Skip to main content
SpringerLink
Log in

“Mathematical knowledge for teaching”: adapting U.S. measures for use in Ireland

  • Published:
Journal of Mathematics Teacher Education Aims and scope Submit manuscript

Abstract

This article describes a study in which measures of mathematical knowledge for teaching developed in the United States were adapted to measure mathematical knowledge for teaching in Ireland. When adapting the measures it was not assumed that the mathematical knowledge used by Irish and U.S. teachers is the same. Instead psychometric and interview-based methods were used to determine a correspondence between the constructs being measured, and ensure the integrity of item performance in the Irish context. The study found overlap between the knowledge that is used to teach in both Ireland and the United States, and that the items tapped into this knowledge. However, specific findings confirm the usefulness of conducting extensive checks on the validity of items used in cross-national contexts. The process of adaptation is described to provide guidance for others interested in using the items to measure mathematical knowledge for teaching outside the United States. The process also enabled the authors to raise questions about the assumptions that lie behind the practice-based construct of mathematical knowledge for teaching.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Notes

  1. The lead author spent 11 years teaching in Irish primary schools, where he taught all class levels and worked for almost 2 years as a resource teacher with specific responsibility for mathematics.

  2. We use the terms “translate” and “adapt” interchangeably throughout this article.

  3. Some will argue that assuming mathematics to be universal is mistaken and that it is culturally bound (e.g., see Jaworski and Phillips 1999).

  4. See http://www.standards.nctm.org/

  5. See http://www.entfemp.ie/press/2000/210700.htm (accessed on October 7th 2006).

  6. For example see http://www.ed.gov/news/pressreleases/2003/11/11192003.html (accessed on October 7th 2006).

  7. Some item stems have three or more items attached and others have just one.

  8. These numbers refer to teachers who took the pre-test only and not to teachers for whom pre- and post-test data were available.

  9. At least 93 of the teachers have spent 1 year teaching mathematics (and all other subjects) at primary school level and most have spent longer. Seventy-nine teachers received their teacher certification from a pre-service programme in the Republic of Ireland, 13 from Great Britain or Northern Ireland, one from the United States. One is certified to teach secondary school mathematics, one is qualified to teach other subjects in secondary school and five did not supply information about their qualifications or experience.

  10. Many of the items used in the test are not released and therefore cannot be reproduced in this article. Released items can be accessed at the website: http://www.sitemaker.umich.edu/lmt/files/LMT_sample_items.pdf.

  11. Although this change would probably reduce the challenge of the word problem for primary school students (Greer 1987), we do not know if it would affect the difficulty of the item for teachers.

  12. It is interesting to note that “congruent” is one of the terms which TIMSS was requested to change. “Same shape and size” was considered an acceptable translation but “equal” was considered too imprecise (Mullis et al. 1996, pp. 1–6).

Abbreviations

CCK:

Common content knowledge

IRT:

Item Response Theory

KCS:

Knowledge of content and students

KCT:

Knowledge of content and teaching

LMT:

Learning Mathematics for Teaching

MKT:

Mathematical knowledge for teaching

NCTM:

National Council of Teachers of Mathematics

PISA:

Program for International Student Assessment

SCK:

Specialized content knowledge

TIMSS:

Trends in International Mathematics and Science Study

U.S.:

United States

References

  • 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, 145–172.

    Article  Google Scholar 

  • Andrews, P., & Hatch, G. (2000). A comparison of Hungarian and English teachers’ conceptions of mathematics and its teaching. Educational Studies in Mathematics, 43(1), 31–64.

    Article  Google Scholar 

  • Ball, D. L. (1988). Knowledge and reasoning in mathematical pedagogy: Examining what prospective teachers bring to teacher education. Unpublished doctoral dissertation, Michigan State University, East Lansing.

  • Ball, D. L. (1990). The mathematical understandings that prospective teachers bring to teacher education. The Elementary School Journal, 90(4), 449–466.

    Article  Google Scholar 

  • Ball, D. L., & Bass, H. (2000). Interweaving content and pedagogy in teaching and learning to teach. In J. Boaler (Ed.), Multiple perspectives on the teaching and learning of mathematics (pp. 83–104). Westport, CT: Ablex.

    Google Scholar 

  • Ball, D. L., & Bass, H. (2003a). Making mathematics reasonable in school. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 27–44). Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • Ball, D. L., & Bass, H. (2003b). Toward a practice-based theory of mathematical knowledge for teaching. Paper Presented at the Proceedings of the 2002 Annual Meeting of the Canadian Mathematics Education Study Group, Edmonton, AB.

  • Ball, D. L., Bass, H., Hill, H., Sleep, L., Phelps, G., & Thames, M. H. (2006). Knowing and using mathematics in teaching. Paper Presented at the Learning Network Conference, Teacher Quality, Quantity, and Diversity, Washington DC, January 30–31. Retrieved on January 9th 2008 from http://www-personal.umich.edu/~dball/presentations/013106_NSF_MSP.pdf

  • Ball, D. L., Thames, M. H., & Phelps, G. (2005). Knowledge of mathematics for teaching: What makes it special? Paper Presented at the AERA Annual Conference, Montréal, Canada.

  • Bass, H., & Lewis, J. (2005, April 15). What’s in collaborative work? Mathematicians and educators developing measures of mathematical knowledge for teaching. Paper Presented at the Annual meeting of the American Educational Research Association, Montréal, Canada.

  • 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.

    Article  Google Scholar 

  • Chrostowski, S. J., & Malak, B. (2004). Translation and cultural adaptation of the TIMSS 2003 instruments. In M. O. Martin, I. V. S. Mullis & S. J. Chrostowski (Eds.), TIMSS 2003 technical report. Chestnut Hill, MA: TIMSS & PIRLS International Study Center, Boston College.

    Google Scholar 

  • Cogan, L. S., & Schmidt, W. H. (1999). An examination of instructional practices in six countries. In G. Kaiser, E. Luna & I. Huntley (Eds.), International comparisons in mathematics education (pp. 68–85). London: Falmer Press.

    Google Scholar 

  • Cronbach, L. J., & Shavelson, R. J. (2004). My current thoughts on coefficient alpha and successor procedures. Education and Psychological Measurement, 64(3), 391–418.

    Article  Google Scholar 

  • Eisenhart, M., Borko, H., Underhill, R., Brown, C., Jones, D., & Agard, P. (1993). Conceptual knowledge falls through the cracks: Complexities of learning to teach mathematics for understanding. Journal for Research in Mathematics Education, 24(1), 8–40.

    Article  Google Scholar 

  • Fuson, K. C., Stigler, J. W., & Bartsch, K. (1988). Grade placement of addition and subtraction topics in Japan, Mainland China, the Soviety Union, Taiwan and the United States. Journal for Research in Mathematics Education, 19(5), 449–456.

    Article  Google Scholar 

  • Gorgorió, N., & Planas, N. (2001). Teaching mathematics in multilingual classrooms. Educational Studies in Mathematics, 47(1), 7–33.

    Article  Google Scholar 

  • Government of Ireland. (1999). Primary school curriculum: Mathematics. Dublin, Ireland: The Stationery Office.

    Google Scholar 

  • Government of Ireland. (2005). An evaluation of curriculum implementation in primary schools: English, mathematics and visual arts. Dublin, Ireland: The Stationery Office.

    Google Scholar 

  • Greer, B. (1987). Nonconservation of multiplication and division involving decimals. Journal for Research in Mathematics Education, 18(1), 37–45.

    Article  Google Scholar 

  • Grisay, A. (2002). Chapter 5. Translation and cultural appropriateness of the test and survey. In R. Adams & M. Wu (Eds.), PISA 2000 Technical report. Paris: Organisation for Economic Co-operation and Development (OECD).

    Google Scholar 

  • Hambleton, R. K., Merenda, P. F., & Spielberger, C. D. (Eds.). (2005). Adapting educational and psychological tests for cross-cultural assessment. Mahwah, New Jersey: Lawrence Erlbaum Associates Inc.

  • Hill, H. C. (2004a). Content knowledge for teaching mathematics measures (CKTM measures) Technical report on number and operations content knowledge items 2001–2003. Ann Arbor, MI: University of Michigan.

    Google Scholar 

  • Hill, H. C. (2004b). Content knowledge for teaching mathematics measures (CKTM measures): Technical report on geometry items—2002. Ann Arbor, MI: University of Michigan.

    Google Scholar 

  • Hill, H. C. (2004c). Technical report on number and operations knowledge of students and content items—2001–2003. Ann Arbor, MI: University of Michigan.

    Google Scholar 

  • Hill, H. C. (2004d). Technical report on patterns, functions and algebra items—2001. Ann Arbor, MI: University of Michigan.

    Google 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.

    Article  Google Scholar 

  • Hill, H. C., Ball, D. L., & Schilling, S. G. (2007). Unpacking “Pedagogical content knowledge”: Conceptualizing and measuring teachers’ topic-specific knowledge of students. Journal for Research in Mathematics Education (in review).

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

    Article  Google Scholar 

  • Jaworski, B., & Phillips, D. (1999). Looking abroad: International comparisons and the teaching of mathematics in Britain. In B. Jaworski & D. Phillips (Eds.), Comparing standards internationally: Research and practice in mathematics and beyond (pp. 7– 22). Oxford: Symposium Books.

    Google Scholar 

  • Keitel, C., & Kilpatrick, J. (1999). The rationality and irrationality of international comparative studies. In G. Kaiser, E. Luna, & I. Huntley (Eds.), International comparisons in mathematics education (pp. 241–256). London: Falmer Press.

    Google Scholar 

  • LeTendre, G. K., Baker, D. P., Akiba, M., Goesling, B., & Wiseman, A. (2001). Teachers’ work: Institutional isomorphism and cultural variation in the U.S., Germany, and Japan. Educational Researcher, 30(6), 3–15.

    Article  Google Scholar 

  • Ma, L. (1999). Knowing and teaching elementary mathematics. Mahwah, New Jersey: Lawrence Erlbaum Associates Inc.

    Google Scholar 

  • Maxwell, B. (1996). Translation and cultural adaptation of the survey instruments. In M. O. Martin & D. L. Kelly (Eds.), Third international mathematics and science study (TIMSS) technical report (Vol. 1, pp. 8-1–8-10). Chestnut Hill, MA: Boston College.

    Google Scholar 

  • Mayer, R. E., Sims, V., & Tajika, H. (1995). A comparison of how textbooks teach mathematical problem solving in Japan and the United States. American Educational Research Journal, 32(2), 443–460.

    Google Scholar 

  • Meyer, J. W., Ramirez, F. O., & Soysal, Y. N. (1992). World expansion of mass education. Sociology of Education, 65(2), 128–149.

    Article  Google Scholar 

  • Mullis, I. V. S., Kelly, D. L., & Haley, K. (1996). Translation verification procedures. In M. O. Martin & I. V. S. Mullis (Eds.), Third international mathematics and science study: Quality assurance in data collection (pp. 1-1–1-14). Chestnut Hill, MA: Boston College.

    Google Scholar 

  • Murphy, B. (2004). Practice in Irish infant classrooms in the context of the Irish Primary School Curriculum (1999): Insights from a study of curriculum implementation. International Journal of Early Years Education, 12(3), 245–257.

    Article  Google Scholar 

  • National Council for Curriculum and Assessment. (2005). Primary curriculum review, phase 1: Final report. Dublin: NCCA.

    Google Scholar 

  • National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • Santagata, R. (2004). “Are you joking or are you sleeping?” Cultural beliefs and practices in Italian and U.S. tachers’ mistake-handling strategies. Linguistics and Education, 15, 141–164.

    Article  Google Scholar 

  • Schmidt, W. H., Jorde, D., Cogan, L. S., Barrier, E., Gonzalo, I., Moser, U., et al. (1996). Characterizing pedagogical flow: An investigation of mathematics and science teaching in six countries. Dordrecht, NL: Kluwer Academic Publishers.

    Google Scholar 

  • Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 4–14.

  • Silver, E. A., Leung, S. S., & Cai, J. (1995). Generating multiple solutions for a problem: A comparison of the responses of U.S. and Japanese students. Educational Studies in Mathematics, 28(1), 35–54.

    Article  Google Scholar 

  • Stein, M. K., Baxter, J. A., & Leinhardt, G. (1990). Subject-matter knowledge and elementary instruction: A case from functions and graphing. American Educational Research Journal, 27(4), 639–663.

    Google Scholar 

  • Stigler, J. W., & Hiebert, J. (1999). The teaching gap: Best ideas from the world’s teachers for improving education in the classroom. New York: The Free Press.

    Google Scholar 

  • Stigler, J. W., Lee, S.-Y., & Stevenson, H. W. (1987). Mathematics classrooms in Japan, Taiwan and the United States. Child Development, 58(5), 1272–1285.

    Article  Google Scholar 

  • Walsh, T. (2006). A critical analysis of curricular policy in Irish primary schools, 1897–1990. Unpublished doctoral dissertation, National University of Ireland Maynooth, Kildare.

  • Wilson, M., & Xie, Y. (2004). The imperial vs. metric study (IMS). Retrieved May 6th 2006, 2006.

  • Wilson, S. M. (2003). California dreaming: Reforming mathematics education. New Haven, CT: Yale University Press.

    Google Scholar 

  • Wilson, S. M., & Berne, J. (1999). Teacher learning and the acquisition of professional knowledge: An examination of research on contemporary professional development. Review of Research in Education, 24, 173–209.

    Google Scholar 

  • Zimowski, M. F., Muraki, E., Mislevy, R. J., & Bock, R. D. (1996). Bilog-MG 3. Chicago: Scientific Software International.

    Google Scholar 

Download references

Acknowledgements

The research reported in this article was supported in part by grants from the U.S. Department of Education to the Consortium for Policy Research in Education (CPRE) at the University of Pennsylvania (Grant #OERI-R308A60003) and the Center for the Study of Teaching and Policy at the University of Washington (Grant #OERI-R308B70003); the National Science Foundation’s Interagency Educational Research Initiative (IERI) to the University of Michigan (Grant #s REC-9979863 & REC-0129421), the William and Flora Hewlett Foundation, and the Atlantic Philanthropies, the Department of Education and Science (Ireland) grant R/D7/03 and by Coláiste Mhuire Marino, Dublin. Opinions expressed in this article are those of the authors, and do not reflect the views of the U.S. Department of Education, the National Science Foundation, the William and Flora Hewlett Foundation, the Atlantic Philanthropies, the Department of Education and Science (Ireland) or Coláiste Mhuire Marino, Dublin. The authors would like to thank Hyman Bass, Merrie Blunk, Carolyn Dean, Imani Goffney, Jennifer Lewis, Laurie Sleep, and Mark Hoover Thames for their help in developing aspects of this article. The authors would also like to thank Jeremy Kilpatrick and Larry Ludlow who read and offered feedback on earlier drafts of this article. Thanks also to Dina Tirosh and four anonymous reviewers for helpful comments which improved this article. Errors are the responsibility of the authors.

Author information

Authors and Affiliations

  1. Coláiste Mhuire, Marino Institute of Education, Griffith Avenue, Dublin, 9, Ireland

    Seán Delaney

  2. University of Michigan, Ann Arbor, MI, USA

    Seán Delaney, Deborah Loewenberg Ball, Heather C. Hill, Stephen G. Schilling & Deborah Zopf

Authors
  1. Seán Delaney
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Deborah Loewenberg Ball
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Heather C. Hill
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Stephen G. Schilling
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Deborah Zopf
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Seán Delaney.

Additional information

An earlier draft of this article was presented at the annual meeting of the American Educational Research Association, Montréal, CN, April 15, 2005.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Delaney, S., Ball, D.L., Hill, H.C. et al. “Mathematical knowledge for teaching”: adapting U.S. measures for use in Ireland. J Math Teacher Educ 11, 171–197 (2008). https://doi.org/10.1007/s10857-008-9072-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10857-008-9072-1

Keywords

Search

Navigation