Abstract
In this chapter, we describe reflected abstraction and discuss its implications for empirical research into the teaching and learning of mathematics. We discuss the relationship between reflected abstraction and other theoretical constructs within Piaget’s genetic epistemology, and the mathematics education research literature generally. With this background established, we outline various implications of reflected abstraction for students’ learning of mathematics and teachers’ construction of pedagogical content knowledge. We then consider strategies for promoting reflected abstractions and propose explicit standards of evidence for supporting valid claims about whether an individual has constructed schemes at the level of reflected thought. We conclude by outlining pertinent questions that mathematics education researchers might pursue regarding either the necessary conditions for promoting reflected abstractions or the implications of reflected schemes for students’ mathematical activity and teachers’ instructional practices.
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Notes
- 1.
Because Piaget neither considered cognitive development a process determined by environmental pressures nor biological maturation, his theory is often referred to as a middle-ground position, third possibility, or tertium quid (Gallagher & Reid, 2002, p. 23).
- 2.
We note that pseudo-empirical, reflecting, reflected, and metareflection are non-mutually exclusive forms of abstraction.
- 3.
Ellis, Lockwood, and Paoletti (this volume) provide thorough descriptions of empirical, pseudo-empirical, and reflecting abstraction.
- 4.
A reflected abstraction is a scheme constructed at the level of reflected thought.
- 5.
See the discussion of reflecting abstraction later in this chapter.
- 6.
We use symbol to refer to any object that evokes in an individual an “abstracted representation” of experience and associated meanings (von Glasersfeld, 1995, p. 99). The object may be a stimulus in one’s sensory field (i.e., exogeneous) or a representation of experience (i.e., endogenous).
- 7.
Consider, for example, the common abstraction that a product is always greater than its factors (i.e., “multiplication makes bigger”).
- 8.
Piaget encapsulates differentiation within the projective aspect of reflecting abstraction. We make differentiation explicit because it is one of the key distinctions between reflecting abstraction and pseudo-empirical abstraction.
- 9.
Throughout this chapter, we use “representation” in a manner consistent with von Glasersfeld’s (1995) interpretation of how Piaget applied the term: “For Piaget, representation is always the replay, or re-construction from memory, of a past experience and not a picture of something else, let alone a picture of the real world” (p. 59).
- 10.
For this to be an appropriate method for comparing the “size” (i.e., non-quantitative comparison) of the angles, the student must recognize that the selected point on the terminal must be the same distance away from the vertex of each angle—a fact that the student had the opportunity to pseudo-empirically abstract while engaged in the activity displayed in Figure 8.4.
- 11.
Regarding the décalage, or temporal delay, between reflecting and reflected abstraction, Piaget (2001) explained, “The subject becomes conscious of the result of his acts—which requires a simple static read-off—before becoming conscious of their mechanism and their exact unfolding—which requires the reconstruction of a process” (p. 191).
- 12.
B and R, respectively, represent the length of the blue and red walls; b represents the length of a blue block and r represents the length of a red block.
- 13.
Piaget and Duckworth’s (1968) book On the Development of Memory and Identity necessitates this qualification.
- 14.
Throughout this chapter, we use “represent” (without the hyphen) to refer to the action of reifying a re-presentation of experience in the form of a symbol.
- 15.
Piaget referred to symbols as “internalized imitations,” which are necessarily grounded in the knowing subject’s actions (Ginsburg & Opper, 1988, p. 74).
- 16.
Tillema (this volume) provides a thorough description of scheme from a Piagetian perspective.
- 17.
It is possible for figurative schemes to emerge via empirical and pseudo-empirical abstractions, but such schemes are, by definition, non-operative (see the discussion of figurative and operative knowledge structures below). Logico-mathematical knowledge, on the other hand, must be operative, meaning that the schemes it entails are composed of reversible mental actions that can be applied to a generic class of objects without regard for an initial state.
- 18.
The terminal point of an angle in standard position is the intersection of the terminal ray and a circle centered at the angle’s vertex.
- 19.
Thompson, Byerley, and O’Bryan (this volume) provide a thorough description of imagery from a Piagetian perspective.
- 20.
Also, see the next section where we discuss figurative thought and operative thought.
- 21.
Moore, Stevens, Tasova, and Liang (this volume) provide a detailed description of figurative and operative modes of thought.
- 22.
A ratio is a multiplicative comparison of the measures of two constant (non-varying) quantities, while a rate defines a proportional relationship between varying quantities’ measures (Thompson & Thompson, 1992). Constructing a rate therefore involves images of smooth continuous variation (Thompson & Carlson, 2017), as well as the expectation that as two quantities covary, multiplicative comparisons of their measures remain invariant.
- 23.
Dubinsky (1991) used the term “reflective abstractions,” and so we follow his lead in this section. Piaget used “reflective abstraction” generally to describe abstraction that is not empirical or pseudo-empirical. Its use does not distinguish between reflecting abstractions, the construction of reflected schemes, and meta-reflection. Dubinsky does not himself distinguish between these more nuanced characterizations of abstraction in his theory, although we would argue that his use of “reflective abstraction” might be better interpreted as “reflecting abstraction” consistent with how we’ve described the construct in this chapter.
- 24.
The terms “schemes,” “schemas,” and “schemata” have appeared in English translations of Piaget’s work as plurals of “scheme.”
- 25.
This example is related to quantitative reasoning and covariational reasoning (see the next section).
- 26.
Boyce (this volume) provides a detailed description of quantitative reasoning.
- 27.
See our example in the previous section.
- 28.
Steffe and Thompson (2000) distinguished first- and second-order observers as follows: “first-order observers address what someone understands, while second-order observers address what they understand about what the other person could understand” (p. 303, italics in original).
- 29.
Thompson (2002) defined an epistemic student not as “anyone in particular. Rather, we speak generically of a person, as in ‘Suppose a person understands fractions as ‘so many out of so many …” Images of this type allow us to propose ways of thinking that are not specific to any one person, yet they are not ‘disembodied cognizing’” (p. 195). These images are what we mean by epistemic ways of understanding. Develo** images of multiple ways of understanding students in a class could have is highly useful in hel** teachers react productively in the moment by hel** them recognize the potential way of understanding a student could have that is manifesting in particular behaviors.
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Acknowledgements
Much of what we express in this chapter has its origins in conversations we have had with Patrick Thompson. Throughout this chapter, we acknowledge where our writing was informed by or connects with Pat Thompson’s published work. However, these references do not adequately represent his influence on our understanding of Piaget’s theory. We are indebted to Pat for his many years of hel** us navigate the complexities of Piaget’s genetic epistemology. This chapter would not have been possible without our benefiting from his expertise.
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Tallman, M.A., O’Bryan, A.E. (2024). Reflected Abstraction. 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_8
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