PARTITIONING:

A GROUNDED THEORY INVESTIGATION

OF INSTRUCTOR MATHEMATICS

PHILOSOPHY SHAPING COMMUNITY

COLLEGE MATHEMATICS

M. Joanne Kantnerjkantner@kishwaukeecollege.edu

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)

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.

Purpose of the Study

To theorize in what ways instructor beliefs about their subject shape

the practice in a community college environment.

Research Questions

Primary Question

How are faculty’s philosophies of mathematics shaping mathematics instruction in community colleges?

Research Questions-2-

Supplementary Questions

RQ1: What are community college instructor’s self identified beliefs about the nature of mathematics?

RQ2: How do community college instructors view the intentions of their instruction?

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?

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.

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

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%)

Review of Literature

Conceptual Framework

METHODGrounded Theory

METHODOLOGY

Interactional Constructivism

PEDAGOGYExploratory

COGNITIVE PSYCHOLOGYCognitive-Constructivism

PHILOSOPHYRelativism, Postpositivism,

Fallibilism

Theoretical Framework

Philosophy-Ernest,Personal Knowledge

Acculturation,Intercultural Learning,

Discourse-Kalathil

Socialization,Teaching Orientation,

Subject goals,Community College

Beliefs, Knowledge,Structure-Beliefs,

Beliefs Identification

RESEARCH

Gaps In Literature

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 individual’s 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.

Methodology

Constructivist Grounded Theory

Charmaz, 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.

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

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

Data Collection and Analysis

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?

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 instructor’s 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?

Results

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 doPA: Patterns PA: ProblemsDB: 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

Mathematics is axiomatic system Abstract structure Mathematics is a study of relationships Predictions

Generalizations Patterns Modeling Problem

PB: Mathematics is an axiomatic system PB: Mathematics is the study of relationshipsPA: Axiomatic system PA: RelationshipsDB: Mathematics is rules of logic, patterns, theorems, DB: Mathematics is explorations, cause-effect, generalizations, abstract modeling, problems, predictionsDA: Logic, theorems, generalizations, abstract DA: Predictions

Morris Frank

Valorizationthe action of instructors assigning more worth, merit or

importance to certain practices over other practices

VALORIZATION

Definition

Values

Valuation EU-Valorization Valorization

Mathematics

Rationalism

GeneralizabilitySituated

Education

Curriculum:AtomizationConnected

Teaching: ExplanatoryExploratory

Future Need:TerminalLifelong

InstructorIdentity

Institutional:SecondaryUniversity

Professional:Scholar

Educator

Bridging Discoursesan instructor created transitional bridge which converges learners

to a mathematics subculture

Bridging Discourses

Pattern Types

Questioning

Listen-Responding

Problem-framing

Flexibility Explanatory

Factual Guiding Probing

Evaluative Interpretive Hermeneutic

Individual Global

Course Track ContentCourse Track

DepthTeacher Socialization

Investigatory

Orientation to Learning

Voices of Authoritythe voluntary submission to policies which becomes an

intervening condition controlling content and pedagogical knowledge

VOICES OF

AUTHORITY

Mandating

Exercising

InstructorCredentialing

Use of Textbooks

ImplementingPre-requisites

AdministrativeArticulationAgreement

AssessmentProcedures

Transferring Expertise to Textbook

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.

Theory of Partitioning

Instructor beliefs separate mathematics discourses intosubcultures 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.

Theory of Partitioning Education Values

Mathematics Values Instructor Identities

Patterns Mandating

Questioning Problem-framing Exercising

Listening-Responding Transferring

Philosophy of

Mathematics

VALORIZATION

BRIDGING DISCOURSE

VOICES OFAUTHORITY

Intranigent Connections

Absolutist:

Study Patterns

VALORIZATION:Textbook as Expert

PROBLEMS:Isolated

computations Examination

practice

AUTHORITY:Mandated Textbook

Partitioning: Accommodating

Community College’s Multiple Missions

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 doesn’t prepare these future workers for the complex changing knowledge needed in their lifespan.

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.

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.

Partitioning: Implications for Future

Research

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.

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.

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?

Mathematics Education -2-

What conclusions can be drawn from integrating a theory of partitioning with Paul Ernest’s 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?

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?

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?

Discussion

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