Abstraction in Mathematics Learning - merga

20-9 MERJ v.19 n.2 final 2/11/07 9:46 AM Page 1 

Mathematics Education Research Journal 2007, Vol. 19, No. 2, 1–9 

Editorial 

Abstraction in Mathematics Learning 

Michael Mitchelmore and Paul White 

Abstraction has been a frequent discussion topic since the days of Aristotle and 

Plato. Constructivist theories often espouse the notion of levels of abstraction. 

Hiebert and Lefevre (1986, pp. 4-5) put it this way: 

It is useful to distinguish between two levels at which relationships between 

pieces of mathematical knowledge can be established. One level we will call 

primary. At this level the relationship connecting the information is constructed 

at the same level of abstractness (or at a less abstract level) than that at which 

the information itself is represented. That is, the relationship is no more abstract 

than the information it is connecting. 

Some relationships are constructed at a higher, more abstract level than the 

pieces of information they connect. We call this the reflective level. 

Relationships at this level are less tied to specific contexts. They often are 

created by recognising similar core features in pieces of information that are 

superficially different. The relationships transcend the level at which the 

knowledge currently is represented, pull out the common features of differentlooking 

pieces of knowledge, and tie them together. 

Of particular interest are the assertions that relationships at the “higher, more 

abstract level” are “created by recognising similar core features” and “are less 

tied to specific contexts”. These two features are often referred to as the 

hierarchical and decontextualisation views of abstraction. 

Why, then, did Hiebert and Carpenter (1992) write a whole chapter on 

learning mathematics with understanding in a major treatise on mathematics 

education without even mentioning abstraction They defined understanding as 

“making connections between ideas, facts, or procedures” (p. 67) and discussed 

two methods for doing this: investigating similarities and differences, and 

establishing inclusion relationships — both of which are crucial to learning by 

abstraction. Clearly they had the notion of abstraction in mind, but did not dare 

speak its name. Carpenter (personal communication, May 1999) informed us that 

the term “brought too much baggage with it.” 

Abstraction had gained a bad reputation (in some circles) because of the 

criticisms expressed by the situated cognition movement. According to this 

movement, “the primary concern of schools often seems to be the transfer of ... 

abstract, decontextualized concepts” (Brown, Collins, & Duguid, 1989, p. 32), 

based on the theory that “knowledge acquired in ‘context-free’ circumstances is 

supposed to be available for general application in all contexts” (Lave, 1988, p. 

9). However, “much of what is taught turns out to be almost useless in practice”


2 Mitchelmore & White 

(Brown et al., 1989, p. 32), consonant with “the meagre evidence for transfer 

garnered from a very substantial body of work [in cognitive psychology]” (Lave, 

1988, p. 32). 

Objections continued to be made to the traditional approach to abstraction. 

Noss and Hoyles (1996) argued against the hierarchical and decontextualisation 

views of abstraction. They characterised “abstraction as a process of connection 

rather than ascension” (p. 48) and sought to break the word abstract “free from 

its dehumanising connotations” (p. 49). They introduced the idea of a webbing 

as “the presence of a structure that learners can draw up and reconstruct for 

support — in ways that they can choose as appropriate for their struggle to 

construct meaning for some mathematics” (p. 108), and introduced the term 

situated abstraction to describe “how learners construct mathematical ideas by 

drawing on the webbing of a particular setting which, in turn, shapes the way the 

ideas are expressed” (p. 122). 

Van Oers (1998) also criticised the idea of decontextualisation as the basis for 

abstraction, arguing that context is always relative to an individual so that 

decontextualisation suggests removal of the individual. He has also argued (van 

Oers, 2001) that removing context must impoverish a concept rather than enrich 

it. 

What these examples show is that the notion of abstraction cannot be 

ignored, but is as controversial as ever. Interest in learning by abstraction has 

grown in the past ten years. The International Journal of Educational Research 

published a special issue on “Approaches to abstraction” in 1997 and the 

Cognitive Science Quarterly followed suit with a special issue on “Abstraction in 

context” in 2001. A research forum on “Abstraction: Theories about the 

emergence of knowledge structures” took place at the annual PME conference in 

2002 (Boero et al., 2002), and a symposium on “Rethinking abstraction and 

decontextualization in relationship to the transfer dilemma” was held at the 

Annual Meeting of the American Educational Research Association in 2004. The 

discussion groups on learning by abstraction that we organised at the 2005 and 

2006 PME conferences eventually led to this special issue of the Mathematics 

Education Research Journal. 

Empirical Abstraction 

The foundational work on hierarchical abstraction theory was done by the late 

Richard Skemp, partly building on the earlier work of Zoltan Dienes (1961). The 

second edition of his 1971 book The psychology of learning mathematics, published 

in 1986, had a strong influence on our own work but did not have a widespread 

impact in the face of the situated cognition movement. However, a more recent 

resurgence of interest in his work is evidenced by a commemorative volume (Tall 

& Thomas, 2002) that has chapters contributed from authors in the United 

Kingdom, New Zealand, Australia, United States, Israel, The Netherlands and 

Canada. We were honoured to be asked to contribute one of the Australian 

chapters.


Editorial 3 

Skemp’s conception of abstraction consists of similarity recognition 

followed by embodiment of the similarity in a new mental object: 

Abstracting is an activity by which we become aware of similarities ... among 

our experiences. Classifying means collecting together our experiences on the 

basis of these similarities. An abstraction is some kind of lasting change, the 

result of abstracting, which enables us to recognise new experiences as having 

the similarities of an already formed class. ... To distinguish between abstracting 

as an activity and abstraction as its end-product, we shall ... call the latter a 

concept. (Skemp, 1986, p. 21; italics in original) 

Skemp’s conception can be called empirical abstraction because it is based on 

experience. But it is much deeper than the empirical abstraction described by 

Piaget, where the term denotes the extraction of common properties of objects 

and extensional generalisations such as colour and weight (Piaget et al., 1977). 

The similarity Skemp talks about is not in terms of superficial appearances but of 

underlying structure. Moreover, similarity recognition is not some random 

haphazard identification of things which look alike but is usually occasioned by 

purposefully directed selective attention. For example, a parent may show a 

young child various hands and feet while saying the words hand and foot with 

the specific intention of teaching the child these words. In the mathematics 

classroom, the direction would usually come from the teacher. Hence, the 

embodiment of the similarity (often called reification) is a constructive process 

based on generalising purposefully identified similarities. 

Our interest in the abstraction process dates back to the early nineties with 

a presentation (White & Mitchelmore, 1992) at the annual conference of the 

Mathematics Education Research Group of Australasia (where, by chance, 

Richard Skemp was a keynote speaker). Much of our early work consisted of 

applying Skemp’s theory to the development of young children’s understanding 

of the angle concept (Mitchelmore & White, 2000a), but we also investigated 

children’s understanding of rates of change (White & Mitchelmore, 1996) and 

decimals (Mitchelmore, 2002). 

Because empirical abstraction focuses on the identification of underlying 

general features, we call empirical concepts abstract-general (Mitchelmore & 

White, 1995; White & Mitchelmore, 2002). Our concern is that students often 

learn abstract concepts in isolation, without engaging in an abstraction process. 

Such abstract-apart concepts are poorly understood, easily forgotten, and rarely 

applicable — the very characteristics that the situated cognition movement 

criticised so strongly. The advantage of empirical abstraction theory is that it 

leads to a theory of teaching that could develop deeper understanding. We have 

been developing a theory of what we call Teaching for Abstraction (Mitchelmore & 

White, 2000b, 2004a), where students: 

• familiarise themselves with the structure of a variety of relevant contexts; 

• recognise the similarities between these different contexts; 

• reify the similarities to form a general concept, and then 

• apply the concept in new situations.



Investigations of the classroom implementation of this approach in the 

teaching of percentages (White & Mitchelmore, 2005; White, Wilson, Faragher, & 

Mitchelmore, 2007) and ratios and rates (Mitchelmore, White, & McMaster, 2007) 

have been promising, but have revealed many practical implementation problems. 

Several of the authors represented in this special issue compare their 

approach to empirical abstraction but there is no paper which expounds the theory 

in sufficient detail for the uninformed reader to judge the significance of these 

comparisons. It is for this reason that we have taken the liberty to discuss empirical 

abstraction at length and we trust the reader will forgive us the indulgence. 

Theoretical Abstraction 

Often contrasted to empirical abstraction is the idea of theoretical abstraction. Its 

genesis owes a great deal to Soviet psychologists, especially Vygotsky and Davydov. 

In essence, theoretical abstraction consists of the creation of concepts to fit 

into some theory. Davydov (1972/1990) notes that theoretical concepts are 

“produced on the basis of a mental and systemic analysis of the relations and 

connections among objects” (p. 25) and that, whereas “elementary concepts 

basically provide for the identification and classification of objects and 

phenomena, theoretical ones additionally permit the explanation of various 

manifestations of certain qualities of objects” (p. 27). 

Vygotsky (1934/1987) made a corresponding distinction between everyday 

and scientific concepts. Everyday concepts are formed by empirical abstraction, 

but the formation of scientific concepts has three features: The establishment of a 

system of relations among concepts, an awareness of one’s own mental activity, 

and “penetration to the object’s essence ... an enrichment rather than an 

impoverishing of the reality presented in the concept” (p. 185). “A theoretical 

idea or concept should bring together things that are dissimilar, different, 

multifaceted, and not coincident, and should indicate their proportion in the 

whole. ... Such a concept, in contrast to an empirical one, does not find something 

identical in every particular object in a class, but traces the interconnection of 

particular objects within the whole, within the system in its formation” (p. 255). 

Theoretical abstraction is seen as quite different from empirical abstraction. 

“Scientific knowledge is not a simple extension, intensification, and expansion of 

people’s everyday experience. It requires the cultivation of particular means of 

abstracting, a particular analysis, and generalization, which permits the internal 

connections of things, their essence, and particular ways of idealizing the objects 

of cognition to be established” (Davydov, 1972/1990, p. 86). An important phase 

of theoretical abstraction is the identification of the essence of an idea. “The 

essence of a thing is none other than the basis (included in itself) for all of the 

changes that occur with it in interaction with other things” (Rubinstein, cited by 

Davydov, 1972/1990, p. 194). After theoretical abstraction, an object is “mentally 

replaced by another — its model” (Davydov, 1972/1990, p. 250). 

Some examples could clarify the difference between empirical or theoretical 

abstraction. How do children learn the basic geometrical concept of a line 

According to empirical abstraction theory, children recognise an underlying


Editorial 5 

similarity between real objects that are only roughly linear and have been 

produced by a variety of means (folding, pulling, planing, etc.). According to 

theoretical abstraction theory, theoretical ideas of points and lines are needed to 

express generalisations arising from spatial investigations (e.g., two lines meet at 

a point). A second example would contrast learning by examining several 

worked exercises (empirical abstraction) to the deep analysis of one problem in 

order to identify its essential variables and relationships (theoretical abstraction). 

There is little conclusive empirical evidence on which to base a judgement as to 

which is the more satisfactory explanation or the more effective method of 

learning mathematics, so research goes on. 

The most recent example of an attempt to explain theoretical abstraction in 

mathematics learning is the RBC model (Recognising, Building-With, 

Constructing) first proposed by Hershkowitz, Schwarz, and Dreyfus (2001). 

Since several of the papers in this issue adopt this model as their theoretical 

framework, we do not need to describe it here. 

In This Issue 

There are five articles in this edition, with the authors coming from five 

countries: United Kingdom, Israel, Turkey, Netherlands, and Australia. As such, 

the papers represent a variety of theoretical perspectives on the role of 

abstraction in mathematics learning. 

Van Oers and Poland look at schematising activities among 5 and 7 year old 

Dutch children as they make representational drawings, showing that they lead 

to abstract thinking. The authors argue that “the general is not the end result of 

an abstraction process; rather, some general principle is always the beginning. 

An abstract concept then is not so much a reproduction of reality, but actually 

establishes a point of view that guides our thinking.” In other words, an 

abstraction does not involve the recognition of a new, previously unnoticed 

general characteristic but constitutes an attribute that is added to the object in 

our thinking. This approach appears to cast a shadow on empirical abstraction, 

but closer inspection shows that there need be no inconsistency. For it can be 

argued that the point of view students take determines the underlying 

mathematical similarities they look for. Moreover, this point of view is imposed 

by teachers as they propose representational tasks to students. 

Gray and Tall cover a spectrum of mathematical topics (whole number, 

fractions, algebra, and functions) and a wide variety of countries (England, 

Malaysia, Brazil, and Turkey). They describe how mathematical abstraction 

arises through a natural mechanism of the human brain in which complicated 

phenomena are compressed into what they call thinkable concepts. Their 

framework suggests that, to improve long-term conceptual learning, the whole 

curriculum must be framed in such a way as to produce thinkable concepts at 

every stage. This result can only be achieved if educators are aware of the 

abstraction process. The authors give several examples where students acquire 

what we have called abstract-apart concepts when educators would prefer they 

be abstract-general.



These first two articles, if they do not explicitly espouse the theory of 

empirical abstraction as we have outlined it, are both concerned with describing 

how children learn to interpret their experience. The remaining three articles are 

more concerned with theoretical abstraction, and they are all based on the RBC 

model. 

Hershkowitz, Hadas, Dreyfus, and Schwarz investigate the development of 

understanding of probability among groups of Grade 8 students in Israel. Using 

an extended RBC+C model (RBC plus Consolidating), they analyse students’ 

learning in considerable detail and demonstrate convincingly that their model 

allows the description of theoretical abstraction in terms of epistemic actions. The 

criticism that they have voiced in discussion groups, that empirical abstraction 

theory does not provide a similar level of detail, is probably well justified. We are 

looking forward to research showing how such a detailed analysis can be used 

by mathematics educators to improve student learning. 

Ozmantar and Monaghan take a dialectical materialist approach to 

abstraction, following Marxist theory, in investigating the learning of absolute 

value function among Turkish Grade 10 students. This paper is particularly 

interesting because it explicitly considers the role of the teacher in student 

learning. In coming to grips with the first section of the paper, the reader should 

note that “in dialectical materialism, it has been agreed to call concrete this 

objective integrity that exists in the connection of individual things” (Davydov, 

1972/1990, p. 256). So the term “concrete” does not have the same meaning in 

this paper as it usually does elsewhere. The theory is quite right to reject the type 

of abstraction which leaves ideas disconnected (abstract-apart), but empirical 

abstraction as we have described it would also produce ideas that are connected 

both to each other and to the world of experience (abstract-general). Learning 

about absolute value functions cannot involve empirical abstraction because it is 

an entirely theoretical construct. In this case, learning is much better described by 

the RBC model of theoretical abstraction. 

Williams uses the RBC Model to examine a phenomenon that all teachers 

dream of — spontaneous student learning. Her data come from interviews with 

two Grade 8 students, one from the USA and one from Australia, working on 

linear functions. Using ideas first advanced by Krutetskii (1969/76), she pulls 

apart their thinking processes and refines the Building-With and Constructing 

epistemic actions to create a Spontaneous Abstracting Model. Noting that both 

case study students demonstrated visible positive affect and achieved a 

remarkable depth of understanding in such a short time, Williams poses the 

fascinating question of how teachers can promote spontaneous abstracting. 

One Theory of Abstraction 

The papers in this issue, together with our own exposition of Empirical 

Abstraction above, represent a variety of theories of abstraction. Is there one 

theory which encompasses them all Boero et al. (2002, p. 135) asked “Do we 

need one theory of abstraction” We believe it would be useful, but it would be 

a theory that takes account of students’ level of mathematical development. We


Editorial 7 

would argue that empirical abstraction describes well the initial stage at which 

students form fundamental mathematical concepts of number and space, but 

that the process of learning to manipulate symbols (as in algebra and calculus) 

requires a theoretical abstraction approach such as the RBC+C model (Hassan & 

Mitchelmore, 2006). The relation between empirically and theoretically formed 

abstractions also needs consideration (Mitchelmore & White, 2004b), and it is 

also here that proceptual learning may occur (Gray & Tall, 1994). 

Tall’s (2004) three mathematics worlds may provide the foundation for 

building such a unified theory. We note that his conceptualisation includes a 

formal world of reasoning from definitions and axioms, where abstraction may 

be very different again (Dreyfus, 1991). Research on the relation between what 

we might provisionally call formal abstraction and theoretical (and even 

empirical) abstraction could prove valuable to tertiary mathematics educators. 

Acknowledgements 

We are indeed fortunate that several researchers who have already made 

significant contributions to the continuing debate on the nature of abstraction in 

mathematics learning are represented in this issue. We thank all the authors for 

their patience as we have sought to knit the papers into a whole for this issue. We 

hope that the result will in turn make a contribution that is greater than the sum 

of its parts. In particular, we hope that this issue will stimulate research that seeks 

to reconcile and combine the various theories. 

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Editorial 9 

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Guest Editors 

Michael C. Mitchelmore, Australian Centre for Educational Studies, Macquarie 

University NSW 2109. Email: 

Paul White, School of Education (NSW), ACU National, 25A Barker Road, 

Strathfield NSW 2135. Email:

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