Supporting Rigorous Teaching and Learning of Mathematics

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Supporting Rigorous Teaching and Learning of
Mathematics

• Selecting and Sequencing Students Solution Paths
  to Maximize Student Learning

Disciplinary Literacy September 1, 2009
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Overview of Activities

• Discuss mathematical goals and expectations for a
  lesson.
• Analyze samples of student work and link the work
  to the instructional goals for the lesson.
• Select and sequence the student work for the
  Share, Discuss and Analyze phase of the lesson.
• Identify Rules of Thumb for selecting and
  sequencing student work.

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Rationale
Teachers provoke students reasoning about
mathematics through the tasks they provide and
the questions they ask (NCTM, 1991). Asking
questions that reveal students knowledge about
mathematics allows teachers to design instruction
that responds to and builds on this knowledge
(NCTM, 2000). Orchestrating discussions that
build on students thinking places significant
pedagogical demands on teachers and requires an
extensive and interwoven network of knowledge.
Teachers often feel that they should avoid
telling students anything, but are not sure what
they can do to encourage rigorous mathematical
thinking and reasoning. The five practices help
you make student-centered instruction more
manageable by moderating the degree of
improvisation that you have to do when leading a
discussion.
Stein, Engle, Smith, Hughes, 2008
In this session we will focus on two of the five
practices, selecting and sequencing student work
so you can assist and advance student learning
during the Share, Discuss and Analyze phase of
the lesson.
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Pythagorean Theorem Task

• Complete the table on page 32 of Investigation 3.

• For each row of the table
• Draw a right triangle with the given leg lengths
  on dot paper.
• Draw a square of each side of the triangle.
• Find the areas of the squares and record the
  results.

• Recall that a conjecture is your best guess about
  a mathematical relationship. It is usually a
  generalization about a pattern you think might be
  true, but that you do not yet know for sure is
  true.
• For each triangle, look for a relationship among
  the areas of the three squares. Make a
  conjecture about the areas of squares drawn on
  the sides of any right triangle.

• Draw a right triangle with side lengths that are
  different than those given in the table. Use
  your triangle to test your conjecture from
  Question B.

Connected Mathematics, The Pythagorean Theorem
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Five Practices for Orchestrating Productive
Mathematical Discussions
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The Five Practices

• Anticipating
• Monitoring
• Selecting
• Sequencing
• Connecting

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1. Anticipating

• likely student responses to mathematical
  problems.

• It involves developing considered expectations
  about
• How students might interpret a problem
• The array of strategies they might use
• How those approaches relate to the math they are
  to learn

• It is supported by
• Doing the problem in as many ways as possible
• Doing so with other teachers
• Drawing on relevant research
• Documenting student responses year to year

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Anticipating Student Responses

• Use the student work to help you anticipate
  possible student responses to the Pythagorean
  Theorem Task.

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Prior Learning Experiences

• The whole investigation has been provided. How
  does the investigation develop from 3.1 3.4?
  What will students prior learning be when you
  start investigation 3.1?

• Explore distances on a coordinate grid.
• Connect properties of quadrilaterals to
  coordinate representations.
• Develop strategies for finding areas of squares
  and irregular figures on a grid.
• Understand square root geometrically, as the side
  length of a square with known area.

What instructional goals must be accomplished in
3.1?
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Thinking Through a Lesson Protocol Part 1

• At your tables, use the CMP2 lesson, the
  Curriculum Guide, and the initial planning
  questions on the TTLP, Part 1 Selecting and
  Setting Up a Mathematical Task, to set up the
  lesson.
• Analyze the student work from the Pythagorean
  Theorem Task to help you in this process.

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Planning for a Lesson Focus on Key Aspects of a
Lesson

• Part 1 Selecting and Setting Up a Mathematical
  Task

• What are your mathematical goals for the lesson
  (i.e., what is it you want students to know and
  understand as a result of this lesson)?
• In what ways does the task build on students
  previous knowledge? What definitions, concepts,
  or ideas do students need to know in order to
  begin to work on the task? What questions will
  you ask to help students access their prior
  knowledge?
• What are all the ways the task can be solved?
  Which methods do you think your students will
  use? What misconceptions might students have?
  What errors might students make?
• What will you see or hear that lets you know that
  students in the class understand the mathematical
  ideas that you intended for them to learn?

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2. Monitoring

• students actual responses during independent
  work.

• It involves
• Circulating while students work on the problem
• Recording interpretations, strategies, other ideas

• It is supported by
• Anticipating student responses beforehand
• Carefully listening and asking probing questions
• Using recording tools

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Pennsylvania Anchors and Unit Keys

• The Pennsylvania Assessment Anchors and the Unit
  Keys appear on the green sheet in your folder
• How do the goals that you identified link to the
  Pennsylvania Assessment Anchors and the Unit Keys?

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The Five Practices

• Anticipating
• Monitoring
• Selecting
• Sequencing
• Connecting

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Preparing for the Discussion The Share, Discuss
and Analyze Phase of the Lesson
Examine the solutions produced by groups of
students for the Pythagorean Theorem Task.

• Determine which solution paths you would want to
  share during the class discussions, and keep
  track of your rationale for selecting the pieces
  of student work.
• Determine the order that the work will be shared.
  Keep track of your rationale for choosing a
  particular order for sharing the work.

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Selecting and Sequencing

• Listen to each groups rationale for selecting
  and sequencing student work.
• As you listen to the rationale, come up with a
  general Rule of Thumb that can be used to guide
  you when selecting and sequencing work for the
  Share, Discuss and Analyze phase of the lesson.

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3. Selecting
student responses to feature during discussion.

• It involves
• Choosing particular students to present because
  of the mathematics available in their responses
• Gaining some control over the content of the
  discussion
• Giving the teacher some time to plan how to use
  responses

• It is supported by
• Anticipating and monitoring
• Planning in advance which types of responses to
  select

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4. Sequencing
student responses during the discussion.

• It involves
• Purposefully ordering presentations to facilitate
  the building of mathematical content during the
  discussion
• Need for empirical work to compare sequencing
  methods

• It is supported by
• Anticipating, monitoring and selecting
• Considering how possible student responses are
  mathematically related during anticipation work

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The Fifth Practice Connecting
Some teachers selected and sequenced student work
with the goal of having students make connections.

• What kinds of connections are important?
• Why is this important?

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The Rules of Thumb for Selecting and Sequencing
Student Work
What are the benefits of using the Rules of
Thumb as a guide when selecting student work for
the Share, Discuss and Analyze phase of the
lesson?