# Prospective Teachers’ Pedagogical Considerations of Mathematical Connections: A Framework to Motivate Attention to and Awareness of Connections

## Keywords:

attention, awareness, mathematical connections, noticing, teacher education## Abstract

Research findings and reform-oriented standards emphasise the importance of mathematical connections in support of students’ conceptual development. Previous research on teachers attending to mathematical connections has tended to focus on expert teachers’ practice. Complementing previous research, this study describes how a cohort of twelve prospective mathematics teachers attended to and made sense of mathematical connections that arose when working with secondary students in small-group instruction. Results indicated prospective teachers were able to attend to mathematical connections during instruction and made several pedagogical considerations around such connections. We present a framework, the *Pedagogical Considerations of Mathematical Connections* (PCMC) framework, which offers mathematics teacher educators a new model to expand prospective teachers’ attention to and awareness of mathematical connections. The study contributes to the existing literature on teacher noticing by providing a new kind of theme-specific noticing (i.e., mathematical connections) and informing mathematics teacher educators of how prospective secondary teachers attend to mathematical connections.

## References

Aguirre, J. M., Turner, E. E., Bartell, T. G., Kalinec-Craig, C., Foote, M. Q., McDuffie, A. R., & Drake, C. (2013). Making connections in practice: How prospective elementary teachers connect to children’s mathematical thinking and community funds of knowledge in mathematics instruction. Journal of Teacher Education, 64(2), 178–192.

Australian Curriculum Assessment and Reporting Authority. (n.d.). The Australian Curriculum: Foundations to Year 10. Retrieved from https://www.australiancurriculum.edu.au/f-10-curriculum/

Ball, D. L. (1993). With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. The Elementary School Journal, 93(4), 373–397.

Boaler, J., & Humphreys, C. (2005). Connecting Mathematical Ideas: Middle School Video Cases to Support Teaching and Learning. Heinemann.

Borko, H., & Livingston, C. (1989). Cognition and improvisation: Differences in mathematics instruction by expert and novice teachers. American Educational Research Journal, 26(4), 473–498.

Braun, V., & Clarke, V. (2006). Using thematic analysis in psychology. Qualitative Research in Psychology, 3(2), 77–101.

Cobb, P. (1988). The tension between theories of learning in mathematics education. Educational Psychologist, 23, 87–103.

Confrey, J. (1990). Chapter 8: What Constructivism Implies for Teaching. Journal for Research in Mathematics Education. Monograph, 4, 107–210.

Dreher, A., & Kuntze, S. (2015). Teachers’ professional knowledge and noticing: The case of multiple representations in the mathematics classroom. Educational Studies in Mathematics, 88(1), 89–114.

Even, R., Tirosh, D., & Robinson, N. (1993). Connectedness in teaching equivalent algebraic expressions: Novice versus expert teachers. Mathematics Education Research Journal, 5(1), 50–59.

Fernández, C., Llinares, S., & Valls, J. (2012). Learning to notice students’ mathematical thinking through on-line discussions. ZDM, 44(6), 747–759.

Goodwin, C. (1994). Professional vision. American Anthropologist, 96(3), 606–633.

Hiebert, J., Gallimore, R., Garnier, H., Givvin, K. B., Hollingsworth, H., Jacobs, J., … Stigler, J. (2003). Teaching Mathematics in Seven Countries: Results from the TIMSS 1999 Video Study. National Center for Education Statistics.

Hiebert, J., & Grouws, D. A. (2007). The effects of classroom mathematics teaching on students’ learning. In F. K. Lester, Jr. (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (pp. 371–404). Information Age Publishing.

Hill, H. C., & Charalambous, C. Y. (2012). Teaching (un)connected mathematics: Two teachers’ enactment of the pizza problem. Journal of Curriculum Studies, 44(4), 467–487.

Hufferd-Ackles, K., Fuson, K. C., & Sherin, M. G. (2004). Describing levels and components of a math-talk learning community. Journal for Research in Mathematics Education, 35(2), 81–116.

Jacobs, V. R., Lamb, L. L. C., & Philipp, R. A. (2010). Professional noticing of children’s mathematical thinking. Journal for Research in Mathematics Education, 41(2), 169–202.

Jacobs, V. R., & Spangler, D. A. (2017). Research on core practices in K-12 mathematics teaching. In J. Cai (Ed.), Compendium for Research in Mathematics Education (pp. 766–792). National Council of Teachers of Mathematics.

Krupa, E. E., Huey, M., Lesseig, K., Casey, S., & Monson, D. (2017). Investigating secondary preservice teacher noticing of students’ mathematical thinking. In E. O. Schack, M. H. Fisher, & J. A. Wilhelm (Eds.), Teacher Noticing: Bridging and Broadening Perspectives, Contexts, and Frameworks (pp. 49–72). Springer International Publishing.

Lampert, M. (2001). Teaching Problems and the Problems of Teaching. Yale University Press.

Leinhardt, G. (1989). Math lessons: A contrast of novice and expert competence. Journal for Research in Mathematics Education, 20(1), 52–75.

Livingston, C., & Borko, H. (1990). High school mathematics review lessons: Expert-novice distinctions. Journal for Research in Mathematics Education, 21(5), 372–387.

Lobato, J., Clarke, D., & Ellis, A. B. (2005). Initiating and eliciting in teaching: A reformulation of telling. Journal for Research in Mathematics Education, 36(2), 101–136.

Mason, J. (1998). Enabling teachers to be real teachers: Necessary levels of awareness and structure of attention. Journal of Mathematics Teacher Education, 1(3), 243–267.

Mason, J. (2002). Researching your own practice: The discipline of noticing. Routledge.

Monson, D., Krupa, E., Lesseig, K., & Casey, S. (2020). Developing secondary prospective teachers’ ability to respond to student work. Journal of Mathematics Teacher Education, 23(2), 209–232.

National Council of Teachers of Mathematics (Ed.). (2000). Principles and Standards for School Mathematics. Authors.

Russ, R. S., Sherin, B. L., & Sherin, M. G. (2016). What constitutes teacher learning? In D. H. Gitomer & C. A. Bell (Eds.), Handbook of Research on Teaching (5th Edition, pp. 391–438).

Schack, E. O., Fisher, M. H., Thomas, J. N., Eisenhardt, S., Tassell, J., & Yoder, M. (2013). Prospective elementary school teachers’ professional noticing of children’s early numeracy. Journal of Mathematics Teacher Education, 16(5), 379–397.

Sherin, M. G., Russ, R. S., & Colestock, A. A. (2011). Accessing mathematics teachers’ in-the moment noticing. In M. G. Sherin, V. R. Jacobs, & R. A. Philipp (Eds.), Mathematics Teacher Noticing: Seeing Through Teachers’ Eyes (pp. 79–94). Routledge.

Sherin, M. G., & van Es, E. A. (2008). Effects of video club participation on teachers’ professional vision. Journal of Teacher Education, 60(1), 20–37.

Singletary, L. M. (2012). Mathematical connections made in practice: An examination of teachers’ beliefs and practices (Unpublished doctoral dissertation). University of Georgia, Athens, GA.

Smith, M. S., & Stein, M. K. (2011). 5 Practices for Orchestrating Productive Mathematics Discussions. National Council of Teachers of Mathematics; Corwin.

Star, J. R., & Strickland, S. K. (2008). Learning to observe: Using video to improve preservice mathematics teachers’ ability to notice. Journal of Mathematics Teacher Education, 11(2), 107–125.

Stein, M. K., Engle, R. A., Smith, M. S., & Hughes, E. K. (2008). Orchestrating productive mathematical discussions: Five practices for helping teachers move beyond show and tell. Mathematical Thinking and Learning, 10(4), 313–340.

Stein, M. K., Grover, B. W., & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33(2), 455–488.

Stevens, R., & Hall, R. (1998). Disciplined perception: Learning to see in technoscience. In M. Lampert & M. L. Blunk (Eds.), Talking Mathematics in School: Studies of Teaching and Learning (pp. 107–150). Cambridge University Press.

van Es, E. A., & Sherin, M. G. (2008). Mathematics teachers’ “learning to notice” in the context of a video club. Teaching and Teacher Education, 24(2), 244–276.

von Glasersfeld, E. (1995). Radical Constructivism: A way of knowing and learning. The Falmer Press.

Walkoe, J. (2015). Exploring teacher noticing of student algebraic thinking in a video club. Journal of Mathematics Teacher Education, 18(6), 523–550.