The Mathematics Curriculum K

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The Mathematics Curriculum K12

• A STANDARDS-BASED Search for Coherence
• John A. Dossey
• Illinois State University
• Jeremy Kilpatrick
• University of Georgia

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Curriculum

• Not just a syllabus or textbook
• Includes
• Aims
• Content
• Methods
• Assessment procedures
• Depends on ones position
• Plan
• Experience
• A linear path through a multi-dimensional domain

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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

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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)

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Source of Goals
Society/Workforce
Psychology
Mathematics Education
Discipline
Assessments
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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.

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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)

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Previous Periods of Curricular Reform

• Early 1900s Commissions
• ....
• Late 1950slate 60s New math
• Late 1980s90s Standards
• Early 2000s Based on assessments

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Today Recent Past

• NCLB and state curriculum reforms
• College Boards Standards for College Success
  Mathematics and Statistics (2009)
• NCTM Curriculum Focal Points (Pre-K8) (2006)
  Focus in High School Mathematics Reasoning and
  Sense Making (912) (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/

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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
• Connection
• Linearity
• Differentiation
• Choice

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Providing Structure
Content Domains
Cognitive Processes
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Content Domains

• 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!

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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) Dont take
  your countrys curriculum and approach to
  teaching algebra for granted and dont assume all
  other educational jurisdictions operate in a
  similar waythey conspicuously do not.
• Striking differences in
• 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

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Interconnected Content Domains Cognitive
Processes
Relationships
Content Domains
Concepts
Procedure
Facts
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Recommendation 2

• Keep the lists of content and processes focused
  and concise
• Do not stray greatly from current models without
  strong justificationit just makes work for
  teachers and costs moneylittle real change
• Make sure statements of goals, standards, and
  outcomes have clarity and reflect a range of
  cognitive processes

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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.

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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.

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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.

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Coherence

• Above alla 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.

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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