Title: Fostering Algebraic Thinking
1Fostering Algebraic Thinking
2A core belief underlying Fostering Algebraic
Thinking is that good mathematics teaching
begins with understanding how mathematics is
learned.
3While 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.
4Goals
- Become familiar with the Fostering Algebraic
Thinking materials. - Examine activities that may be challenging to
facilitate. - Develop plans for implementation at your sites.
5We 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.
6Fostering Algebraic Thinking Modules
- Analyzing Student Written Work
- Listening to Students
- Asking Questions of Students
- Documenting Patterns of Student Thinking
7Agenda 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
8Group 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.
9Agenda
- 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
10Agenda
- 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
11Introductory 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.
12Think 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?
13Look 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.
14Look 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?
15Postage 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.
16Postage 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?
18Mathematical 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.
19Group 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?
20Analyzing 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.
21Algebraic Habits of Mind (A-HOMs)
- Doing-Undoing
- Building Rules to Represent Functions
- Abstracting from Computation
22Doing-Undoing Features
- Input from output
- Working backward
23Building Rules to Represent Functions Features
- Organizing information
- Predicting patterns
- Chunking the information
- Describing a rule
- Different representations
- Describing change
- Justifying a rule
24Abstracting from Computation Features
- Computational shortcuts
- Calculating without computing
- Generalizing beyond examples
- Equivalent expressions
- Symbolic expressions
- Justifying shortcuts
25Crossing 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.
26Crossing 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?
27MTRCrossing 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.
28Crossing the River Example Papers
- Follow the instructions in Activity 1, pages 5-14.
29Group 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?
30For 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.