Abstraction in Mathematics Education
KeywordsProcesses of abstraction Recontextualization Concretion Empirical abstraction Reflecting abstraction Objectification Reification Procept Structural abstraction Shift of attention APOS Learning through activity Webbing Situated abstraction Abstraction in Context
An abstraction, to most mathematicians, is an object, such as a vector space, which incorporates a structure – elements and relationships between them – common to many instances appearing in diverse contexts. Mathematics educators, on the other hand, are more interested in the processes that lead learners to grasp a structure than in the structure itself. Hence, for mathematics educators, abstraction is a process rather than an object, and they tend to identify abstraction with construction of knowledge.
The example of vector space illustrates the above. Instances of vector spaces appearing in different contexts include Euclidean 3-space, the complex plane, the set of solutions to a system of linear equations with real coefficients, and the space of states of a quantum mechanical system. The nature of the elements that serve as vectors in different contexts may be different: An element of Euclidean 3-space is a point; an element of the complex plane is a complex number; a solution of a system of linear equations is an n-tuple of (real) numbers; and a state of a quantum mechanical system is represented by a function. Nevertheless, if one ignores or “abstracts from” these contextual differences, in each case the vectors can be added and multiplied by scalars (numbers) according to exactly the same rules, and each of the spaces is closed under these two operations. Focusing on operations with and relationships between vectors while ignoring the specific nature and properties of the vectors in each context, the mathematician obtains the abstract vector space. Hence, to mathematicians, abstraction is closely linked to decontextualization.
When mathematics educators use the term abstraction, however, they usually mean the processes by which learners attempt, succeed, or fail to reach an understanding of the structure of mathematical notions such as concepts, strategies, and procedures. Mathematics educators also study conditions, situations, and tasks that facilitate or constrain such processes. Most mathematical notions have structure – relationships and connections between their elements – and hence these processes are relevant to the notions usually learned in schools such as addition, the algorithm for multiplying multidigit numbers, negative number, ratio, rate of change, sample space, and the integral. Moreover, since learners are usually taught these notions in a specific context, context is an important factor to be taken into consideration when investigating processes of abstraction. In this respect, van Oers (1998) has made the point that context is constitutive of meaning and abstraction is a process of continuous progressive recontextualization – rather than of decontextualization.
Wilensky (1991) has remarked that “concretion” might be a more appropriate term than abstraction for what mathematics educators intend to achieve: Attaining an understanding of structure means establishing connections, and “[t]he more connections we make between an object and other objects, the more concrete it becomes for us” (p 198). Hence, the goal is to make notions that are considered abstract by mathematicians more concrete for learners.
Piaget (2001/1977) may have been the first to attend to the issue of abstraction as a cognitive process in mathematics and science learning, in particular young children’s learning; his distinction between empirical and reflecting abstraction and his work on reflecting abstraction have been enormously influential. Empirical abstraction is the process of a learner recognizing properties common to objects in the environment. Reflecting abstraction is a two-stage process of (i) projecting properties of a learner’s actions onto a higher level where they become conscious and reorganizing them at the higher level so they can be connected to or integrated with already existing structures. As Campbell, the editor, remarks, projecting refers to the optical meaning of reflecting, whereas reorganizing refers to its cognitive meaning, and the term “reflecting” is more accurate than the usually used “reflective.”
Mitchelmore and White (1995) focus on and further develop Piaget’s notion of empirical abstraction. They build on Skemp’s (1986) elaboration of empirical abstraction as lasting change allowing the learner to recognize similarities between new experiences and an already formed class and propose a theory of teaching for abstraction that links the lasting nature of the change to the learner’s connections between different contexts.
A number of researchers have further developed Piaget’s thinking about reflecting abstraction and applied it to school age learners. Thompson (1985) has proposed a theoretical framework in which mathematical knowledge is characterized in terms of processes and objects. The central issue is how a learner can conceptualize a process such as counting, multiplication, or integration as a mathematical object such as number, product, or integral. The learner usually first meets a notion as a process and is later asked to act on the object corresponding to this process. The transition from process to object has been called objectification. The notion of reification, proposed by Sfard (1991), is closely related to objectification; the relationship has been discussed in the literature (Thompson and Sfard 1994).
Gray and Tall (1994) have pointed out that mathematical understanding and problem solving requires the learner to be able to flexibly access both, the process and the object. They proposed the term procept to refer to the amalgam of three components: a process, which produces a mathematical object, and a symbol, which is used to represent either process or object. Given this background, Tall (2013) and Scheiner (2016) later proposed the notion of structural abstraction from objects, according to which individuals interpret new concepts in terms of their prior knowledge; this allows them to move from simple to complex knowledge structures by successive stages of increasing sophistication.
The notions of process and object are central to learning mathematics, and it is very important for mathematics educators to gain insight into learners’ processes of objectification and into how such processes can be encouraged and supported. Mason (1989) proposed to consider abstraction as a delicate shift of attention, and the essence of the process of abstraction as coming to look at something differently than before. The shift from a static to a dynamic view of a function graph may be an example; the shift from seeing an (algebraic) expression as an expression of generality to seeing it as an object is another one. Researchers investigating processes of abstraction only gradually took Mason’s perspective seriously, possibly because of the heavy investment in time and effort it implies. Indeed, in order to gain insight into learners’ shifts of attention and hence processes of abstraction, microanalytic analyses of learning processes are required. Such analyses have been carried out by several teams of researchers and are at the focus of the remainder of this article.
Dubinsky and his collaborators (Dubinsky and Mcdonald 2002) observed undergraduate students’ learning process by means of the theoretical lens of schemas composed of processes and objects; they did this for case of mathematical induction, predicate calculus, and several other topics. For each topic, the analysis led to a genetic decomposition of the topic and to conclusion on the design for instruction supporting conceptual thinking.
Simon et al. (2004, 2018) propose a theoretical framework called Learning Through Activity, according to which mathematical concepts are always the result of reflective abstraction. They elaborate a mechanism for conceptual learning that interprets how students build more advanced concepts from prior concepts in terms of higher level actions being the result of the coordination of lower-level actions, a coordination that includes student awareness of the logical necessity involved in a particular mathematical relationship.
Taking up Wilensky’s (1991) theme, Noss and Hoyles (1996) stress the gain of new meanings (rather than a loss of meaning) in the process of abstraction and hence consider this process as experiential, situated, activity-based, and building on layers of intuition, often in a technology-rich learning environment. They introduce the metaphor of webbing, where local connections become accessible to learners, even if the global picture escapes them. Recognizing that in each instance such webbing is situated in a particular setting, they coin the term situated abstraction. Pratt and Noss (2010) discuss design heuristics implied by this view of abstraction.
Another characteristic of situated abstraction, possibly because of the authors’ focus on the use of computers, is a strong role of visualization in processes of abstraction. This has led Bakker and Hoffmann (2005) to propose a semiotic theory approach according to which learners proceed by forming “hypostatic abstractions,” that is, by forming new mathematical objects which can be used as means for communication and further reasoning.
Whereas the latter approaches to abstraction are situated, the ones discussed earlier are cognitive in nature. Abstraction in Context (Hershkowitz et al. 2001; Dreyfus et al. 2015) bridges cognition and situatedness by providing tools for analyzing learners’ processes of abstraction as they occur in a mathematical, social, historical, and physical context, as well as a specific learning environment. This wide interpretation of context corresponds to van Oers’ view of abstraction as opposed to decontextualization.
Scheiner (2016) recently pointed to the dialectic between abstraction from actions (e.g., Dubinsky above) and abstraction from objects (e.g., Tall above). While both share the image of abstraction as a process of knowledge compression, learners abstracting from actions extract meaning from actions by reflective abstraction, whereas learners abstracting from objects bestow meaning on objects by structural abstraction.
- Dreyfus T, Hershkowitz R, Schwarz B (2015) The nested epistemic actions model for abstraction in context – theory as methodological tool and methodological tool as theory. In: Bikner-Ahsbahs A, Knipping C, Presmeg N (eds) Approaches to qualitative research in mathematics education: examples of methodology and methods. Advances in mathematics education series. Springer, Dordrecht, pp 185–217Google Scholar
- Gray E, Tall D (1994) Duality, ambiguity and flexibility: a proceptual view of simple arithmetic. J Res Math Educ 26:115–141Google Scholar
- Mason J (1989) Mathematical abstraction as the result of a delicate shift of attention. Learn Math 9(2):2–8Google Scholar
- Piaget J (2001) Studies in reflecting abstraction (ed and trans: Campbell RL). Psychology Press, London (Original work published 1977)Google Scholar
- Simon MA, Kara M, Placa N, Avitzur A (2018) Towards an integrated theory of mathematics conceptual learning and instructional design: The Learning Through Activity theoretical framework. J Math Behav. https://doi.org/10.1016/j.jmathb.2018.04.002
- Skemp R (1986) The psychology of learning mathematics, 2nd edn. Penguin, HarmondsworthGoogle Scholar
- Thompson PW (1985) Experience, problem solving, and learning mathematics: considerations in developing mathematics curricula. In: Silver EA (ed) Learning and teaching mathematical problem solving: multiple research perspectives. Erlbaum, Hillsdale, pp 189–236Google Scholar
- Thompson PW, Sfard A (1994) Problems of reification: representations and mathematical objects. In: Kirshner D (ed) Proceedings of the annual meeting of the international group for the psychology of mathematics education – North America, vol 1. Louisiana State University, pp 1–32Google Scholar
- Wilensky U (1991) Abstract meditations on the concrete and concrete implications for mathematics education. In: Harel I, Papert S (eds) Constructionism. Ablex, Norwood, pp 193–203Google Scholar