As a teacher, curiosity around students’ mathematical thinking was the driving force behind the teaching and learning in my classroom. To better understand what they were thinking, I needed to not only have great, accessible problems but also create opportunities for students to openly share their ideas with others. It only makes sense that when I learned about routines that encouraged students to share the many ways they were thinking about math such as Number Talks, Notice and Wonder, and Which One Doesn’t Belong?, I was quick to go back to the classroom and try them with my students. It didn’t matter which unit we were in or lesson I had planned for that day, I plopped them in whenever and wherever I could because I was so curious to hear what students would say. Continue reading
Each month, teachers choose their Learning Lab content focus for our work together. Most months, 1/2 of the grade level teachers choose to have a Math Learning Lab while the other 1/2 work with Erin, the reading specialist in an ELA Learning Lab. This month, however, we decided to mesh our ELA and Math Labs to do some mathematizing around children’s literature in Kindergarten and 1st grade! This idea was inspired by a session at NCTM last year, led by Allison Hintz, that left me thinking more about how we use read-alouds in our classrooms and the lenses by which students listen as we read.
In The Reading Teacher, Hintz and Smith describe mathematizing as, “…a process of inquiring about, organizing, and constructing meaning with a mathematical lens (Fosnot & Dolk, 2001). By mathematizing books commonly available in classroom collections and reading them aloud, teachers provide students with opportunities to explore ideas, discuss mathematical concepts, and make connections to their own lives.” Hintz, A. & Smith, T. (2013). Mathematizing Read Alouds in Three Easy Steps. The Reading Teacher, 67(2), 103-108.
Erin and I have literally been talking about this idea all year long based on Allison’s work. We discussed the ways we typically see read-alouds used, such as having a focus on a particular text structure or as a counting book in math.
As Erin was reading Kylene Beers & Robert Probst’s book, Reading Nonfiction she pointed me to a piece of the book on disciplinary literacy which automatically had me thinking about mathematizing.
Beers refers to McConachie’s book Content Matters (2010), in which she defines disciplinary literacy as, “the use of reading, reasoning, investigating, speaking, and writing required to learn and form complex content knowledge appropriate to a particular discipline.” (p.15) She continues to say, “…disciplinary literacy “emphasizes the unique tools that experts in a discipline use to engage in that discipline” (Shanahan and Shanahan 2012, p.8).
As I read this section of the book, my question became this…(almost rhetorical for me at this point)
Does a student’s lens by which they listen and/or read differ based on the content area class they are sitting in?
For example, when reading or listening to a story in Language Arts class, do students hear or look for the mathematical ideas that may emerge based on the storyline of the book or illustrations on the page? or Do students think about a storyline of a problem in math class or are they simply reading through the lens of “how am I solving this?” because they are sitting in math class?
Mathematizing gets at just this. To think about this more together, Erin and I decided to jump right into the children’s book The Doorbell Rang by Pat Hutchins. Erin talked about the ideas she had for using this in an ELA class, I talking through the mathematical ideas that could emerge in math class, and then we began planning for our K/1 Learning Lab where we wanted teachers to think more about this idea with us! We were so fortunate to have the opportunity to chat through some of our thoughts and questions with Allison the day before we were meeting with the teachers. (She is just so wonderful;)
The first part of our Learning Lab rolled out like this…
We opened with this talking point on the board:
“When you change the way you look at things, the things you look at change.”
Everyone had a couple of minutes to think about whether they agreed, disagreed, or were unsure about the statement. As with all Talking Points activities, each teacher shared as the rest of us simply listened without commenting. The range of thoughts on this was so interesting. Some teachers based it on a particular content focus, some on personal connections, while I thought there is a slight difference between the words “look” and “see.”
After the Talking Point, Erin read The Doorbell Rang to the teachers and we asked them to discuss what the story was about with a partner. This was something Allison brought up that Erin and I had not thought about in our planning. I don’t remember her exact wording here, but the loose translation was, “Read for enjoyment. We want students to read for the simple joy of reading.” While Erin and I were so focused on the activity of exploring the text through a Math or ELA lens, we realized that the teachers first just needed to enjoy the story without a purpose.
For the second reading of the book, we gave each partner a specific lens. This time, one person was listening with an ELA lens while, the other, a Math lens. We asked them to jot down notes about what ideas could emerge through these lenses with their classes. You may want to go back and watch the video again to try this out for yourself before reading ahead!
Here are some of their responses:
Together we shared these ideas and discussed how the ELA and Math lenses impacted one another. A question we asked, inspired by Allison, was “Could a student attend to the math ideas without having a deep understanding of the story?”
Many questions came up:
- Could we focus on text structures and the math in the same lesson?
- Could we start with an activity before reading the book, like a probable passage?
- Would an open notice/wonder after the first reading allow the lens to emerge from the students? Do they then choose their own focus or do we focus on one?
- How could focusing on the problem and solution get at both the ELA and Math in the book?
- How could we use the pictures to think about other problems that arise in the book?
- How do we work the materials part of it? Do manipulatives and white boards work for K/1 while a story is being read or is it too much distraction?
- What follow-up activities, maybe writing, could we think about after the book is read?
Unfortunately, our time together ended there. On Tuesday, we meet again and the teachers are going to bring some new books for us to plan a lesson around! So excited!
Last week, the kindergarten students solved a problem about Jack and his building blocks. It went something like this:
Jack was building with blocks. He used 4 blocks to build a wall and 2 blocks to build a bridge. How many blocks did Jack use altogether?
The teachers posed the problem without the question, did a notice/wonder, and then gave the students time to answer how many blocks altogether. We looked at this student work in our planning for their upcoming lesson.
We were curious to see what the students would do without the numbers in the problem, so we planned for a numberless story problem during last week’s Learning Lab. Three kindergarten teachers and I had a chance to be in the same room to see it in action today.
Nicole, the teacher, posed the following story to her students:
Susie is building with blocks. She used some blocks to build a wall. She used some blocks to build a bridge.
She asked the students what they noticed/wondered and the very first notice was there were no numbers to tell us how many blocks, awesome. They did some wondering about how many blocks she used and compared this story to Jack’s building from last week.
We planned to have the students choose the number of blocks they wanted Susie to use in her building and then find how many she used altogether. Their number choices were so interesting and left me wondering when students begin to explain the usefulness of 10? I know some of them know 10 is a great number to add after the activity today, but I am wondering the questions to ask to make it clear to them because they just “know it.”
Here were some examples of their work…so much cooler than 4+2 in Jack’s problem!
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: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.
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?
Yesterday, I had the chance to teach the 1st grade lesson I planned here. It was so much fun and SUCH a learning experience for me! After all of the conversation in the comments and on Twitter, I decided to start with the open, one sentence Notice/Wonder. Only having 45 minutes and this being the students first time doing a N/W, I decided not to begin with a number talk/routine (which I usually always do).
The students started on the carpet, I put up the sentence, read it and asked, “What do you notice and wonder about this sentence?” Just then a student exclaims that he just noticed that “Notice” was not, “Not Ice.” At that moment, I began thinking maybe my question had them looking at the physical pieces of the sentence/words so I quickly rephrased, “I would love to hear what you notice and wonder about what is happening in the sentence.” They used their Number Talk signals, thumbs up when they had a notice or wonder and then used their fingers to indicate more than one. I was so impressed by all of their thoughts, but I did realize that is it hard to end their wonderings! The amazing thing was how all of their wonderings really could turn this sentence into a story in their ELA class because they were all really important details they could add to it. Here was how the board ended.
I asked which wonder we could work on together today in class and there was a unanimous vote for “How many kids are on the bus?” however there were a few that suggested, “How many student can the bus hold?” because “math is counting things and we could count the seats.” I starred the wonder “How many kids are on the bus?” and told them next time they get on their bus I would love to hear how many seats and students they found are there. We discussed whether they know how many students were on the bus by reading our sentence and they said no, they only know that there were 3 stops. I asked, what they would want to know and they wanted to know how many kids were at the stops. I wrote that at the top.
When I told them they would get to choose how many students were at each stop, they were so excited! I gave them a paper with the sentence at the top, let them choose a partner and sent them on their way.
As I walked around and asked students why they chose the numbers they did, I quickly wondered how much I should have helped organize their work for them. I found so many with numbers everywhere and it was hard to see where their bus stop numbers were, let alone their total. Should I have put Bus Stop 1____, Bus Stop 2____, and Bus Stop 3_______ to have a clearer picture while also modeling for students how we can make our math work more clear? Quite a few looked like this…
There were so many interesting papers, so I love WordPress’ new tiling feature for pictures to make it look less cumbersome!
Top Row, Pic 1: This student had 24 as two stops. When I asked him how many stops we noticed in the beginning, I got a “Oh gosh” and he wrote Bus stop 1, 2, and3. He then stuck with the 24 and when I came back he had 8, 8, and 8. I didn’t see this until after class so I am curious how he arrived at that answer. I also realized that these 1st graders move fast and it is SO easy to miss the cool things they do so quickly!!
Top Row, Pic 2: They said 3 and 22 were easy to add and then they just chose another small number. The interesting thing here that I need to find out more about is the 5×6 with the one box shaded in. I loved the commutativity showing up here!
Top Row, Pic 3: This was so interesting because I had never thought that a student would first think of how many students were on the bus and divide it up from there. They thought 30 students would fit on a bus so they made the stops fit that information. (They saw the error on the last one during the share).
Bottom Row, Pic 1: This student said that because there were 3 stops, there were 3 students at each one and ended with 9.
Bottom Row, Pic 2 &3: This student wanted big numbers so his first response, after he insisted on re-writing the sentence, was 1,000 and 1,000,000 and 4. Then on the back of his paper he wanted 6 stops and chose 6 new numbers. This led to some great conversation during the share.
This student figured that if there were 1 or 2 at a seat then there would be 55 students on the bus. I love all of this work so much! Then when I asked her about the students at each stop, she said, 30, 20 and 5.
We shared as a group back on the carpet and I tried to capture why they chose the numbers they did:
I then gave them the original problem and asked them to solve it individually. After seeing them work on this problem, I think there are so many interesting conversations that could happen Monday morning!
This is where I had so many questions as to how we get the younger students to make their thinking more visible. I found so much of it happens on fingers, 100s chart, and number line on their desk that I was getting an equation on the paper. It is great when I am sitting there asking, but that cannot obviously happen when they are done so quickly and there a bunch of them! Is this something that comes with practice? I did find that once I asked them if they could explain to me on the paper how they solved it, they did a great job. My next question is, would taking the 100s charts and number lines off the desks help push students to look for friendlier numbers? I found the majority of them went left to right, counting on instead of using the 6 and 4 first. This is something that I think a structured share out on Monday could bring to light for those who never thought of it.
Here are a sampling of the papers I look forward to chatting with the teacher, Lisa, about on Monday. We can chat about how we can structure this share out.
Lisa, through number talks and investigations, has been working a lot on having students noticing number patterns leading to generalizations. It was neat to see this work of adjusting addends and keeping the same sum showing up here too. It seemed to show up most after they had their answer and were playing around with the numbers, which I love!!
I am happy to have started with the open notice/wonder because I learned so much about how they think about problems and I think the opportunity to choose their own numbers got them thinking about the context over solving for an answer to an addition problem. I am, however, extremely curious how it would have changed the work if I had given students the problem with the 13 given and the other two missing? Would I have seen more about how they choose numbers to make the 13 easier to add a third number? I am hoping to get into another 1st grade classroom to try this out with another teacher but I would love it if any other 1st grade teachers would there would love to try it out and report back!!
I am so looking forward to Jamie’s post on this because her student work looked amazing on Twitter yesterday!!
Yeah, Jamie’s post is up! Check it out here! Cannot wait for our Google Hang Out tomorrow to chat all about it!
Every day, I notice and wonder things about both the students thinking and the mathematics in my classroom. Over the past couple of weeks, however, there are a couple of things I have noticed that still have me wondering….
We do Number Puzzles in our first Investigations unit, serving as a review of properties of numbers that students have worked with over the years. For those who have not done number puzzles, this is an example:
At the end of this activity (14 puzzles), I have the students design their own number puzzle, trade with another pair of students and talk about the solutions. It is a great formative assessment of their learning and it is fun to see some students creating puzzles that are impossible or have more than one possible answer. The conversations are amazing. However, this year, a student completely stumped me. I wish I had saved his puzzle, but it was two of his four clues that got me wondering. First clue: My number is odd. Second clue: My number becomes even when you make it a decimal. Hmmmm. Thankfully I do not have to recap the conversation here because Christopher Storified our conversation here: https://storify.com/Trianglemancsd/it-s-even-when-i-make-it-a-decimal
My first thought, after I got done reeling over his thinking, was, when was the last time the students have revisited even/odd numbers? I know in second grade, students investigate odd/even numbers in terms of being able to break a number (positive integers) into two equal groups or share between two people. The exact math focus points are…
So, they establish that a number is even if it can be broken into two equal groups and if there is a leftover, it is odd. But, at this point, they have only dealt in whole numbers. Do we ever revisit that when they begin working with rational numbers or negative integers? Do we just assume that students keep the understanding with positive integers and don’t try to apply it to other numbers? What an interesting thing I have never thought about before! It blows my mind that after 19 years of teaching (12 of those years in 5th grade) that I have never had this conversation before with a student. This whole conversation has me digging back into the math introduced in earlier years to see if there are other things that we never revisit in our work as we move into decimals, fractions, negatives…etc. Just for some added fun, Christopher said he thinks he could master “Billy’s” even/odd quiz. Billy made a quiz here: pic.twitter.com/p24gMG4Uwd and you can check your answers here: https://www.dropbox.com/s/2chzj2n3534stjo/billy.parity.answer.key.pdf?dl=0
My second wondering stems from my students’ work yesterday with Volume. Up to this point, we did a lot of building with unit cubes and in Minecraft of rectangular prisms in route to formalizing our strategies for finding volume as l x w x h or area of b x h. Students were very comfortable with finding the volume of rectangular prisms, with or without the cubes drawn in the diagrams. They could talk about the arrays in each layer they saw, and how many layers they saw “stacked” in finding the volume. I was feeling so confident that these understandings would follow us right into our work with two rectangular prisms combined to form a solid figure. It was looking great as we worked with the gridded diagrams. Here is an example of one student’s work:
Then hit the unmarked models…
Even anticipating there would be some confusion, I was honestly blown away by the number of students struggling to break apart the picture and think about the lengths, widths, and heights they needed to reason about layers of cubes. I found that some of the students felt the need to use every number in the picture in their solution while others were making no connection to their reasoning about arrays and layers. They could describe how they broke the shape apart into two pieces, however when I asked how many cubes were in the bottom layer of the prism to the right, they could not see the 3 x 2 as 6 cubes in that layer, and if they did (and had broken it horizontally) would do 9 layers instead of the 5.
I was struggling with how to help them go back to their amazing work with cubes. Why was this visualization disappearing? In hopes of having them have a better visualization, I asked them to build the above picture in Minecraft. Done. Every single struggling student automatically laid down a 2 x 3 array, made 5 stacks of them before moving into four 12 x 2 arrays on top of that. What?!? This has me completely perplexed. How can they see it to build it without a second thought, but cannot communicate about it otherwise? More importantly, where do I go from here? They obviously cannot build every figure they need simply because when the numbers are larger, it is completely time consuming and I want them to connect their concrete building to a more abstract reasoning. I am thinking from here, I will have some continue to build until they don’t need it, pushing them to abstract with questions such as, “If you were to build this right now, describe it to me.” Pausing occasionally to ask how many cubes that would be.
These two wonderings, although so obvious at times, have seemed so complex to me over the past weeks. I love every moment of looking into my students’ reasonings’ because it challenges me to be a better teacher and look deeper into my own understandings about how children learn. Great stuff.