Thank You Math Mistakes

Every year I dread teaching exponents like nothing else. I still do. It's one of those concepts (units) I have a tough time relating to, and if I have a tough time, imagine my students. For me, exponent rules and properties have been reduced to nothing more than a good puzzle. Not sure admitting that here is wise, but I do enjoy puzzles. However, to me, expressions with exponents don't necessarily lend themselves to having applied meaning for middle schoolers, or at least no simple contextual application I can relate with. This is where you jump straight to the comments and say something like,
"Andrew, exponents are seen in [insert awesome idea] and here's a link." or
"Mr. Stadel, students can relate to exponents when they [insert other awesome idea]." or
"Stades, I'm surprised you haven't done a 3 Act lesson on exponents using [insert other fabulous idea]."
"This year will be different." You ever mutter that to yourself? Well this year is and will continue to be. I shamelessly request you to temporarily stop reading this blog and visit Michael Pershan's website Math Mistakes. Seriously, type into your url address. The more I research teaching strategies and content, the more I'm starting to see the benefit of students learning math by identifying mistakes, correcting them, and justifying their reasoning while doing so. Inspired by Michael's post on exponents, here's the handout I threw at my 8th grade Algebra students today with the following directions:

The following statements are all INCORRECT.
  1. Identify the mistake(s).
  2. Correct.
  3. Justify (show) your reasoning.
Me: I'll give you guys 3-5 minutes of individual time to work through as many questions as possible. Then you'll share and discuss your ideas with your group followed by a whole class discussion.
Students worked individually on the questions for a few minutes. When most students were at least 75% complete with the handout, I told them they had until the end of today's estimation song Can't Buy Me Love (in its entirety) to share their solutions and reasoning with their group members. I heard some great stuff. Next we did our Fish Bowl where students come up to the front, walk us through their work and reasoning on the board while we listen and watch both intently and quietly waiting to ask questions. Here's the four we got through today.

Students in the Fish Bowl identified the mistake as multiplying two by five, producing ten. Students also correctly explained how two to the fifth power is thirty-two. The fun part (for me) was seeing multiple representations. 
  1. Common: 2*2 = 4, 4*2 = 8, 8*2 = 16, and 16*2 = 32.
  2. Grouping two's so 2*2 = 4 twice. 4*4 = 16 and 16*2 = 32.

The mistake here was explained a few different ways, but mainly revolved around the student forgetting the negative.  So far, questions 1 and 2 have a lower entry point for today's task, considering these students haven't seen exponents since last year. Multiple representations looked like this:
  1. Common: -2*-2*-2 = -8
  2. Preferred: (-2)(-2)(-2) = -8
The classic. I always enjoy this example. Students explained that the mistake was made by multiplying -6 and -6, producing positive 36. Many students pointed out that the exponent is "attached" to the closest term (6) and not the entire expression (-6). Multiple representations included:
  1. Common: -(6*6) = -36
  2. Common: -6*6 = -36
  3. Extra: -1*6*6 = -36
This was an extremely fun conversation to have in class. In fact, for some periods, we never established a conclusive answer to the question before the bell rang. Here's what they came up with. Some students simplified it to thirty-seven and many were convinced of this at first. Very few simplified it to one, but couldn't convince anyone why. Some students offered the following:
Joey: My fifth grade teacher told me anything to the zero power is one.
Raquel: There's some rule that says it's one. It's just the rule. 
Me: Who's convinced by their reasoning? No one? 
Here are the multiple student representations we saw:
  1. Pattern: 37^2 = 37*37,  37^1 = 37, so 37^0 = 1 (this didn't convince many).
  2. "Anything over itself is 1," some said. Therefore, 37^5 over 37^5 is one and you subtract the exponents (5-5) to get 37^0. Therefore, 37^0 = 1 (this convinced many).
I forgot to mention that before we did Fish Bowl, I asked students what was different about what they initially did with the handout. Here are a few things they said:
  1. You gave us individual time instead of just going straight to group work.
  2. We had to make corrections.
  3. We had to try and figure out the questions without you telling us any rules.
The last observation was my favorite. This activity gave students a desire to listen to each other and want to know the answer to these questions. I wasn't at the front of the room blabbing out rules, properties, their names, and examples. I didn't provide students with guided practice. This combination of activities and strategies felt right. More to come tomorrow, but we can't ignore that there's something to this mistake idea. What do you think?


*UPDATE: The next day does like this.