Category Archives: 2nd grade

Number Talk: Which Numbers Are Helpful?

I think Number Talks are such a powerful routine in developing students’ fluency and flexibility with operations, but maybe not for the reason most think. One of the most highlighted purposes of a Number Talk is the ability to elicit multiple strategies for the same problem, however, an even more important goal for me during a Number Talk is for students to think about the numbers they are working with before they begin solving. And then, as they go through their solution path, think about what numbers are helpful in that process and why.

The struggle with trying to dig deeper into that thinking is simply, time. If the opportunity arises, I ask students about their number choices during the Talk but often students just end up re-explaining their entire strategy without really touching on number choices. Not to mention the other 20ish students start losing interest if they take too long. I do think it is a particularly tough question if students are not used to thinking about it and when the thinking happens so quickly in their head, they don’t realize why they made particular choices.

Last week in 2nd grade I did a Number Talk with two problems, one addition and one subtraction. During the addition talk, I noticed students using a lot of great decomposition to make friendly numbers (the term they use to describe 10’s and 100’s).

IMG_4374

During the subtraction problem, I saw the same use of friendly numbers, however in this one I actually got 100 as an answer. My assumption was because the student knew he was using 100 instead of 98, but got stuck there so went with 100 as the answer. I was really impressed to see so many strategies for this problem since subtraction is usually the operation teachers and I talk endlessly about in terms of where students struggle. I find myself blogging on and on about subtraction all of the time!

IMG_4373

When the Number Talk ended, I looked at the board and thought if my goal was to elicit a lot of strategies, then I was done – goal met. However, I chose the numbers in each problem for a particular reason  and wanted students to dig more into their number choices.

This is where I find math journals to be so amazing. They allow me to continue the conversation with students even after the Number Talk is finished.

I went back to the 100, circled it and told the class that I noticed this number came up a lot in both of our problems today. I asked them to think about why and then go back to their journal to write some other problems where 100 would be helpful.

Some used 100 as a number they were trying to get to, like in this example below. I really liked the number line and the equations that both show getting to the 100, but in two different ways.

IMG_4434

This student got to 100 in two different ways also. I thought this was such a clear explanation of how he decomposed the numbers to also use 10’s toward the end of their process as well.

IMG_4435

This student used the 100 in so many ways it was awesome! She got to 100, subtracted by 100 and adjusted the answer, and then added up to get to 100.

IMG_4428

While the majority of the students chose to subtract a number in the 90’s, this student did not which I find so incredibly interesting. I would love to talk to him more about his number choices!

IMG_4432

I didn’t give a clear direction on which operation I wanted them to use, so while most students chose subtraction because that was the problem we ended on, this one played around with both, with the same numbers. I would love to ask this student if 100 was helpful in the same or different way for the two problems.

IMG_4433

As I said earlier, this is a really tough thing for students to think about because it is looking deeper into their choices and in this case apply it to a new set of numbers. This group was definitely up for the challenge and while I love all of the work above, these two samples are so amazing in showing the perseverance of this group.

In this one, you can see the student started solving the problem and got stuck so she drew lines around it and went on to subtract 10’s until she ran out of time. I love this so much.

FullSizeRender 7

This student has so much interesting work. It looks as if he started with an addition problem involving 84, started adding, then changed it to subtraction and got stuck.

IMG_4436

This is what I call continuing the conversation. They wrote me notes to let me know Hey, I am not done here yet and I am trying super hard even though there are mistakes here. That is so powerful for our learners. So while there was no “right” answer to my prompt, I got a glimpse into what each student was thinking after the Number Talk which is often hard to do during the whole-group discussion.

If you want to check out how I use journals with other Number Routines, they are in the side panel of all of my videos on Teaching Channel. 

Asking Better Questions

I am sure we have all seen it at one time or another – those math questions that make us cringe, furrow our brow, or just plain confuse us because we can’t figure out what is even being asked. Sadly, these questions are in math programs more often than they should be and even though they may completely suck, they do give us, as educators, the opportunity to have conversations about ways we could adapt them to better learn what students truly know. These conversations happen all of the time on Twitter and I really appreciate talking through why the questions are so bad because it pushes me to have a more critical lens of the questions I ask students. Through all of these conversations, I try to lead my thinking with three questions:

  • What is the purpose of the question?
  • What does the question tell students about the math?
  • What would I learn about student thinking if they answered correctly? Incorrectly?

Andrew posted this question from a math program the other day on Twitter….

Screen Shot 2017-02-09 at 7.13.35 PM.png

I tried to answer my three questions…

  • What is the purpose of the question? I am not sure. Are they defining “name” as an expression? Are they defining “name” as the word? What is considered a correct answer here?
  • What does the question tell students about the math? Math is about trying to interpret what a question is asking and/or trick me because “name” could mean many things and depending on what it means, some of these answers look right. 
  • What would I learn about student thinking if they answered correctly? Incorrectly? Correctly? I am not sure I even know what that is because I don’t know what “name” means in this case. Is it a particular way the program has defined it?

On Twitter, this is the conversation that ensued, including this picture from, what I assume to be, the same math program:
Screen Shot 2017-02-09 at 6.56.47 PM.png

When a program gives problems like this, we not only miss out on learning what students know because they get lost trying to navigate the wording, but we also miss out on all of the great things we may not learn about their thinking. For example, even if they got the problem correct, what else might they know that we never heard?

The great thing is, when problems like this are in our math program, we don’t have to give them to students as is. We have control of the problems we put in front of students and can adapt them in ways that can be SO much better. These adaptations can open up what we learn about student thinking and change the way students view mathematics.

For example, if I want to know what students know about 12, I would just ask them. I would have them write in their journal for a few minutes individually so I had a picture of what each student knew and then would share as a class to give them the opportunity to ask one another questions.

After I saw those the problem posted on Twitter, I emailed the 2nd grade teachers in my building and asked them to give their students the following prompt:

Tell me everything you know about 12. 

20170208_092159 (1).jpg

Ms. Thompson’s Class

IMG_5487 (1).jpg

Mrs. Leach’s Class

IMG_5381.JPG

Mrs. Levin’s Class

Look at all of the things we miss out on when we give worksheets from math programs like the one Andrew posted. I do believe having a program helps with coherence, but also believe it is up to us to use good professional judgement when we give worksheets like that to students. While it doesn’t help us learn much about their thinking it also sends a sad message of what learning mathematics is.

I encourage and appreciate conversations around problems like the one Andrew posted. I think, wonder, and reflect a lot about these problems. To me, adapting them is fun…I mean who doesn’t want to make learning experiences better for students?

Looking for more like this? I did this similar lesson with a Kindergarten teacher a few years ago. Every time I learn so much and they are so excited to share what they know!

Subtraction in 2nd Grade

From my own experience teaching 5th grade and a lot of conversations with teachers in grades 1-4, subtraction always seems to be such an area of concern with students. After the introduction of subtraction as take-away, students tend to live in that land forever. This idea was spinning in my head the entire time I read Zak’s article in NCTM’s Teaching Children Mathematics. The article came at an ideal time as I had just planned and taught a subtraction lesson in two second grade classrooms.

As we were planning, the teachers and I discussed two reasons we think students struggle with relational thinking when dealing with subtraction:

  • The lack of variety in contexts we often give students to think about subtraction in different ways.
  • The lack of explicitness in connecting the ways students solve various subtraction contexts.

Based on the problems they were doing in their Investigations unit, we wanted to take the same numbers and create two different contexts for subtraction…one as a take-away context and one as a distance context and then have the students think about what made them subtract in one and what made them add in another. (After reading Zak’s article I want to try another structure where we keep the same context, but change the numbers to see if that impacts their solution process.)

First we did a notice/wonder on the following scenario:

Mike is saving money to buy a game. He already has some money saved.

After collecting the things they noticed and wondered (in black ink), I asked them what question they think we could ask and solve based on this story. They gave the question in red ink and then I asked them what information they would need to solve it and I put a red star next that information. I told them the game cost $22 and he had $8 saved and sent the on their way.

IMG_3637.jpg

Every single student added up, as we anticipated.

We shared some strategies and asked students how they were similar and different. We wanted to hear if students were using the number to solve or represent their thinking. We also wanted those students using the number to solve to see how they could also represent their thinking with equations.

Then we gave the students a second problem to solve: same numbers, different context:

Bobby had 22 toy cars in his book bag. He gave 8 of his toy cars to his friend Becky. How many cars did Bobby have left?  

They all took away the 8 except for one student. The variety of strategies was great and we were actually surprised to see a student solve it the same exact way he had used in the previous context.

The lone addition solution…

fullsizerender-63

We shared some strategies and asked them to compare the two problems. They noticed they were the same numbers and same answer but one student actually exclaimed, “How can they be the same answer when we added in one and subtracted in the other?!” What a great jumping off point for future lessons!

Time was up for this class period, but I made sure I stopped by to chat with the student who added in both because I was interested in his thinking. He said he knew that he had to subtract in that problem because it was taking away and he knew that you can add up to solve a subtraction problem. So, how does that happen? He has been in our school with all of these other students K-2, so how does he know this? What experience did he have, inside or outside of school, that allowed him to decide this was subtraction, know he can add up, and solve it? and then, Is there ever a context where he WOULD take away or has he learned this as to never have to take anything away?

After this lesson and Zak’s article, I am still reeling with my thoughts on subtraction. There has also been a lot of great conversation on Twitter about sequencing student work and I wonder how a set of student work like this could/would/should play in to students seeing subtraction more relationally? Lots of great stuff to think about!

 

Measuring Tools in 2nd Grade

Last week, the 2nd grade team and I planned for a measurement lesson. Their measurement unit falls at the end of the year, so this was actually the first lesson of their unit.

We focused on work on the first of these two standards, anticipating the other would be a natural part of the work as well:

Screen Shot 2016-05-03 at 10.37.50 AM.png

We put out the following measuring tools: square tiles, inch bricks (unlabeled ruler from Investigations), a ruler with inches and cm, and a tape measure in cm.

IMG_2305

The teacher launched the lesson by introducing the “Land of Inch,” a context that Investigations uses in the measurement unit. The introduction involved showing a picture of the 4 places in the Land of Inch: the castle, a cottage, apple orchard, and stable. The students discussed why they thought each one was in the Land of Inch.

Screen Shot 2016-05-03 at 11.22.16 AM.png

On a piece of paper, partners were asked to put the places of the Land where they thought they belonged and measure the distance from the castle to each, choosing whichever tool they thought was appropriate. The only stipulations were that there must be a path from the castle to each and each must be a different distance from the castle.

There were some really cool things that came up as we watched them working:

  • Every group took only the straightedge ruler and tape measure.
  • All of the straight lines were measured with the straightedge.
  • They all noticed the unit difference. We did not state what the unit of each tool was beforehand to see if they noticed.
  • They labeled 12 inches as 1 foot.
  • Students measured the curved paths using both the straightedge and tape measure.
  • Some students wanted to change centimeters to inches because it was the Land of Inch so they lined up the tape measure with the straightedge.

IMG_2307

  • One group recorded their measurements in ranges. They had no interest in starting at the end of the ruler. They just put the ruler down and wrote the two measurements it fell between.

IMG_2314

We wrapped up the lesson asking students to talk about why they chose their measuring tools. We had planned for them to share these ideas before they did a different journal prompt we designed last week. However, as they were sharing, there were one or two students doing a lot of talking (great stuff, but a lot) so we decided to have them reflect on their own before having this conversation.

IMG_2312.jpg

 

This student did a great job of explaining when they used one tool over another:

FullSizeRender 48

This student discussed why they chose to use the ruler but not the square inch tiles at all because it would take too long. So while both tools were the same unit, one tool has connected units versus individual units that need to be put together.

FullSizeRender 47

This group noticed that the centimeters (on the tape measure) would take them longer than the straightedge because there were more centimeters than there would be inches.

FullSizeRender 49

There were a couple instructional prompts we are revising for the next time this lesson gets taught by one of the other 2nd grade teachers that were there:

  • We didn’t let them know the paths didn’t have to be straight until after we saw them get started that way. Need to launch with that.
  • We didn’t have out meter or yard sticks, oops, need those next time. Talked about it during our planning, but we completely forgot.
  • We didn’t do a poster share which I think we want to incorporate next time because they all wanted to share. So maybe just two groups explaining their choices.
  • Wondering about the writing connection as they all had interesting reasoning behind where their places were located. Could they write a description about the placement and reasoning for their poster and then have other partners try to match them up?

Next up, reading Inch by Inch and the lesson inspired by the TCM article Inch by Inch in the most recent publication.

 

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!

Formative Assessment

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:

Screen Shot 2016-04-06 at 2.43.31 PM.png

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

Obsessed With Counting Collections

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!

IMG_1713.jpg

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!

IMG_1711.jpg

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.

IMG_1707.jpg

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!