Paul Harrison looks at a new book which challenges current maths practice
This timely book takes a hard-headed look at some of the hallowed assumptions on which much contemporary maths pedagogy is based, particularly those aspects related to notions of "child-centred" and "discovery" learning. It pleads not for a return swing of any crude "traditional" versus "progressive" pendulum, but for a move towards what might be considered the more formal aspects of education.
Three major contributory factors to the current decline in numeracy emerge: individualised learning; the belief that mathematical development in all children is a matter of moving them through the same set of hierarchical stages; and the cult of the concrete - the belief that for children to understand a numerical concept necessarily involves the manipulation of concrete objects.
In her introductory chapter, Ruth Merttens traces individualised learning back to the Plowden Report and the influence of the theories of Piaget. The argument was simple: children learn at different rates so they must learn individually. Out went maths learning as a shared experience, with its linguistic, oral and mental aspects; in came a lonely ploughing through pages of commercial maths schemes.
Merttens does not want a return to traditional teaching ("teaching as lining up children and telling them things") but instead calls for a greater emphasis on teaching as an interactive process, combining elements such as straightforward instruction, modelling or demonstration, explanation and dialogue.
The belief that mathematical development involves children travelling through a prescribed set of hierarchical stages is firmly rooted in Piaget. Although largely discredited, the notion still widely persists. Hence the almost sacrosanct belief that all children, regardless, must engage in pre-number activities such as sorting and matching before they work with numbers; or that children must be able to conserve number before they can count.
Helen Williams, in a chapter on the development of numeracy in the early years, argues that children come to school with a considerable range and depth of mathematical understanding which does not fit into neat hierarchical packages. For example, they often arrive with an everyday familiarity and understanding of numbers far greater than those they have to work through systematically at school: "not all children under seven live in houses with numbers less than 10". Further, children arrive at school with complex ways of thinking - although often immature and different to those of adults.
Rather than pushing children through developmental activity-based hoops, she asserts, teachers should be building on children's existing understanding, and valuing and nurturing their existing mathematical thinking, particularly through talk between teachers and children.
Finally, the cult of the concrete, again with origins in Piaget and Plowden, and given a further boost in the more recent Cockcroft Report. Ruth Merttens suggests that the importance indiscriminately bestowed on the use of concrete materials (and implicitly, the cardinal rather than ordinal aspect of number) has been at the expense of the amount of attention paid to the mental, semantic and linguistic aspects of mathematical understanding, If children's numerical fluency as well as understanding is to blossom, then the balance needs to be shifted towards the latter. Take a simple example: 4 + 3. The author asks us to question whether counting out sets of four and three counters, combining the sets and then counting the total is necessarily the sole approach and the one most conducive to the development of numerical fluency. She suggests the earlier introduction to the ordinal aspect of numbers and thus to number lines, with their ready assimilation as mental imagery.
This seminal, highly-practical book includes chapters on calculators, play, bilingual children and assessment. It should be required reading in all teacher training institutions.
Paul Harrison is co-author of Nelson Mathematics and Maths 2000, both published by Thomas Nelson