connecting arithmetic to algebra

Connecting Arithmetic to Algebra

An On-line Course for Teachers

Virginia Bastable

Deborah Schifter TDG 2011

Susan Jo Russell

Foundations of Algebra project

A collaboration of the Education Research Collaborative at TERC, the Education Development Center (EDC), and SummerMath for Teachers (Mt. Holyoke College), and 25 teachers

Funded in part by the National Science Foundation

5-year project

Connecting Arithmetic and Algebra

Today’s Session

Goals and structure of the on-line course What do CAA participants do? What do CAA course facilitators do? What are we learning from this experience?

Integrating algebraic thinking into arithmetic instruction involves:

Investigating, describing, and justifying general claims about how an operation behaves

A shift in focus from solving individual problems to looking for regularities and patterns across problems

Representations of the operations as the basis for proof

The operations themselves become objects of study

What do teachers need to learn?

the mathematics content how to recognize opportunities instructional strategies how a range of students can engage with

this content

Key Structures of CAA A year-long course in three 6-week parts Every participant is part of a local team; each

section includes approximately 20 participants Assignments are posted on a course web board;

participants respond to one another’s postings Each 6-week part includes two 2-hour webinars

to allow for “live” discussions and interactions Each 6-week part includes 3 Student Thinking

assignments; participants receive personal responses from facilitators on these.

CAA Book Chapters and Course Sections

Part One

1. Generalizing in arithmetic--noticing

2. Generalizing in arithmetic, getting started

3. Generalizing in Arithmetic with the range of learners

4. Articulating general claims

Part Two

5. Developing Mathematical Arguments Part I

6. Focus on the Range of Learners

7. Learning Algebraic notation

Part Three

8. Developing Mathematical Arguments Part II

9. Looking ahead to middle grades

10. Building across the school year

What participants do: Read and respond to each chapter of the

course text Respond to each other on the web board Do math activities with their team Carry out lessons with their class and write

about them (“student thinking assignments”) Participate in 6 2-hour webinars

Focus Questions Reading One

Identify passages that help you make connections to your own students.

Identify passages that illustrate a specific question or action of the teacher that interests you. What was the impact of this teacher move on the students’ thinking?

What facilitators do: Read and take notes on web board responses Adapt webinar plans to react to participants’

postings Co-teach the webinars Respond to participants’ student thinking

assignments individually

Focus Questions Reading Two

What do you learn about the participants by reading their responses?

What ideas might you want to bring out at the next webinar as a result?

Is this number sentence true?

35 + 19 = 34 + 20

Explain how you know it is true or not

without calculating


35 - 19 = 34 - 20

What general claim is suggested by this set of equations?

6 x 8 = 12 x 4 150 x 64 = 300 x 32 35 x 8 = 70 x 4

Develop at least two representations (story situations, diagrams, etc.) to illustrate this claim.

Can you use your representations to talk about the claim without referring to specific numbers.

Analyzing Web board Math Responses

Read the two responses to this assignment (Reading Three)

•What do you learn about the math understandings of the participants?

•What else do you want them to learn?

.

Examples of student arguments to justify this claim

1. What does each argument show that the student understands about proving the general claim?

2. What more would the student need to do to move towards proving this claim?

Student argument #1

I figured out that 2 times 6 equals 4 times 3, and also 8 times 10 equals 4 times 20. So it works.

Student arguments #2 and #3

Argument #2.

I did a story context. I have 2 stacks of books, and each one has 6 books. That’s 12 books. Then I have 4 stacks of books, and each one only has 3 books. That’s 12, too. So they’re the same.

Argument #3. I have 2 stacks of books, and each one has 6 books. But the stacks were too heavy to carry, so I put each stack in half. Now there are 4 stacks and each has 3 books. So when I doubled the number of stacks, there was only half of the books in a stack than there was before.

Student arguments #4 and #5

See this is a 2 by 6, and this is a 4 by 3, and they both have 12.

I cut the 2 by 6 in half, and I put one piece underneath. It’s half across the top, but now it’s twice as long. It’s all the same stuff I started with, like if this was a carpet and I cut it and moved it around.

Argument #4: Argument #5:

Representation-based proof• The meaning of the operation(s) involved in the

claim is represented in diagrams, manipulatives, or story contexts.

• The representation can accommodate a class of instances (for example, all whole numbers).

• The conclusion of the claim follows from the structure of the representation; that is, the representation shows why the statement must be true.

Would any of these arguments hold if the numbers under consideration were not whole?

What participants do: Read and respond to each chapter of the

course text Respond to each other on the web board Do math activities with their team Carry out lessons with their class and write

about them (“student thinking assignments”) Participate in 6 2-hour webinars

What facilitators do: Read and take notes on web board responses Adapt webinar plans to react to participants’

postings Teach the webinars Respond to participants’ student thinking

assignments individually

Early algebra

Notice a regularity about an operation

Articulate the generalization Prove why the claim is true

What we are finding

Elementary grade students are interested in examining generalizations about the behavior of the operations.

Such discussions engage a range of students and support the development of computational fluency.

Visual representations and story contexts provide a mechanism for proof accessible to elementary grade students.

Implications

Elementary school: Studying the behavior of the operations supports the development of computational fluency.

Middle school: “The kids who have the deepest trouble with middle school math are those without a clear and rich set of models for what multiplication is and how it is different from addition.”

Summary On-line/off line Asynchronous/synchronous interaction Individual/team/webinar groupings Responses from

individuals/teams/facilitators Alternate course work focused on their

own math and implementation with students

What’s happened so far Participants are actively engaged in the

mathematics content (in teams, on web board) Tape recording and analyzing class sessions is

powerful (student thinking assignments) A focus on general claims is being integrated

into instruction Students are engaged in significant

mathematical thinking

Data we’re collecting:

Teacher assessments

Teacher evaluations

Student assessments

To receive information regarding CAA options for 2011-2012 school year:

email [email protected]

Include CAAFall11 in the subject line

mailto:[email protected]

Working on math in teams

Adding 1 to a factor in a multiplication expression

“I found that after writing the story and drawing the boxes of erasers, how helpful the story context could be for some of my students. The story context may make the statement easier to relate to for the children that struggle with relating to numbers.”



“Each time we solve a problem or examine a statement as a group, I am amazed at the different ways we all think about the problem. I always solve the problem and think about it before meeting with our group and then I always walk away with a new way of thinking about solving or representing the problem. It reminds me how important it is to have my students share their different strategies with their peers.”



General claim #1: If you add n to factor b in a multiplication expression a x b, you add a x n to the result. I have 3 baskets with 4 apples in each basket. If I add an apple to each basket I added 3 more apples, one for each basket.

General claim #2: If you add n to factor a in a multiplication expression a x b, you add b x n to the result. I have 3 baskets with 4 apples in each basket. If I fill another basket with the same number of apples, I have added 4 more apples.



“What was so interesting was that my partner changed the first factor in the expression and I changed the second factor in the expression. This is how we discovered that we needed to have two general claims that could explain both situations.”

What’s happened so far Participants are actively engaged in the

mathematics content (in teams, on web board) Tape recording and analyzing class sessions is

powerful (student thinking assignments) A focus on general claims is being integrated

into instruction Students are engaged in significant

mathematical thinking

Grade 1


3 + 7 = 7 + 3

Grade 1

Are these the same amount?

3 + 7 7 + 3

Grade 1

My class and I continue to become more and more comfortable with asking “why”, using models to represent ideas, and pushing for articulation. These are connected processes that are hard to look at separately. It is helpful to analyze transcripts of classroom conversation to practice understanding students’ ideas and how to respond to them. . . .

Grade 1

As we move forward with the version of this routine, “Are These the Same Amount?”, I will be interested in hearing more about what it sounds like for first graders to articulate generalizations and what it sounds like if and when teachers press them to leave the specific numbers behind. Perhaps we can also revisit “Is this Number Sentence True?” and check for development in students’ ideas about the meaning of the equal sign.

What’s happened so far

• Participants are actively engaged in the mathematics content (in teams, on web board)

• Tape recording and analyzing class sessions is powerful (student thinking assignments)

• A focus on general claims is being integrated into instruction

• Students are engaged in significant mathematical thinking

Grade 3

6 x 2 = 3 x 4

6 x 2 = 3 x 4

Thad: I have 6 apples and my mom gave me 2 more. My sister had 3 apples and my mom gave her 4 more. We have the same amount of apples.

Teacher: So, Thad, I have 6 apples and I got 2 more, how many do I have?

Thad: 8

Teacher: And if sister has 3 and gets 4 more how many does she have?

Thad: 7. Oh I didn’t do it right. I was adding.

6 x 2 = 3 x 4

Teacher: And where is this story problem showing that doubling or halving that we really wanted to model?

Thad: It doesn’t.

Teacher: So let’s see if we can change your story to show both multiplication and the doubling and halving. We can start with the same 6 apples. But how can we show 6 x 2?

Martin (Thad’s partner who is listening in): Each apple has two worms!

6 x 2 = 3 x 4

I have 6 apples with 2 worms in each apple.

If my sister only has 3 apples and the worms crawl over to her apples, they will have to double up, so there will be 4 worms in each apple.

The teacher reflects:

I walked away from this class session knowing so much about what my students were thinking. I saw a common mistake when writing story problems for multiplication as Thad wrote an addition problem. I think this could be a great story to return to later when I want to compare the behavior of addition and multiplication.

Grade 4

Write a word problem, draw a picture, or another kind of representation to convince or prove to another class of fourth graders that

237 + 195 = 232 + 200

Prove that 237 + 195 = 232 + 200

Two cookie jars

237 cookies 195 cookies

Prove that 237 + 195 = 232 + 200

237 cookies 195 cookies232 200

move 5 cookies

Sam: If you have a cake and you cut it into two pieces even if one of the pieces is bigger than the other and then you put the two pieces together you still have one cake.

Teacher: Oh, what happens if one person gives some of their piece to the other person?

Sam: Then you are just taking a little bit away from one and giving it to the other but you still have one whole cake.

Keith: It’s like x + y - y = x.

Derrick: I don’t understand.

Teacher: Can you help Derrick understand by explaining what you said in numbers and words?

Keith: If you have any number plus any number, to make an equivalent expression you add an amount to the first any number, then you need to take that same amount away from the second any number so they stay the same.

connecting arithmetic to algebra

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