Incoherencies in elementary pre-service teachers’ understanding of calculations in proportional tasks

Surani Joshua; Mi Yeon Lee

doi:10.29333/iejme/12178

Full Text (PDF)

Surani Joshua ¹ , Mi Yeon Lee ¹ ^*

More Detail

¹ Arizona State University, Tempe, AZ, USA^* Corresponding Author

Abstract

In this study we investigated pre-service teachers’ (PSTs) proportional reasoning and how they interpret their calculations in proportional tasks. We administered a written questionnaire to 199 PSTs and used an inductive content analysis approach for data analysis. We found that one item, in which PSTs were asked to interpret the meaning of the results of their calculations, had unusually low coherency, and applying open coding to the responses revealed several common errors. We argue these common errors cannot be dismissed as simple unit or rounding mistakes but rather reflect problems in how respondents think about quantities, story problems, and the nature of mathematics itself. We end with suggestions on how to address these problems.

Keywords

pre-service teacher education
teacher knowledge
proportional reasoning
problem solving
quantitative reasoning

License

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Research Article

INT ELECT J MATH ED, 2022, Volume 17, Issue 4, Article No: em0698

https://doi.org/10.29333/iejme/12178

Publication date: 25 Jun 2022

Article Views: 825

Article Downloads: 542

Open Access Fundings Disclosures References How to cite this article

Funding

No funding source is reported for this study.

Disclosure

Declaration of interest: No conflict of interest is declared by authors.

Data sharing statement: Data supporting the findings and conclusions are available upon request from the corresponding author.

References

Arican, M. (2019). Preservice mathematics teachers’ understanding of and abilities to differentiate proporational relationships from nonproportional relationships. International Journal of Science and Mathematics Education, 17(7), 1423-1443. https://doi.org/10.1007/s10763-018-9931-x
Artut, P. D., & Pelen, M. S. (2015). 6th grade students’ solution strategies on proportional reasoning problems. Procedia-Social and Behavioral Sciences, 197, 113-119. https://doi.org/10.1016/j.sbspro.2015.07.066
Baxter, G., & Junker, B. W. (2001). Designing developmental assessmnets: A case study in proportional reasoning [Paper presentation]. The Annual Meeting of the National Council of Measurement in Education, Seattle, WA, USA.
Beckmann, S., & Izsak, A. (2015). Two perspectives on proportional relationships: Extending complementary origins of multiplication in terms of quantities. Journal for Research in Mathematics Education, 46(1), 17-38. https://doi.org/10.5951/jresematheduc.46.1.0017
Ben-Chaim, D., Keret, Y., & Ilany, B. (2007). Designing and implementing authentic investigative proportional reasoning tasks: The impact on preservice mathematics teachers’ content and pedagogical knowledge and attitudes. Journal of Mathematics Teacher Education, 10, 333-340. https://doi.org/10.1007/s10857-007-9052-x
Byerley, C., & Thompson, P. W. (2017). Teachers’ meanings for measure, slope, and rate of change. Journal of Mathematical Behavior, 48, 168-193. https://doi.org/10.1016/j.jmathb.2017.09.003
Castillo-Garsow, C., Johnson, H. L., & Moore, K. C. (2013). Chunky and smooth images of change. For the Learning of Mathematics, 33(3), 31-37.
Chen, L., Van Dooren, W., Chen, Q., & Verschaffel, L. (2011). An investigation on Chinese teachers’ realistic problem posing and problem-solving ability and beliefs. International Journal of Science and Mathematics Education, 9(4), 919-948. https://doi.org/10.1007/s10763-010-9259-7
Cross, D., Adefope, O., Lee, M. Y., & Perez, A. (2012). Hungry for early spatial and algebraic reasoning. Teaching Children Mathematics, 19(1), 42-49. https://doi.org/10.5951/teacchilmath.19.1.0042
Giorgi, A. J., Roberts, T. G., Estepp, C. M., Conner, N. W., & Stripling, C. T. (2013). An investigation of teacher beliefs and actions. NACTA Journal, 57(3), 2-9.
Glassmeyer, D., Brakoniecki, A., & Amador, J. M. (2021). Identifying and supporting teachers’ robust understanding of proportional reasoning. The Journal of Mathematical Behavior, 62, 100873. https://doi.org/10.1016/j.jmathb.2021.100873
Grbich, C. (2007). Qualitative data analysis: An introduction. SAGE.
Greer, B., Verschaffel, L., & De Corte, E. (2002). “The answer is really 4.5”: Beliefs about word problems. InG. C. Leder, E. Pehkonen, & G. Törner (Eds.), Beliefs: A hidden variable in mathematics education? (pp. 271-292). Springer. https://doi.org/10.1007/0-306-47958-3_16
Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement. American Eeducational Research Journal, 42(2), 371-406. https://doi.org/10.3102/00028312042002371
Hilton, A., & Hilton, G. (2019). Primary school teachers implementing structured mathematics interventions to promote their mathematics knowledge for teaching proportional reasoning. Journal of Mathematics Teacher Education, 22(6), 545-574. https://doi.org/10.1007/s10857-018-9405-7
Hines, E., & McMahon, M. (2005). Interpreting middle school students’ proportional reasoning strategies: Observations from preserivce teachers. School Science and Mathematics, 105(2), 88-105. https://doi.org/10.1111/j.1949-8594.2005.tb18041.x
Inoue, N. (2005). The realistic reasons behind unrealistic solutions: the role of interpretive activity in word problem solving. Learning and Instruction, 15, 69-83. https://doi.org/10.1016/j.learninstruc.2004.12.004
Izsak, A., & Jacobson, E. (2017). Preservice teachers’ reasoning about relationships that are and are not proportional: A knowledge-in-pieces account. Journal for Research in Mathematics Education, 48(3), 300-339. https://doi.org/10.5951/jresematheduc.48.3.0300
Lamon, S. (2007). Rational numbers and proportational 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 Publishing.
Lee, M. Y. (2017a). Generating linear equations based on quantitative reasoning. Mathematics Teaching in the Middle School, 23(2), 112-116. https://doi.org/10.5951/mathteacmiddscho.23.2.0112
Lee, M. Y. (2017b). Pre-service teachers’ flexibility with referent units in solving a fraction division problem. Educational Studies in Mathematics, 96(3), 327-348. https://doi.org/10.1007/s10649-017-9771-6
Lee, M. Y (2021). Using a technology tool to help pre-service teachers notice students’ reasoning and errors on a mathematics problem. ZDM–Mathematics Education, 53(1), 135-149. https://doi.org/10.1007/s11858-020-01189-z
Lee, M. Y., & Francis, D. C. (2018). Investigating the relationship among elementary teachers’ perception about the use of students’ thinking, their professional noticing skills and their teaching practice. Journal of Mathematical Behavior, 51, 118-128. https://doi.org/10.1016/j.jmathb.2017.11.007
Livy, S., & Herbert, S. (2013). Second-year pre-service teachers’ responses to proportional reasoning test items. Australisan Journal of Teacher Education, 38(11), 17-32. https://doi.org/10.14221/ajte.2013v38n11.7
National Governors Association Center for Best Practices. (2010). Common core state standards: Mathematics standards. http://www.corestandards.org/Math/
NCTM. (2000). Principles and standards for school mathematics. National Council of Teachers of Mathematics. https://www.nctm.org/standards/
Pelen, M. S., & Artut, P. D. (2019). Examining the effect of problem cassification and number structures on proportional reasoning. International Journal of Educational Studies in Mathematics, 6(1), 34-43.
Rosales, J., Vicente, S., Chamoso, J. M., Muñez, D., & Orrantia, J. (2012). Teacher–student interaction in joint word problem solving. The role of situational and mathematical knowledge in mainstream classrooms. Teaching and Teacher Education, 28(8), 1185-1195. https://doi.org/10.1016/j.tate.2012.07.007
Son, J. W. (2013). How preservice teachers interpret and respond to student errors: Ratio and proportion in similar rectangles. Educational Studies in Mathematics, 84(1), 49-70. https://doi.org/10.1007/s10649-013-9475-5
Son, J., & Lee, M. Y. (2021). Exploring the relationship between preservice teachers’ conceptions of problem solving and their problem-solving performance. International Journal of Science and Mathematics Education, 19(1), 129-150. https://doi.org/10.1007/s10763-019-10045-w
Thevenot, C. (2017). Arithmetic word problem solving: The role of prior knowledge. In D. C. Geary, D. B. Berch, R. J. Ochsendorf, & K. M. Koepke (Eds.), Acquisition of complex arithmetic skills and higher-order mathematics concepts (pp. 47-66). Academic Press. https://doi.org/10.1016/B978-0-12-805086-6.00003-5
Thompson, P. W. (2011). Quantitative reasoning and mathematical modeling. In S. A. Chamberlin, & L. L. Hatfield (Eds.), New perspectives and directions for collaborative research in mathematics education WISDOMe monographs (pp. pp. 33-57). University of Wyoming Press.
Thompson, P. W., & Carlson, M. P. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 421-456). National Council of Teachers of Mathematics.
Tjoe, H., & de la Torre, J. (2014). On recognizing proportionality: Does the ability to solve missing value proportional problems presuppose the conception of proportional reasoning? The Journal of Mathematical Behavior, 33, 1-7. https://doi.org/10.1016/j.jmathb.2013.09.002
Ucar, Z. T., & Bozkus, F. (2018). Elementary school students’ and prospective teachers’ proportional reasoning skills. International Journal for Mathematics Teaching and Learning, 19(2), 205-222.
Van Dooren, W., de Bock, D., & Verschaffel, L. (2010). From addition to multiplication…and back: The development of students’ additive and multiplicative reasoning skills. Cognition and Instruction, 28(3), 360-381. https://doi.org/10.1080/07370008.2010.488306
Vergnaud, G. (1983). Multiplicative structures. In R. Lesh, & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 127-174). Academic Press.
Verschaffel, L., De Corte, E., & Borghart, I. (1997). Pre-service teachers’ conceptions and beliefs about the role of real-world knowledge in mathematical modelling of school word problems. Learning and Instruction, 7(4), 339-359. https://doi.org/10.1016/S0959-4752(97)00008-X
Weiland, T., Orrill, C. H., Nagar, G. G., Brown, R. E., & Burke, J. (2021). Framing a robust understanding of proportional reasoning for teachers. Journal of Mathematics Teacher Education, 24(2), 179-202. https://doi.org/10.1007/s10857-019-09453-0
Yoshida, H., Verschaffel, L., & De Corte, E. (1997). Realistic considerations in solving problematic word problems: Do Japanese and Belgian children have the same difficulties? Learning and instruction, 7(4), 329-338. https://doi.org/10.1016/S0959-4752(97)00007-8

How to cite this article

APA

Joshua, S., & Lee, M. Y. (2022). Incoherencies in elementary pre-service teachers’ understanding of calculations in proportional tasks. International Electronic Journal of Mathematics Education, 17(4), em0698. https://doi.org/10.29333/iejme/12178

Vancouver

Joshua S, Lee MY. Incoherencies in elementary pre-service teachers’ understanding of calculations in proportional tasks. INT ELECT J MATH ED. 2022;17(4):em0698. https://doi.org/10.29333/iejme/12178

AMA

Joshua S, Lee MY. Incoherencies in elementary pre-service teachers’ understanding of calculations in proportional tasks. INT ELECT J MATH ED. 2022;17(4), em0698. https://doi.org/10.29333/iejme/12178

Chicago

Joshua, Surani, and Mi Yeon Lee. "Incoherencies in elementary pre-service teachers’ understanding of calculations in proportional tasks". International Electronic Journal of Mathematics Education 2022 17 no. 4 (2022): em0698. https://doi.org/10.29333/iejme/12178

Harvard

Joshua, S., and Lee, M. Y. (2022). Incoherencies in elementary pre-service teachers’ understanding of calculations in proportional tasks. International Electronic Journal of Mathematics Education, 17(4), em0698. https://doi.org/10.29333/iejme/12178

MLA

Joshua, Surani et al. "Incoherencies in elementary pre-service teachers’ understanding of calculations in proportional tasks". International Electronic Journal of Mathematics Education, vol. 17, no. 4, 2022, em0698. https://doi.org/10.29333/iejme/12178