Spontaneous Focusing on Quantitative Relations in the Development of Children's Fraction Knowledge

While preschool-aged children display some skills with quantitative relations, later learning of related fraction concepts is difficult for many students. We present two studies that investigate young children's tendency of Spontaneous Focusing On quantitative Relations (SFOR), which may help explain individual differences in the development of fraction knowledge. In the first study, a cross-sectional sample of 84 kindergarteners to third graders completed tasks measuring their spontaneous recognition and use of quantitative relations and then completed the tasks again with explicit guidance to focus on quantitative relations. Findings suggest that SFOR is a measure of the spontaneous focusing of attention on quantitative relations and the use of these relations in reasoning. In the second (longitudinal) study, 25 first graders completed measures of SFOR tendency and a measure of fraction knowledge three years later. SFOR tendency was found to predict fraction knowledge, suggesting that it plays a role in the development of fraction knowledge.</p

Spontaneous focusing on quantitative relations as a predictor of rational number and algebra knowledge

Spontaneous Focusing On quantitative Relations (SFOR) has been found to predict the development of rational number conceptual knowledge in primary students. Additionally, rational number knowledge has been shown to be related to later algebra knowledge. However, it is not yet clear: (a) the relative consistency of SFOR across multiple measurement points, (b) how SFOR tendency and rational number knowledge are inter-related across multiple time points, and (c) if SFOR tendency also predicts algebra knowledge. A sample of 140 third to fifth graders were followed over a four-year period and completed measures of SFOR tendency, rational number conceptual knowledge, and algebra knowledge. Results revealed that the SFOR was relatively consistent over a one-year period, suggesting that SFOR is not entirely context-dependent, but a more generalizable tendency. SFOR tendency was in a reciprocal relation with rational number conceptual knowledge, each being uniquely predictive of the other over a four-year period. Finally, SFOR tendency predicted algebra knowledge three-years later, even after taking into account non-verbal intelligence and rational number knowledge. The results of the present study provide further evidence that individual differences in SFOR tendency may have an important role in the development of mathematical knowledge, including rational numbers and algebra.</p

Spontaneous mathematical focusing tendencies in mathematical development

Children's own spontaneous mathematical activities are crucial for their mathematical development. Mathematical thinking and learning does not only occur in explicitly mathematical situations, such as the classroom. Those children with higher tendencies to recognize and use mathematical aspects of their everyday surroundings, both within the classroom and without, appear to have an advantage in learning formal mathematical skills and knowledge. In this introduction to the special issue, we provide an overview of the existing literature on spontaneous mathematical focusing tendencies. We then provide a brief overview of the contributions to the special issue

Maternal sensitivity in responding during play and children’s pre-mathematical skills: a longitudinal study from infancy to preschool age

This longitudinal study explored how mothers’ sensitivity in responding to their child’s cognitive and emotional needs in infancy and toddlerhood predicts children’s pre-mathematical skills at early preschool age. The sample consisted of 65 mother-child dyads (N = 130 individuals) videotaped during joint play at ages 1;0 and 2;0. The children’s pre-mathematical skills were tested at age 3;0. The path analyses showed that, in infancy, mothers’ autonomy support and scaffolding are more strongly related than emotional support to children’s later performance on spatial and numerical tasks. The findings are discussed in relation to how maternal sensitivity in responding fosters children’s pre-mathematical development in an optimal way.</p

Young children’s recognition of quantitative relations in mathematically unspecified settings

Children have been found to be able to reason about quantitative relations, such as non- symbolic proportions, already by the age of 5 years. However, these studies utilize settings in which children were explicitly guided to notice the mathematical nature of the tasks. This study investigates children&rsquo;s spontaneous recognition of quantitative relations on mathe- matically unspecified settings. Participants were 86 Finnish-speaking children, ages 5&ndash;8. Two video-recorded tasks, in which participants were not guided to notice the mathe- matical aspects, were used. The tasks could be completed in a number of ways, including by matching quantitative relations, numerosity, or other aspects. Participants&rsquo; matching strategies were analyzed with regard to the most mathematically advanced level utilized. There were substantial differences in participants&rsquo; use of quantitative relations, numerosity and other aspects in their matching strategies. The results of this novel experimental set- ting show that investigating children&rsquo;s spontaneous recognition of quantitative relations provides novel insight into children&rsquo;s mathematical thinking and furthers the understand- ing of how children recognize and utilize mathematical aspects when not explicitly guided to do so.</p

Preschool spontaneous focusing on numerosity predicts rational number conceptual knowledge 6 years later

Recent evidence suggests that early natural number knowledge is a predictor of later rational number conceptual knowledge, even though students&rsquo; difficulties with rational numbers have also been explained by the overuse of natural number concepts&mdash;often referred to as the natural number bias. Hannula and Lehtinen (Learn Instr 15:237&ndash;256, 2005) have shown that children&rsquo;s tendency to spontaneously focus on numerosity (SFON) predicts the development of natural number and arithmetic skills. The present study follows 36 children from the age of 6 years to the age of 12 years in order to determine how preschool SFON tendency and number&nbsp; sequence skills are related to rational number conceptual knowledge at the age of 12 years.&nbsp; The results show that children&rsquo;s SFON tendency before school age is a strong predictor of later rational numbers conceptual knowledge, even after controlling for preschool number sequence skills. This finding has implications for the understanding of how the transition from reasoning about natural number concepts to reasoning about rational numbers may be influenced by children&rsquo;s self-initiated practice with numbers in everyday situations.</p

Spontaneous focusing on quantitative relations as a predictor of the development of rational number conceptual knowledge

Many people have serious difficulties in understanding rational numbers, limiting their ability to interpret and make use of them in modern daily life. This also leads to later difficulties in learning more advanced mathematical content. In this study, novel tasks are used to measure 263 late primary school students’ spontaneous focusing on quantitative relations, in situations that are not explicitly mathematical. Even after controlling for a number of known predictors of rational number knowledge, spontaneous focusing on quantitative relations is found to have a strong impact on students’ learning of rational number conceptual knowledge. This finding opens new possibilities for developing pedagogical solutions to one of the most difficult challenges of mathematics education. The findings suggest that students’ own focusing tendency and self-initiated practice may have on important role in the long-term development of complex cognitive skills.</p

Predicting adaptive expertise with rational number arithmetic

Background. Adaptive expertise is a highly valued outcome of mathematics curricula. One aspect of adaptive expertise with rational numbers is adaptive rational number knowledge, which refers to the ability to integrate knowledge of numerical characteristics and relations in solving novel tasks. Even among students with strong conceptual and procedural knowledge of rational numbers, there are substantial individual differences in adaptive rational number knowledge.Aims. We aimed to examine how a wide range of domain-general and mathematically specific skills and knowledge predicted different aspects of rational number knowledge, including procedural, conceptual, and adaptive rational number knowledge.Sample. 173 6th and 7th grade students from a school in the southeastern US (51% female) participated in the study.Methods. At three time points across 1.5 years, we measured students’ domaingeneral and domain-specific skills and knowledge.Weused multiple hierarchal regression analysis to examine how these predictors related to rational number knowledge at the third time point.Result. Prior knowledge of rational numbers, general mathematical calculation knowledge, and spontaneous focusing on multiplicative relations (SFOR) tendency uniquely predicted adaptive rational number knowledge, after taking into account domain-general and mathematically specific skills and knowledge. Although conceptual knowledge of rational numbers and general mathematical achievement also predicted later conceptual and procedural knowledge of rational numbers, SFOR tendency did not.Conclusion. Results suggest expanding investigations of mathematical development to also explore different features of adaptive expertise as well as spontaneous mathematical focusing tendencies.</p

Distinguishing adaptive from routine expertise with rational number arithmetic

Adaptive expertise is a valued, but under-examined, feature of students' mathematical development (e.g. Hatano & Oura, 2012). The present study investigates the nature of adaptive expertise with rational number arithmetic. We therefore examined 394 7th and 8th graders’ rational number knowledge using both variable-centered and person-centered approaches. Performance on a measure of adaptive expertise with rational number arithmetic, the arithmetic sentence production task, appeared to be distinct from more routine features of performance. Even among the top 45% of students, all of whom had strong routine procedural and conceptual knowledge, students varied greatly in their performance on the arithmetic sentence production task. Strong performance on this measure also predicted later algebra knowledge. The findings suggest that it is possible to distinguish adaptive expertise from routine expertise with rational numbers and that this distinction is important to consider in research on mathematical development.</div

Spontaneous focusing on numerosity in preschool as a predictor of mathematical skills and knowledge in the fifth grade

Previous studies in a variety of countries have shown that there are substantial individual differences in children’s spontaneous focusing on numerosity (SFON), and these differences are positively related to the development of early numerical skills in preschool and primary school. A total of 74 5-year-olds participated in a 7-year follow-up study, in which we explored whether SFON measured with very small numerosities at 5 years of age predicts mathematical skills and knowledge, math motivation, and reading in fifth grade at 11 years of age. Results show that preschool SFON is a unique predictor of arithmetic fluency and number line estimation but not of rational number knowledge, mathematical achievement, math motivation, or reading. These results hold even after taking into account age, IQ, working memory, digit naming, and cardinality skills. The results of the current study further the understanding of how preschool SFON tendency plays a role in the development of different formal mathematical skills over an extended period of time.</p