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Supporting Rigorous Mathematics Teaching and Learning

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Supporting Rigorous Mathematics Teaching and Learning

Selecting and Sequencing Based on Essential Understandings

Tennessee Department of Education

Middle School Mathematics

Grade 6

There is wide agreement regarding the value of teachers attending to and basing their instructional decisions on the mathematical thinking of their students (Warfield, 2001).

By engaging in an analysis of a lesson-planning process, teachers will have the opportunity to consider the ways in which the process can be used to help them plan and reflect, both individually and collectively, on instructional activities that are based on student thinking and understanding.

Participants will learn about:

- goal-setting and the relationship of goals to the CCSS and essential understandings;
- essential understandings as they relate to selecting and sequencing student work;
- Accountable Talk® moves related to essential understandings; and
- prompts that problematize or “hook” students during the Share, Discuss, and Analyze phase of the lesson.

Accountable Talk is a registered trademark of the University of Pittsburgh.

“The effectiveness of a lesson depends significantly on the care with which the lesson plan is prepared.”

Brahier, 2000

“During the planning phase, teachers make decisions that affect instruction dramatically. They decide what to teach, how they are going to teach, how to organize the classroom, what routines to use, and how to adapt instruction for individuals.”

Fennema & Franke, 1992, p. 156

The Mathematical Tasks Framework

TASKS

as set up by the teachers

TASKS

as implemented by students

TASKS

as they appear in curricular/ instructional materials

Student Learning

Stein, Smith, Henningsen, & Silver, 2000

The Mathematical Tasks Framework

TASKS

as set up by the teachers

TASKS

as implemented by students

TASKS

as they appear in curricular/ instructional materials

Student Learning

Stein, Smith, Henningsen, & Silver, 2000

Setting Goals

Selecting Tasks

Anticipating Student Responses

- Orchestrating Productive Discussion
- Monitoring students as they work
- Asking assessing and advancing questions
- Selecting solution paths
- Sequencing student responses
- Connecting student responses via Accountable Talk discussions

- MONITOR: Teacher selects
- examples for the Share, Discuss,
- and Analyze Phase based on:
- Different solution paths to the
- same task
- Different representations
- Errors
- Misconceptions

Set Up of the Task

Set Up the Task

The Explore Phase/Private Work Time

Generate Solutions

The Explore Phase/

Small Group Problem Solving

Generate and Compare Solutions

Assess and Advance Student Learning

SHARE: Students explain their

methods, repeat others’ ideas,

put ideas into their own words,

add on to ideas and ask

for clarification.

REPEAT THE CYCLE FOR EACH

SOLUTION PATH

COMPARE: Students discuss

similarities and difference

between solution paths.

FOCUS: Discuss the meaning

of mathematical ideas in each

representation

REFLECT: Engage students

in a Quick Write or a discussion

of the process.

Share, Discuss, and Analyze Phase of the Lesson

1. Share and Model

2. Compare Solutions

Focus the Discussion on Key

Mathematical Ideas

4. Engage in a Quick Write

Imagine that you are working with a group of students who have the following understanding of the concepts:

- 70% of the students need to make sense of what it means to represent rational numbers on a number line. (6.NS.C.6, C.6a, C.6c)
- 20% of the students need additional work understanding when values in context should be represented with negative numbers (6.NS.C.5). These students also need opportunities to struggle with and make sense of the problem. (MP1)

- 5% of the students are consistently able to represent positive and negative rational numbers as points on the number line and are working on understanding absolute value as distance from 0. (6.NS.C.7c)
- 5% of the students struggle to pay attention and their understanding of numbers and operations is two grade levels below sixth grade.

Common Core State Standards, 2010, p. 43, NGA Center/CCSSO

Common Core State Standards, 2010, p. 43, NGA Center/CCSSO

Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO

- Make sense of problems and persevere in solving them.
- Reason abstractly and quantitatively.
- Construct viable arguments and critique the reasoning of others.
- Model with mathematics.
- Use appropriate tools strategically.
- Attend to precision.
- Look for and make use of structure.
- Look for and express regularity in repeated reasoning.

Solve the task.

Discuss the possible solution paths to the task.

Dia’Monique and Yanely picnicked together. Then Dia’Monique hiked 17 miles. Yanely hiked 14 miles in the opposite direction. What is their distance from each other? Draw a picture to show their distance from each other.

Does the task provide opportunities for students to access the Standards for Mathematical Content and Standards for Mathematical Practice that we have identified for student learning?

Study the essential understandings associated with the Number System Common Core Standards.

Which of the essential understandings are the goals of the Hiking Task?

Essential Understandings (Small Group Discussion)

Analyze the student work.

Identify what each group knows related to the essential understandings.

Consider the questions that you have about each group’s work as it relates to the essential understandings.

Assume that you have circulated and asked students assessing and advancing questions.

Study the student work samples.

- Which pieces of student work will allow you to address the essential understanding?
- How will you sequence the student’s work that you have selected? Be prepared to share your rationale.

In your small group, come to consensus on the work that you select, and share your rationale. Be prepared to justify your selection and sequence of student work.

What order did you identify for the EUs and student work?

What is your rationale for each selection?

Group A

Group B

Group C

What order did you identify for the EUs and student work?

What is your rationale for each selection?

A teacher must always be assessing and advancing

student learning.

A lesson is academically rigorous if student learning related to the essential understanding is advanced in the lesson.

Accountable Talk discussion is the means by which teachers can find out what students know or do not know and advance them to the goals of the lesson.

Recall what you know about the Accountable Talk features and indicators. In order to recall what you know:

- Study the chart with the Accountable Talk moves. You are already familiar with the Accountable Talk moves that can be used to Ensure Purposeful, Coherent, and Productive Group Discussion.
- Study the Accountable Talk moves associated with creating accountability to:
- the learning community;
- knowledge; and
- rigorous thinking.

Accountability to the Learning Community

- Active participation in classroom talk.
- Listen attentively.
- Elaborate and build on each others’ ideas.
- Work to clarify or expand a proposition.
Accountability to Knowledge

- Specific and accurate knowledge.
- Appropriate evidence for claims and arguments.
- Commitment to getting it right.
Accountability to Rigorous Thinking

- Synthesize several sources of information.
- Construct explanations and test understanding of concepts.
- Formulate conjectures and hypotheses.
- Employ generally accepted standards of reasoning.
- Challenge the quality of evidence and reasoning.

Accountable Talk Moves

Accountable Talk Moves (continued)

From the list of potential EUs and its related student work, each group will select an essential understanding to focus their discussion.

Identify a teacher in the group who will be in charge of leading a discussion with the group after the Accountable Talk moves related to the EU have been written.

Write a set of Accountable Talk moves on chart paper so it is public to your group for the next stage in the process.

The Focus Essential Understanding

Positive and Negative Numbers Can be Used to Represent Real-World Quantities

Positive numbers represent values greater than 0 and negative numbers represent values less than 0. Many real-world situations can be modeled with both positive and negative values because it is possible to measure above and below a baseline value (often 0).

Group B Group G

- Group B, how did your group determine where to place the point for each girl?
- Who understood what she said about the opposite directions? (Community)
- Can you say back what he said about the sign of the numbers? (Community)
- Signed numbers can be used to represent positions in opposite directions from a set point. (Revoicing)
- Group G didn’t use negative numbers. How does your model represent the problem situation? (Knowledge)
- What do the 14 and 17 in Group G’s model represent? Is it possible to walk a negative distance? What does the -14 represent? (Rigor)

Using the list of essential understandings identified earlier, write Accountable Talk discussion questions to elicit from students a discussion of the mathematics.

Begin the discussion with a “hook” to get student attention focused on an aspect of the mathematics.

The Focus Essential Understanding

Positive and Negative Numbers Can be Used to Represent Real-World Quantities

Positive numbers represent values greater than 0 and negative numbers represent values less than 0. Many real-world situations can be modeled with both positive and negative values because it is possible to measure above and below a baseline value (often 0).

Group B Group G

- You can’t walk negative 14 miles, can you? Is it okay to use a negative number to represent Yanely? Do you have to use negative numbers? (Hook)
- Group B, how did your group determine where to place the point for each girl?
- Who understood what she said about the opposite directions? (Community)
- Can you say back what he said about the sign of the numbers? (Community)
- Signed numbers can be used to represent positions in opposite directions from a set point. (Revoicing)
- Group G didn’t use negative numbers. How does your model represent the problem situation?

Revisit your Accountable Talk prompts.

Have you problematized the mathematics so as to draw students’ attention to the mathematical goal of the lesson?

If you have already problematized the work, then underline the prompt in red.

If you have not problematized the lesson, do so now. Write your problematizing prompt in red at the bottom and indicate where you would insert it in the set of prompts.

We will be doing a Gallery Walk after we role-play.

- You will have 15 minutes to role-play the discussion of one essential understanding.
- Identify one observer in the group. The observer will keep track of the discussion moves used in the lesson.
- The teacher will engage you in a discussion. (Note: You are well-behaved students.)
The goals for the lesson are:

- to engage all students in the group in developing an understanding of the EU; and
- to gather evidence of student understanding based on what the student shares during the discussion.

The observer has 2 minutes to share observations related to the lessons. The observations should be shared as “noticings.”

Others in the group have 1 minute to share their “noticings.”

Now that you have engaged in role-playing, what are you now thinking about regarding Accountable Talk discussions?

Do a Gallery Walk. Read each others’ problematizing “hook.”

What do you notice about the use of hooks? What role do “hooks” play in the lesson?

What have you learned today that you will apply when planning or teaching in your classroom?

Participants:

- identify goals for instruction;
- Align Standards for Mathematical Content and Standards for Mathematical Practice with a task.
- Select essential understandings that relate to the Standards for Mathematical Content and Standards for Mathematical Practice.

- prepare for the Share, Discuss, and Analyze Phase of the lesson.
- Analyze and select student work that can be used to discuss essential understandings of mathematics.
- Learn methods of problematizing the mathematics in the lesson.