# Teachers’ conceptions of prior knowledge and the potential of a task in teaching practice

## Abstract

In this study, we specify teacher’s knowledge and beliefs that influence their enactment of mathematics tasks. The use of tasks in lessons requires teachers to consider students’ mathematical thinking and to know how to effectively implement tasks. Among the many aspects of teachers’ understanding of students and ability to implement tasks for learning, this study focuses on teachers’ knowledge and beliefs of students’ prior knowledge and the potential to develop new knowledge while solving mathematical tasks; we exemplify these knowledge and beliefs through the practices of three Algebra I teachers. Three distinct conceptions of prior knowledge and two conceptions of the potential of a task emerged. From combinations of these conceptions, we defined three types of teaching. Two types of teaching reduced potential new knowledge to the prior context, whereas one type of teaching promoted prior knowledge to develop new knowledge. In particular, the type of teaching we call “reviewing” explains why teachers repeat the way they taught a concept the first time. Our study suggests that it is important for teachers to conceive students’ prior knowledge as the knowledge to be developed and the task as having potential for developing new knowledge in order to teach for coherence.

## Keywords

Teacher knowledge Teacher conception Prior knowledge Mathematical potential of tasks Teaching practice Mathematical knowledge used in teaching## Notes

### Acknowledgements

This article is based on a professional development project funded by a Grant authorized under Title II Part A Subpart 3 and Title II Part B of the Elementary and Secondary Education Act, administered at the federal level by the US Department of Education and at the state level by the Washington Student Achievement Council and Office of Superintendent of Public Instruction, respectively. These federal and state agencies are not responsible for statements made in the paper

## References

- Askew, M. (2008). Mathematical discipline knowledge requirements for prospective primary teachers, and the structure and teaching approaches of programs designed to develop that knowledge. In P. Sullivan & T. Wood (Eds.),
*Knowledge and beliefs in mathematics teaching and teaching development*(pp. 13–35). Rotterdam: Sense publishers.Google Scholar - Askew, M., & Brown, M. (1997).
*Effective teachers of numeracy in UK primary schools: Teachers’ beliefs, practices and pupils’ learning*. Paper presented at the European conference on educational research (ECER 97), Johann Wolfgang Goethe Universitat, Frankfurt am Main.Google Scholar - Askew, M., Brown, M., Rhodes, V., Johnson, D., and William, D. (1997).
*Effective teachers of numeracy.*Final report. London: King’s College.Google Scholar - Ball, D. L., Lubienski, S. T., & Mewborn, D. S. (2001). Research on teaching mathematics: The unsolved problem of teachers’ mathematical knowledge. In V. Richardson (Ed.),
*Handbook of research on teaching*(4th ed., pp. 433–456). Washington, DC: American Educational Research Association.Google 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 - Bennett, N., & Carre, C. (Eds.). (1993).
*Learning to teach*. London: Routledge.Google Scholar - Bennett, N., & Desforges, C. (1988). Matching classroom tasks to students’ attainments.
*Elementary School Journal,**88*(3), 221–234.CrossRefGoogle Scholar - Bishop, J. P., Lamb, L. C., Philipp, R. A., Whitacre, I., Schappelle, B. P., & Lewis, M. L. (2014). Obstacles and affordances for integer reasoning from the analysis of children’s thinking and the history of mathematics.
*Journal for Research in Mathematics Education,**45*(1), 19–61.CrossRefGoogle Scholar - Borko, H., Eisenhart, M., Brown, C. A., Underhill, R. G., Jones, D., & Agard, P. C. (1992). Learning to teach hard mathematics: Do novice teachers and their instructors give up too easily?
*Journal for Research in Mathematics Education,**23*(3), 194–222.CrossRefGoogle Scholar - Brousseau, G. (1997).
*Theory of didactical situations in mathematics*(N. Balacheff, M. Cooper, R. Sutherland, & V. Warfield, Eds., Trans.). Dordrecht: Kluwer.Google Scholar - Brown, M., Askew, M., Baker, D., Denvir, H., & Millett, A. (1998). Is the national numeracy strategy research-based?
*British Journal of Educational Studies,**46*(4), 362–385.CrossRefGoogle Scholar - Calderhead, J. (1996). Teachers: Beliefs and knowledge. In D. C. Berliner & R. C. Calfee (Eds.),
*Handbook of educational psychology*(pp. 709–725). New York: Simon & Schuster Macmillan.Google Scholar - Carlson, M., & Oehrtman, M. (2005)
*Key aspects of knowing and learning the concept of function*. Research sampler series, MAA notes online.Google Scholar - Carpenter, T., Fennema, E., & Franke, M. L. (1996). Cognitively guided instruction: A knowledge base for reform in primary mathematics instruction.
*The Elementary School Journal,**97*(1), 3–20.CrossRefGoogle Scholar - Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (1999).
*Children’s mathematics: Cognitively guided instruction*. Portsmouth, NH: Heinemann.Google Scholar - Charalambous, C. Y., (2008). Mathematical knowledge for teaching and the unfolding of tasks in mathematics lessons: Integrating two lines of research. In O. Figuras, J. L. Cortina, S. Alatorre, T. Rojano, & A. Sepulveda (EDS.),
*Proceedings of the 32nd annual conference of the international group for the psychology of mathematics education*, (Vol. 2, pp. 281–288).Morelia: PME.Google Scholar - Charmaz, K. (2002). Qualitative interviewing and grounded theory analysis. In J. Gubrium & J. A. Holstein (Eds.),
*Handbook of interview research*(pp. 675–694). Thousand Oaks: Sage.Google Scholar - Clark, C. M., & Peterson, P. L. (1986). Teachers’ thought processes. In M. C. Wittrock (Ed.),
*Handbook of research on teaching*(3rd ed., pp. 255–296). New York: Macmillan.Google Scholar - Cohen, D. K. (1990). A revolution in one classroom: The case of Mrs. Oublier.
*Educational Evaluation and Policy Analysis,**12*(3), 311–329.CrossRefGoogle Scholar - Cohen, D. K., & Ball, D. L. (1990). Policy and practice: An overview.
*Educational Evaluation and Policy Analysis,**12*(3), 233–239.CrossRefGoogle Scholar - Common Core State Standards Initiative (CCSSI). 2010
*Common core state standards for mathematics*. Retrieved December 10, 2015, from http://www.corestandards.org/wp-content/uploads/Math_Standards.pdf. - Desforges, C., & Cockburn, A. (1987).
*Understanding the mathematics teachers: A study of practice in first schools*. London: The Palmer Press.Google Scholar - Doyle, W. (1988). Work in mathematics classes: The context of students’ thinking during instruction.
*Educational Psychologist,**23*(2), 167–180.CrossRefGoogle Scholar - Fennema, E., Carpenter, T. P., Franke, M. L., Levi, L., Jacobs, V. R., & Empson, S. B. (1996). A longitudinal study of learning to use children’s thinking in mathematics instruction.
*Journal for Research in Mathematics Education,**27*(4), 403–434.CrossRefGoogle Scholar - Franke, M. L., Kazemi, E., & Battey, D. (2007). Mathematics teaching and classroom practice. In F. K. Lester Jr. (Ed.),
*Second handbook of research on mathematics teaching and learning*(pp. 225–256). Greenwich, CT: Information Age.Google Scholar - Gibson, J. J. (1986).
*The ecological approach to visual perception*. Hillsdale, NJ: Erlbaum.Google Scholar - Heinz, K., Kinzel, M., Simon, M. A., & Tzur, R. (2000). Moving students through steps of mathematical knowing: An account of the practice of an elementary mathematics teacher in transition.
*Journal of Mathematical Review,**19,*83–107.Google Scholar - Henningsen, M., & Stein, M. K. (1997). Mathematical tasks and student cognition: Classroom based factors that support and inhibit high-level mathematical thinking and reasoning.
*Journal for Research in Mathematics Education,**28*(5), 424–549.CrossRefGoogle Scholar - Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.),
*Handbook of research on mathematics teaching and learning*(pp. 65–97). New York: Macmillan.Google Scholar - Hiebert, J., & Stigler, J. W. (2000). A proposal for improving classroom teaching: Lessons from the TIMSS video study.
*Elementary School Journal,**101,*3–20.CrossRefGoogle Scholar - Hill, H. C., Ball, D. L., & Schilling, S. G. (2008a). 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. C., Blunk, M. L., Charalambous, C. Y., Lewis, J. M., Phelps, G. C., Sleep, L., et al. (2008b). Mathematical knowledge for teaching and the mathematical quality of instruction: An exploratory study.
*Cognition and Instruction,**26*(4), 430–511.CrossRefGoogle Scholar - Hill, H., Rowan, B., & Ball, D. L. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement.
*American Educational Research Journal,**42*(2), 371–406.CrossRefGoogle Scholar - Klug, J., Bruder, S., Kelava, A., Spiel, C., & Schmitz, B. (2013). Diagnostic competence of teachers: A process model that accounts for diagnosing learning behavior tested by means of a case scenario.
*Teaching and Teacher Education,**30,*38–46. doi: 10.1016/j.tate.2012.10.004.CrossRefGoogle Scholar - Knuth, E. J. (2000). Student understanding of the Cartesian connection: An exploratory study.
*Journal for Research in Mathematics Education,**31*(4), 500–507.CrossRefGoogle Scholar - Leikin, R., & Zazkis, R. (2010). Teachers’ opportunities to learn mathematics through teaching. In R. Leikin & R. Zazkis (Eds.),
*Learning through teaching mathematics: Development of teachers’ knowledge and expertise in practice*(pp. 3–21). New York: Springer.CrossRefGoogle Scholar - Liljedahl, P., Rolka, K., & Rösken, B. (2007). Affecting affect: The re-education of preservice teachers’ beliefs about mathematics and mathematics learning and teaching. In M. Strutchens & W. Martin (Eds.)
*69th NCTM Yearbook.*NCTM.Google Scholar - Ma, L. (1999).
*Knowing and teaching elementary mathematics*. Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar - National Board for Professional Teaching Standards. (2001).
*NBPTS early childhood generalist standards*(2nd ed.). Arlington, VA: National Board for Professional Teaching Standards.Google Scholar - National Council of Teachers of Mathematics. (1991).
*Professional standards for teaching mathematics*. Reston, VA: NCTM.Google Scholar - Peterson, P. L. (1990). Doing more in the same amount of time: Cathy Swift.
*Educational Evaluation and Policy Analysis,**12,*261–280.CrossRefGoogle Scholar - Philipp, R. A. (2007). Mathematics teachers’ beliefs and affect. In F. K. Lester Jr. (Ed.),
*Second handbook of research on mathematics teaching and learning*(pp. 257–315). Charlotte, NC: Information Age.Google Scholar - Polly, D., McGee, J. R., Wang, C., Lambert, R. G., Pugalee, D. K., & Johnson, S. (2013). The association between teachers’ beliefs, enacted practices, and student learning in mathematics.
*The Mathematics Educator,**22*(2), 11–30.Google Scholar - Polly, D., Neale, H., & Pugalee, D. K. (2014). How does ongoing task-focused mathematics professional development influence elementary school teacher’s knowledge, beliefs and enacted pedagogies?
*Early Childhood Education Journal*. doi: 10.1007/s10643-013-0585-6.Google Scholar - Schwartz, D. L., Sears, D., & Chang, J. (2007). Reconsidering prior knowledge. In M. C. Lovett & P. Shah (Eds.),
*Thinking with data*(pp. 319–344). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar - Simon, M. A. (1993). Prospective elementary teachers’ knowledge of division.
*Journal for Research in Mathematics Education,**24*(3), 233–254.CrossRefGoogle Scholar - Simon, M. A., & Schifter, D. (1991). Towards a constructivist perspective: An intervention study of mathematics teacher development.
*Educational Studies in Mathematics,**22,*309–331.CrossRefGoogle Scholar - Simon, M. A., & Schifter, D. (1993). Toward a constructivist perspective: The impact of a mathematics teacher inservice program on students.
*Educational Studies in Mathematics,**25,*331–340.CrossRefGoogle Scholar - Simon, M. A., & Tzur, R. (1999). Explicating the teacher’s perspective from the researchers’ perspectives: Generating accounts of mathematics teachers’ practice.
*Journal for Research in Mathematics Education,**30*(3), 252–264.CrossRefGoogle Scholar - Steffe, L. P. (2002). A new hypothesis concerning children’s fractional knowledge.
*Journal of Mathematical Behavior,**20,*267–307.CrossRefGoogle Scholar - Stein, M. K., Remillard, J., & Smith, M. S. (2007). How curriculum influences student learning. In F. Lester (Ed.),
*Second handbook of research on mathematics teaching and learning*(pp. 319–370). Greenwich, CT: Information Age Publishing.Google Scholar - Strauss, A., & Corbin, J. (1990).
*Basics of qualitative research: Grounded theory procedures and techniques*(2nd ed.). Newbury Park, CA: Sage.Google Scholar - Sullivan, P. (2008). Knowledge for teaching mathematics. In P. Sullivan & T. Wood (Eds.),
*Knowledge and beliefs in mathematics teaching and teaching development*(Vol. 1, pp. 1–9). Dordrecht: Sense Publishers.Google Scholar - Swan, M. (2006). Designing and using research instruments to describe the beliefs and practices of mathematics teachers.
*Research in Education,**75,*58–70.CrossRefGoogle Scholar - Swan, M. (2007). The impact of task-based professional development on teachers’ practices and beliefs: A design research study.
*Journal of Mathematics Teacher Education*. doi: 10.1007/s10857-007-9038-8.Google Scholar - Sztajn, P., Confrey, J., Wilson, P. H., & Edgington, C. (2012). Learning trajectory based instruction toward a theory of teaching.
*Educational Researcher,**41*(5), 147–156.CrossRefGoogle Scholar - Thompson, A. G. (1984). The relationship of teachers’ conceptions of mathematics and mathematics teaching to instructional practice.
*Educational Studies in Mathematics,**15,*105–127.CrossRefGoogle Scholar - Thompson, A. G. (1992). Teachers’ beliefs and conceptions: A synthesis of the research. In D. A. Grouws (Ed.),
*Handbook of research on mathematics teaching and learning*(pp. 127–146). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Thompson, P. W. (1994). The development of the concept of speed and its relationship to concepts of rate. In G. Harel & J. Confrey (Eds.),
*The development of multiplicative reasoning in the learning of mathematics*(pp. 181–234). Albany, NY: SUNY Press.Google Scholar - Thompson, P. W. (2015). Researching mathematical meanings for teaching. In L. English & D. Kirshner (Eds.),
*Third handbook of international research in mathematics education*(pp. 435–461). London: Taylor and Francis.Google Scholar - Tzur, R. (2008). Profound awareness of the learning paradox: A journey towards epistemologically regulated pedagogy in mathematics teaching and teacher education. In B. Jaworski & T. Wood (Eds.),
*The international handbook for mathematics teacher education: The mathematics teacher educator as a developing professional*(Vol. 4, pp. 137–156). Rotterdam: Sense.Google Scholar - Tzur, R. (2010). How and what might teachers learn through teaching mathematics: Contributions to closing an unspoken gap. In R. Leikin & R. Zazkis (Eds.),
*Learning through teaching mathematics: Development of teachers’ knowledge and expertise in practice*(pp. 49–67). New York: Springer.CrossRefGoogle Scholar - Watson, A., & Mason, J. (2007). Taken-as-shared: A review of common assumption about tasks in teacher education.
*Journal of Mathematics Teacher Education,**10*(4), 205–215.CrossRefGoogle Scholar - Watson, A., & Ohtani, M. (2015).
*Task design in mathematics education: An ICMI study 22*. Berlin: Springer.CrossRefGoogle Scholar - Wilson, P. H., Sztajn, P., Edgington, C., & Confrey, J. (2014). Teachers’ use of their mathematical knowledge for teaching in learning a mathematics learning trajectory.
*Journal of Mathematics Teacher Education,**17*(2), 149–175.CrossRefGoogle Scholar