on the board, and then Grace projected the actual graph on the board. In case you were predicting yourself, here is what y=10x looks like:

After this lesson, Grace then asked students to write a generalization regarding the patterns they noticed, and students shared their generalization. For instance, one student wrote, "When the number in front of the X gets bigger, the line gets steeper."

Okay, that was the lesson. So how did this lesson require students to use comprehension processes?

To me, the biggest comprehension process that students used was "make predictions," which Buehl listed under "making inferences" (p. 5). Students had to use the existing texts (graphs) to extrapolate information beyond what was explicitly stated within the text.

Another comprehension process that students used was synthesizing when they had to summarize the pattern at the end of the lesson. Buehl said that synthesizing includes "constructing generalizations," which is exactly what these students did.

I think this lesson might also have used "visualizing," in the sense that students visualized patterns through line graphs.

This mathematics teacher had the highest end-of-year test scores in her whole school district, and to me, this lesson exemplified some of her good teaching. At the same time, I think that to extend this lesson, the teacher could have encouraged students to ask questions such as, "Does this pattern always apply?" I don't think that students had learned about negative integers yet, but that could lead to other interesting questions. However, I think for the purpose of this lesson, given the stage of students' mathematical development, this lesson did require students to apply several comprehension processes in ways that made them think more deeply about patterns in mathematics.