Common Core State Standards © 2011



Using the CCSS to Develop Mathematically Proficient Students:

Fluency with Whole Numbers: Day 1

Professional Development

Facilitator Handbook

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Pearson School Achievement Services

Using the CCSS to Develop Mathematically Proficient Students: Fluency with Whole Numbers: Day 1

Facilitator Handbook

Published by Pearson School Achievement Services, a division of Pearson, Inc.

1900 E. Lake Ave., Glenview, IL 60025

© 2013 Pearson, Inc.

All rights reserved.

Printed in the United States of America.

ISBN 118990

Facilitator Agenda

Using the CCSS to Develop Mathematically Proficient Students:

Fluency with Whole Numbers: Day 1

|Section |Time |Agenda Items |

|Introduction |15 minutes |Slides 1–4 |

| | |Welcome |

| | |Agenda |

| | |Outcomes |

| | |Activity: One Up, One Down |

|1: Along the Way to Mathematical Proficiency |60 minutes |Slides 5–16 |

| | |Section Introduction/Big Question |

| | |Making Sense of Mathematics |

| | |Video: “John Van de Walle Teacher Workshop” |

| | |Activity: Number Sense and Rectangular Area |

| | |Long-Standing Importance |

| | |Mathematical Proficiency: Focus on Fluency |

| | |College and Career Readiness |

| | |Mathematics Required in Today’s Classroom: CCSS |

| | |Mathematics Required in Today’s Classroom: Standards for Mathematical Practice |

| | |Reflection: Two Statements and One Question |

|Break |15 minutes | |

|2: Teaching through Problem Solving in the CCSS-Based |105 minutes |Slides 17–26 |

|Mathematics Classroom | |Section Introduction/Big Question |

| | |Task Selection and Focal Areas of the CCSS |

| | |Grades K–2 Activity: Writing Word Problems (+,−) |

| | |Grades 3–5 Activity: Writing Word Problems (×,÷) |

| | |Task Selection and Focal Areas of the CCSS (continued) |

| | |Task Selection and Problem-Based Tasks |

| | |Grades K–2 Activity: Estimating on a Number Line |

| | |Grades 3–5 Activity: Fish Bowl |

|Lunch |30 minutes | |

|Section |Time |Agenda Items |

|2: Teaching through Problem Solving in the CCSS-Based |80 minutes |Slides 27–35 |

|Mathematics Classroom (continued) | |What is it about teaching that influences student learning? |

| | |Activity: Card Sort |

| | |Three-Phase Lesson Structure |

| | |Before the Task |

| | |Video: “Before (or Opening): Hundreds Chart Patterns” |

| | |During the Task |

| | |Video: “During (or Work Time): Dominoes Elementary Workshop” |

| | |After the Task |

| | |Video: “After (or Closing): Rates” |

| | |Reflection: 360 Degrees |

|Break |15 minutes | |

|3: Teaching as Learning in the CCSS-Based Mathematics Classroom|80 minutes |Slides 36–48 |

| | |Section Introduction/Big Question |

| | |Connecting the Content to the Practices: Points of Intersection |

| | |Learning to Notice Differently |

| | |Attending |

| | |Interpreting |

| | |Deciding How to Respond |

| | |Thematic Stories |

| | |Video: “Gretchen’s Story” |

| | |Video: “Sean’s Story” |

| | |Reflection: Staying on Target |

|Reflection and Closing |20 minutes |Slides 49–54 |

| | |Review Video |

| | |Final Reflection: Next Steps |

| | |Evaluation |

|Total |6 hours | |

Preparation and Background

Workshop Information

Big Ideas

▪ The pressures that influence school mathematics have become much more complex with the rigorous problem solving required by the CCSSM. The ideas that follow create the foundation for teachers to effectively teach in a problem-solving environment, so that all students will be college and career ready.

▪ Exploring what it means to know and do mathematics is at the very heart of teaching through problem solving—that is, teaching from the perspective of a student who must develop his or her own ideas and understanding. Teachers must come to believe that experts do not need to explain mathematics but the mathematics can emerge from carefully planned instruction.

▪ If knowledge is the possession of concepts and ideas, then understanding is a measure of how well new concepts, facts, and procedures are integrated with or connected to other existing ideas. Another way to explain understanding of the content is by understanding the extent to which students demonstrate the CCSSM’s Standards for Mathematical Practice for a particular topic.

▪ Teaching through problem solving means that teachers teach content by posing carefully planned problems through which students develop important concepts. Guidelines for selecting tasks are critical to teaching through problem solving.

▪ Problem solving promotes active reflective thought and student involvement in mathematics and in turn enhances students’ understanding of the content.

▪ Whether it is a ten-minute activity, a full-period problem, or a multiday task, a simple way to think about organizing instruction is to think of it in three phases: Before, During, and After. For each of these parts, there are separate agendas and corresponding teaching actions to meet these agendas. You cannot overemphasize the importance of the After portion of the lesson. In order to make teaching with problem-based tasks effective, there must be a significant period in which students share and discuss their results and justify them to the class (not to the teacher).

▪ Planning a lesson based on the three-phase lesson structure helps teachers develop the mental processes for thinking through lesson planning while they keep students’ learning at the forefront. This process can serve as a guide for taking activities and turning them into effective lessons that integrate content and processes. The more experience teachers have with this process, the more likely they are to internalize the process of connecting objectives, assessment, and lesson activities.

▪ Perhaps the most important benefit of a problem-solving approach to instruction is that it has the potential to meet the needs of a wide range of learners. In a problem-solving approach, every student can approach tasks with his or her own ideas and strategies.

▪ Using video clips that show children’s mathematical thinking in professional development contexts can motivate and promote growth in teachers’ content knowledge of mathematics. It can also hone their abilities to guide each child’s foundational knowledge development and deep understandings as outlined in the CCSSM.

▪ Class discussion provides the opportunity for teachers and students to make explicit relationships between facts, ideas, and procedures in an intentional and public way and often provides clarification.

▪ Teachers need to “dig below the surface” of correct answers to determine whether correct answers are associated with rich reasoning.

▪ Teachers need to understand and expect that students can correctly answer a problem and still have mathematical misconceptions that teachers must address.

▪ Teachers’ professional noticing of children’s mathematical thinking has three components (Jacobs, Lamb, and Philipp quoted in Philipp and Schappelle 2012, 8): attending, interpreting, and deciding how to respond.

▪ Teachers can learn to focus closely on their students’ mathematical thinking and develop a “students’-thinking” lens to begin to see the mathematical content in two ways—as a teacher and as a student.

▪ When teachers navigate student learning, they often transition from listening to students to determining whether they are right to listening to determine whether they have used correct reasoning.

▪ Teachers are responsible for supporting students’ mathematical learning. To that end, teachers must focus on students’ reasoning and be flexible enough for it to make a difference in classroom instructional decisions.

Big Questions

▪ What are the foundations and perspectives for developing mathematical proficiency through process and content?

▪ How do you use the task selection guidelines, lesson features, and classroom structures to provide opportunities for students to develop mathematical proficiency?

▪ How can you navigate student learning to focus on the mathematical reasoning that the content and processes demand?

Outcomes

▪ Articulate a structure for teaching through problem solving that incorporates the CCSSM and the Standards for Mathematical Practice.

▪ Connect the design of a lesson to the opportunity for focused instruction and ongoing formative assessment.

▪ Identify strategies for scaffolded instruction as a means of supporting students until they can apply new knowledge and skills independently.

▪ Articulate daily classroom structures that build independence of learning.

Section 2: Teaching through Problem Solving in the CCSS-Based Mathematics Classroom (Slides 17–35)

Time: 185 minutes

Big Question

• How do you use the task selection guidelines, lesson features, and classroom structures to provide opportunities for students to develop mathematical proficiency?

Training Objectives

• Identify the primary focal areas and connection to the instructional support these areas provide to teachers.

• Identify the three features of a problem-based task, and articulate how the task is critical to the agendas of each of the lesson phases.

• Articulate the important points of the summary of the meta-analysis on how teaching influences student learning.

• Observe and reflect on the behaviors of teachers and students during the three-phase lesson structure.

• Link the attributes of the lesson structure to the consideration of time, focus, and energies for effective mathematics instruction.

Materials per Section

• Participant Workbook, pages 13–24

• CCSSM (CCSSI_Math Standards.pdf found in the Additional Resources folder)

• Enhancing Instruction in Mathematics: Teaching with Understanding in High School (Hiebert_and_Grouws_Facilitator_Copy.pdf found in the Additional Resources folder)

• Chart paper

• Index cards

• Pens or pencils

|Three-Phase Lesson Structure |Transition to Slide 28 by making the point that when teachers teach through problem solving and |[pic] |

| |attend to making mathematical relations explicit, as well as allow students to struggle with |PW: Page 20 |

| |important mathematics, the structure of the instructional block becomes increasingly important. | |

| |The structure of the three-phase lesson plan provides an excellent palette for teachers to design| |

| |an efficient and effective instructional block. | |

|Before the Task |Display Slide 29. |[pic] |

| |The key to helping students make sense of problems and persevere in solving them is to have a |PW: Page 21 |

| |framework that will give the teacher the opportunity to plan for specific strategies within a | |

| |lesson. Start by having participants further examine the three-phase structure of a lesson, one | |

| |phase at a time—beginning with the Before phase. | |

| | | |

| |Notice the title “Getting Ready.” Who is getting ready? For what are they getting ready? | |

| |(Van de Walle, Karp, and Bay-Williams 2013,49) | |

| |Display Slide 30. |[pic] |

| |Video: “Before (or Opening): Hundreds Chart Patterns” |PW: Page 21 |

| |At this point, play the video segment of the “Before (or Opening): Hundreds Chart Patterns | |

| |Elementary Workshop.” Ask participants to make note of how the teacher gets students ready to | |

| |work on the task at hand. | |

| |Analyzing the Before Phase | |

| |How does the teacher activate prior knowledge and connect the task to prior student experiences? | |

| |Anticipated responses could include the following: The teacher activated prior knowledge by | |

| |providing a similar task to the one the students would be working on in the lesson that the | |

| |students were familiar with. She used a large hundreds chart and asked students to complete | |

| |various patterns. She discussed the patterns with the students. | |

| |How does the teacher ensure that the students understand the problem, and how does she decide it | |

| |is time to move on to exploring the task? | |

| |Anticipated responses could include the following: The teacher continued asking students | |

| |questions. When it was apparent that most students understood the directions and the concept that| |

| |patterns repeat, she moved on. She provided directions verbally as well as wrote them out in | |

| |steps on the board. She worked through the beginning of the task, using a highlighter, exactly | |

| |like the students would be doing to complete their task. She moved on once students had no more | |

| |questions. | |

| |Are expectations clear? How so? | |

| |Anticipated responses could include the following: Expectations are clear. When the teacher | |

| |provided directions for the task, she continued to stress her expectations that students would be| |

| |working as partners and sharing jobs, as highlighter and recorder. She modeled how she wanted the| |

| |recorder to write numbers down, using commas to separate the numbers. She also told the students | |

| |they would be sharing before the class. She provided very clear directions and repeated them when| |

| |necessary. | |

| |Three Features of a Problem and Teaching through Problem Solving | |

| |In what ways does the teacher’s practice support our assumptions and the three features of a | |

| |problem? | |

| |In what ways does her practice support teaching through problem solving? | |

| |(Van de Walle, Karp, and Bay-Williams 2013, 35) | |

| |Background Information | |

| |This video has a running time of ten minutes and fifty-nine seconds. It shows a first-grade | |

| |class. The task is for students to understand that skip counting (fives and tens) patterns occur | |

| |with various start numbers of one through ten. | |

|During the Task |Display Slide 31. |[pic] |

| |The key to the work time is independent work—individual and/or cooperative—and the teacher’s |PW: Page 22 |

| |ability to let go so that students have the opportunity to struggle with important mathematics. | |

| |Notice the title “Students Work.” Who should be making sense of the mathematics and persevering | |

| |in working through the content? What should the teacher do? | |

| |(Van de Walle, Karp, and Bay-Williams 2013, 52) | |

| |Display Slide 32. |[pic] |

| |Video: “During (or Work Time): Dominoes Elementary Workshop” |PW: Page 22 |

| |At this point, play the video segment, “During (or Work Time): Dominoes Elementary Workshop.” Ask| |

| |participants to make note of what the teacher does and what the students do during the work time.| |

| |Analyzing the During Phase | |

| |How does the teacher let go? | |

| |Anticipated responses could include the following: Even when the teacher talks with the students | |

| |about the math of the task, she did not provide any answers. She only provided clarification in | |

| |most instances. | |

| |How does she ensure that students work independently in groups or individually? | |

| |Anticipated responses could include the following: She told them to each pick out the three | |

| |dominoes that they would use for the task. She had students work individually but let them know | |

| |they could ask their partners if they needed help. | |

| |What does she do to monitor and assess students’ thinking? | |

| | | |

| |Anticipated responses could include the following: The teacher moved between groups. She listened| |

| |to the students. She asked questions, such as, “Have you noticed anything that keeps repeating?” | |

| |She provided some hints like, “So you say you see there are addition problems, do you notice | |

| |anything else?” She listened and gave the students time to think. She let students know if she | |

| |wanted them to share their solutions during the closing phase of the lesson by writing numbers on| |

| |the board and assigning students to a number (an area of the board in which to show work). | |

| |(Van de Walle, Karp, and Bay-Williams 2013, 52) | |

| | | |

| |Background Information | |

| |The “During (or Work Time): Dominoes Elementary Workshop” video clip has a running time of seven | |

| |minutes and forty-nine seconds. The class is a second-grade class taught by Ms. Pittman. Ms. | |

| |Pittman introduced the task to the students, who lead them to discover the part-part-whole | |

| |relationship represented in a domino. Prior to the beginning of the clip, there were | |

| |approximately eight additional minutes where the students independently worked through writing | |

| |facts associated with each domino. During that time, a few students began to write down their | |

| |observations regarding what they noticed. This video begins where she is more active in helping | |

| |them discover the big idea about the part-part-whole relationship and that all fact families have| |

| |two different operations and four related facts. | |

|After the Task |Display Slide 33. |[pic] |

| |The key to the class discussion is the opportunity to make explicit the relationships between |PW: Page 23 |

| |facts, ideas, and procedures—in an intentional and public way. (Van de Walle, Karp, and | |

| |Bay-Williams 2013, 54) | |

| |Display Slide 34. |[pic] |

| |Video: “After (or Closing): Rates” |PW: Page 23 |

| |Play the segment of the video, “After (or Closing): Rates.” Ask participants to make note of what| |

| |the teacher does and what the students do during the closure. | |

| |Analyzing the After Phase | |

| |How does the teacher listen actively without evaluation? | |

| |Anticipated responses could include the following: He had the students present their charts to | |

| |the class, and he asked clarifying questions regarding why they used the numbers they did and | |

| |what exactly the numbers they used referred to. He asked more for clarification than evaluation. | |

| |How does he promote a community of mathematical thinkers? | |

| |Anticipated responses could include the following: He asked, “Is everyone responsible for | |

| |presentations?” He reviewed the expectation that everyone should be able to answer, even though | |

| |there would be only one spokesperson. He also had students applaud for one another. | |

| |Were there any examples of clarification? Were there any examples of making mathematical | |

| |relationships explicit? | |

| |Anticipated responses could include the following: The teacher asked clarifying questions, such | |

| |as, “Why will we use repeated addition?” The teacher made connections between addition and | |

| |multiplication. When he heard students mention money, he reminded the students that it is a good | |

| |strategy to use something you know, like money, in order to estimate. The teacher asked, “So are | |

| |we using repeated addition or multiplication?” He clarified by saying, “We can use multiplication| |

| |or repeated addition.” The teacher asked, “What do we have to know to get our daily rate?” The | |

| |teacher was very specific about the units that the students used. He worked with small groups and| |

| |provided clarification. He asked, “Is it 12 × 900 or is it 12 × the rate?” He also asked, “What | |

| |does 375 stand for?” He asked the students that had made the connection to money to share their | |

| |connection with the group. He asked several final questions: “What happens when the hourly rate | |

| |goes up, what happens to all of it?” “What happens when the rate goes down, if it is the only | |

| |thing that changes?” | |

| |Background Information | |

| |The “After Rates” lesson is a fourth-grade lesson taught by Keith Kalway, who is a Math Coach. He| |

| |teaches at Achievement Plus Elementary in St. Paul, Minnesota. The running time for this video is| |

| |seven minutes and fifty-one seconds. | |

| |In this fourth-grade math workshop taught by the school's Math Coach, the lesson focuses on | |

| |calculating simple rates using repeated addition and/or multiplication and using one rate to find| |

| |another rate. In this video, the teacher and students review the standards together, graph a | |

| |scenario that they create, work in groups to graph the rates of a word problem, and finally | |

| |present their graphs to the class in the closing meeting. | |

|Reflection: 360 Degrees |Display Slide 35. |[pic] |

| |Whether it is a ten-minute activity, a full-period problem, or a multiday task, a simple way to |PW: Page 24 |

| |think about organizing instruction is around the three phases: Before, During, and After. | |

| |Given the circle provided in the Participant Workbook, ask participants to think about what seems| |

| |reasonable to them—in terms of degrees—for the sector of time allowed for each of the three | |

| |phases. | |

| |While participants sketch their views of the 360 degrees, group participants to share and discuss| |

| |their ideas around the reflection questions in the Participant Workbook. Use a grouping technique| |

| |to get participants moving around the room to talk to someone other than their shoulder partner | |

| |or group. | |

| |Share participants’ reflections as time permits. | |

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