Mathematical structure is central to the interconnectedness of numerical, spatial or logical relationships, but it is not known how well teachers understand this concept or implement it in pedagogical practice. In this paper, we examine three junior secondary mathematics teachers’ understanding of mathematical structure and observe whether they apply it in their teaching. Case study data comprised teacher interviews and mathematics lesson observations. Analysis of interviews and teacher-directed communication (utterances) in lessons utilised an emergent framework for categorising mathematical structure: connections to other learning (C), recognising patterns (R), identifying similarities and differences (I) and generalising and reasoning (G). Findings from the interviews indicated that teachers supported the development of mathematical structure, but the interview responses were not generally reinforced by their utterances in mathematics lesson observations. Analysis of teacher utterances revealed superficial understanding and use of mathematical structure although there was some evidence of CRIG components in the teacher-directed communication for individual cases.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Albert, L. R. (2012). Rhetorical ways of thinking: Vygotskian theory and mathematical learning. New York: Springer Science and Business Media.
Attard, C. (2010). Students’ experiences of mathematics during the transition from primary to secondary school. In L. Sparrow, B. Kissane, & C. Hurst (Eds.), Shaping the future of mathematics education: Proceedings of the 33rd annual conference of the Mathematics Education Research Group of Australasia (pp. 53–60). Fremantle: MERGA.
Attard, C. (2011). “My favourite subject is maths. For some reason no-one really agrees with me”: student perspectives of mathematics teaching and learning in the upper primary classroom. Mathematics Education Research Journal, 23, 363–377.
Australian Curriculum, Assessment and Reporting Authority (ACARA) (2012). Mathematics proficiencies. https://www.australiancurriculum.edu.au/resources/mathematics-proficiencies. Accessed 30 March 2017.
Australian Curriculum, Assessment and Reporting Authority (ACARA). (2018). My School. http://www.myschool.edu.au. Accessed 30 April 2019.
Ball, D., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: what makes it special? Journal of Teacher Education, 59, 389–407.
Barnard, T. (1996). Structure in mathematics and mathematical thinking. Mathematics Teaching, 150, 6–10.
Bass, H. (2017). Designing opportunities to learn mathematics theory-building practices. Educational Studies in Mathematics, 95, 229–244.
Bishop, J. P., Lamb, L. L., Philipp, R. A., Whitacre, I., & Schappelle, B. P. (2016). Leveraging structure: logical necessity in the context of integer arithmetic. Journal of Mathematical Thinking and Learning, 18, 209–232.
Blömeke, S., Suhl, U., & Kaiser, G. (2011). Teacher education effectiveness: quality and equity of future primary teachers’ mathematics and mathematics pedagogical content knowledge. Journal of Teacher Education, 2, 154–171.
Boaler, J. (2015). What’s math got to do with it?: how teachers and parents can transform mathematics learning and inspire success. New York: Penguin.
Board of Studies NSW. (2012). Mathematics K-10 Syllabus. https://syllabus.nesa.nsw.edu.au/mathematics. Accessed 21 May 2017.
Borko, H., & Livingston, C. (1989). Cognition and improvisation: differences in mathematics instruction by expert and novice teachers. American Educational Research Journal, 4, 473–498.
Buchholtz, N. F. (2017). The acquisition of mathematics pedagogical content knowledge in university mathematics education courses: results of a mixed methods study on the effectiveness of teacher education in Germany. ZDM—The International Journal on Mathematics Education, 49, 249–264.
Cavanagh, M. (2006). Mathematics teachers and working mathematically: Responses to curriculum change. In P. Grootenboer, R. Zevenbergen, & M. Chinnappan (Eds.), Identities, cultures and learning space: Proceedings of the 29th Annual Conference of the Mathematics Education Research Group of Australasia (pp. 115–122). Canberra, Australia: MERGA.
Choy, B. H., & Dindyal, J. (2017). Noticing affordances of a typical problem. In B. Kaur, W. K. Ho, T. L. Toh, & B. H. Choy (Eds.), Proceedings of the 41st Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 249–256). Singapore: PME.
Cohen, L., Manion, L., & Morrison, K. (2013). Research methods in education. London: Routledge.
Cragg, L., Keeble, S., Richardson, S., Roome, H. E., & Gilmore, C. (2017). Direct and indirect influences of executive functions on mathematics achievement. Cognition, 162, 12–26.
Dinham, S. (2013). The quality teaching movement in Australia encounters difficult terrain: a personal perspective. Australian Journal of Education, 57(2), 91–106.
Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.), Advanced mathematical thinking (pp. 95–126). Dordrecht: Kluwer.
Fischbein, E., & Muzicant, B. (2002). Richard Skemp and his conception of relational and instrumental understanding: open sentences and open phrases. In D. Tall & M. Thomas (Eds.), Intelligence, learning and understanding in mathematics: a tribute to Richard Skemp (pp. 49–77). Flaxton: Pressed Publishers.
Gray, E. M., & Tall, D. O. (1994). Duality, ambiguity, and flexibility: a “proceptual” view of simple arithmetic. Journal for Research in Mathematics Education, 25, 116–140.
Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: an introductory analysis. In J. Hierbert (Ed.), Conceptual and procedural knowledge: the case of mathematics. Hillsdale: Erlbaum.
Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42, 371–406.
Ivars, P., Fernández-Verdú, C., Llinares, S., & Choy, B. H. (2018). Enhancing noticing: using a hypothetical learning trajectory to improve pre-service primary teachers’ professional discourse. Eurasia Journal of Mathematics, Science and Technology Education, 14, 11–16.
Jacobs, V., Lamb, L., & Philipp, R. (2010). Professional noticing of children’s mathematical thinking. Journal for Research in Mathematics Education, 4, 169–202.
Jones, D., & Bush, W. S. (1996). Mathematical structures: answering the “why” questions. The Mathematics Teacher, 716–722.
Kaiser, G., Blömeke, S., König, J., Busse, A., Döhrmann, M., & Hoth, J. (2017). Professional competencies of (prospective) mathematics teachers—cognitive versus situated approaches. Educational Studies in Mathematics, 94, 161–182.
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). Cham: Springer.
Lokan, J., McRae, B., & Hollingsworth, H. (2003). Teaching mathematics in Australia: results from the TIMSS 1999 video study. https://research.acer.edu.au/timss_video/4. Accessed 30 June 2017.
Mason, J. (2002). Researching your own practice: the discipline of noticing. London: Routledge Falmer.
Mason, J. (2004). Doing ≠ construing and doing + discussing ≠ learning: the importance of the structure of attention. Copenhagen: Tenth International Congress on Mathematics Education http://math.unipa.it/~grim/YESS5/ICME%2010%20Lecture%20Expanded.pdf. Accessed 21 November 2017.
Mason, J. (2008). Making use of children’s powers to produce algebraic thinking. In J. Kaput, D. Carraher, & M. Blanton (Eds.), Algebra in the early grades (pp. 57–94). Mahwah: Lawrence Erlbaum.
Mason, J. (2011). Noticing: Roots and branches. In M. Sherin, V. Jacobs, & R. Philipp (Eds.), Mathematics teacher noticing: seeing through teachers' eyes (pp. 65–80). New York: Routledge.
Mason, J. (2017). Probing beneath the surface of experience. In E. Schack, M. Fisher, & J. Wilhelm (Eds.), Teacher noticing: bridging and broadening perspectives, contexts, and frameworks. Research in Mathematics Education (pp. 1–17). Springer International Publishing.
Mason, J. (2018). Structuring structural awareness: a commentary on Chapter 13. In M. G. Bartolini & H. X. Sun (Eds.), Building the foundation: whole numbers in the primary grades (pp. 325–340). Cham: Springer.
Mason, J., Burton, L., & Stacey, K. (1982). Thinking mathematically (2nd ed.). Harlow: Pearson Education Limited.
Mason, J., Stephens, M., & Watson, A. (2009). Appreciating mathematical structure for all. Mathematics Education Research Journal, 21, 10–32.
Miller, K. F. (2011). Situation awareness in teaching: what educators can learn from video-based research in other fields. In M. Sherin, V. Jacobs, & R. Philipp (Eds.), Mathematics teacher noticing: Seeing through teachers’ eyes (p. 81). 95: Routledge.
Mulligan, J., & Mitchelmore, M. (2009). Awareness of pattern and structure in early mathematical development. Mathematics Education Research Journal, 21, 33–49.
Mullis, I. V. S., Martin, M. O., Goh, S., & Cotter, K. (2015). TIMSS 2015 encyclopedia, education policy and curriculum in mathematics and science. In TIMSS 2015 Encyclopedia, Education Policy and Curriculum in Mathematics and Science: TIMSS & PIRLS international study center, Boston College.
QSR International NVivo 12 [Computer software]. (2018). http://www.qsrinternational.com. Accessed 30 January 2019.
Richland, L. E., Stigler, J. W., & Holyoak, K. J. (2012). Teaching the conceptual structure of mathematics. Educational Psychologist, 47, 189–203.
Saeki, E., Pendergast, L., Segool, N. K., & von der Embse, N. P. (2015). Potential psychosocial and instructional consequences of the common core state standards: implications for research and practice. Contemporary School Psychology, 19(2), 89–97.
Scheiner, T. (2016a). Teacher noticing: Enlightening or blinding? ZDM —The International Journal on Mathematics Education, 48, 227–238.
Scheiner, T. (2016b). New light on old horizon: Constructing mathematical concepts, underlying abstraction processes, and sense making strategies. Educational Studies in Mathematics, 91, 165–183.
Scheiner, T. (2020). Dealing with opposing theoretical perspectives: Knowledge in structures or knowledge in pieces? Educational Studies in Mathematics, 104(1), 127–145.
Scheiner, T., & Pinto, M. M. (2019). Emerging perspectives in mathematical cognition: contextualizing, complementizing, and complexifying. Educational Studies in Mathematics, 101(3), 1–16.
Schmidt, W., Houang, R., & Cogan, L. (2002). A coherent curriculum: the case of mathematics. Journal of Direct Instruction, 4(1), 13–28.
Schwarz, B. B., Dreyfus, T., & Hershkowitz, R. (2009). The nested epistemic actions model for abstraction in context. In B. B. Schwarz, T. Dreyfus, & R. Hershkowitz (Eds.), Transformation of knowledge through classroom interaction (pp. 11–41). New York: Routledge.
Sherin, M. G., Jacobs, V. R., & Philipp, R. A. (2011). Mathematics teacher noticing: seeing through teachers' eyes. New York: Routledge.
Shulman, L. S. (1987). Knowledge and teaching: foundations of the new reform. Harvard Educational Review, 57(1), 1–22.
Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20–26.
Smith, P., Ladewig, M., & Prinsley, R. (2018). Improving the mathematics performance of Australian students. Canberra: Commonwealth of Australia https://mawainc.org.au/wp2/wp-content/uploads/2018/07/OCS_Occassional_Paper_July_2018.pdf. Accessed 30 September 2018.
Stephens, M. (2008). Designing questions to probe relational or structural thinking in arithmetic. Paper presented at the 5th Annual Conference of the International Society for Design and Development in Education, Cairns, Australia.
Stephens, M., & Armanto, D. (2010). How to build powerful learning trajectories for relational thinking in the primary school years. In L. Sparrow, B. Kissane, & C. Hurst (Eds.), Shaping the future of mathematics education: proceedings of the 33rd annual conference of the Mathematics Education Research Group of Australasia (pp. 523–530). Fremantle: MERGA.
Taylor, H. E., & Wade, T. L. (1965). On the meaning of structure in mathematics. The Mathematics Teacher, 58, 226–231.
Thomson, S., De Bortoli, L., Underwood, C., & Schmid, M. (2019). PISA 2018: Reporting Australia’s Results, Volume I Student Performance. Camberwell, VIC: Australian Council for Educational Research (ACER). Retrieved from https://research.acer.edu.au/ozpisa/35. Accessed 15 December 2019.
Vale, C. (2013). Equivalence and relational thinking: opportunities for professional learning. Australian Primary Mathematics Classroom, 18(2), 34–40.
Vale, C., McAndrew, A., & Krishnan, S. (2011). Connecting with the horizon: developing teachers’ appreciation of mathematical structure. Journal of Mathematics Teacher Education, 14(3), 193–212.
Vale, C., Widjaja, W., Doig, B., & Groves, S. (2019). Anticipating students’ reasoning and planning prompts in structured problem-solving lessons. Mathematics Education Research Journal, 31, 1–25.
van Es, E. (2011). A framework for learning to notice student thinking. In M. Sherin, V. Jacobs, & R. Philipp (Eds.), Mathematics teacher noticing: seeing through teachers’ eyes (pp. 134–151). New York: Routledge.
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.
Vincent, J., & Stacey, K. (2008). Do mathematics textbooks cultivate shallow teaching? Applying the TIMSS video study criteria to Australian eighth-grade mathematics textbooks. Mathematics Education Research Journal, 20, 82–107.
Watson, A., & Mason, J. (2005). Mathematics as a constructive activity: learners generating examples. Mahwah: Lawrence Erlbaum Associates, Inc..
Woodward, A., Beswick, K., & Oates, G. (2017). The four proficiency strands plus one? Productive disposition and the Australian curriculum: Mathematics. Paper presented at the Mathematics Association of Victoria 54th Annual Conference, Melbourne, Victoria.
This study was supported by the NSW Catholic Education Commission through the Br. John Taylor Education Research Fellowship awarded to the first author in November 2014. Parts of this analysis were reported in the paper, Teachers’ understanding and use of mathematical structure (Gronow, Mulligan, & Cavanagh, 2017), 40 years on: We are still learning!: Proceedings of the 40th Annual Conference of the Mathematics Education Research Group of Australasia (pp. 286–292). Melbourne, MERGA.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Gronow, M., Mulligan, J. & Cavanagh, M. Teachers’ understanding and use of mathematical structure. Math Ed Res J (2020). https://doi.org/10.1007/s13394-020-00342-x
- Mathematical structure
- Professional learning