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Supporting Mathematics Teachers’ Diagnostic Competence Through the Use of One-to-One, Task-Based Assessment Interviews

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Diagnostic Competence of Mathematics Teachers

Part of the book series: Mathematics Teacher Education ((MTEN,volume 11))

Abstract

In this chapter, the important role that one-to-one, task-based assessment interviews can play in developing inservice and preservice mathematics teachers’ diagnostic competence is presented. We argue that the use of such interviews builds competence through enhancing teachers’ knowledge of individual and group understanding of mathematics, including misconceptions and preferred strategies, while providing an understanding of the typical learning paths in various mathematical domains. The use of such interviews also provides a model for teachers’ interactions and discussions with children in classrooms, building both pedagogical content knowledge and subject matter knowledge.

This chapter draws substantially on a previous paper by the authors (Clarke, Clarke, & Roche, 2011. Building teachers’ expertise in understanding, assessing and developing children’s mathematical thinking: the power of task-based, one-to-one interviews. ZDM Mathematics Education, 43(6), 901–913.)

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References

  • Behr, M., & Post, T. (1992). Teaching rational number and decimal concepts. In T. Post (Ed.),Teaching mathematics in grades K-8: Researchbased methods (2nd ed., pp. 201–248). Boston, MA: Allyn and Bacon.

    Google Scholar 

  • Behr, M. J., Lesh, R., Post, T. R., & Silver, E. A. (1983). Rational number concepts. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 91–126). New York, NY: Academic Press.

    Google Scholar 

  • Behr, M. J., Wachsmuth, I., & Post, T. R. (1985). Construct a sum: A measure of children’s understanding of fraction size. Journal for Research in Mathematics Education, 16(2), 120–131.

    Article  Google Scholar 

  • Bobis, J., Clarke, B. A., Clarke, D. M., Gould, P., Thomas, G., Wright, R., & Young-Loveridge, J. (2005). Supporting teachers in the development of young children’s mathematical thinking: Three large scale cases. Mathematics Education Research Journal, 16(3), 27–57.

    Article  Google Scholar 

  • Brown, R. I., & Semple, L (1970). Effects of unfamiliarity on the overt verbalisation and perceptual motor behaviour of nursery school children. British Journal of Educational Psychology, 40, 291–298.

    Google Scholar 

  • Clarke, B. A. (2004). A shape is not defined by its shape: Developing young children’s geometric understanding. Journal of Australian Research in Early Childhood Education, 11(2), 110–127.

    Google Scholar 

  • Clarke, B. A. (2008). A framework of growth points as a powerful teacher development tool. In D. Tirosh & T. Wood (Eds.), The international handbook of mathematics teacher education, Tools and processes in mathematics teacher education (Vol. 2, pp. 235–256). Rotterdam, Netherlands: Sense Publishers.

    Google Scholar 

  • Clarke, B. A. (2015). Assessing young children’s mathematical understanding: Opportunities and expectations at the transition to school. In B. Perry, A. MacDonald, & A. Gervasoni (Eds.), Mathematics and transition to school (pp. 31–46). Singapore: Springer.

    Google Scholar 

  • Clarke, B. A., & Clarke, D. M. (2004). Using questioning to elicit and develop children’s mathematical thinking. In G. W. Bright & R. N. Rubenstein (Eds.), Professional development guidebook for perspectives on the teaching of mathematics (pp. 5–10). Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • Clarke, B. A., Clarke, D. M., & Cheeseman, J. (2006). The mathematical knowledge and understanding young children bring to school. Mathematics Education Research Journal, 18(1), 81–107.

    Article  Google Scholar 

  • Clarke, D., Clarke, B., & Roche, A. (2011). Building teachers’ expertise in understanding, assessing and developing children’s mathematical thinking: The power of task-based, one-to-one interviews. ZDM Mathematics Education, 43(6), 901–913.

    Article  Google Scholar 

  • Clarke, D. M. (2001). Understanding, assessing, and developing young children’s mathematical thinking: Research as a powerful tool for professional growth. In J. Bobis, M. Mitchelmore, & B. Perry (Eds.),Numeracy and beyond (Proceedings of the 24th Annual Conference of the Mathematics Education Research Group of Australasia, pp. 9–26). Sydney, Australia: MERGA.

    Google Scholar 

  • Clarke, D. M., & Roche, A. (2009). Students’ fraction comparison strategies as a window into robust understanding and possible pointers for instruction. Educational Studies in Mathematics, 72, 127–138.

    Article  Google Scholar 

  • Clarke, D. M., Cheeseman, J., Gervasoni, A., Gronn, D., Horne, M., McDonough, A., … Rowley, G. (2002). Early Numeracy Research Project final report. Melbourne, Australia: Mathematics Teaching and Learning Centre, Australian Catholic University.

    Google Scholar 

  • Clarke, D. M., Roche, A., Mitchell, A., & Sukenik, M. (2006). Assessing student understanding of fractions using task-based interviews. In J. Novotna, H. Moraova, M. Kratka, & N. Stehlikova (Eds.), Proceedings of the 30th Conference of the International Group of Psychology of mathematics Education (Vol. 2, pp. 337–344). Prague, Czech Republic: PME.

    Google Scholar 

  • Clements, D. H., Swaminathan, S., Hannibal, M. A. Z., & Sarama, J. (1999). Young children’s conceptions of space. Journal for Research in Mathematics Education, 30(2), 192–212.

    Article  Google Scholar 

  • Clements, M. A., & Ellerton, N. (1995). Assessing the effectiveness of pencil-and-paper tests for school mathematics. In B. Atweh & S. Flavel (Eds.),Galtha: MERGA 18 (Proceedings of the 18th Annual Conference of the Mathematics Education Research Group of Australasia, pp. 184–188). Darwin, Australia: MERGA.

    Google Scholar 

  • Cohen, D. K., & Ball, D. L. (1999). Instruction, capacity, and improvement. Philadelphia, PA: Consortium for Policy Research in Education, University of Pennsylvania.

    Book  Google Scholar 

  • Department of Education & Training. (2001). Early numeracy interview booklet. Retrieved from https://www.eduweb.vic.gov.au/edulibrary/public/teachlearn/student/mathscontinuum/onlineinterviewbklet.pdf

  • Erickson, F. (1986). Qualitative methods in research on teaching. In M. C. Whittrock (Ed.), Handbook of research on teaching (pp. 119–161). New York, NY: Macmillan.

    Google Scholar 

  • Faragher, R., & Clarke, B. A. (Eds.). (2014). Educating learners with down syndrome: Research, theory and practice with children and adolescents. London, UK: Routledge

    Google Scholar 

  • Fosnot, C. T., & Dolk, M. (2002). Young mathematicians at work: Constructing fractions, decimals and percents. Portsmouth, NH: Heinemann.

    Google Scholar 

  • Ginsburg, H. (2009). The challenge of formative assessment in mathematics education: Children’s minds, teachers’ minds. Human Development, 52, 109–128.

    Article  Google Scholar 

  • Ginsburg, H., Klein, A., & Starkey, P. (1998). The development of children’s mathematical thinking: Connecting research with practice. In I. E. Siegel & K. A. Renninger (Eds.), Handbook of child psychology, Child psychology in practice (Vol. 4, 5th ed., pp. 23–26). New York, NY: John Wiley & Sons.

    Google Scholar 

  • Graeber, A., & Tirosh, D. (2008). Pedagogical content knowledge. In P. Sullivan & T. Wood (Eds.),Knowledge and beliefs in mathematics teaching and teaching development (pp. 117–132). Rotterdam, The Netherlands: Sense Publishers.

    Google Scholar 

  • den Heuvel-Panhuizen, V. (2001). Children learn mathematics. Utrecht, The Netherlands: Freudenthal Institute.

    Google Scholar 

  • Hill, H., Ball, D., & Schilling, S. (2008). Unpacking pedagogical content knowledge: Conceptualising and measuring teachers’ topic-specific knowledge of students. Journal for Research in Mathematics Education, 39, 372–400.

    Google Scholar 

  • Horne, M, & Rowley, G. (2001). Measuring growth in early numeracy: Creation of interval scales to monitor development. In M van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education (pp. 3-161–3-168). Utrecht, The Netherlands: Freudenthal Institute.

    Google Scholar 

  • Lamon, S. (2007). Rational numbers and proportional reasoning. In F. K. Lester (Ed.), Second handbook on research on mathematics teaching and learning (pp. 629–668). Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • Lehrer, R., & Chazan, D. (1998). Designing learning environments for developing understanding of geometry and space. Mahwah, NJ: Lawrence Erlbaum.

    Google Scholar 

  • McDonough, A., Clarke, B. A., & Clarke, D. M. (2002). Understanding assessing and developing young children’s mathematical thinking: The power of the one-to-one interview for preservice teachers in providing insights into appropriate pedagogical practices. International Journal of Educational Research, 37, 107–112.

    Article  Google Scholar 

  • Miles, M. B., & Huberman, A. M. (1994). Qualitative data analysis: An expanded sourcebook. Beverley Hills, CA: Sage.

    Google Scholar 

  • Pearn, C., & Stephens, M. (2004). Why you have to probe to discover what year 8 students really think about fractions. In I. Putt, R. Faragher, & M. McLean (Eds.), Mathematics education for the third millennium: Towards 2010 (Proceedings of the 27th Annual Conference of the Mathematics Education Research Group of Australasia) (pp. 430–437). Townsville, Austraia: MERGA.

    Google Scholar 

  • Post, T., Behr, M. J., & Lesh, R. (1986). Research-based observations about children’s learning of rational number concepts. Focus on Learning Problems in Mathematics, 8(1), 39–48.

    Google Scholar 

  • Post, T., & Cramer, K. A. (2002). Children’s strategies in ordering rational numbers. In D. L. Chambers (Ed.), Putting research into practice in the elementary grades (pp. 141–144). Reston, VA: National Council of the Teachers of Mathematics.

    Google Scholar 

  • Roche, A. (2005). Longer is larger—Or is it? Australian Primary Mathematics Classroom, 10(3), 11–16.

    Google Scholar 

  • Roche, A., & Clarke, D. M. (2004). When does successful comparison of decimals reflect conceptual understanding? In I. Putt, R. Farragher, & M. McLean (Eds.), Mathematics education for the third millennium: Towards 2010 (Proceedings of the 27th annual conference of the Mathematics Education Research Group of Australasia, pp. 486–493). Townsville, Australia: MERGA.

    Google Scholar 

  • Shulman, L. (1987). The wisdom of practice: Managing complexity in medicine and teaching. In D. C. Berliner & B. V. Rosenshine (Eds.), Talks to teachers: A festschrift for N. L. Gage (pp. 369–387). New York, NY: Random House.

    Google Scholar 

  • Sleep, L., & Boerst, T. A. (2012). Preparing beginning teachers to elicit and interpret students’ mathematical thinking. Teaching and Teacher Education, 28, 1038–1048.

    Article  Google Scholar 

  • Sowder, J. (2007). The mathematics education and development of teachers. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 157–223). Charlotte, NC: Information Age Publishing & National Council of Teachers of Mathematics.

    Google Scholar 

  • Stake, R. (1995). The art of case study research. Thousand Oaks, CA: Sage.

    Google Scholar 

  • Steinle, V., & Stacey, K. (2003). Grade-related trends in the prevalence and persistence of decimal misconceptions. In N. Pateman, B. Dougherty, & J. Zilliox (Eds.), Proceedings of the 2003 Joint Meeting of PME and PMENA (Vol. 4, pp. 259–266). Honolulu, Hawaii: International Group for the Psychology of Mathematics Education.

    Google Scholar 

  • Sullivan, P., Clarke, D. M., Cheeseman, J., & Mulligan, J. (2001). Moving beyond physical models in learning multiplicative reasoning. In M. van den Heuvel-Panhuizen (Ed.). Proceedings of the 25th annual conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 233–240). Utrecht, The Netherlands: Freundenthal Institute.

    Google Scholar 

  • Swan, P. A. J. (2002). The computation choices made by students in Years 5 to 7. Unpublished doctoral dissertation, Edith Cowan University, Western Australia.

    Google Scholar 

  • Webb, N., & Romberg, T. A. (1992). Implications of the NCTM Standards for mathematics assessment. In T. A. Romberg (Ed.), Mathematics assessment and evaluation: Imperatives for mathematics education (pp. 37–60). Albany, NY: State University of New York Press.

    Google Scholar 

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Clarke, D.M., Roche, A., Clarke, B. (2018). Supporting Mathematics Teachers’ Diagnostic Competence Through the Use of One-to-One, Task-Based Assessment Interviews. In: Leuders, T., Philipp, K., Leuders, J. (eds) Diagnostic Competence of Mathematics Teachers. Mathematics Teacher Education, vol 11. Springer, Cham. https://doi.org/10.1007/978-3-319-66327-2_9

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  • DOI: https://doi.org/10.1007/978-3-319-66327-2_9

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