May 19-22, 2008

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

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

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

 Monday AM Introductory Session  Monday PM Analyzing Written Student Work  Tuesday AM Analyzing Written Student Work  Tuesday PM Listening to Students  Wednesday AM Asking Questions of Students  Wednesday PM Planning for Implementation  Thursday AM Documenting Patterns of Student Thinking  Thursday PM Closing Session

 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.

 9:00-9:30Announcements  9:30-9:55Introduction to A-HOMs  9:55-10:25Postage Stamp Problem  10:25-10:40Break  10:40-11:20 Postage Stamp Discussion  11:20-11:45Making a Mathematical Thinking Record  11:45-12:00Group Process Discussion Announcements

 1:00-1:30A-HOMs Discussion  1:30-2:15Crossing the River Problem  2:15-2:30Break  2:30-3:00Crossing the River Discussion  3:00-4:00Crossing the River Example Papers

 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.

 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?

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.

 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?

 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 group’s work.

 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?”

 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?

 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.

 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?

 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.

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

 Input from output  Working backward

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

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

 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 group’s work.

 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?

 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.

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

 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?

 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”.