The Development of Teachers’ Knowledge of the Nature of Mathematical Modeling Scale

Year 2020, , 236 - 254, 13.06.2020

Reuben Asempapa

https://doi.org/10.21449/ijate.737284

Cited By: 3

Abstract

This study addresses a gap in the literature on mathematical modeling education by developing the mathematical modeling knowledge scale (MMKS). The MMKS is a quantitative tool created to assess teachers’ knowledge of the nature of mathematical modeling. Quantitative instruments to measure modeling knowledge is scare in the literature partially due to the lack of appropriate instruments developed to assess such knowledge among teachers. The MMKS was developed and validated with a total sample of 364 K–12 teachers from several public-schools using three phases. Phase 1 addresses content validity of the scale using reviews from experts and interviews with knowledgable teachers. Initial psychometric properties and piloting results are presented in phase 2 of the study, and phase 3 reports on the findings during the field test, factor structure, and factor analyses. The results of the factor analyses and other psychometric measures supported a 12-item, one-factor scale for assessing teachers’ knowledge of the nature of mathematical modeling. The reliability of the MMKS was moderately high and acceptable (α = .84). The findings suggest the MMKS is a reliable, valid, and useful tool to measure teachers’ knowledge of the nature of mathematical modeling. Potential uses and applications of the MMKS by researchers and educators are discussed, and implications for further research are provided.

Keywords

Nature of modeling, MMKS, Factor analysis, Scale deveopment, Teachers’ knowledge

References

• Ang, K. C. (2015). Mathematical modelling in Singapore schools: A framework for instruction. In N. H. Lee & K. E. D. Ng (Eds.), Mathematical modelling: From theory to practice (pp. 57–72). Singapore: World Scientific.
• Asempapa, R. S., & Foley, G. D. (2018). Classroom assessment of mathematical modeling tasks. In M. Shelley & S. A. Kiray (Eds.), Education Research Highlights in Mathematics, Science, and Technology 2018 (pp. 6–20). Ames; IA. International Society for Research in Education and Science (ISRES).
• Ball, D. L. (1990). The mathematical understandings that prospective teachers bring to teacher education. Elementary School Journal, 90, 449–466.
• Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59, 389–407.
• Benjamin, T. E., Marks, B., Demetrikopoulos, M. K., Rose, J., Pollard, E., Thomas, A., & Muldrow, L. L. (2017). Development and validation of scientific literacy scale for college preparedness in STEM with freshmen from diverse institutions. International Journal of Science and Mathematics Education, 15(4), 607–623.
• Blum, W. (2015). Quality teaching of mathematical modelling: What do we know, what can we do? In S. J. Cho (Ed.), Proceedings of the 12th International Congress on Mathematical Education: Intellectual and attitudinal challenges (pp. 73–96). New York, NY: Springer.
• Blum, W., & Borromeo Ferri, R. (2009). Mathematical modelling: Can it be taught and learnt? Journal of Mathematical Modelling and Application, 1, 45–58.
• Blum, W., & Leiss, D. (2007). How do students and teachers deal with modelling problems? In C. Haines, P. Galbraith, W. Blum, & S. Khan (Eds.), Mathematical modelling: Education, engineering and economics–ICTMA 12 (pp. 222–231). Chichester, United Kingdom: Horwood.
• Boaler, J. (2001). Mathematical modelling and new theories of learning. Teaching Mathematics and Its Applications, 20, 121–127.
• Borromeo, F. R. (2018). Learning how to teach mathematical modeling in school and teacher education. Picassoplatz, Switzerland: Springer. https://doi.org/10.1007/978-3-319-68072-9
• Brown, T. A. (2015). Confirmatory factor analysis for applied research (2nd ed.). New York, NY: Guilford Press.
• Consortium for Mathematics and Its Applications [COMAP] and Society for Industrial and Applied Mathematics [SIAM] (2016). Guidelines for assessment and instruction in mathematical modeling education. Retrieved from http://www.comap.com/Free/GAIMME/
• Converse, J. M., & Presser, S. (1986). Survey questions: Handcrafting the standardized questionnaire. Beverly Hills, CA: Sage.
• Costello, A. B., & Osborne, J. W. (2005). Best practices in exploratory factor analysis: Four recommendations for getting the most from your analysis. Practical Assessment, Research & Evaluation, 10, 1–9. https://doi.org/10.1.1.110.9154
• Cronbach, L. J. (1988). Five perspectives on validity argument. In H. Wainer & H. I. Braun (Eds.), Test validity (pp. 3–17). Hillsdale, NJ: Erlbaum.
• Crouch, R. & Haines C. (2004). Mathematical modeling: Transitions between the real world and the mathematical model. International Journal of Mathematics Education Science Technology, 35(2), 197–206.
• DeVellis, R. F. (2017). Scale development: Theory and applications (4th ed.). Thousand Oaks, CA: Sage.
• Doerr, H. M., Ärlebäck, J. B., & Costello Staniec, A. (2014). Design and effectiveness of modeling‐based mathematics in a summer bridge program. Journal of Engineering Education, 103(1), 92–114.
• Downing, S. M. (2003). Validity: on the meaningful interpretation of assessment data. Medical Education, 37, 830–837.
• English, L. D., Fox, J. L., & Watters, J. J. (2005). Problem posing and solving with mathematical modeling. Teaching Children Mathematics, 12, 156–163.
• English, L., & Sriraman, B. (2010). Problem solving for the 21st century. In B. Sriraman & L. D. English (Eds.), Theories of mathematics education: Seeking new frontiers–advances in mathematics education (pp. 263–290). New York, NY: Springer.
• Field, A. (2009). Discovering statistics using SPSS. Los Angeles, CA: Sage.
• Fowler, F. J. (2014). Survey research methods (5th ed.). Thousand Oaks, CA: Sage.
• Furr, R. M., & Bacharach, V. R. (2014). Psychometrics: An introduction. (2nd ed.). Thousand Oaks, CA: Sage. Geiger, V. (2015). Mathematical modelling in Australia. In N. H. Lee & K. E. D. Ng (Eds.), Mathematical modelling: From theory to practice (pp. 73–82). Singapore: World Scientific.
• Gould, H. T. (2013). Teachers’ conceptions of mathematical modeling. Retrieved from http://academiccommons.columbia.edu/item/ac:16149
• Hill, H. C., Schilling, S. G., & Ball, D. L. (2004). Developing measures of teachers’ mathematics knowledge for teaching. The Elementary School Journal, 105, 11–30.
• Hu, L. T., & Bentler, P. M. (1999). Cutoff criteria to fit indexes in covariance structure analysis: Conventional criteria versus new alternatives. Structural Equation Modeling: A Multidisciplinary Journal, 6, 1–55.
• Ikeda, T. (2015). Mathematical modelling in Japan. In N. H. Lee & K. E. D. Ng (Eds.), Mathematical modelling: From theory to practice (pp. 83–96). Singapore: World Scientific.
• Johanson, G. A., & Brooks, G. P. (2010). Initial scale development: Sample size for pilot studies. Educational and Psychological Measurement, 70(3), 394–400.
• Kaiser, G., Blum, W., Borromeo Ferri, R., & Stillman, G. (2011). Trends in teaching and learning of mathematical modelling–Preface. In G. Kaiser, W. Blum, R. Borromeo Ferri, & G. Stillman (Eds.), Trends in teaching and learning of mathematical modelling: ICTMA14 (pp. 1–8). New York, NY: Springer.
• Kaiser, G., Schwarz, B., & Tiedemann, S. (2010). Future teachers’ professional knowledge on modeling. In R. Lesh, P. L. Galbraith, C. R. Haines, & A. Hurford, (Eds.), Modeling students’ mathematical modeling competencies: ICTMA 13 (pp. 433–444). New York, NY: Springer.
• Kaiser, G., & Sriraman, B. (2006). A global survey of international perspectives on modelling in mathematics education. ZDM–The International Journal on Mathematics Education, 38(3), 302–310.
• Kenny, D. A. (2015). Measuring model fit. Retrieved from davidakenny.net/cm/fit.htm
• Kline, P. (2000). The handbook of psychological testing (2nd ed.). New York, NY: Routledge.
• Kline, R. B. (2016). Principles and practice of structural equation modeling (4th ed.). New York, NY: Guilford Press.
• Leong, R. K. E. (2012). Assessment of mathematical modeling. Journal of Mathematics Education at Teachers College, 3, 61–65.
• Lesh, R. (2012). Research on models & modeling and implications for common core state curriculum standards. In R. L. Mayes & L. L. Hatfield (Eds.), WISDOMe monograph: Quantitative reasoning and mathematical modeling: A driver for STEM integrated education and teaching in context (Vol. 2, pp. 197–203). Laramie, WY: University of Wyoming.
• Lesh, R., & Doerr, H. M. (Eds). (2003). Beyond constructivism: Models and modelling perspective on mathematics problem solving, learning, and teaching. Mahwah, NJ: Erlbaum.
• Lortie, D. C. (2002). School teacher (2nd ed.). Chicago, IL: University of Chicago Press.
• Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Erlbaum.
• Messick, S. (1995). Validity of psychological assessment: validation of inferences from persons’ responses and performances as scientific inquiry into score meaning. American Psychologist, 50, 741–749.
• Messick, S. (1998). Test validity: A matter of consequences. Social Indicators Research, 45, 35–44.
• Meyers, L. S., Gamst, G., & Guarino, A. J. (2013). Applied multivariate research: Design and interpretation (2nd ed.). Los Angeles, CA: Sage.
• National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author.
• National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common core state standards for mathematics. Washington, DC: Author. Retrieved from http://corestandards.org/assets/ CCSSI_Math%20Standards.pdf
• O’Connor, B. P. (2000). SPSS and SAS programs for determining the number of components using parallel analysis and Velicer’s MAP test. Behavior Research Methods, Instruments, & Computers, 32(3), 396–402.
• Organisation for Economic Co-operation and Development (2003). The PISA 2003 assessment framework—mathematics, reading, science and problem solving, knowledge, and skills. Paris, France: OECD Press.
• Osterlind, S. J. (2010). Modern measurement: Theory, principles, and applications of mental appraisal. Upper Saddle River, NJ: Pearson.
• Paolucci, C., & Wessels. H. (2017). An examination of preservice teachers’ capacity to create mathematical modeling problems for children. Journal of Teacher Education, 68(3), 330–344.
• Perrenet, J., & Zwaneveld, B. (2012). The many faces of the mathematical modeling cycle. Journal of Mathematical Modelling and Application, 1(6), 3–21.
• Pollak, H. O. (2011). What is mathematical modeling? Journal of Mathematics Education at Teachers College, 2, 64.
• Ponte, J. P., & Chapman, O. (2008). Preservice mathematics teachers’ knowledge and development. In L. D. English (Ed.), Handbook of international research in mathematics education (2nd ed., pp. 223–261). New York, NY: Routledge.
• Preacher, K. J., & MacCallum, R. C. (2003). Repairing Tom Swift’s electric factor analysis machine. Understanding Statistics, 2(1), 13–43.
• Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15, 4–14.
• Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57, 1–22.
• Sokolowski, A., & Rackley, R. (2011). Teaching harmonic motion in trigonometry: Inductive inquiry supported by physics simulations. Australian Senior Mathematics Journal, 24(2), 45–54.
• Sriraman, B., & English, L. D. (2010). Surveying theories and philosophies of mathematics education. In B. Sriraman & L. D. English (Eds.), Theories of mathematics education: Seeking new frontiers—advances in mathematics education (pp. 7–34). New York, NY: Springer.
• Tourangeau, R., Rips, L. J., & Rasinski, K. (2000). The psychology of survey response. New York, NY: Cambridge University Press.
• Warner, R. M. (2013). Applied statistics: From bivariate through multivariate techniques (2nd ed.). Thousand Oaks, CA: Sage.
• Wolfe, N. B. (2013). Teachers’ understanding of and concerns about mathematical modeling in the common core standards (Doctoral dissertation). Retrieved from https://search.proquest.com/docview/1432193702
• Ziebarth, S., Fonger, N., & Kratky, J. (2014). Instruments for studying the enacted mathematics curriculum. In D. Thompson, & Z. Usiskin (Eds.), Enacted mathematics curriculum: A conceptual framework and needs (pp. 97–120). Charlotte, NC: Information Age.