# Today’s Number: Making Connections

The Investigations curriculum and Jessica Shumway’s book, Number Sense Routines contain so many wonderful math routines. Routines designed to give students access to the mathematics and elicit many ways of thinking about the same problem. One of the more open routines, is Today’s Number. In Today’s Number, a number is posed to the class and the teacher can ask students for questions about that number, expressions that equal that number, or anything they know about that number. I love this routine, and while it is more commonly used in the primary grades, I used it often in my 5th grade classroom. While I would capture so much amazing student thinking, I always felt like all of that great thinking was left hanging out there. I could see some students were using what they knew about operations and properties to generate new expressions for the given number, however I wondered how many saw each expression as individual, unconnected ideas.

After I read Connecting Arithmetic to Algebra, I had a different ending to Today’s Number, an ending that pushed students to look explicitly at relationships between expressions. I tried it out the other day in a 3rd grade classroom.

I asked students for expressions that equaled Today’s Number, 48. I was getting a lot of addition, subtraction and multiplication expressions with two numbers, so I asked students if they could think of some that involved division or more than two numbers. I ran out of room so I moved to a new page and recorded their ideas.

After their thinking was recorded, I asked the students which expressions they saw a connection between. This is where my recording could improve tremendously, but I drew arrows between the two expressions as students explained the connection.

In case the mess is hard to see, these are some connecting ideas that arose:

Commutative Property: 3 x 16 and 16 x 3, 6 x 8 and 8 x 6, 12 x 4 and 4 x 12

Fraction and Fraction addition: 48/1 and 24/1 + 24/1 and 24 + 24

Subtracting from 100 and 1000: 100-52 and 1000-952

Multiplication and Repeated Addition: 4 x 12 and 12+12+12+12

Adjusting Addends in similar ways: 38+10 and 18 + 30, 40 + 8 and 48+0

Other ideas that I don’t particularly know how to categorize:

10 x 4 + 8 = 10 x 4 + 4 x 2

58 x 1 – 10 = 58 – 10

The second page got even more interesting:

“Groups of” and Decomposition: 7 x 4 + 2 + 18 = 14 + 14 + 2 + 10 + 8 . This student saw the two 14s as two groups of 7 and then the 18 decomposed into 10 + 8.

Halving and Halving the Dividend and Divisor: 192÷4 = 96÷2. This student actually used the 192 to get the expression with 96.

Another variation of the one above was 200 ÷ 4 – 2 = 100 ÷ 2 – 2.

Other cool connection:

96 ÷2 = (48 + 48) ÷ 2; This student saw the 96 in both expressions since they were both dividing by 2.

I think asking students to look for these connections pushes them to think about mathematical relationships so expressions don’t feel like such individual ideas. I can imagine the more this is done routinely with students, the more creative they get with their expressions and connections. I saw a difference in the ways students were using one expression to get another after just pushing them try to think of some with more than 2 numbers and some division.

# 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).

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!

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.

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.

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.

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!

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.

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.

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.

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.

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….

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:

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.

Ms. Thompson’s Class

Mrs. Leach’s Class

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!

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.

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.

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:

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.

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.

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.

• 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.

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.

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

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.

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.

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?
• 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:

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: Counting

# 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!

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.

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!

# Sorting Data in 2nd Grade

Today, I met the Yekttis.

While our intention today was to plan for the lesson after these crazy, fabled, Investigations characters, this activity quickly became the center of our conversation. It seemed the more we talked, the more tangled we got in our own thinking around the math itself, in addition to how to pose the activity to students and what questions to ask as they sorted. It felt like wording was a big deal here. How were we using the words: attribute, category, rule? Were they interchangeable? Would they make a difference in the way student thought about it? Do they make a difference in how we think about it? What is this mathematically and where is it going? While I was planning with three other teachers, only one of the teachers had taught this lesson before and she expressed how difficult it was for students once they were asked to sort based on two rules. We were ready to rethink the whole thing and kept asking ourselves if it was worth what the students would get out of it. But, because of all the questions and confusion in our own thinking, we were really intrigued to see how students would think about it.

Feeling a little like I jumped into the middle of a series of lessons, the teachers were great about filling me in on the students’ work prior to this activity. They had played a game called “Guess My Rule” which I was knew from 5th grade. In this activity, the teacher secretly chooses a rule, points out a few students who fit the rule, others who do not, and students try to guess the rule used to sort. They were really successful with this and enjoyed it.

Now, enter the Yekttis. They are a bunch of cards like the ones above. They have different shaped faces, eyes, and antennae. We decided to give them some time to play with the Yektti cards today and ask them how we could sort the Yekttis. I am hoping Tara, Lauren, and Kristin comment on here so they can go into depth about what the students did because I had to be 5th grade while they taught this lesson. When I caught up with Lauren toward the end of the day to recap, she noticed that the students, at first, looked at sorting as organizing the Yekttis in patterns rather than by attributes. They finally got to what attributes they could use, but when asked if they could sort based on a second rule, they were stumped. They could say “has this, but not this” type of sorts, but were seeing that as two rules because they were creating two groups…the haves and the have nots. As her and I talked, we realized how difficult it was to ask students to sort by two rules vs only one.

Since I left school, I have been thinking about this and have reread the lesson (I will post that at the bottom, after my questions). To me, it feels really difficult for students to sort by two rules and create a Venn diagram based on that sort. Choosing the categories is the stickies part because up until this point, they have experience only choosing categories that are mutually exclusive.

I find the really cool part of this whole thing is students realizing what categories will have an overlap versus those that will not. For this reason, I don’t want to walk students through this, but I feel there are some questions to ask in the process that could be pretty important. This is where I am struggling. What do I ask that does not put the answer right in from of them or become just another process of representing data. My thought for tomorrow is to play Guess My Rule with the Yektiis. Put a few Yektti cards inside and outside of the circle and ask students what the rule could be.  Once they guess the rule, I will label the circle and place the rest of the cards accordingly. Next, and this is the question I don’t know is the right one, I will ask “Is there another rule we can use to sort the Yekttis in our circle?” For example, I could choose “Has a Square Face” as my rule, we sort by placing all of the square faces in the circle and the others out. Now, let’s say the students say our second rule could be, “Has two antennae.” How do we proceed from here? Do I draw in the second circle that overlaps? Do I ask if the circles will overlap? Why does it then feel weird to then pull cards that were once outside of the circle back into the new circle?

After coming up this idea, I looked at the book to realize they handle it quite differently:

I don’t know how I feel about this and need to re-read it in the morning when I am not also thinking about a 3rd and 5th grade lesson for tomorrow! I feel it takes a bit of the “sorting power” out of the students hands? I would love any thoughts on this!

# 2nd Grade Learning Lab: Data

Last week in our Learning Lab, the second grade team and I planned for a lesson within the data work they are currently doing in Investigations. We spent a lot of time the previous week revisiting the Learning Progressions  and the focus by grade level document at Achieve the Core while also discussing the addition work, involving grouping, from their most recent math unit.

Since the students have been doing a lot of work constructing bar graphs, we wanted to move past the polling and construction piece that their unit spends a lot of time on, and make more connections to all of their recent number work.

We chose this image to be the focus of the lesson:

I found this graph on Brian Bushart’s awesome blog

We chose this image for a few reasons:

• The rain was in groups of 2 which we thought related really nicely to their most recent addition work.
• The half box was really interesting and we wanted to see how students dealt with it.
• The bars were horizontal as opposed to the vertical bars they have been using in their bar graphs.
• It lent itself to a variety of questions involving comparisons with larger numbers than their classroom graphs they have been doing.

Now, what to do with this image? As we talked about different questions we would want the students to be able to answer about the graph, I threw out the possibility of having students generate the questions after they do some noticing. It was such a fun teacher conversation as we looked at the graph through the eyes of a student and brainstormed questions that could be elicited from the graph. During our brainstorming, we paid careful attention to the type of problem the questions would elicit:

• Join problems involving combining numbers within one bar. This would be a nice connection to the adding by groups they have been working on in class. For example, how much rain did Waco get? Students could count by 2’s or count five boxes as 10.
• Join problems involving multiple bars. For example, how  much rain did all of the cities get altogether?
• Comparison problems involving two bars. For example, how much more rain did Austin get than San Antonio?
• Most and least questions. For example, who got the most rain?
• Combination of Join and Compare problems. For example, how much more rain did Georgetown and Waco get than Austin and San Antonio? (This may be a stretch;)

The day of the lesson, Lauren launched the lesson with just me in the room and the other teachers were scheduled to join us during the question-generating time. We thought that would be the most interesting section to see since we only can find coverage for @20 minutes for the teachers.

The students did great noticings in their groups and Lauren and I were feeling really confident that the students could use these noticings to generate questions to match them.

Blank stares.

We were a completely surprised because we though for sure they could work their way backwards from their noticings to create the question that it would answer. At this point we had the entire team of second grade teachers in the room and we began discussing how to clarify the directions. After one teacher prompted the students to think about “question words,” we decided to let them start working in their groups.

This is the point of the lesson where I realized a component I needed to add to our Learning Lab planning, teacher role during group work. This was our first time having everyone enter during the group work portion of the lesson and while there were great conversations around the room, it was hard to tell how much was students interacting with one another or with the teacher at the table. I think this came about because we could have done better in planning our directions for the students so, as a result, everyone was trying to clarify the directions at the table with the students.  In the end, Lauren’s students did finish with a lot of the same questions we anticipated and many questions they could solve the following day:

We had planned for students to choose one of their questions and show how they would arrive at their answer in their journal, but the question generating took a bit longer than expected!

Two things I am left wondering:

• In regards to Learning Lab planning, how would we have defined teacher interaction within the groups? Would we just be taking notes on what students were saying/doing? Would be asking students to clarify their thinking? Would we be answering questions they tried to ask us? Should we all be doing the same thing to be consistent in our debrief?
• In regards to the math, how do students work backwards to generate questions for a given image? Would rephrasing the directions help them think about it differently? If we asked them to create a quiz for the teachers based on the graph, would that have helped? How is wondering about an image different than generating questions for it?