# Expert mathematics teacher educators’ purposes and practices for providing prospective teachers with opportunities to develop pedagogical content knowledge in content courses

- 497 Downloads
- 1 Citations

## Abstract

Enhancing teachers’ pedagogical content knowledge (PCK) is essential to improving the teaching and learning of mathematics. Mathematics teacher educators (MTEs) need to help prospective teachers enhance PCK (Marks in J Teach Educ 41(3):3–11, 1990. doi: 10.1177/002248719004100302; Mason 2008). However, we know very little about the practices of MTEs, especially in mathematics content courses, as these practices are not widely researched or disseminated (e.g., Bergsten and Grevholm, in: Jaworski, Wood (eds) The international handbook of mathematics teacher education, vol 4, Sense Publishers, Rotterdam, pp 223–246, 2008; Floden and Philipp, in: Lester, Ferrini-Mundy (eds) Proceedings of the NCTM research catalyst conference, National Council of Teachers of Mathematics, Reston, pp 171–176, 2003). This phenomenographical study offers empirical findings on commonly identified purposes across ten *expert* MTEs who provided K-8 prospective teachers with opportunities to develop PCK in their mathematics content courses. Furthermore, our emergent findings indicated that *expert* MTEs also provided opportunities for prospective teachers to develop orientations toward teaching the subject, which prompted framework adaptations and articulations of “orientations” as a construct discussed in the broader literature outside of mathematics teacher education research (Magnusson et al., in: Gess-Newsome, Lederman (eds) Examining pedagogical content knowledge, Kluwer, Dordrecht, pp 95–132, 1999). Research and practitioner implications from this study provide specific PCK-related learning opportunities of prospective teachers through the lenses of *expert* MTEs’ (personal and professional) purposes and reflections on teaching, as a foundation on which the field can continue building future research and MTEs can continue building their practice in mathematics content courses.

## Keywords

Mathematics teacher educator Prospective teacher Mathematics content course Pedagogical content knowledge Course purposes## References

- Akerlind, G. (2005). Variation and commonality in phenomenographic research methods.
*Higher Education Research & Development,**24*(4), 321–334.CrossRefGoogle Scholar - Ames, C. (1992). Classrooms: Goals, structures, and student motivation.
*Journal of Educational Psychology,**84*(3), 261–272.CrossRefGoogle Scholar - An, S., Kulm, G., & Wu, Z. (2004). The pedagogical content knowledge of middle school, mathematics teachers in China and the U.S.
*Journal of Mathematics Teacher Education,**7*(2), 145–172.CrossRefGoogle Scholar - Anderson, C. W., & Smith, E. L. (1985). Teaching science. In J. Koehler (Ed.),
*The educator’s handbook: A research perspective*(pp. 84–111). New York: Longman.Google Scholar - Australian Curriculum and Assessment Reporting Authority [ACARA]. (2014).
*Foundation to year 10 curriculum: Mathematics.*Retrieved from http://www.australiancurriculum.edu.au/mathematics/aims. - Ball, D. L., & Cohen, D. K. (1999). Developing practices, developing practitioners. In L. Darling-Hammond & G. Sykes (Eds.),
*Teaching as the learning profession: Handbook of policy and practice*(pp. 3–32). San Francisco: Jossey-Bass.Google Scholar - Ball, D. L., Sleep, L., Boerst, T. A., & Bass, H. (2009). Combining the development of practice and the practice of development in teacher education.
*The Elementary School Journal,**109*(5), 458–474.CrossRefGoogle Scholar - Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special?
*Journal of Teacher Education,**59*(5), 389–407.CrossRefGoogle Scholar - Bass, H. (2005). Mathematics, mathematicians, and mathematics education.
*Bulletin of the American Mathematical Society,**42*(4), 417–430.CrossRefGoogle Scholar - Baumert, J., Kunter, M., Blum, W., Brunner, M., Voss, T., Jordan, A., et al. (2010). Teachers’ mathematical knowledge, cognitive activation in the classroom, and student progress.
*American Educational Research Journal,**47,*133–180.CrossRefGoogle Scholar - Bergsten, C., & Grevholm, B. (2008). Knowledgeable teacher educators and linking practices. In B. Jaworski & T. Wood (Eds.),
*The international handbook of mathematics teacher education*(Vol. 4, pp. 223–246). Rotterdam: Sense Publishers.Google Scholar - Blömeke, S., Buchholtz, N., Suhl, U., & Kaiser, G. (2014). Resolving the chicken-or-egg causality dilemma: The longitudinal interplay of teacher knowledge and teacher beliefs.
*Teaching and Teacher Education,**37,*130–139.CrossRefGoogle Scholar - Bloom, B. S. (1956).
*Taxonomy of educational objectives. Vol. 1: Cognitive domain*. New York: McKay.Google Scholar - Boeije, H. (2002). A purposeful approach to the constant comparative method in the analysis of qualitative interviews.
*Quality & Quantity,**36*(4), 391–409.CrossRefGoogle Scholar - Borko, H., & Putnam, R. T. (1996). Learning to teach. In D. C. Berliner & R. C. Calfee (Eds.),
*Handbook of educational psychology*(pp. 673–708). New York: Simon & Schuster Macmillan.Google Scholar - Bowden, J. A., & Walsh, E. (2000).
*Phenomenography*. Melbourne: RMIT University.Google Scholar - Boyd, D., Grossman, P. L., Hammerness, K., Lankford, R. H., Loeb, S., McDonald, M., et al. (2008). Surveying the landscape of teacher education in New York City: Constrained variation and the challenge of innovation.
*Educational Evaluation and Policy Analysis,**30*(4), 319–343.CrossRefGoogle Scholar - Bransford, J. D., Brown, A. L., & Cocking, R. R. (Eds.). (2000).
*How people learn*(expanded ed.). Washington, DC: National Academy Press.Google Scholar - Burton, M., Daane, C. J., & Giessen, J. (2008). Infusing mathematics content into a methods course: Impacting content knowledge for teaching.
*Issues in the Undergraduate Mathematics Preparation of School Teachers,**1,*1–12.Google Scholar - Capraro, R. M., Capraro, M. M., Parker, D., Kulm, G., & Raulerson, T. (2005). The mathematics content knowledge role in developing preservice teachers’ pedagogical content knowledge.
*Journal of Research in Childhood Education,**20,*108–124.CrossRefGoogle Scholar - Carpenter, T. P., Fennema, E., Peterson, P. L., & Carey, D. A. (1988). Teachers’ pedagogical content knowledge of student’s problem solving in elementary arithmetic.
*Journal for Research in Mathematics Education,**19,*385–401.CrossRefGoogle Scholar - Chapin, S. H., O’Connor, C., & Anderson, N. C. (2009).
*Classroom discussions: Using math talk to help students learn, Grades K-6*. Sausalito: Math Solutions.Google Scholar - Cochran, K. F., DeRuiter, J. A., & King, R. A. (1993). Pedagogical content knowing: An integrative model for teacher preparation.
*Journal of Teacher Education,**44*(4), 263–272.CrossRefGoogle Scholar - Common Core State Standards Initiative. (2010).
*Common Core State Standards for Mathematics (CCSSM)*. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.Google Scholar - Conference Board of the Mathematical Sciences. (2001).
*The mathematical education of teachers*(Issues in mathematics education, Vol. 11). Providence, RI: American Mathematical Society.Google Scholar - Conference Board of the Mathematical Sciences. (2012).
*The mathematical education of teachers II*(Issues in mathematics education, Vol. 17). Providence, RI: American Mathematical Society.Google Scholar - Corbin, J., & Strauss, A. (2008).
*Basics of qualitative research: Techniques and procedures for developing grounded theory*. Thousand Oaks, CA: Sage.CrossRefGoogle Scholar - Darling-Hammond, L. (2000). Reforming teacher preparation and licensing: Debating the evidence.
*Teachers College Record,**102*(1), 28–56.CrossRefGoogle Scholar - Darling-Hammond, L., Macdonald, M. B., Snyder, J., Whitford, B. L., Rusco, G., & Fickel, L. (2000).
*Studies of excellence in teacher education: Preparation at the graduate level*. New York, NY: AACTE Publications.Google Scholar - Doyle, W. (1986). Classroom organization and management.
*Handbook of Research on Teaching,**3,*392–431.Google Scholar - Elbaz, F. (1983).
*Teacher thinking. A study of practical knowledge. Croom Helm curriculum policy and research series*. New York, NY: Nichols Publishing Company.Google Scholar - European Agency for Development in Special Needs Education. (2011).
*Teacher education for inclusion across Europe: A synthesis of policy and practice in 25 countries*. Østre: Author.Google Scholar - Even, R. (2008). Facing the challenge of educating educators to work with practicing mathematics teachers. In B. Jaworski & T. Wood (Eds.),
*The international handbook of mathematics teacher education*(Vol. 4, pp. 57–73). Rotterdam: Sense Publishers.Google Scholar - Feimen-Nemser, S. (2001). From preparation to practice: Designing a continuum to strengthen and sustain teaching.
*Teachers College Record,**103*(6), 1013–1055.CrossRefGoogle Scholar - Floden, R. E., & Philipp, R. A. (2003). Report of working group 7: Teacher preparation. In F. K. Lester & J. Ferrini-Mundy (Eds.),
*Proceedings of the NCTM research catalyst conference*(pp. 171–176). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Florian, L., & Black-Hawkins, K. (2011). Exploring inclusive pedagogy.
*British Educational Research Journal,**37*(5), 813–828.CrossRefGoogle Scholar - Friedrichsen, P. J., Abell, S. K., Pareja, E. M., Brown, P. L., Lankford, D. M., & Volkmann, M. J. (2009). Does teaching experience matter? Examining biology teachers’ prior knowledge for teaching in an alternative certification program.
*Journal of Research in Science Teaching,**46*(4), 357–383.CrossRefGoogle Scholar - Ghousseini, H., & Herbst, P. (2016). Pedagogies of practice and opportunities to learn about classroom mathematics discussions.
*Journal of Mathematics Teacher Education,**19*(1), 79–103.CrossRefGoogle Scholar - Goldrick-Rab, S. (2007).
*Promoting academic momentum at community colleges: Challenges and opportunities*. New York: Columbia University, Teachers College, Community College Research Center.Google Scholar - Goodell, J. (2006). Using critical incident reflections: a self-study as a mathematics teacher educator.
*Journal of Mathematics Teacher Education,**9*(3), 221–248.CrossRefGoogle Scholar - Greenberg, J., & Walsh, K. (2008).
*No common denominator: The preparation of elementary teachers in mathematics by America’s education schools*. Washington, DC: National Council on Teacher Quality.Google Scholar - Grossman, P. L. (1990).
*The making of a teacher: Teacher knowledge and teacher education*. New York: Teachers College Press.Google Scholar - Grossman, P., Hammerness, K., & McDonald, M. (2009). Redefining teaching, re-imagining teacher education.
*Teachers and Teaching,**15*(2), 273–289.CrossRefGoogle Scholar - Grossman, P. L., & McDonald, M. (2008). Back to the future: Directions for research in teaching and teacher education.
*American Educational Research Journal,**45*(1), 184–205.CrossRefGoogle Scholar - Grubb, N. W., & Associates. (1999).
*Honored but invisible: An inside look at teaching in community colleges*. New York: Routledge.Google Scholar - Hallett, D., Nunes, T., & Bryant, P. (2010). Individual differences in conceptual and procedural knowledge when learning fractions.
*Journal of Educational Psychology,**102*(2), 395–406.CrossRefGoogle Scholar - Hiebert, J., & Lefevre, P. (2013). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.),
*Conceptual and procedural knowledge: The case of mathematics*(pp. 1–28). Hillside, NJ: Lawrence Erlbaum Associates Inc.CrossRefGoogle Scholar - Hiebert, J., & Morris, A. K. (2009). Building a knowledge base for teacher education: An experience in K-8 mathematics teacher preparation.
*The Elementary School Journal,**109*(5), 475–490.CrossRefGoogle Scholar - Hiebert, J., Morris, A. K., & Glass, B. (2003). Learning to learn to teach: An “experiment’’ model for teaching and teacher preparation in mathematics.
*Journal of Mathematics Teacher Education,**6*(3), 201–222.CrossRefGoogle Scholar - Hill, H. C., & Ball, D. L. (2004). Learning mathematics for teaching: Results from California’s mathematics professional development institutes.
*Journal for Research in Mathematics Education,**35*(5), 330–351.CrossRefGoogle Scholar - Hill, H., Ball, D. L., & Schilling, S. G. (2008). Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers’ topic-specific knowledge of students.
*Journal for Research in Mathematics Education,**39*(4), 372–400.Google Scholar - Hill, H., & Grossman, P. (2013).
*Learning from teacher evaluations: Challenges and opportunities*(pp. 371–384). Cambridge: Harvard Education Press.Google Scholar - Hill, H. C., Sleep, L., Lewis, J. M., & Ball, D. L. (2007). Assessing teachers’ mathematical knowledge: What knowledge matters and what evidence counts.
*Second Handbook of Research on Mathematics Teaching and Learning,**1,*111–155.Google Scholar - Hodgson, B. (2001). The mathematical education of school teachers: Role and responsibilities of university mathematicians. In D. Holton (Ed.),
*The teaching and learning of mathematics at university level*(pp. 501–518). Dordrecht: Kluwer.Google Scholar - Jaworski, B. (2008). Mathematics teacher educator learning and development: An introduction. In B. Jaworski & T. Wood (Eds.),
*The international handbook of mathematics teacher education*(Vol. 4, pp. 1–13). Rotterdam: Sense Publishers.Google Scholar - Kaur, B., Kwon, O., & Leong, Y. (2017).
*Professional development of mathematics teachers. Mathematics education—An Asian perspective*. Singapore: Springer.CrossRefGoogle Scholar - Kazemi, E., Lampert, M., & Franke, M. (2009). Developing pedagogies in teacher education to support novice teacher’s ability to enact ambitious instruction. In R. Hunter, B. Bicknell, & T. Burgess (Eds.),
*Crossing divides: Proceedings of the 32nd annual conference of the Mathematics Education Research Group of Australasia*(Vol. 1, pp. 12–30). Palmerston North, NZ: MERGA.Google Scholar - Levine, A. (2006).
*Educating school teachers*. New York: The Education Schools Project.Google Scholar - Lutzer, D. J., Rodi, S. B., Kirkman, E. E., & Maxwell, J. W. (2007).
*Statistical abstract of undergraduate programs in the mathematical sciences in the United States*. Washington, DC: American Mathematical Society.Google Scholar - Ma, L. (1999).
*Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States*. Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar - Magnusson, S., Krajcik, J., & Borko, H. (1999). Nature, sources, and development of pedagogical content knowledge for science teaching. In J. Gess-Newsome & N. G. Lederman (Eds.),
*Examining pedagogical content knowledge*(pp. 95–132). Dordrecht: Kluwer.Google Scholar - Marks, R. (1990). Pedagogical content knowledge: From a mathematical case to a modified conception.
*Journal of Teacher Education,**41*(3), 3–11. doi: 10.1177/002248719004100302.CrossRefGoogle Scholar - Marton, F. (1981). Phenomenography—Describing conceptions of the world around us.
*Instructional Science,**10*(2), 177–200.CrossRefGoogle Scholar - Marton, F. (1986). Phenomenograpy: A research approach to investigating different understandings of reality.
*Journal of Thought*,*21*(3), 28–49.Google Scholar - Marton, F., & Booth, S. A. (1997).
*Learning and awareness*. Hove: Psychology Press.Google Scholar - Masingila, J. O., Olanoff, D. E., & Kwaka, D. K. (2012). Who teaches mathematics content courses for prospective elementary teachers in the United States? Results of a national survey.
*Journal of Mathematics Teacher Education,**15*(5), 347–358.CrossRefGoogle Scholar - Mason, J. (2008). PCK and beyond. In P. Sullivan & T. Wood (Eds.),
*The international handbook of mathematics teacher education*(Vol. 1, pp. 301–322). Rotterdam: Sense Publishers.Google Scholar - McDuffie, A. R., Drake, C., & Herbel-Eisenmann, B. A. (2008). The elementary mathematics methods course. In B. Jaworski & T. Wood (Eds.),
*The international handbook of mathematics teacher education*(Vol. 4, pp. 247–264). Rotterdam: Sense Publishers.Google Scholar - Moss, P. A. (2011). Analyzing the teaching of professional practice.
*Teachers College Record,**113*(12), 2878–2896.Google Scholar - National Council of Teachers of Mathematics. (2000).
*Principles and standards for school mathematics*. Reston, VA: Author.Google Scholar - National Council of Teachers of Mathematics. (2014).
*Principles to actions: Ensuring mathematical success for all*. Reston, VA: Author.Google Scholar - National Research Council. (2001).
*Adding it up: Helping children learn mathematics*. Washington, DC: National Academy Press.Google Scholar - OECD. (2005).
*Teachers matter: Attracting, developing and retaining effective teachers*. Paris: OECD Publications.CrossRefGoogle Scholar - OECD. (2010).
*Educating teachers for diversity: Meeting the challenge*. Paris: OECD Publications.CrossRefGoogle Scholar - Patton, M. Q. (2002).
*Qualitative research & evaluation methods*(3rd ed.). Thousand Oaks: Sage.Google Scholar - Porter, E. J., & Cohen, M. Z. (2013). Phenomenology. In A. A. Trainor & E. Graue (Eds.),
*Reviewing qualitative research in the social sciences*(pp. 180–196). New York, NY: Routledge.Google Scholar - Sherin, M. G. (2001). Developing a professional vision of classroom events. In T. Wood, B. S. Nelson, & J. Warfield (Eds.),
*Beyond classical pedagogy: Teaching elementary school mathematics*(pp. 75–93). Hillsdale, NJ: Erlbaum.Google Scholar - Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching.
*Educational Researcher,**15*(2), 4–14.CrossRefGoogle Scholar - Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform.
*Harvard Educational Review,**57,*1–21.CrossRefGoogle Scholar - Sternberg, R. J., & Horvath, J. A. (1995). A prototype view of expert teaching.
*Educational Researcher,**24*(6), 9–17.CrossRefGoogle Scholar - Stigler, J. W., Givvin, K. B., & Thompson, B. J. (2010). What community college developmental mathematics students understand about mathematics.
*MathAMATYC Educator,**1*(3), 4–16.Google Scholar - Strauss, A., & Corbin, J. (1994). Grounded theory methodology.
*Handbook of Qualitative Research,**17,*273–285.Google Scholar - Superfine, A. C., & Li, W. (2014). Exploring the mathematical knowledge needed for teaching teachers.
*Journal of Teacher Education,**65*(4), 303–314.CrossRefGoogle Scholar - Sztajn, P., Ball, D. L., & McMahon, T. A. (2006). Designing learning opportunities for mathematics teacher developers. In K. Lynch-Davis & R. L. Rider (Eds.),
*The work of mathematics teacher educators: Continuing the conversation*,*AMTE monograph*(Vol. 3, pp. 149–162). San Diego, CA: Association of Mathematics Teacher Educators.Google Scholar - Taylor, C. E. (2013). Facilitating prospective teachers’ knowledge of student understanding: The case of one mathematics teacher educator. In A. M. Lindmeier & A. Heinze (Eds.),
*Proceedings of the 37th conference of the international group for the psychology of mathematics education*(Vol. 4, pp. 273–280). Kiel: PME.Google Scholar - Taylor, P. M., & Ronau, R. (2006). Syllabus study: A structured look at mathematics methods courses.
*AMTE Connections,**16*(1), 12–15.Google Scholar - Thanheiser, E., Browning, C., Edson, A. J., Lo, J. J., Whitacre, I., Olanoff, D., et al. (2014). Mathematical content knowledge for teaching elementary mathematics: What do we know, what do we not know, and where do we go?
*The Mathematics Enthusiast*,*11*(2), 433–448.Google Scholar - Thompson, A. G., Philipp, R. A., Thompson, P. W., & Boyd, B. A. (1994). Calculational and conceptual orientations in teaching mathematics. In A. Coxford (Ed.),
*Professional development for teachers of mathematics. Yearbook of the NCTM*(pp. 79–92). Reston, VA: NCTM.Google Scholar - Tirosh, D., Tsamir, P., Levenson, E., & Tabach, M. (2011). From preschool teachers’ professional development to children’s knowledge: Comparing sets.
*Journal of Mathematics Teacher Education,**14,*113–131.CrossRefGoogle Scholar - Tzur, R. (2001). Becoming a mathematics teacher-educator: Conceptualizing the terrain through self-reflective analysis.
*Journal of Mathematics Teacher Education,**4*(4), 259–283.CrossRefGoogle Scholar - Vale, C. (2010). Supporting “out-of-field” teachers of secondary mathematics.
*Australian Mathematics Teacher,**66*(1), 17–24.Google Scholar - Van Hiele, P. M. (1999). Developing geometric thinking through activities that begin with play.
*Teaching Children Mathematics,**5*(6), 310.Google Scholar - Van Zoest, L. R., Moore, D. L., & Stockero, S. L. (2006). Transition to teacher educator: A collaborative effort. In K. Lynch-Davis & R. L. Rider (Eds.),
*AMTE monograph, Vol. 3: The work of mathematics teacher educators*(pp. 133–148). San Diego: Association of Mathematics Teacher Educators.Google Scholar - Windschitl, M., Thompson, J., Braaten, M., & Stroupe, D. (2012). Proposing a core set of instructional practices and tools for teachers of science.
*Science Education,**96*(5), 878–903.CrossRefGoogle Scholar - Wolters, C. A., & Pintrich, P. R. (1998). Contextual differences in student motivation and self-regulated learning in mathematics, English, and social studies classrooms.
*Instructional Science,**26*(1–2), 27–47.CrossRefGoogle Scholar - Zaslavsky, O. (2007). Mathematics-related tasks, teacher education, and teacher educators.
*Journal of Mathematics Teacher Education,**10*(4), 433–440.CrossRefGoogle Scholar - Zaslavsky, O. (2009). Mathematics educators’ knowledge and development. In R. Even & D. L. Ball (Eds.),
*The professional education and development of teachers of mathematics*(pp. 105–111). Berlin: Springer.CrossRefGoogle Scholar