Functions in the Elementary
Grades: A Context for Work
on Multiplication
Deborah Schifter, EDC, Inc.
Virginia Bastable, Mount Holyoke College
Susan Jo Russell, TERC
Two trends in mathematics
education
Algebra in the elementary and middleschool grades
 The standards movement
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Teams studying elementary and
middle-school algebra
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Jim Kaput and Maria Blanton
Tom Carpenter, Megan Franke, and Linda Levi
David Carraher and Analucia Schliemann
Mark Driscoll
Judy Mumme and Nanette Seago
Peg Smith
Deborah Schifter, Virginia Bastable, and Susan
Jo Russell
Two branches of early algebra
Generalizing and justifying
 Functions
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PD in elementary and middleschool algebra
Thinking Mathematically
 Learning and Teaching Linear Functions
 Fostering Algebraic Thinking
 Improving Instruction in Algebra
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Two new DMI modules
Reasoning Algebraically about Operations
 Patterns, Functions, and Change
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Algebra in Elementary
Curriculum Materials
For example, the second edition of
Investigations in Number, Data, and
Space
 Generalizing and justifying included in
Number and Operations units
 A unit on Patterns and Functions at each
grade, K-5
Standards Movement
1989 publication of the first of the NCTM
Standards series
 State frameworks
 Statewide testing
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Problems we have run into
Viewing the curriculum as a checklist
 A curriculum that is “a mile wide and an
inch deep”
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Curriculum Focal Points
Produced as a response to these
problems
 Recommends three main mathematical
themes for each grade
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Dangers of a narrow reading
Going from a curriculum that is
a mile wide and an inch deep
to one that is
an INCH wide and an inch deep
How will the study of functions
be treated?
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Will it be dropped?
How will the study of functions
be treated?
Will it be dropped?
 Or will it be seen as basic support for the
core content of the elementary grades?
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Staircase Towers
Start with 1
 Add 2 each time
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Create a sequence for the number of cubes in
each tower.
Penny Jar
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There are 5 pennies in the jar.
You add 3 pennies each round.
After one round, there are 8 pennies in the jar.
Create a sequence for the number of pennies in
the jar after 1 round, 2 rounds, 3 rounds, etc.
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What mathematics might first graders
encounter as they engage with these
tasks?
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How might this work connect to
developing ideas about multiplication?
First Grade Students with Kirsten
What is the mathematics that these children
are engaged in?
How does their work connect to their developing
ideas about multiplication?
Cube Train
1
2
3
4
5
6
7
8
9 10 11 12
The pattern of YYOOB cubes continues.
Create a table that identifies the position
of the 1st blue, 2nd blue, 3rd blue, …
Building Buildings
Each floor in a set of buildings has 3 rooms.
Create a table that identifies the number of
rooms in a building with 1 floor, 2 floors,
3 floors, 4 floors, 5 floors, 10 floors.
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What mathematics might second graders
encounter as they engage with these
tasks?

How might this work connect to
developing ideas about multiplication?
Second Grade Students with Mike
What is the mathematics that these children
are engaged in?
How does their work connect to their developing
ideas about multiplication?
Penny Jar with fourth graders
Start with 3
Add 4 each round
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What mathematics might fourth graders
encounter as they engage with these
tasks?
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How might this work connect to
developing ideas about multiplication?
Fourth Grade Students with Nancy
What is the mathematics that these children
are engaged in?
How does their work connect to their developing
ideas about multiplication?