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

 

 

Download Syllabus as a PDF

I.     Overview

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.

 

II.   Requirements

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.