Student teachers’ common content knowledge for solving routine fraction tasks

Anne Tossavainen

doi:10.31129/LUMAT.10.2.1656

Authors

Anne Tossavainen Department of Health, Learning and Technology, Luleå University of Technology, Sweden https://orcid.org/0000-0002-0429-243X

DOI:

https://doi.org/10.31129/LUMAT.10.2.1656

Keywords:

common content knowledge, elementary school, fractions, student teacher, teacher education

Abstract

This study focuses on the knowledge base that Swedish elementary student teachers demonstrate in their solutions for six routine fraction tasks. The paper investigates the student teachers’ common content knowledge of fractions and discusses the implications of the findings. Fraction knowledge that student teachers bring to teacher education has been rarely investigated in the Swedish context. Thus, this study broadens the international view in the field and gives an opportunity to see some worldwide similarities as well as national challenges in student teachers’ fraction knowledge. The findings in this study reveal uncertainty and wide differences between the student teachers when solving fraction tasks that they were already familiar with; two of the 59 participants solved correctly all tasks, whereas some of them gave only one or not any correct answer. Moreover, the data indicate general limitations in the participants’ basic knowledge in mathematics. For example, many of them make errors in using mathematical symbol writing and different representation forms, and they do not recognize unreasonable answers and incorrect statements. Some participants also seemed to guess at an algorithm to use when they did not remember or understand the correct solution method.

References

Ball, D. L. (1990). The mathematical understandings that prospective teachers bring to teacher education. The Elementary School Journal, 90(4), 449–466. https://doi.org/10.1086/461626

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.

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. https://doi.org/10.2307/749118

Charalambous, C. Y., Hill, H. C., Chin, M. J., & McGinn, D. (2020). Mathematical content knowledge and knowledge for teaching: exploring their distinguishability and contribution to student learning. Journal of Mathematics Teacher Education, 23(6), 579–613. https://doi.org/10.1007/s10857-019-09443-2

Charalambous, C. Y., & Pitta-Pantazi, D. (2007). Drawing on a theoretical model to study students’ understanding of fractions. Educational Studies in Mathematics, 64(3), 293–316. https://doi.org/10.1007/s10649-006-9036-2

Cramer, K. A., Post, T. R., & delMas, R. C. (2002). Initial fraction learning by fourth- and fifth-grade students: A comparison of the effects of using commercial curricula with the effects of using the Rational Number Project curriculum. Journal for Research in Mathematics Education, 33(2), 111–144. https://doi.org/10.2307/749646

Ernest, P. (1989). The knowledge, beliefs and attitudes of the mathematics teacher: a model. Journal of Education for Teaching, 15(1), 13–33. https://doi.org/10.1080/0260747890150102

Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed). Conceptual and procedural knowledge: The case of mathematics (pp. 1–27). Lawrence Erlbaum.

Hoover, M., Mosvold, R., Ball, D. L., & Lai, Y. (2016). Making progress on mathematical knowledge for teaching. The Mathematics Enthusiast, 13(1 & 2), 3–34. https://doi.org/10.54870/1551-3440.1363

Häkkinen, K., Tossavainen, T., & Tossavainen, A. (2011). Kokemuksia luokanopettajaksi pyrkivien matematiikan soveltuvuustestistä Savonlinnan opettajankoulutuslaitoksessa. In E. Pehkonen (Ed.), Luokanopettajaopiskelijoiden matematiikkataidoista, Tutkimuksia 328 (pp. 47–64). Department of Applied Educational Science, University of Helsinki.

Jakobsen, A., Ribeiro, C. M., & Mellone, M. (2014). Norwegian prospective teachers' MKT when interpreting pupils' productions on a fraction task. Nordic Studies in Mathematics Education, 19(3-4), 135–150.

Jóhannsdóttir, B., & Gísladóttir, B. (2014). Exploring the mathematical knowledge of prospective elementary teachers in Iceland using the MKT measures. Nordic Studies in Mathematics Education, 19(3-4), 21–40.

Kieren, T. E. (1993). Rational and fractional numbers: From quotient fields to recursive understanding. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research (pp. 49-84). Lawrence Erlbaum Associates.

Lamon, S. J. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. In F. K. Lester, Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 629-667). Information Age.

Lamon, S. J. (2020). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers (Fourth Ed.). Routledge.

Lin, C.-Y., Becker, J., Byun, M.-R., Yang, D.-C., & Huang, T.-W. (2013). Preservice teachers’ conceptual and procedural knowledge of fraction operations: A comparative study of the United States and Taiwan. School Science and Mathematics, 113(1), 41–51. https://doi.org/10.1111/j.1949-8594.2012.00173.x

Löwing, M. (2016). Diamant – diagnoser i matematik: Ett kartläggningsmaterial baserat på didaktisk ämnesanalys [Doctoral dissertation, University of Gothenburg]. http://hdl.handle.net/2077/47607

Ma, L. (2010). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Anniversary Edition. Routledge.

Maciejewski, W., & Star, J. R. (2016). Developing flexible procedural knowledge in undergraduate calculus. Research in Mathematics Education, 18(3), 299–316. https://doi.org/10.1080/14794802.2016.1148626

Marchionda, H. (2006). Preservice teachers’ procedural and conceptual understanding of fractions and the effects of inquiry-based learning on this understanding [Doctoral dissertation, Clemson University]. https://tigerprints.clemson.edu/all_dissertations/37

Moss, J., & Case, R. (1999). Developing children’s understanding of the rational numbers: A new model and an experimental curriculum. Journal for Research in Mathematics Education, 30(2), 122–147. https://doi.org/10.2307/749607

Muir, T., & Livy, S. (2012). What do they know? A comparison of pre-service teachers’ and in-service teachers’ decimal mathematical content knowledge. International Journal for Mathematics Teaching and Learning, 2012, December 5th, 1–15. Retrieved from http://www.cimt.org.uk/journal/muir2.pdf

Newton, K. J. (2008). An extensive analysis of preservice elementary teachers’ knowledge of fractions. American Educational Research Journal, 45(4), 1080–1110. https://doi.org/10.3102/0002831208320851

Olanoff, D., Lo, J.-J., & Tobias, J. (2014). Mathematical content knowledge for teaching elementary mathematics: A focus on fractions. The Mathematics Enthusiast, 11(2), 267–310. https://doi.org/10.54870/1551-3440.1304

Radatz, H. (1979). Error analysis in mathematics education. Journal for Research in Mathematics Education, 10(3), 163–172. https://doi.org/10.2307/748804

Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14.

Siegler, R. S., Duncan, G. J., Davis-Kean, P. E., Duckworth, K., Claessens, A., Engel, M., Susperreguy, M. I., & Chen, M. (2012). Early predictors of high school mathematics achievement. Psychological Science, 23(7), 691–697. https://doi.org/10.1177/0956797612440101

Skolverket [Swedish National Agency for Education]. (2011). Curriculum for the compulsory school, preschool class and school-age educare. Revised 2018. Skolverket.

Skolverket [Swedish National Agency for Education]. (2016). TIMSS 2015. Svenska grundskoleelevers kunskaper i matematik och naturvetenskap i ett internationellt perspektiv. Internationella studier 448. Skolverket.

Skolverket [Swedish National Agency for Education]. (2019). PISA 2018. 15-åringars kunskaper i läsförståelse, matematik och naturvetenskap. Internationella studier 487. Skolverket.

Tirosh, D. (2000). Enhancing prospective teachers’ knowledge of children’s conceptions: The case of division of fractions. Journal for Research in Mathematics Education, 31(1), 5–25. https://doi.org/10.2307/749817

Tirosh, D., Fischbein, E., Graeber A. O., & Wilson, J. W. (1998). Prospective elementary teachers’ conceptions of rational numbers. Retrieved from http://jwilson.coe.uga.edu/Texts.Folder/Tirosh/Pros.El.Tchrs.html

Toluk-Uçar, Z. (2009). Developing pre-service teachers understanding of fractions through problem posing. Teaching and Teacher Education, 25(1), 166–175. https://doi.org/10.1016/j.tate.2008.08.003

Van Steenbrugge, H., Lesage, E., Valcke, M., & Desoete, A. (2014). Preservice elementary school teachers’ knowledge of fractions: a mirror of students’ knowledge? Journal of Curriculum Studies, 46(1), 138–161. https://doi.org/10.1080/00220272.2013.839003

Zhou, Z., Peverly, S.T., & Xin, T. (2006). Knowing and teaching fractions: A cross-cultural study of American and Chinese mathematics teachers. Contemporary Educational Psychology, 31, 438–457. https://doi.org/10.1016/j.cedpsych.2006.02.001

Young, E., & Zientek, L. R. (2011). Fraction operations: An examination of prospective teachers’ errors, confidence, and bias. Investigations in Mathematics Learning, 4(1), 1–23. https://doi.org/10.1080/24727466.2011.11790307