Skip to main content
SpringerLink
Log in

Preservice teachers’ mathematical understanding exhibited in problem posing and problem solving

  • Original Paper
  • Published:
ZDM – Mathematics Education Aims and scope Submit manuscript

Abstract

Fostering conceptual understanding in mathematics classrooms is an important goal in mathematics education. To support this goal, we need to be able to diagnose and assess the extent to which students have conceptual understanding. In this study we employed a problem-posing task and a problem-solving task in order to diagnose and assess preservice teachers’ mathematical understanding of fraction division. The results of the study show that although over 99% of the preservice teachers could correctly perform fraction division, they were much less likely to exhibit conceptual understanding of fraction division either through a problem-solving task involving graphical representation or through a problem-posing task. However, engaging in problem posing appears to more likely than making a graphical representation to elicit preservice teachers’ conceptual understanding. Moreover, a conceptual cue about the meaning of fraction division appears to greatly increase the likelihood that preservice teachers will exhibit conceptual understanding of fraction division in their posed problems and generate conceptually based diagrams of fraction division. The findings highlight the usefulness of problem-posing tasks for diagnosing and assessing preservice teachers’ mathematical understanding.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Notes

  1. This differentiation was to guard against an order effect, although as we show in the results below, task order was not associated with the mathematical understanding exhibited by the preservice teachers.

References

  • Adu-Gyamfi, K., Schwartz, C. S., Sinicope, R., & Bossé, M. J. (2019). Making sense of fraction division: Domain and representation knowledge of preservice elementary teachers on a fraction division task. Mathematics Education Research Journal, 31, 507–528.

    Article  Google Scholar 

  • Ball, D. L. (1990). Prospective elementary and secondary teachers’ understanding of division. Journal for Research in Mathematics Education, 21, 132–144.

    Article  Google Scholar 

  • Behr, M., Harel, G., Post, T., & Lesh, R. (1992). Rational number, ratio and proportion. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 296–333). Macmillan Publishing.

    Google Scholar 

  • Behr, M., Harel, O., Post, T., & Leah, R. (1987). Theoretical analysis: Structure and hierarchy, missing value proportion problems. In J. C. Bergeron et al., (Eds.), Proceedings of the international conference on the psychology of mathematics education (PME-11), Montreal, Canada, July 19–25, 1987 (p. 752).

  • Booth, J. L., Newton, K. J., & Twiss-Garrity, L. K. (2014). The impact of fraction magnitude knowledge on algebra performance and learning. Journal of Experimental Child Psychology, 118, 110–118.

    Article  Google Scholar 

  • Bahr, D. L., & Bossé, M. J. (2008). The State of Balance Between Procedural Knowledge and Conceptual Understanding in Mathematics Teacher Education. Faculty Publications, 924. https://scholarsarchive.byu.edu/facpub/924. Accessed 25 Nov 2008.

  • Brueckner, L. J., & Elwell, M. (1932). Reliability of diagnosis of error in multiplication of fractions. The Journal of Educational Research, 26(3), 175–185.

    Article  Google Scholar 

  • Cai, J. (2000). Mathematical thinking involved in US and Chinese students’ solving of process-constrained and process-open problems. Mathematical Thinking and Learning, 2(4), 309–340.

    Article  Google Scholar 

  • Cai, J., Chen, T., Li, X., Xu, R., Zhang, S., Hu, Y., & Song, N. (2020). Exploring the impact of a problem-posing workshop on elementary school mathematics teachers’ conceptions on problem posing and lesson design. International Journal of Educational Research, 102, 404–415.

    Google Scholar 

  • Cai, J., & Ding, M. (2017). On mathematical understanding: Perspectives of experienced Chinese mathematics teachers. Journal of Mathematics Teacher Education., 20(1), 5–29.

    Article  Google Scholar 

  • Cai, J., & Hwang, S. (2002). Generalized and generative thinking in US and Chinese students’ mathematical problem solving and problem posing. The Journal of Mathematical Behavior, 21(4), 401–421.

    Article  Google Scholar 

  • Cai, J., & Hwang, S. (2020). Learning to teach through mathematical problem posing: Theoretical considerations, methodology, and directions for future research. International Journal of Educational Research, 102, 420–430.

    Google Scholar 

  • Cai, J., Hwang, S., Jiang, C., & Silber, S. (2015). Problem-posing research in mathematics education: Some answered and unanswered questions. In F. M. Singer, N. Ellerton, & J. Cai (Eds.), Mathematical problem posing: From research to effective practice (pp. 3–34). Springer.

    Chapter  Google Scholar 

  • Cai, J., Moyer, J. C., Wang, N., Hwang, S., Nie, B., & Garber, T. (2013). Mathematical problem posing as a measure of curricular effect on students’ learning. Educational Studies in Mathematics, 83(1), 57–69.

    Article  Google Scholar 

  • Canobi, K. H., & Bethune, N. E. (2008). Number words in young children’s conceptual and procedural knowledge of addition, subtraction and inversion. Cognition, 108(3), 675–686.

    Article  Google Scholar 

  • Canobi, K. H., Reeve, R. A., & Pattison, P. E. (2003). Patterns of knowledge in children’s addition. Developmental Psychology, 39(3), 521.

    Article  Google Scholar 

  • Carpenter, T. P., Kepner, H., Corbitt, M. K., Lindquist, M. M., & Reys, R. E. (1980). Results and implications of the second NAEP mathematics assessments: Elementary school. The Arithmetic Teacher, 27(8), 10–47.

    Article  Google Scholar 

  • Charalambous, C. Y., & Pitta-Pantazi, D. (2007). Drawing on a theoretical model to study students’ understandings of fractions. Educational Studies in Mathematics, 64, 293–316.

    Article  Google Scholar 

  • Conference Board of the Mathematical Sciences. (2012). The mathematical education of teachers II. Providence, RI & Washington, DC: American Mathematical Society & Mathematical Association of America.

  • Davidson, A. (2012). Making it in America. The Atlantic, 309, 58–70.

    Google Scholar 

  • Ellerton, N. F., Singer, F. M., & Cai, J. (2015). Problem posing in mathematics: Reflecting on the past, energizing the present, and foreshadowing the future. In F. M. Singer, N. Ellerton, & J. Cai (Eds.), Mathematical problem posing: From research to effective practice (pp. 547–556). Springer.

    Chapter  Google Scholar 

  • Harel, G., Koichu, B., & Manaster, A. (2006). Algebra teachers’ ways of thinking characterizing the mental act of problem posing: The mental act of problem posing. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings of the 30th conference of the international group for the psychology of mathematics education (Vol. 2, pp. 241–248). Charles University.

  • Isik, C., & Kar, T. (2012). An error analysis in division problems in fractions posed by pre-service elementary mathematics teachers. Educational Sciences: Theory and Practice, 12(3), 2303–2309.

    Google Scholar 

  • Izsák, A. (2003). “We want a statement that is always true”: Criteria for good algebraic representations and the development of modelling knowledge. Journal for Research in Mathematics Education, 34(3), 191–227.

    Article  Google Scholar 

  • Kieren, T. E. (1993). Rational and fractional numbers: From quotient fields to recursive understanding. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Studies in mathematical thinking and learning. Rational numbers: An integration of research (p. 49–84). Lawrence Erlbaum Associates, Inc.

  • Lamon, S. J. (2001). Presenting and representing from fractions to rational numbers. In A. A. Cuoco & F. R. Curcio (Eds.), The roles of representation in school mathematics (pp. 146–165). National Council of Teachers of Mathematics.

  • Lortie-Forgues, H., Tian, J., & Siegler, R. S. (2015). Why is learning fraction and decimal arithmetic so difficult? Developmental Review, 38, 201–221.

    Article  Google Scholar 

  • Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Lawrence Erlbaum.

    Book  Google Scholar 

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

  • National Research Council. (2001). Knowing and learning mathematics for teaching. National Academy Press.

    Google Scholar 

  • Ni, Y. (2001). Semantic domains of rational numbers and the acquisition of fraction equivalence. Contemporary Educational Psychology, 26(3), 400–417.

    Article  Google Scholar 

  • Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93(2), 346–362.

    Article  Google Scholar 

  • Schneider, M., Rittle-Johnson, B., & Star, J. R. (2011). Relations among conceptual knowledge, procedural knowledge, and procedural flexibility in two samples differing in prior knowledge. Developmental Psychology, 47(6), 1525–1538.

    Article  Google Scholar 

  • Siegler, R. S. (2016). Magnitude knowledge: The common core of numerical development. Developmental Science, 19(3), 341–361.

    Article  Google Scholar 

  • Siegler, R. S., Duncan, G. J., Davis-Kean, P. E., Duckworth, K., Claessens, A., Engel, M., & Chen, M. (2012). Early predictors of high school mathematics achievement. Psychological Science, 23(7), 691–697.

    Article  Google Scholar 

  • Siegler, R. S., & Lortie-Forgues, H. (2015). Conceptual knowledge of fraction arithmetic. Journal of Educational Psychology, 107(3), 909–918.

    Article  Google Scholar 

  • Siegler, R. S., & Lortie-Forgues, H. (2017). Hard lessons: Why rational number arithmetic is so difficult for so many people. Current Directions in Psychological Science, 26(4), 346–351.

    Article  Google Scholar 

  • Siegler, R. S., & Pyke, A. A. (2013). Developmental and individual differences in understanding of fractions. Developmental Psychology, 49(10), 1994–2004.

    Article  Google Scholar 

  • Silver, E. A. (1994). On mathematical problem posing. For the Learning of Mathematics, 14(1), 19–28.

    Google Scholar 

  • Silver, E. A., & Cai, J. (1996). An analysis of arithmetic problem posing by middle school students. Journal for Research in Mathematics Education, 27(5), 521–539.

    Article  Google Scholar 

  • Singer, F. M., Ellerton, N., & Cai, J. (2013). Problem-posing research in mathematics education: New questions and directions. Educational Studies in Mathematics, 83(1), 1–7.

    Article  Google Scholar 

  • Skemp, R. R. (1978). Relational understanding and instrumental understanding. Arithmetic Teacher, 26, 9–15.

    Article  Google Scholar 

  • Tichá, M., & Hošpesová, A. (2013). Developing teachers’ subject didactic competence through problem posing. Educational Studies in Mathematics, 83(1), 133–143.

    Article  Google Scholar 

  • 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.

  • Toluk-Uçar, Z. (2009). Developing pre-service teachers understanding of fractions through problem posing. Teaching and Teacher Education, 25(1), 166–175.

    Article  Google Scholar 

  • Van Patten, J., Chao, C. I., & Reigeluth, C. M. (1986). A review of strategies for sequencing and synthesizing instruction. Review of Educational Research, 56(4), 437–471.

    Article  Google Scholar 

  • Whitin, D. J., & Whitin, P. (2012). Making sense of fractions and percentages. Teaching Children Mathematics, 18, 490–496.

    Article  Google Scholar 

  • Yang, D. C., Reys, R. E., & Reys, B. J. (2009). Number sense strategies used by pre-service teachers in Taiwan. International Journal of Science and Mathematics Education, 7(2), 383–403.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Hangzhou Normal University, Hangzhou, China

    Yiling Yao

  2. University of Delaware, Newark, USA

    Stephen Hwang & Jinfa Cai

Authors
  1. Yiling Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stephen Hwang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jinfa Cai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yiling Yao.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

1.1 Tasks in the Tests

  1. a.

    Please write the answer to the following expressions.

$$\begin{gathered} 1\frac{3}{4} \div \frac{1}{2} = \quad \quad \quad \quad \frac{5}{6} \div 5 = \quad \quad \quad \quad 1\frac{1}{4} \div \frac{5}{8} = \quad \quad \quad \hfill \\ 1\frac{3}{4} \div 2 = \quad \quad \quad \quad 2 \div \frac{1}{3} = \quad \quad \quad \quad \frac{1}{3} \div \frac{1}{2} = \quad \quad \quad \hfill \\ \end{gathered}$$
  1. b.

    Please make a drawing for the expression showing the solution of 1¾ ÷ ½.

  2. c.

    Please pose as many mathematical problems as possible that show the solution of 1¾ ÷ ½. (Note: only pose mathematical problems, you don’t have to solve them.)

The conceptual cue:

1¾ ÷ ½ means how many one-halves are in 1¾.

Order of tasks in test 1, 2, 3, and 4:

Test 1

Test 2

Test 3

Test 4

a, b, c

a, c, b

a, b (with the conceptual cue), c

a, c, b (with the conceptual cue)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yao, Y., Hwang, S. & Cai, J. Preservice teachers’ mathematical understanding exhibited in problem posing and problem solving. ZDM Mathematics Education 53, 937–949 (2021). https://doi.org/10.1007/s11858-021-01277-8

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11858-021-01277-8

Keywords

Search

Navigation