Fostering Algebraic Thinking

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Fostering Algebraic Thinking

• May 19-22, 2008

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A core belief underlying Fostering Algebraic
Thinking is that good mathematics teaching
begins with understanding how mathematics is
learned.
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While the materials provide activities for
teachers to do with students, the primary focus
is on TEACHER learningin the belief that student
learning will also be served.
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Goals

• Become familiar with the Fostering Algebraic
  Thinking materials.
• Examine activities that may be challenging to
  facilitate.
• Develop plans for implementation at your sites.

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We will focus on

• How students think about mathematics.
• Understanding students thinking through analysis
  of different kinds of data.
• Understanding how algebraic thinking develops.
• Instructional implications.

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Fostering Algebraic Thinking Modules

• Analyzing Student Written Work
• Listening to Students
• Asking Questions of Students
• Documenting Patterns of Student Thinking

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Agenda Overview

• Monday AMIntroductory Session
• Monday PMAnalyzing Written Student Work
• Tuesday AMAnalyzing Written Student Work
• Tuesday PM Listening to Students

• Wednesday AMAsking Questions of Students
• Wednesday PMPlanning for Implementation
• Thursday AMDocumenting Patterns of Student
  Thinking
• Thursday PMClosing Session

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Group Norms

• Begin and end on time.
• Respect your colleagues ideas and opinions.
• Monitor your own participation.
• When working in groups, allow time for group
  members to read and think about the problem
  before beginning your discussion.
• Only one conversation should take place in a
  group at a time.

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Agenda

• 900-930 Announcements
• 930-955 Introduction to A-HOMs
• 955-1025 Postage Stamp Problem
• 1025-1040 Break
• 1040-1120 Postage Stamp Discussion
• 1120-1145 Making a Mathematical Thinking
  Record
• 1145-1200 Group Process Discussion
  Announcements

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Agenda

• 100-130 A-HOMs Discussion
• 130-215 Crossing the River Problem
• 215-230 Break
• 230-300 Crossing the River Discussion
• 300-400 Crossing the River Example Papers

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Introductory SessionGoals

• Build the foundations of a comfortable and
  productive study group.
• Familiarize yourselves with the FAT sessions and
  some of the tools, such as the Mathematical
  Thinking Record (MTR).
• Explore the concepts of algebraic habits of mind.
• Become comfortable working on mathematics
  activities together and sharing mathematical
  ideas.

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Think about the phrase habits of mind.

• Have you heard this phrase before in the context
  of mathematics?
• What does the phrase mean to you?
• What ideas or other phrases does it bring to mind?

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Look at the Algebraic Habits of Mind Diagram and
Table.

• The algebraic habits of mind are a language for
  describing algebraic thinking. We will use this
  language as a tool to understand and talk about
  the kinds of thinking that you and your students
  do about mathematics.

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Look at the features of the different habits of
mind.

• Which of these lines of thought seem familiar to
  you?
• Can you think of things you have seen your
  students do that indicate that they are engaging
  in these productive lines of thought?

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Postage Stamps

• In groups of four people, work on the Postage
  Stamps math activity.
• While working on this problem, think about the
  methods people in your small group tried, the
  questions they asked, the process for coming to a
  deeper understanding, and the different ways of
  thinking about the problem.
• Post your groups work.

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Postage Stamps Discussion

• In what ways is this problem algebraic? How
  does it elicit algebraic thinking?
• You may have noticed yourself working from output
  to input. How did different group members work
  from output to input to answer questions such as
  How can I make 53 worth of postage?

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• What computational shortcuts did group members
  use as they worked on the problem?
• How were these shortcuts useful?
• What rules did group members come up with to help
  them generate postage values of 5 and 7 stamps?

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Mathematical Thinking Record (MTR)Postage Stamps

• What would you like to recall about the different
  strategies and/or solutions used by your
  colleagues? Record the approaches and strategies
  you would like to remember.
• What would you like to recall about the
  algebraic thinking? Record the specific features
  of habits of mind that you have seen in the
  different solutions.
• What would you like to recall about the different
  strategies and/or solutions used by your
  students? Record the mathematical approaches or
  strategies you would like to remember.

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Group Process Discussion

• How does the way the group works help you develop
  a spirit of inquiry and ask questions about
  algebraic thinking or the teaching of algebraic
  thinking?
• How could the group do this better?

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Analyzing Written Student Work Goals

• Explore the Algebraic Habits of Mind.
• Examine algebraic thinking in your own and your
  colleagues written work.
• Use student written work as data during the
  process of exploring algebraic thinking.
• Explore the range of algebraic ideas that can
  occur in students thinking .
• Look for potential in students written work.

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Algebraic Habits of Mind (A-HOMs)

• Doing-Undoing
• Building Rules to Represent Functions
• Abstracting from Computation

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Doing-Undoing Features

• Input from output
• Working backward

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Building Rules to Represent Functions Features

• Organizing information
• Predicting patterns
• Chunking the information
• Describing a rule
• Different representations
• Describing change
• Justifying a rule

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Abstracting from Computation Features

• Computational shortcuts
• Calculating without computing
• Generalizing beyond examples
• Equivalent expressions
• Symbolic expressions
• Justifying shortcuts

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Crossing the River

• Work with the members of your group on the
  Crossing the River activity.
• As you work, think about the strategies you are
  using to solve the problem.
• Post your groups work.

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Crossing the River Discussion

• Did everyone come up with the same solution (or
  partial solution) to the problem? Why or why
  not?
• What aspects of algebraic thinking were involved
  in the various approaches?
• What might the strategies for solving this
  problem indicate about understanding the
  algebraic concept of variable?
• The last question is sometimes difficult for
  students. Why do you think that is?

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MTRCrossing the River

• What would you like to recall about the different
  strategies and/or solutions used by your
  colleagues? Record the approaches and strategies
  you would like to remember.
• What would you like to recall about the
  algebraic thinking? Record the specific features
  of habits of mind that you have seen in the
  different solutions.
• What would you like to recall about the different
  strategies and/or solutions used by your
  students? Record the mathematical approaches or
  strategies you would like to remember.

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Crossing the River Example Papers

• Follow the instructions in Activity 1, pages 5-14.

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Group Papers

• The small group discusses What evidence do you
  see in these papers of the habit of mind Building
  Rules to Represent Functions?How did students
  organize information?In what ways do they
  describe any rules they are building?Do any
  other features of Building Rules to Represent
  Functions play out in the student work?
• What evidence do you see in these papers of
  Doing/Undoing or Abstracting from Computation?

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For Tuesday

• Review Sums of Consecutive Numbers activity.
• Review The Staircase Problem.
• Select one or two examples of student work to
  bring to the group.
• Read Algebraic Thinking Tasks.