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Insight into early precursors of proportional reasoning is necessary to further our theoretical understanding of mathematical development and to guide early interventions. Although several researchers have suggested patterning as a possible precursor for proportional reasoning, there is little empirical evidence to support this assumption, particularly at a young age. To address this gap, the current study explored if patterning in 4- to 5-year-olds (n = 346) is associated with proportional reasoning one and a half years later. Two measures of patterning ability (repeating and growing patterns) and two measures of proportional reasoning (one with discrete quantities and one with a discrete and a continuous quantity) were administered, together with measures addressing general cognitive and numerical abilities. Regression analyses showed that patterning is a unique predictor of proportional reasoning ability over and above sex and general cognitive and numerical abilities. An interaction effect between pattern types and the nature of the quantities was observed: Performance on repeating patterns was uniquely related to performance on proportional reasoning with two discrete quantities, whereas performance on growing patterns was uniquely related to performance on proportional reasoning with a discrete and a continuous quantity. Theoretical implications and suggestions for future studies are discussed.

Young children show large individual differences in their tendency to focus spontaneously on numerical aspects (e.g., numerosities or Arabic number symbols) of their everyday environment. The origins of these individual differences are unclear. Given the role of the home numeracy environment (HNE) in children’s early mathematical development and the assumed link between children’s spontaneous number focusing tendencies and their numerical behavior in everyday life, it is plausible that children’s spontaneous focusing tendencies are related to their HNE. The present study aimed to test this hypothesis by longitudinally investigating children’s spontaneous focusing on numerosity (SFON) and spontaneous focusing on Arabic number symbols (SFONS) in relation to their HNE. Participants were 128 children (4- to 5-year-olds), who were followed from the second until the third year of Flemish kindergarten. In both kindergarten years, children completed a SFON and SFONS Picture task while their parents completed a home numeracy questionnaire. Correlation analyses and structural equation modeling revealed no significant associations between children’s spontaneous number focusing tendencies and their HNE, neither in second nor in third year of kindergarten. This finding suggests that children’s spontaneous number focusing tendencies are not per se related to their HNE. Various possible explanations for this unexpected finding are discussed and directions for further research on this relationship are suggested. ispartof: ZDM Mathematics Education vol:52 issue:4 pages:729-749 status: published

Subtractions of the type M − S = ? can be solved by various strategies, including subtraction by addition. In this study, we investigated children’s use of subtraction by addition by means of reaction time analyses. We presented 106 third to sixth graders with 32 large non-tie single-digit problems in both subtraction (12 − 9 = .) and addition format (9 + . = 12). We examined the fit of three regression models, which represented the consistent use of direct subtraction, of subtraction by addition and of flexibly switching between both strategies based on the relative size of the subtrahend. Findings revealed that children did not switch flexibly between the two strategies, as adults do, but that they rely on direct subtraction for problems presented in subtraction format and on subtraction by addition for problems in addition format. We end with the major theoretical, methodological and educational implications of these results.

Some years ago, Hatano differentiated between routine and adaptive expertise and made a strong plea for the development and implementation of learning environments that aim at the latter type of expertise and not just the former. In this contribution we reflect on one aspect of adaptivity, namely the adaptive use of solution strategies in elementary school arithmetic. In the first part of this article we give some conceptual and methodological reflections on the adaptivity issue. More specifically, we critically review definitions and operationalisations of strategy adaptivity that only take into account task and subject characteristics and we argue for a concept and an approach that also involve the sociocultural context. The second part comprises some educational considerations with respect to the questions why, when, for whom, and how to strive for adaptive expertise in elementary mathematics education.

This study investigates elementary school children’s flexible use of mental calculation strategies on additions and subtractions in the number domain 20–100. Sixty third-graders of three different mathematical achievement levels individually solved a series of 2-digit additions and subtractions in one choice and two no-choice conditions. In the choice condition, children could choose between the compensation (56 + 29 = ?; 56 + 30 = 86, 86 − 1 = 85) and jump strategy (56 + 29 = ?; 56 + 20 = 76, 76 + 9 = 85) on each item. In the two no-choice conditions, children had to solve each item with either the compensation or the jump strategy. The results demonstrated that children of all achievement levels spontaneously applied both the compensation and the jump strategy to solve the items from the choice condition. Furthermore, they all executed the compensation strategy equally accurately, but faster than the jump strategy in the no-choice conditions. Finally, children neither took into account the expected task nor individual strategy efficiency characteristics during the strategy choice process. Results are discussed in terms of recent models of adaptive strategy choices and instructional practices in the number domain 20–100.

This study aimed at analysing traditionally taught children’s acquisition and use of shortcut strategies in the number domain 20–100. One-hundred-ninety-five second, third, and fourth graders of different mathematical achievement levels participated in the study. They were administered two tasks, both consisting of a series of two-digit additions and subtractions that maximally elicit the use of the compensation $$\left( {45 + 29 = \_;45 + 30 - 1 = 75 - 1 = 74} \right)$$ and indirect addition strategy ( $$71 - 68 = \_;\,68 + 2 = 70,\,70 + 1 = 71$$ , so the answer is 2 + 1 or 3). In the first task, children were instructed to solve all items as accurately and as fast as possible with their preferred strategy. The second task was to generate at least two different strategies for each item. Results demonstrated that children of all grades and all achievement levels hardly applied the compensation and indirect addition strategy in the first task. Children’s strategy reports in the second task revealed that younger and lower achieving children did not apply these strategies because they did not (yet) discover these strategies. By contrast, older and higher achieving children appeared to have acquired these strategies by themselves. Results are interpreted in relation to cognitive psychological and socio-cultural perspectives on children’s mathematics learning.

In the research literature several positions concerning the role played by metacognition in adaptive strategy choice can be distinguished. While many authors adhere so-called metacognitive models of strategy choice and strategy change, others have questioned the extent to which metacognitive factors are associated with strategy choice and task performance and have proposed alternative theoretical frameworks wherein strategy choices are described in terms of associative models. In the present article we report data coming from a larger research project on the development of children’s numerosity judgement strategies and skills. The experimental task involved judging numerosities of colored blocks presented in a rectangular grid. Participants were 59 second grade and 50 sixth grade children, whose strategic performance data — obtained by means of a systematic analysis of their response-time patterns — were compared with interview data collected at the end of the experiment. The major result of this comparison is that not only the children from the oldest age group, but also the children from the youngest age group showed clear evidence of metacognitive awareness and understanding of different aspects of their strategic performance.

In this study we investigated the variability, frequency, efficiency, and adaptiveness of young children’s strategy use in the domain of simple addition by means of the choice/no-choice method. Seventy-seven beginning second-graders, divided in 3 groups according to general mathematical ability, solved a series of 25 simple additions in 3 different conditions. In the first condition, children could choose whatever strategy they wanted to solve each problem. In the second and third condition, the same children had to solve all problems with one particular strategy, respectively adding up to 10 and retrieval. The results demonstrate that second-graders as a whole choose adaptively between retrieval, decomposition, and counting strategies when solving simple additions, and that they use these strategies neither equally frequently nor equally efficiently. Furthermore, our results indicate that children with different mathematical ability use generally the same strategies to solve these problems, but differ in the frequency, accuracy and adaptiveness with which they apply these strategies. Finally, this study documents the value of the choice/no-choice method to assess the adaptiveness of young children’s strategy use in the domain of early arithmetic.