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Understanding and promoting students’ mathematical thinking: a review of research published in ESM

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Abstract

In this paper, we offer a comparative review of research on understanding and promoting students’ mathematical thinking. The sources for the review are papers that were published in Educational Studies in Mathematics (ESM) during two windows of time: 1994–1998 and 2014–2018. Selection of these two time periods enables us to comment on the “state of the art” in research as well as identify changes over the past 25 years. The review is guided by an analysis of conceptualizations of “mathematical thinking” proposed in the research literature, selected curriculum documents, and international assessment programs such as the OECD’s Programme for International Student Assessment (PISA). The review not only documents salient features of research studies, such as the country of origin of the authors, educational level of the participants, research aims, theoretical perspectives, and methodological approaches, but also identifies the contribution to knowledge made by this body of work as well as future research directions and opportunities.

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Notes

  1. This paper extends the review of research on mathematical thinking presented by the first author at the 2018 Regional Conference of the International Group for the Psychology of Mathematics Education (PME). The conference theme was Understanding and promoting mathematical thinking.

References

  • Adler, J., Ball, D., Krainer, K., Lin, F.-L., & Novotna, J. (2005). Reflections on an emerging field: Researching mathematics teacher education. Educational Studies in Mathematics, 60, 359–381.

    Google Scholar 

  • Alberracin, L., & Gorgorio, N. (2014). Devising a plan to solve Fermi problems involving large numbers. Educational Studies in Mathematics, 86, 79–96.

    Google Scholar 

  • Anderson, J., & White, P. (2004). Problem solving in learning and teaching mathematics. In B. Perry, G. Anthony, & C. Diezmann (Eds.), Research in mathematics education in Australasia 2000–2003 (pp. 127–150). Flaxton, Australia: PostPressed.

    Google Scholar 

  • Australian Curriculum, Assessment and Reporting Authority (ACARA) (n.d.). Mathematics proficiencies. Retrieved 12 October 2018 from https://www.australiancurriculum.edu.au/ resources/ mathematics-proficiencies/

  • Bakker, A. (2019). A virtual special issue to celebrate 100 volumes of Educational Studies in Mathematics: Former editors’ choices. Educational Studies in Mathematics, 101, 1–5.

    Google Scholar 

  • Beckers, D. (2019). Why to publish on mathematics education so as to be useful? Educational Studies in Mathematics and its founder Hans Freudenthal. Educational Studies in Mathematics, 101, 7–17.

    Google Scholar 

  • Blanton, M., Brizuela, B. M., Gardiner, A. M., Sawrey, K., & Newman-Owens, A. (2017). A progression in first-grade children’s thinking about variable and variable notation in functional relationships. Educational Studies in Mathematics, 95, 181–202.

    Google Scholar 

  • Bruce, C., Davis, B., Sinclair, N., McGarvey, L., Hallowell, D., Drefs, M., … Woolcott, G. (2017). Understanding gaps in research networks: Using “spatial reasoning” as a window into the importance of networked educational research. Educational Studies in Mathematics, 95, 143–161.

    Google Scholar 

  • Campbell, K. J., Collis, K., & Watson, J. (1995). Visual processing during mathematical problem solving. Educational Studies in Mathematics, 28, 177–194.

    Google Scholar 

  • Chinnappan, M. (1998). Schemas and mental models in geometry problem solving. Educational Studies in Mathematics, 36, 201–217.

    Google Scholar 

  • Dawkins, P., & Weber, K. (2017). Values and norms of proof for mathematicians and students. Educational Studies in Mathematics, 95, 123–142.

    Google Scholar 

  • Downton, A., & Sullivan, P. (2017). Posing complex problems requiring multiplicative thinking prompts students to use sophisticated strategies and build mathematical connections. Educational Studies in Mathematics, 95, 303–328.

    Google Scholar 

  • English, L. (1997). The development of fifth-grade children’s problem-posing abilities. Educational Studies in Mathematics, 34, 183–217.

    Google Scholar 

  • English, L., & Gainsburg, J. (2016). Problem solving in a 21st-century mathematics curriculum. In L. English & D. Kirshner (Eds.), Handbook of international research in mathematics education (3rd ed., pp. 313–335). New York, NY: Routledge.

    Google Scholar 

  • Fan, L., Qi, C., Liu, X., Wang, Y., & Lin, M. (2017). Does a transformation approach improve students’ ability in constructing auxiliary lines for solving geometric problems? An intervention-based study with two Chinese classrooms. Educational Studies in Mathematics, 96, 229–248.

    Google Scholar 

  • Fiallo, J., & Gutierrez, A. (2017). Analysis of the cognitive unity or rupture between conjecture and proof when learning to prove on a grade 10 trigonometry course. Educational Studies in Mathematics, 96, 145–167.

    Google Scholar 

  • Gabel, M., & Dreyfus, T. (2017). Affecting the flow of a proof by creating presence—A case study in Number Theory. Educational Studies in Mathematics, 96, 187–205.

    Google Scholar 

  • Garfield, J., Le, L., Zieffler, A., & Ben-Zvi, D. (2015). Developing students’ reasoning about samples and sampling variability as a path to expert statistical thinking. Educational Studies in Mathematics, 88, 327–342.

    Google Scholar 

  • Goos, M., & Galbraith, P. (1996). Do it this way! Metacognitive strategies in collaborative mathematical problem solving. Educational Studies in Mathematics, 30, 229–260.

    Google Scholar 

  • Harel, G., & Sowder, L. (2007). Towards comprehensive perspectives on the learning and teaching of proof. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 805–842). Reston, VA: NCTM.

    Google Scholar 

  • Heino, K. (2015). Comparing multiple solutions in the structured problem solving: Deconstructing Japanese lessons from learner’s perspective. Educational Studies in Mathematics, 90, 121–141.

    Google Scholar 

  • Hilton, A., Hilton, G., Dole, S., & Goos, M. (2016). Promoting middle school students’ proportional reasoning skills through an ongoing professional development programme for teachers. Educational Studies in Mathematics, 92, 193–219.

    Google Scholar 

  • Hitt, F., Saboya, M., Zavala, C. C., & GRUTEAM. (2017). Rupture or continuity: The arithmetico-algebraic thinking as an alternative in a modelling process in a paper and pencil and technology environment. Educational Studies in Mathematics, 94, 97–116.

    Google Scholar 

  • International Commission on Mathematical Instruction. (2017). School mathematics curriculum reforms: Challenges, changes and opportunities. ICMI Study 24 Discussion Document. In Y. Shimizu & R. Vithal (Eds.), Proceedings of the 24 thICMI Study (pp. 571–587). Tsukuba, Japan: University of Tsukuba.

    Google Scholar 

  • Jaworski, B., Chapman, O., Clark-Wilson, A., Cusi, A., Esteley, C., Goos, M., … Robutti, O. (2017). Mathematics teachers working and learning through collaboration. In G. Kaiser (Ed.), Proceedings of the 13 thInternational Congress on Mathematical Education, ICME-13 Monographs (pp. 261–276). https://doi.org/10.1007/978-3-319-62597-3_17 Springer Nature.

    Chapter  Google Scholar 

  • Jeannotte, D., & Kieran, C. (2017). A conceptual model of mathematical reasoning for school mathematics. Educational Studies in Mathematics, 96, 1–16.

    Google Scholar 

  • Jiang, C., Hwang, S., & Cai, J. (2014). Chinese and Singaporean sixth-grade students’ strategies for solving problems about speed. Educational Studies in Mathematics, 87, 27–50.

    Google Scholar 

  • Johnson, H., & McClintock, E. (2018). A link between students’ discernment of variation in unidirectional change and their use of quantitative variational reasoning. Educational Studies in Mathematics, 97, 299–316.

    Google Scholar 

  • Jones, G., Langrall, C., Thornton, C., & Mogill, A. T. (1997). A framework for assessing and nurturing young children’s thinking in probability. Educational Studies in Mathematics, 32, 101–125.

    Google Scholar 

  • Jones, I., & Inglis, M. (2015). The problem of assessing problem solving: Can comparative judgement help? Educational Studies in Mathematics, 89, 337–355.

    Google Scholar 

  • Jurdak, M. E., & El Mouhayar, R. R. (2014). Trends in the development of student level of reasoning in pattern generalization tasks across grade level. Educational Studies in Mathematics, 85, 75–92.

    Google Scholar 

  • Lerman, S. (2006). Theories of mathematics education: Is plurality a problem? ZDM Mathematics Education, 38, 8–13.

    Google Scholar 

  • Lesh, R., & Zawojewski, J. (2007). Problem solving and modeling. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 763–804). Reston, VA: NCTM.

    Google Scholar 

  • Markovits, Z., & Hershkowitz, R. (1997). Relative and absolute thinking in visual estimation processes. Educational Studies in Mathematics, 32, 29–47.

    Google Scholar 

  • Mason, J., Burton, L., & Stacey, K. (1985). Thinking mathematically (rev. ed.). Wokingham, UK: Addison-Wesley.

    Google Scholar 

  • Mata-Pereira, J., & da Ponte, J.-P. (2017). Enhancing students’ mathematical reasoning in the classroom: Teacher actions facilitating generalization and justification. Educational Studies in Mathematics, 96, 169–186.

    Google Scholar 

  • Mesa, V., & Wagner, D. (2019). Behind the door: A critical look at the process of publication in Educational Studies in Mathematics. Educational Studies in Mathematics, 101, 301–324.

    Google Scholar 

  • Ministry of Education. (2012). Mathematics syllabus secondary one to four: Express course, normal (academic) course. Singapore: Ministry of Education.

    Google Scholar 

  • Miyakawa, T. (2017). Comparative analysis on the nature of proof to be taught in geometry: The cases of French and Japanese lower secondary schools. Educational Studies in Mathematics, 94, 37–54.

    Google Scholar 

  • National Council for Curriculum and Assessment, Ireland (2015). Leaving certificate mathematics syllabus: Foundation, ordinary and higher level. Retrieved 10 October 2018 from https://curriculumonline.ie/getmedia/f6f2e822-2b0c-461e-bcd4-dfcde6decc0c/ SCSEC25_Maths_syllabus_examination-2015_English.pdf

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

    Google Scholar 

  • National Research Council. (2001). In J. Kilpatrick, J. Swafford, & B. Findell (Eds.), Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.

    Google Scholar 

  • Organisation for Economic Cooperation and Development (OECD) (2017). PISA 2021 mathematics: A broadened perspective. Retrieved 21 September 2019 from https://www.oecd.org/pisa/pisaproducts/PISA2021_Mathematics_StrategicDirectionPaper.pdf

  • Organisation for Economic Cooperation and Development (OECD) (2018). PISA 2021 mathematics framework (draft). Retrieved 19 September 2019 from http://www.oecd.org/pisa/publications/

  • Prediger, S., Bikner-Ahsbahs, A., & Arzarello, F. (2008). Networking strategies and methods for connecting theoretical approaches: First steps towards a conceptual framework. ZDM Mathematics Education, 40, 165–178.

    Google Scholar 

  • Ramos, J., & Weber, K. (2014). Why and how mathematicians read proofs: Further evidence from a survey study. Educational Studies in Mathematics, 85, 161–173.

    Google Scholar 

  • Robutti, O., Cusi, A., Clark-Wilson, A., Jaworski, B., Chapman, O., Esteley, C., … Joubert, M. (2016). ICME international survey on teachers working and learning through collaboration. ZDM Mathematics Education, 48, 651–690.

    Google Scholar 

  • Samper, C., Perry, P., Camargo, L., Sáenz-Ludlow, A., & Molina, Ó. (2016). A dilemma that underlies an existence proof in geometry. Educational Studies in Mathematics, 93, 35–50.

    Google Scholar 

  • Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 334–370). New York: McMillan.

    Google Scholar 

  • Schukajlow, S., Krug, A., & Rakoczy, K. (2015). Effects of prompting multiple solutions for modelling problems on students’ performance. Educational Studies in Mathematics, 89, 393–417.

    Google Scholar 

  • Selden, J., & Selden, A. (1995). Unpacking the logic of mathematical statements. Educational Studies in Mathematics, 29, 123–151.

    Google Scholar 

  • Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses, and mathematizing. New York: Cambridge University Press.

    Google Scholar 

  • Simpson, A. (2015). The anatomy of a mathematical proof: Implications for analyses with Toulmin’s scheme. Educational Studies in Mathematics, 90, 1–17.

    Google Scholar 

  • Stillman, G., & Galbraith, P. (1998). Applying mathematics with real world connections: Metacognitive characteristics of secondary students. Educational Studies in Mathematics, 36, 157–189.

    Google Scholar 

  • Törner, G., Schoenfeld, A., & Reiss, K. M. (2007). Problem solving around the world: Summing up the state of the art. ZDM Mathematics Education, 39(5–6), 353.

    Google Scholar 

  • United Nations Statistical Division (2019). Methodology: Standard country or area codes for statistical use (M49). Retrieved 24 September 2019 from https://unstats.un.org/unsd/methodology/m49/

  • Watson, J., & Moritz, J. (1998). The beginning of statistical inference: Comparing two data sets. Educational Studies in Mathematics, 37, 145–168.

    Google Scholar 

  • Weber, K., & Leikin, R. (2016). Recent advances in research on problem solving and problem posing. In A. Gutiérrez, G. Leder, & P. Boero (Eds.), The second handbook of research on the psychology of mathematics education (pp. 353–382). Rotterdam, the Netherlands: Sense Publishers.

    Google Scholar 

  • Williams, S., & Leatham, K. (2017). Journal quality in mathematics education. Journal for Research in Mathematics Education, 48, 369–396.

    Google Scholar 

  • Xie, J., & Masingila, J. (2017). Examining interactions between problem posing and problem solving with prospective primary teachers: A case of using fractions. Educational Studies in Mathematics, 96, 101–118.

    Google Scholar 

  • Yackel, E., & Hanna, G. (2003). Reasoning and proof. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to Principles and Standards for School Mathematics (pp. 227–236). Reston, VA: NCTM.

    Google Scholar 

  • Yusof, Y. M., & Tall, D. (1999). Changing attitudes to university mathematics through problem solving. Educational Studies in Mathematics, 37, 67–82.

    Google Scholar 

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  1. University of Limerick, Limerick, Ireland

    Merrilyn Goos & Sila Kaya

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Goos, M., Kaya, S. Understanding and promoting students’ mathematical thinking: a review of research published in ESM. Educ Stud Math 103, 7–25 (2020). https://doi.org/10.1007/s10649-019-09921-7

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