# True or False Equations: Kindergarten

Yesterday, the Kindergarten math routine videos I recorded were posted. This set of routines were so incredibly fascinating to me and completely out of my comfort zone – such an incredible learning experience. In addition to the 6 videos that are posted, there are about 12 more in my Google Drive that I had to chose between, each with its own unique student responses that would be so interesting to discuss.

With all of the videos and associated student work, I am just finding some of the work I thought I forgot to collect after the routine. This particular set of work is from the True or False Equation routine.  This is probably my favorite routine in the set because it really pushes me to think about the language, recording, and understandings students have around the meaning of the equal sign.

The final equation, as anticipated, caused a bit of a controversy. Since the class was split on whether 2+3 equaled 1+4, I asked the students to explain their reasoning in their journal.

This response is reflective of the student’s experience with equations. How much do we record, or ask students to record, equations that only have three numbers? I would guess many students only see equations with two addends the majority of the time.

I liked the “same thing” and “I used my fingers” here. This is the language piece I find so interesting. What does it mean to be the same? In this case it could mean the same amount or looks the same. She could find the amount on her fingers or the two ways of showing the expressions on each hand would look the same in the end. A small, but important distinction I think.

This student appears to have related this to the first problem in the string, seeing both as the same since 5=5.

I really like this one because of the arrangement of dots. This student seems to think of exactly the same as the same amount since the dots look different in the way they are drawn. The dots are great because they look the way a student would easily subitize an arrangement.

This student broke the equation apart and set each side equal to 5 and showed with circles that each side was 5.

I was inspired by this number talk to dive even further into what it means to the be the same. I brainstormed some thoughts here and then tried an activity about what it means to be the same that ended in this work: (I haven’t gotten around to a blog on this one, but will soon!)

# The Equal Sign

True or False?

5 = 5

5 = 4 + 2

2 + 3 = 1 + 4

After reading so much about the meaning of the equal sign in books such as Thinking Mathematically and About Teaching Mathematics , I anticipated students may think each was false for different reasons…..

5 = 5: There is no operation on the left side.

5 = 4 + 2: The sum comes first or 4+2 is not 5.

2 + 3 = 1 + 4: there is an operation on both sides or because 2+3 =5 (and ignore the 4) or because 2+3+1 ≠ 4.

While I anticipated how students may respond, I was so intrigued by the number of students (probably about 75%) that said false for 5=5. They were about split on the second one, but for many reasons – not many of them being that 5 ≠ 6. The final one left many confused, in fact one student said, “Well now you are just trying to confuse people by putting two plus signs.” So cute.

As they explained their reasoning, my mind was reeling….

• What questions do I ask to get them to:
• Think about what the symbols mean?
• Talk about what is the same?
• Realize the equal sign in the first one is not a plus sign, so there is no answer of 10?
• See the equal sign to not mean “the answer is next”?
• What wording do I use for the equal sign?
• “The same as” felt wrong because the sides do not look the same in both cases….so, is “Is the same amount” a helpful way for them to think about it?

I got back to my room and starting thinking about what learning experiences would be helpful for students in building their understanding of the equal sign? I talked through it with some colleagues at school and reached out to those outside of school, I needed some serious help!

I started playing around with some cubes and realized how interestingly my thinking changed with each one. I didn’t take a pic of those cubes so I recreated them virtually to talk thru my thinking here.

The first set represents 5 = 5. I can see here where “the same as” works for the equal sign because there are 5 and they are all yellow. But what if I put 5 yellows on the left and 5 red on the right? Then they are the same quantity, but do not look the same.

The second set represents 2+3=5 and is definitely the one students are most comfortable seeing and representing as an equation. It looks and feels like composition to me so I can definitely see why student think the equal sign means “makes” or “the total is.” It looks like 2 and 3 more combine to make 5.

Something interesting happened with the green set. I made two sets of 5 and then broke one set to make the right side – felt like decomposition. I can see why it would feel differently to students. I also realized that when I look at them, I look left to right and much of that lends itself to the way I was thinking about what was happening.

The last set I made by taking my 2 sets of 5 connected cubes and breaking each set differently. Again, “the same as” doesn’t work for me here really well either because they don’t look the same.

Still thinking of next steps because I always like to put context into play with these types of things, but I am finding that very difficult without forcing the way students represent their thinking which I don’t want to do.

Right now, things I am left thinking about before planning forward:

• What do students attend to when we ask if things are the same?
• Our language and recording is SO incredibly important.
• How can these ideas build in K-1 to be helpful in later grades?
• If I am thinking of moving students from a concrete to more abstract understanding, how does that happen? Is it already a bit abstract in the way the numbers are represented?
• Do we take enough time with teachers digging into these ideas? [rhetorical]

I look forward to any thoughts! So much learning to do!

# Asking Better Questions

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

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

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

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!

# 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!

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!

# Literature & Algebraic Reasoning

I read the book One is a Snail, Ten is a Crab to two Kindergarten classes this week. If you have not read the book, Marilyn Burns does a great post about it here. After reading aloud, making predictions and doing a notice/wonder, I placed 10 tiles in the middle of our circle in an arrangement like this:

I asked the students who could be standing on the beach and they quickly guessed a dog, two people and two snails. I asked if they could give me an equation for the feet they see and they said 4 + 2 + 2 + 1 + 1 = 10. I asked if anyone had a different equation and they switched the order of the numbers, but agreed there was still 10. I did a few more arrangements before sending them back to their groups to investigate with the tiles. The directions were for one student to put out the arrangement, the groupmates guess who was standing on the beach, and write an equation for what they see. Their equations were all so different but the ways they were composing and decomposing the tiles to make new arrangements was really interesting!

We brought them back to the carpet and asked what they noticed about all of their equations. They said they all ended in 10 and equaled 10, so I asked if that meant we could write the equations so they were equal to each other? I asked two groups to share one of their equations and I put them equal to one another on the board. I asked if that was true? How could be we prove it? Their first answer was like, duh, they both equaled 10 so yes. I asked if they could combine or break apart any of the numbers like they did with their tiles to prove it. One student talked about combing the 1’s circled in the picture below.

They had to leave for lunch so I left them with an equation to talk about when they got back!

This lesson was such an amazing way of allowing students the space to think about equality and the meaning of the equal sign. It took one student talking about the ways he combined the numbers to open up the conversation and possibilities for future equations. I would love to see what the students could do if I wrote one equation on the board and asked students to write all of the different ways they could fill in the other side of the equal sign.

# 100 Hungry Ants: Math and Literature

This week the Kindergarten and 1st grade teachers planned with Erin, the reading specialist, and I for an activity around a children’s book. This planning was a continuation of our previous meeting about mathematizing. We jumped right into our planning by sharing books everyone brought, discussing the mathematical and language arts ideas that could arise in each. I made a list of the books the teachers shared here.

We chose  the book One Hundred Hungry Ants and planned the activity for a Kindergarten class. We decided the teacher would read the story and do a notice/wonder the day before the activity. We thought doing two consecutive readings may cause some students to lose focus and we would lose their attention. Based on Allison Hintz’s advice, we wanted the students to listen and enjoy the story for the first read-through. Here is an example from one classroom:

So many great problem and solutions, cause and effects, illustration and mathematical ideas were noticed by the students.

The following day, the teacher revisited the things students noticed and focused the students’ attention on all of the noticings about the ants. She told the students she was going to read the story one more time but this time she wanted them to focus on what was happening with the ants throughout the story. We had decided to give each student a clipboard and blank sheet of paper to record their thoughts.

We noticed a few great things during this time..

• Some students like to write a lot!
• After trying to draw the first 100 ants, students came up with other clear ways to show their thinking. I love the relative size of each of the lines in these!
• A lot of students had unique ways of recording with numbers. Here is one that especially jumped out at me because of the blanks:

Students shared their recordings at the end of the reading and it was great to hear so many students say they started the story by trying to draw all of the ants, but changed to something faster because 10o was a lot!

After sharing, we asked students, “What could have happened if they had 12 or 24 ants?” We put out manipulatives and let them go! So much great stuff!

–>

Next time I do this activity, I would like to see them choose their own number of ants.

Just as I was telling Erin that I could see this book being used in upper elementary grades when looking at generalizations about multiplication, I found some great posts by Marilyn Burns on this book for upper elementary and middle school:

Excited to do this in a 1st grade classroom today!

# Mathematizing Learning Lab

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

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

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

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

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

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

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

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

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

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

We opened with this talking point on the board:

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

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

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

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

Here are some of their responses:

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

Many questions came up:

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

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