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?

Show Your Work vs Show Your Thinking; 4th Grade Division

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The Planning

Last week, the 4th grade team and I planned during our Learning Lab for their first division lesson in Unit 3 of Investigations. The book opened with the following problem:

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As we talked during the lab, one of the main concerns expressed by the teachers was student comprehension of the context. This was not necessarily in reference to this specific problem, but story problems in general. This is not surprising and definitely something, I would say, we as teachers face quite often. This is why I love Notice/Wonders and offered that as an idea to take that “number grabbing and compute” feel out of the problem.

As we read the problem aloud, we anticipated how students may respond in two scenarios:

Keeping the numbers in the problem and taking the question out.
Taking out both the numbers and the question.

We opted for the second one with the thought that we could ask them to anticipate what a question may be for this problem and what information they would need from us. It felt like a bit of a mash up of noticing/ wondering and a 3 act task.

The Launch

I really want to get to the student thinking here so I will not go into the details of the N/W, but here is what we ended our conversation with…

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After this, we gave them the original problem and sent them off too work on the problem. After 5 minutes of individual time, they came together as small groups to share their work and create a poster with all of their strategies.

*We did debate that last sentence for a bit. We didn’t know whether the rows being the same size would arise and/or if it would send the problem in a different direction. Another 4th grade teacher tried it without that sentence in a different class period and it didn’t seem to change the focus of the work so I wonder if student make that assumption? Then I wondered if we give too many problems that are arrays and maybe don’t play around more with the “extra” pieces that could be added on?

Show Your Work vs Show Your Thinking

After the lesson, I wasn’t surprised by the students’ strategies as much as I was left questioning how we posed the prompt for students to show their strategies. The teacher asked them to solve the problem and show their work. This piece is not something we talked about at all in our planning because it is what we all say, but does that make a difference in how students show their solution path? Do we ever make explicit the way we can show our thinking clearly and explicitly either pictorially or numerically?

As I walked around, I would have thought at first glance that over 50% of the students were “passing out the apples” on their paper to arrive at 14, however after asking them to explain their thinking to me, I would have completely assumed wrong.

For example, this student’s work (on the left) was just the rows and dots underneath it when I walked by. I assumed he had “passed out” 56 apples to each of the 4 rows, however when I asked him to explain his thinking to me, it was so much more than that. He said he “gave each row 10 because that would be 40 and then there were 16 left, so half of that is 8 and half of that is 4, so each row got 4 more. I asked him if we could take what he did to write an equation for it? He said 10+10+10+10 and then wrote 10×4=40 and added the 4×4=16 underneath. I asked where the 56 apples appeared in his work and where the apples in each row were. So, the thinking he explained made his understandings so much more clear than the work he had shown. I wondered if he really needed to show all of that work or he just thought he did for the teacher to see “work”? Next, I partnered him with his neighbor (work on the right) to talk about how his thinking could be show within her multiple tower.

When I walked up to this student, she had everything on her paper but the equations on the right. I asked her to explain her thinking and she explained that she divided up the 56 into 4 rows. By the way it looked to me, her x’s were written going down each column which did not indicate to me that she has passed them out so I asked her if she solved using those x’s and she quickly responded, “No, I just got bored…see all of my decorations!” I pressed her a bit more to explain how she got the 14 and she said she knew 8 x 7 = 56 and she used that but was having trouble explaining it. I asked her if she thought it may help her remember if she wrote down the 8×7=56 on her paper and quickly after she wrote it she explained she had halved the 8 and doubled the four. Again, he work would not have matched her thinking.

Going in the other direction, this student had the 56 circles with 28 labeled at the top, 14 labeled on the right and the equations written at the bottom when I walked up. I assumed she had split the 56 in half and then in half again, using what she knew from her equations. However when I asked her to explain her thinking, I found she had drawn the circles one at a time in rows of 4. She said she split it into two groups, counted the 28 on one side and then counted how many were in one column. This was interesting to me and I am kicking myself about not asking more about the 4 rows because she counted by rows of 4 but then found her 4 rows of apples actually in the 4 columns. In this case I would have assumed things that didn’t happen from her work that really didn’t match what she was thinking.

Here are some other great examples of student thinking: IMG_1289 (1)

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Show your work versus show your thinking…do we say both? do we make explicit what we mean by them? I am not sure, but what I do know is that we never learn more about student thinking than we do when we talk to students about their work!

2nd Grade – Even and Odd

5th Grade Fraction Multiplication

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Yesterday, in my planning, I was bouncing around between a couple of ideas for the lessons I was teaching today. I decided to go with this Illustrative task that was the basis for the lesson study project I did last year with The Teaching Channel and Illustrative. I though it would be a great formative assessment as students move from thinking about fraction of a fraction on a fraction bar to an area model using a square unit.

I opened with a choral count in which the students counted by 3/4. I played around with ending each row on a whole number (12/4, 24/4, 36/4…etc) or ending on something not as “nice.” I opted for the second, but in hindsight, probably should have gone with the first. I really wanted more conversation around when the whole number occurred and why and possibly the distributive property (4 x 3/4) + (1 x 3/4) = 15/4, but not as obvious as I felt the first option made it, but it didn’t happen. The count looked like this (I didn’t get a chance to take a pic of the board):

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I asked students what they noticed and I got a variety of responses. I got many variations of equations such as 3/4 + 6/4 = 9/4 and 12 x 3/4 = 36/4, and with each one, I asked how we could show that within the count. One student said she could see 20 x 3/4 =60/4 in the count. Just when I thought she was going to explain by saying we counted by 3/4 twenty times, she surprised me in her wording. She said, “I knew the count was a 4 by 5 array, which is 20 numbers and then each one was 3/4, so I got 60/4.” This struck me different than the skip counting or repeated addition of 3/4 that others were doing and reminded me a a bit of this diagram from the learning progressions:

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I think it was the way she described the size of each number in the count as 3/4 that drew me to this diagram.

Next I gave them the task to engage in alone for 5 minutes before I let them move into group work. I was really impressed by the way they jumped right in and easily could find 1/3 of 1/4. I was not surprised that many ignored the “square pan” piece and went with fraction bars. Much of the work looked like this, cut vertically: IMG_1262 (1).jpg

While there were also quite a few students who quartered it by cutting vertically and horizontally in half and then splitting that quarter into thirds:

This is something for me to think (learn) more about because if we are thinking area model, the dimensions of the piece, to me, looks like 1/2 x 1/6. That seems like it could be problematic to me when the square has dimensions in units of length.

Another interesting thing that always comes up in this work is the difference between dividing by a fraction and dividing it up into fractional parts. I saw those equations sneaking there way in like this…

That will be something to keep in mind in future lessons. What it means to divide by a number versus divide into parts? Is it different when we are thinking about whole numbers versus fractions? Cool convos to be had around that!

There was one student who was not working with the original whole in her work. She was working with the 1/4 in the first part and then the 1/2 in the second part. When I asked her about how she was determining 12ths, she said it was just like her phone, she took the whole thing and just zoomed in (she did the fingers swipes as you would do on your phone) on the part she needed.

There was some really great proportional reasoning going on with the cost of the cornbread pieces. When the pan price changed from $12 to $24, $6, and $18, student used great reasoning in relation to $12. In the example above you can see that work as well as here:

I left them with a question about the denominator, why is not ending up in any of the denominators we are using the problem? I only had a chance to snap one pic before I headed to another classroom.

Lots of great stuff to keep in mind and I think comparing the ways in which the students divided the models would be a really interesting conversation. Then after they move to an area work, I wonder if it would be great to bring that back for a comparison.

Also, I forgot to add that the substitute who was in for Leigh today was so incredible about taking notes and talking to the students because he was going to be in charge of teaching this same lesson to the next math class that came in. He said he learned so much and I thought this could be such a powerful way to have substitutes involved in learning more about the way in which math is taught.

~Kristin

Planning K-5, literally

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Tomorrow I have the opportunity to teach a Kindergarten, 2nd and 5th grade class! It is so exciting and interesting to be thinking across all of the grade levels in one day of lesson planning! The most interesting part for me, in thinking through this, is the connections across all of the grades. There is so much potential for conjecture and claim-making supported by their development of proofs.

Background: The 5th and 2nd grade teachers are out at a state math teacher leader meeting so I am teaching instead of the substitute. The kindergarten teacher and I will be teaching it together. I have met with each teacher to chat about where they are within their units and what they have been seeing students do within the current work. I invited teachers both at those grade levels and at other grade levels to pop in if they have the time. I thought it would be great having more people to reflect with after the lessons!

5th Grade: They have just started working with finding a fraction of a fraction using bar models. The initial work is unit fraction of a unit fraction and then moves to non-unit. (My post on that from a couple of years ago on this work, I wish I had done that better, so here is a chance to try something new;) Leigh, the teacher, says they have been really successful in partitioning the bars and arriving at the correct answer. I am thinking about starting with a number routine of either a choral count or a number talk string like 1/2 of 12 = __ of 24… As far as the lesson, I could continue work with this and have students look at noticings after and explore them deeper.They have done these noticings with whole number times a fraction or mixed number, so this could be a revisiting of similarities or differences. OR I could do this cornbread task as a formative assessment as the next piece they will move into is an area model. It may be really helpful for Leigh to see how they are thinking about this before they jump into the work. This is my least planned because I keep bouncing all around with ideas.

2nd Grade: They have been working with even and odd numbers and counting by groups of 2’s, 5’s, and 10’s. All of this work is within contexts of break a group of students into equal teams or everyone having a partner. Tara, the classroom teacher, said the students are really great at determining whether a number is odd or even, however when asked how many would be on each team, a lot of students struggle. They are great if they know a related double fact, however if they don’t they resort to “passing out” by tallies or drawing the picture and physically dividing the number of things in half. For example, if they do not know 11+11 is 22, then finding the number of on each team become passing out 22 things into two groups to find 11. While they are successful in this, Tara and I were wondering why they do not say 10+10=20 and 1+1=2 so 11+11=22. They are able to add 11 and 11 but unable to decompose it as fluently.

In thinking about this, I am inclined to want to connect that addition to halving. I am thinking a counting collection would be fabulous for this. Give students a collection of things to count. Share how we counted them because I am positive they will not count them by 1’s given a large set. We can share as a whole group, record ways in which we counted and determined if our number was even or odd. Then, put the collection back together, switch with a partnering team and then split the collection into two groups. The share would be, “Could you make two equal groups?” “Was your number even or odd? How did you know?” Record strategies. Ask for noticings/wonderings about how they counted and how they divided into two groups.

Kindergarten: The students in this class have been doing a lot of work with ten frames, dot images, counting jars, etc and having students counting and adding to compose a number. They have just begun working on decomposition of number so I immediately thought about the mice activity in Thinking Mathematically. Linda, the teacher, and I planned to do this activity with the students. In preparation, we read NCTM TCM’s article by Zachary M. Champagne, Robert Schoen, and Claire M. Riddell, Variations in Both Addends Unknown Problems. We are going to use 6 bunnies and see how students show all of the ways the bunnies can be inside and outside in a pen. Instead of just giving a context, I was imagining that the students may need a visual of the rabbit pen so I created this image to launch with a quick notice and wonder:

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We will then let the students work on finding the different ways in partners and then come back for a whole group share and record the ideas on the board. We are really looking to see two things….1-how they organize their information and 2- the strategies they use. The students will do a notice/wonder about the recorded information. If there is time it would be great to see if students, when given a different number, would apply any of the strategies and/or organizational tools shared.

Going for a run to think through this a bit more! Would love any thoughts/suggestions, as always!

~Kristin

1st Grade: 1st Day of Subtraction

4th Grade Multiple Towers – Pt 2

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Due to a schedule change today because of an assembly, Malorie and I did not get a chance to sit and plan together before teaching the lesson. Luckily, she is always so prepared and had read my thoughts on the blog I did yesterday about it so we went into the lesson with a common vision and then talked our way through the shortened class period with the students.

We started with choral counting by 3 and then 30. We chose to include 5 multiples in each row in hoping the students would see a pattern in the “friendly numbers” if they struggled to see patterns in other places. After the list was up, they took a few minutes to write down any patterns they saw in their journals and we shared out. This was the count and then notices Malorie typed as they were sharing:

I could have stayed with these patterns all day long, but we decided to let students revisit these patterns over the course of this unit. There are some really great ones up there like adding the digits and that number is also a multiple of three and if you add any of the numbers up there, the answer will also be a multiple of 3. They made some predictions about numbers that would show up later in the list and talked about connections between the two lists. Multiplying by 10 and the zero “put on the end” was definitely the most popular noticing of the class period, which was a nice lead into our stacking of the boxes of oranges.

We explained the boxes of oranges held 30 oranges and asked them to estimate how many boxes (represented by the post-its) would be needed to stack up to Mya’s shoulder. Malorie and I quickly tried to figure out if we wanted to ask about number of boxes or number of oranges and decided to give them the option of either. It worked out beautifully because the majority of students told us boxes along with how many oranges it would be as well. As Malorie stacked the post-its, the students counted along. Some were counting post its while I heard others counting oranges. We stopped approximately halfway, took adjustments on their estimates and continued. We finished with 480 oranges, 30 boxes. As we were running out of time, we decided to end by asking if there were any equations they could think of to represent the oranges and boxes.

I was surprised to get division first…480 ÷ 30 = 16 and then 16 x 30 = 480.

This was unfortunately where we had to leave off because the classes were shortened and then next class was waiting at the door. So we met during her planning later that day to regroup and find our starting point for the next day.

Tomorrow we have decided to lead with a talk about the orange tower we built today. This is where we had a big discussion around what we want to really have students think about….the book seems to really focus on the multiplying by a multiple of 10, like what would be the 20th multiple, the 30th multiple, etc but we want to play around with the properties a bit more here.

So, we decided to ask students what equations or expressions they could use based on the tower to arrive at the 330 post it. We are thinking we may get some things like this:

30 x 11

(3 x 10) x 11

(3 x 11) x 10

(30 x 10) + 30

(30 x 10) + (1 x 30)

We want to set them equal to one another and ask how they could prove are the same answer. We are hoping to see the associative and distribute properties come out. We definitely could get some division too and that could make it really interesting!

We decided to go with this because Malorie says that she often sees students “putting one zero” on the end of any number when they multiply by any multiple of 10. For example, when multiplying 40 x 30, they will just get in the habit of putting one zero because they don’t see that 10 x10 happening. This is where being able to think about 40 x 3 x 10 is helpful for students.

Next, they will make their multiple towers based on these numbers we will assign to partners: 15, 16, 32, 28, 24, 48, 18, 36, 35, 70, 45, 14. Since Doubling/halving is coming up, we thought this could bring out some of those ideas. Then we will ask them to come up with expressions for any number they choose in their tower. We had planned for that to be in their journal, but I wonder if that is something they could hang up next to their tower for a notice/wonder walk around the room? Hmmm…have to ask Mal in the morning!

~Kristin

Follow Up…today we did a choral count 6 and then 60 and the pattern-finding was even better then yesterday. It was always amazing to me how students notice new things and then based on how we record. This time Malorie did four multiples in each row so in addition to finding new patterns, they also started comparing it back to what they noticed yesterday during the 3 and 30 count. Even though the 3 and 30 count was not up on the board anymore, they still remembered the patterns and numbers like they were right in front of them because they were so excited about them. So cool to see/hear.

Next, we looked back at the multiple of tower to 30 on the wall. I asked the students if they could find an equation with the answer of 330 (which was on the tower) using numbers on the tower and gave them time to jot some ideas in their journal.

As the students started to share, I realized it was a really Badly Worded Question. As they shared I was getting “ways to get 330” but not relating anything to the tower, which we were hoping would happen. We wanted to see the associative and distributive properties emerge, but were getting addition and subtraction equations.

So, as I got “420-90=330 and 270 + 60=330…” etc, I asked “How would would you show that action on the tower….and then they were on a roll. I didn’t want them to think what they had written was not valued so we asked them to share what they had originally written and we built on from there! There is SO much to work with here so I cannot wait to plan for some exciting algebraic work.

I had to plan with another teacher so I left as the students began working on their multiple towers. After they finished their own towers, we were going to ask them write equations for a number on their tower based on moves on their tower. I have to check back in with Malorie in the morning!

~Kristin

Multiple Towers in 4th Grade

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In the Making Number Talks Matter chapter on Division, one of the strategies described was “Make a Tower.”

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This strategy immediately reminded me of the 4th grade lesson in Investigations called just that, Multiple Towers. To make the timing even better, I am planning and teaching that lesson with a 4th grade teacher this week. The goal of this lesson is not necessarily eliciting a division strategy, but there is so much possibility for so many things in this lesson. Because of that, however, I am struggling with how we can leave it open enough for the many ideas to emerge, but focused enough to have a purpose in our planning.

The structure of the lesson, as it is the book, looks roughly like this:

Students count around the classroom by 3’s to 72 and it is recorded on the board. They then count by 30’s to 720 and it is recorded on the other side of the board, so both lists are visible. Asking students what they notice and what relationships they see. We record these noticings and leave this because the investigation of multiplying by a multiple of 10 continues to emerge throughout the unit.

Next, as a class, we introduce multiple towers through a context of stacking boxes of 30 oranges. Using post-its on the wall, students place the post-its one above the other, labeling the amount of oranges as the tower grows until it reaches the shoulder height of one student. We then ask if there is anything in the tower that would help us figure out how many oranges are there without counting each one? They estimate how many to the top of the student’s head, ask how many multiples are there, and record equations for it.

Choosing from the list below, students work with a partner to create their own multiple towers as tall as one of them on a strip of paper:

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After creating their tower, they answer questions in their activity book about the number of multiples, how they could find the 20th multiple without counting, and writing equations for some pieces of their work. They will continue to reference these towers in the following lessons in which they multiply 2-digit numbers and discuss various strategies and representations for multiplication.

I love this lesson, but as always, I like to play around with different ideas when I am lesson planning and these are some things I cannot wait to chat with Malorie, the 4th grade teacher, about on Monday:

Instead of a number talk do we open with the count around the classroom to save time for the multiple towers?
How do we record the multiples? Will different recordings draw out different noticings?
Do we start with estimation of how many cases of oranges will it be before we just start the tower? Could that draw out their use of multiplication combinations they know?
Do narrow the list of starting numbers to choose from so there are some really great relationships that emerge, like one factor that is 1/2 of another so the towers have a relationship? or a factor that is 1/3? or do we let them choose their own off of that list within a range?
Do we put the towers up, walk around and do a notice/wonder before they jump into the activity book pages?
Do we want to end with a journal prompt? The answer is obviously yes, but what do we want it to be?
How will this leave her for the next day’s lesson?

Feel free to chime in with thoughts/suggestions and I will post the plan we decide upon after our planning on Monday!

~Kristin

Follow Up Post

MathMinds

by Kristin Gray.

Writing About Writing

Both Addends Unknown in Kindergarten

2nd Grade Learning Lab: Data

Show Your Work vs Show Your Thinking; 4th Grade Division

2nd Grade – Even and Odd

5th Grade Fraction Multiplication

Planning K-5, literally

1st Grade: 1st Day of Subtraction

4th Grade Multiple Towers – Pt 2

Multiple Towers in 4th Grade