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Much Ado about Watermelons

Day 2 of #summermathcamp

To feed our math brains at 8 am on a Tuesday morning in the summer, we showed this photo of some watermelons in the hopes of generating some conversation.

How many watermelons are there? How do you know?

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http://ntimages.weebly.com/

 

This was just going to be 10 to 15 minutes of a notice and wonder conversation. Yea. We were wrong. Fifty minutes later we were still chatting about watermelons. Who knew a pile of cut up watermelons could keep 45 educators engrossed. Really, what is there to talk about—it’s just a bunch of watermelons, people.

Car full of watermelons

Image from: weknowmemes.com

 

So, here’s what we chatted about.

We predicted that the most common answer would be to cut up two of the one-half sized pieces to make the missing one-fourth pieces, slide those around to make 4 whole watermelons. Then, add the other two halves to make another whole watermelon. So, the sum of 4 whole watermelons and the other whole make 5 watermelons. We thought that some version of that idea would be a good start to the day.

And that’s exactly what happened. One of the campers shared her version and just about every person in the room said, “Yup, I thought about it that way, too.”

answer #1

 

And then Shannon said, “I saw it another way. I saw 4 groups of 3/4 of a watermelon and then I added the 4 one-half sized pieces. So, 3 whole watermelons and two more means that there are 5 watermelons in the picture.” The conversation then moved to connecting Shannon’s use of the algorithm to the photo and then adding in the notation.

answer #2

Since many of the campers were happy to think about the quantity of watermelons using the algorithm in Shannon’s explanation (4 x 3/4), we thought we would pause here and push on the idea of equivalent representations. [We much preferred thinking of the watermelons as 3 groups of 1—written as 3 x 4/4.]

This provoked lots of conversation on how come we can do this.

  • How does the image show (4 x 3/4) = (3 x 4/4)?
  • How does the picture show Shannon’s equation: (4 x 3/4) + (4 x 1/2) = 5?

 

The third way that surfaced was to simply add all the pieces of the watermelons visible in the photo. We recorded it like this: ¾ + ½ + ¾ + ½ +¾ + ½ + ¾ + ½

answer #3

 

Here are some of our takeaways.

If it’s important to the kids it MUST be important to us

  • We need to listen to what our kids are saying and not be on the lookout for the answer listed in the TE or for the student response that matches our preferred strategy.

Questions and their power

  • Your questions need to offer ALL kids a place in the conversation. So, in this situation we could have asked a couple of questions.
    • How many watermelons are there?
    • How many whole watermelons are there?

           Which question lets the kid who says there are 8 watermelons be in the conversation?

Dots on ten frames and Photos of watermelons, almonds, and tangram puzzles

  • Math is an active subject—it’s interesting, irritating, perplexing, confusing and invigorating. It makes your head hurt when you are in the midst of the struggle and then you get to embrace the high fives when that last piece falls into place and the connection appears as a result of your hard work.

 

Reflections on Summer Fun

The last few days spent in classes have been lovely. I knew they were going to be. Spending time in classes with kiddos and their teachers trying out some ideas that were new to my colleagues, and then chatting about what we saw and heard, and what we learned was just wonderful.

1.  Kids (and their teachers) are super interested in containers filled with stuff. It is nearly impossible to resist joining in the conversation about “how much stuff” is in the container, especially when the stuff is the lovely orange cheese balls that you can buy in the mega-container at Target.

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It offers kiddos who really, really, really struggle with math anxiety a way to join in the conversation. Everyone made guesses and built number lines and laughed and shared ideas.IMG_2161

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These are the sights and sounds of what learning should be for our students.

2.  Games provide an awesome arena for kids to explore numbers and their structure, especially the game we shared on Monday morning, What Fits Between?  We taught them how to play the game. They had to figure out how the three numbers created are related (largest number, smallest number, and the number that fits between) and organize the results.

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Then, we put the students into groups, distributed the record sheets and the sets of UNO cards, and they were off and running. We listened in on their conversations and watched to see what decisions they made about their numbers. Occasionally, we answered questions about situations that arose in their games that hadn’t in the few rounds we modeled. There were lots of conversations: kiddos asking each other for help, doing the if I had made 48 with my cards instead of 84, my number would have been in the middle and I would have won, and deciding what to do if two of the three players in a group made the same number. (They gave each player one point for the round because no one had created a number that fit between the other two.)

  • We learned that if we leave them alone to figure things out, they do. The adage, be less helpful, comes to mind.
  • Kids are able to explore ideas more deeply when they engage in conversation with classmates than when they engage, individually, with worksheets.
  • It’s great fun to have a group of teachers in a classroom when you want to try out new things especially when you are able do some planning together before hand.
  • It’s really nice to have a couple of days to work on new ideas and have time for people to talk together on each of the days.
  • Kids are amazing. If we choose the right tasks, they aren’t even nervous and anxious about the math part of their day.

3.  Using games as assessment is an informative practice. Notes and photos and conversation bring the next instructional steps to light.

  • Of the 12 kiddos playing, nine were ready to move onto playing the next level of the game where each player must build a three-digit number and determine who had created the number this fits between the other two. Three students needed additional opportunities to play the original version of the game.
  • The “top” math student got to play and interact with the ideas just like everyone else, as did the kiddos who are often not interested in math. All 12 students were invested in the game. The students who usually need to get a drink, sharpen a pencil, or use any number of distractions to help them get through math did not need to employ any of those strategies.
  • During the reflection time at the end of the class period the kids decided that they wanted to change the criteria that determined who won the game. Now, in their class, the winner is determined after each player counts up the number of rounds s/he won. The player who has the number of wins that is between the other two, wins.

How do we use the learning we saw today to support tomorrow’s instruction?

The collaborative work started when we noticed that kids were having a difficult time estimating height in an estimation warm-up. We decided to address the manner in which they worked with the concept of estimation. We selected a graphic organizer for them to use, determined that the quantity of objects for them to estimate would be less than 100, provided some boundaries to the quantity, and put the cheese balls in clear, plastic containers so the students could see them. With these adjustments the students were more comfortable in making estimates and discussing their ideas. After the team talked about the work the students did that day, it was decided our next move would be to use a series of photographs from Estimation180.com. We selected the group that asks kids to estimate the number of cheese balls on plates and trays. The students will play What Fits Between? and do some other math work.

I am looking forward to hearing how the day went, and what the team is going to tackle next.

Summer Fun

One of the highlights of the summer is being part of a project with a math team at an independent school. We are focusing on collaboration and open dialogue in instruction to support student learning. The project is structured so that there are daily opportunities to experiment with lesson design and pedagogy.

The first day I was on campus, the head teacher and I spent time in each of the math classrooms. We were interested in learning what instructional practices and teacher moves are used during mathematics instruction, and the connection of these elements to the work students were doing and the mathematical ideas they were investigating.

In one class we visited that morning, the day’s learning opened with a question from estimation180.com. The students were asked to determine Mr. Kraft’s height. As we listened in on the conversation we noticed the students’ guesses about Mr. Kraft’s height were very random. The explanations contained little comparative thinking or relational thinking. This notice was a priority in our team planning conversation.

During our planning conversation, we determined the students were more confident with estimating quantities of objects rather than the measurement situation they were asked to investigate using a photograph from the website. They are working on how to use given information to create reasonable estimates and are building deeper understanding of magnitude.

So, here’s our plan for Monday.

PART 1: Estimation–how many plastic ninjas are in the middle container?

The three containers shown in the photo will be on the front table. Students will estimate how many plastic ninjas are in the middle container.

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They will record the information about the containers and their estimate in a specific way. The number of plastic ninjas in container 1 and container 3 are written in the boxes. The estimate of the number of plastic ninjas goes on the number line.

Screen Shot 2016-06-26 at 10.08.50 PM(The number line is adapted from Graham Fletcher’s 3-Act Task recording sheet.)

The students will then discuss the information they used to create their estimates.

PART 2: Estimation--how many cheese balls are in the middle container?

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The second round of estimation uses cheese balls from the mega-size container of cheese balls from Target. There will be 10 cheese balls in container #1 and 75 cheese balls in container #3. Students will estimate how many cheese balls are in container #2.

They will record the information about the containers and their estimates in the same way they did for the plastic ninjas. The number of cheese balls in container 1 and container 3 are written in the boxes. The estimate of the number of cheese balls in container 2 goes on the number line. A class discussion on the information used to create the estimate will wrap up this part of the day’s work. 

Part 3:  We’ll play an adaptation of a game from Box Cars and One-eyed Jacks called What fits between? 

In this game, students work in groups of three. Each player takes two cards from the stack of UNO cards (only contains cards 1 -9), placed face down, and creates a two-digit number. When all three players in the group have created their two-digit numbers, they each lay down their cards and read their number. As a group, they decide who has created the largest number and who has the smallest number. The player who has created the number that fits between the largest and the smallest numbers earns the point for that round. The player who has the most points at the end of the work time, wins the game.

We will end this part of class with a conversation about how they decided to create the numbers in each round, and what they noticed about the game.

Final Thoughts: We are exploring how the use of a specific structure in the lesson supports students as they build a more comprehensive sense of number. Offering the “too low” and “too high” in an estimation task is intended to provide some reference points for the students to use as they create an estimate. The plan is to use this structure and its variations, and then gradually remove the two estimates. Our goal is to ensure that students can construct a sense of the quantity that is represented by the written form of a number and to use this sense of quantity in other situations. We liken it to what happens when we read a word and we create an image. We want that to happen for our students when they are working with number.

I can’t wait for school to start this morning.

 

 

The marvelousness that is the #MTBoS

Because of #MTBoS, I met @AlexOverwijk at Twitter Math Camp in Jenks, OK in July 2014. One afternoon, Alex, @gwaddellnvhs, and I spent time talking about Lesson Study.

Because of #MTBoS, I am on my way to Ottawa, Ontario to spend the next two days participating in a Lesson Study at Glebe Collegiate Institute; otherwise known as Alex’s school.

Lesson Study offers educators an incredible opportunity to learn from and with other educators. The collaboration, the conversation about the important work we do, the opportunity to explore ideas and try stuff out and see what happens, and to examine how all of the layers, pieces, and components of instruction move students’ learning during a lesson is something more of us should have the chance to do.

It’s going to be AWESOME.

 

 

 

containers and estimation

Quite awhile ago, I facilitated a day-long workshop for grade 2 teachers. During the morning break, I had the opportunity to chat with one of the participants about the work she was doing with her students. She talked about a routine she was using to open up conversations in her math class.

Fast forward about 10 months.

While looking through my photos, I came across the set of pictures I had taken with the intention of writing up this post. The following is my version of the really nice task she shared.

Step 1:  Share this stack of containers.

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Ask the students what they notice. Record their ideas. Next, ask them to share their wonderings and record their ideas.

Step 2:  Tell them that the bottom container holds 5 crayons and the top container has 32 crayons. Ask them to think about how many crayons might be in the middle container, and prepare a statement that would convince the group that their idea could be true.

Step 3:  Have them share their idea with one other person; a test run to practice their statements before sharing with the whole class.

Step 4:  Ask the class, “Who’s partner had something interesting to say?” Step back and listen. (Totally stole that line from Cathy Fosnot–she uses it so strategically and magically in her workshops to create great conversations.)

Step 5:  Ask for some more ideas. Record the quantities the students share.

I am taking a teacher time out here.

A teacher time out is nice strategy we’ve used in Lesson Study. It is a coaching move that Elham Kazemi has talked about. The purpose of the time out is to literally take time, right in the middle of class, to reflect and decide on the next set of teacher moves you are going to use. It is incredibly powerful to have the option to talk in the moment rather than saying after the lesson is over, “I wish I had…” 

The intent of the teacher time out I inserted here (much, much later, long after the lesson was over) is to consider the idea that I am currently wrestling with: what to pursue. This idea is courtesy of Tracy Zager Johnson’s blog post Which Mistake to Pursue? Looking back on this conversation with the second graders, I want to think about how to pursue the ideas of quantity and magnitude.

How formal should this be?

Should we just open up the middle container and count them because this is the first time we have used this structure to pursue the idea of how quantities relate to each other? (The thinking behind selecting how many crayons were in each container was to be sure and use quantities of crayons kids could easily imagine.)

Do we have a sequence of teacher moves ready, like those listed below, and see what the kids think and say and do, and trust them to lead the conversation?

Have all kids record, on post-it notes, the quantity they each believe might be in the middle container and stick them up on the board. Talk about the estimates.

Talk about the number of crayons that couldn’t be in the middle container and why that could be true.

Have a few more students share what his/her partner said, and chat about the possibilities.

Have your kiddos do a short writing piece that captures the ideas explored so far:  If you had to choose one of the amounts written on the board (and it can’t be yours), which one you would choose? Why do you think that could be the amount in the middle container?

Maybe I am over-thinking this whole deal and should just enjoy the conversation. (Because when you start to explore all of the possibilities as to how and where the conversation might go, it can get crazy and it starts to take on a life of its own.)

Step 6:  What I actually did was collect a few more estimates, open up the middle container, and count the number of crayons. Lots of ohhing! and ahhing! and groaning! and I was so close!  A very nice way to move into the next part of the day’s math work.

 

For those of you who want to check your estimate, the number of crayons in the middle container was:
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A task for a Wednesday morning: nine and three-fourths

Nine and three-fourths is another task on my list of favorites. Students are always intrigued with it, as are colleagues who tackle it in workshop settings. It has been one of my go-to tasks for quite awhile. I do have to be honest and say that I have no idea where I found it. However, I want to extend my thanks to the author, as well as, my sincere apologies for losing track of whose work it is. It is an awesome task.

It is an awesome task because everyone can play. Everyone can join in the conversation and the learning. Everyone has ideas to investigate, think about, and share with others.

This past June, more than forty K – 5 teachers spent some time with this task during a five-day summer math camp. We used nine and three-fourths to kick off our two-day conversation centered around the big ideas in fractions.

WEDNESDAY MORNING’S TASK

Choose one or more pattern blocks to represent one unit. Based on the unit you selected, create a picture that is worth 9 and three-fourths units.

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  • Identify the unit you selected
  • Trace and color your picture
  • Justify that your picture does have a value of 9 and three-fourths units

The morning session began with an exploration of the patterns blocks and the relationships that exist among the various blocks in the set. The set of pattern blocks we used included all of the blocks shown, as well as, the brown trapezoids from the add-on set. We talked about the task itself; which including a discussion of an example of a picture worth nine and three-fourths, when the yellow hexagon is worth 1. The picture was comprised of nine yellow hexagons and 3 brown trapezoids (not shown in the picture above). We agreed that we would not use the yellow hexagon, or an equivalent, as the unit. And they were off; working, building, talking, re-starting, recording, and considering how to share this with students, and what aspect of the task should be used to begin the class discussion. At the conclusion of the morning’s work time, a few of the workshop participants presented their pictures.

The following are representative of the pieces that were shared.

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Using the 5 Practices for Orchestrating Productive Mathematical Discussion by Margaret Smith and Mary Kay Stein to build our conversation, we talked about the unit each picture was based on, how three-fourths was represented in each drawing, how and why each picture met the criteria of the task, and how each unit was selected. We discussed the teacher moves used throughout the morning session, the impact they had on the learning, and what adjustments might need to be made when sharing the task with students. The morning session concluded with a conversation focused on how the task provides opportunities to discuss and tackle some of the misconceptions students have about the big ideas of fractions.

It was a great way to spend a Wednesday morning.

Race to 20–part 2

As I said in the previous post (which was much longer ago than intended), Race to 20 is one of my favorite activities. Everybody can play. Everybody can talk about what they tried. Everybody can share what they noticed. I like the everybody piece of the activity.

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Here’s a typical sequence I use for rolling out Race to 20:

  1. Give a brief overview of the game: The goal is to be the person who takes the last counter from the pile.
  2. Demonstrate the game: I like to use a document camera so that everyone can see how the game is played. Ask a student ahead of time to be part of the demonstration. I make sure to give him/her a chance to play the game with me, once or twice, before we demonstrate the game in front of everyone.
  3. Review the directions: Have the directions listed on a keynote or powerpoint slide, on a piece of butcher paper or chart paper posted up on the wall, or written on a white board or on an interactive white board.
  4. Demonstrate the game again
  5. Take questions, but do not discuss strategy
  6. Make sure everyone knows where the directions are posted
  7. Create pairs, pass out the counters, and set the timer for 5 or 6 minutes
  8. Wander and roam: I want to listen in on conversations and see how the groups are progressing.
  9. When the timer goes off, ask what did you notice: This question is asked to give everyone the opportunity to hear what other groups are thinking, what ideas have been tried, and what they are speculating about or thinking of trying. The purpose of this conversation is NOT to share your answer to this is what you need to do every time so you can take the last counter in the pile.
  10. Change groups: One partner stands up and shifts over two or three groups. I want the groups to share ideas and try new things. Changing groups helps. Set the timer for another 4 or 5 minutes.
  11. Wander and roam: This provides the opportunity to monitor how thinking is developing, who is collaborating, what strategies are being tried, and how frustration is being handled (that perseverance piece).
  12. Debrief: When the timer rings, the group chats again about what they notice and what they are wondering.
  13. Change groups again: The partner who didn’t move the first time, moves to a new group, and they are off for another 3 to 4 minute work session.
  14. Debrief: Ask a pair to come up to the document camera and play. I usually check in with a couple of groups during the work session and ask if they are willing to play a round of Race to 20 in front of everyone. I let the group or groups know who I will ask to play, prior to the timer ringing. Being able to watch another group play, after you have had some time to try out ideas is really important. Sometimes, I ask a second group to play, especially if I have quite a few groups who are stuck on one idea. It provides the observers the opportunity to focus on strategy and ideas, rather than their next move. After the round is over, ask the players to talk about what they did. Be sure and ask them to discuss how and why their decisions helped them, or didn’t help them, be the player to take the last counter from the pile.
  15. Return to the original pairs: Play for a few more minutes. It is important for everyone to able to try ideas out right away.
  16. Journal or exit card: Everyone records their current thinking to the question what is your strategy so that you will be able to take the last counter every time you play.
  17. Talk about it again tomorrow: I can pretty much guarantee that your kids will go home and play, look up information, and be ready take on the challenge of being able to take the last counter, every time.

So, now decide how you are going to handle tomorrow. 

Sometimes, we split into two groups; the kids who are ready to see if they have a strategy that will let them take the last counter every time, and the kids who need more time to try out ideas.

Eventually, we come to the conclusion that we need a new game because too many of us (read everybody) know how Race to 20 works.

Race to 20

One of my most favorite activities to use with groups is Race to 20. I have no idea if that is the real name of the game, or when or where I first ran across it. However, what I do know after using it for a really long time and with lots of groups of kiddos and grownups, is that it has an appeal that is pretty much about impossible to resist. I am pretty sure, like about 100%, it belongs to the family of NIM games.

Here’s the set up:

  • You need a partner and you need 20 counters. (unifix cubes, two-color counters, pebbles, square tiles, etc.)
  • Set the counters in a pile in the middle of the table

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  • The object of the game is to be the person who takes the last counter from the pile
  • Each person may remove one or two counters from the pile on each turn

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  • So, Player 1 takes one or two counters
  • Then, Player 2 takes one or two counters

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  • And, Player 1 takes his/her turn
  • Then, Player 2 takes one or two counters

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  • And so on, until all the counters have been taken
  • The person who took the last counter wins that round
  • Play again

The goal is to figure out the strategy so that you can win every time.

Give it a try.

Which One Doesn’t Belong (again)

This is a good week.

No.This is a GREAT week.

A colleague and I have the privilege of spending four days with four groups of teachers: Day 1 with TK and Kindergarten, Day 2 with grades 1 and 2, Day 3 with grades 3, 4, and 5, and Day 4 with MS math teachers (grades 6 – 8). Pretty nice.

The opening task of each day is Which One Doesn’t Belong? It’s a pretty nice way to start the day. We show a photo, similar to this one we used with grades 1 and 2.

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We ask everyone to take a moment to think about and prepare a response to the following questions:

  1. Which domino doesn’t belong? Why?
  2. Find a second or third reason why the domino doesn’t belong?
  3. Find a reason why each of the dominoes would not belong.

Then, everyone has the opportunity to share their ideas in small groups. After a bit, the entire group chats about the reasons why each domino does not belong.

Some more good stuff happens as teachers make their own WODB that can they use next week–which for us in this part of Southern California–is the FIRST week of school. The conversations really gets going: ideas are shared, drawing, planning, arguing, advocating for ideas. Pretty nice.

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We do a Gallery Walk. After, the groups retrieve their posters, review the reasons that were noted on their posters, and make any revisions they want to based on the feedback shared during the gallery walk.

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The best part happens now. We talk about how Which One Doesn’t Belong? is good for kids. We talk about how this is a great opportunity for kids to share ideas, to listen to each other, to talk with each other, to build language skills, and develop ideas about reasoning. We talk and laugh and think. PERFECTION!

Our conversation moves on to discussing the Standards of Mathematical Practice, specifically SMP #3: Create viable arguments and critique the reasoning of others. On Monday, we had the usual teacher talk about how WODB supports Standard of Mathematical Practice #3. On Tuesday, we had a different conversation because Christopher Danielson posted this statement on Twitter:

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Tuesday’s conversation was different. More serious. More focused on our responsibility to help our students develop good habits to use while they work on interesting and perplexing questions, scenarios, tasks, and problems.

Thank you, Christopher, on behalf of our kiddos.

Which One Doesn’t Belong?

Summer Math Camp = Five Days +  Educators + a Focus on Learning

We spent quite a bit of time considering the underpinnings of the frame for the five-day program. After considerable deliberation, we determined the structures we would use to engage the campers in thinking about teaching and learning would be exactly what we want classrooms to be built upon:

  • Building Community
  • Promoting Conversation
  • Encouraging Exploration

We then moved to the next step in our planning: THE HOW?

How do we accomplish this? How do we make it a transparent component of the five days? How do we start camp? The how was following by and connected to the what. What opening task would we choose and how will it support our underlying structures?

The Question: What do you choose to be the opening task?

The Answer: Which One Doesn’t Belong?

We started with this image from Christopher Danielson’s book, Which One Doesn’t Belong? and asked everyone to take a moment to consider which shape doesn’t belong and why.

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The educators then thought about these two statements:

  1. Provide more than one reason that the shape you chose might not belong
  2. Offer a reason why each of the shapes might not belong

We asked that they ensure that each person in their group had enough time to ponder, notice, and formulate ideas prior to engaging in a group conversation.

The Result: LOTS of conversation

We then showed the next image, from the website Which One Doesn’t Belong.Screen Shot 2015-07-17 at 9.19.36 AM

After the table conversations were concluded and a brief whole group discussion was had, chart paper and markers were distributed, and directions to create a WODB that your students could discuss in the opening week of school were posted. The groups set to work creating their posters.

The energy and the conversations the groups generated were amazing. We heard consideration about number size, entry points for all kids, WODB as a formative assessment tool for the Standards for Mathematical Practice, specifically practice #3, and all kinds of important stuff.

After the charts were finished, they were posted around the room.

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Each camper grabbed some post-it notes, selected a chart that was interesting, and off they went on a gallery walk. We used the following protocol for the gallery walk.

  • Take a moment and individually think about the options displayed on each chart
  • Decide which one doesn’t belong
  • When all members of your group are ready, share ideas and explanations
  • Select one of the ideas and the explanation and record it on the wrong side of a post-it note, and attach it to the chart

A side note: If the ideas are written on the wrong side of the post-it note, groups are not influenced by the ideas already shared. If a group wants to check out what others have written, they can always check after posting their idea. The team that developed the chart will have a larger set of original ideas, not just variations on a theme.

WODB.gallerywalk

At the end of the gallery walk, the team retrieved their chart, discussed the information on the post-it notes, and made revisions, as needed.

The Opening Task debrief encompassed the following topics:

  • general thoughts
  • feedback from the gallery walk
  • instructional moves that were used
  • the critical idea of providing an entry point for a task
  • inclusion of all kiddos into the conversation

What I learned:

  1. It is really hard to resist the question “which one doesn’t belong?” even at 8:15 am on a lovely summer morning.
  2. Gallery walks need to be purposefully structured. When organized as such, they become a strong instructional tool that promote mathematical conversations.
  3. When the teacher participates, students emulate the academic modeling.
  4. The closing structure of the opening task needed to be stronger, more focused on the learning and the learner. It needed something similar to the question written by Nicora Placa her blog post on Bridging the Gap.

Doing mathematics: Write a about the activity from the perspective of a learner. Think about the learning processes. What helped you as a learner? What helped you sort out the mathematics?

Over the course of the next two weeks, I’ll be using a similar version of the opening task with a few groups. I am interested in how the use of a different closing reflection might impact the learning and the learners.

Resources: 

  1. Talking Math With Your Kids–a blog hosted by Christopher Danielson
  2. Which One Doesn’t Belong–a blog hosted by Mary Bourassa
  3. Bridging The Gap–a blog hosted by Nicora Placa