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Fraction & Decimal Number Lines

Recently, our 3rd, 4th, and 5th grade teachers had the opportunity to chat math for 2 hours during a Learning Lab held on a professional development day. It was the first time we had done a vertical lab and it felt like perfect timing as 3rd and 4th grade would soon be starting their fraction unit and 5th would be entering their decimal unit. Prior to the meeting, we read the NCTM article, “Identify Fractions and Decimals on a Number Line” by Meghan Shaughnessy, so we started the meeting discussing ideas in the article. We then jumped into playing around with clothesline number lines and double number lines, discussing what they could look like at each grade level based on where students are in the fractional thinking.

I have co-taught number line lessons in both 5th grade and Kindergarten this year, but both were a bit different in not only number, but organization. In 5th grade we used one clothesline with the whole class, while in Kindergarten we used tape on the floor and students worked in small groups. Leigh, 5th grade teacher, was interested in trying the small group number lines on the floor. As we planned the lesson, the one thing we thought would be difficult about having small groups is getting around to each group to hear their conversations, especially when we were planning cards purposefully to address misconceptions and misunderstandings. However, knowing we would have the two of us circulating, as well as two 3rd grade teachers who wanted to see the lesson (yeah!), we knew we had plenty of eyes and ears around the room to hear the math conversations.

During the lesson, in groups of three, students placed 24 cards on their number line. There were two sets of cards, so after placing all of their cards, each group visited a number line with a different set of cards to discuss. Instead of boring you with all of the number choices we made, here are a few of the choices in cards and the reason(s) we chose them:

1/3 and .3: Students often think these two are equivalent so before the decimal unit we were curious to see how they were thinking around that idea and how they used what they knew about fractions or percents to reason about it.

0.3, 0.33, 0.333, 1/3: The 1/3 and .333 were there to think about equivalency, while the others were there to think about what is the same in each and how much more each decimal has to make it larger. Which you can see caused some confusion here:

2.01 and 2.08: We were curious about the distance they put between these two cards.

1 6/9 and 1.6: We wanted to see how students compared fractions and decimals when they can’t easily convert 6/9 into a decimal. Then, if they began thinking 6/9 and 6/10, how would they decide on the larger fraction and then how much distance do they put between them?

The group below practically had them on top of one another:

While this group had a bit of a space between them:

2.8 and 2.80: Tenths and hundredths equivalency. They all seemed to handle this with ease.

.005 and 1/100: Curious to see the placement in relation to the other numbers. This 1/100 is close to 0 but I wonder about it in relation to the .2. Definitely a conversation worth having!

2.8 and 2 7/8: To see how they compared the 7/8 to the 8/10.

After they visited other number lines, they had a chance to meet with that group and discuss card placements they agreed with and placements they did not. Groups then made adjustments accordingly…

Here was a group’s completed number line and my first stab at panoramic on my phone!

The journal entry we left them with was, “Which cards were the most difficult to place on the number line? Why?” Many were just as we suspected.

The conversation as I walked back over to the other building with the 3rd grade teachers was, what does this look like in 3rd grade? Could we use array images to place on the line instead of the fractions? Could the pictures include over 1 whole? What whole numbers would we use? Do we play with equivalent pictures with different partitioning? Being mindful of the students’ second grade fraction exposure, below, we are planning on trying out something very soon! I am thinking the cards like these on Illustrative, with the pictures but no fraction names at this point.

CCSS.MATH.CONTENT.2.G.A.2
Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

CCSS.MATH.CONTENT.2.G.A.3
Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

…and then could the journal could be, “Can you name any of the fractions on your number line? How do you know?”…or something like that!

Counting is Complex

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There are so many interesting things to consider when counting, however having taught upper elementary and middle school, I have never taken the time to consider these complexities. After doing a Counting Collections activity in Kindergarten on Friday, I saw so many foundational ideas being constructed in simply counting a set of things. As I planned and implemented the activity with Kindergarten teachers Jenn and Michele, I felt a lot of the ideas in this post by Joe Schwartz, including the amazing counting conversations, surfacing.

The lesson plan seemed simple: Give the students a set of things to count. Walk around, watch, listen, and ask questions. From that simple plan, our planning conversation was filled with so many questions since this group of students had never done a counting collections activity. After watching this Teaching Channel video, we began asking ourselves: What is important about the set of things when students count? What tools do we make available? How do we keep them from grabbing every tool because they can? What questions do we ask? What numbers are appropriate for each student? How will they record their thinking? What is the end idea we want students to leave thinking about? This post would be so long if I included our decisions on all of this, but I am happy to answer any questions in the comments if you are curious. Not that they were the right or wrong answers, but our decisions.

We decided to keep the sets the same object, same color to avoid students sorting by color. I think it will be an interesting conversation in the future about what matters when we are counting, but for the first time, we wanted to really see what they did with counting without distractions and how they recorded their count. We made available big cups, small cups, ten frames and plates.

These are some of the interesting things we observed students considering when counting and explaining their count…

Choice of Tool(s)

Students had to consider the size of the object and size of the set to decide on the tool they felt would best organize their set. We saw groups switch because their object did not fit in or on what they originally chose. Some groups liked the plates to stack and organize their count because the ten frames took up too much space on the table. I wish I got a picture of one group who used the yellow and blue ten frames in the picture below…they put their full tens on the blue and their ones on the yellow and were able to articulate that statement as their counting strategy.

Counted vs Not Counted

This is where the tool they chose came into play again. Some groups chose to put the uncounted objects in one cup and count by 1’s into another cup. For these groups we saw a lot of losing track of count. We anticipated this in our planning and decided to ask them if there was another way they could count to help keep track of their numbers or if there was a way to organize their objects so we could see how they counted when we walked up.

Another group drew a line on their first plate and said it was for ones they counted and ones they hadn’t, however as the crayons took up too much space, they moved to putting 10 on a plate.

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Do We Even Have to Count?

This group got away with not really doing any counting at all until they combined their end organization. They filled a ten frame and poured it into cups. It was interesting because in the first, smaller set they counted just before this larger one, they filled all 10 frames on their table (in the pic under Choice of Tool). As it took up too much space this time, they switched.

How We Count vs How We Record The Number Of Objects We Count

This group put 10 on each plate and started to label each plate by 10s, however on the second plate one partner wrote “20” because she was counting, 10, 20, 30..etc. Her partner corrected her and said there were not 20 on the plate. You can see the scratched out mark on the plate and hear it here.

Adjusting for Larger Sets

This group kind of blew my mind. Their second set was 225, so they decided to put ten beans in every box of the ten frame. They were then even able to articulate the fact they knew 10 groups of 10 is 100. You can listen to them here. This is my first try with YouTube, so if this doesn’t play, please let me know in the comments;)

Recording Our Counting

This was one of the areas where we were so curious to see what happened! Jenn and Michele do a ton of Number Talks and journal writing, however during the Number Talks, the teachers does the recording. We didn’t know if they would draw everything they counted or be able to record it more abstractly with equations or would they do both? We did see a mix of all of this!

When Counting and Number of Things Counted Gets Jumbled…What To Do?

We had the groups share their first set (which were all in the 50s) and one group picked up on a counting strategy and way to organize that got a bit jumbled in the end. This group put 10 on the first plate, 20 on the second, 30 on the next, etc. It was like counting by 10s but now there were not 10 on each plate. When I asked how many were on each plate, they were able to tell me that the number they recorded way was the number of beans on each plate and when I asked how many were there altogether, they said they didn’t know. This is where they ended:

So, where would you go from here with this group? Our feeling is to pull out the ten frames and put the beans from each plate on them because the students are really great at seeing and counting by 10s in this way. Would this be an interesting thing for the whole class to explore?

So many things to think about when counting! I love to think about how these ideas of counting and combining groups keep showing up in the work I am doing in all of the other grade levels! If we could really do this more and give students space to make sense of groups and how and why they work, wouldn’t it make so much of their future math work so much more accessible? If students really understood these foundational ideas, would we need to spend the time (and money) on intervention programs in later grades that are addressing these very same things?

Counting is Complex but we can structure ways to allow students to be successful in thinking about all of these ideas!

3rd Grade Perimeter Part II

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Last week, I posted about a 3rd grade lesson I planned and taught with Hope and Lori. We did not get to everything we planned so I love that they filled me in on what happened the next day when they continued the work! And when the continuation involves looking at student work, I love it even more! That said, this will be a bit of a student work-heavy post with things I noticed and wondered in steps moving forward from here with the students…

After measuring a piece of paper in the previous lesson, we wanted to ask students how they would find the distance around any-sized piece of paper. In giving them the journal writing, we wanted to have them reflect on the measuring and calculating they did in a more general sense and see how they put the process into words. Most student papers resembled the explanation in piece of work below:

It was really interesting to find most students drew a picture to illustrate their explanation even when not asked to do so. To me, this is a nice mix of show your thinking and show your work. In reading this example below, this student is thinking a lot about conversions and I think, moving forward, the class needs to have a discussion about combining different measurement units.

The mix of units shows up again below. I can see they probably chose centimeters because they didn’t have a smaller unit than the inch and didn’t know how to name the measurement in inches. I love the “not really the size” but I wonder about the border look of the perimeter. Is this student seeing the 6 inches and 3 cm ending where the line is and counting boxes instead of the distance around?

This one is an amazing look at how the formula we all know, and probably had to memorize, arises in third grade. The calculation on the back was equally as nice. This is an example of something during a class share that I would show last in a progression to compare with the previous strategies as it does a nice job of showing the process of finding perimeter in two ways.

This one was so interesting because it involved a square and a circle. The measurements on the back were most intriguing and I have so many questions for this student. Like, how do you know that is a square? (because the sides are not the same length) Where is the 1/2 coming from in your answer? (because I cannot tell where he is stopping and starting his measuring) and Why did you want to cut a circle and a square?

Then, Hope asked them to draw their own ant path and some really interesting things came out that will have to be a blog post in and of itself! There are things we didn’t think about in our question and some things we really need to think about moving forward. Like…

Could this student start thinking about area? Why did the choose to draw a non-rectangular path?

Where are the measurements for each side? Why did you label them where you did? Why did you choose to use inches and centimeters?

When did you choose to use inches and when did you choose centimeters? Could you have measured it all in inches? all in centimeters?

First, the fact the student sent the ant to Walmart is too funny:) I would love to ask this student how he or she added all of those side lengths? and Why was it longer to get home from Walmart than it was to get there? Could the ant have walked the same distance there and back? How?

On this one, we did not anticipate students’ ants taking the same path back that he did out. So this is important to think about distance and versus distance around something.

Oh, an isosceles right triangle, how fun! I would love to ask this student about this perimeter in centimeters because of the diagonal cuts in the boxes. A lot of students counted the diagonals as 1 unit like they did for the sides of the boxes, so would that work out if you measured it with a ruler in centimeters? Why?

My question is where to go with a student who is here? All teachers face this, right? There are some students who conceptually and computationally have a grasp on an idea. This student can obviously find perimeter and is very comfortable with the computation piece of it, so what do you ask him from here? Do you give him things to measure that closer to a quarter and see how he works with the fractions? Do you ask him if his strategy will work for every shape? (I lean toward this one) Do you ask him about non-rectangular polygon areas? Do you do anything with area at this point? So much to think about!

~Kristin

Two Things I Am Wondering…

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It is an interesting perspective moving out of the classroom into a coaching position. I have had more face-to-face teacher math conversations this year than ever in my career and it is wonderful. This position also lets me take a step back from the daily lesson planning and think about things I see across all of the grade levels. Most times, my thoughts are about the trajectory of mathematical ideas, however over the past couple of weeks I find myself thinking about two things I saw as norms when I taught, but now wonder more about…

1 – Is there a such thing as an addition, subtraction, multiplication or division problem?

I am sure we all can relate to the stories of students struggling with story problems. We see them be successful with Notice/Wonders and 3-Act Math tasks, however when given a story problem some “number grab” and compute without thinking about reasonableness. Why? While I think there are many factors at play here, I have another theory that has led me to question problem types. I could be completely off, but as I look through the curriculum and think about the progression in which I taught in 5th grade, I wonder if there is something to teaching “types of problems” within a unit. For example, in Unit 1, Investigation 1 could be my multiplication lessons while Investigation 2 could be my division lessons. While we don’t explicitly say, “this is how you solve a multiplication problem” and we explore various strategies to make connections between the operations, the header of the activity book pages say things such as, “Division Stories” or “Multiplication Stories.” Also, the majority of the work that week is the specific operation and applications.

From there I began to wonder, is there really a such thing as a specific operation problem? I would think that any division story could also be thought of as a multiplication problem. Do we lead students to think there are certain types of problems even if we make clear all of the strategies to find solutions? I love how CGI talks about problem types and wondering why more curriculum are not set up that way instead of keying students into operation-specific problems?

I asked some 4th graders about this exact idea. I gave them some multiplicative compare problems and asked them if they thought about each as multiplication, division or both. Then we talked about why.

2 – What makes students attach meaning to a vocabulary word? Do they need to?

Every year in 5th grade, I was confident that all of my students could find area and perimeter of rectangles. However, I was also confident that there would also be a handful of students who could find area and perimeter but didn’t know which was which. After much work with area and perimeter, they would have it by the end of the unit, but did they remember when they got to 6th grade? I am not sure.

Now, seeing all of the work they are doing with this beginning in 3rd grade, and talking to 3rd and 4th grade teachers who are seeing the same thing, I am left wondering why this is? What makes students attach meaning to vocabulary? This question is then followed by the my very next question…when do they have to?

I wonder if students should ever be given a problem where the context would not allow the students to figure out which one, area or perimeter, the problem was asking. For example, if Farmer Brown is buying fencing he would need the perimeter where if he was buying something to cover a piece of ground, it would be area. Should we ever give them just naked perimeter or area problems with no context where knowing the meaning of the word impacted their ability to solve it?

And then, after they do all of this work with both measurements, why do they forget which word is which year after year? I know the teachers do investigations with the work and use the vocabulary daily during the unit, as I did, but students still don’t hold on to it. What makes it become part of their vocabulary? Is it just too long between when they use it? Is it

These are just two things I am wondering about….

Perimeter in 3rd Grade

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I am in the unique position over the next few weeks to see perimeter and area work in 3rd, 4th and 5th grade. It is so incredible to see the overlap across all three grade levels and, being a 5th grade teacher for so long, it is great for me to see where this work begins.

After planning with Hope and her student teacher, Lori, last week, we taught the lesson introducing perimeter today. On Friday, the students measured things around the room in different units of measure, having discussions about most appropriate units. For example, when measuring the length of the room would we use the same unit as we would for the width of our pencil eraser? Why?

Since I was not there for the lesson on Friday, I was super curious to just hear what students thought about when they heard the word “measuring.” I wrote the word on the board and away we went. They were very quick with benchmarks, equivalents and different dimensions we can measure. I did a terrible job with my picture, but I got a couple really interesting questions like, “Can we measure anything? Air?” and “Can we measure the corner angles of things like the carpet?” Also, after a student had shared that one yard is the same as your hip to your ankle, students questioned if that was true because of the different heights of people. All of these things are great for students to explore at later times!

Hope introduced the Investigations problem of an ant traveling around the edge of a piece of paper. To be honest, we were not thrilled with the context, but at the time we could not come up with anything snappy or original, so we went with it. We thought it was nice because, in inches, we could see if students measured to the half inch and also how they worked with the half inch when combining to find the perimeter. In hindsight, I am thinking a city map might have worked, however then the scale comes into play, so maybe not?? We let them choose the unit they thought would be appropriate, put them with a partner and they went off to work together. We were surprised to see most students using inches and when asked, thought that it would be “too many centimeters.” They seemed to chose units based on the biggest unit that still fits the object, but not thinking about precision and getting the smallest unit for that.

This is where I am continually amazed by what students know and intuitively do with mathematics.

It was interesting to see some pairs not know how to deal with the half,”not quite 9,” but know they only had to measure one side and then put “11” on the opposite side.

While another group had the 8 and one half written exactly like they said it “8 and 1 (one).5(half) inches.” Although written incorrectly, they dealt with it beautifully in their computation. However, I would want to bring up the equal sign in future share outs so they 8×2=16+1=17 would be written correctly. Does anyone use arrows in the elementary grades for this? 8×2–>16+1–>17? Or is it more appropriate for separate lines at this age?

When I walked up to this group I asked where the ant was walking because of their lines through the middle of the paper. They said around the outside but it is the same no matter where you draw the line. I asked them to show me the 8 inches and I left them to talk about the 1/2 inch.

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Some students did not deal with the 1/2 inch but seeing the ways they found the perimeter and wrote their equations, I was able to see the formula for perimeter coming to life.

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As students got finished with their first unit choice, we had them find the perimeter in another unit. It was nice to see the multiplication from their previous unit showing up a lot.

When I saw this one, I didn’t really know what to do with it. What do you with a 3rd grader using .5 as half? I asked them what .5 meant and they quickly said one half. They said one can be broken into .5 and .5 just like it can be broken into 1/2 and 1/2. That is so interesting to me and I would have loved to explore that conversation more, but with a whole class that is not ready to go there, I wrote it on the board and moved on.

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As always, there is not enough time in a class period for me to talk about math with the kiddos. Tomorrow morning, students will journal about their strategy to find the distance the ant traveled. Since the majority of the class only measured two sides, we want to make explicit, through student sharing, why they didn’t have to measure all four sides in this case.

They next part of the lesson, which Hope and Lori will continue tomorrow, includes the students creating their own ant path on grid paper and finding the perimeter of that path. We are not going to dictate that the path must be a rectangle, but the ant must stay on the grid lines. We are hoping that this generates the conversation of when we can double the two sides and add them and when we can’t, assuming students draw irregular shape paths.

Sorting Data in 2nd Grade

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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:

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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!

Decomposition of Number in Kindergarten

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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.”

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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.

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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!

Writing About Writing

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I have learned a lot about myself as a writer in just a few short years of blogging. I have noticed I love over-using commas, I am horrible with verb tenses, I jump around a lot in my thinking, and I am much more inclined to write when inspired by something, than on a given topic. In addition to the logistics of my writing, I have personally learned that writing publicly about teaching can be a very vulnerable place in which to reflect and learn with others.

Lately, there have been a lot of conversations on Twitter around teacher-writers. Recently I engaged in (more like butted in) on a conversation with Chris and Michael, only to realize one more interesting thing about myself as a writer: I never really think about how I am writing when I am writing. I typically sit and start typing my thoughts, questions, things I saw, heard and learned. When Michael suggested we compare the styles of Saunders in his piece entitled “The Perfect Gerbil,” which I had never read, and Lampert, who I am reading now, I was intrigued. Before I go on, let me highly recommend both of these works!

As I read Saunders work, which unbeknownst to me was not about math but instead story-telling, I realized that I probably never thought about my style of writing as a writer because I have never really considered this when I read. Do I not consider it because when I read fiction I fixate on the storyline to escape thinking about the details and moves of the author, or when I read math ed books I am seeking out an authority on a topic or possibly because when I read blogs I focus on my connection to the content? Now, given two completely different works to compare, I struggle with how to think about this within a common viewpoint as Michael does so beautifully here.

As I began thinking about the two works in terms of style, I had a few questions: Is it more natural to write about someone else’s work from a place of writer and reader as peers than it is about one’s one work? Does an expert/authority point of view naturally come with writing about a nonfiction experience versus writing about writing fiction? What implications for teacher-writing come from each of the different styles? The best way for me to work through these is to compartmentalize and deal with each question individually. I don’t know if by the end I have any real point or even address the intention of our reading assignment, but here it goes…

Is it more natural to write about someone else’s work who you appreciate from a place of writer and reader as peers than it is about one’s one work?

In Saunders’ piece, I felt his appreciation for Donald Barthelme’s writing and it was as if we were co-reading it. He was masterful in pulling me into the art of storytelling with him and I absolutely loved his writing style. However, would that have felt the same if he did not have an appreciation for that piece of work? Would his style have changed and would I have then felt badly for the subject of his writing and therefore not felt a camaraderie with Saunders? Would we no longer have felt like peers? And, although he was not talking about his own work, I still felt as if I was on this mutual reading journey with an expert. He incorporated analogies such as the one below that made me realize he knows things about writing that I don’t and he made me want to learn more. It left me with questions for him.

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This same expert type of feeling comes in a different way when I read Lampert’s book. Like Saunders, she talks about the problems of practice and impact of the moves we make, but instead of being about another person’s teaching or how one writes, it is about her own teaching of math. However, while I initially thought this would position her differently in terms of writer and reader, I felt just as much of a peer feeling with her as I did with Saunders. I loved her framing of the purpose of the book in the beginning and that set the tone for me as a reader.

“My intention in writing this book is not to argue in favor of a particular approach to teaching or to have the last word on the nature of the teaching practice, but to contribute to the conversation about the nature of the work that schoolteachers do.”

I felt that although she is an expert to me in terms of teaching, we were still peers. This could be because she is speaking my 5th grade language, asking so many of the questions I have had myself, and the fact that the book feels like a blogging journal to me. I am only on chapter 5, but I feel like I am on this teaching journey with her. Would I feel the same way if it were not the curriculum work I have been invested in for over 10 years? I also wonder how her work would sound if it were about someone else’s classroom where she was a fan of the teaching?

Does an expert point of view naturally come with writing about a nonfiction experience versus writing about writing fiction? Does fiction create more of a reader/writer relationship?

At first, I completely thought Saunders’ piece felt more like writer/reader as peers because he was talking about writing fiction. The freedom of fiction characters and their actions feels like something we all can talk about because it is changeable. In teaching, the characters, both students and ourselves, are not as easily changed. For that reason I thought the expert position happened naturally because you need to know about the students and, in Lampert’s case, the math involved.

However, the more I think about this and detach the content (which is hard for me) it is not so much about the fiction vs nonfiction but more what things are happening vs control over what things might happen next. This is where I do feel more of a peer feeling with Saunders and expert feeling with Lampert, both of which I completely understand and appreciate within their purposes for their writings.

For example, Saunders writes as if he wonders with us who the characters are and what their next moves will be, with neither of us having control over it. He shares these wonders with lines such as,

“Will they do it?” and “A few lines ago we didn’t even know Helen existed.”

As I read, I felt like we were both on this ride together, where we could be surprised, happy, or disappointed by what happened next in the story. However, while this felt like a writer/reader moment, I did know in the back of my mind that he would experience these emotions based on a much more expert view of literary writing than myself. Either way, it wasn’t about either of us in those moments, it was about Barthelme’s choice.

Lampert, writes about her thoughts, reflections and observations where she obviously is the authority. She has control over what is happening next in her planning and teaching so there are no mutual surprises or wonderings happening next between the writer and reader because she is in control of her questions and moves. I know in reading her work that I am taking what I learn and applying to my own about teaching and learning. I have no control over what happens next in her classroom, just as she would have no control over what I take and apply in my classroom. This is what I see mainly happening in teacher-writing, my own included, which leads me to my last question…

What implications for teacher-writing come from each of the different styles?

After all of this rambling, I think I have come to the non-surprising conclusion that there are important places for both of these styles in teacher-writing. The more important question is, how do we make this happen?

I personally learn a lot from just sitting, reflecting and writing about what happened in class each day around various pieces of a lesson. I wouldn’t say these posts, however, engage others in the learning with me. And while I would never call myself an authority on anything I blog, I am wondering if there are ways to draw others into the learning with me? Wouldn’t it be amazing if we moved from “learning something from a blog” to “learning with people based on a blog ?”

In looking back at my original questions, I am left with these new questions for us as teacher-writers…

Could writing about a common piece of work in relation to our own work help connect readers and writers and empower more readers to write?
Could writing more with unknown pieces or mutual wonderings encourage readers and writers to connect more? For example, if we didn’t know the next moves and possible impact on student thinking, could we put ideas out there and learn together?
How can we take the problems that Lampert does such a beautiful job writing about in her book and highlight their realities in our own classrooms? Could we write these classroom stories as Saunders does, inviting others to work with us in solving these problems openly, publicly in conversations more than 140 characters?

So, as with the ending to many of my blog posts, I don’t know if I really addressed anything or was even on track with the original conversation Michael and I had! But, I did get introduced to Saunders’ work, who I cannot wait to read more from, read a few more chapters in Lampert’s book which is awesome and gained a new perspective on how I write. All in all a great learning experience!

Both Addends Unknown in Kindergarten

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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).

2nd Grade Learning Lab: Data

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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.

After sharing as a whole group, Lauren prompted the students to begin thinking about what questions they could ask about this graph.

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?

MathMinds

by Kristin Gray.

Category Archives: Uncategorized

Fraction & Decimal Number Lines

Counting is Complex

3rd Grade Perimeter Part II

Two Things I Am Wondering…

Perimeter in 3rd Grade

Sorting Data in 2nd Grade

Decomposition of Number in Kindergarten

Writing About Writing

Both Addends Unknown in Kindergarten

2nd Grade Learning Lab: Data