Connecting Arithmetic to
Algebra
An On-line Course for Teachers
Virginia Bastable
Deborah Schifter
Susan Jo Russell
TDG 2011
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
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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:
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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
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Key Structures of CAA
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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
Part Two
5. Developing
1. Generalizing in
Mathematical
arithmetic--noticing
Arguments Part I
2. Generalizing in
arithmetic, getting
6. Focus on the Range of
started
Learners
3. Generalizing in
Arithmetic with the
range of learners
7. Learning Algebraic
notation
4. Articulating general
claims
Part Three
8. Developing
Mathematical
Arguments Part II
9. Looking ahead to
middle grades
10. Building across
the school year
What participants do:
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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
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Identify passages that help you make
connections to your own students.
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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:
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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
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What do you learn about the participants
by reading their responses?
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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
Is this number sentence true?
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?
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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
Argument #4:
See this is a 2 by 6, and
this is a 4 by 3, and
they both have 12.
Argument #5:
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.
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:
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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:
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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
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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.
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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.”
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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
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What’s happened so far
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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:
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Teacher assessments
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Teacher evaluations
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Student assessments
To receive information regarding CAA options for
2011-2012 school year:
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email vbastabl@mtholyoke.edu
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Include CAAFall11 in the subject line
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.”
Working on math in teams
Adding 1 to a factor in a multiplication expression
“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.”
Working on math in teams
Adding 1 to a factor in a multiplication expression
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.
Working on math in teams
Adding 1 to a factor in a multiplication expression
“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
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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
Is this number sentence true?
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
6x2=3x4
6x2=3x4
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.
6x2=3x4
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!
6x2=3x4
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
move 5 cookies
237 cookies
232
195 cookies
200
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.