# Developing prospective teachers’ conceptions with well-designed tasks: explaining successes and analyzing conceptual difficulties

## Abstract

Several researchers have documented prospective teachers’ (PTs’) conceptions of various mathematical topics. However, less is known about how PTs’ conceptions develop. To address this gap, I designed two tasks with the goals of addressing the PTs’ initial conceptions of multidigit whole numbers and helping them develop more sophisticated ones. I examined how PTs’ conceptions changed while working on these tasks in two settings (a teaching experiment with 6 PTs and a mathematics methods course with 33 PTs) and modified the tasks on the basis of the results. Consistent with prior findings, this study showed that PTs entered with limited conceptions. This study showed further that (a) well-designed tasks (addressing the PTs’ incoming conceptions as well as focusing on the desired conceptions) can help PTs develop content knowledge, (b) conceptual difficulties may persist even with well-designed tasks, and (c) artifacts of children’s mathematical thinking can be used to develop mathematical content knowledge. Instructional implications are discussed.

## Keywords

Prospective teacher Teacher education Content knowledge Whole number Place value Task design## Notes

### Acknowledgments

I would like to thank my mentor and advisor Randy Philipp for his continued interest in my work and for the walks during which we brainstorm ideas. I would also like to thank Bonnie Schappelle, who has also continuously supported my work by sharing my interest and giving me feedback. In addition, I would like to thank Signe Kastberg, who helped with the enactment of the teaching experiment, as well as Briana Mills, Krista Stand, and Jodi Fasteen, who work(ed) with me tirelessly to understand prospective teachers better.

## References

- Ambrose, R. (1998). A classroom study of invented subtraction strategies. Doctoral dissertation, 1998.
*Dissertation Abstracts International, 59*(05), 1497.Google Scholar - Ball, D. (1988a).
*Knowledge and reasoning in mathematical pedagogy: Examining what prospective teachers bring to teacher education (doctoral dissertation, 1988).*50 doctoral dissertation, Michigan State University, Ann Arbor.Google Scholar - Ball, D. (1988b).
*The subject-matter preparation of prospective mathematics teachers: Challenging the myths*. Lansing: Michigan State University.Google Scholar - Ball, D. L. (1991). Research on teaching mathematics: Making subject-matter knowledge part of the equation. In J. Brophy (Ed.),
*Advances in research on teaching: Teachers’ knowledge of subject matter as it relates to their teaching practice*(Vol. 2, pp. 1–48). Greenwich, CT: JAI Press.Google Scholar - Ball, D., & Bass, H. (2000). Interweaving content and pedagogy in teaching and learning to teach: Knowing and using mathematics. In J. Boaler (Ed.),
*Multiple perspectives on mathematics teaching and learning*(pp. 83–104). Westport, CT: Greenwood Publishing.Google Scholar - Bennett, A. B., & Nelson, L. T. (2007).
*Mathematics for elementary teachers: A conceptual approach*. Boston, MA: McGraw Hill.Google Scholar - Bransford, J. D., Brown, A. L., & Cocking, R. R. (Eds.). (1999).
*How people learn*. Washington, DC: National Academy Press.Google Scholar - Brown, S., Cooney, T., & Jones, D. (1990). Mathematics teacher education. In W. Houston (Ed.),
*Handbook of research on teacher education*(pp. 87–109). New York, NY: Macmillan.Google Scholar - Cobb, P., Wood, T., Yackel, E., & McNeal, B. (1992). Characteristics of classroom mathematics traditions: An interactional analysis.
*American Educational Research Journal,**29*(3), 573–604.CrossRefGoogle Scholar - Comiti, C., & Ball, D. (1996). Preparing teachers to teach mathematics: A comparative perspective. In A. Bishop, K. Clements, C. Keitel, J. Kilpatrick, & C. Laborde (Eds.),
*International handbook of mathematics education*(pp. 1123–1153). Dordrecht, The Netherlands: Kluwer.Google Scholar - Conference Board of Mathematical Sciences. (2001).
*The mathematical education of teachers*. Washington, DC: Mathematical Association of America.Google Scholar - Ell, F., Hill, M., & Grudnoff, L. (2012). Finding out more about teacher candidates’ prior knowledge: Implications for teacher educators. [Article].
*Asia-Pacific Journal of Teacher Education,**40*(1), 55–65. doi: 10.1080/1359866x.2011.643760.CrossRefGoogle Scholar - Freudenthal, H. (1973).
*Mathematics as an educational task*. Dordrecht: Reidel.Google Scholar - Fuson, K. C. (1990). A forum for researchers: Issues in place-value and multidigit addition and subtraction learning and teaching.
*Journal for Research in Mathematics Education,**21*, 273–280.CrossRefGoogle Scholar - Fuson, K. C., Wearne, D., Hiebert, J. C., Murray, H. G., Human, P. G., Olivier, A. I., et al. (1997). Children’s conceptual structures for multidigit numbers and methods of multidigit addition and subtraction.
*Journal for Research in Mathematics Education,**28*, 130–162.CrossRefGoogle Scholar - Graeber, A. O., Tirosh, D., & Glover, R. (1989). Preservice teachers’ misconceptions in solving verbal problems in multiplication and division.
*Journal for Research in Mathematics Education,**20*(1), 95–102.CrossRefGoogle Scholar - Harkness, S. S., & Thomas, J. (2008). Reflections on “multiplication as original sin”: The implications of using a case to help preservice teachers understand invented algorithms.
*Journal of Mathematical Behavior,**27*(2), 128–137.CrossRefGoogle Scholar - Hiebert, J. (1997).
*Making sense: Teaching and learning mathematics with understanding*. Portsmouth, NH: Heinemann.Google Scholar - Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.),
*Conceptual and procedural knowledge: The case of mathematics*(pp. 1–27). Hillsdale, NJ: Erlbaum.Google Scholar - Hiebert, J., & Morris, A. K. (2012). Teaching, rather than teachers, as a path toward improving classroom instruction.
*Journal of Teacher Education, 63*(2), 92–102.Google Scholar - Hiebert, J., & Wearne, D. (1996). Instruction, understanding, and skill in multidigit addition and subtraction.
*Cognition and Instruction,**14*, 251–283.CrossRefGoogle Scholar - Kamii, C. (1986). Place value: An explanation of its difficulty and educational implications for the primary grades.
*Journal of Research in Childhood Education,**1*(2), 75–86.CrossRefGoogle Scholar - Kamii, C. (1994).
*Young children continue to reinvent arithmetic—3rd-grade-implications of Piaget’s theory*. New York: Teachers College, Columbia University Press.Google Scholar - Kamii, C., & Dominick, A. (1998). The harmful effects of algorithms in grades 1–4. In L. J. Morrow & M. J. Kenney (Eds.),
*The teaching and learning of algorithms in school mathematics. 1998 yearbook of the national council of teachers of mathematics*(pp. 130–140). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Kamii, C., Lewis, B. A., & Livingston, S. J. (1993). Primary arithmetic: Children inventing their own procedures.
*Arithmetic Teacher,**41*, 200–203.Google Scholar - Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001).
*Adding it up*. Helping children learn mathematics. Washington, DC: National Academy Press.Google Scholar - Liljedahl, P., Chernoff, E., & Zazkis, R. (2007). Interweaving mathematics and pedagogy in task design: A tale of one task.
*Journal of Mathematics Teacher Education,**10*(4–6), 239–249.CrossRefGoogle Scholar - Lo, J.-J., Grant, T. J., & Flowers, J. (2008). Challenges in deepening prospective teachers’ understanding of multiplication through justification.
*Journal of Mathematics Teacher Education,**11*(1), 5–22.CrossRefGoogle Scholar - Ma, L. (1999).
*Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in china and the United States*. Mahwah, NJ: Erlbaum.Google Scholar - McClain, K. (2003). Supporting preservice teachers’ understanding of place value and multidigit arithmetic.
*Mathematical Thinking and Learning,**5*(4), 281–306.CrossRefGoogle Scholar - Murphy, C. (2011). Comparing the use of the empty number line in england and the Netherlands.
*British Educational Research Journal,**37*(1), 147–161.CrossRefGoogle Scholar - National Governors Association, & Council of Chief State School Officers. (2010). Common core standards initiative. Retrieved February, 10, 2012, from http://www.corestandards.org/.
- Nugent, P. M. (2007). Lattice multiplication in a preservice classroom.
*Mathematics Teaching in the Middle School,**13*(2), 110–113.Google Scholar - Overbay, S. R., & Brod, M. J. (2007). Magic with mayan math.
*Teaching Mathematics in the Middle School,**12*, 340–347.Google Scholar - Pesek, D. D., & Kirshner, D. (2000). Interference of instrumental instruction in subsequent relational learning.
*Journal for Research in Mathematics Education,**31*, 524–540.CrossRefGoogle Scholar - Philipp, R., Schappelle, B., Siegfried, J., Jacobs, V., & Lamb, L. (2008).
*The effects of professional development on the mathematical content knowledge of k-3 teachers*. NY: Paper presented at the Annual Meeting of the American Educational Reserach Association New York.Google Scholar - San Diego State Foundation, Philipp, R., Cabral, C., & Schappelle, B. (2012).
*Imap: Integrating mathematics and pedagogy: Searchable collection of children’s mathematical thinking video clips and facilitator’s guide*. Allyn & Bacon.Google Scholar - Simon, M. (1993). Prospective elementary teachers’ knowledge of division.
*Journal for Research in Mathematics Education,**24*(3), 233–254.CrossRefGoogle Scholar - Simon, M. (2006). Key developmental understandings in mathematics: A direction for investigating and establishing learning goals.
*Mathematical Thinking & Learning,**8*(4), 359–371.CrossRefGoogle Scholar - Simon, M., & Blume, G. W. (1992).
*Mathematization as a component of the concept of ratio*-*as*-*measure: A study of prospective elementary teachers*. Paper presented at the Annual Meeting of the American Educational Research Association, San Francisco, CA. http://stats.lib.pdx.edu/proxy.php?url=http://search.ebscohost.com/login.aspx?direct=true&db=eric&AN=ED349175&site=ehost-live. - Smith, M. S., & Stein, M. K. (1998). Selecting and creating mathematical tasks: From research to practice.
*Mathematics Teaching in the Middle School,**3*(5), 344–350.Google Scholar - Southwell, B., & Penglase, M. (2005). Mathematical knowledge of pre-service primary teachers. In H. L. Chick & J. L. Vincent (Eds.),
*Proceedings of the international group for the psychology of mathematics education*(Vol. 4, pp. 209–216). Melbourne, Australia: University of Melbourne.Google Scholar - Sowder, J., & Schappelle, B. (1994). Research into practice: Number sense-making.
*Arithmetic Teacher,**41*, 342–345.Google Scholar - Steffe, L. P., & Cobb, P. (1988).
*Construction of arithmetical meanings and strategies*. New York: Springer-Verlag.CrossRefGoogle Scholar - Steffe, L., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In A. E. Kelly & R. A. Lesh (Eds.),
*Handbook of research design in mathematics and science education*(pp. 267–306). Mahwah, NJ: Erlbaum.Google Scholar - Stein, M. K., & Lane, S. (1996). Instructional tasks and the development of student capacity to think and reason: An analysis of the relationship between teaching and learning in a reform mathematics project.
*Educational Reseach and Evaluation,**2*, 50–80.CrossRefGoogle Scholar - Thanheiser, E. (2009a).
*Developing preservice teachers’ conceptions of numbers in our base-ten numeration system*. San Diego, CA: Paper presented at the Annual Meeting of the American Educational Reserach Association.Google Scholar - Thanheiser, E. (2009b). Preservice elementary school teachers’ conceptions of multidigit whole numbers.
*Journal for Research in Mathematics Education,**40*(3), 251–281.Google Scholar - Thanheiser, E. (2010). Investigating further preservice teachers’ conceptions of multidigit whole numbers: Refining a framework.
*Educational Studies in Mathematics,**75*(3), 241–251.CrossRefGoogle Scholar - Thanheiser, E. (2012). Understanding multidigit whole numbers: The role of knowledge pieces, connections, and context in understanding regrouping 3 + digit numbers.
*Journal of Mathematical Behavior,**31*(2), 220–234.CrossRefGoogle Scholar - Thanheiser, E., & Philipp, R. (2012).
*Using interviews with preservice teachers as a tool to motivate them to learn mathematics*. Paper presented at the AMTE.Google Scholar - Thanheiser, E., Browning, C. A., Lo, J. J., Kastberg, S., & Edson, A. J. (2013a). Building a knowledge base: Understanding prospective elementary school teachers' mathematical content knowledge.
*International Journal for Mathematics Teaching and Learning*. Retrieved from http://www.cimt.plymouth.ac.uk/journal/thanheiser.pdf. - Thanheiser, E., Philipp, R., Fasteen, J., Strand, K., & Mills, B. (2013b). Preservice-teacher interviews: A tool for motivating mathematics learning.
*Mathematics Teacher Educator,**1*(2), 137–147.CrossRefGoogle Scholar - Thanheiser, E., & Rhoads, K. (2009). Exploring preservice teachers’ conceptions of numbers via the Mayan number system. In S. Swars, D. Stinson, & S. Lemons-Smith (Eds.),
*North American chapter of the international group for the psychology of mathematics education*(Vol. 5, pp. 1220–1227). Atlanta, GA: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.Google Scholar - Tirosh, D., & Graeber, A. O. (1989). Preservice elementary teachers’ explicit beliefs about multiplication and division.
*Educational Studies in Mathematics,**20*(1), 79–96.CrossRefGoogle Scholar - Valeras, M., & Becker, J. (1997). Children’s developing understanding of place value: Semiotic aspects.
*Cognition and Instruction,**15*, 265–286.CrossRefGoogle Scholar - Verschaffel, L., Greer, B., & De Corte, E. (2007). Whole number concepts and operations. In F. K. Lester & M. National Council of Teachers (Eds.),
*Second handbook of research on mathematics teaching and learning: A project of the national council of teachers of mathematics*(pp. 577–628). Charlotte, NC: Information Age Pub.Google Scholar - Watson, A., & Mason, J. (2007). Taken-as-shared: A review of common assumptions about mathematical tasks in teacher education.
*Journal of Mathematics Teacher Education,**10*(4–6), 205–215.CrossRefGoogle Scholar - Weber, K. (2001). Student difficulties in constructing proofs: The need for strategic knowledge.
*Educational Studies in Mathematics,**48*, 101–119.CrossRefGoogle Scholar - Yackel, E., Underwood, D., & Elias, N. (2007). Mathematical tasks designed to foster a reconceptualized view of early arithmetic.
*Journal of Mathematics Teacher Education,**10*(4–6), 351–367.CrossRefGoogle Scholar - Zaslavsky, O. (2007). Mathematics-related tasks, teacher education, and teacher educators: The dynamics associated with tasks in mathematics teacher education.
*Journal for Mathematics Teacher Education,**10*(4–6), 433–440.CrossRefGoogle Scholar - Zazkis, R., & Campbell, S. (1996). Divisibility and multiplicative structure of natural numbers: Preservice teachers’ understanding.
*Journal for Research in Mathematics Education, 27*, 540–563.Google Scholar