Pre-Service Mathematics Teachers’ Web of Knowledge Recalled for Mathematically Rich and Contextually Realistic Problems

Serife Sevinc 1 * , Richard Lesh 2
More Detail
1 Mathematics and Science Education, Middle East Technical University, Ankara, TURKEY
2 Emeritus Professor, Counseling and Educational Psychology, Indiana University, Bloomington, USA
* Corresponding Author
EUR J SCI MATH ED, Volume 10, Issue 4, pp. 471-494. https://doi.org/10.30935/scimath/12250
Published: 25 July 2022
OPEN ACCESS   1022 Views   625 Downloads
Download Full Text (PDF)

ABSTRACT

This study aimed to elicit middle school preservice mathematics teachers’ self-reported web of knowledge recalled in generating mathematically rich and contextually realistic problems. We designed this study as multi-tier design research incorporated into two teacher education courses in which 40 preservice teachers enrolled in total. Preservice teachers worked in small groups and recorded the characteristics of mathematically rich and contextually realistic problems. They were also asked to produce webs of knowledge recalled in this process. Preservice teachers’ individual reflection papers, audio records of their group discussions and interviews were analyzed to understand how different types of knowledge in their web of knowledge function in relation to the mathematical richness and contextual realness aspects of the problems. The findings indicated that preservice teachers could identify various characteristics for mathematical richness and contextual realistic aspects of the problems. In relation to those characteristics, the preservice teachers’ self-reported web of knowledge produced three core knowledge types for ensuring mathematical richness (i.e., knowledge of content, curriculum, and pedagogy) and two aspects of realistic contexts (i.e., real life knowledge and interdisciplinary knowledge). Furthermore, although they included common knowledge types, the webs of knowledge were in different shapes and indicated various relationships (i.e., hierarchical, categorical, influential, and holistic.). Considering the various relations indicated by the webs of knowledge, we claimed that teachers needed an interconnected knowledge base for mathematically rich and contextually realistic problems, the implications of which we discussed for mathematics teacher education.
Keywords: teacher knowledge, mathematically rich problems, contextual mathematics problems, multi-tier design research

CITATION

Sevinc, S., & Lesh, R. (2022). Pre-Service Mathematics Teachers’ Web of Knowledge Recalled for Mathematically Rich and Contextually Realistic Problems. European Journal of Science and Mathematics Education, 10(4), 471-494. https://doi.org/10.30935/scimath/12250

REFERENCES

  • Agathangelou, S. A., & Charalambous, C. Y. (2021). Is content knowledge pre-requisite of pedagogical content knowledge? An empirical investigation. Journal of Mathematics Teacher Education, 24(5), 431-458. https://doi.org/10.1007/s10857-020-09466-0
  • Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching what makes it special? Journal of Teacher Education, 59, 389-407. https://doi.org/10.1177/0022487108324554
  • Bas-Ader, S., Erbas, A. K., Cetinkaya, B., Alacaci, C., & Cakiroglu, E. (2021). Secondary mathematics teachers’ noticing of students’ mathematical thinking through modeling-based teacher investigations. Mathematics Education Research Journal. https://doi.org/10.1007/s13394-021-00389-4
  • Bonotto, C. (2007). How to replace word problems with activities of realistic mathematical modelling. In W. Blum, P. Galbraith, M. Niss, & H.-W. Henn (Eds.), Modelling and applications in mathematics education. The 14th ICMI study (New ICMI; Studies Series) (Vol. 10, pp. 185-192). Springer.
  • Brown, J. P. (2019). Real-world task context: Meanings and roles. In G. A. Stillman & J. P. Brown, (Eds.), Lines of inquiry in mathematical modelling research in education (pp. 53-81). Springer.
  • Carlson, J., & Daehler, K. (2019). The refined consensus model of pedagogical content knowledge in science education. In A. Hume, R. Cooper, & A. Borowski (Eds.), Repositioning pedagogical content knowledge in teachers’ knowledge for teaching science (pp. 77-92). Springer. https://doi.org/10.1007/978-981-13-5898-2_2
  • Chamberlin, M. (2005). Teachers’ discussions of students’ thinking: Meeting the challenge of attending to students’ thinking. Journal of Mathematics Teacher Education, 8, 141-170. https://doi.org/10.1007/s10857-005-4770-4
  • Chapman, O. (2013). Mathematical-task knowledge for teaching. Journal of Mathematics Teacher Education, 16(1), 1-6. https://doi.org/10.1007/s10857-013-9234-7
  • Charmaz, K. (2006). Constructing grounded theory: A practical guide through qualitative analysis. Sage.
  • Cross, D., & Lepareur, C. (2015). PCK at stake in teacher–student interaction in relation to students’ difficulties. In M. Grangeat (Ed.), Understanding science teachers’ professional knowledge growth (pp. 47-61). Springer.
  • Csíkos, C., & Szitányi, J. (2020). Teachers’ pedagogical content knowledge in teaching word problem solving strategies. ZDM Mathematics Education, 52(1), 165-178. https://doi.org/10.1007/s11858-019-01115-y
  • English, L., & Sriraman, B. (2010). Problem solving for the 21st century. In B. Sriraman & L. English (Eds.), Theories of mathematics education (pp. 263-290). Springer. https://doi.org/10.1007/978-3-642-00742-2_27
  • Even, R. (1993). Subject-matter knowledge and pedagogical content knowledge: Prospective secondary teachers and the function concept. Journal for Research in Mathematics Education, 24(2), 94-116. https://doi.org/10.5951/jresematheduc.24.2.0094
  • Gainsburg, J. (2008). Real-world connections in secondary mathematics teaching. Journal of Mathematics Teacher Education, 11(3), 199-219. https://doi.org/10.1007/s10857-007-9070-8
  • Gravemeijer, K., & Doorman, M. (1999). Context problems in realistic mathematics education: A calculus course as an example. Educational Studies in Mathematics, 39(1-3), 111-129. https://doi.org/10.1023/A:1003749919816
  • Greefrath, G., Siller, HS., Klock, H., & Wess, R. (2022) Preservice secondary teachers’ pedagogical content knowledge for the teaching of mathematical modelling. Educational Studies in Mathematics, 109, 383-407. https://doi.org/10.1007/s10649-021-10038-z
  • Grossman, P. L. (1990). The making of a teacher: Teacher knowledge and teacher education. Teachers College Press.
  • Grossman, P. L., & Richert, A. E. (1988). Unacknowledged knowledge growth: A re-examination of the effects of teacher education. Teaching and Teacher Education, 4(1), 53-62. https://doi.org/10.1016/0742-051X(88)90024-8
  • Hill, H. C., & Ball, D. L. (2004). Learning mathematics for teaching: Results from California’s mathematics professional development institutes. Journal for Research in Mathematics Education, 35(5), 330-351. https://doi.org/10.2307/30034819
  • Hill, H. C., Blunk, M. L., Charalambous, C. Y., Lewis, J. M., Phelps, G. C., Sleep, L., & Ball, D. L. (2008). Mathematical knowledge for teaching and the mathematical quality of instruction: An exploratory study. Cognition and Instruction, 26(4), 430-511. https://doi.org/10.1080/07370000802177235
  • Hill, H. C., Schilling, S. G., & Ball, D. L. (2004). Developing measures of teachers’ mathematics knowledge for teaching. Elementary School Journal, 105(1), 11-30. https://doi.org/10.1086/428763
  • Kind, V., & Chan, K. K. (2019). Resolving the amalgam: connecting pedagogical content knowledge, content knowledge and pedagogical knowledge. International Journal of Science Education, 41(7), 964-978. https://doi.org/10.1080/09500693.2019.1584931
  • Koellner-Clark. K., & Lesh, R. (2003). A modeling approach to describe teacher knowledge. In R. Lesh & H. Doerr (Eds.), Beyond constructivism: A models and modeling perspective (pp. 159-173). Lawrence Erlbaum.
  • Leavy, A., & Hourigan, M. (2020). Posing mathematically worthwhile problems: Developing the problem posing skills of prospective teachers. Journal of Mathematics Teacher Education, 23(4), 341-361. https://doi.org/10.1007/s10857-018-09425-w
  • Lee, J. E. (2012). Prospective elementary teachers’ perceptions of real-life connections reflected in posing and evaluating story problems. Journal of Mathematics Teacher Education, 15(6), 429-452. https://doi.org/10.1007/s10857-012-9220-5
  • Lee, Y., Capraro, R. M., & Capraro, M. M. (2018). Mathematics teachers’ subject matter knowledge and pedagogical content knowledge in problem posing. International Electronic Journal of Mathematics Education, 13(2), 75-90. https://doi.org/10.12973/iejme/2698
  • Lesh, R. (2006). New directions for research on mathematical problem solving. In P. Grootenboer, R. Zevenbergen, & M. Chinnappan (Eds.), Identities, cultures and learning spaces, Proceedings of the 29th annual conference of the mathematics education research group of Australasia, Canberra (Vol. 1, pp. 15-34). MERGA.
  • Lesh, R., & Doerr, H. M. (2003). Foundations of a models and modeling perspective on mathematics teaching, learning, and problem solving. In R. Lesh & H. M. Doerr (Eds.), Beyond constructivism: Models and modeling perspectives on mathematics problem solving learning, and teaching (pp. 3-34). Lawrence Erlbaum.
  • Lesh, R., Hoover, M., Hole, B., Kelly, A., & Post, T. (2000). Principles for developing thought-revealing activities for students and teachers. In A. Kelly & R. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 113-149). Lawrence Erlbaum.
  • Lesh, R., & Lehrer R. (2003). Models and modeling perspectives on the development of students and teachers. Mathematical Thinking and Learning, 5(2-3), 109-129. https://doi.org/10.1080/10986065.2003.9679996
  • Lincoln, Y. S., & Guba, E. G. (1985). Naturalistic inquiry. Sage.
  • Ma, L. (1999). Knowing and teaching elementary mathematics: teachers’ understanding of fundamental mathematics in China and the United States. Lawrence Erlbaum.
  • Norton, S. (2019). The relationship between mathematical content knowledge and mathematical pedagogical content knowledge of prospective primary teachers. Journal of Mathematics Teacher Education, 22(5), 489-514. https://doi.org/10.1007/s10857-018-9401-y
  • Park, S. & Oliver, J. S. (2008). Revisiting the conceptualization of pedagogical content knowledge (PCK): PCK as a conceptual tool to understand teachers as professionals. Research in Science Education, 38(3), 261-284. https://doi.org/10.1007/s11165-007-9049-6
  • Rossouw, L., & Smith, E. (1998). Teachers’ pedagogical content knowledge of geometry. In A. Olivier & K. Newstead (Eds.), Proceedings of 22nd PME international conference, 4, 57-63.
  • Sevinc, S. (2022). Knowledge-in-action for crafting mathematics problems in realistic contexts. Journal of Mathematics Teacher Education. https://doi.org/10.1007/s10857-022-09541-8
  • Sevinc, S. & Lesh, R. (2018). Training mathematics teachers for realistic math problems: A case of modeling-based teacher education courses. ZDM Mathematics Education, 50, 301-314. https://doi.org/10.1007/s11858-017-0898-9
  • Sevinc, S. & Lesh, R. (2021). Preservice mathematics teachers’ conceptions of mathematically rich and contextually realistic problems. Journal of Mathematics Teacher Education. https://doi.org/10.1007/s10857-021-09512-5
  • Sevis, S. (2016). Unpacking teacher knowledge for bridging in-and out-of-school mathematics using mathematically-rich and contextually-realistic problems (UMI No: 10143631) [Doctoral dissertation, Indiana University]. ProQuest Dissertations and Theses Global.
  • Shahbari, J. A. (2018). Mathematics teachers’ conceptions about modelling activities and its reflection on their beliefs about mathematics. International Journal of Mathematical Education in Science and Technology, 49(5), 721-742. https://doi.org/10.1080/0020739X.2017.1404650
  • Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14. https://doi.org/10.3102/0013189X015002004
  • Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57(1), 1-23. https://doi.org/10.17763/haer.57.1.j463w79r56455411
  • Simon, M. A. (1997). Developing new models of mathematics teaching: An imperative for research on mathematics teacher development. In E. Fennema & B. S. Nelson (Eds.), Mathematics teachers in transition (pp. 55-86). Lawrence Erlbaum.
  • Stillman, G., & Brown, J. P. (2011). Preservice Secondary Mathematics Teachers’ Affinity with Using Modelling Tasks in Teaching Years 8–10. In G. Kaiser, W. Blum, R. B. Ferri, & G. Stillman (Eds.), Trends in teaching and learning of mathematical modelling, international perspectives on the teaching and learning of mathematical modelling (pp. 289-298). Springer. https://doi.org/10.1007/978-94-007-0910-2_29
  • Tamir, P. (1991). Professional and personal knowledge of teachers and teacher educators. Teaching and Teacher Education, 7(3), 263-268. https://doi.org/10.1016/0742-051X(91)90033-L
  • Thornberg, R., & Charmaz, K. (2012). Grounded theory. In S. D. Lapan, M. T. Quartaroli, & F. J. Riemer (Eds.), Qualitative research: An introduction to methods and designs (pp. 41-67). Jossey-Bass.
  • Treffers, A. (1987). Three dimensions. A model of goal and theory description in mathematics education: The Wiskobas project. D. Reidel Publishing.
  • Verschaffel, L., Greer, B., & de Corte, E. (2000). Making sense of word problems (Contexts of Learning Series). Swets & Zeitlinger.
  • Zawojewski, J., Lesh, R., & English, L. (2003). A models and modeling perspective on the role of small group learning activities. In R. Lesh & H. M. Doerr (Eds.), Beyond constructivism: A models and modelling perspective on mathematics problem solving; learning and teaching (pp. 337-358). Lawrence Erlbaum.