Related essay: Stringing together equations
At the top of page 1-6 in Module 2, Writing Expressions from Tables, Dimitri wrote “… so n is 3n-1”. Of course, it’s reasonable to assume that he meant “When the input is n, the output is 3n-1.” However, imprecise language, even an innocent misstatement like this one, can lead to misconceptions later.
The teacher made the decision NOT to address the incorrect language in the moment. She did not want to break the flow of the discussion to focus on a misstatement that students did not even notice. In addition, English is not Dimitri’s first language, and the teacher was concerned that pointing out a mistake might make him self-conscious and less likely to share his thoughts during future discussions.
However, the teacher did not just drop the issue. After the discussion described in the module, she reminded students of the importance of being able to communicate clearly about mathematical ideas using precise language. She wrote the statement “n is 3n-1” on the board and asked the class what that means to them. Since the teacher has created a safe environment for sharing ideas, Dimitri quickly replied “Oh, I wrote that! I just meant the nth term in the table was 3n-1.” Another student added, “Yeah, he meant the output is 3n-1 when the input is n.” Dmitri spoke up again, “At first, I forgot I wrote that and when I saw it, I started to solve the equation n = 3n-1, and got n = 1/2, which makes no sense! I see what you mean about being careful about communicating our ideas – I almost tricked myself!”
Later in the class discussion, Vince proposed it was necessary to “times the input by 4.” This is a common “slang” way to say “multiply the input by 4,” but it is still imprecise. The teacher did not discuss this statement since she realized that all the students understood what Vince meant. Although she always attempts to use conventional language herself (and encourages her students to do so, as well), she thought this might be an example of “community accepted terminology,” considering that one day in the future she might address the issue that class-based vocabulary might be confusing to others outside the class. But that was a discussion for another day.