A Problem-based Curriculum
Tom Sallee
University of California, Davis
Outline
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How do we get more students to learn algebra 1?
Math Goals. Attitude Goals.
Learning Approach.
Examples of how approach was implemented.
How did we approach writing the books?
Lessons learned.
If you want to try this yourself.
Questions
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CPM Fast Facts
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CPM has developed curriculum for 19 years.
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Is a non-profit organization, and has
curriculum for grades 6 through 12.
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Was started with an Eisenhower grant…
not NSF
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Written by 6-12 teachers, mostly from
California
• Was heavily influenced by the 1985 and 1992
California Frameworks and the 1989
NCTM Standards
• Has evolved significantly
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Primary Focus
Getting more students to learn algebra 1,
retain their knowledge, and be able to
transfer it, not just “cover the material”.
Originally the first year of a three-year
sequence.
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Central Issue
Difficulties of most students
are more about
Learning
than about Mathematics
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Math Goals for Students
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Understand the Big Ideas as a connected set
of concepts
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Be able to move among different
representations of the same concept:
written, tabular, graphical, symbolic.
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Use Problem Solving techniques as both a
solution tool and a learning tool
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Attitude Goals for Students
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I can figure out most problems without being
told by the teacher.
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I want to learn math.
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I want to understand what I learn.
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Big Ideas
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Representing functions with equations,
graphs, tables, and contextual situations,
and making connections among these
representations.
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Writing equations from contexts (word
problems)
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Solving equations, systems of equations, and
inequalities
•Symbolic manipulation, using equivalence
•Proportionality
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Learning Approaches
• Math is not a spectator sport
Work matters.
Engagement matters.
• Solving problems is the best way to learn new
ideas.
• Talking about mathematics with others will
help you understand new ideas.
• Connecting abstract concepts (like factoring
trinomials) with concrete experiences (like
manipulating building rectangles with
algebra tiles) helps integrate your
knowledge.
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Learning Approaches
• You will retain ideas better if practice is spaced
over weeks or months
• It takes a long time to learn a big idea.
• There are mathematical ways of thinking (such
as generalizing, justifying, connecting) that
take time and practice to develop.
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Multiple Representations
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Use of Algebra Tiles
Symbolic manipulation is developed through
use of concrete tools
• “Legend” reminds
students and teachers
which tiles are positive
and negative
• “Minus” region negates
the tiles in that region,
helping students
represent the opposite
of a negative.
2x  x 1 (2  3)
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Intro to simultaneous equations
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Student tasks for problem
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Guidance as necessary
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Introduction of a new idea
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End of the problem
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Setting up equations
A rectangle is 3 cm longer than it is wide and has a
perimeter of 54 cm. What are its dimensions? Write
an equation that will allow you to solve this problem.
Guess side
Other side
Perimeter
=60?
10
13
46
low
15
18
66
high
14
17
62
high
x
x+3
2x+2(x+3)
=60
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Setting up equations
A rectangle is 3 cm longer than it is wide and has a
diagonal of 30 cm. What are its dimensions? Write an
equation that will allow you to solve this problem.
Guess
side
Other
side
10
13
15
18
20
23
x
x+3

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Diagonal
100 169 16.4
225  324  23.43
400  529  30.48
x 2  x  3
2
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=30?
low
low
high
=30
How did we approach
writing the book?
Constrained optimization problem.
Have talked about math goals and
attitude goals for students.
What were constraints?
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Constraints
Assumptions need to be made about
• Students
• Teachers
• Schools
• States
• Parents
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Examples of our Assumptions
Students
Most think math is something to be memorized.
Many of those we are most anxious to reach will
not have a place to do homework.
Teachers
A course that requires more work will not be kept.
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Examples of our Assumptions
About Constraints
Schools
Generally a new program must co-exist with the old.
States
Standards, frameworks, accountability.
Parents (Mostly parents of high-ability students)
I need to be able to help my child.
There needs to be plenty of practice.
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Generation 2
Algebra Connections
Major differences
• Clearer storylines of the mathematics
• More transparent daily structure--eg what is
homework?
• Made the mathematical goal of problems, lessons,
and chapters explicit.
• Teacher has the option of presenting problems
with less scaffolding, so teamwork is more
necessary.
• Much more extensive teacher notes
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Biggest Things We Learned
1. Can’t just write a book with a new approach
Need LOTS of professional development to go
with it. Current model--eight days (free)
inservice for each course plus classroom visits
2. Politics matters.
3. In judging effectiveness, facts matter a lot
less than personal prejudices.
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If you want to do this yourself
• Think very hard about the students for whom
you are writing the books and your goals
for them.
• Trust yourself on the math.
• Trust teachers on pedagogy. Don’t ever think
you know more about what will work in a
classroom than a good teacher.
• Go sit in classrooms and find out what the
reality is before you begin.
• Iterate your efforts.
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Get Involved in K-12 Math
•There is a need,
•There is funding, and
•It is the most fun you will
ever have.
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