# Adding & Subtracting: Tools and Representations

There is always a lot of talk about students using an algorithm, process or set of rules, for addition and subtraction. Whether talking about “any algorithm” or “the algorithm,” I am certain, in most cases, people are talking about a process that is absent of tools such as a 100 charts, number lines or base ten blocks. But, what happens when we see the tool becoming an algorithm in and of itself? Can moving left or right on a number line, making jumps of 10s and 1s, writing problems vertically, or jumping rows and columns become an algorithm where students lose sight of the numbers themselves because the process is one more thing to learn?

This was the exact conversation I had yesterday with two 3rd grade teachers as I was leaving school. The students had been playing a game called Capture 5 and struggled making various jumps on the 100 chart. The teacher, understandably, was concerned students were confused about adding and subtracting tens and ones. After more conversation, we began to wonder if the students saw the 100 chart as a set of rules to follow in order to add and subtract instead of a place to look for patterns and structure as we add and subtract. Were they getting caught up in the left, right, up, down movement and losing sight of what was actually happening to the number?

As I thought more about it last night, I wondered about other tools and representations  students learn that could easily turn themselves into an unhelpful set of procedures. I also wondered how often we make connections between these tools and representations explicit. Like, when is one helpful over another? How are they the same? How are they different?

I emailed the teacher my notes (below) and we decided we would try this out this morning.

If you can’t interpret from my notes, the plan was to have each student in a group using a different tool or representation as I called out a series of operations to carry out. After the series of addition and subtraction, they compared their answers and discussed any differences. They rotated seats after each series so they had a chance to try out each of the roles.

We came back together to discuss their favorite one. The recording is below…what do I have against writing horizontally, really?? I found this entire conversation SO incredibly interesting!

• They found the base ten blocks to be “low stress” because they were easy to count, move and trade, but did agree that bigger numbers would be really hard with them because there would be too many.
• They really did not like adding and subtracting on the number line with multiple jumps. It got messy.
• They liked mental math because there was nothing (tool) to distract them and they could focus but they didn’t like that you couldn’t check your answer.
• The 100 chart’s only perk was they didn’t have to write the numbers in, they were already there for them.
• I really loved that they mentioned the equations were they only way they could track their work. So if someone in their group messed up, the equation person was the only one that could help them retrace their steps easily.

I am not sure what I learned today, I am still thinking a lot about this. I know that I loved having them compare the tools and representations and that the teacher felt much better about their ability to add 10s and 1s. I feel like there are so many other cools things to do here, but my brain is fried today so that will have to wait!:)

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.

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!

# Decomposition of Number in Kindergarten

This post has been sitting in my drafts just waiting to be written for weeks now, thank goodness for a vacation to get all caught up!

This lesson came about during the same Kindergarten planning session as the Both Addends Unknown (BAU) lesson. As the team and I talked about the dot images they had recently been using during number talks and the decomposition of number standard, we were curious how students would do with a context in which a number is broken down into more than two addends. We knew it wasn’t exactly matching the standard, however we were interested in seeing how the ideas that emerged were similar or different from the BAU problem.

The first piece of our planning was developing a context so the students would have a visual of something moving from one place to another as the addends changed. For no better reason than the fact that Jodi, the classroom teacher, had counting bears, we decided upon polar bears as our context. We launched the problem with an image of 6 polar bears swimming at a zoo, all in the same pool. We asked the students what would happen if the zoo had six different pools for the bears to choose from? Could they all be in the same pool? Could they each be in a different pool? How many different ways could these 6 bears be arranged in the pools? The students did some talking about how they could swim together or by themselves.

I then showed them the muffin tin below and asked if this could be the pools for us to work with today since we didn’t have the actual bears or zoo with us. They counted and agreed it could be the pools since there were 6 spaces, but we had to also agree that it was “not big enough for the real bears.”

Jodi and I knew recording freely in a journal would have been a bit tricky without something to match the tin, so I printed the image below and we put a stack of them on each table for students to use. We did have a conversation on the carpet about recording, because our goal was for students to have multiple ways to decompose the group of 6 and we didn’t want time wasted drawing bears. I asked them how we could show the bears if we didn’t draw each one and dots and circles were the most agreed upon way.

I felt like this whole introduction took way too long. I don’t know how to make it quicker, but I would have loved to have had more time at the end of the lesson connecting the representations than in the launch. Perhaps just giving them 6 bears and asking how they could be in the tin, recording it on the board, asking them to change it, recording those, and then comparing?

From there, we set them off with a partner, 6 bears and a muffin tin. I was so impressed by the way they worked together. In so many groups, one student moved the bears in the tin while the other recorded and then they switched. As they got the hang of moving the bears around, a lot of them began to look like they were on a race, cranking out a ton of different recordings. We did not have to give them more than 10-15 minutes before they had at least six or more ways. We stopped them from working, asked them to put their papers out on the table in front of them, and talk to their partner about ones that seemed the same and ones that seemed different.

As they spread them out on their tables and chatted, I saw and heard SO much possibility but not enough time. So many patterns, so many interesting ways of composing and decomposing groups, and so much commutativity.  However, they were leaving for recess soon and we wanted to wrap it up with a whole class notice/wonder before they left.

I strategically chose sets like the ones in the pictures above and asked students what they noticed/wondered. This is the point where you could see a bit of the hour-long math class fatigue setting in. A lot of noticing of 6 total bears and patterns such as 2,1,2,1 and 2,2,1,1, however we did not hear any talk of how the bears regrouped. For example in the 1,1,1,1,1,1 and the 2,2,2, I was wondering if students may say the two ones each made a two or any type of movement like that. I wonder if I asked how they were the same or different if I would have gotten a different response? Not sure.

Jodi and I chatted after the class and agreed we wanted to revisit this lesson. We wanted to revisit because we did not get to writing equations for each picture, as we had planned. We were curious to see what they would do with that and if any other similarities and/or differences would arise. We also thought this could be a great activity for a math center, but we are just not sure what angle to take with it yet. Could it be about arranging them three ways and then comparing? Could it be practice at writing equations for their model? Could it be eventually knowing the combinations without manipulating the bears? Could it possibly be a mix of all of this? I am not sure…I am learning everyday in Kindergarten!

# Both Addends Unknown in Kindergarten

A few weeks ago, I planned a lesson with one of our Kindergarten teachers, Linda. The beginning planning stages and readings behind this lesson are described in this previous post. Based on the work she had been doing with dot image number talks, she was seeing students combining addends to arrive at a sum and also writing equations to match their thinking. After reading the NCTM article, she was curious to see how students would think about the addends when they weren’t right in front of them, as they were with the dots. Also, since we have been reading Connecting Arithmetic to Algebra recently, a lot of our work has been thinking about how students make conjectures and prove claims. This would also be before their Investigations activity called Toss the Chips so I was really interested to see if the movement of bunnies gave a different visual for students than flipping the chips over to different colors?

We posed the image of the bunny cage and 6 bunnies (in my previous post) and did a quick noticing. After noticing the bunny ears and explaining how we need to take really good care of them, the class noticed some really important things around the math: there were 6 bunnies (we counted to be sure), the cage had an inside and outside part, and there was a door for them to go inside and outside.

We explained that with their partner, they were going to see how many different ways these bunnies could be in the cage. Then I asked,”Since we don’t have the bunnies here with us, what could we use instead to help us?” After a suggestion of building a bunny, a couple students suggested the connecting cubes they had on the shelf, perfect. Each pair took their cubes, paper, and pencil and went to work. We purposely didn’t tell them or model how to show or organize their work because we were curious to see how they would do it on their own.

Things I noticed as I circulated:

• There were so many amazing ways students organized their information!
• A lot of partners started with 3 and 3.
• The commutative property was not showing itself at all, so possibly they saw 4 and 2 the same as 2 and 4?
• Many partners were moving the cubes as if they were the bunnies to start a really cool pattern but stopping when they got to 6 and 0.

Here are some pieces of work that I thought highlighted my noticings:

This one looked like they were going in a particular order but then jumped to 3 and 3. I loved the labeling on both the drawing and the list! So clear!

This group’s work is not quite as organized as the one above, but they definitely were showing a movement of a bunny. They believed they were finished at 6 and 0, as many did, which leads me to believe the commutative property feels like the same combination to them.

Again, had a pattern going and then jumped back to 3 and 3 after we asked them if they had all of the possible ways.

After walking around the room while they were working, we brought the students back to the carpet to talk about what they did. We started with 3 and 3 and I recorded it every single way I saw it being recorded. This is where I shifted from my initial goal of eliciting all of the combinations to the question,

How are these representations the same or different?

From here it was really nice to hear that “and” is the same as the adding because, “We had 3 and 3 more, so it is 6.” A lot of students easily connected the first and second examples above and we talked about how we could tell which bunnies were inside/outside. I didn’t ask which one was clearer to understand, for fear of making the students feel like their work wasn’t amazing, however the partners of the second example quickly said they could have written in and out over their numbers and it would have been the same.  We ended by listing all of the combinations and the commutative property did come out from one group so we ended with them thinking about whether 6 and 0 was the same as 0 and 6.

Things I am left wondering…

•  Should we have worked with an even number? As I walked around and began looking for any use of the commutative property, I began to wonder how I would have answered questions about 3 and 3. Technically it is the same exact equation, however in context, it would have been 3 different bunnies, so it is not. Would I have written 3+3 twice? I think I would have if the argument came up, but is that something to support in future work?
• From here, the students move to Toss The Chips. Do the red and yellow chips reinforce this work or without the context is it different? I know mathematically it can elicit the same discussions, but does the movement of the bunny (as something that moves itself) better support the conjectures of adjusting the addends? Does it not make a difference?
• I am so curious to see if the sharing of their organization structures transfer to their Toss the Chips activity. I would like to see them play the game without the table at first to see what they do with it!
• How does this thinking change with you play around with more than two addends? (I was so curious about this one that I planned an activity with a K teacher on just this question…that is my next post).

Yesterday, I had the chance to teach the 1st grade lesson I planned here. It was so much fun and SUCH a learning experience for me! After all of the conversation in the comments and on Twitter, I decided to start with the open, one sentence Notice/Wonder. Only having 45 minutes and this being the students first time doing a N/W, I decided not to begin with a number talk/routine (which I usually always do).

The students started on the carpet, I put up the sentence, read it and asked, “What do you notice and wonder about this sentence?” Just then a student exclaims that he just noticed that “Notice” was not, “Not Ice.” At that moment, I began thinking maybe my question had them looking at the physical pieces of the sentence/words so I quickly rephrased, “I would love to hear what you notice and wonder about what is happening in the sentence.” They used their Number Talk signals, thumbs up when they had a notice or wonder and then used their fingers to indicate more than one. I was so impressed by all of their thoughts, but I did realize that is it hard to end their wonderings! The amazing thing was how all of their wonderings really could turn this sentence into a story in their ELA class because they were all really important details they could add to it. Here was how the board ended.

I asked which wonder we could work on together today in class and there was a unanimous vote for “How many kids are on the bus?” however there were a few that suggested, “How many student can the bus hold?” because “math is counting things and we could count the seats.” I starred the  wonder “How many kids are on the bus?” and told them next time they get on their bus I would love to hear how many seats and students they found are there. We discussed whether they know how many students were on the bus by reading our sentence and they said no, they only know that there were 3 stops.  I asked, what they would want to know and they wanted to know how many kids were at the stops. I wrote that at the top.

When I told them they would get to choose how many students were at each stop, they were so excited! I gave them a paper with the sentence at the top, let them choose a partner and sent them on their way.

As I walked around and asked students why they chose the numbers they did, I quickly wondered how much I should have helped organize their work for them. I found so many with numbers everywhere and it was hard to see where their bus stop numbers were, let alone their total. Should I have put Bus Stop 1____, Bus Stop 2____, and Bus Stop 3_______ to have a clearer picture while also modeling for students how we can make our math work more clear? Quite a few looked like this…

There were so many interesting papers, so I love WordPress’ new tiling feature for pictures to make it look less cumbersome!

Top Row, Pic 1: This student had 24 as two stops. When I asked him how many stops we noticed in the beginning, I got a “Oh gosh” and he wrote Bus stop 1, 2, and3. He then stuck with the 24 and when I came back he had 8, 8, and 8. I didn’t see this until after class so I am curious how he arrived at that answer. I also realized that these 1st graders move fast and it is SO easy to miss the cool things they do so quickly!!

Top Row, Pic 2:  They said 3 and 22 were easy to add and then they just chose another small number. The interesting thing here that I need to find out more about is the 5×6 with the one box shaded in. I loved the commutativity showing up here!

Top Row, Pic 3: This was so interesting because I had never thought that a student would first think of how many students were on the bus and divide it up from there. They thought 30 students would fit on a bus so they made the stops fit that information. (They saw the error on the last one during the share).

Bottom Row, Pic 1: This student said that because there were 3 stops, there were 3 students at each one and ended with 9.

Bottom Row, Pic 2 &3: This student wanted big numbers so his first response, after he insisted on re-writing the sentence, was 1,000 and 1,000,000 and 4. Then on the back of his paper he wanted 6 stops and chose 6 new numbers. This led to some great conversation during the share.

This student figured that if there were 1 or 2 at a seat then there would be 55 students on the bus. I love all of this work so much! Then when I asked her about the students at each stop, she said, 30, 20 and 5.

We shared as a group back on the carpet and I tried to capture why they chose the numbers they did:

I then gave them the original problem and asked them to solve it individually. After seeing them work on this problem, I think there are so many interesting conversations that could happen Monday morning!

This is where I had so many questions as to how we get the younger students to make their thinking more visible. I found so much of it happens on fingers, 100s chart, and number line on their desk that I was getting an equation on the paper.  It is great when I am sitting there asking, but that cannot obviously happen when they are done so quickly and there a bunch of them! Is this something that comes with practice? I did find that once I asked them if they could explain to me on the paper how they solved it, they did a great job. My next question is, would taking the 100s charts and number lines off the desks help push students to look for friendlier numbers? I found the majority of them went left to right, counting on instead of using the 6 and 4 first. This is something that I think a structured share out on Monday could bring to light for those who never thought of it.

Here are a sampling of the papers I look forward to chatting with the teacher, Lisa, about on Monday. We can chat about how we can structure this share out.

Lisa, through number talks and investigations, has been working a lot on having students noticing number patterns leading to generalizations. It was neat to see this work of adjusting addends and keeping the same sum showing up here too. It seemed to show  up most after they had their answer and were playing around with the numbers, which I love!!

I am happy to have started with the open notice/wonder because I learned so much about how they think about problems and I think the opportunity to choose their own numbers got them thinking about the context over solving for an answer to an addition problem. I am, however, extremely curious how it would have changed the work if I had given students the problem with the 13 given and the other two missing? Would I have seen more about how they choose numbers to make the 13 easier to add a third number? I am hoping to get into another 1st grade classroom to try this out with another teacher but I would love it if any other 1st grade teachers would there would love to try it out and report back!!

I am so looking forward to Jamie’s post on this because her student work looked amazing on Twitter yesterday!!

~Kristin

Yeah, Jamie’s post is up! Check it out here! Cannot wait for our Google Hang Out tomorrow to chat all about it!

Tomorrow I go into a 1st grade classroom to teach a lesson on addition and subtraction story problems. This Investigations lesson for the day centers on students solving these 6 problems, however I am looking to change it up a bit.

While reading my CGI book, Children’s Mathematics, to learn more about the trajectory in which students solve these types of problems, I found this diagram really helpful and interesting….

I went into this planning thinking I was going to be looking for how students combined numbers in the context of the diagram above. From there, I was planning to have students do a structured share of their strategies, comparing and contrasting along the way. However, as I got ideas from Jamie (@JamieDunc3) on Twitter, I started to think how much more I would learn about their thinking in talking about their noticings, wonderings, and number choices. My goal has now changed to looking at not only their strategies for combining but how they choose numbers in which they will have to combine.

So…I took the second question, removed the actual question and made it a notice/wonder:

Assuming the wonder of how many students were on the bus arose, I would see how students combined the numbers. Would they look for friendly combinations? Would they count all? Model it? Count on? or any combination of those?

Then, I thought I could keep the 13 and leave the other two numbers blank to see what numbers they chose.

Did they pick a combination that was easy to add to 13, like 5 and 5? or would they keep adding onto the 13? how would they add with the 13, would they choose 7 to make 20 and then another 1 digit number? would they choose all 2-digit numbers to challenge themselves?

But then, I thought what could happen if I took all three numbers out?

For some reason, without the numbers it seems more “wordy” to me. I don’t know why that is? So THEN, I went to this last option….

I really love this one, although, I must admit, I feel a bit out of control of the course of the lesson in choosing this one over the others. But, I think that is what makes it such a beautiful choice. After taking noticings and wonderings, I am thinking of having the students work in pairs to create their own story and solution for one of the wonderings.

In creating their stories, I am concerned that students will choose numbers such as 0 at two stops and 1 at the third and I won’t be able to get a picture of how they combine numbers, however I will have a possible picture of their number comfort level. If they do this and finish quickly, I will be ready with the second choice above to see how they deal with now having the 13 in the problem.

In their journals I will ask them to tell me why they chose the numbers they did for the problem.

~Kristin

A couple of weeks ago, I blogged about my planning with a first grade teacher here.  After teaching the lesson, the students did an amazing job with the dot images we chose to use. Some students moved the dots to make the dice look the same on both sides of the equal sign while others solved both sides. On the last image they easily decomposed the 4 into the two 2’s to prove both sides were equal so that was something we were hoping to see transfer into the dot image activity.

We walked around, recorded the expressions we saw students writing, and asked students questions about their strategies for choosing cards. As I do with many lessons, in thinking about their strategies beforehand, I referred to the Learning Progressions to see how students progress through algebraic reasoning.  If they didn’t know the the addition expression from memory, like 3+3 or 5+5, this clip from the progressions best describes how I was seeing students arrive at the first expression written for each given sum. Because the commutative property was the way most students found the second expression for each sum the day before, this particular day we told the students they had to use different cards than their partner in thinking about writing their expression.

I especially loved this passage in the Progressions about counting on…I had never thought of counting on as seeing the first addend embedded in the total, although it makes complete sense now! I wonder how understanding that could impact the way in which I question students about their thinking when adding?

What we were looking for as we walked around in particular was how students were using either this Level 2 method above or, what the progressions would call it, Level 3:

It is hard to convey all of the conversations we heard, however here are some of the game boards I captured after the finished playing the game. (Some boards were 6,9,10,15 and others were 8,9,12,16)

These partners seemed to think individually about their expressions on the left and right sides of the board. The student on the left appears to use facts they know such as 7+3 to arrive at 4+3+3 (since there were no 7 cards). I love the use of the equal sign between the two columns!

The other two pairs appears to have done the same thing…

The two groups below, I remember talking to because I was so interested in how closely their sides were related. After the student on the left had written their expression, the student on the right either combined or decomposed numbers to write an equivalent expression. I would love to talk to both groups about the sum for 12 because I am curious if they are decomposing and making a “new” number based on what they are “taking from” another number.

After playing the game, we put the equations we saw for each of the sums on the board and asked students what they noticed. Some noticed relationships between the expressions for a given sum while others looked at expressions for various sums. For example, when looking at the expressions for 10 and 15, they noticed that each expression added 5. Then we discussed whether that 5 was always a 5 and students were really comfortable saying that it could be a 2 and 3 or a 4 and 1. They could have shared their noticings for quite a while so we asked them to go back to their journals and describe something they were noticings among any of the equations.

It was at this moment when I started to detach myself from the math for a quick second and began seeing how journaling really begins. I found I take it for granted that when I say write in your journal about something, that they understand how we explain our mathematical thinking. I know that writing at various grade levels differs based on so many things such as vocabulary, writing experience, and just how they write words in general. However, one thing I did not think so much about is how students view writing in math. I did not realize until I saw this student showing all of his compensation in numbers by connecting the numbers that were staying the same with lines and showing the number that was “one less” by writing -1 when going from an expression that totals 10 to a sum of 9. He explained it so beautifully but was having trouble communicating that on paper. When he finished talking a girl next to him, asked me, “Can we use words too?” <—- that is when I had an aha! Do students think about writing in math as only communicating numerically? Do we ever explicitly tell them it is ok to write about math in numbers, words, or we can use both numbers and words? I think I have always assumed they knew.

Then I came back later and the very same girl had written all of this wonderful thinking…

This student showed a wonderful connection to what was happening when he went from 6 to 9 and then from 10 to 15:

After they had finished journaling, the students moved to recess, however this student sat for another 20 minutes explaining to me all of the wonderful thoughts he had in his journal. The arrows were movement of numbers that were changing however being able to clearly communicate that in his writing was not something he was able to capture clearly. THIS is the power of writing in math I think…learning to take all of the amazing thoughts and communicate it clearly because the more he talked it out to me, the more arrows he drew, the more he elaborated on his thoughts.

Moving forward from here there is so much to think about for me….in addition to moving students thinking about addition and relating that to subtraction, how do I begin to think more about journaling in math, how does it really start?

For Dot Addition game I am wondering if we could allow some students the option to use subtraction? Make the range of card choices larger to allow for students to play around with that relationship. It is something that I thought about as I looked at the table in the Learning Progressions..

So much to think about each time I leave a classroom!

~Kristin