PARTITIONING:

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PARTITIONING

• A GROUNDED THEORY INVESTIGATION OF INSTRUCTOR
  MATHEMATICS PHILOSOPHY SHAPING COMMUNITY COLLEGE
  MATHEMATICS
• M. Joanne Kantner
• jkantner_at_kishwaukeecollege.edu

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Identifying Beliefs

• To introduce philosophical considerations into
  a discussion of education has always been
  dynamite. Socrates did it, and he was promptly
  given hemlock.
• (von Glasersfeld, 1983)

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Introduction to the Problem

• Response to the changing demands of the workplace
  and to concerns coming with the emerging
  knowledge economy.
• The multiple missions create a complex teaching
  environment for its faculty.
• Expected to service educational programs with
  different and often conflicting mathematics
  needs.

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Purpose of the Study

• To theorize in what ways instructor beliefs
  about their subject shape the practice in a
  community college environment.

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Research Questions

• Primary Question
• How are facultys philosophies of mathematics
  shaping mathematics instruction in community
  colleges?

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Research Questions-2-

• Supplementary Questions
• RQ1 What are community college instructors
  self identified beliefs about the nature of
  mathematics?
• RQ2 How do community college instructors view
  the intentions of their instruction?

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Research Questions-3-

• Supplementary Questions
• RQ3 In what ways do instructors see their view
  of mathematics as shaping their teaching?
• RQ4 In what ways do instructors see their
  mathematics philosophies shaping their course
  decisions?

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Significance

• Design programming, pedagogical strategies,
  assessments and faculty professional development
  which supports adult mathematics learning.
• Because little attention is given to occupational
  students need for continuous lifelong learning.
  
• For research problemizing in community college
  higher education and adult mathematics education
  literature.

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Definition of Terms

• Articulation Initiative Agreements (IAI)- state
  agreements for the transfer of community college
  credits to baccalaureate institutions
• College-level Courses-general education, teacher
  preparation and mathematics intensive courses
  taught within the mathematics department

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Definition of Terms -2-

• Developmental Courses-courses below the first
  college-level courses containing content
  equivalent to secondary institutions (67)
• Transfer Courses-general education courses
  covered by the IAI (32)
• Vocational Courses-a college level course with
  content towards a specific technical field (1)

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• Review of Literature

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Conceptual Framework
METHOD Grounded Theory
METHODOLOGY Interactional Constructivism
PEDAGOGY Exploratory
COGNITIVE PSYCHOLOGY Cognitive-Constructivism
PHILOSOPHY Relativism, Postpositivism, Fallibilism
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Theoretical Framework
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• Gaps In Literature

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Gaps in Literature

• Few works have been done investigating in-service
  instructor beliefs at any level of education
  with none found specifically
• studying community college faculty.
• It fails to address the influence of the
  constructed knowledge of a group (as represented
  by the instructor) on the individuals
  constructed knowledge.
• Quantitative studies measure the forced choices
  of individuals and disregard measuring collective
  beliefs of a social group.
• Few studies exist which combine measurement with
  observation to provide an understanding of the
  interaction between beliefs, intentions and the
  actions connected to a belief.

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• Methodology

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Constructivist Grounded TheoryCharmaz, 2006

• To offer an interpretation (not exact picture) of
  the studied world
• To study how and why participants and actions are
  constructed.
• To learn how, when and to what extent the
  phenomena are embedded in larger and hidden
  positions.
• To recognize grounded theorizing is a social
  action that researchers construct with others in
  specific places and times.

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Setting

• Rural Mid-west Community College
• District population of 100,000
• Served 10,100 students per year
• 229 full-time faculty (9 mathematics)
• Class sizes 10-40 seats
• Median age of 23 in credit courses

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Participants

• Instructor Undergraduate Graduate
  Prior Teaching Professional
  Degree Degree
  Experience Self-Identity
• Morris A.S./B. S. Math M.S.
  Math University Teaching
  Mathematician
• 
  Doctoral Path Graduate Assistant
• Frank A.S./B.S. Math M. S. Ed. Math
  Secondary Teaching Educator
• Secondary Cert
• Jamie B.S. Math M.S.
  Math University Teaching
  Algebraist
• 
  Doctoral Path Graduate Assistant
• J. P. B.S. Math
  M.A.T. Math Secondary Teaching
  Mathematician
• Secondary Cert

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• Data Collection and Analysis

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Interview Prompts

• 1. What is mathematics?
• 2. Where does mathematical
  knowledge come from?
• 3. How is mathematical knowledge formed?
• 4. How do you know mathematics is true?
• 5. What is the value of mathematics?

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MemoCultural/Intercultural Learning
Liberal ed paradox? How come culture/communal so
tied to self-knowledge/solitary learning? Before
one can multi-culturalize math, teacher must
recognize it as cultural. Student brings own
culture? multiple cultures? past mathematical
culture? Does this have more influence over math
knowledge formed or does instructors definition
of mathematical culture dominate the learning?
Need to get distinctions enculuration,
acculturation, assimilate, cultural domination,
(is there mathematical genocide?) cultural
competence, cultural negotiation, cultural
conflict, culture shock, and intercultural
understanding, cultural teacup/cultural torus?
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• Results

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PhilosophiesChapman, 2002

• Order
  Memorization Algebra
  Problems
• 
  
• Mathematics is ? a study of Patterns ?
  Repetition Mathematics is ? what
  Mathematicians do ? Problem-solving
• 
  Computation
  Geometry Real Analysis
• PB Mathematics is a study of patterns
  PB Mathematics is what mathematicians
  do
• PA Patterns PA Problems
• DB Mathematics is order, memorized,
  computation, and repetition DB
  Mathematics is solving problems in algebra,
• DA order, memorized, computation, repetition
  geometry and real analysis
• DA Problem-solving,
  domains, problems
• Jamie J. P.
• Logic Theorems
  Explorations Cause-effect

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Valorizationthe action of instructors assigning
more worth, merit or importance to certain
practices over other practices
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Bridging Discoursesan instructor created
transitional bridge which converges learners to a
mathematics subculture
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Voices of Authoritythe voluntary submission to
policies which becomes an intervening condition
controlling content and pedagogical knowledge
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Core Category

• Partitioning
• In mathematics, Partitioning is the
  decomposition of a set into a family of
  non-empty, disjoint sets where the union of the
  sets equals the original set.

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Theory of Partitioning

• Instructor beliefs separate mathematics
  discourses into
• subcultures of workplace, applied and academic
• mathematics communities. Under the constraints of
  
• political and administrative authority, the
  faculty
• valorize certain instructional practices when
  creating the
• classroom discourses to bridge learners to the
  course
• content. The knowledge between the subcultures
  can
• be incommensurable which makes mobility between
• subcultures problematic and prevents adults from
• acquiring needed skill updates in the future.

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Theory of Partitioning

• 
  
  Education Values
• 
  Mathematics Values
  Instructor Identities
• Patterns
  
  Mandating

VALORIZATION
Philosophy of Mathematics
BRIDGING DISCOURSE
VOICES OF AUTHORITY
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Intranigent Connections
VALORIZATION Textbook as Expert
Absolutist Study Patterns
PROBLEMS Isolated computations Examination
practice
AUTHORITY Mandated Textbook
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• Partitioning
• Accommodating Community Colleges Multiple
  Missions

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Integrating Academic and Vocational Content

• It would be reasonable to expect vocational
• students to have as great a need in the future,
• if not a greater need, to build upon a
• mathematics foundation as students pursuing
• transfer degrees. Partitioning vocational
• mathematics as a surface level course doesnt
• prepare these future workers for the complex
• changing knowledge needed in their lifespan.

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Recruitment of Community College Faculty

• Valorizations of beliefs concerning the
• nature of mathematics influenced the partition
• in instructors professional identity as a
• discipline scholar or pedagogue. To meet the
• mission of the community college, the
• instructors have a special need for balance
• between pedagogical content knowledge and
• subject knowledge.

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Partitioning and Position of the Research Domain
Adults Mathematics Education

• Teaching adults in community colleges is a
• specialization with its own unique context within
  
• the fields of higher education, adult education
• (including vocational education) and mathematics
• education. A consilience of knowledge between
• adult education, mathematics education, and
• higher education could form new comprehensive
• theories.

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• Partitioning
• Implications for Future
• Research

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Higher Education

• Does a theory of partitioning provide
  explanations for the shaping of instruction in
  other general studies disciplines such as
  English, economics, or the psychology?
• Community college instructors self identified as
  mathematicians or as educators of mathematics.
  More research into instructors perceptions of
  self when teaching across and within each
  partitioned subculture is needed.

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Higher Education -2-

• Does a theory of partitioning provide
  understanding of mathematics instruction
  occurring in non-credit or contracted
  instruction?
• Future research on community college curriculum
  design, articulation policies, classroom
  instruction, recruitment, retention, and program
  completion recognizing the partitioning process
  is needed.

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Mathematics Education

• Mathematics is not universal across cultures and
  nations so it is reasonable to believe
  mathematical philosophies regarding the nature of
  the subject also vary by culture. What are the
  implications of culture to the partitioning
  process?
• Many academic biological and physical science
  courses contain mathematics content. Can the
  theory of partitioning explain mathematics
  instruction taught by faculty within
  non-mathematics courses?

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Mathematics Education -2-

• What conclusions can be drawn from integrating a
  theory of partitioning with Paul Ernests model
  of mathematics instruction?
• How does a partitioning framework affect the NCTM
  standards for teacher preparation and best
  practices of mathematics teaching in vocational,
  developmental and general education courses?

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Adult Education

• Research into community college instructors
  awareness, identifications, and images of
  themselves as adult educators would provide
  theoretical problemizing into an adult education
  practice which is inside community colleges but
  outside ESL, ABE, and ASE programs.
• How does the partitioning theory apply to
  instructors recruited from secondary education
  practices, four-year institutions, and also
  outside the field of education?

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Adult Education -2-

• What are the long-term implications of
  partitioning to the workplace? Does a theory of
  partitioning occur in informal and formal
  workplace learning?
• In what way does partitioning mathematics prevent
  subgroups entry into occupations- denying them
  full social and political participation?

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Discussion

• Thank you for attending