PARTITIONING:

1. PARTITIONING: A GROUNDED THEORY INVESTIGATION OF INSTRUCTOR MATHEMATICS PHILOSOPHY SHAPING COMMUNITY COLLEGE MATHEMATICS M. Joanne Kantner jkantner@kishwaukeecollege.edu

2. 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)

3. 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.

4. Purpose of the Study To theorize in what ways instructor beliefs about their subject shape the practice in a community college environment.

5. Research Questions Primary Question How are faculty’s philosophies of mathematics shaping mathematics instruction in community colleges?

6. 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?

7. 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?

8. 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.

9. 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

10. 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%)

11. Review of Literature

12. Conceptual Framework METHOD Grounded Theory METHODOLOGY Interactional Constructivism PEDAGOGY Exploratory COGNITIVE PSYCHOLOGY Cognitive-Constructivism PHILOSOPHY Relativism, Postpositivism, Fallibilism

13. Theoretical Framework

14. Gaps In Literature

15. 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.

16. Methodology

17. 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.

18. 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

19. 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

20. Data Collection and Analysis

22. 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?

23. 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?

24. Results

25. 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 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 relationships PA: Axiomatic system PA: Relationships DB: Mathematics is rules of logic, patterns, theorems, DB: Mathematics is explorations, cause-effect, generalizations, abstract modeling, problems, predictions DA: Logic, theorems, generalizations, abstract DA: Predictions Morris Frank

26. Valorizationthe action of instructors assigning more worth, merit or importance to certain practices over other practices

27. Bridging Discoursesan instructor created transitional bridge which converges learners to a mathematics subculture

28. Voices of Authoritythe voluntary submission to policies which becomes an intervening condition controlling content and pedagogical knowledge

29. 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.

30. 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.

31. Theory of Partitioning Education Values Mathematics Values Instructor Identities Patterns Mandating Questioning Problem-framing Exercising Listening-Responding Transferring VALORIZATION Philosophy of Mathematics BRIDGING DISCOURSE VOICES OF AUTHORITY

32. Intranigent Connections VALORIZATION: Textbook as Expert Absolutist: Study Patterns PROBLEMS: Isolated computations Examination practice AUTHORITY: Mandated Textbook

33. Partitioning: Accommodating Community College’s Multiple Missions

34. 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.

35. 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.

36. 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.

37. Partitioning: Implications for Future Research

38. 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.

39. 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.

40. 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?

41. 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?

42. 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?

43. 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?

44. Discussion Thank you for attending