# Mediating primary mathematics: theory, concepts, and a framework for studying practice

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## Abstract

In this paper, we present and discuss a framework for considering the quality of primary teachers’ mediating of primary mathematics within instruction. The “mediating primary mathematics” framework is located in a sociocultural view of instruction as mediational, with mathematical goals focused on structure and generality. It focuses on tasks and example spaces, artifacts, inscriptions, and talk as the key mediators of instruction. Across these mediating strands, we note trajectories from error and a lack of coherence, via coherence localized in particular examples or example spaces, towards building a more generalized coherence beyond the specific example space being worked with. Considering primary mathematics teaching in this way foregrounds the nature of the mathematics that is made available to learn, and we explore the affordances of attending to both coherence and structure/generality. Differences in ways of using the framework when either considering the quality of instruction or working to develop the quality of instruction are taken up in our discussion.

## Keywords

Primary mathematics teaching Sociocultural theory Mediation South Africa Artifacts Inscriptions Instructional quality## References

- Adler, J., & Pillay, V. (2007). An investigation into mathematics for teaching: Insights from a case.
*African Journal of Research in Mathematics, Science and Technology Education*,*11*(2), 87–108.CrossRefGoogle Scholar - Adler, J., & Ronda, E. (2015). A framework for describing mathematics discourse in instruction and interpreting differences in teaching.
*African Journal of Research in Mathematics, Science and Technology Education*,*19*(3), 237–254.CrossRefGoogle Scholar - Adler, J., & Venkat, H. (2014). Teachers’ mathematical discourse in instruction: Focus on examples and explanations. In H. Venkat, M. Rollnick, J. Loughran, & M. Askew (Eds.),
*Exploring mathematics and science teachers’ knowledge: Windows into teacher thinking*(pp. 132–146). Abingdon, Oxon: Routledge.Google Scholar - Alexander, R. (2000).
*Culture and pedagogy: International comparisons in primary education*. Oxford: Blackwell.Google Scholar - Andrews, P. (2009). Mathematics teachers’ didactic strategies: Examining the comparative potential of low inference generic descriptors.
*Comparative Education Review*,*53*(4), 559–582.CrossRefGoogle Scholar - Anghileri, J. (1995). Language, arithmetic and the negotiation of meaning.
*For the Learning of Mathematics*,*15*(3), 10–14.Google Scholar - Arzarello, F. (2006). Semiosis as a multimodal process. Relime, Numéro Especial, 267–299.Google Scholar
- Askew, M., Brown, M., Rhodes, V., Johnson, D. C., & Wiliam, D. (1997).
*Effective teachers of numeracy. Report of a study carried out for the teacher training agency 1995–96 by the School of Education, King’s College London*. London: Teacher Training Agency.Google Scholar - Askew, M., Venkat, H., & Mathews, C. (2012). Coherence and consistency in South African primary mathematics lessons. In T. Y. Tso (Ed.),
*Proceedings of the 36th Conference of the International Group for the Psychology of Mathematics Education*(Vol. 2, pp. 27–34). Taipei, Taiwan: PME.Google Scholar - Bakhurst, D. (1991).
*Consciousness and revolution in Soviet philosophy: From the Bolsheviks to Evald Ilyenkov*. Cambridge: Cambridge University Press.CrossRefGoogle Scholar - Cole, M. (1996).
*Cultural psychology: A once and future discipline*. Cambridge: Mass Harvard University Press.Google Scholar - DoE. (2008).
*Foundations for learning campaign. Government gazette. Letter to Foundation Phase and intermediate Phase teachers*. Pretoria: DoE.Google Scholar - Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics.
*Educational Studies in Mathematics*,*61*(1–2), 103–131.CrossRefGoogle Scholar - Edwards, D., & Mercer, N. (1987).
*Common knowledge: The development of understanding in the classroom*. London: Routledge.Google Scholar - Ekdahl, A.-L., Venkat, H., & Runesson, U. (2016). Coding teaching for simultaneity and connections: Examining teachers’ part-whole additive relations instruction.
*Educational Studies in Mathematics.*,*93*(3), 293–313.CrossRefGoogle Scholar - Ensor, P., Hoadley, U., Jacklin, H., Kuhne, C., Schmitt, E., Lombard, A., & Van den Heuvel-Panhuizen, M. (2009). Specialising pedagogic text and time in Foundation Phase numeracy classrooms.
*Journal of Education*,*47*, 5–30.Google Scholar - Goldenberg, P., & Mason, J. (2008). Shedding light on and with example spaces.
*Educational Studies in Mathematics*,*69*, 183–194.CrossRefGoogle Scholar - Gravemeijer, K. (1997). Mediating between the concrete and the abstract. In T. Nunes & P. Bryant (Eds.),
*Learning and teaching mathematics: An international perspective*. Hove: Psychology Press.Google Scholar - Graven, M. (2014). Poverty, inequality and mathematics performance: The case of South Africa’s post-apartheid context.
*ZDM*,*46*, 1039–1049.CrossRefGoogle Scholar - Hill, H., Blunk, M. L., 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*, 430–511.CrossRefGoogle Scholar - Hoadley, U. (2006). Analysing pedagogy: The problem of framing.
*Journal of Education*,*40*, 15–34.Google Scholar - Hoadley, U. (2012). What do we know about teaching and learning in South African primary schools?
*Education as Change*,*16*(2), 187–202.CrossRefGoogle Scholar - Hughes, M. (1986).
*Children and number: Difficulties in learning mathematics*. London: Blackwell Publishing.Google Scholar - Kozulin, A. (2003). Psychological tools and mediated learning. In A. Kozulin, B. Gindis, V. S. Ageyev, & S. S. Miller (Eds.),
*Vygotsky’s educational theory in cultural context*(pp. 15–38). Cambridge: Cambridge University Press.CrossRefGoogle Scholar - Leinhardt, G. (1990).
*Towards understanding instructional explanations. Washington: Office of Educational Research and Improvement*. Retrieved November 1, 2016, from http://files.eric.ed.gov/fulltext/ED334150.pdf - Ma, L. (1999).
*Knowing and teaching elementary mathematics: Teachers’ understandings of fundamental mathematics in China and the United States*. Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar - Marton, F. (2014).
*Necessary conditions of learning*. Oxford: Routledge.Google Scholar - Marton, F., & Booth, S. (1997).
*Learning and awareness*. Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar - Mason, J. (2002). Generalisation and algebra: Exploiting children’s powers. In L. Haggerty (Ed.),
*Aspects of teaching secondary school mathematics: Perspectives on practice*(pp. 105–120). London: RoutledgeFalmer.Google Scholar - Mason, J., & Pimm, D. (1984). Generic examples: Seeing the general in the particular.
*Educational Studies in Mathematics.*,*15*(3), 277–289.CrossRefGoogle Scholar - Mason, J., & Spence, M. (1999). Beyond mere knowledge of mathematics: The importance of knowing-to act in the moment.
*Educational Studies in Mathematics.*,*38*(1-3), 135–161.Google Scholar - Mathews, C. (2014). Teaching division: The importance of coherence in what is made available to learn. In H. Venkat, M. Rollnick, J. Loughran, & M. Askew (Eds.),
*Exploring mathematics and science teachers’ knowledge: Windows into teacher thinking*(pp. 84–95). London: Routledge.Google Scholar - Moyer, P. (2001). Are we having fun yet? How teachers use manipulatives to teach mathematics.
*Education Studies in Mathematics*,*47*(2), 175–197.CrossRefGoogle Scholar - Pritchett, L., & Beatty, A. (2015). Slow down, you’re going too fast: Matching curricula to student skill levels.
*International Journal of Educational Development*,*40*, 276–288.CrossRefGoogle Scholar - Rowland, T., Turner, F., Thwaites, A., & Huckstep, P. (2009).
*Developing primary mathematics teaching: Reflecting on practice with the knowledge quartet*. London: Sage Publications.CrossRefGoogle Scholar - Rowland. (2013). Learning lessons from instruction: Descriptive results from an observational study of urban elementary classrooms.
*Sisyphus Journal of Education*,*1*(3), 15–43.Google Scholar - Sfard, A. (2008).
*Thinking as communicating*. Cambridge: Cambridge University Press.CrossRefGoogle Scholar - Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000).
*Implementing standards-based instruction: A casebook for professional development*. New York: Teachers College Press.Google Scholar - Stigler, J. W., & Hiebert, J. (1999).
*The teaching gap: Best ideas from the world’s teachers for improving education in the classroom*. New York: Free Press.Google Scholar - Tabulawa, R. (2013).
*Teaching and learning in context: Why pedagogical reforms fail in sub-Saharan Africa*. Dakar, Senegal: CODESRIA.Google Scholar - Taylor, N. (2011).
*The National School Effectiveness Study (NSES): Summary for the synthesis report*. Johannesburg: Joint Education Trust.Google Scholar - Venkat, H. (2013, June).
*Curriculum development minus teacher development ≠ mathematics education*. Paper presented at the proceedings of the 19th Annual National Congress of the Association for Mathematics Education of South Africa, Cape Town, University of the Western Cape.Google Scholar - Venkat, H., & Askew, M. (2012). Mediating early number learning: Specialising across teacher talk and tools?
*Journal of Education*,*56*, 67–90.Google Scholar - Venkat, H., & Naidoo, D. (2012). Analyzing coherence for conceptual learning in a grade 2 numeracy lesson.
*Education as Change*,*16*(1), 21–33.CrossRefGoogle Scholar - Venkat, H., & Spaull, N. (2015). What do we know about primary teachers’ mathematical content knowledge in South Africa? An analysis of SACMEQ 2007.
*International Journal of Educational Development*,*41*, 121–130.CrossRefGoogle Scholar - Vygotsky, L. (1987). Thinking and speech. In R. W. Rieber & A. S. Carton (Eds.),
*The collected works of L.S. Vygotsky, volume 1: Problems of general psychology*(N. Minick, Trans.). New York: Plenum.Google Scholar - Walkerdine, V. (1988).
*The mastery of reason: Cognitive development and the production of rationality*. London: Routledge.Google Scholar - Watson, A., & Mason, J. (2005).
*Mathematics as a constructive activity: Learners generating examples*. New York: Lawrence Erlbaum Publishers.Google Scholar - Watson, A., & Mason, J. (2006a). Seeing an exercise as a single mathematical object: Using variation to structure sense-making.
*Mathematical Thinking and Learning*,*8*(2), 91–111.CrossRefGoogle Scholar - Watson, A., & Mason, J. (2006b). Variation and mathematical structure.
*Mathematics Teaching (incorporating Micromath)*,*194*, 3–5.Google Scholar - Wertsch, J. V. (1991).
*Voices of the mind: A sociocultural approach to mediated action*. Cambridge, MA: Harvard University Press.Google Scholar - Wertsch, J. V. (1998).
*Mind as action*. New York: Oxford University Press.Google Scholar