Spatial Proportional Reasoning Is Associated With Formal Knowledge About Fractions

Proportional reasoning involves thinking about parts and wholes (i.e., about fractional quantities). Yet, research on proportional reasoning and fraction learning has proceeded separately. This study assessed proportional reasoning and formal fraction knowledge in 8- to 10-year-olds. Participants (N = 52) saw combinations of cherry juice and water in displays that highlighted either part–whole or part–part relations. Their task was to indicate on a continuous rating scale how much each mixture would taste of cherries. Ratings suggested the use of a proportional integration rule for both kinds of displays, although more robustly and accurately for part–whole displays. The findings indicate that children may be more likely to scale proportional components when being presented with part–whole as compared with part–part displays. Crucially, ratings for part–whole problems correlated with fraction knowledge, even after controlling for age, suggesting that a sense of spatial proportions is associated with an understanding of fractional quantities.     

Numerical approximation abilities correlate with and predict informal but not formal mathematics abilities

Previous research has found a relationship between individual differences in children’s precision when nonverbally approximating quantities and their school mathematics performance. School mathematics performance emerges from both informal (e.g., counting) and formal (e.g., knowledge of mathematics facts) abilities. It remains unknown whether approximation precision relates to both of these types of mathematics abilities. In the current study, we assessed the precision of numerical approximation in 85 3- to 7-year-old children four times over a span of 2 years. In addition, at the final time point, we tested children’s informal and formal mathematics abilities using the Test of Early Mathematics Ability (TEMA-3). We found that children’s numerical approximation precision correlated with and predicted their informal, but not formal, mathematics abilities when controlling for age and IQ. These results add to our growing understanding of the relationship between an unlearned nonsymbolic system of quantity representation and the system of mathematics reasoning that children come to master through instruction.     

Editorial: The role of reasoning in mathematical thinking

ABSTRACT

Research into mathematics often focuses on basic numerical and spatial intuitions, and one key property of numbers: their magnitude. The fact that mathematics is a system of complex relationships that invokes reasoning usually receives less attention. The purpose of this special issue is to highlight the intricate connections between reasoning and mathematics, and to use insights from the reasoning literature to obtain a more complete understanding of the processes that underlie mathematical cognition. The topics that are discussed range from the basic heuristics and biases to the various ways in which complex, effortful reasoning contributes to mathematical cognition, while also considering the role of individual differences in mathematics performance. These investigations are not only important at a theoretical level, but they also have broad and important practical implications, including the possibility to improve classroom practices and educational outcomes, to facilitate people's decision-making, as well as the clear and accessible communication of numerical information.     

American and Chinese parental involvement in young children's mathematics learning

This study compared the involvement of American and Chinese mothers in their 5- and 7-year-old children's number learning in their everyday experience and during mother–child interaction on mathematics tasks pertaining to proportional reasoning. Results indicated that Chinese mothers of both the 5- and 7-year-old children were more likely to teach mathematics calculation in their everyday involvement with children's number learning than their American counterparts. No differences were found in maternal instruction between American and Chinese mothers during the mother–child interaction on mathematics tasks. However, maternal instruction was related to Chinese children's learning of proportional reasoning but negligibly related to American children's learning of proportional reasoning.     

Abstracting Sequences: Reasoning That Is a Key to Academic Achievement

The ability to understand sequences of items may be an important cognitive ability. To test this proposition, 8 first-grade children from each of 36 classes were randomly assigned to four conditions. Some were taught sequences that represented increasing or decreasing values, or were symmetrical, or were rotations of an object through 6 or 8 positions. Control children received equal numbers of sessions on mathematics, reading, or social studies. Instruction was conducted three times weekly in 15-min sessions for seven months. In May, the children taught sequences applied their understanding to novel sequences, and scored as well or better on three standardized reading tests as the control children. They outscored all children on tests of mathematics concepts, and scored better than control children on some mathematics scales. These findings indicate that developing an understanding of sequences is a form of abstraction, probably involving fluid reasoning, that provides a foundation for academic achievement in early education.     

明确嵌套集合关系对贝叶斯推理的促进效应

以经典的乳癌问题作为实验任务,通过两个实验分别探讨了有助于明确嵌套集合关系的逐步提问、树图表征等外部表征方式以及元认知调控和被试类型等因素对贝叶斯推理的影响。结果发现：(1)逐步提问对改善贝叶斯推理的成绩没有显著作用;(2)完整和不完整的树图表征显著地促进了推理成绩,但简约的树图表征的促进作用不显著;(3)叙述理由引发的元认知监控显著地促进了推理成绩。(4)文科和理科两组被试的推理成绩没有显著差异     

Proportional Reasoning in 5- to 6-Year-Olds

There have been mixed results in studies investigating proportional reasoning in young children. The current study aimed to examine whether providing visual scaling cues and structuring the reasoning process can improve proportional reasoning in 5- to 6-year-old children. In a series of computerized tasks, children compared the sweetness of 2 mixtures. Each mixture was represented by a juice rectangle stacked on top of a water rectangle. Two rectangles shared the same width but were of same or different heights. The mixtures were scaled by either changing their widths or their heights. In Experiment 1, children’s performance was poor when judging equivalent proportions. In Experiment 2, the 2 mixtures were individually previewed to encourage individual estimation of each mixture and thereby allow participants to strategically reason about the relative proportions. Children performed significantly better than in Experiment 1. In Experiment 3, children explicitly rated the sweetness of each preview mixture. Performance did not improve relative to Experiment 2. Throughout all 3 experiments, children were more sensitive in detecting equivalence when scaling occurred along the width compared with the height, demonstrating the effectiveness of visual-spatial scaling cues. Together, these experiments suggested that visuospatial scaling cues and structuring the 2-step reasoning process using previews can improve 5- to 6-year-olds’ proportional reasoning with certain limitations.     

Children’s and Adults’ Predictions of Black,White, and Multiracial Friendship Patterns

Cross-race friendships can promote the development of positive racial attitudes, yet they are relatively uncommon and decline with age. In an effort to further our understanding of the extent to which children expect cross-race friendships to occur, we examined 4- to 6-year-olds’ (and adults’) use of race when predicting other children’s friendship patterns. In contrast to previous research, we included White (Studies 1 and 2), Black (Study 3), and Multiracial (Study 4) participants and examined how they predicted the friendship patterns of White, Black, and Multiracial targets. Distinct response patterns were found as a function of target race, participant age group, and participant race. Participants in all groups predicted that White children would have mostly White friends and Black children would have mostly Black friends. Moreover, most participant groups predicted that Multiracial children would have Black and White friends. However, White adults predicted that Multiracial children would have mostly Black friends, whereas Multiracial children predicted that Multiracial children would have mostly White friends. These data are important for understanding beliefs about cross-race friendships, social group variation in race-based reasoning, and the experiences of Multiracial individuals more broadly.     

Descartes's Regulae, mathematics, and modern psychology: "the Noblest example of all" in Light of Turing's (1936) On computable Numbers

There are surprisingly strong connections between the philosophy of mind and the philosophy of mathematics. One particular important example can be seen in the Regulae (1628) of Descartes. In "the noblest example of all," he used his new abstract understanding of numbers to demonstrate how the brain can be considered as a symbol machine and how the intellect's algebraic reasoning can be mirrored as operations on this machine. Even though his attempt failed, it is illuminating to explore it because Descartes launched 2 traditions--mechanistic philosophy of mind and abstract mathematics--that would diverge until A. Turing (1936) approached symbolic reasoning in a similar "symbol machine-existence proof" way. Descrates's and Turing's thought experiments, which mark the beginning of modern psychology and cognitive science, respectively, indicate how important the development of mathematics has been for the constitution of the science of mind.     

Theory of mind and emotion understanding predict moral development in early childhood

The current study utilized longitudinal data to investigate how theory of mind (ToM) and emotion understanding (EU) concurrently and prospectively predicted young children's moral reasoning and decision making. One hundred twenty‐eight children were assessed on measures of ToM and EU at 3.5 and 5.5 years of age. At 5.5 years, children were also assessed on the quality of moral reasoning and decision making they used to negotiate prosocial moral dilemmas, in which the needs of a story protagonist conflict with the needs of another story character. More sophisticated EU predicted greater use of physical‐ and material‐needs reasoning, and a more advanced ToM predicted greater use of psychological‐needs reasoning. Most intriguing, ToM and EU jointly predicted greater use of higher‐level acceptance‐authority reasoning, which is likely a product of children's increasing appreciation for the knowledge held by trusted adults and children's desire to behave in accordance with social expectations.     

Development of intuitive rules: Evaluating the application of the dual-system framework to understanding children’s intuitive reasoning

Theories of adult reasoning propose that reasoning consists of two functionally distinct systems that operate under entirely different mechanisms. This theoretical framework has been used to account for a wide range of phenomena, which now encompasses developmental research on reasoning and problem solving. We begin this review by contrasting three main dual-system theories of adult reasoning (Evans & Over, 1996; Sloman, 1996; Stanovich & West, 2000) with a well-established developmental account that also incorporates a dual-system framework (Brainerd & Reyna, 2001). We use developmental studies of the formation and application of intuitive rules in science and mathematics to evaluate the claims that these theories make. Overall, the evidence reviewed suggests that what is crucial to understanding how children reason is the saliency of the features that are presented within a task. By highlighting the importance of saliency as a way of understanding reasoning, we aim to provide clarity concerning the benefits and limitations of adopting a dual-system framework to account for evidence from developmental studies of intuitive reasoning.     

The contribution of logical reasoning to the learning of mathematics in primary school

It has often been claimed that children's mathematical understanding is based on their ability to reason logically, but there is no good evidence for this causal link. We tested the causal hypothesis about logic and mathematical development in two related studies. In a longitudinal study, we showed that (a) 6‐year‐old children's logical abilities and their working memory predict mathematical achievement 16 months later; and (b) logical scores continued to predict mathematical levels after controls for working memory, whereas working memory scores failed to predict the same measure after controls for differences in logical ability. In our second study, we trained a group of children in logical reasoning and found that they made more progress in mathematics than a control group who were not given this training. These studies establish a causal link between logical reasoning and mathematical learning. Much of children's mathematical knowledge is based on their understanding of its underlying logic.     

Hume on the social construction of mathematical knowledge

Mathematics for Hume is the exemplary field of demonstrative knowledge. Ideally, this knowledge is a priori as it arises only from the comparison of ideas without any further empirical input; it is certain because demonstration consist of steps that are intuitively evident and infallible; and it is also necessary because the possibility of its falsity is inconceivable as it would imply a contradiction. But this is only the ideal, because demonstrative sciences are human enterprises and as such they are just as fallible as their human practitioners. According to the reading suggested here, Hume develops a radical sceptical challenge for mathematics, and thereby he undermines the knowledge claims associated with demonstrative reasoning. But Hume does not stop there: he also offers resources for a sceptical solution to this challenge, one that appeals crucially to social practices, and sketches the social genealogy of a community-wide mathematical certainty. While explaining this process, he relies on the conceptual resources of his faculty psychology that helps him to distinguish between the metaphysics and practices of mathematical knowledge. His account explains why we have reasons to be dubious about our reasoning capacities, and also how human nature and sociability offers some remedy from these epistemic adversities.

     

The Influence of Relational Knowledge and Executive Function on Preschoolers’ Repeating Pattern Knowledge

Children’s knowledge of repeating patterns (e.g., ABBABB) is a central component of early mathematics, but the developmental mechanisms underlying this knowledge are currently unknown. We sought clarity on the importance of relational knowledge and executive function (EF) to preschoolers’ understanding of repeating patterns. One hundred twenty-four children aged 4 to 5 years old were administered a relational knowledge task, 3 EF tasks (working memory, inhibition, set shifting), and a repeating pattern assessment before and after a brief pattern intervention. Relational knowledge, working memory, and set shifting predicted preschoolers’ initial pattern knowledge. Working memory also predicted improvements in pattern knowledge after instruction. The findings indicated that greater EF ability was beneficial to preschoolers’ repeating pattern knowledge and that working-memory capacity played a particularly important role in learning about patterns. Implications are discussed in terms of the benefits of relational knowledge and EF for preschoolers’ development of patterning and mathematics skills.     

RECENT COGNITIVE THEORIES APPLIED TO SEQUENTIAL LENGTH MEASURING KNOWLEDGE IN YOUNG CHILDREN

Summary . This research was designed to determine sequential length measuring knowledge in children aged 3–7 years. Sequences were predicted in advance logically from measurement theory, from a review of the literature, and from the information processing demand of the tasks. A sample of 80 children from mixed socio-economic backgrounds was tested on measures of capacity to process information and 15 main measurement tasks. Analysis of the data showed that the empirical sequence of length measuring knowledge was most like that predicted from analyses of the information processing demand of the tasks. It is asserted that mathematics curriculum content could be sequenced on the basis of similar information processing analysis.     

Rule transition on the balance scale task: a case study in belief change

For various domains in proportional reasoning cognitive development is characterized as a progression through a series of increasingly complex rules. A multiplicative relationship between two task features, such as weight and distance information of blocks placed at both sides of the fulcrum of a balance scale, appears difficult to discover. During development, children change their beliefs about the balance scale several times: from a focus on the weight dimension (Rule I) to occasionally considering the distance dimension (Rule II), guessing (Rule III), and applying multiplication (Rule IV; Siegler, 1981). Because of the detailed empirical findings the balance scale task has become a benchmark task for computational models of proportional reasoning. In this article, we present a large empirical study (N = 420) of which the findings provide a challenge for computational models. The effect of feedback and the effect of individually adapted training items on rule transition were tested for children using Rule I or Rule II. Presenting adapted training items initiates belief revision for Rule I but not for Rule II. The experience of making mistakes (by providing feedback) induces a change for both Rule I and Rule II. However, a delayed posttest shows that these changes are preserved after 2 weeks only for children using Rule I. We conclude that the transition from Rule I to Rule II differs from the transition from Rule II to a more complex rule. Concerning these empirical findings, we will review performance of computational models and the implications for a future belief revision model.

It is one Thing, to show a Man that he is in an Error, and another, to put him in possession of Truth. John Locke

     

工作记忆训练及对数学能力的迁移作用

工作记忆缺陷会影响个体数学能力发展。通过记忆策略、广度、刷新、转换等功能的工作记忆训练,可以改善个体认知功能。然而,工作记忆训练对个体的阅读、数学、流体智力等方面的远迁移效果并不一致。研究表明,工作记忆训练可以改善数感、视觉空间能力、推理能力等数学一般技能;也会通过改善语音工作记忆以及空间能力促进数学计算能力,或者通过改善中央执行系统,提升数学问题表征、模式识别、解题迁移、策略选择等复杂的过程,从而促进数学问题解决能力。因此,区分不同数学任务的认知过程,可以获得工作记忆训练对数学能力迁移效果的进一步证据。今后,神经影像学的证据或许也是未来工作记忆训练对数学能力提高的又一佐证。     

Arithmetic knowledge from the spontaneous attention to relations

Children's knowledge of arithmetic principles is a key aspect of early mathematics knowledge. Knowledge of arithmetic principles predicts how children approach solving arithmetic problems and the likelihood of their success. Prior work has begun to address how children might learn arithmetic principles in a classroom setting. Understanding of arithmetic principles involves understanding how numbers in arithmetic equations relate to another. For example, the Relation to Operands (RO) principle is that for subtracting natural numbers (A ? B = C), the difference (C) must be smaller than the minuend (A). In the current study we evaluate if individual differences in arithmetic principle knowledge (APK) can be predicted by the learners' spontaneous attention to relations (SAR) and if feedback can increase their attention to relations. Results suggest that participants’ Spontaneous Attention to Number (SAN) does not predict their knowledge of the RO principle for symbolic arithmetic. Feedback regarding the attention to relations did not show a significant effect on SAR or participants’ APK. We also did not find significant relations between reports of parent talk and the home environment with individual differences in SAN. The amount of parent's talk about relations was not significantly associated with learner's SAR and APK. We conclude that children's SAR with non‐symbolic number does not generalize to attention to relations with symbolic arithmetic.     

The role of planning in different mathematical skills

Despite the fact that planning has been found to be a significant predictor of reading (particularly of reading comprehension), much less is known about its contribution to mathematics. The purpose of this study was to examine the role of two levels of planning (operation planning and action planning) in three mathematical skills (calculation fluency, math problem-solving, and math reasoning). Eighty Grade 2 children from Shanghai, China were assessed on measures of nonverbal cognitive ability (nonverbal matrices), working memory (digit span backwards and N-back), operation planning (matching numbers, planned codes, and planned search), action planning (crack the code), and mathematics (calculation fluency, math problem-solving, and math reasoning). The results of regression analyses showed that both levels of planning accounted for unique variance in mathematics over and above the effects of nonverbal cognitive ability and working memory. The effects of action planning were particularly strong in math problem-solving. These findings suggest that measures of planning could be used along with measures of working memory to detect children at-risk for mathematics disabilities and that intervention programmes targeting planning could be developed to boost children's mathematics performance.     

Choosing between hearts and minds: Children's understanding of moral advisors

Moral development research has often focused on the development of moral reasoning without considering children's understanding of moral advisors. We investigated how children construe sources of moral advice by examining the characteristics that children deem necessary for reasoning about moral or scientific problems. In two experiments, children in grades K, 2, and 4 were presented with dilemmas of a moral nature or scientific nature and chose between two advisors. Second and fourth graders chose advisors differentially based on their expertise, while kindergartners did not discriminate between advisors. In a third experiment, older children indicated that only certain characteristics are needed to solve moral or scientific problems, and they endorsed these characteristics differentially based on the problem to be solved. Thus, by middle childhood, children construe moral knowledge as distinct from scientific knowledge and select advisors in each area accordingly.