This post from Growing Exponentially starts with an awesome insight on how students interpret solving equations. But then the test reveals a stubborn misconception and some procedural understanding (or difficulty with mathematical explanations in general?). Can you help?

Most of my students correctly selected Kristen; however, I was extremely disappointed in their explanations. I expected their explanations to be more in depth after the many discussions we had about solving for the x-intercepts. They mostly went with the procedural explanation of setting it equal to zero. I wanted them to explain WHY we set it equal to zero. I don’t know how to ask that without directly giving away which student’s work is correct in the first place. There’s also a problem with the many responses claiming Kristen is correct because she found two solutions. This tells me I need to do more examples with only one.

At this point in the year we’re moving onto exponentials, but I’ll be thinking about this problem for a while. Any advice would be greatly appreciated!

I went back and read the original post (and I’m now following this blog) and I took away one important idea. I think that our kids find the ‘set it equal to zero’ idea some magical trope that we math teachers spread around. Even our best need to be reminded WHY this is magical. I try to lead them through a conversation along these lines – If I told you that a * b = 0, then what do you know about these mystery numbers? How about if I told you that a * b = 12? Do you know much about a or b in this case? They are pretty convinced that an equation with a finite number of solutions is preferable to one with an infinite number. I think that a periodic repetition of this conversation really helps seal the deal for them.

Thanks for following, Mr. Dardy! I like how you call a and b mystery numbers, we talked about the zero product property but that gives it a little twist. And switching 0 to 12 would definitely make them rethink their explanations. I’m definitely going to have to try this!

I found that I could get away from the “just set it equal to zero” by never accepting that as the end of any conversation. I constantly, every single day, every single student asked “why would I set it equal to zero? why not 1? 2?” At first it’s annoying. Then it’s funny. Then it gets annoying again. And I think by the end of the year, there was only one student still stuck on a procedural explanation.

I never thought to follow up my, “Why zero?” with a different number, that’s a great suggestion. I can see how the repeated questioning might get annoying to them but in the end they’ll have it all figured out and should really be able to explain their decision. Thanks so much!

I love the repeated questioning model (esp the annoying/funny/annoying timeline) I think that the hidden advantage is htat you provide a model of behavior that the students can use when they are struggling alone (either at home or on an assessment) Perhaps the habit becomes so ingrained that they can have an internal dialogue that allows them to work through problems on their own by asking the same sort of chain of questions that you would ask them yourself.