THE ROLE OF CONCRETE MATERIALS IN CONSTRUCTING MATHEMATICAL MEANING
Effie Maclellan (1997) University of Strathclyde
ABSTRACT

The emphasis on concretisation in primary school mathematics is largely derived from orthodox and neo-Piagetian theory.  Such theory builds on the constructivist assumptions that mathematical knowledge, like all knowledge, is not directly absorbed by the learner from the teacher but is actively constructed by each individual learner.  The realisation that for learning to be effective, it had to be active, manifested itself in a veritable explosion of ‘activity methods’ and ‘learning by doing’ in the belief that unless children were physically acting on concrete materials they could not be learning.  But the available evidence suggests that concrete materials have effects on learners which are different from what teachers intended.  There is therefore a pressing need to develop pedagogical practices which overtake the limitations of concrete materials and allow mathematical meaning to be constructed.

The Context

During the last thirty years or so, since translations of his work have become available, the primary mathematics curriculum has been influenced by Piaget’s contribution to our understanding of children’s thinking.   In 1967 the Central Advisory Council for Education made recommendations (in the form of the Plowden Report) for changes in primary education practices in England and Wales. These recommendations were heavily influenced by Piaget’s views on children’s thinking and, as a consequence, common educational parlance began to incorporate terms such as developmental stages, readiness and active learning.   Government approval for change was also to be found in Scotland with the publication of the Primary Memorandum (1965), the contents of which were not inconsistent with those of the Plowden Report.
Piaget’s influence on the mathematical curriculum was particularly noteworthy.  He argued that the child’s stage of intellectual development set limits on what the child could do and that no matter how careful the teacher was to prime the child for a particular piece of learning, it was impossible to introduce concepts dependent upon a particular stage of development before the child had reached that stage.  In response to this claim there were attempts to invoke pedagogical practices which reflected Piagetian thinking: Copeland (1970, 1976, 1979) stressed that children should be enabled to discover mathematics for themselves through experience of the classical Piagetian tasks; the Nuffield Mathematics Project (1970) emphasised the importance of concrete experiences as precursors to mathematical understanding; while Schwebel and Raph (1974) argued that since children do not necessarily learn what teachers explicitly teach, it would be more beneficial to support children through their own explorations and discoveries rather than defining for children what they ought to know.  There was, then, a fairly influential body of opinion which was suggesting that the mathematics curriculum in schools could be improved by taking account of Piagetian theory.

This concern to improve the mathematics curriculum was not misplaced.   Since the turn of the century (McIntosh, 1981) there had been reports which suggested that while children were mechanically accurate in computation they were unable to apply their skills in contextualised settings.  This disjunction between what children can do and what is deemed that they should do has been recognised by many (such as Skemp, 1971; Lovell, 1972;  Brown, 1979; Biggs, 1983, 1985; Bryant, 1985; Hughes, 1986; Orton, 1987) and is typically attributed to pedagogical practices which emphasise symbol manipulation at the expense of conceptual learning.  The difficulty is summarised by Skemp (1971) when he points out that learning to manipulate symbols to achieve the approved correct answer  “is very hard to distinguish, in its early stages, from conceptual learning”.   However since symbol manipulation is not the same as conceptual learning, children may experience a diet of procedural drills which are memorised by rote but which are largely meaningless to them.  In other words, they may achieve little or no conceptual understanding.  As a result, they cannot apply such procedures as they have learned to any real world situation.   It was this sorry state of affairs that teachers hoped to redress through pedagogical changes.

The Problem

Of particular significance for the primary mathematics curriculum is Piaget’s stage of concrete operations.  Reasoning at this stage enables the child to conserve, to understand one-to-one correspondence and to focus on several aspects of a situation simultaneously.  The child can now perform powerful, internal manipulations.  Mentally, the child can combine, separate, order and transform objects and actions.  Such operations are considered concrete because they are carried out, at least initially, in the presence of the objects and events being thought about.  That concrete operations refer to reasoning which has not yet separated from its empirical context found a particular manifestation in the teaching of mathematics.   Teaching typically involved the use of manipulative materials which would make a vivid (and concrete) representation of the abstract mathematical idea.  Not only did teachers demonstrate abstract mathematical relationships through concrete materials but (and more importantly according to Piaget, 1974) the children themselves were encouraged to be active in their own learning through the exploration of manipulative materials.   Because the child’s thinking was oriented to objects and events in the immediate present, it was assumed that experience of concrete materials would enable the child to construct a mental representation of the concept which was being exemplified by the materials.  In turn, it was assumed that the construction of some ‘mental meaning’ from the physical embodiment of the concrete materials would obviate the manipulation of symbols in a rote and mechanical fashion. 

Disappointingly, however, the use of concrete materials has not necessarily produced the intended outcome of meaningful links between procedural and conceptual knowledge (Hart, 1989; Boulton-Lewis, 1993; Boulton-Lewis and Tait, 1994) and as recently as 1993 Ofsted (p 5) reported:
A large majority of pupils of all ages could perform basic skills adequately: if they knew which operation (i.e. addition, subtraction, multiplication or division) was required they could usually do the calculation.  Problems arose if the question was in a context and pupils had to decide which operation to use.

While this is disappointing finding it is perhaps not a surprising one given the nature of school mathematics and what this implies for teaching.  In order to develop mathematical understanding, two kinds of experience are needed (Ginsburg and Opper, 1988). First of all the child needs physical experience of the observable, perceptual features of objects.  Thus it is through physical observation and/or exploration of objects that the child comes to know about the shape, colour and size of objects.  Next the child needs (in Piagetian parlance) logicomathematical experience.  This second and very different kind of experience enables the child to consider, think about or reflect on his/her own actions in relation to a number of previous physical experiences.  An example may illustrate the difference between the two kinds of experience.  The child explores a cube and through perceptual activity comes to realise that the cube has edges, corners and a number of faces.  Further the child may experience the cube to be heavy.  All of this is physical knowledge.  If, however, the child lifts two cubes and notices that one is heavier than the other, the child has made a comparison and noted a difference.  This difference does not inhere directly in the cubes themselves but in the action of comparing the two particular cubes.   This is logicomathematical knowledge, knowledge which cannot be gained by directly perceiving objects.  Similarly the child explores a set of buttons and notices that some buttons are round, some are square, some have two holes and some have four.  Such observations constitute physical knowledge.  But when the child knows that he/she can count the set of buttons in any order and arrive at the same cardinal value, the child is evidencing logicomathematical knowledge, which does not reside in the buttons per se but in the size of the set.  This is knowledge which has been constructed by the child as a consequence of the his/her actions of repeated enumerations.  

The relationship between physical knowledge and logicomathematical knowledge is important but possibly not well explained in the literature. Physical knowledge provides the basis from which further reflection can occur.  The child cannot estimate that one object is heavier than another without physically handling both objects; the child cannot determine numerical equivalence without accurate enumeration of the countables.  Physical knowledge is a necessary condition for logicomathematical knowledge to develop but of itself is not enough.  Of much greater significance for the development of mathematical understanding is logicomathematical experience and knowledge.  Being able to think about the similarities and differences in relation to physical experiences, being able to generalise from a variety of specifics into a somewhat more abstract, superordinate category is the essence of mathematical thought and it is the logicomathematical experience which allows this more powerful thinking. Without some accompanying mental activity to reflect on the purpose and/or significance of the physical activity, concrete materials will not actually enable the child’s mathematical understanding to develop.
The Way Forward
Since the mathematical ideas do not actually reside in the concrete materials themselves, it follows that it is not possible to ‘discover’ mathematics through the physical exploration of concrete materials.  This, in turn, has implications both for the role of concrete materials and for the role of the teacher.

The role of concrete materials

Doubtless, concrete materials can have a useful role to play in the development of mathematical understanding.  However the use of concrete materials in any mathematical situation and for any one child needs to be carefully monitored.   There are two reasons for this.   Firstly, while concrete materials are intended to model mathematical idea(s), the children may not appreciate that the materials are mere representations of hypothetical problems but instead may attribute an undeserved veracity to the actual materials.  Solomon (1989) argues that children may not differentiate between a hypothetical arithmetical problem and real life; with erroneous consequences.   For example, the child is provided with a set of ten buttons and told ‘There are eight buttons in a bag. Jane takes two buttons out of the bag to sew on a dress. How many buttons are left in the bag?’  Typically the child disregards the ‘given hypothetical’ of there being eight buttons and operates with the actual, ten buttons.  A second reason for advocating caution in the use of concrete materials (which may be related to the first) is the evidence which suggests that allowing children to rely on concrete materials inhibits their achievement of more sophisticated understandings.  For example, Carpenter and Moser (1982) noticed that young children who could competently use the sophisticated strategy of counting on would revert to the more primitive strategy of counting all (despite the extra time and effort involved) when concrete materials were made available.   That children may correctly solve an arithmetical problem when no concrete materials are provided is also an observation made by Solomon (1989).  However they are used, concrete materials should not be used unthinkingly.

The role of the teacher

The abstract nature of mathematics means that unlike other domains of knowledge, in which the young child can construct mental representations as a result of sensory and motor experiences of the physical environment, the child is actually very dependent upon teachers who will mediate between what the child already knows and whatever new aspect of mathematics is deemed appropriate for the child to learn.  Just because the child is presented with some concrete materials, it does not follow that the child will abstract from those materials the mathematical properties which may have been intended by the teacher.  In order for the child to learn what was intended there has to be a discourse between the child and the teacher which will allow the child to bridge the gap between the concrete materials and the abstract ideas.  Heddens (1986) suggests that careful questioning by the teacher can enable children develop their mathematical thinking.  Specifically Heddens (1986) proposes that teachers systematically model the kinds of questions which children should be asking themselves and each other.   The point of such teaching is to promote children’s reflective awareness of mathematics:  children should notice what they are doing mathematically in order that they become alert to how mathematics can help them to solve real life problems.  The need for children to have conscious control over mathematics has of course been recognised for a long time.  What is much more problematic and what remains yet to be solved is how to help children gain this control.   Appealing as Heddens’ suggestions are, they are not new.  Desforges (1989), draws attention to the finding that in spite of the rhetoric which lauds mathematical discussion, there is little empirical evidence of its occurrence in classrooms.   He attributes this, in part, to a mathematics curriculum which emphasises breadth of coverage rather than depth of treatment.  His suggestion to reduce the breadth of the mathematics curriculum may be worthy of some serious attention. 

Unlike the ubiquitous advice which was being offered to teachers twenty years ago, it now seems clear that the teacher does indeed have a crucial and central role to play in the delivery of the mathematics curriculum.  As Dearden (1967) argued at the time, the teacher needs to question, discuss, hint, suggest and instruct children as to what they must do in order that they can have any chance of abstracting the mathematical idea which the teacher intends them to develop.   It is within this context of enabling children to master powerful but abstract ideas that concrete materials may have a role.   What learning is to be achieved through the use of concrete materials must, however, be understood by teachers and children.

On the face of it, it seems that if children’s mathematical understanding is no more sophisticated now than it was thirty years ago, then Piaget’s views have little or nothing to offer the primary teacher who wants to deliver a more effective mathematics curriculum.  Such a conclusion would be simplistic, and unhelpful.  Rather, the explanations for children’s lack of conceptual learning may be more to do with our incomplete understanding of what Piaget really did have to offer.

First of all, while Piaget was interested in learning (and he defined learning in very particular ways) he was much less concerned with teaching.  His principal interest was in the formation and meaning of knowledge and to the extent that he outlined a theory of how individuals construct their own knowledge on the basis of their own experiences, he gave primacy to the child’s efforts and progress in making sense of his/her environment rather than to the environmental factors (one of which could, arguably, be teaching) themselves.  

Secondly, insofar as Piaget was concerned with teaching, it was at the level of principle only.  Piaget (Groen and Kieran, 1983) saw the teacher’s task as twofold.  One task was to help the child to move from thinking about real objects and events to thinking about hypothetical objects and events or, in other words, to help the child to move to thinking about thinking.  The other task was for the teacher to be aware of the child’s level of cognitive functioning (because this will not be the same for all children in any given group) in order to try to understand the significance of a particular experience for any one child. 

Thirdly, although Piaget’s empirical work produced a vast number of studies on the child’s understanding of number and of geometry, there is no explicit connection between Piaget’s findings and the conventional school curriculum in mathematics.  So, for example, while the ability to conserve number is essential for a comprehensive understanding of counting, conservation of number would now not usually be taught in schools, although counting would be.  Moreover, because of the focus of Piaget’s interests, he made no pretence to provide any correspondence between his findings and pedagogical practices.   

What Piaget had to offer was an explanation for how people construct new knowledge.  Since it was never his intention that his theories should have direct ‘application’ in educational settings, it is not helpful to try to make facile (but spurious) connections between Piagetian stages and performance in school mathematics.  However, what Piaget did provide was an account of different types of knowledge.   He distinguished between physical knowledge and logicomathematical knowledge: a distinction to which we should be sensitive.  If physical knowledge is best constructed through the exploration of objects and if logicomathematical knowledge is best constructed through reflection on actions (as Piaget claimed) we need to develop pedagogical practices which are appropriate for each type of knowledge.  At the moment we seem reasonably proficient in helping children to develop physical knowledge through the provision of concrete materials.  However, this strategy, of itself, cannot help the child to develop logicomathematical knowledge.  It seems to be of some considerable urgency that we develop a pedagogy which treats logicomathematical knowledge appropriately if we are to break the cycle of poor conceptual learning in mathematics.

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