# Why are mathematics teachers “not sure”?

## Keywords:

. teachers’ mathematical knowledge for teaching . measurement . multiple-choice items## Abstract

Researchers have widely adopted the mathematical knowledge for teaching (MKT) measures. This paper aims at investigating why teachers select “I’m not sure” as a suggested solution in MKT items. In this study, in-service teachers responded to multiple-choice MKT items, they submitted written responses to open-ended questions, and they discussed these items in group discussions. We analyse the different kinds of responses from three teachers in depth. A framework of cognitive types of teacher knowledge—distinguishing between Type 1, Type 2 and Type 3 knowledge—was applied in the analysis. One teacher selected “I’m not sure” due to insufficient MKT, whereas the other two teachers appeared to have Type 1 and Type 2 knowledge. These findings provide existence proof that the knowledge teachers utilize in responses to open-ended questions and group discussions does not necessarily mirror the knowledge used when selecting a particular multiple-choice response.## References

Adler, J., & Patahuddin, S. M. (2012). Recontexualising items that measure mathematical knowledge for teaching into scenario based interviews: an investigation. Journal of Education (56), 17–43.

Alvesson, M. & Karreman, D. (2011). Qualitative research and theory development: Mystery as method. London: Sage Publications.

Ball, D. L. & Hill, H. C. (2008). Mathematical knowledge for teaching (MKT) measures. Mathematics released items 2008. Retrieved January 11, 2016, from http://sitemaker.umich.edu/lmt/files/LMT_sample_items.pdf.

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. DOI 10.1177/0022487108324554.

Beswick, K., Callingham, R. & Watson, J. (2012). The nature and development of middle school mathematics teachers’ knowledge. Journal of Mathematics Teacher Education, 15(2), 131–157. DOI 10.1007/s10857-011-9177-9.

Blömeke, S. & Delaney, S. (2012). Assessment of teacher knowledge across countries: A review of the state of research. ZDM – The International Journal on Mathematics Education, 44(3), 223–247. DOI 10.1007/s11858-012-0429-7.

Davis, B. & Simmt, E. (2006). Mathematics-for-teaching: An ongoing investigation of the mathematics that teachers (need to) know. Educational Studies in Mathematics, 61(3), 293–319. Retrieved February 13, 2015, from http://www.jstor.org/stable/25472073.

Fauskanger, J. (2015). Challenges in measuring teachers’ knowledge. Educational Studies in Mathematics, 90(1), 57–73. DOI 10.1007/s10649-015-9612-4.

Fauskanger, J. & Mosvold, R. (2015a). En metodisk studie av innholdsanalyse – med analyser av matematikklæreres undervisningskunnskap som eksempel. [English: A methodolodical study of content analysis – with analyses of teachers’ mathematical knowledge for teaching as an example]. Nordic Studies in Mathematics Education, 20(2), 79–96.

Fauskanger, J., & Mosvold, R. (2015b). The difficulty of measuring types of mathematics teachers’ knowledge. In Silfverberg H., Kärki T. & Hannula M. S. (Eds.), Nordic research in mathematics education – Proceedings of NORMA14, Turku, June 3-6, 2014. Studies in Subject Didactics 10 (pp. 71–80). Turku: The Finnish Research. Association for Subject Didactics.

Fauskanger, J. & Mosvold, R. (2014). Studying teachers’ knowledge by the use of multiple-choice items. The case of “I’m not sure”. Nordic Studies in Mathematics Education, 19(3-4), 41–55.

Fauskanger, J. & Mosvold, R. (2012). “Wrong, but still right”. Teachers reflecting on MKT items. I L. R. Van Zoest, J.-J. Lo & J. L. Kratky (Eds.), Proceedings of the 34th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 423–429). Kalamazoo, MI: Western Michigan University.

Fauskanger, J. & Mosvold, R. (in press). Why are Laura and Jane “not sure”? Proceedings of CERME 9, 9th Congress of European Research in Mathematics Education.

Fauskanger, J., Mosvold, R., Bjuland, R. & Jakobsen, A. (2011). Does the format matter? How the multiple-choice format might complicate the MKT items. Nordic Studies in Mathematics Education, 16(4), 45–67.

Fauskanger, J., Jakobsen, A., Mosvold, R. & Bjuland, R. (2012). Analysis of psychometric properties as part of an iterative adaptation process of MKT items for use in other countries. ZDM – The International Journal on Mathematics Education, 44(2), 387–399. DOI 10.1007/s11858-012-0403-4.

Haertel, E. (2004). Interpretive argument and validity argument for certification testing: Can we escape the need for psycological theory? Measurement: Interdisciplinary Research and Perspectives, 2(3), 175–178.

Haladyna, T. M. (2004). Developing and validating multiple-choice test items (3rd ed.). New Jersey, NJ: Lawrence Earlbaum Associates Inc.

Hill, H. C. (2007). Introduction to MKT scales. Mathematical Knowledge for Teaching (MKT) measures. University of Michigan.

Hill, H. C., Blunk, M., Charalambous, C. Y., Lewis, J. M., Phelps, G., Sleep, L. & Al., E. (2008). Mathematical knowledge for teaching and the mathematical quality of instruction: An exploratory study. Cognition and Instruction, 26(4), 430–511. DOI 10.1080/07370000802177235.

Hill, H. C., Rowan, B. & Ball, D. L. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42(2), 371–406. DOI 10.3102/00028312042002371.

Hill, H. C., Schilling, S. G. & Ball, D. L. (2004). Developing measures of teachers’ mathematical knowledge for teaching. The Elementary School Journal, 105(1), 11–30.

Hill, H. C., Sleep, L., Lewis, J. M. & Ball, D. L. (2007). Assessing teachers’ mathematical knowledge. What knowledge matters and what evidence counts? In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 111–156). Charlotte, NC: Information Age Publishing.

Hill, H. C., Umland, K., Litke, E. & Kapitula, L. R. (2012). Teacher quality and quality teaching: Examining the relationship of a teacher assessment practice. American Journal of Education, 118(4), 489–519.

Hoover, M., Mosvold, R. & Fauskanger, J. (2014). Common tasks of teaching as a resource for measuring professional content knowledge internationally. Nordic Studies in Mathematics Education, 19(3–4), 7–20.

Hsieh, H.-F. & Shannon, S. E. (2005). Three approaches to qualitative content analysis. Qualitative Health Research, 15(9), 1277–1288.

Jones, G. A., Thornton, C. A., Putt, I. J., Hill, K. M., Mogill, T. A., Rich, B. S. & Van Zoest, L. R. (1996). Multidigit number sense: A framework for instruction and assessment. Journal for Research in Mathematics Education, 27(3), 310–336. DOI 10.2307/749367.

Kazima, M., Jakobsen, A., & Kasoka, D.N. (2016). Use of Mathematical Tasks of Teaching and the Corresponding LMT Measures in the Malawi Context. The Mathematics Enthusiast, 13(1&2), 172–187.

Kwon, M., Thames, M. H., & Pang, J. (2012). To change or not to change: adapting mathematical knowledge for teaching (MKT) measures for use in Korea. ZDM – The International Journal on Mathematics Education, 44(3), 371–385. DOI 10.1007/s11858-012-0397-y.

Ng, D., Mosvold, R., & Fauskanger, J. (2012). Translating and adapting the mathematical knowledge for teaching (MKT) measures: The cases of Indonesia and Norway. The Montana Mathematics Enthusiast, 9(1-2), 149–17.

Orrill, C. H., Ok-Kyeong, K., Peters, S. A., Lischka, A. E., Jong, C., Sanchez, W. B., & Eli, J. A. (2015). Challenges and Strategies for Assessing Specialised Knowledge for Teaching. Mathematics Teacher Education and Development, 17(1), 12–29.

Osterlind, S. J. (1997). Constructing test items: Multiple-choice, constructed-response, performance and other formats (2nd ed.). Hingham, MA: Kluwer Academic Publishers.

Schilling, S. G., Blunk, M. & Hill, H. C. (2007). Test validation and the MKT measures: Generalizations and conclusions. Measurement: Interdisciplinary Research and Perspectives, 5(2-3), 118–128.

Schoenfeld, A. H. (2007). Commentary: The complexities of assessing teacher knowledge. Measurement: Interdisciplinary Research and Perspectives, 5(2), 198–204.

Silverman, D. (2006). Interpreting qualitative data: Methods for analysing talk, text and interaction (3rd ed.). Thousand Oaks, CA: Sage Publications Inc.

Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20–26.

Tchoshanov, M. A. (2011). Relationship between teacher knowledge of concepts and connections, teaching practice, and student achievement in middle grades mathematics. Educational Studies in Mathematics, 76(2), 141–164. DOI 10.1007/s10649-010-9269-y.