One of the greatest benefits to my current teaching position is its small class sizes, which affords me a significant amount of one-on-one tutorial time. I know fairly well how I think about numbers; I don’t know how other people, particularly students who struggle with mathematics, do. I feel that it’s crucial that mathematics teachers take time to listen to student reasoning as often as possible.

There were two cases last week involving the same problem where each student’s reasoning was revealing. Had I simply cut them off without exploring, I would have lost an opportunity to learn and improve my own teaching strategies.

The problem involved reading a graph. At birth, a girl is ten inches tall. At six years old, she’s fifty inches tall. The graph’s y-axis has tick marks every five inches. How much has she grown over six years?

### Student 1: Right, but sounds wrong

When prompted, the first student correctly identified the relevant data (10 inches at birth, 50 inches at six). I then asked, “Okay, so how do you figure out how much she’s grown over that time?”

The student responded, “You add.”

I paused for a moment, ready to correct her. Instead, I sallied forth: “Okay. And what does that give you?”

She looked at the numbers again and said, “Forty inches.”

I was surprised, but I explored further: “How did you get that number?”

“Because ten plus forty is fifty.”

I had made the mistake of thinking that she was setting it up in the “traditional” way: 50 operator 10 equals…?

### Student 2: Wrong, with “correct” reasoning

As with the first student, the second student correctly identified the relevant data from the graph. In response to my question about the difference, she said we should use subtraction.

I asked her the difference. She looked at the graph and, after a moment, said, “She grew 8 inches.”

I asked her how she’d gotten that. She pointed to the scale and said, “It’s a number line. I counted.” She then counted out the tick marks: 1, 2, 3, 4, 5, 6, 7, 8.

I’ve never been a fan of number lines, personally, but they seem a significant part of modern mathematics teaching. This incident illustrates a potential problem with number lines: It has to be stressed then stressed again that counting off on number lines has to be adjusted if the tick mark scale is anything other than 1. This student was using reasoning to the best of her education.

As soon as I told her that each tick mark was worth 5 inches, she realized she needed to multiply eight by five, and got the correct answer.