I’m new to the blogosphere…shoved here ever so gently by my brilliant math coach Sarah Caban (mathontheedge.wordpress.com)! I’ve decided she is the only person in the district who could add one more thing to my teaching life and make me feel good about it. She bribed me to join this ~~crazy~~ awesome PD adventure with chocolate and wine (which I have yet to receive) and then, before I knew it, she had me hopping and skipping through the halls of school musing to myself about the wonderful world of Twitter and the blogosphere. This is what makes her so brilliant…she is a very clever and subtle math attack-er.

So, here goes my first attempt at finding my blog voice 🙂

Testing is done, routines and procedures have been taught, a math community has been built and it’s time to get down to the nitty gritty. I always start a new math unit (or any unit for that matter) with mixed feelings. I am always excited, but almost certainly nervous and anxious as well. There is pre-test data to sort through and planning to be done. I want to find the perfect entry point, target my small groups just so and plan jaw-dropping, aha-moment inspiring lessons for each day! These expectations and, let’s face it, my control issues, tend to delay the start of any unit, but eventually I convince myself to just jump in.

Enter 4.NBT.1, my entry point into unit 1. It’s a cool standard, but a difficult one for sure. For those unfamiliar, it requires that my fourth grade students understand the relationships between digits in a number, in particular that they recognize that as you move left on the place value chart, the number will become ten times greater and if you move back, it will become ten times less. Now, all of my students came to me knowing how to multiply a number by ten. “That’s easy Ms. Miner!” They all knew how to expand numbers using their place values and they even had a conceptual understanding of regrouping, though they have not been taught the addition/subtraction algorithms yet (yay!). They have been primed for success.

So off we went! Diving deep into this standard. We counted around the circle by tens, hundreds, thousands, etc. I recorded patterns and they talked about what they noticed. Really, I was blown away when they discovered on their own that the value was growing by ten times. They practiced and I reflected. Some students started adding ten to numbers rather then multiplying by ten. So we compared and contrasted ten times and ten more. We practiced some more, counted some more! We worked on the place value chart, used the place value disks, busted out the pennies, dimes and dollars. We worked with the language frame “______ is ten times greater than ______.” It was awesome! I felt like they had it. They were excited and using the terms and answering questions and explaining their thinking. Time for a little formative assessment…

A quick exit slip that asked students to consider the value of a collection of 826 coins. What is the value if they are pennies? Dimes? Dollars? Ten dollars? What is happening to the value of the collection? Explain with words, pictures or numbers. “This will be easy,” I thought, “We’ve been using money like this.” I just needed to document their understanding, and then off to the next concept we would go!

I’m sure you know what’s coming. I reviewed the exit slips and I was stunned. 2 of my 17 got it. TWO! So of course I looked more closely at the exit slips. I asked myself “what do they know?” (Thanks to Sarah). They almost all got the value correct when the collection was of pennies and of dollars. But when it came time think about the value in terms of dimes and ten dollar bills, they almost all got the value incorrect. Okay… “Do I see any patterns in their work?” AND THERE IT WAS! Nearly every student in the class took the value of the pennies and doubled it to get the value of the collection if it were dimes. Then they took the value of the collection of dollars and doubled it to get the value of the collection as ten dollar bills. It was strange, and I don’t know where it came from and I still can’t figure it out. When does “ten times greater” become “two times greater” in a kiddo’s head? What connection are they making that makes this make sense to them? I encountered this same issue last year and when it showed up again I was baffled. I had definitely done a better job teaching “ten times greater” this year but there it was again.

So now I ask you, Blogosphere, for some help! Has anyone else noticed this misunderstanding with their own students? Do you have any suggestions for next steps? And rather than send you an email, Sarah, I wrote a blog post…just like you asked 🙂

Chrissy Miner, I loved reading this post! It is so reflective. This trend is fascinating. I really like this exit slip because it gets to the heart of whether or not the students can apply the concept. Do you think they got pennies and dollars correct because they are using their procedural understanding of how to write the value of coins and/or dollars? I am wondering if the procedural understanding could be masking a shallow conceptual understanding of the ten times relationship between pennies and dollars. What would happen if you juxtaposed a doubling situation with a ten times situation? I wonder if using a numberless word problem would help. For instance show the students this problem – one section at a time -and ask them what they notice and wonder:

1. (First section) Chrissy had ___________ pennies. Sarah had the same amount of dimes.

(Second section) How much money could each of them have?

I am not sure about the next one. It seems like a big leap, but it could get at the associative property (pennies x ten x two)??

2. (First section) Chrissy had _______ pennies. Sarah had twice as many dimes.

(Second section) How much money could each of them have?

OR what about a Which One Doesn’t Belong: 2, 4, 8, 20

I am trying to think of ways to juxtapose doubling and ten times. I think the doubling is a natural place to be as they transition from thinking additively to thinking multiplicatively. We just have to push the disequilibrium.

I love this post and I am so very happy to have you as a colleague. Thanks for sharing!

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The numberless word problem would be an interesting place to take this. I will think more about it and let you know what I find!

Today, I did do the penny activity with a small group of boys and it was AWESOME! First we looked at the doubling example and they talked about what they noticed. I pushed them to define what it means to double something. And they decided it meant the same number twice. So we wrote it out as an addition equation and as a multiplication equation. Then I showed them the pennies growing by 10x. They quickly determined what was happening and when trying to explain it, I got some good insight on their thinking! They wanted to say that each step was “doubling by 10.” So I really pushed them to be more precise with their language. I referred them back to what they had used as a definition for doubling, and after several attempts they finally determined that they had “multiplied by 10.” It was really interesting to listen to that struggle.

When talking about doubling, students said it was “two of the same number.” And I think it is that idea combined with their inclination to say “doubling by 10” that makes them sometimes say that the 100s place is “20x greater than the 10s place.” That’s the same number (10) two times. I am so glad I got to see this today! I wish I had recorded it to watch with you! I was sad that you left my room before we got to it during learning rounds. Let me know what you think! I think you were right about juxtaposing what it looks like to double a number v. make it 10x greater.

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