The Development of Proportional Reasoning: Effect of Continuous Versus Discrete Quantities

This study examines the development of children's ability to reason about proportions that involve either discrete entities or continuous amounts. Six-, 8- and 10-year olds were presented with a proportional reasoning task in the context of a game involving probability. Although all age groups failed when proportions involved discrete quantities, even the youngest age group showed some success when proportions involved continuous quantities. These findings indicate that quantity type strongly affects children's ability to make judgments of proportion. Children's greater success in judging proportions involving continuous quantities appears to be related to their use of different strategies in the presence of countable versus noncountable entities. In two discrete conditions, children—particularly 8- and 10-year-olds—adopted an erroneous counting strategy, considering the number of target elements but not the relation between target and nontarget elements, either in terms of number or amount. In contrast, in the continuous condition, when it was not possible to count, children may have relied on an early developing ability to code the relative amounts of target and nontarget regions.     

小学低年级儿童的等值分数概念发展及干预

等值分数概念的发展要以相对量概念和乘法思维为基础。实验1将小学一至三年级儿童等值分数概念的发展划分为3个阶段：整体量概念、数量化的相对量概念、正式的等值分数概念, 结果表明一年级儿童尚未获得数量化的相对量概念, 二年级儿童尚未发展起成熟的乘法思维。基于此, 依据最近发展区原理设计了干预实验。实验2的干预方法是在一年级儿童的整体量概念基础上促进其数量化的相对量概念的发展, 实验3的干预方法是通过熟悉的任务情境来促进二年级儿童对乘法关系的实际意义的理解, 从而促进其乘法思维的发展。这些干预方法达到了预期效果, 为开展等值分数的早期教学提供了借鉴。     

Generalising principles in spite of procedural differences: Children's understanding of division

Between ages 5 and 7, children are known to be quite good at sharing discrete quantities but very bad at sharing continuous quantities. Our aim was to find whether they can transfer their understanding of logical relations from discrete to continuous quantities though the procedures used in sharing these quantities are markedly different.Two samples of 5- to 7-year-olds participated in two studies. In the first study, the items involved partitive division; in the second, quotitive division tasks. In both studies, the children solved tasks with discrete and continuous quantities.Performance varied significantly across age level and logical principle (equivalence between different rounds of sharing versus inverse relation between the divisor and the quotient) but not across type of quantity (discrete versus continuous). There was a very strong relation between performance across type of quantity. We conclude that children can generalise reasoning principles in division across type of quantity in spite of the difference in sharing procedures.     

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.     

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.     

Relational and Representational Aspects of Early Number Development

Two experiments examined the role of representations of numerosity in children's reasoning about relations between two quantities. In the first experiment, 3- and 4-year-old children were able to solve matching-to-sample problems on the basis of both the numerosity of a focal set (number matches) and the correspondence relation between two sets (relational matches); but in a conflict condition, the younger children used only the relational information. In the second experiment, both the 3- and the 4-year-olds showed very high levels of success in inferring numerical equivalence from commutativity relations between two pairs of sets. Children were successful even when one of the sets in each pair was covered, so that children could not directly enumerate the quantities to be compared. Both findings support the notion that relational reasoning originates independently of processes for representing numerosity.     

Belief bias is stronger when reasoning is more difficult

Three studies examine the influence of varying the difficulty of reasoning on the extent of belief bias, while minimising the possibility that the manipulation would influence the way participants approach the task. Specifically, reasoning difficulty was manipulated by making variations in problem content, while maintaining all other aspects of the problems constant. In Study 1, 191 participants were presented with consistent and conflict problems varying in two levels of difficulty. The results showed a significant influence of problem difficulty on the extent of the belief bias, such that the effect of belief was more pronounced for difficult problems. This effect was stronger in Study 2 (73 participants) where the difference in the difficulty of the problems was purposely accentuated. The results of both studies stress the importance of controlling for problem difficulty when studying belief bias. Study 3 examined one consequence of this, i.e., the classic belief vs. logic interaction could be eliminated by manipulating problem difficulty. Theoretical implications for dual-process accounts of belief bias are also discussed.     

Conceptions of career choice and attainment: Developmental levels in how children think about careers

Children’s Conceptions of career choice and attainment were evaluated in two studies to test whether reasoning levels varied by grade level (Studies 1 and 2) and perspective-taking complexity (Study 2). Results indicated that younger children (Grade K) were more likely to use reasoning strategies associated with fantasy and magical thinking and older children (Grade 6) were more likely to consider personal interests, abilities, and job requirements. Study 2 replicated these results and also found that children evaluated as able to use more complex perspective-taking reported higher reasoning levels when discussing their Conceptions of career choice and attainment.     

Child proportional scaling: Is 1/3 = 2/6 = 3/9 = 4/12?

The current experiments examined the role of scale factor in children's proportional reasoning. Experiment 1 used a choice task and Experiment 2 used a production task to examine the abilities of kindergartners through fourth-graders to match equivalent, visually depicted proportional relations. The findings of both experiments show that accuracy decreased as the scaling magnitude between the equivalent proportions increased. In addition, children's errors showed that the cost of scaling proportional relations is symmetrical for problems that involve scaling up and scaling down. These findings indicate that scaling has a cognitive cost that results in decreasing performance with increasing scaling magnitude. These scale factor effects are consistent with children's use of intuitive strategies to solve proportional reasoning problems that may be important in scaffolding more formal mathematical understanding of proportional relations.     

How Mathematics Propels the Development of Physical Knowledge

Three studies examined whether mathematics can propel the development of physical understanding. In Experiment 1, 10-year-olds solved balance scale problems that used easy-to-count discrete quantities or hard-to-count continuous quantities. Discrete quantities led to age typical performances. Continuous quantities caused performances like those of 5-year-olds. In Experiment 2, 11-year-olds solved problems with feedback. They were encouraged to use math or words to justify their answers. Children who used math developed an understanding superior to most adults, whereas children who used words did not. In Experiment 3, 9-year-olds solved problems with or without prompts to use math. Children encouraged to use math exhibited greater qualitative understanding, even though they were unable to discover metric proportions. The results indicate it is possible to design symbolic experiences to propel the development of physical understanding, thereby relating developmental psychology to instructional theory.     

Does half a pizza equal half a box of chocolates?: Proportional matching in an analogy task

There is now a considerable literature demonstrating analogical reasoning in children as young as 3 and 4 years of age. Here, we used analogy as a sensitive measure of proportional understanding in young children. In two experiments, we examined whether children's performance in a proportional analogy task would be affected when concrete models evoking different kinds of conceptual referents were used as the basis for the analogies. We chose two different conceptual referents (pizza and chocolates) of the kind typically used in fractions instruction. In both experiments, children were shown a base substance by the experimenter (e.g., a whole pizza) from which a proportion was then removed (e.g., a half pizza). Children were asked to complete the analogy by removing an equivalent proportion of their own target set (e.g., a whole box of chocolates changed to half a box of chocolates). This proportional matching paradigm resulted in analogy problems of the form: 8/8 pizza: 4/8 pizza: 4/4 box of chocolates:2/4 box of chocolates. Results indicated that 3- to 4-year-old children do have an emergent understanding of proportional equivalence, even when the materials to be matched are not isomorphic.     

Talking about proportion: Fraction labels impact numerical interference in non‐symbolic proportional reasoning

Across two experiments, we investigated how verbal labels impact the way young children attend to proportional information, well before the introduction of formal fraction education. Five‐ to seven‐year‐old children were introduced to equivalent non‐symbolic proportions labeled in one of three ways: (a) a single, categorical label for multiple fractions (both 3/4 and 6/8 referred to as “blick”), (b) labels that focused on the numerator [e.g., 3/4 labeled as “three blicks” (Experiment 1) or “three‐fourths” (Experiment 2)], or (c) labels that had a complete part‐whole structure (“three‐out‐of‐four”). Children then completed measures of non‐symbolic proportional reasoning that pitted whole‐number information against proportional information for novel proportions. Across both experiments, children who heard the categorical labels were more likely to match non‐symbolic displays based on proportion than children in any of the other conditions, who demonstrated higher levels of numerical interference. These findings suggest that fraction labels have the potential to shape children's attention to proportional information even in the context of non‐symbolic part‐whole displays and for children who are not familiar with formal fraction symbols. We discuss these findings in terms of children's developing understanding of proportional reasoning and its implications for fraction education.     

Toddler subtraction with large sets: further evidence for an analog-magnitude representation of number

Two experiments were conducted to test the hypothesis that toddlers have access to an analog-magnitude number representation that supports numerical reasoning about relatively large numbers. Three-year-olds were presented with subtraction problems in which initial set size and proportions subtracted were systematically varied. Two sets of cookies were presented and then covered. The experimenter visibly subtracted cookies from the hidden sets, and the children were asked to choose which of the resulting sets had more. In Experiment 1, performance was above chance when high proportions of objects (3 versus 6) were subtracted from large sets (of 9) and for the subset of older participants (older than 3 years, 5 months; n = 15), performance was also above chance when high proportions (10 versus 20) were subtracted from the very large sets (of 30). In Experiment 2, which was conducted exclusively with older 3-year-olds and incorporated an important methodological control, the pattern of results for the subtraction tasks was replicated. In both experiments, success on the tasks was not related to counting ability. The results of these experiments support the hypothesis that young children have access to an analog-magnitude system for representing large approximate quantities, as performance on these subtraction tasks showed a Weber's Law signature, and was independent of conventional number knowledge.     

Inferences from decision difficulty

We propose that people infer the relative attractiveness of the choice alternatives from decision difficulty. A difficult decision signifies that the alternatives are close to each other in attractiveness, and an easy decision signifies that the alternatives are remote from each other in attractiveness. In Study 1, observers used reported decision difficulty to infer preferences of the decision maker. Studies 2-4 showed that inferences about the source of one’s own decision difficulty may affect a decision maker’s preferences. Study 4 integrates the notion of inferences from decision difficulty with dissonance theory, showing that in repeatable decisions difficulty reduces post-decisional spreading of alternatives, as predicted by our model, whereas with one-time decisions, difficulty enhances post-decisional spreading of alternatives, as predicted by dissonance theory.     

Teleo‐functional constraints on preschool children's reasoning about living things

These studies explore the degree to which preschool children employ teleological‐functional reasoning – reasoning based on the assumption of function and design – when making inferences about animal behavior. Using a triad induction method, Study 1 examined whether a sensitivity to biological function would lead children to overlook overall similarity and instead attend to relevant functional cues (in the presence of overall dissimilarity), as a basis for generalizing behavioral properties to unfamiliar animals. It found that, between 3 and 4 years of age, children, with increasing consistency, attend to functional features rather than overall similarity when drawing inferences about animal behavior. Children's ability to describe the relevance of functional adaptations to animal behavior also increased with age. Study 2 explored whether Study 1 findings might result from stimulus biases in favor of the function‐based choice. It found that children's attention shifted from functional features to overall similarity when generalizing labels rather than behaviors with the same triads. These results are discussed in relation to the development of biological knowledge.     

Choice behavior of rats in a concurrent-chains schedule: Amount and delay of reinforcement

Rats were exposed to concurrent-chains schedules in which the terminal links were equal fixed-interval schedules terminating in one or three food pellets. Choice proportions for large reward increased with increases in delay intervals programmed on fixed-interval schedules and supported the predictions derived from a general choice model originally formulated by Fantino and later developed by Navarick and Fantino. In addition, a functional equivalence of two alternatives was established by increasing delay intervals with large reward, whereas delay intervals for small reward were held constant. Functionally equivalent delay intervals with large reward, for each delay interval with small reward, can be described by a power function with exponent smaller than 1.0. A better prediction of choice proportions resulted when this function was used to derive predicted choice proportions.     

IMPULSIVITY IN STUDENTS WITH SERIOUS EMOTIONAL DISTURBANCE: THE INTERACTIVE EFFECTS OF REINFORCER RATE,DELAY, AND QUALITY

We conducted two studies extending basic matching research on self-control and impulsivity to the investigation of choices of students diagnosed as seriously emotionally disturbed. In Study 1 we examined the interaction between unequal rates of reinforcement and equal versus unequal delays to reinforcer access on performance of concurrently available sets of math problems. The results of a reversal design showed that when delays to reinforcer access were the same for both response alternatives, the time allocated to each was approximately proportional to obtained reinforcement. When the delays to reinforcer access differed between the response alternatives, there was a bias toward the response alternative and schedule with the lower delays, suggesting impulsivity (i.e., immediate reinforcer access overrode the effects of rate of reinforcement). In Study 2 we examined the interactive effects of reinforcer rate, quality, and delay. Conditions involving delayed access to the high-quality reinforcers on the rich schedule (with immediate access to low-quality reinforcers earned on the lean schedule) were alternated with immediate access to low-quality reinforcers on the rich schedule (with delayed access to high-quality reinforcers on the lean schedule) using a reversal design. With 1 student, reinforcer quality overrode the effects of both reinforcer rate and delay to reinforcer access. The other student tended to respond exclusively to the alternative associated with immediate access to reinforcers. The studies demonstrate a methodology based on matching theory for determining influential dimensions of reinforcers governing individuals' choices.     

Predicting the difficulty of complex logical reasoning problems

The aim of the present research was to develop a difficulty model for logical reasoning problems involving complex ordered arrays used in the Graduate Record Examination. The approach used involved breaking down the problems into their basic cognitive elements such as the complexity of the rules used, the number of mental models required to represent the problem, and question type. Weightings for these different elements were derived from two experimental studies and from the reasoning literature. Based on these weights, difficulty models were developed which were then tested against new data. The models had excellent predictive validity and showed the relative influence of rule based factors and factors relating to the number of underlying models. Different difficulty models were needed for different question types, suggesting that people used a variety of approaches and, at a wider level, that both mental models and mental rules may be used in reasoning.     

Variation of Preference Inconsistency When Applying Ratio and Interval Scale Pairwise Comparisons

Several studies on numerical rating in discrete choice problems address the tendency of inconsistencies in decision makers' measured preferences. This is partly due to true inconsistencies in preferences or the decision makers' uncertainty on what he or she really wants. This uncertainty may be reflected in the elicited preferences in different ways depending on the questions asked and methods used in deriving the preferences for alternatives. Some part of the inconsistency is due to only having a discrete set of possible judgments. This study examined the variation of preference inconsistency when applying different pairwise preference elicitation techniques in a five‐item discrete choice problem. The study data comprised preferences of five career alternatives elicited applying interval scale and numerically and verbally anchored ratio scale pairwise comparisons. Statistical regression technique was used to analyse the differences of inconsistencies between the tested methods. The resulting relative residual variances showed that the interval ratio scale comparison technique provided the greatest variation of inconsistencies between respondents, thus being the most sensitive to inconsistency in preferences. The numeric ratio scale comparison gave the most uniform preferences between the respondents. The verbal ratio scale comparison performed between the latter two when relative residual variances were considered. However, the verbal ratio scale comparison had weaker ability to differentiate the alternatives. The results indicated that the decision recommendation may not be sensitive to the selection between these preference elicitation methods in this kind of five‐item discrete choice problem. The numeric ratio scale comparison technique seemed to be the most suitable method to reveal the decision makers' true preferences. However, to confirm this result, more studying will be needed, with an attention paid to users' comprehension and learning in the course of the experiment. Copyright © 2013 John Wiley & Sons, Ltd.