Making Sense of Mathematics for Teaching Grades K-2

Copyright © 2017 by Solution Tree Press

Materials appearing here are copyrighted. With one exception, all rights are reserved. Readers may reproduce only those pages marked “Reproducible.” Otherwise, no part of this book may be reproduced or transmitted in any form or by any means (electronic, photocopying, recording, or otherwise) without prior written permission of the publisher.

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Solution Tree Press

President: Douglas M. Rife

Editorial Director: Tonya Maddox Cupp

Managing Production Editor: Caroline Weiss

Senior Editor: Amy Rubenstein

Media Editor : Peggie Howard

Copy Chief: Sarah Payne-Mills

Copy Editor: Ashante K. Thomas

Proofreader: Elisabeth Abrams

Cover and Text Designer : Abigail Bowen

Compositor: Laura Cox

Editorial Assistant: Jessi Finn

Writers: Juli K. Dixon, Edward C. Nolan, Thomasenia Lott Adams, Lisa A. Brooks, Tashana D. Howse

Notes to the Facilitator

This workshop provides participants with the opportunity to engage in mathematics as learners in order to make sense of that mathematics for teaching. This workshop is intended to support teachers and teacher leaders to situate this knowledge within learning progressions, classrooms, and the TQE (tasks, questioning, and evidence) process. The TQE process describes three key aspects of the teacher’s role: (1) selecting, adapting, or creating worthwhile tasks to address a specific learning goal; (2) using targeted and appropriate questioning strategies to support all learners; and (3) collecting evidence of students’ conceptual understanding and misconceptions related to the learning goal.

This workshop addresses three specific goals.

1. To explore meaningful tasks as learners of mathematics for teaching

2. To make sense of the TQE process

3. To create a shared vision of classrooms where teachers are supporting the TQE process and students are engaged in meaningful mathematics learning experiences

These goals are accomplished through the use of challenging tasks for teachers, questions related to using those tasks in classrooms, and classroom video where the tasks are modeled during instruction. Series authors introduce classroom video by highlighting important aspects of this workshop specifically and making sense of mathematics for teaching in general.

The workshop is divided into seven segments, and the corresponding video clip titles are listed in parentheses.

1. Welcome and Opening: This segment describes the series’ organization around a common structure—The Challenge, The Progression, The Mathematics, The Classroom, and The Response. Participants discuss the importance of making sense of mathematics for teaching in their own teaching and related roles.

2. Collaborating in Planning (Representing Two-Digit Numbers): In this segment, participants discuss collaboration’s importance in the planning process while exploring multiple ways to represent two-digit numbers. They will learn how teachers learn from one another in their collaborative teams, sharing ideas and strategies to best meet their students’ needs. Participants discuss how the teacher should ask students questions rather than giving away the answers. The example of making sense of two-digit numbers is used as a topic for collaborative planning. Participants discuss their responses to the task of representing the number 24 and how they might support it during instruction. A video of students engaging in the same task follows this discussion. Participants debrief regarding what they expected to see in the classroom episode and what actually occurred.

3. Using the TQE Process (Solving a Word Problem Where the Change Is Unknown): This segment focuses on using strategies to solve a word problem. Participants anticipate how students might solve a word problem representing a join (change unknown) problem. This problem represents a task that many students find challenging. Participants discuss how they expect students to solve the problem, and they anticipate questions the teacher might ask to elicit students’ strategies. They predict the type of evidence the teacher might collect from using questioning related to the task. Participants then watch the video of a Making Sense of Mathematics for Teaching series’ author as she defines and shares the importance of the TQE process. The participants go on to watch the classroom video using the same task they just explored. Discussion is supported following the video connecting what the participants observed to the TQE process and the importance of planning for the TQE process.

4. Creating a Shared Vision of Classroom Instruction (Determining Number Pairs to Ten): This segment starts with participants viewing an author video discussing the importance of creating a shared vision of classroom discussion. Participants then anticipate how kindergarten students might respond to a task that requires them to determine number pairs for 10.

5. Questioning to Facilitate Learning (Using the Make-a-Ten Strategy With Addition): This segment provides participants with the opportunity to plan for questioning to support students to engage in strategy use. Once participants plan for how to use a task during instruction, they compare the questions they prepared to those used in the classroom. Participants connect the questions asked to the evidence collected as part of the TQE process.

6. Preparing for the Mathematics to Come (Comparing Plane Shapes): This segment examines describing and comparing shapes. Participants explore supporting student thinking in kindergarten and how that work translates to supporting student thinking in grade 3. Participants discuss how the learning experiences are related and fit into a learning progression that crosses grades.

7. Closing: In this segment, participants reflect on what they have learned in the workshop and plan for next steps.

Conducting the Workshop

This workshop is designed to last about eight hours. It can be scheduled for a single day consisting of two sessions or be scheduled over two days. All the professional development materials you need to conduct this workshop—the facilitator’s guide with detailed teaching suggestions, and the video resources— are provided in this package.

To conduct a successful learning event, please consider the important issues that follow.

„ Preparation: Please view the entire video program, read all materials, and complete all activities yourself before leading the workshop.

„ Location: The workshop should take place in an area that is large enough for individual, team, and whole-group work.

„ Equipment: You will need a DVD player, a projector, and one or more monitors. You will also need a computer projector to show PowerPoint slides.

„ Reproducible handouts and PowerPoint presentation: Reproducible handouts are included with this guide (starting on page 16 and on the CD). The handouts should be duplicated before the workshop begins and be distributed to participants according to the workshop instructions. A PowerPoint presentation is also included (on the CD).

„ Additional equipment: You will also need flip charts, chalkboards, or whiteboards and appropriate writing materials (blank paper and markers, pens, or pencils) to conduct the workshop.

„ Refreshments: The workshop’s agenda should include one or more breaks at which beverages are offered. Snacks and lunch are optional, but water should be available throughout the workshop.

Video Program

This workshop incorporates a video program that is approximately thirty-eight minutes in length. The video will be used throughout the workshop. It is used to introduce some segments and to close others. Most often, the video provides both a suggestion for what to look for during planning and instruction as explained by authors as well as an example of what model instruction looks like in the classroom through authentic classroom video. It might be helpful to replay certain video portions for participants and also to pause so participants can reflect on what they have seen.

Other Resources

Print

Making Sense of Mathematics for Teaching Grades 3–5

Making Sense of Mathematics for Teaching Grades 6–8

Making Sense of Mathematics for Teaching High School

Video

Making Sense of Mathematics for Teaching Grades 3–5: The TQE Process

Making Sense of Mathematics for Teaching Grades 6–8: The TQE Process

Making Sense of Mathematics for Teaching High School: The TQE Process

Workshop Overview at a Glance

15–30

45–60

60–90

Welcome and Opening

Collaborating in Planning (Representing TwoDigit Numbers)

Using the TQE Process (Solving a Word Problem Where the Change Is Unknown)

Slides 1–2

Slides 3–6

“Task 1: Collaborating in Planning”

“Classroom Video Analysis Tool”

Slides 7–17

“Clock Face”

“Task 2: Using the TQE Process”

“TQE Analysis Tool”

45–60

Creating a Shared Vision of Classroom Instruction (Determining Number Pairs to Ten)

60–90

Questioning to Facilitate Learning (Using the Make-a-Ten Strategy With Addition)

Slides 18–22

“Task 3: Creating a Shared Vision of Classroom Instruction”

“Classroom Video Analysis Tool”

Slides 23–28

“Task 4: Questioning to Facilitate Learning— Using Strategies to Add”

“Classroom Video Analysis Tool”

90–120

Preparing for the Mathematics to Come (Comparing Plane Shapes)

30–45 Closing

Slides 29–38

“Task 5: Preparing for the Mathematics to Come” “Classroom Video Analysis Tool”

Slides 39–41

Workshop Steps and Teaching Suggestions: Statement of Purpose

It is important for participants to engage in productive struggle around high-cognitive-demand tasks as learners, and then to plan for how those same tasks might be supported during instruction through the use of appropriate questioning as teachers. Use the videos to explore how evidence can be collected as part of the formative assessment process during instruction.

Learning Objectives

After viewing the video and participating in the workshop activities, participants will be able to:

„ Identify high-cognitive-demand tasks

„ Adapt existing low-cognitive-demand tasks to make them high cognitive demand

„ Plan questions to support students to engage in productive struggle around high-cognitivedemand tasks

„ Collect evidence of student conceptions and misconceptions through productive questioning while students work to solve tasks

Program Overview

The video associated with this workshop offers perspectives from Juli K. Dixon, Edward C. Nolan, Thomasenia Lott Adams, and Tashana D. Howse—all authors of Making Sense of Mathematics for Teaching Grades K–2 —related to mathematics content knowledge for teaching, the TQE process, and progressions for learning mathematics. Juli, Edward, and Thomasenia model best practices through authentic classroom video to create a shared image of students engaged in meaningful experiences related to mathematics learning.

Materials

„ Video program: Making Sense of Mathematics for Teaching Grades K–2: The TQE Process

„ PowerPoint presentation and slides

„ Reproducible handouts:

‹ “Task 1: Collaborating in Planning”

‹ “Classroom Video Analysis Tool”

‹ “Clock Face”

‹ “Task 2: Using the TQE Process”

‹ “TQE Analysis Tool”

‹ “Task 3: Creating a Shared Vision of Classroom Instruction”

‹ “Task 4: Questioning to Facilitate Learning—Using Strategies to Add”

‹ “Task 5: Preparing for the Mathematics to Come”

„ Flip charts, chalkboards, or whiteboards and appropriate writing materials

Activities

It is best to follow the activities as outlined in the workshop teaching suggestions and pause the video when prompted. After showing each video segment, allow participants time to comment or ask questions about the material. If requested, you can replay video portions as participants consider the questions and activities.

Welcome and Opening

1. Welcome participants to the workshop, and introduce yourself and anyone else serving as a workshop host, cofacilitator, or organizer.

2. If participants do not know one another well, conduct an icebreaker. Ask participants to form pairs and interview each other for about five minutes. Then ask the pairs to introduce each other to another pair, stating the person’s name, something interesting or different about the person, and what the person hopes to gain from the workshop.

3. Begin the workshop showing PowerPoint slide 1, then begin the DVD as Juli Dixon discusses the need for making sense of mathematics for teaching. Juli describes how the text is organized as well as how teachers experience mathematics tasks within the text as learners and then as facilitators of mathematics learning. Pause the video at the “Pause for group work” screen. Have the participants provide their perspectives on their role in this process. Use the following questions as a guide.

i. When was the last time you experienced mathematics as a learner?

ii. Where are you in understanding the progression of the K–2 standards?

iii. What should those standards look like in a classroom? Who decides?

iv. How do you respond to students when students do or do not understand?

v. What questions do you ask to assess students’ understanding in relation to a specific standard?

4. Show slide 2 to provide a general overview of the session goals. Within this session, participants will explore various mathematics tasks as learners, make sense of those tasks through the TQE process, and create a shared image of classrooms where teachers use the TQE process to ensure students are engaged in meaningful mathematics learning experiences.

Collaborating in Planning

This segment is designed to help participants see the importance of planning in a collaborative team.

1. Show slide 3. Then click on “Next Segment” to watch Edward Nolan discussing ways in which the book can be used to support mathematics instruction. Pause the video at 44 seconds.

2. Show slide 4. Pair participants into small groups and ask them to come up with a task to support students in making sense of two-digit numbers. Examples may include making various manipulatives available; tools such as the open number line, hundreds chart, and ten frames; and the use of context. Give each group about five to ten minutes to create a task. Have each group share its task with the whole group. On a whiteboard, flipchart, or chalkboard write the list of the tasks shared.

3. Show slide 5 and distribute the handout “Task 1: Collaborating in Planning.” Help participants to anticipate student actions in response to this task. Ask “What do you think first graders will do to complete this task?” You can also ask the follow-up question “How will you respond to each student’s response?” Allow five to ten minutes for groups to provide a response to those two questions.

4. Distribute the “Classroom Video Analysis Tool” and ask participants to make note of what the teacher and students are doing.

5. Play the video clip “Representing Two-Digit Numbers.”

6. Pause the video at 1:13, after Thomasenia asks students “Show me 24.” Ask participants to discuss the task’s open-ended nature. Point out that students are provided with their own manipulatives and that they have the ability to make a choice in how they represent the number. Tie this back to collaboration in planning by highlighting the need to provide this type of meaningful experience for students and working together as a collaborative team to make the process easier. Collaborative teams will have the opportunity to share anticipated challenges students may have and can also share their lesson results with one another. The availability of manipulatives and providing students with easy access to the manipulatives of their choosing are other discussion topics.

7. Continue the video and stop at 3:25 after Thomasenia displays the two whiteboards and asks students, “What do you think?” Ask participants how the planning process might have influenced her choice of boards. Responses should mention that her objective included that students would be able to model a number in different ways and make sense of being able to show the same number in more than one way. Highlight that while we want students to be the ones making sense of the concepts, we always want to keep our objective in mind and much of what we do when enacting the lesson is responding to students in such a way that brings them back to our stated objective.

8. Play the video and pause at the “Pause for group work” screen. Then provide groups with three to five minutes to share their notes and discuss what stood out to them.

9. Show slide 6. Ask participants if the classroom video was what they anticipated. Discuss the importance of using classroom video to develop a shared understanding and vision of the mathematics and how to support it in the classroom. After this discussion and participants’ questions, ask participants “How did the teacher support students to engage with the task?” Participants should indicate that the teacher did not give away the answers but rather assessed for understanding and then asked probing questions to guide students to make sense of representing a two-digit number in flexible ways.

10. Discuss why supporting students who struggle to make sense of the 10-to-1 relationships between places in two-digit numbers is important. Allow a few minutes for participants, in small groups, to describe the types of struggles they would anticipate for this task. Allow the small groups to share with the whole group.

Using the TQE Process

This segment is designed to help participants see the value of using the TQE process by being intentional about selecting tasks, asking students targeted questions, and collecting evidence of students’ conceptions and misconceptions.

1. Show slide 7. Discuss the importance of using challenging tasks with young learners. Show slide 8. Ask participants, “What is the value of this practice?” and “How can teachers use tasks to scaffold students’ understanding to a specific learning goal?”

2. Distribute the reproducible handout “Clock Face.” Ask participants “What time is it when the hour hand is pointing at the twenty-fourth minute mark?” They will likely answer 4:24 or 4:45. They may also say it is around that time or close to 5:00. Tell them that they need to provide an exact time.

3. Explain that this is a good task because it requires them to think about the clock and the passing of time in a different way. It also provides an opportunity for you as the facilitator to use questioning to elicit their thinking and to scaffold their attempts. If you have a participant that figures it out quickly, you can have them use that opportunity to practice their questioning and facilitation to assist others without giving too much away.

4. Distribute the handout “Task 2: “Using the TQE Process” and display slide 9. Ask participants, “Consider this task with the TQE process in mind. How would you introduce this task?”

5. Display slide 10. How do you anticipate grade 1 students would solve this join (change unknown) problem? Look for participants to say that the teacher would model the problem first—this is not the behavior we seek in the classroom around tasks like this. Layers of facilitation is a more appropriate approach to use.

6. Display slide 11. Ask “What questions might you ask to engage students in this task?” The goal is to support participants to make sense of the TQE process. They explored the task, and now they are exploring the questions they might use with students. Eventually, they will watch the video to see what evidence was collected.

Workshop Steps and Teaching Suggestions: Statement of Purpose

7. Display the overview of the TQE process on slide 12.

8. Show slide 13 and click on “Next Segment” of the video showing Thomasenia Adams discussing the TQE process.

9. Pause the video at 2:06. Show slide 14, then distribute the “TQE Analysis Tool” and introduce the classroom video by asking the participants to anticipate the TQE process. Use the “TQE Analysis Tool” to have participants take notes on how the teacher incorporates the TQE process while watching the video. This helps to guide participants in what to look for with the video.

10. Show the video clip “Solving a Word Problem Where the Change Is Unknown.”

11. Pause the video throughout to discuss teacher moves to develop student thinking, such as allowing students to feel safe to make errors, develop understanding without directly telling, and having students make sense of other students’ thinking. Have participants discuss what they noticed after the video is over. The following examples show where the facilitator can pause the video to discuss teacher moves.

i. Pause at 3:23: Notice how Juli unpacks the students’ thinking in relation to the task. She asks questions to assess the students’ understanding and collects evidence by having the student demonstrate how she determined that the answer was 8, hence the TQE process.

ii. Pause at 3:31: When Juli refocuses the students back to the front of the class, she talks about how she observed the students solve the task in different ways. At this time, Juli has walked around the classroom, gathered evidence, and determined which students she would like to share out as well as when they share. Notice how strategic she is in selecting students to share.

iii. Pause at 5:53: Notice that Juli does not tell the student he was wrong; instead she engages the student in metacognition, where he thinks about his own thinking. Think about the questions she asks of him and how she collects evidence. Also notice how she uses his mistake as a learning opportunity. By doing so, she introduces the make-a-ten strategy and allows the student to experience success.

iv. Pause at 6:39: Notice how Juli asks the student about another strategy. This teacher move is necessary especially when it is a part of the learning goal. This question had the students think of another strategy, doubles, to solve the word problem. When a student responds to the question, notice how Juli questions the student to clarify and collect evidence.

12. Pause the video at the “Pause for group work” screen. Display the overview of the TQE process on slide 15. Ask participants if the classroom video was what they anticipated. Show slide 16. After this discussion and after entertaining participants’ questions, ask participants “How did the teacher incorporate the TQE process in instruction?” Ask “Why did the teacher sequence the tasks the way she did?”

13. Show slide 17. Within groups of three to four, have participants discuss the questions on the slide related to students’ conceptions and misconceptions. After the group discussions, allow the groups to share their thoughts with the whole group.

i. Participants should see that students used the concepts developed from the first task to make sense of the second task.

ii. Participants should see that the teacher used questions to access the thinking of different levels of students.

iii. The evidence that the teacher collected was that some students began with misconceptions but then came to a good understanding. Some students used early strategies for solving problems but need exposure to more advanced strategies. Finally, some students made sense of the challenging task in advanced ways.

Creating a Shared Vision of Classroom Instruction

This segment is designed to help participants see the value of using classroom video to create a shared image of classroom instruction.

1. Show slide 18.

2. Click on “Next Segment” to watch Juli explain the purpose for using classroom video. Pause the video at 42 seconds.

3. Distribute the handout “Task 3: Creating a Shared Vision of Classroom Instruction” and display the problem on slide 19 and have participants anticipate how kindergarten students would solve it. How might students solve this task?

4. Show slide 20. What would instruction look like? Facilitate a discussion so that participants can compare their visions for classroom discussion to see if they are the same. They are likely not the same.

5. Play the video clip “Determining Number Pairs to Ten.”

6. You may pause the video throughout to discuss teacher moves to develop student thinking, such as allowing students to feel safe to make errors, develop understanding without directly telling, and having students make sense of other students’ thinking. Have participants discuss what they noticed after the video is over. Suggested pauses:

i. Pause at 1:41— Notice Juli’s response after the student stated that 6 + 4 = 8. Rather than telling the student he was wrong, she had him discover his mistake.

ii. Pause at 2:08— Notice how Juli gets the students to critique each other’s reasoning, which is Mathematical Practice 3. This practice doesn’t always happen naturally in the classroom; teachers must initiate and facilitate this type of engagement.

iii. Pause at 4:02— Notice how Juli modifies the task. It is evident that the student experienced some cognitive dissonance; however, Juli connected the follow-up question to the context to support the student in thinking about how to solve the new task.

iv. Pause at 4:50— Notice how Juli writes the number sentence on the board. This teacher action supports flexibility in thinking. This is important as students are understanding the equal sign and will need to be exposed to seeing number sentences written in multiple ways.

v. Pause at 5:15—This is after a female student provides a wrong number sentence. How the teacher responds is critical, so have participants think about how they would respond to this student before continuing the video. Have them share with one another and then continue the video. Juli writes the wrong answer on the board as if it is correct and asks the class if they agree. This is another example of using a mistake as an opportunity for learning. This is also an example of initiating and facilitating Mathematical Practice 3, students critiquing the reasoning of others.

vi. Pause at 6:15— Discuss how Juli returns to that student to experience success.

7. Play the video to the “Pause for group work” screen.

8. Display the questions on slide 21 and distribute the “Classroom Video Analysis Tool” handout. Review the Standards for Mathematical Practice (SMPs) on slide 22 and discuss each practice. The “Classroom Video Analysis Tool” guides participants in what to look for in a video. After each video, take some time to have participants share their thoughts and discuss the SMPs that they identified.

9. Ask participants if the classroom video was what they anticipated. After this discussion and after entertaining participants’ questions, ask about the TQE process and what questions the teacher used and what evidence she was able to collect.

10. Within groups, have participants discuss what they think the teacher should do next. Allow groups to share their thoughts with the whole group.

11. Discuss how viewing the video together helps in having a productive discussion regarding next steps.

Questioning to Facilitate Learning

This segment is designed to help explore the use of questioning to support students to engage in strategy use.

1. Show slide 23. Click on the “Next Segment” option on the video and watch Tashana discuss linking the content and practice standards. Pause the video at 36 seconds.

2. Display slide 24 to engage the group in a discussion as to why teachers ask questions. Discuss that questions can be for clarifying, furthering learning, addressing misconceptions, promoting alternative thinking, and so on. Encourage participants to go beyond the expected or usual questions.

3. Distribute the handout “Task 4: Questioning to Facilitate Learning—Using Strategies to Add” and display the task on slide 25. Lead a discussion regarding expectations for how young learners would solve this problem. Lead them to discuss strategies students might use. If participants don’t come up with the make-a-ten strategy, then suggest it so that participants discuss how it might be used. It is unlikely that participants will suggest

combining 5 from the 7 and 5 from the 8 to make 10. If they don’t suggest it don’t bring it up yet as it will be addressed during the classroom video.

4. Show slide 26 then lead a discussion regarding how to set up this first-grade problem for instruction. The goal is to let participants come up with the ideas. Once they have discussed the questions, ask what evidence they would look for to indicate that students have made sense of this strategy.

5. Distribute the “Classroom Video Analysis Tool” handout.

6. Display slide 27. This helps to guide participants in what to look for in the video.

7. Resume the video with the clip “Using the Make-a-Ten Strategy With Addition.”

8. Pause the video at 3:23. Ask the participants to make sense of what the students did and possibly have them recreate the student’s model on a piece of paper. Explain that students are successful when they have multiple strategies and that they can engage with problems in creative ways when they learn to think flexibly about numbers. The way we can encourage this type of creative and flexible thinking is by asking students to provide other ways to solve problems. When we value creative thinking, students recognize that we are not just asking questions and looking for them to come up with the correct answers. Our questioning strategies should highlight student thinking rather than bring students to our solution methods.

9. Resume the video.

10. Stop the video at 4:49. Have participants use a piece of paper to draw two ten frames. Create participant trios and give the following directions: “Take turns with each role. One of you will be the student using the ten frames to model the problem. One of you will be the teacher who will be practicing questioning the student. One of you will be the observer who will provide observations and feedback on the exchange.” Alternatively, you can have participants partner up and have one participant demonstrate a ten-frame strategy and have the other partner practice questioning. Circulate the room and look for participants to ask questions that would cause the other to show a different way. You could make a list of open-ended questions asked by the teachers such as, “What did you do?” and “What do these represent?” and then share with the group when they are finished. Give groups the opportunity to share any thoughts from the activity.

11. Resume the video and pause at the “Pause for group work” screen.

12. Ask participants “Was the classroom video what you anticipated?” Show slide 28. Ask “What teacher moves stood out to you?” “What questions did the teacher ask?” “How did those questions help to collect evidence of student learning?” Discuss the importance of using classroom video to develop a shared understanding and vision of the mathematics and how to support it in the classroom. After this discussion and after entertaining participants’ questions, ask participants how the teacher supported students to engage in the task. Participants should indicate that the teacher did not give away the answers but rather assessed

for understanding and then asked probing questions to guide students to begin to make sense of the make-a-ten strategy.

13. Within groups, have participants discuss what they would expect the teacher to do next.

Preparing for the Mathematics to Come

This segment is designed to help explore teaching with coherence and progressions.

1. Show slide 29. Click on “Next Segment.” Edward shares the importance of learning progressions. Pause the video at 23 seconds.

2. Distribute the handout “Task 5: Preparing for the Mathematics to Come.” Show slide 30. Support participants to discuss these questions. They will watch a video that shows kindergarten students at varying levels describing and comparing shapes. This will be linked to work to come in a third-grade class where students are classifying quadrilaterals.

3. Resume the video to watch the clip “Comparing Plane Shapes” and have participants complete the handout “Classroom Video Analysis Tool.”

4. Pause the video at the “Pause for group work” screen and show slide 31. Ask participants

“What did you notice?” and “Was the classroom video what you anticipated?”

5. Ask “How might this relate to third-grade geometry work?”

6. Click on “Next Segment.” Edward discusses the teacher as learner. Pause the video at 34 seconds. Share that the next part of the workshop will continue to explore how teachers need to think about the importance of the learners’ roles.

7. Direct participants to draw a square. (See slide 32 for reference.) Then have them compare what they drew to what others drew.

8. Display slide 33 and ask “Which shape most closely resembles the square you drew?”

Participants will likely have drawn a square with a horizontal base. Indicate that students may not recognize shapes A and B as squares.

9. Show slide 34. Direct participants to draw a rectangle and then have them compare what they drew to what others drew.

10. Display slide 35 and ask “Which shape most closely resembles the rectangle you drew?”

Participants will likely have drawn a rectangle like shape A but some draw one like B and C. Remind them of the need to provide examples other than those with horizontal bases. Then ask participants if any of them stopped after drawing the square because they already had a rectangle.

11. Ask them “Do you think third-grade students know the relationship between rectangles and squares?” In small groups have participants design a task to find out. This should result in conversation among participants. Be sure to link the conversation to types of experiences that should occur in the primary grades. Then ask what they think third-grade students would do if they were asked to make a rectangle when they had already made a square (if the participants don’t suggest this).

12. Display slide 36. Distribute the handout “Classroom Video Analysis Tool.” This helps guide participants in what to look for with the video.

13. Resume the video showing the clip “Defining and Classifying Squares and Rectangles.”

14. Pause the video at 4:07. Juli describes that a rectangle can have two long sides and two short sides, but it doesn’t have to have two long sides and two short sides. The look on the students’ faces should help you lead the discussion here. Point out the importance of using correct definitions and examples from the students’ earliest experiences. This alleviates the need to reteach important concepts. Ask them to think about posters they may have on their walls or other documents they may use, such as store-bought name tags that may perpetuate misconceptions with geometry. Resume the video.

15. Pause at 5:12 after the students describe the relationship between a square and a rectangle. At this point some participants may be saying that the teacher was lucky to have a student actually say something like that. However, it is important to let participants know that after the teacher asked the groups to discuss, she walked around listening to the group discussions and allowed a student that had the concept explain it to the class.

16. Pause at 5:44 after Juli asks students to make a square that is not a rectangle. Notice that this task was not prepared ahead of time. The teacher asks all students to “make a square that is not a rectangle” based on a student response. A teacher’s ability to design tasks on the spot directly comes from his or her own deep understanding of the content as well as knowing the common errors that arise in students’ thinking. Participating in workshops like this and reading and interacting with books like the Making Sense of Mathematics for Teaching series allow teachers to build their content knowledge as well as learn those common errors and ways to respond to them. Bringing these experiences to their school-based professional learning communities enables a teacher to continue to build on their understanding and share with, as well as learn from, their peers.

17. Resume the video and play to the “Pause for group work” screen.

18. Show slide 37. Ask participants the questions and ask “Was the classroom video what you anticipated?” Discuss the importance of using classroom video to develop a shared understanding and vision of the mathematics and how to support it in the classroom. After this discussion and after entertaining participants’ questions, ask participants how the teacher supported students to engage in a challenging task. Participants should indicate that the teacher did not give away the answers but rather assessed for understanding and then asked probing questions to guide students to this important relationship with quadrilaterals that is often challenging for students to grasp.

19. Show slide 38 and in small groups, have participants discuss the questions. After the group discussions, allow the groups to share their thoughts with the whole group.

i. Participants should see that many students had a faulty definition for rectangle—that it must have two long sides and two short sides. This led students to struggle with the relationship between squares and rectangles. Be sure to stress the importance of

defining shapes carefully from the start in kindergarten. For example, teachers should never say that rectangles have two short sides and two long sides because that excludes the square from being included as a rectangle.

ii. Participants should conclude that students had a solid definition for squares.

It is important to expose students to many examples of rectangles including those with different orientations and those that are squares. The teacher should anticipate and address similar misconceptions with the relationship between rectangles and parallelograms, rhombuses and parallelograms, and rhombuses and squares.

Closing

This segment reviews the TQE process.

1. Show slide 39. Have participants discuss each step of the process and provide an opportunity for them to ask questions. Share resources that will be helpful for identifying quality tasks. Discuss the importance of maintaining the complexity of tasks. We want them to avoid doing the work for their students.

2. Show slide 40 and revisit the session goals.

3. Click on “Next Segment.” Edward connects the experiences from the workshop to the book series.

4. Show slide 41 and ask participants “What is your biggest takeaway from the workshop, and how will you use what you learned?” Ask them to think about their mathematics instruction and to identify the following: one thing they will stop doing, one thing they will start doing, and one thing they will continue doing.

5. Ask the participants what kind of support they might need as they implement the things they have learned during the workshop.

REPRODUCIBLE HANDOUTS

REPRODUCIBLE

Task 1: Collaborating in Planning

Solve the problem: Grade 1 students are asked to show the number 24. They are given whiteboards, markers, and base ten blocks. How do you think they will respond?

REPRODUCIBLE 21

Classroom Video Analysis Tool

Describe the student discourse.

Describe the teacher actions that support student engagement.

Which Mathematical Practices are emphasized?

What expectations (spoken or understood) are communicated to the students?

Task 2: Using the TQE Process

Solve the problem: Stefan has 7 stickers. How many more stickers does he need to have 15 stickers altogether?

TQE Analysis Tool

Task 3: Creating a Shared Vision of Classroom Instruction

Solve the problem: Jasmine has 10 marbles. Some of them are red, and the rest are yellow. How many marbles could be red, and how many marbles could be yellow?

Task 4: Questioning to Facilitate Learning— Using Strategies to Add

How would you expect young learners to solve this problem: Kim went to a county fair. She played a game and won 7 rubber bracelets. She played another game and won 8 bouncing balls. How many prizes did Kim win at the fair?

Task 5: Preparing for the Mathematics to Come

Answer the following questions.

„ What vocabulary is necessary for kindergarten students to use when describing and comparing shapes?

„ How do you facilitate instruction when students are at very different levels in this regard?

Now complete the following tasks.

1. Draw a square.

2. Draw a rectangle.

Making Sense of Mathematics for Teaching Grades 3–5

Juli K. Dixon, Edward C. Nolan, Thomasenia Lott Adams, Jennifer M. Tobias, and Guy Barmoha

Develop a deep understanding of mathematics. With this user-friendly resource, grades 3–5 teachers will explore strategies and techniques to effectively learn and teach significant mathematics concepts and provide all students with the precise, accurate information they need to achieve academic success.

BKF696

Making Sense of Mathematics for Teaching Grades 6–8

Edward C. Nolan, Juli K. Dixon, George J. Roy, and Janet B. Andreasen

Develop a deep understanding of mathematics. With this user-friendly resource, grades 6–8 teachers will explore strategies and techniques to effectively learn and teach significant mathematics concepts and provide all students with the precise, accurate information they need to achieve academic success.

BKF697

Making Sense of Mathematics for Teaching High School

Edward C. Nolan, Juli K. Dixon, Farshid Safi, and Erhan Selcuk Haciomeroglu

Develop a deep understanding of mathematics. With this user-friendly resource, high school teachers will explore strategies and techniques to effectively learn and teach significant mathematics concepts and provide all students with the precise, accurate information they need to achieve academic success.

BKF698

Making Sense of Mathematics for Teaching Grades 3–5: The TQE Process

Juli K. Dixon, Edward C. Nolan, Thomasenia Lott Adams, Jennifer M. Tobias, and Guy Barmoha

This multimedia workshop helps create a shared vision of classrooms where teachers and students are engaged in meaningful mathematics learning experiences.

DVF068

Making Sense of Mathematics for Teaching Grades 6–8: The TQE Process

Edward C. Nolan, Juli K. Dixon, George J. Roy, and Janet B. Andreasen

This multimedia workshop helps create a shared vision of classrooms where teachers and students are engaged in meaningful mathematics learning experiences.

DVF069

DIG DEEP INTO CONTENT

DIXON NOLAN ADAMS

MATHEMATICS

Bring

Dixon Nolan Adams Mathematics experts to your school

Our Services

1. Big-Picture Shifts in Content and Instruction

Janet B. Andreasen

Guy Barmoha

Lisa Brooks

Kristopher Childs

Craig Cullen

Brian Dean

Lakesia L. Dupree

Jennifer Eli

Erhan Selcuk Haciomeroglu

Tashana Howse

Stephanie Luke

Amanda Miller

Samantha Neff

George J. Roy

Farshid Safi

Jennifer Tobias

Taylar Wenzel

3. Implementation Workshops

2. Content Institutes

Introduce content-based strategies to transform teaching and advance learning.

Build the capacity of teachers on important concepts and learning progressions for grades K–2, 3–5, 6–8, and 9–12 based upon the Making Sense of Mathematics for Teaching series.

Evidence of Effectiveness

Support teachers to apply new strategies gained from Service 2 into instruction using the ten high-leverage team actions from the Beyond the Common Core series.

4. On-Site Support

Discover how to unpack learning progressions within and across teacher teams; focus teacher observations and evaluations on moving mathematics instruction forward; and support implementation of a focused, coherent, and rigorous curriculum.

The River Ridge High School Geometry PLC went from ninth out of fourteen high schools in terms of Geometry EOC pro ciency in 2013–2014 to rst out of fourteen high schools in Pasco County, Florida, for the 2014–2015 school year.” Katia

Juli K. Dixon

Thomasenia

Lott Adams

Edward C. Nolan