Tuesday, April 16, 2013

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 mathmistakes.org 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 goes like this.


  1. I'm so glad you've posted this. I am in a weird situation with exponent props right now. Starting 2.5 weeks ago:

    1. We burned through the properties in a few days.

    2. Spring break hit us - prior to Spring Break I assigned a project on the properties. Here's an example of what I've received: http://mrwardteaches.wordpress.com/2013/04/10/exponent-properties-as-explained-by-star-wars-characters/

    3. The entire week after Spring Break, I had less than half of my class (the class is 8th and 9th grade, 8th graders were on class trip to D.C.), so we've been in a holding pattern and I let the 9's finish up their projects.

    4. Tomorrow is the first day with the entire class back together. I've set aside the 65 minute period to review the props.

    I was going to do a practice and share activity where students work through examples individually and then each student has to volunteer to "present" one of the problems - they'll kind of self-differentiate with me pushing the more advanced kids to volunteer to do the tougher problems.

    However, now I'm wondering. I might swipe your mistakes worksheet and start with that, then move onto the practice problems. Or do I start with the practice and then attack the mistakes. I'm leaning towards the former, but it's an interesting dilemma.

    1. I'd go former as well. I can't describe the level of engagement my students had when tasked with finding and correcting the mistakes. The entry point is so low. They aren't intimidated with expressions that they have to simplify. Here, they know that if they get the "wrong" answer they can try another theory in solving.
      The engagement level was head and shoulders above any previous year where I too "burned" through the properties and kids get practice exercises in front of them, immediately saying, "Mr. Stadel, I forget what to do. I don't get this. Do you add or multiply?"
      Exponent properties aren't necessarily perplexing. That's fine. My goal was to increase the level of engagement and encourage students to have a better understanding through exploration instead of direct instruction. We're getting there.

  2. Love this post, and the idea of having students explain the minor math errors they see around them. I think the 37^0 problem opens up nice discussions about concepts in math we need to establish via rule or definition, vs concepts we can understand through tangible means. Sometimes we need to have a toolbox of established rules, which then provide a framework for everything which follows. It feels like Euclid's postulates: ideas which "seem" correct, would be logically impossible to prove, yet are critical for everything that comes after.

    1. Yes, those student conversations were wonderful. I agree, we need more of those.

  3. Love the strategy of presenting mistakes. To add to the middle-school drama, I created a "Find 'n Fix" template where I do 4 hand-written problems, claiming the work has been done by "other students". They always want to know whose work they're analyzing. Sometimes all 4 problems are wrong, and sometimes there's a mix of correct and incorrect problems. Students actually "grade" the paper, providing comments and advice, as well as corrected work for the problems that are wrong (in red ink, of course). They love owning the role of the teacher who gets to grade the paper. It's always fascinating when they're convinced a problem that is actually wrong is correct, and vice-versa. Activities like this truly give the teacher a direct peek inside the brain!

    1. Hi Cathy,
      Thanks for checking in and sharing your "Find 'n Fix" activity. I'll have to try that some time.

  4. I'm at that point in Math 8 where we have spent a week on exponent rules and I loathe this unit. Something that makes sense mathematically to me... makes sense when we talk about it... but they can't keep it straight in their brains. I get depressed. I wonder what I'm doing wrong. And the cycle continues. Can't wait to try this out today! Thanks dood.

    1. I can relate! Been there! Please let me know how it goes. Making connections and demonstrating the relevance of exponent rules to anyone, let alone middle schoolers, is a challenge.

  5. This is a rock star post, Mr. Stadel. Kids need to be given a chance to figure things out on their own, not just be told rules again and again. You might enjoy "My Favorite No" at http://www.youtube.com/watch?v=Rulmok_9HVs, which is similar to your idea above. Rock on!

    1. Thanks. I do enjoy "My Favorite No" and briefly mention it here:
      I morphed it into "My Favorite Yes/No" since there are times when I have a favorite "yes."

  6. Very cool. And timely. I'm going to be diving in to dreaded exponents soon. Have you ever used a visual proof of power of zero=1? It's my favorite. Students fold their paper in half repeatedly. Because they are folding in half, the base is two. The number of folds is the exponent. To visually 'evaluate' the exponent expression, you unfold the paper and count the number of sections. We do this up to 2^3. Then I hold up a clean, unfolded sheet of paper and say, "So what if I fold it zero times?" and then I can see about 35 lightbulbs turn on. It's great.

    1. That is AMAZING! I will share this with my class tomorrow! Thank you MaryAnn!

  7. Hello! I know I am a year late to reading this but I just discovered your blog last night. Coincidentally I am starting the exponent unit today and guess what ... I am stepping way out of my comfort zone and doing THIS!!! I will let you know how it goes. Thank you!!!

  8. Welcome Sandra. I hope it goes well and I look forward to your report.

  9. This is coming a year or more late, also...but...who cares! I've been teaching for 26 years and a colleague sent out the link to this post. I LOVE IT. Error analysis is such a fantastic skill and shows depth of knowledge. Keep these great ideas coming!!! Thank you!!

    1. Thanks Mr. Clements. Exponents were a big thorn in my side. With your 26 years of teaching, I'm sure you have some gems to share too regarding error analysis. What wisdom would you be able to share with us all?

  10. This is a fantastic post, and I will certainly use this the next time I tackle the exponents unit! Thank you!

    1. Right on. I'd love to hear how it goes!