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2015-05-01 12.16.12

Last week I got to visit three wonderful schools in Michigan to think and learn with teachers about math talk, reasoning and number sense. We got to plan, teach and debrief together, which is one of my favorite ways of learning with a group of teachers!

With one group of 3rd grade teachers, we decided to think more about choral counting routines and how we could use them to get kids talking, reasoning, and looking for patterns and structure.

The team chose to count by 3s starting at 21. Before teaching, we thought about what mathematical ideas and patterns this count might bring out and how we could record the count to support those ideas. This group of teachers also wanted to think more about students journaling around math, so we thought about taking time at a certain point in the count to have kids write about their thinking.

In the above picture are the numbers we said and recorded in our count. Students immediately noticed how the digit in the ones place repeated at a certain point in the count, as well as patterns related to how many times numbers in a particular decade would appear. Students noticed that while we were skip counting by 3s, there was a jump of +30 between the numbers in the same column. When we asked students to predict the number that would go in the orange box below 60, some counted by 3s from 78 (the last number we said together), but many used the +30 noticing.

But things got really interesting when one student spoke up and said “Hey, and 30 + 60 is 90! You can add the first two numbers in the column together to get the one below.”

“That does work here, doesn’t it?” I said, writing it down. “I wonder if that works everywhere or just here? I wonder how we could figure that out.”

In many ways I wish we had had more conversation here, especially since the students were new to this routine and to the idea of pursuing and investigating claims and conjectures, but for the sake of time and getting to the journaling we decided to move ahead with our plan to have them write.

We asked students to write the number they thought would go in the second orange box (the blank box in the picture) and write about the patterns they noticed that helped them figure this out.

As a group of teachers we read through the students’ journal entries. Some kids wrote about skip counting from 90. Some kids wrote about how they knew the ones place would be a 2 and the number would be 100 and that 2. But a good number of kids wrote that the number would be 114 because 42+72=114. This was fascinating because they ignored the previous patterns they had noticed in favor of following this claim set forth by a student. This claim had power! (I think the idea that you could notice something that would lead to a claim was powerful for this group.) The students wanted to make a generalization, find a pattern, find a rule so much that they ignored evidence against their claim. It was fascinating and a beautiful error, in an inaccurate sort of way. :-)

This count and journaling also provided amazing material for teacher discussion. The thoughtful teachers I worked with talked about how they might push students towards ways of investigating conjectures and claims, how you might prove or disprove this claim, and the importance of this kind of reasoning for wider mathematical understanding.

All this great thinking, teaching, and debriefing in a brief little time with this team of teachers. I was only sad that I didn’t get to stay for next class meeting to see where students would take this next!

But it has pushed my thinking as well. First, it made me reach for Russell, Schifter and Bastable’s Connecting Arithmetic to Algebra: Strategies for Building Algebraic Thinking in the Elementary Grades. It also made me reach out to Twitter-thinkers Kristin Gray (@mathminds), Mike Flynn (@MikeFlynn55), Simon Gregg (@Simon_Gregg) and Elham Kazemi (@ekazemi) to clarify my thinking on claims and conjectures (which I’m still working on!). I can’t wait to think and teach more.

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I just love counting routines. I’ve learned a tremendous amount about counting routines from Jessica Shumway’s Number Sense Routines and the teachers featured in Elham Kazemi and Allison Hintz’s Intentional Talk(Several of these teachers are featured in videos you can view online for free. This one is my favorite!)

Today I got to do a Count Around the Circle with a twist in a 3rd grade class launching its fraction unit. While 3rd graders have a good amount of experience with fractions, this group’s teacher was wondering about their understanding of the part/whole relationship and relationship among unit fractions. We decided to do this count to investigate these ideas!

After we gathered students in the counting circle, we told them that today we’d be counting around the circle in a different way. We’d be counting brownies today and each person would have one whole brownie. We explained that we’d be laying out the brownies (post-its!) on a big imaginary plate and we wondered how many brownies we’d have after we had each contributed one brownie to the plate. The students quickly sized up that we would have twenty brownies–one for each person in the circle. As students counted and placed their brownies in the middle of the circle, I helped them organize the brownies onto the plate in rows of five. I knew this would help them see some important relationships once we had counted by halves.

Next we got ready to count by half brownies. These students already had experience looking at different ways that two people could share one brownie equally, so they quickly suggested a way for cutting brownies into halves. Once each student had their half brownie in their hand we said, “We’re going to put these half brownies on another imaginary plate. Do you think we’ll have more or fewer brownies on the plate after we all count around the circle?” We first asked for a quick thumbs up or thumbs down to signify more or fewer brownies. The class was split! These were some of the ideas they shared after a quick turn and talk.

“It’s the same. There’s still twenty people and we still each have one piece to put in the circle.”

“More. This time we’ll have 40 because that’s double the twenty brownies.” (This one fascinated me! Playing around with doubling and halving, but still developing that fraction number sense!)

“Fewer. It will take two people to make a whole brownie.” (Some people had an idea of what number we’d land on. Others just knew fewer.)

As we counted, I again helped students place their brownies in rows of five. After the brownies were laid out we simply asked, “What do you notice? What do you wonder?”

There was a lot of clarifying talk about halving and doubling, two halves equally one whole, and twenty halves being equivalent to ten.

We moved on to our next task by saying, “Let’s save our plate of whole brownies and half brownies. I wonder what would happen if we counted by quarter/one-fourth of a brownie?”  (Which we will do tomorrow! Can’t wait for the discussion of the relationship between the denominator and “size” of the pieces in unit fractions.)

Working with halves and wholes might seems overly simply for 3rd graders. But I’m reminded again that sometimes “simple” numbers offer an opportunity to explore big ideas and important relationships.

[Special thanks to 3rd grade teacher, Ellen Rogers, for planning, teaching, and reflecting on fraction counting circles with me.]


Math in Life Photo Series

The grocery store is one of my favorite places to think about math. Aron, my husband, has gotten used to me saying, “Oh, I’ve just got to take a picture of this!” on our weekly grocery shopping trips. Today I’m sharing a picture I took at the grocery store this week.

This is the first photo I’m sharing as a part of a Math in Life photo series I’m starting. In my book, Math Exchanges, I encourage teachers to find their own rich mathematical life, share that life with their students, and make space in their classroom communities (and beyond!) for students to live their own rich mathematical lives. Sharing photos has always been a part of this practice for me. Include a math in life picture you take (or borrow mine!) in your morning message and ask students a couple of questions that make sense for the photo:

  • What do you notice?
  • What do you wonder?
  • What math does this make you think of?
  • What does this picture make you want to figure out?

I took the photo I’m sharing today because I thought the array were particularly interesting. But it might speak to you and your students differently.

2015-02-22 19.06.13

When I saw these chicks, I wondered how the 3rd graders I work with would mathematize this picture using multiplication.

Would they think of the rows–8 groups of 6 (8×6)?

Would they group the chicks by color and column?–6 groups of 8 (6×8)?

Would they see each individual box first–2 groups of 6 (2×6) and then four of those boxes 4(2×6)? (What a nice what to start playing with the distributive property!)

Tune in soon for more Math in Life photos!

2015-01-13 11.11.34

The first ten minutes of math workshop are often my favorite part of the day. This is a time when we use number sense routines to build math understanding and also to build communities in which students are talking about and playing with math. We try to make this time a time when every single student feels a sense of belonging, efficacy and participation in the mathematical community.

As our 5th graders have begun thinking about addition of fractions, we want students to learn more than procedures for fraction computation. We want them to reason about fractional equivalencies and relationships as they solve problems.

In this version of Over or Under, students are asked to consider whether the sum will be Over or Under 1. Students are presented problems one at a time (we usually did three problems each day). Students have time to think about each problem on their own. No one raises hands yet. We just think in silence (this takes practice!). Then we either have partners do a turn and talk or share ideas as a whole group. Finally, for one of the problems, we  have students share their thinking through drawing and writing in their notebooks.

Here are the three problems we presented on the first day of this routine and some of the students’ thinking. (I admit we didn’t capture a lot of the thinking on our chart paper because we were just really into discussion! We want to get better at this though.)

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Question 1:

2 1/2 + 1/5

We decided to start each day with a problem that would seem fairly easy to most students, but would open conversation up about an important idea. In this case we wanted students to think about mixed numbers and how the whole numbers can help us make reasonable estimates.

Question 2:

1/12 + 1/6

We’ve been doing a lot of work helping students think about which fractions are close to 0, close to 1/2 and close to 1. We want them to create mental images that include area and linear models. Some of the student thinking that came out of this was that 1/12 + 1/6 were both “kind of near zero” and “wouldn’t even be 1/2.”

Question 3:

8/9 + 1/4

This was my favorite question for eliciting fascinating student thinking. Many students talked about what an area model of 8/9 would look like and what the “missing piece” (1/9) would look like. Many said that 1/4 was too big to fit in this “missing piece” and thus the sum would be more than 1.

Here are some of the other problems we used throughout the week. We recorded some student ideas as they talked, but we got so into the talk at times, we didn’t record everything.

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We did notice that students still need more experience thinking about linear models. This week we started playing the awesome Investigations game “Fraction Track” which encourages just this kind of thinking. I’ll be interested to see how experience with this game shifts some of their thinking and reasoning about fraction computation.

Today students took a district-wide multiple choice assessment. One of the questions asked students to solve 4/5+1/3. Only one the answer choices was over one! Many of the students wrote that they didn’t need to solve the whole problem or worry about finding common denominators because they knew that 4/5 + 1/5=1 and since 1/3 >1/5, the sum would be more that 1. Boo ya! Way to outsmart a test question with reasoning and number sense, kids!

Besides building number sense, this routine really built community. The game aspect of the routine (Is it over or under?!) really appealed to the kids. They loved arguing for their ideas. Also, not having to arrive at an exact answer was very inclusive of all students. Even students just developing strategies for addition of fractions were able to share their hunches. “I think it might be over because…” The risk of sharing this idea was low and students refined important ideas about fractions even around the “easy” problems.

I’m looking forward to extending this routine into subtraction with fractions!

How else would you use this routine?

2014-12-10 10.13.02

I want to share a question with you that has been clanking around my brain for a while. At first I wasn’t even sure if I wanted or was even “allowed” to share and write about this question, but it’s one I keep coming back to. It’s a question that makes me feel somewhat vulnerable. And it’s one I’d love to hear your thoughts about, so please feel free to leave a comment even if, like me, your ideas around this question aren’t fully developed either.

Is there room for math that isn’t hard?

One reason I wasn’t sure I wanted to write about this topic is because I really believe in teaching students to work hard to figure out mathematical ideas. I believe that math can be challenging and also enjoyable. I teach teachers and students that doing challenging work is what helps us construct new understanding. I really believe in the power of the “make sense of problems and persevere in solving them” (CCSS.Math.Practice.MP1) I believe that much of math learning will occur in this way. And yet, I keep coming back to this question. Should all math be rigorous? Should we persevere through all tasks? Or is there a room for something else too?

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When imagining strong math communities, I like to consider analogies to literacy communities. In reading workshops, teachers spend time a good amount of time engaging children in read-alouds in which a primary purpose is learning to love books and enjoy reading. As a kindergarten teacher I often read Mo Willems’ Piggie and Elephant books not because they were rigorous, but because it’s fun to read the speech bubbles using the voices we imagine Piggie and Elephant would use. We lingered to laugh for a few moments on the page of David Shannon’s No David that shows David’s naked bottom as he runs down the street naked. In reading workshops book browsing is encouraged. Children squirrel away Lego encyclopedias in their book boxes to pore over with friends. They linger over photographs of animals and come up with their own wonderings like, “Why are reptiles bumpy?” “Why do pigs have curly tails?”

In reading workshop, children should not spend the majority of their independent reading time engaged in books that are difficult for them. Richard Allington’s research indicates that children should be spending the great majority of their reading time engaged in books at their independent level, books they understand and can read accurately. Children need to read and re-read. (This is one quick Allington article, but his work on the topic is much more extensive than this.)

So, I’m wondering, is there a math equivalent of this? If children subsist on a diet that consists only of difficult math, will they learn to enjoy it? Will they learn to pursue it beyond the walls of the classroom? I say that with the huge caveat that I believe challenging and difficult math can be engaging and fun. But I also believe there is space for math that is not terribly difficult, but is very enjoyable.

Strategy games (like Rush Hour, for example) can be extremely challenging. While playing Railroad Rush Hour recently, 5th grader Kevin told me, “I’m only on the first challenge. And it’s already hard. I can’t do it.” He stuck with it though, and figured it out. Kevin moved on to other Rush Hour challenge cards, but a few days later I noticed he was working on that first card again.

“Kevin, didn’t you do that one the other day”” I asked.

“Yes,” replied Kevin matter-of-factly. “But I like to go back to ones I’ve already done and think about it again. It’s fun.”

Was what Kevin was doing the problem solving equivalent of re-reading a favorite book?

2014-12-13 11.55.51

This weekend I spent some time at the grocery store with my daughter watching an employee make a giant batch of guacamole. I watched with fascination. “I wonder how many avocados are in that bowl. How many containers of guacamole will that make?” I considered a few ideas in my mind. How much space would ten avocados take up? So, how many groups of ten avocados? I engaged lightly and playfully with the idea. The avocados offered an invitation, rather than a specific task to complete.

I held my almost-two-year-old up to see the giant batch of guacamole. “How many avocados are in there, Lulu?” (She loves avocados!)

“One and one and one and one!” (That’s what her counting sounds like now.) “Dos!” (That means any quantity more than one in her world.”)

It wasn’t hard. For me or for Lulu. But it was kind of fun to think about. I love looking for math in my life. I hope to inspire that in her and the teachers and students with whom I work. I hope to offer invitations to engage in mathematics. Sometimes in difficult mathematics and sometimes in something less serious.

So, is there space for this in our classrooms? Time for engaging, but not terribly difficult math? What other opportunities do children have to just wonder about math? To browse math? To engage in math play? To relax into math, the way one relaxes into a book. Is this important? Is this a thing in math? Is it something professional mathematicians do? Is it something everyone does as they wonder about their world?

I’d love to hear your thoughts.

~Kassia

#ElemMathChat

2014-02-18 12.10.07

I’m dusting off the ol’ blog to share that I’ll be hosting this week’s #elemmathchat on Twitter this Thursday, December 11, 2014 at 9:00PM EST! I always find the #elemmathchats thoughtful, supportive and inspiring, so I hope you’ll join us.

The topic this week will be building mathematical communities, a topic near and dear to my hear. Check out the questions below for you to mull over this week.

Q1: How do you define mathematical community? What do strong mathematical communities look like/sound like/feel like?

Q2: What big and small things do you do to help build these strong mathematical communities?

Q3: What is a misconception/roadblocks you run into that impacts the development of strong math communities? How have you dealt with it?

Q4: Describe a moment/interaction in your math community that made you feel like “Yes! This is good stuff of mathematical community!” Why was it important?

Q5: If your students walk away from your class with one big idea about what math is all about, what do you want it to be? How would you and your math community have helped them construct this idea?

Q6: Share a picture or piece of student work that shows your mathematical community (and its values) at work.

Looking forward to “seeing” and chatting with some of you on Thursday!

~Kassia

2014-04-22 11.51.15

I’ve been thinking a lot about math communities, mindset, and how children define mathematics. I got the chance to write about some of these thoughts, and share some stories from a great 5th grade classroom in my blog post, “On Being a Mathematician,” over at Stenhouse’s Summer Blogstitute page.

While you’re there, check out all of the summer Blogstitute posts, especially, “Teaching Through and For Discussion,” written by Elham Kazemi (@ekazemi) and Allison Hintz (@AllisonHintz124). They’re the authors of one of my favorite new math books, Intentional Talk. If you haven’t checked it out yet, you should!

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