**Research into the Development of Mathematical Ideas**

Graduate School, New Brunswick

Rutgers University,
College Avenue Campus

Spring 2005

16:300:563

Mondays, 4:50 to 7:30

GSE, Room 211

**Instructor:** Arthur B.
Powell

Associate Professor, Department of
Urban Education, Newark Campus

973.353.3530 (office), abpowell@andromeda.rutgers.edu

Associate Director, Robert B.
Davis Institute for Learning, GSE

eCompanion online course supplement: http://rutgersonline.net

The purpose of this course is to enable you to develop further your understanding and experience in conducting video-facilitated fieldstudy in mathematics education. Fieldstudy is a research genre in the social sciences that is also known, among other labels, as naturalistic research, ethnography, or qualitative study. A specific arena in which to hone your capacity to do ethnography will be investigating the development of students' mathematical ideas and reasoning. This is a line of research within the Robert B. Davis Institute for Learning (RBDIL) of the Graduate School of Education, Rutgers University, that enjoys international recognition. Over the years, through many fieldstudies, an ethnographic methodology for the use of videodata has evolved within the RBDIL. It has been documented and is receiving increased attention among researchers in mathematics education. You will study documents—dissertations and journal articles—of video-facilitated research that have emerged from the work of the RBDIL. You will also read and summarize documents that report on video-facilitated fieldstudy conducted by investigators from other institutions.

Besides studying the ethnographic work of investigators in mathematics education, you will engage two of three components of fieldstudy. In general, ethnographic investigations contain three non-linear, overlapping, and interweaving components: (1) gathering or collecting and assembling data, (2) focusing or asking questions about these data, and (3) analyzing or developing and presenting evidence-based interpretations of these data. Despite having to gather data before asking questions and analyzing them, it is also the case that researchers unavoidably pose questions about and interpret their data as they gather them. In this course, you will study and implement the second and third components of conducting fieldstudy.

1. Successful completion
of Human Subjects Certification Program of the Institutional Review Board for
the Protection of Human Subjects in Research. This can be done either online

http://orsp.rutgers.edu/HSCPLetter.asp
or equivalently attending both parts of the Certification Film in its
entirety. The Film consists of two
videotapes, each approximately an hour and 15 minutes in length. The first videotape, "Basics of Human
Subjects Research", provides a general overview of the regulations and ethical
considerations that must be addressed for such research. The second videotape, "Advanced
Topics", covers regulatory and ethical guidance for vulnerable research
populations, such as pregnant women and fetuses, children, and prisoners.

2. Complete
all readings and associated assignments.

3. Initiate
and contribute to threaded discussions at the course's eCompanion site <http://rutgersonline.net>.

4. Read,
summarize, and report on three doctoral dissertations that involve the use of
video-facilitated fieldstudy. The
specific format of the dissertation summary and report of method will be
detailed in class. The report will
focus on methodological issues related to the investigator's use of
videorecordings for gathering, focusing, and analyzing data. Dissertations are available at the
following Web site: http://wwwlib.umi.com/dissertations/. You will also have access to many dissertations
through our course Web site: http://rutgersonline.net.

5. Code
and analyze video-portfolio data.
You will have access to video-portfolio data from a current,
longitudinal project, "Research on Informal Mathematical Learning" (IML) of the
RBDIL, which is supported by a research grant from the National Science
Foundation (REC-0309062).

6. Write
a paper detailing your focusing and analyzing processes and the results of your
analysis of video-portfolio data.

*Required Readings*

* *

Charmaz, K., & Mitchell, R. G. (2001).
Grounded theory in ethnography. In P. Atkinson, A. Coffey & S. Delamont
(Eds.), *Handbook of ethnography* (pp.
160-174). London: Sage.

Lofland, J., & Lofland, L. H. (1995). *Analyzing
social situations: A guide to qualitative observation and analysis* (Third ed.). Belmont, CA: Wadsworth. (Chapter 6:
Thinking Topics; and Chapter 7: Asking Questions)

Davis, R. B., Maher, C., & Martino, A.
(1992). Using videotapes to study the construction of mathematical knowledge of
individual children working in groups. *Journal of Science, Education, and
Technology, 1*(3), 177-189.

Maxwell, J. A. (2005). *Qualitative
research design: An interactive approach*
(2nd ed.). Thousand Oaks: Sage. (Chapter 6: Methods: What will you actually
do?)

Pirie, S. (1998). Working toward a design
for qualitative research. In A. R. Teppo (Ed.), *Qualitative research methods
in mathematics education* (Monograph Number
9, pp. 79-97). Reston, VA: National Council of Teachers of Mathematics.

Powell, A. B., Francisco, J. M., &
Maher, C. A. (2003). An analytical model for studying the development of
mathematical ideas and reasoning using videotape data. *Journal of
Mathematical Behavior, 22*(4), 405-435.

Stohl, H., & Tarr, J. E. (2002). Developing
notions of inference with probability simulation tools. *Journal of
Mathematical Behavior, 21*(3), 319-337.

* *

* *

*Bibliography of Video-Facilitated Dissertation*

*From Rutgers University*

Bellisio, C. W. (1999). *A study of elementary
students' ability to work with algebraic notation and variables.* Unpublished doctoral dissertation, Rutgers, The
State University of New Jersey, New Brunswick.

Bulgar, S. (2002). *Through a teacher's
lens: Children's constructions of division of fractions.* Unpublished doctoral dissertation, Rutgers, The
State University of New Jersey, New Brunswick.

Francisco, J. M. (2004). *Students'
reflection on mathematical learning: Results from a longitudinal study.* Unpublished doctoral dissertation, Rutgers, The
State University of New Jersey, New Brunswick.

Glass, B. H. (2001). *Mathematical
problem solving and justification with community college students.* Unpublished doctoral dissertation, Rutgers, The
State University of New Jersey, New Brunswick.

Kiczek, R. D. (2000). *Tracing the
development of probabilistic thinking: Profiles from a longitudinal study.* Unpublished doctoral dissertation, Rutgers, The
State University of New Jersey, New Brunswick.

Martino, A. M. (1992). *Elementary
students' construction of mathematical knowledge: Analysis by profile.* Unpublished doctoral dissertation, Rutgers, the
State University of New Jersey, New Brunswick.

Muter, E. M. (1999). *The development of
student ideas in combinatorics and proof: A six-year study.* Unpublished doctoral dissertation, Rutgers, The
State University of New Jersey, New Brunswick.

Powell, A. B. (2003). *"So let's
prove it!" Emergent and elaborated mathematical ideas and reasoning in the
discourse and inscriptions of learners engaged in a combinatorial task.* Unpublished doctoral dissertation, Rutgers, The
State University of New Jersey, New Brunswick.

Steencken, E. P. (2001). *Tracing the
growth of understanding of fraction ideas: A fourth grade case study.* Unpublished doctoral dissertation, Rutgers, The
State University of New Jersey, New Brunswick.

Tarlow, L. D. (2004). *Tracing students'
development of ideas in combinatorics and proof.* Unpublished doctoral dissertation, Rutgers, The State University of
New Jersey, New Brunswick.

Uptegrove, E. B. (2005). *To symbols from meaning:
Students' long-term investigations in counting.* Unpublished doctoral dissertation, Rutgers, The State University of
New Jersey, New Brunswick.

Walter, J. G. (2004). *Tracing
mathematical inquiry: High school students mathematizing a shell.* Unpublished doctoral dissertation, Rutgers, The
State University of New Jersey, New Brunswick.

Warner, L. B. (2005). *Behaviors that indicate
mathematical flexible thought.* Unpublished
doctoral dissertation, Rutgers, The State University of New Jersey, New
Brunswick.

*From other universities*

Choppin, J. M. (2004). *How teachers' discourse
practices affect student engagement in the context of mathematics reform.* Unpublished doctoral dissertation, The University of
Wisconsin, Madison.

Doyle, J. A. (2003). *Student voice: The
influence of complex instruction on fifth grade students' mathematical problem
solving performance.* Unpublished doctoral
dissertation, Boston College, Boston.

Goos, M. (1999). *Metacognition in context: A study
of metacognitive activity in a classroom community of mathematical inquiry.* Unpublished doctoral thesis, University of
Queensland.

Herbst, P. G. (1998). *What works as
proof in the mathematics class.* Unpublished
doctoral dissertation, University of Georgia.

Horn, I. S. (2002). *Learning on the job:
Mathematics teachers' professional development in the contexts of high school
reform.* Unpublished doctoral dissertation,
University of California, Berkeley.

John, A. S. (2001). *Generalizing in interaction:
Students making and using mathematical generalizations in design projects.* Unpublished doctoral dissertation, University of
California-Berkeley, Berkeley.

Larsen, S. P. (2004). *Supporting the
guided reinvention of the concepts of group and isomorphism: A developmental
research project.* Unpublished doctoral
dissertation, Arizona State University.

Magidson, S. (2002). *Teaching, research, and
instructional design: Bridging communities in mathematics education.* Unpublished doctoral dissertation, University of
California, Berkeley.

Martin, L. C. (1999). *The nature of the
folding back phenomenon within the Pirie-Kieren theory for the growth of
mathematical understanding and the associated implications for teachers and
learners of mathematics.* Unpublished
doctoral dissertation, University of Oxford, Oxford, England.

Raman, M. J. (2002). *Proof and
justification in collegiate calculus.*
Unpublished doctoral dissertation, University of California, Berkeley.

Seymour, J. R. (2004). *Tracing the evolution of
pedagogical content knowledge as interanimated discourses.* Unpublished doctoral dissertation, The University of
Wisconsin-Madison, Madison.

Sherin, M. G. (1996). *The nature and dynamics of
teachers' content knowledge.* Unpublished
doctoral dissertation, University of California, Berkeley.

Smith, S. P. (1999). *Children, learning theory,
and mathematics: An analysis of the role of language and representations in
children's mathematical reasoning.*
Unpublished doctoral dissertation, Michigan State University, East Lansing.

*Comparing Units of Analyses & Issues of
Reform-Oriented Teaching and Equity*

Boaler, J. (1998). Open and closed mathematics:
Student experiences and understandings. *Journal for Research in Mathematics
Education, 29*(1), 41-62.

Boaler, J. (2002). Learning from teaching: Exploring
the relationship between reform curriculum and equity. *Journal for Research
in Mathematics Education, 33*(4), 239-258.

Lubienski, S. T. (2000). Problem solving as a means
toward mathematics for all: An exploratory look through the class lens. *Journal
for Research in Mathematics Education, 31*(4),
454-482.

*Articles Involving Video-Facilitated Research in
Mathematics Education*

Goos, M. (2004). Learning mathematics in a classroom
community of inquiry. *Journal for Research in Mathematics Education, 35*(4), 258-291.

Jacobson, C., & Lehrer, R. (2000). Teacher
appropriation and student learning of geometry through design. *Journal for
Research in Mathematics Education, 31*(1),
71-88.

Simon, M. A., Tzur, R., Heinz, K., Kinzel, M., & Schwan
Smith, M. (2000). Characterizing a perspective underlying the practice of
mathematics teachers in transition. *Journal for Research in Mathematics
Education, 31*(5), 579-601.

Stohl, H., & Tarr, J. E. (2002). Developing
notions of inference with probability simulation tools. *Journal of
Mathematical Behavior, 21*(3), 319-337.

*Methodology Bibliography*

"μ-Group",
T. (2001). Theory, video and mathematical understanding: An examination of what
different theoretical perspectives can offer. In R. Speiser, C. A. Maher &
C. N. Walter (Eds.), *Proceedings of the twenty-third annual meeting of the
North American Chapter of the International Group for the Psychology of
Mathematics Education (Snowbird, Utah)*
(Vol. I, pp. 343-379). Columbus, OH: ERIC Clearinghouse for Science,
Mathematics, and Environmental Education.

Alasuutari, P. (1996). Theorizing in
qualitative research: A cultural studies perspective. *Qualitative Inquiry, 2*(4), 371-384.

Berg, B. L. (2004). *Qualitative research
methods for the social sciences* (5th ed.).
Boston: Pearson.

Bottorff, J. L. (1994). Using videotaped
recordings in qualitative research. In J. M. Morse (Ed.), *Critical issues in
qualitative research methods* (pp. 244-261).
Thousand Oaks, CA: SAGE.

Carspecken, P. F., & Apple, M. (1992).
Critical qualitative research: Theory, methodology, and practice. In M. D.
LeCompte, W. L. Millroy & J. Preissle (Eds.), *The handbook of
qualitative research in education* (pp.
508-553). San Diego: Academic Press.

Charmaz, K., & Mitchell, R. G. (2001).
Grounded theory in ethnography. In P. Atkinson, A. Coffey & S. Delamont
(Eds.), *Handbook of ethnography* (pp.
160-174). London: Sage.

Cobb, P., & Whitenack, J. W. (1996). A
method for conducting longitudinal analysis of classroom videorecordings and
transcripts. *Educational Studies in Mathematics, 30*, 213-228.

Corbin, J., & Strauss, A. (1990).
Grounded theory research: Procedures, canons, and evaluative criteria. *Qualitative
Sociology, 13*(1), 3-21.

Creswell, J. W. (1998). *Qualitative
inquiry and research design: Choosing among five traditions*. Thousand Oaks, CA: Sage.

Davis, R. B., Maher, C., & Martino, A.
(1992). Using videotapes to study the construction of mathematical knowledge of
individual children working in groups. *Journal of Science, Education, and
Technology, 1*(3), 177-189.

Denzin, N. K., & Lincoln, Y. S. (Eds.). (2000). *Handbook
of qualitative research* (2nd ed.). Thousand
Oaks, CA: Sage.

Erickson, F. (1992). The Interface between
ethnography and microanalysis. In M. D. LeCompte, W. L. Millroy & J.
Preissle (Eds.), *The handbook of qualitative research in education* (pp. 201-225). San Diego: Academic Press.

Glaser, B. G., & Strauss, A. L. (1967).
*The discovery of grounded theory: Strategies for qualitative research*. New York: Aldine.

Gubrium, J. F., & Holstein, J. A.
(2000). Analyzing interpretive practice. In N. K. Denzin & Y. S. Lincoln
(Eds.), *Handbook of qualitative research*
(2nd ed., pp. 487-508). Thousand Oaks, CA: Sage.

Hall, R. (2000). Videorecording as theory.
In A. E. Kelly & R. Lesh (Eds.), *Handbook of research data design in
mathematics and science education* (pp.
647-664). Mahwah, NJ: Lawrence Erlbaum.

Hiebert, J., Gallimore, R., Garnier, H.,
Givvin, K. B., Hollingsworth, H., Jocabs, J., et al. (2003). *Teaching
mathematics in seven countries: Results from the TIMSS 1999 video study, NCES
(2003-13)*. Washington, DC: U.S. Department
of Education, National Center for Education Statistics.

Kincheloe, J. L., & McLaren, P. (2000).
Rethinking critical theory and qualitative research. In N. K. Denzin & Y.
S. Lincoln (Eds.), *Handbook of qualitative research* (2nd ed., pp. 279-313). Thousand Oaks, CA: Sage.

LeCompte, M. D., & Preissle, J. (1992).
Toward an ethnology of student life in schools and classrooms: Synthesizing the
qualitative research tradition. In M. D. LeCompte, W. L. Millroy & J.
Preissle (Eds.), *The handbook of qualitative research in education* (pp. 816-859). San Diego: Academic Press.

Lesh, R., & Lehrer, R. (2000).
Iterative refinement cycles for videotape analyses of conceptual change. In A.
E. Kelly & R. Lesh (Eds.), *Handbook of research data design in
mathematics and science education* (pp. 665--708).
Mahwah, NJ: Lawrence Erlbaum.

Lincoln, Y. S., & Guba, E. G. (2000).
Paradigmatic controversies, contradictions, and emerging confluences. In N. K.
Denzin & Y. S. Lincoln (Eds.), *Handbook of qualitative research* (2nd ed., pp. 163-188). Thousand Oaks, CA: Sage.

Lofland, J., & Lofland, L. H. (1995). *Analyzing
social situations: A guide to qualitative observation and analysis* (Third ed.). Belmont, CA: Wadsworth.

Jordan, B., & Henderson, A. (1995). Interaction
analysis: Foundations and practice. *Journal of the Learning Sciences, 4*(1), 39-103.

Maxwell, J. A. (2005). *Qualitative
research design: An interactive approach*
(2nd ed.). Thousand Oaks: Sage.

Miles, M. B., & Huberman, A. M.
(1994a). Early steps in analysis. In *Qualitative data analysis: An expanded sourcebook* (2nd ed., pp. 50-89). Thousand Oaks: Sage.

Miles, M. B., & Huberman, A. M.
(1994b). *Qualitative data analysis: An expanded sourcebook* (2nd ed.). Thousand Oaks, CA: Sage.

National Research Council. (2002). *Scientific
research in education*. Committee on
Scientific Principles for Education Research. Shavelson, R. J. and Towne, L.
(Eds.). Center for Education, Division of Behavioral and Social Science and
Education. Washington, DC: National Academy Press.

Patton, M. Q. (1990). Qualitative
interviewing. In *Qualitative evaluation and research methods* (pp. 277-368). Newbury Park: Sage.

Pirie, S. (1998a). Toward a definition for
research. In A. R. Teppo (Ed.), *Qualitative research methods in mathematics
education* (Vol. Monograph Number 9, pp.
17-21). Reston, VA: National Council of Teachers of Mathematics.

Pirie, S. (1998b). Where do we go from
here? In A. R. Teppo (Ed.), *Qualitative research methods in mathematics
education* (Vol. Monograph Number 9, pp.
156-163). Reston, VA: National Council of Teachers of Mathematics.

Pirie, S. (1998c). Working toward a design
for qualitative research. In A. R. Teppo (Ed.), *Qualitative research methods
in mathematics education* (Vol. Monograph
Number 9, pp. 79-97). Reston, VA: National Council of Teachers of Mathematics.

Pirie, S. E. B. (1996). What are the data?
An exploration of the use of video-recording as a data gathering tool in the
mathematics classroom. In E. Jakubowski, D. Watkins & H. Biske (Eds.), *Proceedings
of the eighteenth annual meeting of the North American Chapter of the
International Group for the Psychology of Mathematics Education (Florida State
University, Panama City)* (Vol. II, pp.
553-559). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and
Environmental Education.

Pirie, S. E. B. (2001). Analysis, lies, and
videotape. In R. Speiser, C. A. Maher & C. N. Walter (Eds.), *Proceedings
of the twenty-third annual meeting of the North American Chapter of the
International Group for the Psychology of Mathematics Education (Snowbird,
Utah)* (Vol. I, pp. 346-350). Columbus, OH:
ERIC Clearinghouse for Science, Mathematics, and Environmental Education.

Powell, A. B., Francisco, J. M., &
Maher, C. A. (2003). An analytical model for studying the development of
mathematical ideas and reasoning using videotape data. *Journal of
Mathematical Behavior, 22*(4), 405-435.

Powell, A. B., Francisco, J. M., &
Maher, C. A. (2004). Uma abordagem anàlise de dados de vídeo para investigar
o desenvolvimento das idéias matemáticas e do raciocínio de estudantes [An
analytical model for studying the development of mathematical ideas and
reasoning using videotape data]. *BOLEMA: O Boletim de Educação Matemática
[BOLEMA: The Bulletin of Mathematics Education]*(21), 81-140.

Quantz, R. A. (1992). On critical
ethnography (with some postmodern considerations). In M. D. LeCompte, W. L.
Millroy & J. Preissle (Eds.), *The handbook of qualitative research in
education* (pp. 448-505). San Diego:
Academic Press.

Roschelle, J. (2000). Choosing and using
video equipment for data collection. In A. E. Kelly & R. Lesh (Eds.), *Handbook
of research data design in mathematics and science education* (pp. 709-731). Mahwah, NJ: Lawrence Erlbaum.

Schwandt, T. A. (2000). Three
epistemological stances for qualitative inquiry: Interpretivism, hermeneutics,
and social constructionism. In N. K. Denzin & Y. S. Lincoln (Eds.), *Handbook
of qualitative research* (2nd ed., pp.
189-213). Thousand Oaks, CA: Sage.

Seidman, I. (1998). A structure for
in-depth phenomenological interviewing. In *Interviewing as qualitative research:
A guide for researchers in education and the social science* (pp. 9-21). New York: Teachers College.

Stigler, J. W., Gonzales, P., Kawanaka, T.,
Knoll, S., & Serrano, A. (1999). *The TIMSS Videotape Classroom Study:
Methods and Findings from an Exploratory Research Project on Eighth-Grade
Mathematics Instruction in Germany, Japan, and the United States* (Research and Development Report No. NCES 99-074).
Washington, DC: U.S. Department of Education, National Center for Education
Statistics.

Suchman, L. A., & Trigg, R. H. (1991).
Understanding practice: Video as a medium for reflection and design. In J.
Greenbaum & M. Kyng (Eds.), *Design at work: Cooperative design of
computer systems* (pp. 65-89). Hillsdale,
NJ: Lawrence Erlbaum and Associates.

Teppo, A. R. (Ed.). (1997). *Qualitative research
methods in mathematics education* (Vol.
Monograph Number 9). Reston: National Council of Teachers of Mathematics.

Weitzman, E. A. (2000). Software and
qualitative research. In N. K. Denzin & Y. S. Lincoln (Eds.), *Handbook
of qualitative research* (2nd ed., pp.
803-820). Thousand Oaks, CA: Sage.

Wolcott, H. F. (1994). Description,
analysis, and interpretation in qualitative inquiry. In *Transforming
qualitative data: Description, analysis, and interpretation* (pp. 9-54). Thousand Oaks, CA: Sage.