Abstract
This chapter explores Piaget’s use of formal logic to model children’s reasoning. This is related to, but not the same as, Piaget’s studies of the logic of children’s reasoning. The distinction lies between considering when and how children’s reasoning fits an existing expert model and how to describe the structure of the ways children draw inferences of various types. Logic occupies a large portion of Piaget’s writings, but has not been widely taken up in mathematics education research. This is because logic is less studied in mathematics education (with good reason) and because Piaget’s use of these models was particularly challenging to follow. This chapter tries to elucidate some of what Piaget did in this line of research, drawing particular attention to the role of task design and the subtle ways Piaget reformulates the logical models for the purpose of his modeling of student reasoning. This provides an opportunity to consider the role of expert models more generally in the work of studying student mathematics and to observe the pitfalls and challenges inevitably imposed by this kind of work. One reason I propose for logic’s marginal place in mathematics education research is the gap between student reasoning, which is usually contextual and content-specific, and the features of formal logical models, which are decontextualized and content general. This fundamental gap both increases the likelihood of miscommunicating with such models and for tension between the model and what is being modeled.
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.
Here, I shall use “Piaget” often to refer to Piaget and his colleagues. This is intended for convenience as I do not mean to downplay his colleagues’ contributions or intellectual merit in the shared work. In the case of Beth and Piaget (1966), the book was written in two separate sections by the two authors, though each gave feedback and advice on the refinement of the other section. Though in quotes from that book I may actually mean Piaget as the primary author, I shall usually use his name to stand for the research group.
- 2.
It is beyond the scope of this chapter to cover the variety of logical tools and their conformity to this discussion. I shall focus on the logic most closely associated with Piaget’s own modeling, which is largely first-order logic. Of this, Schroyens (2010) notes, “The degree of idealization in classic first-order logic is the prime reason for anti-psychologism. Contemporary logics are much less idealized (i.e., there is less abstraction, generalization, and/or simplification involved).” (p. 79). The extent to which other logics may be more suitable as modeling tools of human reasoning (as explored by Stenning & van Lambalgen, 2008) is irrelevant to my discussion of Piaget’s work.
- 3.
By which we may mean (1) those aspects of reasoning that generalize across semantic context, (2) reasoning sensitive to syntactic form, or (3) performance on tasks whose intended solution is understood to depend upon logical structure or principles.
- 4.
The bar is a conventional representation of negation, so \( \overline{q} \) is typically read “not q.”
- 5.
He did point out features of tasks that distinguished them from others such as the presence of irrelevant variables as in a pendulum task in Chapter 4 of Inhelder and Piaget (1958).
- 6.
Since these letters represent sets or classes, the prime notation may be interpreted as a complement set.
- 7.
We reproduce the statement as it appears in our translation of the text, though clearly the explanation was meant to have arrows above F and L.
- 8.
Inhelder and Piaget (1958) used italics to quote children and non-italics for exposition and interviewer quotes.
- 9.
In this book, the authors often use a period for “and” or conjunction.
- 10.
As above, this is not what a child said, but rather the rendering of the child’s inference into the language of logic.
- 11.
Indeed, many later scholars have shown how logic models can be misapplied to studies of human reasoning.
References
Beth, E. W., & Piaget, J. (1966). Mathematical epistemology and psychology. D. Reidel Publishing Company.
Burman, J. T. (2016). Piaget’s neo-Gödelian turn: Between biology and logic, origins of the new theory. Theory & Psychology, 26(6), 751–772.
Evans, J. S. B. T. (2002). Logic and human reasoning: An assessment of the deduction paradigm. Psychological Bulletin, 128(6), 978–996.
Evans, J., & Feeny, A. (2004). The role of prior belief in reasoning. In J. P. Leighton & R. J. Sternberg (Eds.), The nature of reasoning (pp. 78–102). Cambridge University Press.
Evans, J., & Over, D. E. (2004). If. Oxford University Press.
Inglis, M. (2006). Dual processes in mathematics: Reasoning about conditionals (Doctoral dissertation). University of Warwick.
Inhelder, B., & Piaget, J. (1958). The growth of logical thinking from childhood to adolescence: An essay on the construction of formal operational structures. Basic Books.
Inhelder, B., & Piaget, J. (1964). The early growth of logic in the child. W. W. Norton.
Markovits, H. (2004). The development of deductive reasoning. In J. P. Leighton & R. J. Sternberg (Eds.), The nature of reasoning (pp. 313–338). Cambridge University Press.
Piaget, J. (1950). The psychology of intelligence. Routledge Classics.
Piaget, J. (1953). Logic and psychology. Manchester University Press.
Piaget, J., & Garcia, R. (1991). Toward a logic of meanings. Lawrence Erlbaum Associates.
Quine, W. V. O. (1960). Letter to Beth, regarding Quine’s reluctance to join the advisory board of Piaget’s International Center for Genetic Epistemology W. V. Quine Papers (MS Am 2587 [852], Box 31). Houghton Library, Harvard University, Cambridge, MA.
Schroyens, W. (2010). Logic and/in psychology: The paradoxes of material implication and psychologism in the cognitive science of human reasoning. In M. Oaksford & N. Chater (Eds.), Cognition and conditionals: Probability and logic in human thinking (pp. 69–84). Oxford University Press.
Stenning, K., & van Lambalgen, M. (2008). Human reasoning and cognitive science. The MIT Press.
Thompson, P. (2013). In the absence of meaning. In K. Leatham (Ed.), Vital directions for research in mathematics education. Springer.
Wason, P. C. (1966). Reasoning. In B. M. Foss (Ed.), New horizons in psychology 1. Pelican.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Dawkins, P.C. (2024). Logic in Genetic Epistemology. In: Dawkins, P.C., Hackenberg, A.J., Norton, A. (eds) Piaget’s Genetic Epistemology for Mathematics Education Research. Research in Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-031-47386-9_10
Download citation
DOI: https://doi.org/10.1007/978-3-031-47386-9_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-47385-2
Online ISBN: 978-3-031-47386-9
eBook Packages: EducationEducation (R0)