Goals Become familiar with the Fostering Algebraic Thinking materials. Examine activities that may be challenging to facilitate. Develop plans for implementation at your sites.
While the materials provide activities for teachers to do with students, the primary focus of this training is teacher learning- in the belief that the student learning will also be served.
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.
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.
What is Algebraic Thinking? Take 5 minutes to discuss in your group what you think Algebraic Thinking really is. Video
What does algebraic thinking really mean? Two components of algebraic thinking, the development of mathematical thinking tools and the study of fundamental algebraic ideas, have been discussed by mathematics educators and within policy documents (e.g., NCTM, 1989, 1993, 2000; Driscoll, 1999). Mathematical thinking tools are analytical habits of mind. They include problem solving skills, representation skills, and reasoning skills. Fundamental algebraic ideas represent the content domain in which mathematical thinking tools develop. Within this framework, it is understandable why conversations and debates occur within the mathematics community regarding what mathematics should be taught and how mathematics should be taught. In reality, both components are important. One can hardly imagine thinking logically (mathematical thinking tools) with nothing to think about (algebraic ideas). On the other hand, algebra skills that are not understood or connected in logical ways by the learner remain "factoids" of information that are unlikely to increase true mathematical understanding and competence. From Shelley Kriegler's project "Mathematics Content Programs for Teachers," UCLA Department of Mathematics, January 2000.
A core belief underlying Fostering Algebraic Thinking is that good mathematics teaching begins with understanding how mathematics is learned.
Consider the following: It's a hot summer day, and Eric the Sheep is at the end of a line of sheep waiting to be shorn. There are 50 sheep in front of him. Being an impatient sort of sheep, though, every time the shearer takes a sheep from the front of the line to be shorn, Eric sneaks up two places in line. Without working out the entire problem, predict how many sheep will get shorn before Eric.
Describe the strategies you used to find the answer to the problem and how you could predict the answer for any number of sheep in the line. Is your method for predicting "algebraic"? Why or why not?
Eric gets more and more impatient. Explore how your rule changes if Eric sneaks past 3 sheep at a time. How about 4 sheep at a time? 10 sheep at a time?
When someone tells you how many sheep there are in front of Eric and how many sheep at a time he can sneak past, describe how you could predict the answer.
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?
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.
Features of the 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?
Postage Stamp Problem In groups of six 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. Wait for instructions to post your group’s work.
Postage Stamp Problem -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?”
Postage Stamp Problem -Discussion 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?
Mathematical Thinking Records - Postage Stamp 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.
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?
Algebraic Habits of Mind (A-HOMs) Doing-Undoing Building Rules to Represent Functions Abstracting from Computation
Doing and Undoing -Features Input from output Working backward
Doing and Undoing - Questions How is this number in the sequence related to the one that came before? What if I start at the end? Which process reverses the one I am using? Can I decompose this number or expression into helpful components?
Building Rules to Represent Functions -Features Organizing information Predicting patterns Chunking the information Describing a rule Different representations Describing change Justifying a rule
Building Rules to Represent Functions -Questions Is there a rule or relationship here? How does the rule work, and how is it helpful? Why does the rule work the way it does? How are things changing? Is there information here that lets me predict what’s going to happen? Does my rule work for all cases? What steps am I doing over and over?
Abstracting from Computation -Questions How is this calculating situation like/unlike that one? How can I predict what’s going to happen without doing all the calculation? What are my operation shortcut options for getting from here to there? When I do the same thing with different numbers, what still holds true? What changes?
Do your students seem to have an easier time with one of the habits of mind than with the others? What aspects of algebraic thinking have you found to be more difficult or less difficult for your students to express in writing?
Crossing the River Problem 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. Wait for instructions to post your group’s work.
Crossing the River Problem -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?
Crossing the River Problem -MTR 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.
Homework Read Article, “Algebraic Thinking Tasks” Try one of the problems with kids Bring 2 samples of student work back Be ready to share
See you December 2! Janna Smith firstname.lastname@example.org (409) 923-5488