# Competences of Mathematics Teachers in Diagnosing Teaching Situations and Offering Feedback to Students: Specificity, Consistency and Reification of Pedagogical and Mathematical Discourses

## Abstract

In the study we report in this chapter, we investigate the competences of mathematics pre- and in-service teachers in diagnosing situations pertaining to mathematics teaching and in offering feedback to the students at the heart of said situations. To this aim we deploy a research design that involves engaging teachers with situation-specific tasks in which we invite participants to: solve a mathematical problem; examine a (fictional yet research-informed) solution proposed by a student in class and a (fictional yet research-informed) teacher response to the student; and, describe the approach they themselves would adopt in this classroom situation. Participants were 23 mathematics graduates enrolled in a post-graduate mathematics education programme, many already in-service teachers. They responded to a task that involved debating the identification of a tangent line at an inflection point of a cubic function through resorting to the formal definition of tangency or the function graph. Analysis of their written responses to the task revealed a great variation in the participants’ diagnosing and addressing of teaching issues – in this case involving the role of visualisation in mathematical reasoning. We describe this variation in terms of a typology of four interrelated characteristics that emerged from the data analysis: *consistency* between stated beliefs/knowledge and intended practice, *specificity* of the response to the given classroom situation, *reification of pedagogical discourses,* and *reification of mathematical discourses.* We propose that deploying the theoretical construct of these characteristics in tandem with our situation-specific task design can contribute towards the identification – as well as reflection upon and development – of mathematics teachers’ diagnostic competences in teacher education and professional development programmes.

## Notes

### Acknowledgement

This study is partially supported by the grant of an annual *Erasmus Teaching Staff Mobility* programme that has been in place between our institutions since 2002.

## References

- Artigue, M., & Perrin-Glorian, M.-J. (1991). Didactic engineering, research and development tool: Some theoretical problems linked to this duality.
*For the Learning of Mathematics*,*11*(1), 3–17.Google Scholar - Ball, D., Thames, H. M., & Phelps, G. (2008). Content knowledge for teaching.
*Journal of Teacher Education*,*59*(5), 389–407.CrossRefGoogle Scholar - Biza, I., Christou, C., & Zachariades, T. (2008). Student perspectives on the relationship between a curve and its tangent in the transition from Euclidean Geometry to Analysis.
*Research in Mathematics Education*,*10*(1), 53–70.CrossRefGoogle Scholar - Biza, I., Nardi, E., & Joel, G. (2015). Balancing classroom management with mathematical learning: Using practice-based task design in mathematics teacher education.
*Mathematics Teacher Education and Development*,*17*(2), 182–198.Google Scholar - Biza, I., Nardi, E., & Zachariades, T. (2007). Using tasks to explore teacher knowledge in situation-specific contexts.
*Journal of Mathematics Teacher Education*,*10*, 301–309.CrossRefGoogle Scholar - Biza, I., Nardi, E., & Zachariades, T. (2009). Teacher beliefs and the didactic contract on visualization.
*For the Learning of Mathematics*,*29*(3), 31–36.Google Scholar - Brousseau, G. (1997).
*Theory of didactical situations in mathematics*. Dordrecht, The Netherlands/Boston, MA/London, UK: Kluwer Academic Publishers.Google Scholar - Carter, K. (1999). What is a case? What is not a case? In M. A. Lundeberg, B. B. Levin, & H. L. Harrington (Eds.),
*Who learns what from cases and how? The research base for teaching and learning with cases*. Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar - Castela, C. (1995). Apprendre avec et contre ses connaissances antérieures: Un example concret, celui de la tangente.
*Recherches en Didactiques des Mathématiques*,*15*(1), 7–47.Google Scholar - Charmaz, K. (2000). Grounded theory: Objectivist and constructivist methods. In K. Denzin & Y. Lincoln (Eds.),
*Handbook of qualitative research*(pp. 509–535). Thousand Oaks, CA: Sage.Google Scholar - Christiansen, B., & Walter, G. (1986). Task and activity. In B. Christiansen, A.-G. Howson, & M. Otte (Eds.),
*Perspectives on mathematics education: Papers submitted by members of the Bacomet Group*(pp. 243–307). Dordrecht, The Netherlands: D. Reide.CrossRefGoogle Scholar - Dreher, A., Nowinska, E., & Kuntze, S. (2013). Awareness of dealing with multiple representations in the mathematics classroom – A study with teachers in Poland and Germany. In A. M. Lindmeier & A. Heinze (Eds.),
*Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education*(Vol. 2, pp. 249–256). Kiel, Germany: PME.Google Scholar - Erens, R., & Eichler, A. (2013). Belief systems’ change – from preservice to trainee high school teachers on calculus. In A. M. Lindmeier & A. Heinze (Eds.),
*Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education*(Vol. 2, pp. 281–288). Kiel, Germany: PME.Google Scholar - Herbst, P., & Chazan, D. (2003). Exploring the practical rationality of mathematics teaching through conversations about videotaped episodes: The case of engaging students in proving.
*For the Learning of Mathematics*,*23*(1), 2–14.Google Scholar - 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.CrossRefGoogle Scholar - Leatham, K. R., Peterson, B. E., Stockero, S. L., & Van Zoest, L. R. (2015). Conceptualizing mathematically significant pedagogical opportunities to build on student thinking.
*Journal for Research in Mathematics Education*,*46*(1), 88–124.CrossRefGoogle Scholar - Leont'ev, A. (1975).
*Dieyatelinocti, soznaine, i lichynosti [Activity, consciousness, and personality]*. Moskva, Russia: Politizdat.Google Scholar - Mamolo, A., & Pali, R. (2014). Factors influencing prospective teachers’ recommendations to students: Horizons, hexagons, and heed.
*Mathematical Thinking and Learning*,*16*(1), 32–50.CrossRefGoogle Scholar - Markovits, Z., & Smith, M. S. (2008). Cases as tools in mathematics teacher education. In D. Tirosh & T. Wood (Eds.),
*The international handbook of mathematics teacher education*,*Tools and processes in mathematics teacher education*(Vol. 2, pp. 39–65). Rotterdam, The Netherlands: Sense Publishers.Google Scholar - Mason, J., & Johnston-Wilder, S. (2006).
*Designing and using mathematical tasks*. New York, UK: QED Press.Google Scholar - Nardi, E., Biza, I., & Zachariades, T. (2012). Warrant’ revisited : Integrating mathematics teachers’ pedagogical and epistemological considerations into Toulmin’s model of argumentation.
*Educational Studies in Mathematics*,*79*(2), 157–173.CrossRefGoogle Scholar - Nardi, E., Healy, L., Biza, I., & Hassan AhmadAli Fernandes, S. (2016). Challenging ableist perspectives on the teaching of mathematics through situation-specific tasks. In C. Csíkos, A. Rausch, & J. Szitányi (Eds.),
*Proceedings of the 40th Conference of the International Group for the Psychology of Mathematics Education (PME)*(Vol. 3, pp. 347–354). Szeged, Hungary: PME.Google Scholar - Sfard, A. (2008).
*Thinking as communicating. Human development, the growth of discourse, and mathematizing*. New York, NY: Cambridge University Press.CrossRefGoogle Scholar - Shulman, L. S. (1992). Toward a pedagogy of cases. In J. H. Shulman (Ed.),
*Case methods in teacher education*(pp. 1–29). New York, NY: Teachers College Press.Google Scholar - Sierpinska, A. (1987). Humanities students and epistemological obstacles related to limits.
*Educational Studies in Mathematics*,*18*, 371–397.CrossRefGoogle Scholar - Sierpinska, A. (2003). Research in mathematics education: Through a keyhole. In E. Simmt & B. Davis (Eds.),
*Proceedings of the Annual Meeting of Canadian Mathematics Education Study Group*. Acadia University. Wolfville, Nova Scotia: CMESG/GCEDMGoogle Scholar - Speer, M. N. (2005). Issues of methods and theory in the study of mathematics teachers’ professed and attributed beliefs.
*Educational Studies in Mathematics*,*58*(3), 361–391.CrossRefGoogle Scholar - Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity.
*Educational Studies in Mathematics*,*12*, 151–169.CrossRefGoogle Scholar - Thompson, A. (1992). Teachers’ beliefs and conceptions: A synthesis of the research. In D. A. Grouws (Ed.),
*Handbook of research on mathematics teaching and learning*(pp. 122–127). New York, NY: Macmillan.Google Scholar - Tirosh, D., & Wood, T. (Eds.). (2009).
*The international handbook of mathematics teacher education*(Vol. 2). Rotterdam, The Netherlands: Sense publishers.Google Scholar - Turner, F., & Rowland, T. (2011). The knowledge quartet as an organizing framework for developing and deepening teachers’ mathematics knowledge. In T. Rowland & K. Ruthven (Eds.),
*Mathematical knowledge in teaching*(pp. 195–212). London, UK/New York, NY: Springer.CrossRefGoogle Scholar - Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In D. Tall (Ed.),
*Advanced mathematical thinking*(pp. 65–81). Dordrecht, The Netherlands: Kluwer Academic Publishers.Google Scholar - Vosniadou, S., & Verschaffel, L. (2004). Extending the conceptual change approach to mathematics learning and teaching. In L. Verschaffel & S. Vosniadou (Guest Eds.),
*The conceptual change approach to mathematics learning and teaching, Special Issue of Learning and Instruction*,*14*, 445–451.Google Scholar - Watson, A., & Mason, J. (2005).
*Mathematics as a constructive activity: Learners generating examples*. Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar - Watson, A., & Mason, J. (2007). Taken-as-shared: A review of common assumptions about mathematical tasks in teacher education.
*Journal of Mathematics Teacher Education*,*10*(4–6), 205–215.CrossRefGoogle Scholar - Zachariades, T., Nardi, E., & Biza, I. (2013). Using multi-stage tasks in mathematics education: Raising awareness, revealing intended practice. In A. M. Lindmeier & A. Heinze (Eds.),
*Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education (PME)*(Vol. 4, pp. 417–424). Kiel, Germany: PME.Google Scholar - Zaslavsky, O., & Sullivan, P. (Eds.). (2011).
*Constructing knowledge for teaching: Secondary mathematics tasks to enhance prospective and practicing teacher learning*. New York, NY: Springer.Google Scholar - Zaslavsky, O., Watson, A., & Mason, J. (2007). Special issue: The nature and role of tasks in mathematics teachers’ education.
*Journal of Mathematics Teacher Education*,*10*(4–6), 201–440.Google Scholar - Zazkis, R., Sinclair, N., & Liljedahl, P. (2013).
*Lesson play in mathematics education: A tool for research and professional development*. New York, NY: Springer.CrossRefGoogle Scholar