Development Of Proportional Reasoning Research Articles

Connecting symbolic fractions to their underlying proportions using iterative partitioning.

Fractions are a challenging mathematics topic for many elementary and middle school students, and even for adults. However, a growing body of developmental research suggests that young children can reason about visually presented proportions, well before fraction instruction, providing insight into how fractions might be introduced to improve learning. We designed a card game to teach first and second grade children (N = 195, including a racially and economically diverse sample from the United States) about fractions in one of three ways. In the Actively Divided condition we iteratively divided an area model into equal-sized units, in the Predivided condition we used an area model with the end-state of the Actively Divided condition, and in the Nondivided condition we used a continuous representation of the fraction magnitude that was not divided into unit-sized parts. Children in the actively divided condition demonstrated larger improvements matching symbolic fractions and visual fractions (i.e., pie charts) than children in the other two conditions. Posthoc analyses of children's gameplay revealed that the actively divided condition may have provided a more optimal level of difficulty for young children than the predivided condition, which was particularly difficult, and the nondivided condition, which was trivially easy. These differences in gameplay performance provide insights into possible mechanisms for our results. We discuss open research questions highlighted by this work and implications of these findings for both the development of proportional reasoning and fraction learning. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

Supplemental Material for The Early Development of Proportional Reasoning: A Longitudinal Study of 5- to 8-Year-Olds

Supplemental Material for The Early Development of Proportional Reasoning: A Longitudinal Study of 5- to 8-Year-Olds

Using partial knowledge to inform the creation of learning progressions

The aim of this study was to show that some of the errors made by students when responding to mathematics assessment items can indicate progress in the development of conceptual understanding. By granting partial credit for specific incorrect responses by early secondary students, estimates of the difficulty of demonstrating full and partial knowledge of skills associated with the development of proportional reasoning were determined using Rasch analysis. The errors were confirmed as indicators of progress, and hence partial knowledge, when the thresholds of achievement followed a logical order: The greater the proficiency of the students, the more likely they were to receive a higher score. Consideration of this partial knowledge can enhance the descriptions of the likely behaviours of students at the various levels of learning progressions and this can be informative for teachers in their planning of learning activities.

The Effects of Integrating LEGO Robotics Into a Mathematics Curriculum to Promote the Development of Proportional Reasoning

The Effects of Integrating LEGO Robotics Into a Mathematics Curriculum to Promote the Development of Proportional Reasoning

Progression Trend of Scientific Reasoning from Elementary School to University: a Large-Scale Cross-Grade Survey Among Chinese Students

This study examines progression trends of Chinese students’ scientific reasoning skills across grade levels from elementary school to university. A large-scale survey using the Classroom Test of Scientific Reasoning (CTSR) was conducted with 2669 Chinese students at 13 grade levels (grades 4–16). The construct validity of the CTSR was first examined using Rasch analysis to verify that the various reasoning sub-skills targeted by the test (hypothetical-deductive reasoning, proportional reasoning, correlation reasoning, probabilistic reasoning, and control of variables) form a cohesive cognitive construct. Based on an affirmative result, we proceeded to fit grade-level averages to the best-fit logistic regression model and generated trend lines for the reasoning sub-skills measured by the CTSR. Results show that the cross-grade progression trends generally follow continuous paths through increasingly greater levels of improvement. While all sub-skills display a noticeable growth during middle and high school years, they progress across the grades with different rates. Specifically, the development of proportional reasoning is fast-paced and large, but the progression of hypothetical-deductive reasoning and control of variables is markedly tardy and small. Formal science instruction, particularly the emergence of multiple domain-specific science courses in middle and high school, can influence the cross-grade progression trends. Possible causes and future improvement in science education are proposed. This study contributes to our understanding about the development of scientific reasoning and offers important implications for science instruction and research at different grades.

Supporting Transgender and Gender-Nonconforming Youth Through Teaching Mathematics for Social Justice

Supporting transgender and gender-nonconforming youth in schools involves changes at all levels of education. Gender-complex education, or education that takes into consideration the existence and experiences of transgender and gender-nonconforming people, should be a basic and pervasive part of curricula and should be seen as critical for students of all gender identities and presentations; it must be infused within every content area, including mathematics. This article synthesizes perspectives on gender-complex education, teaching mathematics for social justice, and research on students’ development of proportional reasoning and statistical concepts, and then proposes a mathematics project for middle schoolers to facilitate their agency in challenging transphobia and gender oppression in their schools.

Icelandic 5th-grade girls’ developmental trajectories in proportional reasoning

Understanding of ratio and proportion is critical to the development of higher level mathematical skills. Following Carpenter, Gomez, et al.’s (1999) proposal of a four-level trajectory in the development of proportional reasoning, a 12-week investigation was undertaken of the developmental trajectory of proportional reasoning of girls in two 5th-grade classes in Iceland. Students in these classes were accustomed to instructional practices that encouraged them to devise and explain their own solutions to mathematical problems. Results of the study confirm the learning trajectory with the addition of a further distinct level of development between Level 2 and Level 3. Results showed that girls moved easily, with minimum scaffolding, from Level 1 to 2 and from Level 2 to 3. The transition to Level 4, which involves explicit awareness of ‘within’ and ‘between’ multiplicative relationships, took greater time and effort. Teacher awareness of the four-level learning strategy, with the new emerging Level 3, assists in the design of appropriate problems, class structure, and teaching strategies. Building on Lamon’s notions of unitizing and norming as by Carpenter et al.’s (1999) developmental model, this study contributes to our comprehension of students’ understanding of proportionality and how it develops.

Development of proportional reasoning: where young children go wrong.

Previous studies have found that children have difficulty solving proportional reasoning problems involving discrete units until 10 to 12 years of age, but can solve parallel problems involving continuous quantities by 6 years of age. The present studies examine where children go wrong in processing proportions that involve discrete quantities. A computerized proportional equivalence choice task was administered to kindergartners through 4th-graders in Study 1, and to 1st- and 3rd-graders in Study 2. Both studies involved 4 between-subjects conditions that were formed by pairing continuous and discrete target proportions with continuous and discrete choice alternatives. In Study 1, target and choice alternatives were presented simultaneously; in Study 2, target and choice alternatives were presented sequentially. In both studies, children performed significantly worse when both the target and choice alternatives were represented with discrete quantities than when either or both of the proportions involved continuous quantities. Taken together, these findings indicate that children go astray on proportional reasoning problems involving discrete units only when a numerical match is possible, suggesting that their difficulty is due to an overextension of numerical equivalence concepts to proportional equivalence problems.

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.

Relative and absolute thinking in visual estimation processes

This study has two main goals: (1) to investigate the processes involved in visual estimation (part I of the study), and (2) to investigate the processes of judgment in visual estimation situations, which mostly involved proportional reasoning (part II). The study was conducted with 9-year old children in the third grade. Four strategies were expressed by the children in visual estimation situations. Exposure to a unit in the Agam project, designed to enhance visual estimation capabilities resulted in changes in the children's strategies. These changes reflected the processes by which children overcame their limited ability to process visual information. The development of proportional reasoning was investigated through a series of judgment situations. Although, as was expected, most of the children showed an additive behavior, these situations stimulated some children towards qualitative proportional reasoning, where easy/difficult considerations played an important role.

Components of Understanding in Proportional Reasoning: A Fuzzy Set Representation of Developmental Progressions

The development of proportional reasoning was examined using a temperature mixture task. Each individual's task understanding was assessed by components measuring understanding of various principles of the task. Age differences were found in the mean component scores. More important, different patterns of components were found depending on whether the task was presented numerically or nonumerically. Component patterns also depended on whether the task was presented such that subjects predicted the outcome of combining 2 containers of water at different temperatures (prediction task) or such that subjects inferred 1 of the 2 initial temperatures given the final temperature (reverse task). The results show the importance of distinguishing between intuitive knowledge and formal computational knowledge of proportional concepts and provide a new perspective on how intuitive and computational knowledge are related during development. Finally, the results also led to a new conceptualization of developmental levels as categories with fuzzy boundaries. Under this conceptualization, individuals can have different degrees of membership in “fuzzy developmental levels.” This new concept preserves individual differences but also describes the sequence of development.

The Development of Proportional Reasoning in Qatar

Abstract The development of proportional reasoning in the Arabian Gulf State of Qatar and the United States was examined. The sample consisted of 554 children from Qatar and the U.S. from the fifth, sixth, and seventh grades. The children were administered two paper and pencil tests: the Orange Juice Test developed by Noelting (1980), which measured proportional reasoning, and the Raven's Standard Progressive Matrices Test, which was used as a nonverbal measure of general intelligence. Significant age group, sex, and country main effects and a significant Sex × Country interaction for proportional reasoning were determined even after controlling for nonverbal intelligence.

A Comparison of the Proportional Reasoning Abilities of Learning Disabled and Non-Learning Disabled Children and Adolescents

Noelting's (1980a, 1980b) three parallel instruments on proportional reasoning—two presenting problem-solving tasks in the ratio and division interpretations of fractions and one presenting the tasks in the purely symbolic form of numerical fractions—were administered to 6 female and 41 male learning disabled students, grades four through eight. Performances on the instruments were then compared to the performances of 120 non-learning disabled students in grades five through nine of the same school district. The purpose of the study was to determine whether learning disabled students differed in their development of proportional reasoning and whether their disability was in the use of symbols and language and not in their ability to solve proportional problems. Developmental scalograms, PPR>0.93, resulted in support of the hypothesis that the proportional reasoning abilities of the learning disabled student are developmental and thus not unlike those of the non-learning disabled student. A comparison of the three means for the two groups revealed a reversal in performance with the learning disabled students more successful at problem solving and the non-learning disabled students more successful at the purely symbolic form of numerical fractions. Unlike the non-learning disabled students, the learning disabled students' inability to express a strategy did not indicate an inability to solve the problem.

The development of proportional reasoning and the ratio concept Part II?problem-structure at successive stages; problem-solving strategies and the mechanism of adaptive restructuring

The Orange Juice Experiment, consisting of 23 items of varying degrees of complexity bearing on the concept of ratio, was given to subjects between 6 and 16 years. Stages had been differentiated through statistical analysis and described in Part I of this paper. These stages had been interpreted in terms of the Genevan scale of development. In Part II, given here, the strategies applied at each stage are first analysed, and a second order analysis made to determine the possible processes for passing from one stage to another. A process of “adaptive restructuring” finally evolved, bearing on two periods of development—construction of the ratio concept, and construction of the Common Denominator algorithm.

The development of proportional reasoning and the ratio concept Part I ? Differentiation of stages

Two related problems have to be solved before we can have a clearer picture of cognitive development: (i) Is development hierarchical, leading to higher-order systems controlling lower-order subsytems? (ii) If so, what are the mechanisms involved in a process of development?