Learning support pedagogy for children who struggle to develop the concepts underlying the operations of addition and subtraction of numbers: the ‘Calculia’ programme

Authors: Annemarie Fritz-Stratmann, Antje Ehlert and Gabriele Klüsener

 The authors of this article argue that it is important for pre-service teachers to know about the psychology of learning mathematics (i.e. how children develop concepts) in order for them to be able to help children who encounter problems. Therefore, this article explores how a remedial programme, based on a conceptual model of mathematical knowledge development in young children, can be used in a structured remedial programme for such children – either individually or in groups?

The article deals broadly with some background on concept development in mathematics; followed by a description of the model of development that is used. The article then discusses a programme designed to assist young learners with the mathematical operations of addition and subtraction.

The reason for using a conceptual model is because conceptual reasoning  is  the  first  step  children take to recognise a problem. The strategies children use to solve the problem originate from their conceptual knowledge (and the reasoning that it elicits). This principle, the authors say, if often ignored in pedagogy.

Many teachers are satisfied with the teaching of operations (procedural knowledge), perhaps not realising that these may obscure the conceptual knowledge that has to precede it. If children do  not  learn  to  see  mathematics  conceptually, they may struggle from the early grades onwards, because although they may learn procedures from instruction and even ‘master’ these, this does not guarantee that they understand what they are doing.

The findings of research about the mathematical performance prognosis of children who experience challenges early in their learning path show the importance of support at an early stage (at preschool and primary school) to counteract negative developmental pathways in a preventive and compensatory way.

We describe a model for the development of numerical concepts and arithmetic skills in children between the ages of four and eight years, and then discuss how their ‘Calculia’ intervention is based on this model. The intervention presents ideas for teaching sessions that facilitate a step-by-step understanding of the mathematical operations of addition and subtraction by focusing on the concept of parts making up a whole number.

The authors recommend that this intervention be incorporated into pre-service teacher education programmes. This will allow students to practice these teaching strategies in their practicum work at schools under the guidance of their mentor teachers and lecturers. In this way, students will be able to explore and test their own knowledge of how children learn and how they can be given suitable learning support, which should be done before these students enter the classroom.

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