Algebraic knowledge for teaching test: An adaptation study

Abstract

In this study, the Mathematical Knowledge for Teaching-Elementary Patterns Functions and Algebra-Content Knowledge (MKT-PFA) test, originally developed in English as part of the "Learning Mathematics for Teaching Project" at Michigan University, was adapted into Turkish. The test comprises two equivalent forms, A and B, each translated into Turkish and culturally adapted through consultations with two mathematics education academics and five secondary school math teachers pursuing doctoral studies. A total of 328 pre-service teachers at a Turkish public university's elementary school mathematics teaching department were administered form A (14 questions, 29 items) and form B (12 questions, 27 items) at a one-week interval. Psychometric analyses revealed high reliability (KR-20: A=0.712, B=0.735; Lord reliability: A=0.733, B=0.756), and strong correlations (rpbi) with the original English forms, indicating suitable adaptation. Item difficulties analyzed using a one-parameter Item Response Theory model showed a normal distribution, affirming the tests' validity for assessing pre-service teachers' algebra teaching knowledge in Türkiye.

Keywords

• Algebra teaching
• Algebraic knowledge
• Adaptation study
• Preservice Teachers
• MKT-PFA.

References

1. 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(2), 145-172.
2. Aryadoust, V., Ng, L.Y., & Sayama, H. (2021). A comprehensive review of Rasch measurement in language assessment: Recommendations and guidelines for research. Language Testing, 38(1), 6-40. https://doi.org/10.1177/0265532220927487
3. Baker, F.B. (2001). The basics of item response theory (2nd ed). (ED458219). https://eric.ed.gov/?id=ED458219
4. Baker, F.B., & Kim, S.H. (2017). The basics of item response theory using R (Vol. 969). Springer.
5. Ball, D.L. (1990). Prospective elementary and secondary teachers' understanding of division. Journal for Research in Mathematics Education, 21(2), 132-144.
6. Ball, D.L., & Hill, H.C. (2008). Measuring teacher quality in practice. In D. H. Gitomer (Ed.), Measurement Issues and Assessment for Teaching Quality, pp. 80-98. SAGE.
7. Ball, D.L., Hill, H.C., & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American Educator (Fall 2005), 14-46.
8. 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. https://doi.org/10.1177/0022487108324554

9. Brennan, R.L., & National Council on Measurement in Education (NCME). (2006). Educational measurement. Praeger.
10. Borsboom, D. (2005). Measuring the mind: Conceptual issues in contemporary psychometrics. Cambridge University.
11. Boykin, A.A., Ezike, N.C., & Mysers, A.J. (2023). Model-data fit evaluation: Posterior checks and Bayesian model selection. International Encyclopedia of Education (4th Edition), 279-289.
12. Burnham, K.P., & Anderson, D.R. (2002). Model selection and multimodal inference: a practical information-theoretic approach. Springer.
13. Charalambous, C.Y. (2008). Prospective teachers’ mathematical knowledge for teaching and their performance in selected teaching practices: Exploring a complex relationship. (Doctoral dissertation) University of Michigan.
14. Chou, Y.T., & Wang, W.C. (2010). Checking dimensionality in item response models with principal component analysis on standardized residuals. Educational and Psychological Measurement, 70(5), 717-731. https://doi.org/10.1177/0013164410379322
15. Cohen, I. (2011). Teacher-student interaction in classrooms of students with specific learning disabilities learning English as a foreign language. Journal of Interactional research in communication disorders, 2(2), 271-292. https://doi.org/10.1558/jircd.v2i2.271
16. Council of Higher Education, (CoHE (Yükseköğretim Kurulu), 2018). New teacher training programs, reasons for updating the programs, innovations and implementation principles [In Turkish]. https://www.yok.gov.tr/Documents/Kurumsal/egitim_ogretim_dairesi/Yeni-Ogretmen-Yetistirme-Lisans_Programlari/AA_Sunus_%20Onsoz_Uygulama_Yonergesi.pdf [In Turkish]
17. Cole, Y. (2012). Assessing elemental validity: The transfer and use of mathematical knowledge for teaching measures in Ghana. ZDM, 44(3), 415-426. https://doi.org/10.1007/s11858-012-0380-7
18. Crocker, L., & Algina, J. (1986). Introduction to classical and modern test theory. (ED312281). ERIC. https://eric.ed.gov/?id=ED312281
19. Cronbach, L.J., & Shavelson, R.J. (2004). My current thoughts on coefficient alpha and successor procedures. Educational and Psychological Measurement, 64(3), 391-418.
20. Çelen, Ü. (2008). Comparison of validity and reliability of two tests developed by classical test theory and item response theory. Elementary Education Online, 7(3), 758-768.
21. de Ayala, R.J. (2013). The theory and practice of item response theory. Guilford.
22. Delaney, S., Ball, D.L., Hill, H.C., Schilling, S.G., & Zopf, D. (2008). Mathematical knowledge for teaching: Adapting US measures for use in Ireland. Journal of Mathematics Teacher Education, 11(3), 171-197. https://doi.org/10.1007/s10857-008-9072-1
23. Driscoll, M. (1999). Fostering algebraic thinking: a guide for teachers grades 6-10. NH: Heinemann.
24. Edelen, M.O., & Reeve, B.B. (2007). Applying item response theory (IRT) modeling to questionnaire development, evaluation, and refinement. Quality of life research, 16, 5-18. https://doi.org/10.1007/s11136-007-9198-0
25. Embretson, S.E., & Reise, S.P. (2013). Item response theory. Psychology.
26. Esendemir, O., & Bindak, R. (2019). Adaptation of the test developed to measure mathematical knowledge of teaching geometry in Turkey. International Journal of Educational Methodology, 5(4), 547-565. https://doi.org/10.12973/ijem.5.4.547
27. Fan, J., & Bond T. (2019). Applying Rasch measurement in language assessment: Unidimensionality and local independence. In V. Aryadoust & M. Raquel (Eds.), Quantitative data analysis for language assessment, Vol. I: Fundamental techniques (pp. 83–102). Routledge. https://doi.org/10.4324/9781315187815
28. 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, 44, 387–399. https://doi.org/10.1007/s11858-012-0403-4
29. Frary, R.B. (1989). Partial credit scoring methods for multiple choice Tests. Applied Measurement in Education, 2(1), 79-96.
30. Hambleton, R.K, Swaminathan, H., & Rogers, H.J. (1991). Fundamentals of item response theory. SAGE Publications.
31. Hambleton, R.K., & Swaminathan, H. (1985). Item response theory: principles and applications. Academic Publishers Group.
32. Hambleton, R.K. (1994). Guidelines for adapting educational and psychological tests: A progress report. European Journal of Psychological Assessment, 10, 229–240.
33. Han, H. (2022). The effectiveness of weighted least squares means and variance adjusted based fit indices in assessing local dependence of the rasch model: Comparison with principal component analysis of residuals. PloS ONE, 17(9). https://doi.org/10.1371/journal.pone.0271992
34. Hill, H.C. (2007). Mathematical knowledge of middle school teachers: Implications for the no child left behind policy initiative. Educational Evaluation and Policy Analysis, 29(2), 95–114. https://doi.org/10.3102/0162373707301711
35. 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.
36. 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. https://doi.org/3699380
37. Hill, H.C., Schilling, S.G., & Ball, D.L. (2004). Developing measures of teachers’ mathematics knowledge for teaching. The Elementary School Journal, 105(1), 11 30. https://doi.org/10.1086/428763
38. Hill, H., & Ball, D.L. (2009). The curious and crucial case of mathematical knowledge for teaching. Phi Delta Kappan, 91(2), 68-71. https://doi.org/10.1177/00317217090910021
39. Holmes, F. & Brian F.F. (2019). A comparison of estimation techniques for IRT models with small samples, Applied Measurement in Education, 32(2), 77 96. https://doi.org/10.1080/08957347.2019.1577243
40. Huang, R., & Kulm, G. (2012). Preservice middle grade mathematics teachers’ knowledge of algebra for teaching. The Journal of Mathematical Behavior, 31(4), 417-430. https://doi.org/10.1016/j.jmathb.2012.06.001
41. Kieran, C., Kieran, C., & Ohmer. (2018). Teaching and learning algebraic thinking with 5-to 12-year-olds (pp. 79-105). Springer.
42. Kim, Y. (2016). Interview prompts to uncover mathematical knowledge for teaching: focus on providing written feedback. The Mathematics Enthusiast, 13(1), 71 92. https://doi.org/10.54870/1551-3440.1366
43. Kim, Y. (2020). Korean teachers’ mathematical knowledge for teaching in algebraic reasoning. Journal of Educational Research in Mathematics, (Special Issue), 185 198. https://doi.org/10.29275/jerm.2020.08.sp.1.185
44. Knipping, C. (2003). Learning from comparing. Zentralblatt für Didaktik der Mathematik, 35(6), 282-293.
45. Kline, P. (1994) An Easy Guide to Factor Analysis. Routledge.
46. Koellner, K., Jacobs, J., Borko, H., Schneider, C., Pittman, M.E., Eiteljorg, E., & Frykholm, J. (2007). The problem-solving cycle: A model to support the development of teachers' professional knowledge. Mathematical Thinking and Learning, 9(3), 273-303. https://doi.org/10.1080/10986060701360944
47. 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, 44, 371–385. https://doi.org/10.1007/s11858-012-0397-y
48. Langrall, C.W. & Swafford J.O. (1997). Grade six students' use of equations to describe and represent problem situation. Paper presented at the American Educational Research Association, Chicago, IL.
49. Lew, H.C. (2004). Developing algebraic thinking in early grades: Case study of Korean elementary school mathematics. The Mathematics Educator, 8(1), 88-106.
50. Lord, F.M. & Novick, M.R. (1968). Statistical theories of mental test scores. Addison-Wesley Pub. Co.
51. Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Lawrence Erlbaum Associates, Inc.
52. Marcinek, T., & Partová, E. (2016). Exploring cultural aspects of knowledge for teaching through adaptation of U.S.-developed measures: Case of Slovakia. Paper presented at the 13th International Congress on Mathematical Education. Hamburg, Germany.
53. Marcinek, T., Jakobsen, A., & Partová, E. (2022). Using MKT measures for cross-national comparisons of teacher knowledge: case of Slovakia and Norway. Journal of Mathematics Teacher Education, 1-31. https://doi.org/10.1007/s10857-021-09530-3
54. Morris, A.K., Hiebert, J., & Spitzer, S.M. (2009). Mathematical knowledge for teaching in planning and evaluating instruction: What can preservice teachers learn?. Journal for research in mathematics education, 40(5), 491-529. https://doi.org/40539354
55. Mosvold, R., & Fauskanger, J. (2009). Challenges of translating and adapting the MKT measures for Norway. Paper presented at the American Educational Research Annual Meeting in San Diego, CA.
56. National Mathematics Advisory Panel (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. U.S. Department of Education.
57. Ng, D. (2012). Using the MKT measures to reveal Indonesian teachers’ mathematical knowledge: Challenges and potentials. ZDM, 44(3), 401 413. https://doi.org/10.1007/s11858-011-0375-9
58. Ng, D., Mosvold, R., & Fauskanger, J. (2012). Translating and adapting the mathematical knowledge for teaching (MKT) measures: The cases of Indonesia and Norway. The Mathematics Enthusiast, 9(1), 149-178. https://doi.org/10.54870/1551-3440.1238
59. Özdemir, D. (2004). A comparison of psychometric characteristics of multiple choice tests based on the binaries and weighted scoring in respect to classical test and latent trait theory. Hacettepe University Journal of Education, 26, 117-123.
60. Pekmezci, F.B., & Avşar, A.Ş. (2021). A guide for more accurate and precise estimations in Simulative Unidimensional IRT Models. International Journal of Assessment Tools in Education, 8(2), 423-453. https://doi.org/10.21449/ijate.790289
61. Reyhanlıoğlu, Ç., & Doğan, N. (2020). An analysis of parameter invariance according to different sample sizes and dimensions in parametric and nonparametric item response theory. Journal of Measurement and Evaluation in Education and Psychology, 11(2), 98-112. https://doi.org/10.21031/epod.584977
62. Şahin, A., & Anıl, D. (2017). The effects of test length and sample size on item parameters in item response theory. Educational Sciences: Theory & Practice, 17(1), 321-33. https://doi.org/10.12738/estp.2017.1.0270
63. Schmittau, J. (2005). The development of algebraic thinking. Zentralblatt für Didaktik der Mathematik, 37(1), 16-22.
64. Sheng, Y. (2013). An empirical investigation of Bayesian hierarchical modeling with unidimensional IRT models. Behaviormetrika, 40(1), 19 40. https://doi.org/10.2333/bhmk.40.19
65. Shulman, L.S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14.
66. Sireci, S.G., Thissen, D., & Wainer, H. (1991). On the reliability of test let-based tests. Journal of Educational Measurement, 28(4), 237-247.
67. Smith, R.M., & Miao, C.Y. (1994). Assessing unidimensionality for Rasch measurement. Objective Measurement: Theory into Practice, 2, 316-327.
68. Stiegler, J.W., & Hiebert, J. (1999). The Teaching Gap. Best ideas from the world’s teachers for improving education in the classroom. The Free.
69. Strand, K., & Mills, B. (2014). Mathematical content knowledge for teaching elementary mathematics: A focus on algebra. The Mathematics Enthusiast, 11(2), 385-432. https://doi.org/10.54870/1551-3440.1307
70. Tabachnick, B., & Fidell, L. (2012). Using multivariate statistics. Pearson.
71. Welder, R.M., & Simonsen, L.M. (2011). Elementary Teachers' Mathematical Knowledge for Teaching Prerequisite Algebra Concepts. Issues in the Undergraduate Mathematics Preparation of School Teachers, 1.
72. Wilson, L., Andrew, C., & Sourikova, S. (2001). Shape and structure in primary mathematics lessons: A comparative study in the North‐east of England and St Petersburg, Russia‐some implications for the daily mathematics lesson. British Educational Research Journal, 27(1), 29-58.
73. Wright, B.D. (1996). Local dependency, correlations and principal components. Rasch Measurement Transactions, 10(3), 509–511. https://www.rasch.org/rmt/rmt103b.htm
74. Yang, S. (2007). A comparison of unidimensional and multidimensional RASCH models using parameter estimates and fit indices when assumption of unidimensionality is violated [Doctoral dissertation]. The Ohio State University.
75. Yen, W.M. (1993). Scaling performance assessments: Strategies for managing local item dependence. Journal of Educational Measurement, 30(3), 187–213.
76. Zazkis, R., & Liljedahl, P. (2002). Generalization of patterns: The tension between algebraic thinking and algebraic notation. Educational Studies in Mathematics, 49(3), 379-402.