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?

Mistakenly,

1227

*UPDATE: The next day goes like this.

Day 1 handout

Day 2 handout

Day 3 handout

Capturing Time (musically)

Recently, I had the idea to do a theme of "song lengths" over at Estimation 180. Inspired by a recent comment from Fawn, I chose Santana's Oye Como Va. At first, I opened up iTunes and took a screen shot of the music player.  I threw in an album cover and edited it to look like this, asking "How long is Santana's Oye Como Va?":

I can get away with directly asking the question at Estimation 180. How would you make your estimate? I'd make my estimate based on the time played so far (1:26) and the location of the playhead in the timeline. I absolutely love that students have to use time here, specifically 60 seconds in a minute. Furthermore, I'm hoping students use some type of spatial reasoning with the timeline, either as a fraction, percentage, proportion, or something else. But that's it. Can we go anywhere else with this? This task feels constrained. This doesn't capture the medium of music correctly. There's got to be more, right?

The more I thought about it, I was curious of better ways (or the best way) to capture time and music. Let me rephrase that. If I were going for a more perplexing approach and wanted to create a 3 Act task to share at 101qs.com, how would I go about doing that? I remembered that I own the djay app and experimented with a really lengthy Jethro Tull song titled, Thick As A Brick. This is where I need your help. I'd appreciate you checking out Act 1 and letting me know the first question that comes to mind. Or watch it here and leave a comment/question in the comments.

Based on some initial questions, I'm thinking of revising Act 1 where the virtual record player looks more like this. (notice the record?)

The virtual record player opens up many possibilities with this task. There's a white tape marker on the record for precise tracking when playing the track. I feel there's a lot more math opportunities here, but at the same time it feels a little contrived?
Am I over-thinking this?
What do you see here?
What are your thoughts?
I need some help. Thanks in advance.

Spin it,
339

Buses [Day 2]

Woah!!! We saw six buses today on our way to preschool. You might want to check out the Buses [Day 1] post from two days ago to understand the context of what's ahead. I was a little late to the bus-counting action on our way to preschool today so here's our first exchange.

Me: Have you seen any buses today?

Son (who turns 3 in a month): Yes. One.

Me: Was it a city or school bus?

Son: City. 

Me: Okay, we'll have to look for more buses today.

We immediately saw another city bus, bringing our total buses to two on the day. A little bit down the road was another city bus heading in the oposite direction.

Me: I see something on the other side of the road coming this way. What is it?

Son: It's another bus!! A city bus!

Me: How many total buses have we seen today? 

Son: Three.

Me: Good.

We continue along and a shorter city bus passes us. I don't point it out, but he spots it.

Son: There's another bus. 

Me: You're right. Wow! Now, how many buses have we seen?

Son: Hmph.

I pause and wait for him to process the question. Keep in mind, this is happening while I'm driving and he's in the backseat so I can't turn around to talk with him.

Son: Hmph

If I get a second "hmph" I know he doesn't have an answer and won't come up with one. Trust me, I've waited for long periods of time and will just continue to hear his cute little "hmph."

Me: Well, we already saw three buses and now we saw one more. What number comes after three?

Son: (whispering to self: one, two, three) Four.

Me: Good. So how many buses have we seen?

Son: Four.

Surprisingly, we hadn't seen a school bus yet. We were at our last red light and there it was in all its glorious mustardy-yellowy paint, a school bus. He exclaims, "A school bus!" Since we're at a red light, I turn around and ask how many buses have we seen. I get the two "hmph" count. I hold up one hand with four fingers up and the other hand with one finger up.

Me: We saw four city buses and now one school bus. How many buses have we seen today?

Son: Three

Did we just have a flashback (regression) to Tuesday?

Me: We did see three buses today, you're right. But, I think we've seen more. Count the fingers.

Son: One,... Two,... Three,... Five.

The light turns green and I have to go. I'm curious about him skipping 'four' and still landing on the correct number of buses. Seriously, what's up with that? I mumble to myself, "That's odd that he skipped four." Right as we're about to pull into the preschool parking lot, another school bus goes whizzing by in the opposite direction. He exclaims, "Another school bus!" Wow! We saw six school buses today, so I'm thinking we park the car and quickly review this last school bus.

Me: Wow! We just saw another school bus. We saw a lot of buses today. We saw five buses and now we saw one more. How many total buses did we see today?

Son: Hmph. (x2)

I hold up one hand with all five fingers up and the other hand with one finger up.

Me: So we saw five buses and we just saw another bus. Count the fingers.

Son: One,... Two,... Three,... Four,... Six.

Me: You're right. What happened to five?

He giggles! I do too because it's contagious. Seriously, what's up with this? He answered the correct number, but skipped the number directly preceding it. TWICE! He's happy he saw so many buses today. I am too. He's content with landing on the correct number. I'm perplexed.

Hmph,
1140

BTW. Thanks Christopher Danielson, for continuing to inspire me to have (and cherish) these conversations with my son.