Intentional Communities Learning Together to Lead Productive Mathematical Discussions

Elham Kazemi University of Washington

ekazemi@uw.edu

Team Work

Magdalene Lampert Megan Franke Hala Ghousseini Heather Beasley Angela Turrou Adrian Cunard Allison Hintz Megan Kelley-Petersen Helen Thouless Lynsey Gibbons Bryan Street Teresa Lind

Laura Mah Anita Lenges Becca Lewis Liz Hartmann Leslie Nielsen Emily Shahan Elizabeth Dutro Morva McDonald Core Practices Consortium Ryan Reilly Jessica Calabrese Staff at Lakeridge Elementary School ECML Colleagues

Children have amazing ways of solving problems. Their often surprising responses make teaching fascinating work

Knowing what to do with students’ ideas and teaching children to meaningfully participate in discussion can be invigorating and challenging

Flow

¤  Principles that support discussion

¤  Different goals, different discussion

¤  Planning and facilitating open and targeted talk

¤  How to learn together to lead productive discussions

Principles that support discussion

¤  Students need to be supported to know what to share and how so their ideas are heard and useful to others

¤  Teachers need to orient students to one another and the mathematical ideas so that every member of the class is involved in achieving the mathematical goal

¤  Teachers must communicate that all children are sensemakers, knowledge generator, invested in school and wanting to succeed. Our curiosity communicates to children that their contributions are valued.

Intentionally orchestrating discussion to meet a mathematical goal

¤  Goal acts as your compass

¤  Helps decide what to listen for, which ideas to pursue, and which to highlight

¤  Distinguishing two types of discussions ¤  Open Strategy Sharing

¤  Targeted Sharing

Open Strategy Share Elicit many different ideas to see the range of possibilities

Generate many different ideas

Building students’ repertoire of strategies

Students listen for and contribute in different ways to solve the same problem

Move across a broad terrain of concepts, procedures, representations, explanations

“Can you explain how you solved?” “Did anybody solve in a different way?”

Ms. Lind’s 4th grade classroom

25 X 18

As we read thru this transcript on the following slides… think about what makes this an open strategy share?

Open Strategy Share

Ms. Lind: OK, Faduma, tell us about what you wrote as you figured out this solution. I want everyone else to think about whether you are understanding what Faduma did and if you used a similar or different strategy.

Faduma: Since I can multiply numbers by 10, I broke up the 18 to a 10 and a 8. I multiplied 25 x 10 and 25 x 8. I got 250 + 200 which is 450.

Ms. Lind: Thank you. I’ve written on the board what I heard Faduma say. And you’re showing me that many of you did the same thing. Who can add on to help us explain why we would split the 18 the way Faduma did?

Jordan: Well, it’s like Faduma said, multiplying by 10 can be easier to do. So since one way of thinking about 25 x 18 is that you have 25, 18 times, you can first do 25 ten times and then you have 8 more 25s.

Ms. Lind: Does anyone have any questions for Faduma?

Marcus: I do. I kind of solved it the same way but I got a different answer. Eighteen is close to 20 so I did 20 x 25 to get 500 but then I subtracted 2 to get to 498. I’m not sure why our answers are different.

Ms. Lind: So you’re really trying to make sense of Faduma’s strategy through your own way. I’m writing your question up here but before we take up your question, let’s see if we can put one more strategy up here and maybe that will help us think about what is going on here.

Celia: I used what I know about quarters. 4 quarters make a $1.00. So 16 makes 400 and then two more makes 450.

Ms. Lind: What do you think Celia means when she says that quarters helped her solve the problem? And if you’re not sure, you can ask her to repeat what she said.

(several students repeat Celia’s idea)

Ms. Lind: We seem to have three different strategies and two different answers. Could you turn and talk to your elbow partner about which strategies convince you and what questions you have?

Open Strategy Share

Select the problem

Anticipate how students will solve

Pose the problem and monitor students at work

Elicit and discuss a range of solutions

Solution 1 Solution 2 Solution 3

Solution 4

Targeted Sharing

A range of possible solutions

Solution 1 Solution 2 Solution 3 Solution 4

Targeted Sharing

A range of possible solutions

Solution 1 Solution 2 Solution 3 Solution 4 Solution 1

Targeted Sharing

A range of possible solutions

Solution 1 Solution 2 Solution 3 Solution 4

Targeted Sharing

?

A range of possible solutions

Solution 1 Solution 2 Solution 3 Solution 4

Targeted Sharing Discussion Structures

¤  Compare and Connect: to compare similarities and differences among strategies

¤  Why? Let’s Justify: to generate justifications for why a particular strategy works

¤  What’s Best and Why?: to determine a best (most efficient) solution in particular circumstances

¤  Define and Clarify: to define and discuss appropriate ways to use mathematical models, tools, vocabulary, or notation

¤  Troubleshoot and Revise: to reason through which strategy produces a correct solution or figuring out where a strategy went awry

Kazemi & Hintz (2014) Intentional Talk: How to Structure and Lead Productive Mathematical Discussions.

Portland, ME: Stenhouse Publishers.

Targeted discussion

Ms. Lind’s 4th grade classroom

Focus on the solution of breaking apart into tens and ones and justify with the array – Why? Let’s Justify

OSS 25 X 18

Focus on the solution of rounding one number and adjusting the product - Troubleshoot and Revise

¤  How are the targeted discussions different from open strategy share discussion?

¤  What goal does the teacher seem to be working toward?

Why? Let’s Justify Ms. Lind: Yesterday as we were listening to people solve 25 x 18, I realized it has been awhile since we worked with arrays. I thought it would be useful to explain what is happening when we break apart numbers to make a problem easier and how to make sure we’ve accounted for 18 groups of 25. Mark up the array in your journal to show how it matches this numerical strategy

Ms. Lind: Celeste has an idea about the array to offer us, please look up to the screen at this drawing of the array for 25 x 18. I want you to see if you can make sense of how she divided up the array. Celeste: I drew Faduma’s idea. She kept the 25 whole and she broke the 18 into 10 and 8. So 25 X 18 is the same as 25 X 10 and 25 X 8. Ms. Lind: Who can use Celeste’s drawing to justify why Faduma’s strategy works?

Troubleshoot & Revise

Marcus: I did 20 x 25 to get 500 and then I subtracted 2 to get 498. I kind of think it should work because 20 is just two more than 18. But I’m not sure why I’m not getting the same answer as Faduma. I think I should be.

Ms. Lind: Marcus had a great way of beginning this problem. By changing the 18 to 20, he started by making the problem easier for himself. It might help us to put these numbers into a story. Let’s imagine Marcus had 25 packs of colored pencils with 18 pencils in each pack.

Andre: Oh I see, when Marcus changed the numbers to 20 x 25, it made it like there were 25 packs of pencils with 20 pencils in each pack. But, we need to have 18 pencils in each pack, not 20!

Ms. Lind: Ok, let’s draw the 25 packs of pencils with 20 pencils in each pack.

Ms. Lind: So, what needs to be removed from each pack to go back to having 18 pencils in a pack instead of 20?

Ms. Lind: Ok we have worked together to figure out why Marcus’s answer was different. Who can say again why his answer was different?

Targeted Discussion

Focal Strategy

What mathematical strategy or idea are we targeting in our discussion?

What is the explanation I want the students to come up with? OR what is the mathematical confusion we want to bring understanding to?

Different goal, different plan

Learning together

What makes for good contexts for adult learning?

Principles of learning

¤  Teaching is highly complex and relational. It requires us to communicate to our students that school is a place where they are known, where they can invest themselves, and where they belong.

¤  Teaching is intellectual work and requires specialized knowledge

¤  Teaching is something that can be learned

¤  Learning to do something requires repeated opportunities to practice

¤  There is value in making teaching public  

Supporting teachers and students in your setting

As you think about your own setting

¤  How are you thinking about planning intentionally ¤  Identify a goal

¤  Use planning templates to imagine the orchestration of the talk

¤  When are different types of discussions useful to students? ¤  Open strategy share

¤  Targeted share

Collaborative classroom visits

¤  A group of people can get together, create or use a common plan and think about something they’d like to try out or learn as a few volunteers initiate the lesson.

Cycles of investigating, planning, trying, and learning

Visiting classrooms together

¤  When we choose and plan together, “our lesson,” we get to try it out with our students together.

¤  Often there are insights we gain by listening to children. Or we have questions or new ideas about questions to ask, or things to try. During the lesson, we can call “teacher time outs” in order to pause instruction briefly and chat with each other about a next step to take, a question to ask, or a way to steer the instruction.

¤  We work together through a learning cycle

What are teacher time outs?

Teacher Time Out (TTO) is enacted when educators co-engineer lessons with students present, pausing regularly within the lesson to think aloud, share decision making with one another and determine where to steer instruction. Educators signal that they have a question to ask one another (e.g., “Should I ask x question next or y?” “Is this a good time to try to represent this students’ ideas?”) or a question they want to ask the students (e.g., “Wait, let’s ask students about …”) or to narrate their instructional decision making (e.g., “I think I’d like to pursue this mathematical idea next with students”). We have found these exchanges open up rich ground for making sense of practice together in the moment.

¤ Pause  the  lesson  to:    ¤ make  sense  of  students’  thinking  ¤ make  sugges4ons  to  the  group  for  considera4on  ¤ narrate  instruc4onal  decision-­‐making  ¤ ask  for  help  for  next  move  

¤ Jump  in  to  ask  students  a  ques4on  

¤ Take  the  pen  ¤ Model  the  mathema4cal  idea  being  discussed  

A starting list of norms for working together

¤  Be willing to take risks with new ideas

¤  Listen actively and generously

¤  Build on others ideas and invite others to participate

¤  Give each other time to think & process ideas

¤  It’s okay to share ideas in progress and revise your thinking

¤  Use specific language to describe what we see students doing, rather than labeling students. Avoid labels such as “low” and “high.”

¤  Treat the lessons and students that we work on together as “our lessons” and “our students”

¤  Cultivate joy for teaching & for working together

An example of a teacher time out

¤  It’s helpful if the teacher who volunteers to take the lead initiates the first teacher time outs. These are not to be read as corrections or evaluations. Remember, we have designed “our lesson.” Instead, these time outs are opportunities to put our heads together as we try to bring to life the lesson we planned.

Introducing teacher time out to children

Context: Fair Sharing Problem 3 chocolate bars shared equally among 4 people

What equations would students right to connect to these situation? Will they write equations that match the original problem? Or their solution?

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Teacher Education by Design: Tools for Learning

Tools for teachers & teacher educations: tedd.org

Thank you!

Elham Kazemi

Email: ekazemi@uw.edu

Twitter: @ekazemi

Allison Hintz

Email: ahintz@uwb.edu

Twitter: @allisonhintz124