"Making it Stick": Math Teacher Richard Chang

Richard Chang’s seventh grade math class is all about possibilities. Today, his students have eagerly accepted the challenge to physically arrange 24 cubes in different configurations in order to minimize the surface area. Chang, of course, could show the students how to do it, but instead he prefers to spark flexible and creative inquiry.

To test his students’ reasoning, Chang asks a seemingly simple question: “How many possibilities can we find?” Hands around the class shoot up.

Henry says five. Marina counters with six. Chang asks, “Does anyone have a different number?” At first, the students appear unsettled. Some can’t sit still, others lean over a neighbor’s desk to rearrange the cubes, and still others focus on their own work while someone else calls ideas from the other side of the room.

Chang, who was named department chair this year, sees the relationship between creative thinking and correct thinking as overlapping, as a Venn Diagram of sorts. As long as a student’s idea works, and can be applied correctly, consistently, and repeatedly, he encourages its use.

This style of teaching math, Chang says, is the inversion of his own experience with the formula-based, plug-in-numbers style of teaching he learned in school. “Memorizing takes too much brain power unless the formulas themselves make sense,” he says. “We do have our students memorize, but it’s the reasoning that makes it stick.”

In the room, a couple of minutes pass and the students’ focus starts to sharpen. Most are sharing ideas with each other, and eventually some begin to call them out. Chang, from the front of the room, is in command of the class energy as he refines their collective offerings: “Great idea, but how would we find this out in an orderly fashion?... How can we do this without skipping something?... Charlie, factors of four?... How else?... That’s helpful, but we need more.”

Marina offers, “I did it differently. I divided using two as much as possible.”

Chang brings her idea to life on the SmartBoard by projecting her arrangement of the cubes and then drawing over the image. He shows the class how the cubes become a diagram, which in turn becomes a formula that the students can use and manipulate themselves.

“Were you confident you didn’t skip anything?” Chang asks. His eyes survey the room. “Anyone have a different way of organizing the information to make sure you didn’t skip anything?”

His attention turns to a desk on the side of the room. “Andrew, come to the board and show how you did it.” Andrew practically tips his chair over to race up to the front of the room. He grabs the pen to illustrate then explain his formula on the SmartBoard. He asks the other kids in the back if they can see and moves slightly to the side so his ideas are visible to everyone. His classmates soak it up. They nod in agreement or offer refinements as he talks. Andrew smiles as he makes revisions to his formula on the board. He’s as curious as anyone about how to make his answer more complete.

Chang seizes on the opportunities presented by Andrew’s ideas. “Charlie, did you do it a different way? What else do we have to do? What’s left?”

Throughout the class, Chang smiles and laughs non-stop. He enjoys the material and his students’ willingness to experiment with it only adds to the fun.

Midway through the class period, Chang keeps the enthusiasm piqued by arranging the students into groups and posing the next challenge: “First, I have never given you a formula for the surface area of a cylinder. But, based on your prior knowledge, I think you can figure it out. Second, can you explain your formula so clearly that even a person who fears and/or hates math would think it makes total sense?”

Chang walks around the class handing each group a cylinder (a cup, a water bottle, a food storage container) for students to hold, examine, move around, and make a “net” for their work. The net, a piece of paper the students fashion to wrap around the entire 3-D object, becomes, when laid out flat on the table, a 2-D representation of the surface area. The tactile process really helps. Students instinctively hand the familiar objects around the group for inspection. In the context of this math class, these old objects have suddenly become new and fascinating.

Soon, the class is buzzing again with conclusions:

“I got it!”

“How does that look?”

“Circumference of what?”

“Ooh, we got it! We got it!”

“You do the top part plus the other part.”

“Pi r squared times two plus the area of the rectangle.”

“So we have it—we have a circle and this other rectangle?”

“Can I go first?”

One by one, the groups come up to the SmartBoard to illustrate and explain their formula. Each group builds on the last until the end-of-class bell finally interrupts them. As most students head for the corridor and their next class, one group stays behind to polish their ideas. After finishing, Ava looks at Chang and asks, “Is that clear?”

Now that’s teaching.