# Primary teachers notice the impact of language on children’s mathematical reasoning

- 1k Downloads
- 2 Citations

## Abstract

Mathematical reasoning is now featured in the mathematics curriculum documents of many nations, but this necessitates changes to teaching practice and hence a need for professional learning. The development of children’s mathematical reasoning requires appropriate encouragement and feedback from their teacher who can only do this if they recognise mathematical reasoning in children’s actions and words. As part of a larger study, we explored whether observation of educators conducting mathematics lessons can develop teachers’ sensitivity in noticing children’s reasoning and consideration of how to support reasoning. In the Mathematical Reasoning Professional Learning Research Program, demonstration lessons were conducted in Australian and Canadian primary classrooms. Data sources included post-lesson group discussions. Observation of demonstration lessons and engagement in post-lesson discussions proved to be effective vehicles for developing a professional eye for noticing children’s individual and whole-class reasoning. In particular, the teachers noticed that children struggled to employ mathematical language to communicate their reasoning and viewed limitations in language as a major barrier to increasing the use of mathematical reasoning in their classrooms. Given the focus of teachers’ noticing of the limitations in some types of mathematical language, it seems that targeted support is required for teachers to facilitate classroom discourse for reasoning.

## Keywords

Language Noticing Reasoning Primary mathematics Mathematical terminology## References

- Adler, J., & Davis, Z. (2011). Modelling teaching in mathematics teacher education and the constitution of mathematics for teaching. In T. Rowland & K. Ruthven (Eds.),
*Mathematical knowledge in teaching*(pp. 139–160). Netherlands: Springer.CrossRefGoogle Scholar - Australian Curriculum Assessment and Reporting Authority [ACARA]. (2013).
*Australian curriculum: mathematics*. Sydney: Author.Google Scholar - Ball, D., & Bass, H. (2003). Making mathematics reasonable in school. In J. Kilpatrick, G. Martin, & D. Schifter (Eds.),
*A research companion to the principles and standards for school mathematics*(pp. 27–45). Reston: National Council of Teachers of Mathematics.Google Scholar - Bowden, J., & Marton, F. (1998).
*The university of learning. Beyond quality and competence*. London: Kogan Page.Google Scholar - Bragg, L. A., Vale, C., Herbert, S., Loong, E., Widjaja, W., Williams, G., & Mousley, J. (2013). Promoting awareness of reasoning in the primary mathematics classroom. In A. McDonough, A. Downton, & L. A. Bragg (Eds.),
*Mathematics of the planet earth: proceedings of the MAV 50th annual conference*(pp. 23–30). Melbourne: Mathematical Association of Victoria.Google Scholar - Bragg, L. A., Loong, E. Y.-K., Widjaja, W., Vale, C., & Herbert, S. (2015). Promoting reasoning through the Magic V task.
*Australian Primary Mathematics Classroom, 20*(2), 10–14.Google Scholar - Brodie, K. (2010).
*Teaching mathematical reasoning in secondary school classrooms*. New York: Springer.CrossRefGoogle Scholar - Carpenter, T. P., & Lehrer, R. (1999). Teaching and learning mathematics with understanding. In E. Fennema & T. A. Romberg (Eds.),
*Mathematics classrooms that promote understanding*(pp. 19–32). Mahwah: Lawrence Erlbaum.Google Scholar - Clarke, D. M., Clarke, D. J., & Sullivan, P. (2012). Reasoning in the Australian curriculum: understanding its meaning and using the relevant language.
*Australian Primary Mathematics Classroom, 17*(3), 28–32.Google Scholar - Clarke, D., Roche, A., Wilkie, K., Wright, V., Brown, J., Downton, A., Horne, M., Knight, R., McDonough, A., Sexton, M., & Worrall, C. (2013). Demonstration lessons in mathematics education: teachers’ observation foci and intended changes in practice.
*Mathematics Education Research Journal, 25*(2), 207–230.CrossRefGoogle Scholar - Clarkson, P. C. (2004). Researching the language for rational explanations in mathematics teaching and learning. In
*AARE 2004 conference papers [computer file] from the proceedings of the annual conference of the Australian association for research in education, 28 November – 2 December 2004*. Melbourne: Australian Association for Research in Education.Google Scholar - Department for Education, England (2013).
*National curriculum in England*:*mathematics programmes of study*. Manchester, UK: Author. https://www.gov.uk/government/collections/national-curriculum. - Herbert, S. (2014). A framework for teachers’ knowledge of mathematical reasoning. In J. Anderson, M. Cavanagh, & A. Prescott (Eds.),
*Curriculum in focus: research guided practice. Proceedings of the 37th annual conference of the mathematics education research group of Australasia*(pp. 702–705). Sydney: MERGA.Google Scholar - Herbert, S., Vale, C., Bragg, L. A., Loong, E., & Widjaja, W. (2015). A framework for primary teachers’ perceptions of mathematical reasoning.
*International Journal of Educational Research, 74*, 26–37.CrossRefGoogle Scholar - Hunter, R. (2006). Structuring the talk towards mathematical inquiry. 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*(Vol. 1, pp. 309–317). Adelaide: MERGA.Google Scholar - Hunter, R. (2008). Facilitating communities of mathematical inquiry. In M. Goos, R. Brown, & K. Makar (Eds.),
*Navigating currents and charting directions: proceedings of the 31st annual conference of the mathematics education research group of Australasia*(Vol. 1, pp. 31–39). Adelaide: MERGA.Google Scholar - Kabasakalian, R. (2007). Language and thought in mathematics staff development: a problem probing protocol.
*Teachers College Record, 109*(4), 1–21.Google Scholar - Kaput, J. (1992). Technology and mathematics education. In D. A. Grouws (Ed.),
*Handbook of research on mathematics teaching and learning*(pp. 515–556). New York: Macmillan.Google Scholar - Kaput, J. & Blanton, M. (1999).
*Algebraic reasoning in the context of elementary mathematics: making it implementable on a massive scale*. Paper presented at the annual meeting of the American Education Research Association, Montreal, Quebec, Canada.Google Scholar - Komatsu, K., & Tsujiyama, Y. (2013). Principles of task design to foster proofs and refutations in mathematical learning: proof problem with diagram.
*Task design in mathematics education: Proceedings of ICMI Study, 22*, 471–479.Google Scholar - Lewis, C., Perry, R., & Hurd, J. (2004). A deeper look at lesson study.
*Educational Leadership, 61*(5), 18–23.Google Scholar - Lo, M. L. (2012).
*Variation theory and the improvement of teaching and learning (a publication in the Acta Universitatis Gothoburgensis series)*. Gothenburg: Gothenburg Studies in Educational Sciences.Google Scholar - Loong, E. (2014). A primary teacher’s developing understanding of mathematical reasoning. In J. Anderson, M. Cavanagh, & A. Prescott (Eds.),
*Curriculum in focus: research guided practice. Proceedings of the 37th annual conference of the mathematics education research group of Australasia*(pp. 706–709). Sydney: MERGA.Google Scholar - Loong, E., Vale, C., Bragg, L., & Herbert, S. (2013). Primary school teachers’ perceptions of mathematical reasoning. In V. Steinle, L. Ball, & C. Bardini (Eds.),
*Yesterday, today and tomorrow: proceedings of the 36th annual conference of the mathematics research group of Australasia*(pp. 466–473). Melbourne: MERGA.Google Scholar - Loucks-Horsley, S., Love, N., Stiles, K. E., Mundry, S., & Hewson, P. W. (2003).
*Designing professional development for teachers of science and mathematics*. Thousand Oaks: Sage.Google Scholar - Marton, F. (2006). Sameness and difference in transfer.
*Journal of the Learning Sciences, 15*(4), 499–535.Google Scholar - Marton, F., & Booth, S. (1997).
*Learning and awareness*. Mahwah: Lawrence Erlbaum.Google Scholar - Marton, F., & Tsui, A. B. M. (2004).
*Classroom discourse and the space of learning*. Mahwah: Lawrence Erlbaum Associates.Google Scholar - Marton, F., Runesson, U., & Tsui, A. B. M. (2004). The space of learning. In F. Marton & A. B. M. Tsui (Eds.),
*Classroom discourse and the space of learning*(pp. 3–40). Mahwah: Lawrence Erlbaum Associates.Google Scholar - Mason, J. (1997). Recognising a possibility to act. In V. Zack, J. Mousley, & C. Breen (Eds.),
*Developing practice: teachers’ inquiry and educational change*(pp. 87–101). Geelong: Deakin University.Google Scholar - Mason, J. (2002).
*Researching your own practice: the discipline of noticing*. London: Routledge Falmer.Google Scholar - Mason, J. (2011). Noticing: roots and branches. In M. G. Sherin, V. R. Jacobs, & R. A. Philipp (Eds.),
*Mathematics teacher noticing: seeing through teachers’ eyes. Studies in mathematical thinking and learning*(pp. 35–50). New York: Routledge.Google Scholar - McCluskey, C., Mulligan, J. T., & Mitchelmore, M. C. (2016). The role of reasoning in the Australian curriculum: mathematics. In B. White, M. Chinnappan, & S. Trenholm (Eds.),
*Opening up mathematics education research: proceedings of the 39th annual conference of the mathematics education research group of Australasia*(pp. 445–452). Adelaide: MERGA.Google Scholar - Ministry of Education, Province of British Columbia. (2007).
*Mathematics K to 7: integrated resource package*. Vancouver: Author.Google Scholar - Ministry of Education Singapore. (2013).
*Primary mathematics teaching and learning syllabus*. Singapore: Author.Google Scholar - Moss, J. (2010). A partnership in induction and mentoring: noticing how we improve our practice.
*Australian Journal of Teacher Education, 35*(7), 43–53.CrossRefGoogle Scholar - National Council of Teachers of Mathematics [NCTM]. (2000).
*Principles and standards for school mathematics*. Reston: National Council of Teachers of Mathematics.Google Scholar - National Governors Association Center for Best Practices Council of Chief State School Officers. (2010).
*Common core state standards for mathematics*. Washington, DC: Author.Google Scholar - New South Wales Board of Studies. (2002).
*Mathematics K-6 syllabus 2002*. Sydney: NSW Board of Studies.Google Scholar - Nicol, C., Bragg, L. A., & Nejad, M. J. (2013). Adapting the task: what preservice teachers notice when adapting mathematical tasks. In A. M. Lindemeier & A. Heinze (Eds.),
*Mathematics learning across the lifespan. Proceedings of the 37th conference of the international group for the psychology of mathematics education*(Vol. 3, pp. 369–376). Kiel: IGPME.Google Scholar - Nunes, T., Bryant, P., Sylva, K., & Barros, R. (2009).
*Development of maths capabilities and confidence in primary school*. London: Department for Children, Schools and Families.Google Scholar - Palius, M. F., & Maher, C. A. (2013). Teachers learning about student reasoning through video study.
*Mediterranean Journal of Mathematics Education, 12*(1–2), 39–55.Google Scholar - Rogers, K. C., & Steele, M. D. (2016). Graduate teaching assistants’ enactment of reasoning-and-proving tasks in a content course for elementary teachers.
*Journal for Research in Mathematics Education, 47*(4), 372–419.CrossRefGoogle Scholar - Rychen, D. S., & Salganik, L. H. (Eds.). (2003).
*Key competencies for a successful life and well-functioning society*. Göttingen: Hogrefe & Huber Publishers.Google Scholar - Sfard, A. (2001). There is more to discourse than meets the ears: looking at thinking as communicating to learn more about mathematical learning.
*Educational Studies in Mathematics, 46*(1/3), 13–57.CrossRefGoogle Scholar - Sfard, A., & Kieran, C. (2001). Cognition as communication: rethinking learning-by-talking through multi-faceted analysis of students’ mathematical interactions.
*Mind, Culture, and Activity, 8*(1), 42–76.CrossRefGoogle Scholar - Sherin, M. G., Jacobs, V. R., & Philipp, R. A. (Eds.). (2011).
*Mathematics teacher noticing: seeing through teachers’ eyes*. New York: Routledge.Google Scholar - Stacey, K. (2010). Mathematics teaching and learning to reach beyond the basics. In C. Glascodine & K.-A. Hoad (Eds.),
*Teaching mathematics? Make it count: what research tells us about effective mathematics teaching and learning*(pp. 17–20). Camberwell: ACER.Google Scholar - Stein, C. C. (2007). Let’s talk. Promoting mathematical discourse in the classroom.
*Mathematics Teacher, 101*(4), 285–289.Google Scholar - Stein, M., Engle, R., Smith, M., & Hughes, E. (2008). Orchestrating productive mathematical discussions: five practices for helping teachers move beyond show and tell.
*Thinking and Learning, 10*(4), 313–340.CrossRefGoogle Scholar - Stylianides, A. J., & Stylianides, G. J. (2006). Content knowledge for mathematics teaching: the case of reasoning and proving. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.),
*Proceedings 30th conference of the international group for the psychology of mathematics education*(Vol. 5, pp. 201–208). Prague: PME.Google Scholar - Vale, C., Widjaja, W., Herbert, S., Bragg, L. A., & Loong, E. Y.-K. (2016). Mapping variation in children’s mathematical reasoning: the case of ‘what else belongs?’.
*International Journal of Science and Mathematics Education*, 1–22. doi: 10.1007/s10763-016-9725-y. - van Es, E. A. (2011). A framework for learning to notice student thinking. In M. G. Sherin, V. R. Jacobs, & R. A. Philipp (Eds.),
*Mathematics teacher noticing: seeing through teachers’ eyes. Studies in mathematical thinking and learning*(pp. 134–151). New York: Routledge.Google Scholar - Visnovska, J., & Cobb, P. (2009). Learning about building mathematics instruction from students’ reasoning: a professional development study. In R. Hunter, B. Bicknell, & T. Burgess (Eds.),
*Proceedings of the 32nd annual meeting of the mathematics education research group of Australasia*(pp. 547–554). Wellington: MERGA.Google Scholar - Weingrad, P. (1998). Teaching and learning politeness for mathematical argument in school. In M. Lampert & M. L. Blunk (Eds.),
*Talking mathematics in school: studies of teaching and learning*(pp. 213–237). Cambridge: Cambridge University Press.CrossRefGoogle Scholar - Wood, T. (1999). Creating a context for argument in mathematics class.
*Journal for Research in Mathematics Education, 30*(2), 171–191.CrossRefGoogle Scholar - Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics.
*Journal for Research in Mathematics Education, 27*(4), 458–477.CrossRefGoogle 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: NCTM.Google Scholar