# Decimals, Backwards Slashes, and Giggling in Math Class

Recently, I read a blog post by Andrew Gael, Our kids Are Not Swiss Cheese.  Some quotes that stuck with me from Andrew’s blog:

• “Maybe it is not the learners; maybe it is the way that we conceptualize learning…”
• “Learning is complex, multi-leveled, and no one is all the way “filled in.””
• From Megan Franke, “How do we notice and use what students DO know to support them to make progress in their thinking?”

Last week, during our 5th grade collaborative planning session, we discussed how to introduce decimals to our students. We decided we wanted to start by unearthing what the students already understood about decimals.  I was really excited to approach “decimals” as a concept that connects to prior knowledge, instead of a series of disjointed procedures.

So, today, we started our journey. I co-taught with Mrs. G. and Abby, our school based math specialist, co-taught with Mrs. C.  We wrote this question on the board, “What are decimals?”  We told the students we would ask them for their thoughts about this question at the end of class. Then, we counted.

We started counting by ones and tens. Then, we asked them to count by tenths.  That is all we said, “let’s count by tenths. Who wants to start?”  Matt said he wanted to start.

“one tenth.”

I asked, “how would you like Mrs. G to record that?”

“Just write one tenth.”

“Can you tell us what that will look like?”

He went over to the white board and wrote this:

“Okay,” I said. Mrs. G recorded ‘one tenth’ on our chart.  Then, I asked Gary to continue the count.  We continued the count, each time asking the student how they would like us to record what they said.  This is what they told us:

If someone was unsure, we told them we could put a question mark above their suggestion and we could come back to it.  It was so interesting to me how quickly the students began referring to each other as authors of ideas. When asked, “how would you like us to record that?”  They said, “like _______ did.”  When we got to nine tenths, one student told us he would like us to record it ‘the same as six tenths, but with the slash the other way.’  I actually have a voice clip.  Listen.

As I listen to this clip now, I am smiling. I love it. I hear confidence and creativity. The first time I heard it, I was nervous. I wondered, should we put that on the chart?  What if the students think it is an acceptable way to write a decimal?  What if I ruin them forever by supporting this backwards slash business?  I almost panicked. There were so many times during this routine that I almost caved. I almost said, “actually, that is not how we write decimals.” But, I didn’t. I am so glad that I didn’t.  Look at this chart! I mean really look at it.  What do you notice?  What do you wonder?  What do these students know about decimals?  What do these students know about our number system?

Yes. There are definitely some partially formed ideas here. There is no doubt that we need to continue our study of decimals.  Of course we do. It is only day one.

After we finished our count, we told the students that we will continue to look at this anchor chart and we will continue to count by tenths. We also asked them to write down something they noticed and wondered about our chart.

We asked a few kids to share their thinking: B shared his thoughts about 2.5 = 2.50, S shared his question about whether we can write decimals in word form and we confirmed that we can, K asked about writing decimals in exponential form and we told him it is possible, but we would talk about that in more detail later. At the time of the lesson, Mrs. G and I were so bummed that no one noticed the ten in a row pattern. As I write this blog, I realize Molly DID notice it. Arghh!  We will have to ask her to explain her thinking tomorrow.  Maybe we can compare and connect Molly’s, Patrick’s,  and Gabe’s responses.

Next, we split the kids into two small groups.  Mrs. G took half and I took half. When we made our heterogeneous groups, we considered processing time, distractibility, schema, perseverance, expressive and receptive language, etc. We spent less then 5 minutes, but we considered all of these criteria as we tried to form groups that amplified the learning experience.

Mrs. Gordon and I each facilitated a Number Talk using the following images:

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My group had some really interesting conversations.  I was worried that, if we used money, it might limit our conversation to one context, but I don’t think it did. One of the most interesting questions they asked was when they wondered whether 1.50 and \$1.50 were the same or different. What a deceptively simple thing to wonder about. This next clip is really interesting.  I like it beacause I think it is an example of what Andrew discussed in his blog. Can you hear the non-linear complexity of ideas being formed?

To close out the lesson, we asked the students to do two things. First, we asked them to answer the question, “what are decimals?”

We also gave them a question to think about.  We asked them to tell us whether they thought the statement was true or false and explain why. We didn’t expect them to get the answer *correct*.  In fact, we were less interested in correctness and more interested in how they explained what they understood so far. Here is what they came up with:

After the lesson, Abby and Mrs. C check in while Mrs. G brought the kids down to lunch. Then, Abby told me how it went  in the other 5th grade class. Then Mrs. G and I checked in while Abby went to lunch. We were all wondering what to do next. We decided we would continue with the plan we had sketched out last Thursday. Tomorrow, we will introduce the kids to the Zoom in on the Number Line routine. We will try to connect the magnitude of tenths and hundredths as we compare decimals and place them on number lines with varying intervals. We also decided we would re-use our artifacts from the lesson close.  We are going to give the exit ticket and sticky note answers back to the students throughout the week and ask them what they would add and/or change.

Finally, on my way to Wayne Elementary School, I stopped at varying spots to collect artifacts that reminded me about decimals.  Here is what I came up with.

I texted my artifacts to Mrs. G and Mrs. C and asked them to ask the kids to go on their own scavenger hunt for decimal related pictures or conversations that they wonder about.

I got my first response:

So, yeah, it is scary to invite kids to put a bunch of partially formed ideas on the table. It is messy and it will take us awhile to sift through them and make connections, but I think it will be time well spent.

Last week, I was reading Tracy Zager’s book, Becoming the Math Teacher You Wish  You’d Had.  I tweeted her to let her know that it was going to take me years to finish her book because it so rich with provocative ideas.  Here is one that I have been mulling over for days now:

“Above all else, maintain your focus on developing young mathematicians who listen to and refine their internal truth detectors. Encourage them to be skeptical and allow them to remain in doubt until they are genuinely convinced. Do not apply pressure to concede, even, if you’d like to move on.”

I just love that. I would be pretty psyched if, some day, some thirty-something mathematicians tracked me down to thank me for helping them refine their internal truth detectors.  Thanks again for the push Tracy.

## 2 thoughts on “Decimals, Backwards Slashes, and Giggling in Math Class”

1. I love this blogpost so much, and especially the way you recorded the count! OMG! Thank you for always being willing to share your thinking. I learn so much from you.
Tracy

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