Preschool children’s notations for denoting ordinal position and quantity

The research is funded by The Leverhulme Trust (RPG-2019-330), in the UK

Issues in identifying children with specific arithmetic difficulties through standardised testing: a critical discussion of different cases

The paper discusses issues related to the identification of children with extreme difficulties in arithmetic who are regarded as potentially having dyscalculia. We present cases of 7 year old children with different approaches to a standardised computer-based test. We present issues that the cases raise and reflect on how such tests can inform those who use them and whether they enable or not the identification of children’s specific difficulties in arithmetic learning

An analysis of the presentation of the equals sign in Grade 1 Greek textbooks

Young children often develop a partial, operational understanding of the equals sign that refers to completing an action, such as getting the answer to an addition or multiplication question, and fail to develop a relational understanding of the equals sign as a symbol that denotes equivalence. A partial view of the equals sign as an operator can be the result of primary-age pupils’ overexposure to canonical equations such as a+b=c. This paper presents a preliminary analysis of the different syntaxes and formats used to present equality statements in the Grade 1 textbooks in Greece. The quantitative analysis reveals an overemphasis on presenting the equals sign within canonical equations. However, qualitative analysis reveals that the equals sign is first introduced in a context that conveys the idea of equivalence relation and is presented within an interesting mix of symbolic and non-symbolic contexts which may minimise the tendency to interpret the equals sign exclusively as an operator

Changes in young children's strategies when solving addition tasks

The aim of this study is to determine the pathway of changes that occur in the problem solving strategies of 5-6 year old children when they are engaged in solving a specific form of addition task. Karmiloff-Smith’s model of Representational Redescription (RR) suggests that higher conceptualisation and control of the employed strategy develops both before and after the achievement of an efficient solution. Evidence from data reported in this paper tends to support this hypothesis

The process of knowledge re-description as underlying mechanism for the development of children’s problem-solving strategies: an example from arithmetic

This paper reports on a study which aimed at exploring ways by which 5-6 year-old children organise different pieces of knowledge to develop strategies for solving a specific arithmetical task and furthermore, ways by which children move beyond their successful problem solving approaches to the acquisition of increased control over the procedural and conceptual knowledge that supports their problem solving success. The paper considers the emerging theory of Representational-Redescription which supports the idea of ‘success-based’ cognitive change and argues that new knowledge can be constructed by a process of internal exploitation of knowledge that already exists in the cognitive system of the problem solver. In problem solving, the notion of Representational-Redescription has been studied in spatial, physics, linguistic and notational tasks but currently, it is under-researched in mathematics. The paper presents outcomes from a study which focused on ten cases. The microgenetic method was used for the study of changes in children’s problem solving. This entailed the design of a sequence of sessions during which children were individually involved in solving a specific form of additive task, more than once, and after they had been successful in solving it. The microgenetic method was combined with the clinical method of interviewing. The paper presents a specific path of after-success strategy change. This path of change indicated children’s movement from initial success-oriented behaviour to an organisation-oriented phase during which new strategies were introduced or known strategies were evolved procedurally and conceptually. The paper explains the general analytical direction which was followed to reveal different levels of knowledge accessibility and explicitness which supported children’s main strategy during the after-success change process. These findings support the idea that learning follows not only from failure but also from success, and that the Representational-Redescription theory can offer an additional insight to the complex nature of learning processe

Context variation and syntax nuances of the equal sign in elementary school mathematics

Existing research suggests that young children can develop a partial understanding of the equal sign as an operator rather than as a relational symbol of equivalence. This partial understanding can be the result of overemphasis on canonical equation syntaxes of the type a + b = c in elementary school mathematics. This paper presents an examination of context and syntax nuances of relevant sections from the Grade-1 Greek series of textbooks and workbooks. Using a conceptual framework of context variation, the analysis shows qualitative differences between equations of similar syntax and provides a nuanced determination of contextual and structural aspects of ‘variation’ in how the equal sign is presented in elementary mathematics. The paper proposes that since equations have context-specific meanings, context variations should constitute a separate element of analysis when investigating how the equal sign is presented. The implication for practice and future research is that nuanced considerations of equation syntax within varied contexts are needed for elaborating analyses of the equal sign presentation that move beyond dichotomized categorizations of canonical/non-canonical syntaxes

The process of behavioural, representational and conceptual change in young children's strategies when solving arithmetic tasks

This study is situated in the context of projects which, in the field of arithmetic, explore the process of change in young children's thinking and strategies within problem situations. In particular, the study aims at exploring the pathway of changes that occur in 5-6 year old children's problem solving strategies when they are engaged in solving a specific form of additive task. It is hypothesised that higher conceptualisation and control of the employed strategy and of the factors involved in the task, develops after the achievement of an efficient solution. Previous research has shown that group work involving discussion and reflection upon the solution process are effective practices towards this direction. For this study, Karmiloff-Smith's model of Representational Redescription (RR) provides the theoretical and methodological framework, and is used as a basis for the analysis of the changes observed in the behavioural level and of those inferred at the representational and conceptual level. The study focuses on a number of cases. Changes in children's strategies and their progressive movement from procedural success to higher conceptualisation and control of the employed strategies, are studied in a micro-developmental level. This takes place in the course of a sequence of sessions during which children are individually involved in solving a specific form of additive tasks. The micro-developmental method of data collection and analysis is combined with the clinical method of interviewing. The study shows that children move beyond success, and introduce qualitative changes and modifications to their successful strategies. These changes indicate the movement from success-oriented behaviour to an organisation-oriented phase in problem solving during which children, as problem solvers, acquire better control and an increasing conscious access to knowledge which is present in their cognitive system; i.e. knowledge that they already have. The RR model is proved to be a valuable tool for the exploration and analysis of the post-success behaviours which were identified in the particular arithmetic, problem solving situation.</p

Preschool children’s conceptions of the meanings and use of written numerals in everyday life: a phenomenographic study of the nature and structure of qualitative variation

Supporting children’s understanding of the everyday, cultural use of written numerals is highly significant, as it is this understanding that gives meaning to classroom conversations on the purposes of written numbers. This paper presents findings from a phenomenographic study of the qualitatively different ways in which 3–5-year-old children interpret the meanings and use of numerals in everyday contexts. The study involved a volunteer sample of 37 preschool children. With their family’s support, children played a Number Spotting game, taking photographs of numerals in their environments. These photographs were supplemented with other photographs selected by the researchers and used in individual photo-elicitation interviews with children. We collected data on children’s interpretations of a range of examples of numerals used to denote quantity, order and measurement, and numerals used as labels/identifiers. The findings document qualitatively different categories that capture the range of children’s expressed conceptions as well as the critical aspects of variation that underpin how qualitatively different categories of conceptions differ or relate to each other. The study provides original insights into the nature and structure of children’s awareness of the cultural uses of written numerals. The findings can support early mathematics teaching to make meaningful connections between the knowledge that children develop outside school and the new knowledge about written numbers that they develop in formal education

Procedural and conceptual changes in young children’s problem solving

This study analysed the different types of arithmetic knowledge that young children utilise when solving a multiple-step addition task. The focus of the research was on the procedural and conceptual changes that occur as children develop their overall problem solving approach. Combining qualitative case study with a micro-genetic approach, clinical interviews were conducted with ten 5-6 year-old children. The aim was to document how children combine knowledge of addition facts, calculation procedures and arithmetic concepts when solving a multiple-step task, and how children’s application of different types of knowledge and overall solving approach changes and develops when children engage with solving the task in a series of problem solving sessions. The study documents children’s pathways towards developing a more effective and systematic approach to multiple-step tasks through different phases of their problem solving behaviour. The analysis of changes in children’s overt behaviour reveals a dynamic interplay between children’s developing representation of the task, their improved procedures, and gradually their more explicit grasp of the conceptual aspects of their strategy. The findings provide new evidence that supports aspects of the ‘iterative model’ hypothesis of the interaction between procedural and conceptual knowledge and highlight the need for educational approaches and tasks that encourage and trigger the interplay of different types of knowledge in young children’s arithmetic problem solving