Cuisenaire Rods: Fun in 4th

The 4th grade has just started their fraction unit, so I was curious how that may impact their work with the Cuisenaire rods. I started just like I did in Kindergarten and 3rd grade, with a notice and wonder:

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There were two of the ideas that really struck me as things I want to have the students explore further later: First was, “2 of the staircases (the staggered rods in order of size) could make a square.” They had them arranged on their desks like the picture below…but I want them to answer:

  • Is that a square?
  • If not, could we make it a square?

Then I started to wonder, do we call it a square? Should we say square face? Then what about area…would we say, “What is the area of the rectangle?”? That feels wrong because they keep calling the white rod a cube (which it is). But then asking about volume is not 4th grade. BUT, the tiles we use for area in 3rd grade are also 3-dimensional. <–would love thoughts on any of that in the comments!

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One student noticed that the orange rod was the height of the staircase and I thought of area again since it was said right after the comment above. This idea would be really helpful for the students above when they are determining if their figure is a square.

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I loved that one group noticed that any of the rods could be a whole and another group wondered if orange was the whole. Great lead into what I was thinking I wanted them to explore!

I asked them to find values for the rods based on their relationships. Of course the very first group I call on had 2 as the whole, which blew a lot of students minds, so I want to revisit that a bit later and ask them to explain how that works.

All of the other groups had orange as either 1 or 10, so I asked them to find the other values if the orange was 5 and 100. They played with that for a bit and then I began to hear a lot of aha’s, so I set them off to find more and they could have gone on forever.

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I left them with the prompt, “Tell me about the patterns and relationships you notice.” and for those who looked like they were struggling to answer that question, I added, “If you are struggling with that, tell me how you could find the rest of the values if I gave you one of them and which one would you want?”

I loved how this student chose the orange, white and yellow as the easiest end, beginning, and half. I also like the red x 2 is purple, but we need to talk through that notation a bit.

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This was the most common response, seeing the numbers get smaller as the rod got shorter.

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This student’s noticing could be an interesting number choice question to pose: Why do you think groups chose numbers for orange that were doubles or halves of the other numbers we already had?

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This student disagreed with the student who gave the responses in the first column because he is determined the white is 1/10 because the orange is 1. Would be great to pair them up and have them come to an agreement.

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This student is seeing the white value adding to each value above it to get the next. I also love how she writes notes about how neat her handwriting is:)

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I would love to have them play around with this first pattern in this entry! What other relationships could they find after they explored this one?

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So much fun! Cannot wait to get into other grade levels to see if I can begin to find a progression of ideas with these rods!

 

My Beginnings With Cuisenaire Rods

I have never been more intrigued with using Cuisenaire rods in the classroom until I started reading Simon’s blog! I admit, I have read and watched his work from afar…not knowing really where to start with them and was afraid to just jump into another teacher’s classroom and say, “Hey let me try out something!” when I really didn’t know what that something may be. However, after Kassia reached out to Simon on Twitter asking how to get started with Cuisenaire rods and Simon wrote a great blog response, I was inspired to just jump right in!

I am a bit of an over-planner, so not having a really focused goal for a math lesson makes me a bit anxious. I am fairly certain I could anticipate what 4th and 5th graders would notice and wonder about the Cuisenaire rods because of my experience in that grade band, however I wanted to see what the younger students would do, so I ventured into a Kindergarten and 3rd grade classroom with a really loose plan.

Kindergarten (45 minutes)

Warm-up: Let’s notice and wonder!

  1. Dump out the bags of Cuisenaire rods in the middle of each table of 4 students.
  2. Tell them not to touch them for the first round.
  3. Ask what they notice and wonder and collect responses.

Things they noticed:

  • White ones looked like ice cubes.
  • Orange ones are rectangles.
  • End of blue one is a diamond (another student said rhombus)
  • Different colors (green, white, orange..)
  • They can build things (which is why we did no touching the first round:)
  • Orange is the longest.
  • They are different sizes.
  • We can sort them by colors.
  • We can sort them by size.

Things they wondered:

  • What do they feel like?
  • What can we make with them?

Activity 1: Let’s Sort!

  1. Tell the students to sort them by size or color. (they quickly realized it was the same thing)
  2. Discuss their sort/organization and check out how other tables sorted.

I was surprised to see not many sorted them into piles because that is normally how they sort things. I am wondering if the incremental size difference between each rods made them do more of a progression of size than sort into piles? Some groups worked together while others like making their own set with one of each color (and size) and keep making more of those!

Activity 2: Let’s Make an Orange!

Since a lot of students kept mentioning that the orange was the longest, I decided to see if they could build some trains (as Simon calls them) that made an orange.

My time was running out, but it left my mind reeling of where I wanted to go next! My inclination is to ask them if they could assign numbers to some of the rods or if they could build some trains the same length as the different colors? I would love to hear which piece is their favorite piece because a lot of them found the smallest cube really helpful when building the orange.

3rd Grade (60 minutes)

Warm-up: Let’s notice and wonder!

Things they noticed:

  • Groups were the same color and length.
  • Blue and white is the same length as the orange rod.
  • Kind of like adding.
  • White is 1 cm.
  • Go up by one white cube every time.
  • Odd + even numbers
  • 2 yellows + anything will be bigger than 0.
  • 1 white + 1 green = 1 magenta

Things they wondered:

  • Is red 1 inch?
  • How long are the rods altogether? (Prediction of 26 or 27 in wide)
  • Is orange 4 1/2 or 5 inches?
  • Why doesn’t it keep going to bigger than orange?

Activity 1: Let’s build some equivalents!

I found 3rd graders love to stand them up more than Kindergarteners:)

Activity 2: Let’s assign some values!

After they built a bunch, I asked them to assign a value to each color that made sense to them…this was by far my favorite part – probably because it was getting more into my comfort zone!

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Again, time was running out, but next steps I am thinking…

  • What patterns and relationships do you see in the table?
  • What columns have something in common? Which ones don’t have anything in common? Why?
  • What if I told you orange was 1? What are the others?
  • What if orange was 2? What happens then?

Thank you so much Simon for all of inspiration and Kassia for the push into the classroom with these! Reflecting, I was much more structured than Simon and Kassia, but I look forward to a bit more play with these as the year goes on! I look forward to so much more play with the Cuisenaire rods and continuing Cuisenaire Around Ahe World!

Number Talks Inspire Wonder

Often when I do a Number Talk, I have a journal prompt in mind that I may want the students to write about after the talk. I use these prompts more when I am doing a Number String around a specific idea or strategy, however today I had a different purpose in mind.

Today I was in a 4th grade class in which I was just posing one problem as a formative assessment to see the strategies students were most comfortable or confident using.

The problem was 14 x 25.

I purposefully chose 25 because I thought it was friendly number for them to do partial products as well as play around with some doubling and halving, if it arose. When collecting answers, I was excited to get a variety: 370,220, 350 and 300. The first student that shared did, what I would consider, the typical mistake when students first begin multiplying 2-digit by 2-digit. She multiplied 10 x 20 and 4 x 5 and added them together to get 220. Half of the class agreed with her, half did not. Next was a partial products in which the student asked me to write the 14 on top of the 25 so I anticipated the standard algorithm but he continued to say 4 x 25=100 and 10×25=250 and added them to get 350.

One student did double the 25 to 50 and halved the 14 to 7 and then skip counted by 50’s to arrive at 350, instead of the 300 she got the first time. I asked them what they thought that looked like in context and talked about baskets of apples. I would say some were getting it, others still confused, but that is ok for now. We moved on..

Here was the rest of the conversation:

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I felt there were a lot more wonderings out there than there was a need for them to write to a specific prompt, so I asked them to journal about things they were wondering about or wanted to try out some more.

I popped in and grabbed a few journals before the end of the day. Most were not finished their thoughts, but they have more time set aside on Wednesday to revisit them since they had to move into other things once I left.

What interesting beginnings to some conjecturing!

Number Talk Karaoke

It is always so fun when I have the chance to hang out with my #MTBoS friends in person! This summer Max was in town, so I not only got to have lunch with him but also meet his amazing wife and puppy!  Of course, during lunch, we chatted a lot about the math work we are doing with teachers and some of the routines we are finding really valuable in their classrooms. From these two topics of conversation, Number Talk Karaoke emerged.

We both agreed that while Number Talks are invaluable in a classroom, it can be challenging to teach teachers how to use them in the classrooms. As much as we could model Number Talks during PD and show videos of them in action, it is still not the same as a teacher experiencing it for themselves in their classroom with their students. There is so much to be said for practicing all of the components that are so important during the facilitation with your own students.

That conversation then turned into two questions:  What are these important components? and How do we support teachers in these areas?  We discussed the fact that there are many books on mathematical talk in the classroom to support the work of Number Talk implementation, however the recording of student explanations during a Number Talk is often left to chance. What an important thing to leave to chance when students often write mathematics based on what they see modeled. We brainstormed ways teachers could practice this recording piece together, in a professional development setting, where students were not available.

Enter Number Talk Karaoke.

During Number Talk Karaoke, the facilitator:

  • Plays an audio recording of students during a Number Talk.
  • Asks teachers to record students’ reasoning based solely on what they hear students saying.
  • Pair up teachers to compare their recordings.
  • Ask teacher to discuss important choices they made in their recording during the Number Talk.

Max and I decided to get a recording and try it out for ourselves. So, the next week, I found two of the 3rd grade teachers in my building who were willing to give it a go!

They wanted to try out the recording piece themselves, so they asked me to facilitate the Number Talk. They sat in the back of the room, with their backs to the students and SMARTBoard so they could not see what was happening. All they had in front of them was a paper with the string of problems on it.

Before seeing our recording sheets below, try it out for yourself. In this audio clip of the Number Talk, you will hear two students explain how they solved the first problem, 35+35. The first student explains how he got 70 and the second student explains how he got 80.

Think about:

  • What do you think was really important in your recording?
  • What choices did you have to make?
  • What question(s) would you ask the second student based on what you heard?

The talk went on with three more problems that led to many more recording decisions than the ones made in just those two students, but I imagine you get the point. I have to say, when I was facilitating, I tried to be really clear in my questioning knowing that two others were trying to capture what was being said. That makes me wonder how this activity could be branched out into questioning as well!

Here was my recording on the SMARTBoard with the students:

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Here are the recordings from the two teachers in the back of the room:

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We sat and chatted about the choices we made, what to record and how to record certain things. We also began to wonder how much our school/district-based Number Talk PD impacted the way we record in similar ways.

Doesn’t this seem like a lot of fun?!? It can be done in person like mine was, or take the audio and try it with a room of teachers, like Max did! <– I am waiting on his blog for this:) Keep us posted, we would love to hear what people do with this!

 

Writing in Math: After a Number String

Many people ask me when and how I use journals in math class. At those moments,  I always seem to have so many reasons that it is hard to pinpoint just one to focus on during the conversation. And even when I seem to find a coherent way of explaining when and how I use journals, I often forget the reasons that seem to happen naturally in the classroom. The other day I had one of those moments that I think Joan Countryman, author of Writing to Learn Mathematics, would classify as continuing the conversation.

During Number Talks or Number Strings it always seems to happen…one student has a way of solving the problem that, as he or she gets midway through the explanation, the rest of the class begins to disengage either because it is a long explanation or they are lost in what is being said mathematically. Journals help me continue that conversation with the student who is sharing. I attempt to clearly capture what is being said, but ask the student to tell me more in their journal because I am so interested to hear all of their thinking.

This particular string was in a 3rd grade class who has been working with multiplication. I wanted to see how they thought about changing one of the factors in a particular way. This was the string:

3 x 4

4 x 4

6 x 4

12 x 4

The majority of students shared strategies that involved either skip counting or using repeated addition of one of the factors. Some used previous problems (which was my goal) to help them with the new one, however there was one student who started using 5’s for the last two problem instead of either of the factors. He had a very clear way of explaining it, but I could tell many students were beginning to get lost in the explanation. I encouraged the students to ask him some clarifying questions, but that conversation began to stretch this number talk a bit too long time-wise. Not to mention, many had stopped listening at this point.

I was so curious to hear more about his strategy because to be honest, I was getting a bit lost in his explanation of 12 x 4 using 5’s. I told him I wanted to hear all of his thinking but we needed to finish up the number talk to get started with class. I asked him to explain to me what he as doing with 12 x 4 in his journal and I would be sure to check it out later! He went right to work and knocked out this beautifully clear explanation, not just for 12 x 4 but EACH of the problems!

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The thing that I appreciated most was the opportunity if gave me to continue this conversation with him. I could feel he wasn’t done explaining his strategy during the talk and this also gave him the chance to think about how he could clearly communicate it to me in his writing. What a powerful thing for a student to be able to do! It was amazing to me he had done all of that decomposition, adjusting, and adding in his head!

So, if you asked me this week for a reason I have students write in math class, it is to continue the conversations that are not quite ready to end during our class time together.

 

Dot Image with Halves

Yesterday, I was thinking about some new number routines for the 4th and 5th graders who will soon be in their fraction units. I wanted to find something that both grade levels could engage in so the teachers could try them out and have a common talking point when we met to discuss what we learned about student thinking. I tweeted it out and didn’t want to lose all of the great thoughts so I going to compile them here!

This was the image I created:

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Before launching this with the students, I thought the teacher would establish that a full circle was 1 and ask the question we typically use with our dot images, which is

How many dots do you see and how do you see them?

I thought there could be a variety of responses, but I was most anticipating these two responses:

  • I know two halves make 1 and so the first row is 3. There are 4 rows so 4 x 3 =12.
  • I know two halves make 1, so the first column is 2. There are 6 columns so 6 x 2 =12.

I knew questioning would be important to move from those descriptions to what the equation would look like and imagined we would get to some like this:

  • 6 x 1/2 x 4 = 12
  • 4 x 1/2 x 6 = 12

Other interesting things I am now thinking about thanks to others on Twitter:

Why am I stuck on dots? Circles makes so much more sense!

Yes, why not leave it completely open,not tell them that circle is 1 and talk about the unit?

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It is funny that the same image conjures up different student responses, I always love that! I had not anticipate half of an array. I also love the idea of messing around with quarters here!

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Awesome to have them do some scaling of the dots to find how many are there? And then I can see them saying it is half of the array like Michael suggested once it is uncovered!

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I like the idea of moving them around however the cutting and tearing thing never goes well in my room!! I do love changing the value of the whole a lot!

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I debated this part a lot when I was making it! I thought for the 1st one, a whole number would be best and then move to non!

There is the running list so far! Add away in the comments!

 

 

Division: What’s Left Over?

Interpreting remainders is something the 4th grade teachers and I talk about a lot because so many students seem to struggle with it. Students can typically determine if they need to divide and find a way to get the answer, but if the remainder impacts the response it becomes more difficult. I believe the struggle is not so much about the remainder, but more about students making sense of problems. Many students love to compute the numbers in the problem and get an answer quickly, however they rarely revisit the problem to see if their answer makes sense. I found an even more interesting thing in their work today though that left me thinking about how their solution path impacted the way they dealt with the remainder.

I launched the lesson with a story. If you read my post on numberless word problems, this will be very familiar. I posed the following story to students and asked what they noticed and wondered:

Mrs. Gannon is having a picnic and inviting some people. She is going to the grocery store to buy bottles of water and packs of hot dog rolls. 

Since the students were on the carpet in front of the SMARTBoard and there was not much space to stand and write on the board without trampling a kid, I decided to sit off to the side and type their responses.

After they shared the things they noticed and wondered (in black font below). I told them I would give them some information that would answer some of the things they wondered. After typing in the numbers, I asked if what they noticed and wondered now (typed in red font below).

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(Side note: The cost of things is something I would love to weave into a lesson in the future because that came up a lot and will be great to see what they do with some decimals.)

Since they noticed how much water Mrs. Gannon needed, I wanted to see how they dealt with the hot dog rolls because the remainder would make a difference in the answer. I asked them to find how many packs of hotdog rolls she would need.

Some divided and got 4 r 4 as their answer (the skip counting on these pages came after their chatted with their group.

Some skip counted to get the packs of water and hot dogs:

Others used some multiplying up, some right, some interestingly not:

While I could probably talk for a while about all of the interesting things they did in solving the problem, the most interesting thing to me today was looking at who got the correct answer of 5 packs on their first try.

This is what I noticed as I walked around:

  • The students who went straight to dividing said their answer was 4 remainder 4, no reference to the context, no mention of using that remainder for anything.
  • The students who skip counted nailed it on the first try. They said as they counted they knew 32 rolls were not enough for everyone so they needed to keep going to 40 so everyone got one. They mentioned the context throughout their entire explanation.

I continued the conversation with Erin, the reading specialist, when I got back to the room. We started talking about how this contrast could play out in two different scenarios:

  • On a standardized test, given this same context and answer choices of 4 and 5, the students who could efficiently divide may choose 4, while the skip counter would have gotten it correct.
  • On the same test, give a naked division problem, no context, the efficient divider gets it correct but what about the skip counter. Can they think about the problem the same when their is no context or does skip counting make most sense with a context?

Because I thought it would be interesting, before I left, I asked them how many people she could invite so she had no leftovers at all. Fun stuff to end the class…

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And of course, some students are just funny….

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Multiplication: Does Order Really Matter?

Some things I am wondering right now about 3rd grade multiplication…

  • When students notice 4 x 3 is the same product as 3 x 4 and say, “The order doesn’t matter,” how do you answer that question?
  • Is there a convention for writing 4 groups of 3 as 4 x 3?
  • Is there a time, like when moving into division or fraction multiplication and division when the order does matter in solving or in thinking about the context?

Answers I have right now for these questions….

  • Right now, since they are just learning multiplication, I ask them what they think and why.
  • I think there is a bit of a convention in my mind because the picture changes. Three baskets with 2 apples in each is different than 2 baskets with 3 apples in each. Also, when reading the CCSS it seems that way.
  • I am still thinking about division but it makes me think that this would be the difference between partitive and quotative division. I also think when students begin 4th fraction multiplication, they are relating it to what they know about whole number operations, so 4 x 1/2 is 4 groups of 1/2. This seems important.

The 3rd grade teachers and I have been having a lot of conversation about these ideas. The students have been doing a lot of dot images and some feel strongly that the two expressions mean the same thing because they can regroup the dots to match both expressions. Others think they are different because the picture changes. All of this seems great, but then students are taking this reasoning to story problems. For example, given a problem such as, There are 5 shelves with 6 pumpkins on each shelf. How many pumpkins are on the shelves? students will represent that as 5×6 or 6×5. Is that a problem for me, not really if they have a way to get the 30, but should it be? I am not sure.

I went into a 3rd grade classroom to try some stuff out. I told them I was going to tell them two stories and wanted them to draw a picture to represent the story (not an art class picture, a math picture) along with a multiplication equation that matched.

1st story: On a grocery store wall there are 5 shelves. There are 6 pumpkins on each shelf.

2nd story: On another wall there are 6 shelves with 5 pumpkins on each shelf.

I asked them if the stories were the same and we, as I anticipated, got into the conversation about 5×6 vs 6×5 and what it meant in terms of the story. They talked about 5 groups of 6, related the switching of factors to addition and then some talked about 6 rows of 5.

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From this work, many interesting things emerged…

  • Some students had different answers for the two problems. They obviously did not see the two expressions as the same because they struggled with 5 groups of 6 as they tried to count by 6’s and forgot a row.
  • One student said they liked the second problem better because she could count by 5’s easier than by 6’s.

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  • Students skip counted by 5’s but added 6’s when finding the 5 groups of 6. IMG_3127.jpg
  • One student noticed the difference between 5 and 6 and could relate that removing one shelf was just adding a pumpkin to each of the other rows.

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  • One student showed how he used what he knew about one to switch the factors to make it easier to solve. IMG_3150.jpg

 

But they keep asking Which one is right? and I tell them I don’t have an answer for them. I just keep asking them:

Is the answer the same?

Is the picture the same when you hear the story? 

After chatting with Michael Pershan yesterday, I am still in a weird place with my thinking on this and I think he and I are in semi-agreement on a few things (correct me if I am wrong Michael) …Yes, I think “groups of” is important to the context of a story. I want students to know they can find the answer to these types of problems by multiplying. I want students to be able to abstract the expression and change the order of the factors if they know it will make it easier to solve BUT what I cannot come to a clear decision on is…

If we should encourage (or want) students to represent a problem in a way that matches the context AND if the answer is yes, then is that way: a groups of b is a x b?

What Is It About These Questions?

Today, I gave the 4th graders four questions to get a glimpse into how they think about multiplication and division before starting their multiplication and division unit. Michael Pershan had given the array question to his 4th graders last week and shared the work with me. As we chatted about next steps with his students, I became curious if the students think about multiplication differently depending on the type or setup of the problem.

Here were the questions:

After sorting 35 student responses I found the following:

  • 17 students got the area question wrong but the two multiplication problems on the back correct. Not only correct, but with great strategies based on place value.
  • 8 students got all of the problems correct, however the area was found in many ways, some not so efficient with lots of addition.
  • 10 got more than two of them incorrect. Some were small calculation errors on the back.

So, what makes almost half of the students not get the area?

Here is the perfect example of what I saw on the majority of those 17 papers:

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Then I did a Number String with them to hear how they shared their mental strategies. I wanted to get more insight into some of their thinking because a few students had used the algorithm on the back two problems.

They did great. They used the 10 and 20 to help them solve the problems and talked about adding and removing groups of one of the factors. I was surprised on the final problem of 7 x 18 that no one used the 7 x 20 but instead broke the 18 apart to find partial products.

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This makes me think there is something about that rectangle that makes them not use the 10s to help them decompose for partial products. I would love others thoughts and ideas!

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After reading the comments about area and perimeter, I wanted to throw another typical example of what I saw to see what others think of this (when I asked her she could easily explain partial products on the second and third problem)

 

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.

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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.
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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!:)