Volume 9 · Number 3 · Pages 328–339
Constructivist Model Building: Empirical Examples From Mathematics Education

Catherine Ulrich, Erik S. Tillema, Amy J. Hackenberg & Anderson Norton

Download the full text in
PDF (1009 kB)

> Citation > Similar > References > Add Comment

Abstract

Context: This paper outlines how radical constructivist theory has led to a particular methodological technique, developing second-order models of student thinking, that has helped mathematics educators to be more effective teachers of their students. Problem: The paper addresses the problem of how radical constructivist theory has been used to explain and engender more viable adaptations to the complexities of teaching and learning. Method: The paper presents empirical data from teaching experiments that illustrate the process of second-order model building. Results: The result of the paper is an illustration of how second-order models are developed and how this process, as it progresses, supports teachers to be more effective. Implications: This paper has the implication that radical constructivism has the potential to impact practice.

Key words: Second-order model, radical constructivism, teaching, mathematics education

Citation

Ulrich C., Tillema E. S., Hackenberg A. J. & Norton A. (2014) Constructivist model building: Empirical examples from mathematics education. Constructivist Foundations 9(3): 328–339. http://constructivist.info/9/3/328

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

Similar articles

Tillema E. S., Hackenberg A. J., Ulrich C. & Norton A. (2014) Authors’ Response: Interaction: A Core Hypothesis of Radical Constructivist Epistemology
Bednarz N. & Proulx J. (2011) Ernst von Glasersfeld’s Contribution and Legacy to a Didactique des Mathématiques Research Community
Peschl M. F., Bottaro G., Hartner-Tiefenthaler M. & Rötzer K. (2014) Learning How to Innovate as a Socio-epistemological Process of Co-creation: Towards a Constructivist Teaching Strategy for Innovation
Müller K. H. (2010) The Radical Constructivist Movement and Its Network Formations
Schmidt S. J. (2007) God Has Created Reality, We Create Worlds of Experience: A Speech in Honour of Ernst von Glasersfeld to Mark the Award of the Gregory Bateson Prize, Heidelberg, May

References

Barwell R. (2009) Researchers’ descriptions and the construction of mathematical thinking. Educational Studies in Mathematics 72(2): 255–269. ▸︎ Google︎ Scholar
Brodie K. (2010) Teaching mathematical reasoning in secondary school classrooms. Springer, New York. ▸︎ Google︎ Scholar
Brown C. (2011) Narrow mental content. In: Zalta E. N. (ed.) The Stanford Encyclopedia of Philosophy. http://plato.stanford.edu/archives/fall2011/entries/content-narrow/
Clement J. (2000) Analysis of clinical interviews: Foundations and model viability. In: Kelly A. E. & Lesh R. A. (eds.) Handbook of research design in mathematics and science education. Lawrence Erlbaum, Mahwah NJ: 547–590. ▸︎ Google︎ Scholar
Cobb P. & Steffe L. P. (1983) The constructivist researcher as teacher and model builder. Journal for Research in Mathematics Education 14(2): 83–94. ▸︎ Google︎ Scholar
Cobb P. & Yackel E. (1996) Constructivist, emergent, and sociocultural perspectives in the context of developmental research. Educational Psychologist 31(3/4): 175–190. ▸︎ Google︎ Scholar
Cobb P. (2000) Conducting teaching experiments in collaboration with teachers. In: Kelly A. E. & Lesh R. (eds.) Handbook of research design in mathematics and science education. Lawrence Erlbaum, Mahwah NJ: 307–333. ▸︎ Google︎ Scholar
Cobb P., Stephan M., McKlain K. & Gravemeijer K. (2001) Participating in classroom mathematical practices. The Journal of the Learning Sciences 21(1/2): 113–163. ▸︎ Google︎ Scholar
Cobb P., Wood T. & Yackel E. (1990) Classrooms as learning environments for teachers and researchers. In: Davis R. & Noddings N. (eds.) Constructivist views on the teaching and learning of mathematics. National Council of Teachers of Mathematics, Reston VA: 125–146. ▸︎ Google︎ Scholar
Confrey J. & Lachance A. (2000) Transformative teaching experiments through conjecture-driven research design. In: Kelly A. E. & Lesh R. (eds.) Handbook of research design in mathematics and science education. Lawrence Erlbaum, Mahwah NJ: 231–265. ▸︎ Google︎ Scholar
D’Ambrosio B. & Kastberg S. (2012) Giving reason to prospective teachers. For the Learning of Mathematics 32(3): 22–27. ▸︎ Google︎ Scholar
D’Ambrosio B. (1998) Using research as a stimulus for learning. In: Teppo A. (ed.) Qualitative research methods in mathematics education. National Council of Teachers of Mathematics, Reston VA: 144–155. ▸︎ Google︎ Scholar
D’Ambrosio B. (2004) Preparing teachers to teach mathematics within a constructivist framework: The importance of listening to children. In: Watanabe T. & Thompson D. (eds.) The work of mathematics teacher educators. Association of Mathematics Teacher Educators, San Diego CA: 135–150. ▸︎ Google︎ Scholar
Dewey J. (1933) How we think: A restatement of the relation of reflective thinking to the educative process. D. C. Heath, Lexington MA. ▸︎ 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
Dunphy E., Dooley T., Shiel G., Butler D., Corcoran D., Ryan M. & Travers J. (in press) Mathematics in early childhood and primary education (3–8 years): Definitions, theories, development and progression. NCCA Report 17. National Council for Curriculum and Assessment, Dublin. ▸︎ Google︎ Scholar
Glasersfeld E. von & Kelley M. F. (1982) On the concepts of period, phase, stage, and level. Human Development 25: 152–160. ▸︎ Google︎ Scholar
Glasersfeld E. von (1989) Commentaires subjectifs par un observateur. In: Bednarz N. & Garnier C. (eds.) Construction des savoirs. Agence d’Arc, Montreal: 367–371. ▸︎ Google︎ Scholar
Glasersfeld E. von (1995) A constructivist approach to teaching. In: Steffe L. P. & Gale J. (eds.) Constructivism in education. Lawrence Erlbaum, Hillsdale NJ: 3–15. http://www.vonglasersfeld.com/172
Glasersfeld E. von (1995) Radical constructivism. Falmer Press, London. ▸︎ Google︎ Scholar
Glasersfeld E. von (2001) Scheme theory as a key to the learning paradox. In: Philipp A. & Vonèche J. (eds.) Working with Piaget: Essays in honour of Bärbel Inhelder. Psychology Press, London: 139–146. http://www.vonglasersfeld.com/240
Hackenberg A. J. & Tillema E. S. (2009) Students’ whole number multiplicative concepts: A critical constructive resource for fraction composition schemes. The Journal of Mathematical Behavior 28(1): 1–18. ▸︎ Google︎ Scholar
Hackenberg A. J. (2010) Mathematical caring relations in action. Journal for Research in Mathematics Education 41(3): 236–273. ▸︎ Google︎ Scholar
Hackenberg A. J. (2013) The fractional knowledge and algebraic reasoning of students with the first multiplicative concept. Journal of Mathematical Behavior 32(3): 538–563. ▸︎ Google︎ Scholar
Hackenberg A. J. (2014) Musings on three epistemic algebraic students. In: Moore K. C., Steffe L. P. & Hatfield L. L. (eds.) Epistemic algebraic students: Emerging models of students’ algebraic knowing. University of Wyoming, Laramie WY: 81–124. ▸︎ Google︎ Scholar
Herbst P. G. (2006) Teaching geometry with problems: Negotiating instructional situations and mathematical tasks. Journal for Research in Mathematics Education 37(4): 313–347. ▸︎ Google︎ Scholar
Hodkowski N., Tzur R., Johnson H. L. & McClintock E. (in press) Relating student outcomes to teacher development of student-adaptive pedagogy. In: Proceedings of the 38th Conference of the International Group for the Psychology of Mathematics Education. Vancouver BC. ▸︎ Google︎ Scholar
Kitchener R. F. (1993) Piaget’s epistemic subject and science education: Epistemological vs. psychological issues. Science & Education 2: 137–148. ▸︎ Google︎ Scholar
Lampert M. (1990) When the problem is not the question and the solution is not the answer. American Educational Research Journal 27: 29–63. ▸︎ Google︎ Scholar
Langrall C. W., Mooney E. S., Nisbet S. & Jones G. A. (2008) Elementary students’ access to powerful mathematical ideas. In: English L. (ed.) Handbook of international research in mathematics education. Second edition. Routledge, New York: 109–135. ▸︎ Google︎ Scholar
Maheux J. F. (2010) How do we know? An epistemological journey in the day-to-day, moment to-moment, of researching, Teaching and learning in mathematics education. Unpublished doctoral dissertation. Victoria University. ▸︎ Google︎ Scholar
Maturana H. R. & Poerksen B. (2004) From being to doing: The origins of the biology of cognition. Translated by Wolfram K. Köck and Annemarie R. Köck. Carl-Auer, Heidelberg. ▸︎ Google︎ Scholar
Maturana H. R. & Varela F. J. (1992) The tree of knowledge. Revised Edition. Shambhala, Boston. ▸︎ Google︎ Scholar
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. American Society for Cybernetics (ASC) conference workbook. ▸︎ Google︎ Scholar
McClain K. (2005) A methodology of classroom teaching experiments. In: Goodchild S. & English L. (eds.) Researching mathematics classrooms. Information Age Publishing, Charlotte NC: 91–111. ▸︎ Google︎ Scholar
McCloskey A. (2012) Caring in professional development projects for mathematics teachers: An example of stimulation and harmonizing. For the Learning of Mathematics 32(3): 28–33. ▸︎ Google︎ Scholar
Mercer N. & Dawes L. (2008) The value of exploratory talk. In: Mercer N. & Hodgkinson S. (eds.) Exploring talk in school. Sage, London: 55–71. ▸︎ Google︎ Scholar
Nathan M. J., Kim S. & Elian B. (2009) Methodological considerations for the study of intersubjectivity among participants ▸︎ Google︎ Scholar
Norton A. & Boyce S. (2013) A cognitive core for common state standards. Journal for Mathematical Behavior 32: 266–279. ▸︎ Google︎ Scholar
Norton A. & D’Ambrosio B. S. (2008) ZPC and ZPD: Zones of teaching and learning. Journal for Research in Mathematics Education 39: 220–246. ▸︎ Google︎ Scholar
Norton A. & McCloskey A. (2008) Teaching experiments and professional development. Journal of Mathematics Teacher Education 11(4): 285–305. ▸︎ Google︎ Scholar
Norton A. & Wilkins J. L. M. (2012) The splitting group. Journal for Research in Mathematics Education 43(5): 557–583. ▸︎ Google︎ Scholar
of a dialogic mathematics classroom. In: Schwartz B., Dreyfus T. & Hershkowitz R. (eds.) Transformation of knowledge through classroom interaction. Routledge, New York: 244–260. ▸︎ Google︎ Scholar
Pedemonte B. (2007) How can the relationship between argumentation and proof be analysed? Educational Studies in Mathematics 66: 23–41. ▸︎ Google︎ Scholar
Piaget J. (1966) General conclusions. In: Beth E. W. & Piaget J. (eds.) Mathematical epistemology and psychology. Springer, Dordrecht: 305−312. ▸︎ Google︎ Scholar
Piaget J. (1970) Genetic epistemology. Columbia University Press, New York. ▸︎ Google︎ Scholar
Piaget J. (1970) Piaget’s theory. In: Mussen P. H. (ed.) Carmichael’s manual of child psychology. Wiley, New York: 703–732. ▸︎ Google︎ Scholar
Piaget J. (1974) Biology and knowledge: An essay on the relations between organic regulations and cognitive processes. University of Chicago Press, Chicago. ▸︎ Google︎ Scholar
Piaget J. (1987) Possibility and necessity: Volume 1. The role of possibility in cognitive development. University of Minnesota Press, Minneapolis. ▸︎ Google︎ Scholar
Proulx J. & Maheux J.-F. (2012) Épistémologie et didactique des mathématiques: questions anciennes, nouvelles questions. For the Learning of Mathematics 32(2): 41–46. ▸︎ Google︎ Scholar
Putnam H. (1975) The meaning of “meaning.” Minnesota Studies in the Philosophy of Science 7: 131–193. ▸︎ Google︎ Scholar
Riegler A. (2001) Towards a radical constructivist understanding of science. Foundations of Science 6(1): 1–30. http://www.univie.ac.at/constructivism/riegler/20
Ron G., Dreyfus T. & Hershkowitz R. (2010) Partially correct constructs illuminate students’ inconsistent answers. Educational Studies in Mathematics 75(1): 65–87. ▸︎ Google︎ Scholar
Rowland T. (2000) The pragmatics of mathematics education: Vagueness in mathematical discourse. Falmer Press, London. ▸︎ Google︎ Scholar
Rowland T., Huckstep P. & Thwaites A. (2005) Elementary teachers’ mathematics subject knowledge: The knowledge quartet and the case of Naomi. Journal of Mathematics Teacher Education 8(3): 255−281. ▸︎ Google︎ Scholar
Sfard A. (1991) On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics 22: 1−36. ▸︎ Google︎ Scholar
Silverman J. & Thompson P. W. (2008) Toward a framework for the development of mathematical knowledge for teaching. Journal of Mathematics Teacher Education 11(6): 499–511. ▸︎ Google︎ Scholar
Simon M. A. & Tzur R. (1999) Explicating the teacher’s perspective from the researchers’ perspectives: Generating accounts of mathematics teachers’ practice. Journal for Research in Mathematics Education 30: 252–264. ▸︎ Google︎ Scholar
Simon M. A. & Tzur R. (2004) Explicating the role of mathematical tasks in conceptual learning: An elaboration of the hypothetical learning trajectory. Mathematical Thinking and Learning 6(2): 91–104. ▸︎ Google︎ Scholar
Simon M. A. (1994) Learning mathematics and learning to teach: Learning cycles in mathematics teacher education. Educational Studies in Mathematics 26: 71−94. ▸︎ Google︎ Scholar
Simon M. A. (2008) The challenge of mathematics teacher education in an era of mathematics education reform. In: Jaworksi B. & Wood T. (eds.) The mathematics teacher educator as a developing professional. Sense, Rotterdam: 17–29. ▸︎ Google︎ Scholar
Simon M. A. (2013) Developing theory for design of mathematical task sequences:
Conceptual learning as abstraction. In: Margolinas C. (ed.) Task design in mathematics education. Proceedings of ICMI Study 22: 503–510. http://hal.archives-ouvertes.fr/docs/00/83/74/88/PDF/ICMI_STudy_22_proceedings_2013-FINAL_V2.pdf.
Simon M. A. (2013) Issues in theorizing mathematics learning and teaching: A contrast between learning through activity and DNR research programs. The Journal of Mathematical Behavior 32(3): 281–294. ▸︎ Google︎ Scholar
Simon M. A. (2013) Promoting fundamental change in mathematics teaching: A theoretical, methodological, and empirical approach to the problem. ZDM Mathematics Education 45(5): 573–582. ▸︎ Google︎ Scholar
Simon M. A., Saldanha L., McClintock E., Karagoz Akar G., Watanabe T. & Ozgur Zembat I. (2010) A developing approach to studying students’ learning through their mathematical activity. Cognition and Instruction 28(1): 70–112. ▸︎ Google︎ Scholar
Simon M. A., Tzur R., Heinz K. & Kinzel M. (2004) Explicating a mechanism for conceptual learning: Elaborating the construct of reflective abstraction. Journal for Research in Mathematics Education 35(5): 305–329. ▸︎ Google︎ Scholar
Simon M. A., Tzur R., Heinz K., Kinzel M. & Schwan Smith M. (2000) Characterizing a perspective underlying the practice of mathematics teachers in transition. Journal for Research in Mathematics Education 31(5): 579–601. ▸︎ Google︎ Scholar
Simon R. (1985) A frog’s eye view of the world. Structure is destiny: An interview with Humberto Maturana. The Family Therapy Networker 9(3): 32–37, 41–43. ▸︎ Google︎ Scholar
Slavitt D. (1999) The role of operation sense in transitions from arithmetic to algebraic thought. Educational Studies in Mathematics 37(3): 251–274. ▸︎ Google︎ Scholar
Steffe L. P. & Cobb P. (1988) Construction of arithmetical meanings and strategies. Springer, New York NY. ▸︎ Google︎ Scholar
Steffe L. P. & D’Ambrosio B. (1995) Toward a working model of constructivist teaching: A reaction to Simon. Journal for Research in Mathematics Education 26: 146–159. ▸︎ Google︎ Scholar
Steffe L. P. & Olive J. (2010) Children’s fractional knowledge. Springer, New York NY. ▸︎ Google︎ Scholar
Steffe L. P. & Thompson P. W. (2000) Interaction or intersubjectivity? A reply to Lerman. Journal for Research in Mathematics Education 31(2): 191–209. ▸︎ Google︎ Scholar
Steffe L. P. & Thompson P. W. (eds.) (2000) Radical constructivism in action: Building on the pioneering work of Ernst von Glasersfeld. Falmer Press, London. ▸︎ Google︎ Scholar
Steffe L. P. & Tzur R. (1994) Interaction and children’s mathematics. Journal of Research in Childhood and Education 8(2): 99–116. ▸︎ Google︎ Scholar
Steffe L. P. & Ulrich C. (2014) The constructivist teaching experiment. In: Lerman S. (ed.) Encyclopedia of mathematics education. Springer, Berlin. ▸︎ Google︎ Scholar
Steffe L. P. (1990) Adaptive mathematics teaching. In: Cooney T. J. & Hirsch C. R. (eds.) Teaching and learning mathematics in the 1990s. National Council of Teachers of Mathematics, Reston VA: 41–51. ▸︎ Google︎ Scholar
Steffe L. P. (1991) The constructivist teaching experiment: Illustrations and implications. In: Glasersfeld E. von (ed.) Radical constructivism in mathematics education. Kluwer, New York: 177–194. ▸︎ Google︎ Scholar
Steffe L. P. (1992) Schemes of action and operation involving composite units. Learning and Individual Differences 4(3): 259–309. ▸︎ Google︎ Scholar
Steffe L. P. (1994) Children’s multiplying schemes. In: Harel G. & Confrey J. (eds.) The development of multiplicative reasoning in the learning of mathematics. SUNY Press, New York: 3–39. ▸︎ 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. Kluwer, Dordrecht: 79–99. ▸︎ Google︎ Scholar
Steffe L. P. (2000) Perspectives on issues concerning the self, paideia, constraints, viability, and ethics. In: Steffe L. P. & Thompson P. W. (eds.) Radical constructivism in action: Building on the pioneering work of Ernst von Glasersfeld. Routledge, London: 91–102. ▸︎ Google︎ Scholar
Steffe L. P. (2002) A new hypothesis concerning children’s fractional knowledge. Journal of Mathematical Behavior 20(3): 267–307. ▸︎ Google︎ Scholar
Steffe L. P. (2004) On the construction of learning trajectories of children: The case of commensurate fractions. Mathematical Thinking and Learning 6(2): 129–162. ▸︎ Google︎ Scholar
Steffe L. P. (2010) Perspectives on collaborative research in mathematics education with interdisciplinary connections. In: Chamberlin S. A. & Hatfield L. L. (eds.) New perspectives and directions for collaborative research in mathematics education: Papers from a planning conference for WISDOMe. WISDOMe Monograph Volume 1. College of Education, University of Wyoming, Laramie WY: 11–28. http://www.uwyo.edu/wisdome/publications/monographs/PerspectivesOnCollaboration_Steffe.pdf
Steffe L. P., Glasersfeld E. von, Richards J. & Cobb P. (1983) Children’s counting types: Philosophy, theory and application. Praeger, New York NY. ▸︎ Google︎ Scholar
Steffe L. P., Liss D. R. & Lee H. Y. (2014) On the operations that generate intensive quantity. In: Moore K. C., Steffe L. P. & Hatfield L. L. (eds.) Epistemic algebraic students: Emerging models of students’ algebraic knowing. University of Wyoming, Laramie WY: 49–79. ▸︎ Google︎ Scholar
Thompson P. W. (2000) Radical constructivism: Reflections and directions. In: Steffe L. P. & Thompson P. W. (eds.) Radical constructivism in action: Building on the pioneering work of Ernst von Glasersfeld. Falmer Press, London: 291–315. ▸︎ Google︎ Scholar
Thompson P. W. (2011) Quantitative reasoning and mathematical modeling. In: Chamberlain S. A. & Hatfield L. L. (eds.) New perspectives and directions for collaborative research in mathematics education. Volume 1. University of Wyoming Press, Laramie WY: 33–57. ▸︎ Google︎ Scholar
Tillema E. S. (2013) A power meaning of multiplication: Three eighth graders’ solutions of Cartesian product problems. The Journal of Mathematical Behavior 32(3): 331–352. ▸︎ Google︎ Scholar
Tillema E. S. (2014) Students’ power meanings of multiplication. In: Steffe L. P., Moore K. C., Hatfield L. L. & Belbase S. (eds.) Epistemic algebraic students: Emerging models of students’ algebraic knowing. University of Wyoming, Laramie WY: 281–301. ▸︎ Google︎ Scholar
Tzur R. & Lambert M. A. (2011) Intermediate participatory stages as Zone of Proximal Development correlate in constructing counting-on: A plausible conceptual source for children’s transitory “regress” to counting-all. Journal for Research in Mathematics Education 42(5): 418–450. ▸︎ Google︎ Scholar
Tzur R. & Simon M. A. (2004) Distinguishing two stages of mathematics conceptual learning. International Journal of Science and Mathematics Education 2: 287–304. ▸︎ Google︎ Scholar
Tzur R. (1995) Interaction and children’s fractional learning. Unpublished doctoral dissertation, University of Georgia. Dissertation Abstracts International, 56, 3874A. ▸︎ Google︎ Scholar
Tzur R. (2004) Teacher and students’ joint production of a reversible fraction conception. Journal of Mathematical Behavior 23: 93–114. ▸︎ Google︎ Scholar
Tzur R. (2007) Fine grain assessment of students’ mathematical understanding: Participatory and anticipatory stages in learning a new mathematical conception. Educational Studies in Mathematics 66(3): 273–291. ▸︎ Google︎ Scholar
Tzur R. (2008) Profound awareness of the learning paradox (PALP): A journey towards epistemologically regulated pedagogy in mathematics teaching and teacher education. In: Jaworski B. & Wood T. (eds.) The international handbook of mathematics teacher education: The mathematics teacher educator as a developing professional. Volume 4. Sense, Rotterdam: 137–156. ▸︎ Google︎ Scholar
Tzur R., Johnson H. L., McClintock E., Xin Y. P., Si L., Kenney R., Woodward J., Hord C. & Jin X. (2013) Distinguishing schemes and tasks in children’s development of multiplicative reasoning. PNA 7(3): 85–101. ▸︎ Google︎ Scholar
Tzur R., Simon M., Kinzel M. & Heinz K. (2001) An account of a teacher’s perspective on learning and teaching mathematics: Implications for teacher development. Journal of Mathematics Teacher Education 4: 227–254. ▸︎ Google︎ Scholar
Ulrich C. L. (2012) Additive relationships and signed quantities. Unpublished doctoral dissertation, University of Georgia, USA. ▸︎ Google︎ Scholar
Ulrich C. L. (2012) The addition and subtraction of signed quantities. In: Mayes R. & Hatfield L. (eds.) Quantitative reasoning and mathematical modeling. University of Wyoming, Laramie WY: 127–141. ▸︎ Google︎ Scholar
Van de Walle J., Karp K. & Bay-Williams J. M. (2012) Elementary and middle school mathematics: teaching developmentally. Pearson, Upper Saddle River NJ. ▸︎ Google︎ Scholar
Weissglass J. (1990) Constructivist listening for empowerment and change. The Educational Forum 54(4): 351–370. ▸︎ Google︎ Scholar
Wright R. J. (2000) Professional development in recovery education. In: Steffe L. P. & Thompson P. W. (eds.) Radical constructivism in action: Building on the pioneering work of Ernst von Glasersfeld. Falmer Press, London: 134–151. ▸︎ Google︎ Scholar
Wright R. J., Stanger G., Stafford A. K. & Martland J. (2006) Teaching number: Advancing children’s skills and strategies. Sage, London. ▸︎ Google︎ Scholar

Comments: 0

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