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Mathematizing Learning Lab

Each month, teachers choose their Learning Lab content focus for our work together. Most months, 1/2 of the grade level teachers choose to have a Math Learning Lab while the other 1/2 work with Erin, the reading specialist in an ELA Learning Lab. This month, however, we decided to mesh our ELA and Math Labs to do some mathematizing around children’s literature in Kindergarten and 1st grade! This idea was inspired by a session at NCTM last year, led by Allison Hintz, that left me thinking more about how we use read-alouds in our classrooms and the lenses by which students listen as we read.

In The Reading Teacher, Hintz and Smith describe mathematizing as, “…a process of inquiring about, organizing, and constructing meaning with a mathematical lens (Fosnot & Dolk, 2001). By mathematizing books commonly available in classroom collections and reading them aloud, teachers provide students with opportunities to explore ideas, discuss mathematical concepts, and make connections to their own lives.” Hintz, A. & Smith, T. (2013). Mathematizing Read Alouds in Three Easy Steps. The Reading Teacher, 67(2), 103-108.

Erin and I have literally been talking about this idea all year long based on Allison’s work. We discussed the ways we typically see read-alouds used, such as having a focus on a particular text structure or as a counting book in math.

As Erin was reading Kylene Beers & Robert Probst’s book, Reading Nonfiction she pointed me to a piece of the book on disciplinary literacy which automatically had me thinking about mathematizing.

Beers refers to McConachie’s book Content Matters (2010), in which she defines disciplinary literacy as, “the use of reading, reasoning, investigating, speaking, and writing required to learn and form complex content knowledge appropriate to a particular discipline.” (p.15) She continues to say, “…disciplinary literacy “emphasizes the unique tools that experts in a discipline use to engage in that discipline” (Shanahan and Shanahan 2012, p.8).

As I read this section of the book, my question became this…(almost rhetorical for me at this point)

Does a student’s lens by which they listen and/or read differ based on the content area class they are sitting in?

For example, when reading or listening to a story in Language Arts class, do students hear or look for the mathematical ideas that may emerge based on the storyline of the book or illustrations on the page? or Do students think about a storyline of a problem in math class or are they simply reading through the lens of “how am I solving this?” because they are sitting in math class?

Mathematizing gets at just this. To think about this more together, Erin and I decided to jump right into the children’s book The Doorbell Rang by Pat Hutchins. Erin talked about the ideas she had for using this in an ELA class, I talking through the mathematical ideas that could emerge in math class, and then we began planning for our K/1 Learning Lab where we wanted teachers to think more about this idea with us! We were so fortunate to have the opportunity to chat through some of our thoughts and questions with Allison the day before we were meeting with the teachers. (She is just so wonderful;)

The first part of our Learning Lab rolled out like this…

We opened with this talking point on the board:

“When you change the way you look at things, the things you look at change.”

Everyone had a couple of minutes to think about whether they agreed, disagreed, or were unsure about the statement. As with all Talking Points activities, each teacher shared as the rest of us simply listened without commenting. The range of thoughts on this was so interesting. Some teachers based it on a particular content focus, some on personal connections, while I thought there is a slight difference between the words “look” and “see.”

After the Talking Point, Erin read The Doorbell Rang to the teachers and we asked them to discuss what the story was about with a partner. This was something Allison brought up that Erin and I had not thought about in our planning. I don’t remember her exact wording here, but the loose translation was, “Read for enjoyment. We want students to read for the simple joy of reading.” While Erin and I were so focused on the activity of exploring the text through a Math or ELA lens, we realized that the teachers first just needed to enjoy the story without a purpose.

For the second reading of the book, we gave each partner a specific lens. This time, one person was listening with an ELA lens while, the other, a Math lens. We asked them to jot down notes about what ideas could emerge through these lenses with their classes. You may want to go back and watch the video again to try this out for yourself before reading ahead!

Here are some of their responses:

Together we shared these ideas and discussed how the ELA and Math lenses impacted one another. A question we asked, inspired by Allison, was “Could a student attend to the math ideas without having a deep understanding of the story?”

Many questions came up:

Could we focus on text structures and the math in the same lesson?
Could we start with an activity before reading the book, like a probable passage?
Would an open notice/wonder after the first reading allow the lens to emerge from the students? Do they then choose their own focus or do we focus on one?
How could focusing on the problem and solution get at both the ELA and Math in the book?
How could we use the pictures to think about other problems that arise in the book?
How do we work the materials part of it? Do manipulatives and white boards work for K/1 while a story is being read or is it too much distraction?
What follow-up activities, maybe writing, could we think about after the book is read?

Unfortunately, our time together ended there. On Tuesday, we meet again and the teachers are going to bring some new books for us to plan a lesson around! So excited!

Making Sense of Problems: Part 2

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This post is an extension of a previous post. For the background story to this post, it will be helpful to read THIS POST first.

The original Noticing and Wondering from the launch of the lesson:

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Here are some expanded descriptions of the student work:

Chose numbers strategically to make it easier for themselves:

These two girls were great because they wrote out the paragraph first with the blanks left to fill in after they made a decision on their numbers. You can see the erased 5 in the second blank. When I asked them about it, they said 25 in a class seemed like too many but they couldn’t make the class too small to each get more than 1 bar.

These two were concerned with the number in each box. They said they knew 6 usually came in a box so they just did 4 boxes and then wrote the students in last to make it easy division.

These two were done SUPER fast so I gave them 5 more bars to try and decide what they wanted to do with them. They didn’t do any written work, but asked me how they divide something up into 5 pieces because then each student could get a piece. “We know halves and fourths, but that is not 5 pieces.” After playing around with “fiveths” I gave them the word fifths and they wrote down 1/5.

These two partners were so interesting because when I walked by the first time they had chosen their numbers together, but when I went back the second time, their computation was completely different. I absolutely loved that and asked them to explain their strategy to one another and asked how they were the same and how they were different. The difference was more about the look of their work, but they agreed they were the same because it was still how many 30s were in 63.

Chose numbers randomly:

…and then they worked together on breaking the extra 3 into 10 pieces. Because they didn’t know how to name tenths, they went to something they obviously knew something about…percents! We ran out of time to ask how they knew that was 10%, but I have to make a point to go back and ask!

Dealt with the leftovers using fractions:

This one took a while for me to figure out. These two girls finished rather quickly, so I asked them if they could share the leftovers equally among the kids in the class. It looks like they multiplied the 22 by 2 to get how many pieces they would have if they split them in halves. They each person got an extra 1/2 and they were left with 18 halves. They multiplied by 2 to make them wholes again and ended with 9 bars left over. The sense-making in this one was so incredible to me.

Chose numbers strategically to make it harder for themselves:

When I asked these two girls why they chose the numbers they did and they said they wanted to make it hard! There are so many things I think continued to be fabulous after their initial number choice. The partial products for multiplication and then the repeated subtraction were amazing. I asked them why they were subtracting 26 every time. They said each time they subtracted 26, each student got 1, hence the growing list of 1,2,3,4,5… Absolutely awesome and something I would have never seen if I had given the original problem and the sense-making of what is happening whey you repeated subtract to divide just blows my mind.

To me, every one of these examples, along with all of the papers in the class that day, demonstrated to me how we need to look critically at our math textbooks, think deeply about what we learn about students as they do that work, and adapt materials to allow students to make sense of problems and allow us to learn more about their understandings.

Formative Assessment

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Assessment always seems to be such a broad, hot topic There are rubrics to help create assessments, rubrics for reviewing assessments, and tons of reading about the benefit of assessments. While I agree assessment is an important topic of conversation and all of these things can be helpful, I just lose a bit of interest when it becomes so cumbersome. I feel the longer the rubric and steps to create an assessment, the more detached the assessment becomes from student thinking. This could be completely be my short attention span speaking, however the way assessment is discussed feels either like data (a grade or number-type of data) or a huge process with tons of text in rubrics that I really, quite honestly, don’t feel like reading. Not to mention, I just love looking at student writing and listening to student thinking when planning my immediate next steps (formative) or checking in to see what students have learned over a longer period (summative). This is why I find the work we are doing each month in our Learning Labs such a wonderful way to think about formative assessment in an actual classroom context, in real time.

This passage from NCTM’s Principles to Action really captures how I feel about the work we are doing in our Learning Labs:

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In this most recent Learning Lab in 3rd grade, we planned the activity together using the 5 Practices model and reflected after the lesson. Since this blog is always my thoughts about student work, I thought it would be great to hear what the teachers took away from the activities we are doing in terms of the students’ understandings and impact on their future planning, formative assessment.

The teacher mentioned in the blog said, “I was surprised by how quick many of the students defended their responses that 1/2 will always be greater than 1/3, and then proving this response using visual representation of the same whole ( which is an idea that we have made explicit). I was impressed with “skeptics” in the crowd that were looking to deepen their understanding around the concept by asking those “What if” questions. Going forward, I want to create opportunities that push and challenge my student’s thinking. I want them to continue to question and explore math – especially when it uses the word “always.”

Another teacher who taught the same activity after watching it in action in another classroom said, “I learned that almost half of my students assumed they were comparing the same size wholes. They agreed with the statement, and each student gave at least two different ways to prove their thinking (area and number line model were most common). The students that disagreed almost all provided their own context to the problem, such as an example with small vs large pizzas, or a 2 different-length races being run. I found it so interesting that almost all students confidently chose one side or the other, and were able to defend their thinking with examples (and more than one-yeah!) I was excited to see that they could be so flexible in their arguments as to why they felt as they did. Three students responded that they were unsure, and gave reasons to support both sides of the argument. This impacted my instruction by giving me such valuable formative assessment information with a simple, non-threatening prompt. It took about 5 minutes, and gave me tons of information. It was accessible and appropriate for all. Students were comfortable agreeing or disagreeing, and in some cases, saying “unsure-and here is why.” I was most excited about that!”

She also said, “From this activity, I learned that I really needed to revisit the third grade standard to see what is actually expected. It says they should recognize that comparisons are valid only when the two fractions refer to the same whole. My statement didn’t have a context, so how cool that some were at least questioning this! This impacted my planning and instruction by reminding me how thinking/wondering about adding a context to the statement would influence their responses. I am also reminded that I need to stress that students must consider the whole in order to make comparisons accurately.”

Earlier in their fraction unit, the third grade teachers used the talking point below to hear how her students were talking about fractions. (This work is actually from another teacher’s class, but you get the idea;)

A teacher who did this activity reflected, “From this activity, I learned my students had only ever been exposed to a fraction as a part of a whole (and wanted to strictly refer to fractions in terms of pizza). This impacted my instruction by being sure to have the discussion that fractions can represent parts of a whole, but we can also represent whole numbers with fractions.”

To me, these reflections are what assessment should be….the teachers learn about student thinking, the students think about their own thinking, and what we learn helps us plan future lessons with our students’ understandings in mind!

More examples from different grade levels where the teachers and I learned so much about student thinking that impacted future instruction:

Kindergarten: Adding

Kindergarten: Counting

1st Grade: Fractions and Adding

2nd Grade: Counting and Leftovers

4th Grade: Division

5th Grade: Fraction Number Line

3rd Grade: Comparing Fractions

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I was so excited just walking into Jenn Guido’s room today and seeing this awesomeness on the board from the day before:

We chatted with the class a bit about their responses on the board before jumping into our Number Talk. One thing Jenn and I both noticed during this chat was the use of the word “double” when talking about equivalents such as 2/4 and 4/8. We had the chance to ask them what exactly was doubling and kept that in the back of our mind as something to keep revisiting. Even in 5th grade, I would hear the same thing being said each year. I would always have to ask, “What is doubling?” “What is 1/2 doubled?” “What exactly is doubling in the fraction?” “What happens when we double the numerator? denominator?”

After this chat, it was time to move into our planned activity. The class has been doing a lot of work with partitioning (and they used that word:) circles, rectangles and number lines so we planned a Number Talk consisting of a string of fractions for the students to compare. We were curious to hear how they talked about the fractions themselves and how they used benchmarks and equivalents. The string we developed was this:

1/6 or 1/8 – Unit Fractions

5/8 or 3/8 – Same Denominator (same-sized pieces in student terms)

3/8 or 3/4 – Common Numerator, Benchmark to 1/2, or Equivalents

3/3 or 4/3 – Benchmark to 1

The students shared their responses and did an amazing job of explaining their reasoning very clearly. In all of these problems and actually in all of their work thus far, they have always assumed the fractions referred to the same whole. We decided to change that up on them a bit and see what they would do with the statement, “1/2 is always greater than 1/3.” We thought the word “always” would make them second guess the statement, but we could not have been more wrong…they all agreed. A few students shared their responses, and it was great to see such a variety of representations.

This student was interesting because he used 12ths, and although he could not articulate why, it was labeled correctly. I am assuming it was because 1/2 and 1/3 could be placed on 12hs, but I am not sure because his reasoning sounds like he is comparing the 1/2 and 1/3 as pieces not in 12ths.

Jenn, Meghan (another 3rd grade teacher with us in the room) and I chatted while they were working about how to get them to reason about different-sized wholes. A picture would have been a dead giveaway so I just went up and circled the word always and asked, “Does this word bother anyone?” and one lone student said it made him feel like there was a twist. I love those skeptics. I asked them to talk as a table about what the twist could be in this statement, and then we had some great stuff! They talked as tables, and while only two of the tables talked about different wholes (in terms of number lines which was not what I expected either), there was so many great conversations trying to “break the statement.”

This is an example of the number line argument:

This group kept saying it would be a different answer if they were talking about “1/2 of” or “1/3 of”…then said, “Like 1/3 of 1/2” and THEN KNEW IT WAS 1/6 when I asked what that would be! They said 1/2 is 3/6 so 1/3 of that is 1/6. Wow. Then, of course I could not resist asking what 1/2 of 1/3 would be and they kept saying one half thirds, but could figure out how to write it and then questioned if that could even be right.

After having the tables share with the whole group, they all agreed the statement should be sometimes instead of always. Jenn asked them to complete two statements…

“1/2 is greater than 1/3 when….”

“1/2 is not greater than 1/3 when…”

A great day! We are doing the same thing in Meghan’s classroom tomorrow and are changing the first problem in the string to 1/2 and 1/3 so we can revisit that at the end. Can’t wait!

1st Graders Talking and Using Half

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Yesterday, I had the chance to teach a 1st grade math class. The teacher told me they are about to start their fraction lessons so I thought it would be fun to do a quick check in on what they currently think about half and then do a numberless story problem to see if they incorporated anything about half in that work.

I launched with “Tell me everything you know about half.”

One student started by telling me it is like half a piece of pizza, so I asked what that looked like to her and she said it was the whole thing (big circle with hand) and then cut in half (hand straight down vertically). That springboarded into half of lots of things, cookies, strawberries…and many other things. Each time I asked if each half was the same in the different things and they said no, they were different sizes. So, I asked what was the same and someone said they were all cut in the middle. I got some cool sports references, I asked when halftime happened and they said in the middle of the game. Then one girl said “5 is half of 10.” Awesome. I asked how she knew that and she said, “because 5+5=10.” Hands shot up everywhere after that with other numbers and their halves.

Then I posed the following story on the board and read it to the class:

There was a pile of blocks on the table. Jimmy came into the room and took some of the blocks. He gave the rest of the blocks to his friend Kali.

We did a notice/wonder and they wondered the things I had hoped: How many blocks were there? How many did Jimmy take? How many did Kali get? It was really cool that one student noticed there were none left on the table because Kali got “the rest.” I didn’t expect that one!

I let the partners choose their own numbers and as they got their answers, I asked if they could write an equation for their work. My plan for the group share was to have groups share, some who split the blocks in half, others who did not. My backup plan, because I never know what could happen, was to compare addition and subtraction equations for the same story to see if they noticed a relationship. I ended up with enough half/not-half that I went with that!

Here were some of the groups who split the blocks in half. The number choices were really interesting. I would love to put all of them up there for the class to talk about…why did they choose numbers that end in zero? What do we know about those numbers?

Some did not choose a number they could easily split in half. The group on the right noticed the commutative property right away and drew lines to show the same numbers in their equations.

Some really wanted to write as many equations as they could that didn’t necessarily match what was happening in the story but was great mathematical reasoning in their work!

I cannot wait to see what this group does in their fraction unit, so many great thoughts and work about half!

Decimal Addition Card Sort

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When solving problems in Number Talks, the strategies, en route to the solution, are the focus of the discussion. However, not all problems posed during a Number Talk are created equal or solved the same way every time by every student. While I know the majority of students use a particular strategy for one reason or another, whether it be because of the numbers involved or maybe it is the only strategy they are comfortable using, I like to take time and make these choices explicit. I want the students to think about the numbers before just computing and become more metacognitive about their actions.

Last week, the 5th grade teachers and I planned for a card sort to get at just this. The students have been adding decimals and using some great strategies, but we really wanted to hear about the choices they were making. With the help of the Making Number Talks Matter book, we chose problem types for students to think relationally between whole number and decimal operations. While there are no right or wrong answers, these are the problems we chose with strategies we thought went along with each. The expectation was not to have the students solving it the way we had listed, but to hear, and have other students hear the choices being made.

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We gave partners the cards, they sorted, and named the categories whatever they wanted:

While the card sort conversations were really interesting, the class discussion afterwards was amazing! There were so much questioning of one another about how one strategy is any different than another. For example, some groups used rounding for a problem that another group used compensation and another grouped called it using friendly numbers…so groups had the same problem in three differently-named categories. Again, the category was not important, but more the fact they were actually thinking about the numbers they were given.

The other 5th grade teacher and I are planning to do the same activity with multiplication when they get there! Excited!

Kindergarten: Numberless Problems

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Last week, the kindergarten students solved a problem about Jack and his building blocks. It went something like this:

Jack was building with blocks. He used 4 blocks to build a wall and 2 blocks to build a bridge. How many blocks did Jack use altogether?

The teachers posed the problem without the question, did a notice/wonder, and then gave the students time to answer how many blocks altogether. We looked at this student work in our planning for their upcoming lesson.

We were curious to see what the students would do without the numbers in the problem, so we planned for a numberless story problem during last week’s Learning Lab. Three kindergarten teachers and I had a chance to be in the same room to see it in action today.

Nicole, the teacher, posed the following story to her students:

Susie is building with blocks. She used some blocks to build a wall. She used some blocks to build a bridge.

She asked the students what they noticed/wondered and the very first notice was there were no numbers to tell us how many blocks, awesome. They did some wondering about how many blocks she used and compared this story to Jack’s building from last week.

We planned to have the students choose the number of blocks they wanted Susie to use in her building and then find how many she used altogether. Their number choices were so interesting and left me wondering when students begin to explain the usefulness of 10? I know some of them know 10 is a great number to add after the activity today, but I am wondering the questions to ask to make it clear to them because they just “know it.”

Here were some examples of their work…so much cooler than 4+2 in Jack’s problem!

Rhombus vs Diamond

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Every year in 5th grade, when we begin classifying quadrilaterals, students will continually call a rhombus a diamond. It never fails. While doing a Which One Doesn’t Belong in 3rd grade yesterday, the same thing happened, so Christopher’s tweet came at the most perfect time! (On Desmos here: https://t.co/rZQhu2SGnR)

Which one doesn't belong? (Diamond edition)

cc @MathMinds pic.twitter.com/VmMh8rakXg

— Christopher Danielson (@Trianglemancsd) February 9, 2016

Of course I had to pop into the same classroom today and try it out! The lower right was so obviously a diamond to me that I was curious to see if students saw the same thing and if it changed their reasoning about the rhombus as a diamond.

Here are pictures of the SMARTboard after our talk:

After great discussions around number of sides, rotations, decomposition and orientation, they finally got to the naming piece. Honestly, I was surprised names didn’t come up as one of the first things. It started with a student saying the square didn’t belong because it is the only one that doesn’t look like a diamond. The next student said the lower left was the only one “that didn’t have a name.” When I asked him to explain further, he named the square, rhombus, and diamond. Because I knew at the end of our talk I wanted to ask about the diamond vs rhombus, I wrote the names on the shapes. Another classmate added on and said the lower left “may not have a name but it is kite-shaped and looks like it got stuck in a tree sideways.” I asked the class what they thought about the names we had on the board and it was a unanimous agreement on all of them. Funny how quickly they abandoned their idea from yesterday, so I reminded them….they were not getting off the hook that easy;)

“Yesterday you were calling this rhombus a diamond, what changed your mind?”

Students explained that the lower right actually looks like a real diamond and the rhombus doesn’t now that they see them together.

“Can we call both of them a diamond?” I asked. I saw a few thinking that may be a great idea. I had them turn and talk to a neighbor while I listened to them.

We came back and they seemed to agree we couldn’t call them both a diamond because of the number of sides. They were really confident in making the rule that the quadrilateral one had to be a rhombus and the pentagon was the diamond. I pointed to the kite and asked about that one, since it has four sides. “Could we call this a rhombus?” They said no because the sides weren’t equal, so not a rhombus. And because it didn’t have five sides, not a diamond either.

Thank you Christopher! All of these years of trying to settle that rhombus vs diamond debate settled right here with great conversation all around!

Next up, this one from Christopher…

@jacehan @MathMinds Here's v 3.0. pic.twitter.com/3klfsv2fGJ

— Christopher Danielson (@Trianglemancsd) February 10, 2016

Counting Collections Extension

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Today in 1st grade, we did a counting collections activity. One thing I am thinking about as I see this activity happening in K-2 classrooms, is the extension. While choosing questions for students who are struggling is difficult, choosing questions for those who are finished quickly, and correctly, I find just as difficult. What questions can we ask those students who organize, count and can explain their count perfectly?

I have been toying around with this idea for a bit. I have thought about asking them to combine collections, ask how many more they would need to get to another number or mentally adding tens and hundreds to their count.

Today, I tried asking a student how many more to get to another number. It was pretty cool and led to some more ideas. He and his partner ended with 292. I asked how many more to get to 300? 8. 350? 58. He could explain using the 8 to get to 300 and how to move forward from there. Because of other groups I wanted to chat with, I left him with 500, 652, 1,000, and 1,250. He came back with the answers, but no explanation and said, “I don’t feel like writing all of that out.” I asked him to explain how he got to 500 and I would record the equation for it. He said he added 8 to get to 300 and then 200 more to get to 500, so 208. He was shocked to see it as an equation because he thought I meant to explain it all out in words. I asked him to try the next one and he started with adding 10. He said he wanted to keep the 2 ones to make it easier, awesome. When he said that, I had another idea to have students think about what place values are changing as they add to a certain number. I want to ask him why he ended with 8 ones on three of them but zero ones in another?

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I know I have seen tons of people on Twitter using counting collections and would love to hear of other ways we could extend this activity in the comments!

Obsessed With Counting Collections

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If you have seen my recent Twitter feed and blog posts, you can probably tell I am currently obsessed with Counting Collections! Because of this obsession, during our recent K-2 Learning Lab I made it the focus of our conversation. This was our first chance to talk across grade levels during a Lab and to hear the variation in ways we could incorporate counting in each was so interesting! Based on this lab, yesterday, I had the chance to participate in both a 2nd grade and Kindergarten counting collection activity and while there were so many similarities, I left each thinking about two very different ideas!

2nd Grade: Naming A Leftover

Based on our Learning Lab discussions three 2nd grade teachers had the amazing idea to combine their classes for a counting activity. While it was a great way to give students the opportunity to work with students from other classrooms, it also offered the teachers a chance to observe and talk to one another about what they were seeing while the activity was in progress. I was so excited when they sent me their idea and invitation to join in on the fun! I have never seen so much math in an elementary gymnasium before!

There was a lot of the anticipated counting by 2’s, 5’s, 10’s and a bit of sorting:

And while this is so interesting to see students begin to combine their groups to make it easier to count in the end, there were three groups counting base 10 rods that particularly caught my attention:

1st Group (who I missed taking a picture of): Counted each rod as 1 and put them in groups of 10.

2nd Group: Counted each rod as 10 because of the 10 cube markings, making the small cube equal to 1. They had a nice mix of 20’s in their containers!

3rd Group: Counted each rod as 10 but had a mix of rods, small cubes and some larger blocks. It was so neat to see them adjust the way they counted based on size…the rod=10, small cube=1 and the large block=5 (because they said it looked like it would be half a rod if they broke it up). After this beginning picture, they arranged the 10 rods to make groups of 100.

The second and third group ended up with a final count and recorded their thinking, however group 1 could not wrap it up so neatly. When they finished counting they had 141 rods but one small cube left over. Since they were counting each rod as 1, instead of 10, they were left trying to figure out how to name that leftover part. When I asked the group what they were thinking, one boy said, “It is kind of like half but smaller.” I asked him how many he would need for half of the rod and he examined the rod and said 5. I have to admit, I wasn’t sure where to go with this knowing their exposure to fractions is limited to half and fourths at this point. So, I asked him, “How much we do have of one rod” and he said 1. I followed with, “Of how many?” and he answered, “Ten. So we have 141 and 1 out of 10?” Thankfully it was approaching time to clean up so I could think more about this one. I feel like I left that idea hanging out there and would love to bring it back to the whole class to think about, but I am still wondering, what question would have been good there? How would you structure this share out so this idea of how we name 1 is important and impacts our count? How do we name this leftover piece and why didn’t a group counting the same thing not have that problem? Also, I think it will important for these students to think about the question they could ask that their count would answer…For example, how many objects do you have – would that be accurate for the group who counted each rod as 10?

Kindergarten: Why Ten Frames?

Every time I am in Kindergarten I leave with so many things to think about! In this case I left the activity thinking about Ten Frames. I am a huge fan of ten frames, so this is not about do we use them or do we not, but more about….Why do we use them? How do we use them? What is their purpose? What understandings come from their use? What misunderstandings or misconceptions can be derived from their use? and Where do these misunderstanding rear their ugly head later?

To start the lesson, groups of students were given a set to count. With a table of tools available to help them organize their count, ten frames were by far the most popular choice. However, not having enough (purposefully) for everyone’s set pushed them to think of other means… which ended up looking like they were on a ten frame as well!

As the teachers and I went around and chatted with groups, we heard and saw students successfully counting by 10’s (on the frames or look-alike frames) and then ones. This is what we hope happens as students work with the ten frames, right? They see that group of 10 made up of 10 ones and then can unitize that to 1 group. It reminds me of Cathy Fosnot’s comment via Marilyn Burns on Joe’s post, which I had huge reflection on after this lesson too!

I was feeling great about the use of ten frames until a first grade teacher and I were listening to one group count their set. I wish I snagged a pic, but I was so stuck trying to figure out what to ask the girls, that I didn’t even think about it. They had arranged 4o counters on 4 ten frames and had one left over, sitting on the table, no ten frame. We asked how she counted and she said, 10, 20, 30, 40, 50…the 1 leftover was counted as a 10. I immediately thought of Joe’s post. Not knowing exactly what to do next, I tried out some things…

I picked up the one and asked her how many this was, “One” and then pointed and asked how much was on the ten frame, “Ten.” Ok, so can you count for me one more time? Same response.
I filled an extra ten frame pushed it next to her 4 other full ones and asked her to count: 10, 20, 30, 40, 50. I removed 9, saying “I am going to take some off now,” leaving the one on the ten frame and asked her to count again. Same response.
I asked her to count by 1’s and she arrived at 41. So I asked if it could be 41 and 50 at the same time. She was thinking about it for a minute but stuck with “that is what I got when I counted.”
Then I became curious if she had a reason for using the ten frame, I asked. She said it was to put her things on so I began wondering about the usefulness of the 10 frames for her. Was is something, as an object, that represents 10 to her but not able to think about the 10 things that make it up?

I left that class thinking about how complex unitizing is. We hope students are able to count 10 things, know those 10 things are still there even when we start calling a unit, 1 ten, and then combine those units but still know there are 10 in each one of them. WOW, that is a lot! However, they can easily appear successful in counting by 10’s, which is one of the many reasons Counting Collections are so powerful. They bring to light the misunderstandings or missing pieces in students’ thinking.

I then start to think of recent conversations I have had with 4th grade teachers about students who are struggling with multiplying a number by multiples of ten and wonder if this is where we can “catch” those misunderstandings and confusions before they compound?

What to do next with this class? Erin, the teacher, and I quickly discussed this as she was busy transitioning between classes. We were thinking about displaying an amount, lets say 23, with two full ten frames with 3 extra. Say to the class, “Here are two sets, do they look the same? How can you tell? Two groups counted this amount two different ways. One group counted it 10, 20, 21, 22, 23 and the other group counted it 10, 20, 30, 40, 50. Can it be both? If so, how? If not, which one is it?”

Would love any other thoughts. I am heading back to re-read all of the comments on Joe’s post to gain more insight, but I would love your thoughts too!