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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The original version of the task is in Greek. The term didactical in the context of this task, and more broadly in the context of the post-graduate programme attended by the participants, is used to denote pedagogical strategies related to specific mathematical topics (as in, for example, didactics of Calculus). In the programme the term was also used with the sense that it has within the Theory of Didactic Situations (Brousseau, 1997), for example, as in didactic contract.
- 2.
Our use of the term “reification” takes cue from discursive perspectives such as Sfard’s (2008) where reification is defined as the gradual turning of processes into objects. Discourses, Sfard writes, change in a “chain of intermittent expansion and compression” (p. 118). Reification is the key element of compression which can be endogenous – resulting from saming within one particular discourse – and exogenous which “conflates several discourses into one” (p. 122). Reification is a response to what discursive researchers see as our innate “need for closure” (p. 184) in our use of signifiers and brings at least two potent gains: increasing the communicative effectiveness of discourse and increasing the practical effectiveness of discourse. In our analyses, we are interested particularly in the extent, and ways, in which participants’ discourse (for example, as evident in their use of mathematics education terminology) is reified.
- 3.
If f is a function such that there exists f(k)(x0) for every 1 ≤ k ≤ n, the Taylor polynomial of degree n for f at x 0 is the polynomial \( {T}_{n,{x}_0}(x)=\sum \limits_{k=0}^n\frac{f^{(k)}\left({x}_0\right)}{k!}{\left(x-{x}_0\right)}^k. \)
It is the only polynomial (degree at most n) with the property \( \underset{x\to {x}_o}{\lim}\frac{f(x)-{T}_{n,{x}_0}(x)}{{\left(x-{x}_0\right)}^n}=0 \) and it is the best polynomial approximation of f degree n at the point x 0. For n = 1 this polynomial approximation, f(x 0) + f ′(x 0)(x − x 0), is of degree one, therefore a line. The term local straightness, expresses this in visual terms: locally, the linear approximation of the curve and the curve itself are indistinguishable.
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.
Ball, D., Thames, H. M., & Phelps, G. (2008). Content knowledge for teaching. Journal of Teacher Education, 59(5), 389–407.
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.
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.
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.
Biza, I., Nardi, E., & Zachariades, T. (2009). Teacher beliefs and the didactic contract on visualization. For the Learning of Mathematics, 29(3), 31–36.
Brousseau, G. (1997). Theory of didactical situations in mathematics. Dordrecht, The Netherlands/Boston, MA/London, UK: Kluwer Academic Publishers.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Leont'ev, A. (1975). Dieyatelinocti, soznaine, i lichynosti [Activity, consciousness, and personality]. Moskva, Russia: Politizdat.
Mamolo, A., & Pali, R. (2014). Factors influencing prospective teachers’ recommendations to students: Horizons, hexagons, and heed. Mathematical Thinking and Learning, 16(1), 32–50.
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.
Mason, J., & Johnston-Wilder, S. (2006). Designing and using mathematical tasks. New York, UK: QED Press.
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.
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.
Sfard, A. (2008). Thinking as communicating. Human development, the growth of discourse, and mathematizing. New York, NY: Cambridge University Press.
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.
Sierpinska, A. (1987). Humanities students and epistemological obstacles related to limits. Educational Studies in Mathematics, 18, 371–397.
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/GCEDM
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.
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.
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.
Tirosh, D., & Wood, T. (Eds.). (2009). The international handbook of mathematics teacher education (Vol. 2). Rotterdam, The Netherlands: Sense publishers.
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.
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.
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.
Watson, A., & Mason, J. (2005). Mathematics as a constructive activity: Learners generating examples. Mahwah, NJ: Lawrence Erlbaum Associates.
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.
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.
Zaslavsky, O., & Sullivan, P. (Eds.). (2011). Constructing knowledge for teaching: Secondary mathematics tasks to enhance prospective and practicing teacher learning. New York, NY: Springer.
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.
Zazkis, R., Sinclair, N., & Liljedahl, P. (2013). Lesson play in mathematics education: A tool for research and professional development. New York, NY: Springer.
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this chapter
Cite this chapter
Biza, I., Nardi, E., Zachariades, T. (2018). Competences of Mathematics Teachers in Diagnosing Teaching Situations and Offering Feedback to Students: Specificity, Consistency and Reification of Pedagogical and Mathematical Discourses. In: Leuders, T., Philipp, K., Leuders, J. (eds) Diagnostic Competence of Mathematics Teachers. Mathematics Teacher Education, vol 11. Springer, Cham. https://doi.org/10.1007/978-3-319-66327-2_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-66327-2_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-66325-8
Online ISBN: 978-3-319-66327-2
eBook Packages: EducationEducation (R0)