The Mathematics The Mathematics Curriculum K–12Curriculum K–12

A STANDARDS-BASED Search for Coherence

John A. DosseyIllinois State University

Jeremy KilpatrickUniversity of Georgia

Curriculum• Not just a syllabus or textbook• Includes

• 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 must be approached first of allapproached first of all on the basis ofon the basis of a precise and comprehensive formulation of the valid aims and valid aims and purposespurposes 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 GoalsSociety/Workforce

Discipline

Psychology

Mathematics Education

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

• Early 1900s Commissions• ....• Late 1950s–late 60s“New math”• Late 1980s–90s Standards• Early 2000s 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:

• Connection• Linearity• Differentiation• Choice

Providing Structure

Content DomainsCognitiv

eProcesses

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!

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• 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

ContentDomains

Concepts

Relationships

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.”

CoherenceAbove 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