Volume 12 · Number 1 · Pages 59–69
Negotiating Between Learner and Mathematics: A Conceptual Framework to Analyze Teacher Sensitivity Toward Constructivism in a Mathematics Classroom

Philip Borg, Dave Hewitt & Ian Jones

Download the full text in
PDF (1319 kB)

> Citation > Similar > References > Add Comment


Context: Constructivist teachers who find themselves working within an educational system that adopts a realist epistemology, may find themselves at odds with their own beliefs when they catch themselves paying closer attention to the knowledge authorities intend them to teach rather than the knowledge being constructed by their learners. Method: In the preliminary analysis of the mathematical learning of six low-performing Year 7 boys in a Maltese secondary school, whom one of us taught during the scholastic year 2014-15, we constructed a conceptual framework which would help us analyze the extent to which he managed to be sensitive to constructivism in a typical classroom setting. We describe the development of the framework M-N-L (Mathematics-Negotiation-Learner) as a viable analytical tool to search for significant moments in the lessons in which the teacher appeared to engage in what we define as “constructivist teaching” (CT) during mathematics lessons. The development of M-N-L is part of a research program investigating the way low-performing students make mathematical sense of new notation with the help of the software Grid Algebra. Results: M-N-L was found to be an effective instrument which helped to determine the extent to which the teacher was sensitive to his own constructivist beliefs while trying to negotiate a balance between the mathematical concepts he was expected to teach and the conceptual constructions of his students. Implications: One major implication is that it is indeed possible for mathematics teachers to be sensitive to the individual constructions of their learners without losing sight of the concepts that society, represented by curriculum planners, deems necessary for students to learn. The other is that researchers in the field of education may find M-N-L a helpful tool to analyze CT during typical didactical situations established in classroom settings.

Key words: Mathematics teaching, mathematics learning, constructivist teaching, constructivist framework


Borg P., Hewitt D. & Jones I. (2016) Negotiating between learner and mathematics: A conceptual framework to analyze teacher sensitivity toward constructivism in a mathematics classroom. Constructivist Foundations 12(1): 59–69. http://constructivist.info/12/1/059

Export article citation data: Plain Text · BibTex · EndNote · Reference Manager (RIS)

Similar articles

Borg P., Hewitt D. & Jones I. (2016) Authors’ Response: The M-N-L Framework: Bringing Radical Constructivist Theories to Daily Teaching Practices
Kenny V. (2011) Continuous Dialogues II: Human Experience. Ernst von Glasersfeld’s Answers to a Wide Variety of Questioners on the Oikos Web Site 1997–2010
Cifarelli V. V. & Sevim V. (2014) Examining the Role of Re-Presentation in Mathematical Problem Solving: An Application of Ernst von Glasersfeld’s Conceptual Analysis
Borg P. (2016) The University Lecture Room and the School Classroom: Does the Stage Affect the Acting?
Jones I. (2015) Building Bridges that are Functional and Structural


Ackermann E. (1995) Construction and transference of meaning through form. In: Steffe L. P. & Gale J. (eds.) Constructivism in education. Lawrence Erlbaum, Hillsdale: 341–354. ▸︎ Google︎ Scholar
Adey P. (ed.) (2008) Let’s think! handbook: A guide to cognitive acceleration in the primary school. GL Assessment, London. ▸︎ Google︎ Scholar
Adhami M., Johnson D. C. & Shayer M. (1995) Thinking maths: The curriculum materials of the Cognitive Acceleration through Mathematics Education (CAME) project – Teacher’s Guide. CAME Project/King’s College, London. ▸︎ Google︎ Scholar
Adhami M., Robertson A. & Shayer M. (2004) Let’s think through maths! Developing thinking in mathematics with five and six-year-olds. NferNelson, London. ▸︎ Google︎ Scholar
Adhami M., Shayer M. & Twiss S. (2005) Let’s think through maths! 6–9. NferNelson, London. ▸︎ Google︎ Scholar
Barwell R. (2009) Researchers’ descriptions and the construction of mathematical thinking. Educational Studies in Mathematics 72(2): 255–269 http://cepa.info/3731
Berkeley G. (1949) A treatise concerning the principles of human knowledge. In: Luce A. A. & Jessop T. E. (eds.) The works of George Berkeley, bishop of Cloyne. Volume II. Nelson & Co, London. Originally published in 1710. ▸︎ Google︎ Scholar
Bikner-Ahsbahs A. & Prediger S. (2006) Diversity of theories in mathematics education – How can we deal with it? ZDM Mathematics Education 38(1): 52–57. ▸︎ Google︎ Scholar
Borg P. & Hewitt D. (2015) Gaining meaning for expressions with Grid Algebra: Developing the CAPS framework. In: Adams G. (ed.) Proceedings of the British Society for Research into Learning Mathematics 35(2): 1–6. ▸︎ Google︎ Scholar
Brennan K. (2015) Beyond technocentrism: Supporting constructionism in the classroom. Constructivist Foundations 10(3): 289–296 http://constructivist.info/10/3/289
Cicourel A. V. (1973) Cognitive sociology. Penguin, London. ▸︎ Google︎ Scholar
Confrey J. (2000) Leveraging constructivism to apply to systemic reform. Nordisk Mathematik Didaktik 8(3): 7–30 http://cepa.info/3733
Dewey J. (1902) The child and the curriculum. University of Chicago Press, Chicago. ▸︎ Google︎ Scholar
Dooley T. (2010) The construction of mathematical insight by pupils in whole-class conversation. Unpublished doctoral dissertation. University of Cambridge UK. ▸︎ Google︎ Scholar
Engström A. (2014) RC is a theory of learning, not teaching. Constructivist Foundations 9(3): 314–316 http://constructivist.info/9/3/314
Ernest P. (1991) The philosophy of mathematics education. Falmer Press, London. ▸︎ Google︎ Scholar
Even R. & Schwarz B. (2003) Implications of competing interpretations of practice for research and theory in mathematics education. Educational Studies in Mathematics 54: 283–313. ▸︎ Google︎ Scholar
Freire P. (1998) Pedagogy of freedom: Ethics, democracy and civic courage. Rowman & Littlefield, Lanham MD. ▸︎ Google︎ Scholar
Gash H. (2014) Constructing constructivism. Constructivist Foundations 9(3): 302–310 http://constructivist.info/9/3/302
Glasersfeld E. von (1983) Learning as a constructive activity. In: Bergeron J. C. & Herscovics N. (eds.) Proceedings of the 5th Annual Meeting of the North American Group of Psychology in Mathematics Education, Vol. 1. PME-NA, Montreal: 41–101 http://cepa.info/1373
Glasersfeld E. von (1989) Cognition, construction of knowledge, and teaching. Synthese 80(1): 121–140. ▸︎ Google︎ Scholar
Glasersfeld E. von (1989) Constructivism in Education. In: Husen T. & Postlethwaite T. N. (eds.) International encyclopedia of education. Supplement Volume 1. Pergamon Press, Oxford: 162–163 http://cepa.info/1404
Glasersfeld E. von (1990) An exposition of constructivism: Why some like it radical. In: Davis R. B., Maher C. A. & Noddings N. (eds.) Monographs of the Journal for Research in Mathematics Education, #4. NCTM, Reston VA: 19–29 http://cepa.info/1415
Glasersfeld E. von (1990) Distinguishing the observer: An attempt at interpreting Maturana. Methodologia 8: 4 http://cepa.info/1568
Glasersfeld E. von (1991) Abstraction, re-presentation, and reflection – an interpretation of experience and of Piaget’s approach. In: L. P. Steffe (ed.) Epistemological foundations of mathematical. Springer, New York: 45–67 http://cepa.info/1418
Glasersfeld E. von (1991) Introduction. In: Glasersfeld E. von (ed.) Radical constructivism in mathematics education. Kluwer Academic, Dordrecht: xiii−xix. ▸︎ Google︎ Scholar
Glasersfeld E. von (1991) Questions and answers about radical constructivism. In: Pearsall M. K. (ed.) Scope, sequence, and coordination of secondary school science. Volume II: Relevant research. The National Science Teachers Association, Washington DC: 169−182. ▸︎ Google︎ Scholar
Glasersfeld E. von (1995) Radical constructivism: A way of knowing and learning. Falmer Press, London. ▸︎ Google︎ Scholar
Glasersfeld E. von (2001) Radical constructivism and teaching. Published in French in Perspectives 31 (2): 191–204. ▸︎ Google︎ Scholar
Glasersfeld E. von (2009) Partial memories: Sketches from an improbable life. Imprint Academic, Exeter. ▸︎ Google︎ Scholar
Goodchild S. & Sriraman B. (2012) Revisiting the didactic triangle: From the particular to the general. ZDM Mathematics Education 44(5): 581–586. ▸︎ Google︎ Scholar
Henderson D. (1981) Three papers. For the Learning of Mathematics 1(3): 12–15. ▸︎ Google︎ Scholar
Jarvis S. (2004) What is speculative thinking? Revue internationale de philosophie 1(227): 69–83. ▸︎ Google︎ Scholar
Jaworski B. (1994) Investigating mathematics teaching: A constructivist enquiry. Falmer Press, London. ▸︎ Google︎ Scholar
Jaworski B. (2012) Mathematics teaching development as a human practice: Identifying and drawing the threads. ZDM Mathematics Education 44(5): 613‒626. ▸︎ Google︎ Scholar
Jonassen D. (2009) Reconciling a human cognitive architecture. In: Tobias S. & Duffy T. M. (eds.) Constructivist instruction: Success or failure? Routledge, Taylor and Francis, London: 13–33. ▸︎ Google︎ Scholar
Jones I. & Pratt D. (2012) A substituting meaning for the equals sign in arithmetic notating tasks. Journal for Research in Mathematics Education 43(1): 2–33. ▸︎ Google︎ Scholar
Kirschner P. A., Sweller J. & Clark R. E. (2006) Why minimal guidance during instruction does not work: An analysis of the failure of constructivist, discovery, problem-based, experiential, and inquiry-based teaching. Educational Psychologist 41(2): 75–86. ▸︎ Google︎ Scholar
Kolb D. A. (1984) Experiential learning: Experience as the source of learning and development. Prentice-Hall, Englewood Cliffs NJ. ▸︎ Google︎ Scholar
Lave J. & Wenger E. (1991) Situated learning: Legitimate peripheral participation. Cambridge University Press, New York. ▸︎ Google︎ Scholar
Lerman S. (1996) Intersubjectivity in mathematics learning: A challenge to the radical constructivist paradigm. Journal for Research in Mathematics Education 27(2): 133–150 http://cepa.info/2954
Maheux J.-F. & Proulx J. (2015) Doing|mathematics: Analysing data with/in an enactivist-inspired approach. ZDM Mathematics Education 47(2): 211–221. ▸︎ Google︎ Scholar
Maturana H. R (1987) Everything is said by an observer. In: Thompson W. I. (ed.) Gaia: A way of knowing. Lindisfarne Press. Hudson NY: 65–82. ▸︎ Google︎ Scholar
Maturana H. R. & Varela F. J. (1980) Autopoiesis and cognition: The realization of the living. Reidel, Dordrecht. ▸︎ Google︎ Scholar
Maturana H. R. (1978) Biology of language: The epistemology of reality. In: Miller G. & Lenneberg E. (eds.) Psychology and biology of language and thought: Essays in honor of Eric Lenneberg. Academic Press, New York: 27–63 http://cepa.info/549
Maturana H. R. (1988) Ontology of observing: The biological foundations of self-consciousness and the physical domain of existence. In: Donaldson R. E. (ed.) Texts in cybernetic theory: An in-depth exploration of the thought of Humberto Maturana, William T. Powers, and Ernst von Glasersfeld. American Society for Cybernetics conference workbook, Felton CA: IL http://cepa.info/597
Maturana H. R. (1988) Reality: The search for objectivity or the quest for a compelling argument. Irish Journal of Psychology 9(1): 25–82 http://cepa.info/598
Oakeshott M. (1962) The voice of poetry in the conversation of mankind: Rationalism in politics. Basic Books, New York. ▸︎ Google︎ Scholar
Olive J. (1994) Building a new model of mathematics learning. Journal of Research in Childhood Education 8(2): 162–173. ▸︎ Google︎ Scholar
Peschl M. F. (2001) Constructivism, cognition, and science: An investigation of its links and possible shortcomings. Special issue on “The Impact of Radical Constructivism on Science,” edited by A. Riegler. Foundations of Science 6(1–3): 125–161 http://cepa.info/3635
Piaget J. (1985) Equilibration of cognitive structures. University of Chicago Press, Chicago. ▸︎ Google︎ Scholar
Popper K. (1959) Logic of scientific discovery. Hutchinson, London. German original published in 1934. ▸︎ Google︎ Scholar
Postman N. & Weingartner C. (1969) Teaching as a subversive activity. Dell, New York. ▸︎ Google︎ Scholar
Potari D. & Jaworski B. (2002) Tackling complexity in mathematics teacher development: Using the teaching triad as a tool for reflection and enquiry. Journal of Mathematics Teacher Education 5(4): 351–380. ▸︎ Google︎ Scholar
Prediger S., Bikner-Ahsbahs A. & Arzarello F. (2008) Networking strategies and methods for connecting theoretical approaches: First steps toward a conceptual framework. ZDM Mathematics Education 40: 165–178. ▸︎ Google︎ Scholar
Proulx J. (2014) From model building to the observer. Constructivist Foundations 9(3): 341–344 http://constructivist.info/9/3/341
Proulx J. (2015) Mental mathematics and enactment of specific strategies: The case of systems of linear equations. In: Proceedings of PME-NA 37. PME-NA, East Lansing MI: 173–180. ▸︎ Google︎ Scholar
Proulx J. (2015) Mental mathematics with mathematical objects other than numbers: The case of operation on functions. Journal of Mathematical Behavior 39: 156–176. ▸︎ Google︎ Scholar
Rezat S. & Sträßer R. (2012) From the didactical triangle to the socio-didactical tetrahedron: Artefacts as fundamental constituents of the didactical situation. ZDM Mathematics Education 44(5): 641–653. ▸︎ Google︎ Scholar
Richards J. (1991) Mathematical discussions. In: Glasersfeld E. von (ed.) Radical constructivism in mathematics education. Kluwer Academic Publishers, Dordrecht: 13–51. ▸︎ Google︎ Scholar
Scardamalia M. & Bereiter C. (1991) Higher levels of agency for children in knowledge building: A challenge for the design of knowledge media. Journal of the Learning Sciences 1(1): 37–68. ▸︎ Google︎ Scholar
Schoenfeld A. (1983) Problem solving in the mathematics curriculum: A report, recommendations, and an annotated bibliography. The Mathematical Association of America MA Notes No. 1. http://www.maa.org/press/maa-reviews/problem-solving-in-mathematics-curriculum-a-report-recommendations-and-an-annotated-bibliography
Schütz A. (1964) Collected papers II in social theory. Edited and introduced by A. Brodersen. Martinus Nijhoff, Dordrecht. ▸︎ Google︎ Scholar
Shayer M. & Adey P. S. (eds.) (2002) Learning intelligence: Cognitive acceleration across the curriculum from 5 to 15 years. Open University Press, Buckingham. ▸︎ Google︎ Scholar
Simon M. A. (1994) Learning mathematics and learning to teach: Learning cycles in mathematics teacher education. Educational Studies in Mathematics 26(1): 71–94. ▸︎ Google︎ Scholar
Simon M. A. (1995) Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education 26 (2): 114−145 http://cepa.info/3671
Steedman P. H. (1991) There is no more safety in numbers: A new conception of mathematics teaching. In: Glasersfeld E. von (ed.) Radical constructivism in mathematics education. Kluwer Academic Publishers, Dordrecht: 1–12. ▸︎ Google︎ Scholar
Steffe L. P. & Olive J. (2010) Children’s fractional knowledge. Springer, New York. ▸︎ Google︎ Scholar
Steffe L. P. & Tzur R. (1994) Interaction and children’s mathematics. Journal of Research in Childhood Education 8(2): 99–116. ▸︎ Google︎ Scholar
Steffe L. P. & Ulrich C. (2013) Constructivist teaching experiment. In: Lerman S. (ed.) Encyclopedia of mathematics education. Springer, Berlin: 102–109 http://cepa.info/2959
Steffe L. P. & Wiegel H. G. (1996) On the nature of a model of mathematical learning. In: Steffe L. P., Nesher P., Cobb P., Goldin G. A. & Greer B. (eds.) Theories of mathematical learning. Erlbaum, Mahwah NJ: 477‒498. ▸︎ Google︎ Scholar
Steffe L. P. (1991) The constructivist teaching experiment: Illustrations and implications. In: Glasersfeld E. von (ed.) Radical constructivism in mathematics education. Kluwer Academic Publishers, Dordrecht: 177−194 http://cepa.info/2098
Steffe L. P. (1991) The learning paradox: A plausible counterexample. In: Steffe L. P. (ed.) Epistemological foundations of mathematical experience. Springer, New York: 26–44. ▸︎ Google︎ Scholar
Steffe L. P. (1996) Social cultural approaches in early childhood mathematics education: A discussion. In: Mansfield H., Pateman N. A. & Bednarz N. (eds.) Mathematics for tomorrow’s young children: International perspectives on curriculum. Kluwer Academic Publishers, Dordrecht: 79–99. ▸︎ Google︎ Scholar
Steffe L. P. (1999) Intersubjectivity in mathematics learning: A challenge to the radical constructivist paradigm. A reply to Lerman. Chreods 13: 1–16 http://cepa.info/2955
Steffe L. P. (2007) Radical constructivism and school mathematics. In: Larochelle M. (ed.) Key works in radical constructivism. Sense Publishers, Rotterdam: 279–290. ▸︎ Google︎ Scholar
Steffe L. P., Glasersfeld E. von, Richards J. & Cobb P. (1983) Children’s counting types: Philosophy, theory, and application. Praeger Scientific, New York. ▸︎ Google︎ Scholar
Steinbring H. (1998) Elements of epistemological knowledge for mathematics teachers. Journal of Mathematics Teacher Education 1: 157–189. ▸︎ Google︎ Scholar
Suls J. & Wills T. A. (1991) Social comparison: Contemporary theory and research. Lawrence Erlbaum Associates, Hillsdale NJ. ▸︎ Google︎ Scholar
Thompson P. W. (1982) Were lions to speak, we wouldn’t understand. The Journal of Mathematical Behavior 3(2): 147–165 http://cepa.info/3047
Thompson P. W. (1991) Getting ahead, with theories: I have a theory about this. In: Underhill R. & Brown C. (eds.) Proceedings of the annual meeting of the North American chapter, international group for the psychology of mathematics education: Plenary papers. PME-NA, Blacksburg: 240–245 http://cepa.info/3663
Thompson P. W. (2000) Radical constructivism: Reflections and directions. In: Steffe L. P. & Thompson P. W. (eds.) Radical constructivism in action. Routledge Falmer, London: 291–315 http://cepa.info/2971
Thompson P. W. (2013) In the absence of meaning. In: Leatham K. (ed.) Vital directions for research in mathematics education. Springer, New York: 57–93 http://cepa.info/3669
Tobias S. & Duffy T. M. (2009) Constructivist instruction: Success or failure? Routledge, Taylor and Francis, London. ▸︎ Google︎ Scholar
Ulrich C. (2012) Additive relationships and signed quantities. Unpublished Ph.D. dissertation, Department of Mathematics and Science Education, University of Georgia. ▸︎ Google︎ Scholar
van Manen M. (1991) The tact of teaching: The meaning of pedagogical thoughtfulness. SUNY Press, Albany. ▸︎ Google︎ Scholar
Visnovska J. & Cobb P. (2013) Classroom video in teacher professional development program: Community documentational genesis perspective. ZDM Mathematics Education 45: 1017–1029. ▸︎ Google︎ Scholar
Voigt J. (1994) Negotiation of mathematical meaning and learning mathematics. Educational Studies in Mathematics 26(2/3): 275–298. ▸︎ Google︎ Scholar

Comments: 0

To stay informed about comments to this publication and post comments yourself, please log in first.