The Mathematics
Curriculum K–12
A STANDARDS-BASED Search
for Coherence
John A. Dossey
Illinois State University
Jeremy Kilpatrick
University of Georgia
Curriculum
• Not just a syllabus or textbook
• Includes
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Aims
Content
Methods
Assessment procedures
• Depends on one’s position
• Plan
• Experience
• A linear path through a multi-dimensional domain
What is a curriculum?
• A plan of action directing the content and delivery
of a program of mathematics learning for a
specified group of students
• This plan consists of an articulated set of
statements that provide:
• general goals for the plan
• a structure of content domains and cognitive processes
to be developed
• a philosophy concerning the general conditions of
learning expected
• a listing of general standards and specific outcomes of
learning related to levels of schooling germane to the
population for which the curriculum is intended
The Importance of Mission
“A discussion of mathematical education, and of ways
and means of enhancing its value, must be
approached first of all on the basis of a precise and
comprehensive formulation of the valid aims and
purposes of such education. Only on such basis can
we approach intelligently the problems relating to the
selection and organization of material, the methods of
teaching and the point of view which should govern
instruction.” (The Reorganization of Mathematics in
Secondary Education, 1923)
Source of Goals
Society/Workforce
Psychology
Mathematics
Education
Discipline
Assessments
Most Curricula do not answer
Who?, What?, Where?, Why?
• Situating a curriculum
• NCTM and PISA discuss: Personal Life, Workplace,
Cultural and Societal, and Scientific and Technological.
• College Board Standards for College Success:
Mathematics and Statistics: “College Board presents
these standards [as a guide to] provide all students with
the rigorous education that will prepare them for
success in college, opportunity in the workplace, and
effective participation in civic life.”
Recommendation 1
• What is the purpose of the curriculum?
• Who is it to serve?
• What are the intended outcomes in:
• Broad brush strokes (overview)
• Unit level content and processes (teachers and
materials)
• Specific concepts, principles, and skills
(specialist and resource guide level)
Previous Periods of
Curricular Reform
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Early 1900s
....
Late 1950s–late 60s
Late 1980s–90s
Early 2000s
Commissions
“New math”
Standards
Based on
assessments
Today & Recent Past
• NCLB and state “curriculum” reforms
• College Board’s Standards for College
Success: Mathematics and Statistics (2009)
• NCTM Curriculum Focal Points (Pre-K–8)
(2006) & Focus in High School Mathematics:
Reasoning and Sense Making (9–12) (2009)
• NGA & CCSSO release draft of College- and
Career-Readiness Standards (2009), the first
stage of the Common Core State Standards
Initiative -- a move toward a curriculum
http://www.corestandards.org/
Structuring a curriculum
• How are the “levels” laid out?
• How is the “content” divided?
• How are the “processes” described and
interwoven?
• How is knowledge itself related to intended
learning (and to envisioned pedagogy)?
• Problems:
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Connection
Linearity
Differentiation
Choice
Providing Structure
Cognitive
Processes
Content
Domains
Content Domains
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Number & operations
Algebra & functions
Geometry & transformations
Measurement
Data analysis
Chance
• Be careful to:
• Avoid straying too far from a concise list
because that rapidly leads to a lack of focus
• Make sure there are ample areas to connect!
E.g., algebra is algebra?
• Margaret Kendal and Kaye Stacey in The Future of
the Teaching and Learning of Algebra: The 12th
ICMI Study (Stacey et al., 2004): “Don’t take your
country’s curriculum and approach to teaching
algebra for granted and don’t assume all other
educational jurisdictions operate in a similar way—
they conspicuously do not.”
• Striking differences in
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Who takes algebra
Whether integrated or layered across years
Emphasis put on generality and pattern
Attention to symbolism, formalism, and abstraction
Whether approached through functions and multiple representations
Role played by technology
Interconnected Content
Domains & Cognitive
Processes
Relationships
Content
Domains
Concepts
Procedure
Facts
Recommendation 2
• Keep the lists of content and processes
focused and concise
• Do not stray greatly from current models
without strong justification—it just makes
work for teachers and costs money—little real
change
• Make sure statements of goals, standards, and
outcomes have clarity and reflect a range of
cognitive processes
A Philosophy Of Pedagogy
• Reasoning and Sense Making
• Teaching should value and emphasize reflective reasoning
(sense making) on the part of the student in order to promote
construction of new mathematical concepts based on
previously learned concepts and new ideas.
• Teaching should help students see the need for justifying their
work and help students to develop justification strategies.
• Problem Solving and Mathematical Modeling
• Teaching should engage students in solving open-ended
problems, in groups and individually, to illustrate the need for
mathematics and demonstrate the connectedness of
mathematical ideas.
• Teaching with a problem-solving approach should encourage
students to investigate mathematical models as a vehicle for
analyzing problems, making sense of related content and
structures, and finding problem solutions.
A Philosophy Of Pedagogy (cont.)
• Multiple Perspectives
• Teaching should develop students’ capabilities to look at
mathematical situations and outcomes from multiple
perspectives, compare and critique methods, and reflect on
the value of each.
• Teaching mathematics should encourage students to
employ appropriate tools and techniques such as
computing, graphing, and other analytical methods.
• Mathematical Autonomy
• Teaching mathematics should support student
development and ownership of mathematical ideas.
• Teaching mathematics should strengthen students’
capability to illustrate and communicate their
mathematical ideas, imbuing each student with the belief
that they have the foundations and related capabilities
required to succeed in mathematics.
Pedagogical Framework
• While the instructional implementation is a
partially separate matter, a broad statement of
the boundaries and shape of the program
should be included.
• “What you learn reflects how you learned it.”
• Such a framework also reflects the approaches
and conditions for achieving “depth of
understanding.”
Coherence
Above all—a curriculum must have coherence.
Where coherence can be measured as the ratio of
the number of outcomes that link to other
outcomes when compared to the total number of
potential outcome pairs.
In addition, coherence relates to the overall
consistency of expected depth of understanding
and cognitive level consistency.
Recommendation 3: ATTEND
to the Challenges
• Attend to the related teaching models
• Understand difficulty of making “real change”
• Strive for balance: content and processes
• Get buy-in of schools, teachers, parents
• Be responsive to challenges of diversity – ALL CAN
DO IT and IT IS FOR ALL!
• Give it time to grow and prosper
• Provide for systematic revision