Fostering Algebraic Thinking

1. May 19-22, 2008 Fostering Algebraic Thinking

2. A core belief underlying Fostering Algebraic Thinking is that good mathematics teaching begins with understanding how mathematics is learned.

3. While the materials provide activities for teachers to do with students, the primary focus is on TEACHER learning—in the belief that student learning will also be served.

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

5. 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.

6. Fostering Algebraic Thinking Modules • Analyzing Student Written Work • Listening to Students • Asking Questions of Students • Documenting Patterns of Student Thinking

7. 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

8. 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.

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

10. Agenda • 1:00-1:30 A-HOMs Discussion • 1:30-2:15 Crossing the River Problem • 2:15-2:30 Break • 2:30-3:00 Crossing the River Discussion • 3:00-4:00 Crossing the River Example Papers

11. 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.

12. 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?

13. 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.

14. 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?

15. 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 group’s work.

16. 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?”

17. 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?

18. 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.

19. 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?

20. 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.

21. Algebraic Habits of Mind (A-HOMs) • Doing-Undoing • Building Rules to Represent Functions • Abstracting from Computation

22. Doing-Undoing Features • Input from output • Working backward

23. Building Rules to Represent Functions Features • Organizing information • Predicting patterns • Chunking the information • Describing a rule • Different representations • Describing change • Justifying a rule

24. Abstracting from Computation Features • Computational shortcuts • Calculating without computing • Generalizing beyond examples • Equivalent expressions • Symbolic expressions • Justifying shortcuts

25. 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 group’s work.

26. 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?

27. MTR—Crossing 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.

28. Crossing the River Example Papers • Follow the instructions in Activity 1, pages 5-14.

29. 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?

30. 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”.