Piagetian tasks as readiness measures in mathematics instruction: A critical review

The purpose of this review was to synthesize the research on the relationship between performance on Piagetian tasks of concrete operations and performance on mathematics learning tasks in order to determine whether the Piagetian tasks make useful readiness measures. The research consistently indicates that, while there is a positive correlation between performance on Piagetian tasks and mathematics achievement, many school mathematics tasks can be mastered by children who have not yet developed the reasoning abilities measured by Piagetian tasks. This does not diminish the validity of Piaget's theory, but rather suggests that the kinds of reasoning processes identified by Piaget are not essential in solving many school mathematics tasks.     

Relationships among gender,age, and intellectual development

The purpose of this study was to investigate the relationships among gender, age, and intellectual development. Random samples of 70 females and 70 males were selected with each sex group equally divided into a low-age and a high-age group. The low-age group ranged in age from 16.25 years to 16.75 years and the high-age group from 16.76 years to 17.25 years. The Piaget tasks selected to measure cognitive development were: Conservation of Volume, Separation of Variables, and Equilibrium in the Balance and Combination of Colorless Chemical Liquids. Data from this research produced these findings: (1) males demonstrate a higher level of intellectual development than females, (2) males mature intellectually earlier than females, (3) the value of the conservation of volume task as a component of a battery of formal tasks depends upon whether the decisions are to be made on the basis of the total-task results or on individual task performance, and (4) there appear to be factors other than age and gender that are related to the development of formal operational reasoning. These investigators hypothesize that experiences is another important factor.     

Reexamining spatial ability within a Piagetian framework

The goal of this study was to identify and examine results of Piagetian spatial research studies. For inclusion in this review, the studies had to (a) entail individual clinical interviews, (b) involve the manipulation of objects and explanation of answers by subjects, (c) offer written protocols and scoring criteria, and (d) provide documentation of male and female results. Involving over 7,600 individualized clinical interviews, the primary problem focused on whether gender differences existed in Piagetian spatial assessments and, if so, discussion of the significance and implications of the differences. Twenty-three studies assessing for 22 topologic, Euclidean, and projective spatial structures were reviewed and total male and female results compiled. A chi-square test for two independent samples showed that the majority of performance assessments did not demonstrate a significant difference between males and females. Males significantly outperformed females on 8 of the 22 spatial structure tasks. However, as with other gender studies, similarities between male and female performances far outweighed any differences. The clinical interview results provide evidence to support the overall poor spatial ability of both males and females. Educational practices designated to improve spatial abilities should not be a female-only endeavor. © 1996 John Wiley & Sons, Inc.     

The role of physical content in piagetian spatial tasks: Sex differences in spatial knowledge?

Sex-related differences on Piagetian horizontality (water level) and verticality (plumb line) tasks were examined in 64 college students. It was hypothesized that females' difficulties on these Euclidean spatial problems are due not to differences in underlying spatial competence, but rather to differences in knowledge of task specific information about the physical properties of water levels and plumb lines. This was tested by presenting subjects with the standard water level and plumb line problems and also modified problems not requiring knowledge of physical principles (i.e., drawing “straight up and down” or “straight across” lines inside tipped rectangles). While males were expected to outperform females on the standard tasks, no sex differences were expected on the modified tasks. Results of an ANOVA on scores for horizontality and verticality each showed main effects for sex and task version but failed to reveal the hypothesized interaction. However, performance on the Euclidean spatial tasks was also considered in terms of overall success versus failure. While males were more successful than females in the standard format, males and females were equally successful in the modified, nonphysical, format. Hence, college aged males and females generally do not differ in spatial competence although they may be differentially influenced by task content. Findings are discussed in terms of their implications for theory and practice. It is emphasized that science educators must be especially aware of such task influences for females so that performance deficits are not mistaken for competence deficits.     

A Piagetian-Based Programme for Learning “Elements and Symbols”

Piaget's learning theory on cognitive development has had considerable impact on science education (Piaget, 1964; Inhelder and Piaget, 1958; Craig, 1972). The theory classifies cognitive learning into four successive stages: (a) sensory-motor, (b) pre-operational, (c) operational, and (d) formal. Various programmes and instructional strategies have been developed based on the theory (Batt, 1980; Karplus, 1977; Renner and Stafford, 1979; Ryan, et al., 1980; Herron, 1978; Good et al; 1978). An application of this theory for teaching and learning scientific concepts is the Piagetian learning cycle (Karplus, 1977) which is of growing interest among science educators. This article intends to introduce briefly the learning cycle in general and suggest a learning cycle for teaching a topic in chemistry: “Elements and Symbols”.     

Contributions of Piagetian and post-Piagetian theories to education

The aim of this article was to reflect on the contributions of Piagetian and neo-Piagetian theories to education. Topics analyzed included the evolution of Piaget’s and co-workers’ theory, the reaction by the scientific community to the main theoretical and methodological aspects of each period of his work, the educational potentialities of methodological and theoretical aspects of his theory, the criticisms about the potentialities of Piagetian theory for grounding educational practice. Then the emergence of the neo-Piagetian theories was described, as well as their major aims and their educational potentialities. Finally some considerations concerning the strengths and weaknesses of Piagetian and neo-Piagetian theories were presented.     

Role of the epistemic subject in Piaget's genetic epistemology and its importance for science education

According to Piaget, a fundamental epistemological distinction must be made between the psychological and the epistemic subject. The epistemic subject is studied by the genetic epistemologist who charts development through a “common universal rationality, which develops,” whereas the psychological subject is studied by the developmental/cognitive psychologist by focusing on accidental contingencies surrounding particular people and their individual differences. The epistemic subject as compared to the psychological subject is an idealized abstraction, viz., that set of underlying epistemic structures common to everyone at the same level of development. The objective of this study is to investigate the degree to which investigators in science education conceptualize the difference between the epistemic and the psychological subjects. It is argued that just as the ideal gas law (based on the theoretical formulation of Maxwell and Boltzmann) provides a “general model” to which the real gases approximate under different experimental conditions, so we can consider (by abduction) the epistemic subject to be an “ideal knower” to which the real (psychological) subjects approximate to varying degrees. The difference between the epistemic and the psychological subjects, however, cannot be used as an “epistemological shield” in defense of Piagetian theory. Any test of the Piagetian theory must involve psychological or real subjects. Empirical testability, however, need not be equated to being scientific. An analogy is drawn between Galileo's idealization, which led to the discovery of the law of free-fall, and Piaget's epistemic subject. Research conducted in science education shows that at least for some critics the wide variations in the age at which individuals acquire the different Piagetian stages is crucial for rejecting the theory. It is argued that the real issue is not the “proportion of heterogeneity” but the understanding that Piaget, by neglecting individual differences, attempts to build a general model applicable across types of situations/subjects. The distinction between the epistemic and the psychological subjects is important not for defending Piaget's theory (which has serious theoretical flaws) but to understand epistemic transitions, for example, the one between Piaget's epistemic subject and Pascual-Leone's metasubject. It is concluded that failure to understand the distinction between the epistemic and the psychological subjects would lead to misconstruing the significance of our research findings and, what is more serious, to a lack of a historical perspective.     

The occurrence of assimilation and accommodation in learning high school physics

The Piagetian model of intelligence consists of the elements of content, mental structures and mental functioning. While the first two elements are said to be variants, the third element is described as invariant. Mental functioning begins with a process called assimilation which promotes disequilibrium and, in reestablishing equilibrium, the learner accommodates to what was assimilated. The assimilation-disequilibrium-accommodation process has been labeled adaptation which Piaget described as the accord of thought with things. The newly-acquired ideas (or concepts) are next organized with other knowledge; Piaget described that organization process as the accord of thought with thought. While the Piagetian mental functioning model has been available for some time, empirical evidence that the model actually did or did not function was not found. Teaching and curriculum development procedures were extracted from the mental functioning model and a twelfth-grade physics course was so organized and taught. Naturalistic data collected from individual student interviews and in-class discussions were used to evaluate the hypothesis that mental functioning and the factors which compose it are integral parts of the learning process.     

The Development and Acquisition of Concepts in Primary School Children in Singapore

What follows is an account which summarises a study that took 18 months to complete and which examined concept development and acquisition in a sample of Singapore primary schoolchildren. For more details the reader is to refer to Thomas, Lee, Fam, (1984a). The broad aim of the study was to obtain baseline information on children's cognitive development mainly through the application of Piagetian tasks.     

College chemistry and piaget: The relationship of aptitude and achievement measures

Cognitive performance on “An Inventory of Piaget's Developmental Tasks” (IPDT) was related to the Scholastic Aptitude Tests and performance in both college chemistry lecture and laboratory classes. The IPDT is a valid and reliable 72-item, untimed, multiple-choice paper-and-pencil inventory. It has 18 subtests grouped into five problem areas representing different Piagetian tasks. Subjects (n = 225) from two different levels of introductory chemistry courses participated. IPDT scores were significantly correlated with SAT and placement data. In overall Piagetian development, “A” students rated higher than others. Performance in classification and proportional reasoning problem areas correlated with course achievement, particularly for higher-level students. Weakest areas of development were indicated by subtest scores and are described with implications for course performance. Finally, Piagetian tasks were related to learning and instructional activities in introductory chemistry classes and are discussed.     

Gender,Internal Controls,and Academic Dishonesty: Investigating Mediating and Differential Effects

While research has found that males tend to commit more academically dishonest acts than females, we know little about what accounts for the gender differential or whether the same factors can explain academic dishonesty for males and females. This study assesses how internal controls, i.e., self‐control, shame, embarrassment, and moral beliefs, account for the relationship between gender and cheating behavior. Using a sample of students from a southern university in the United States, this study explores three questions. First, do internal controls vary across males and females? Second, can internal controls account for gender differences in test cheating? Third, do internal controls differentially or similarly predict test cheating for males and females? Results show that (a) gender differences among several internal controls as well as cheating behavior, (b) internal controls reduce the gender gap but cannot completely explain why gender differences exist in cheating, and (c) two internal controls, self‐control and moral beliefs, differentially predict male and female test cheating. We discuss limitations of our findings and how research can expand the investigation of gender differences and explanations for engaging in academically dishonest acts.     

Gender differences in inference generation by fourth‐grade students

The purpose of this study was to determine if there are gender differences among elementary school‐aged students in regard to the inferences they generate during reading. Fourth‐grade students (130 females; 126 males) completed think‐aloud tasks while reading one practice and one experimental narrative text. Females generated a larger number and a greater proportion of reinstatement inferences than did males (Cohen's d = .34, p = .01; Cohen's d = .26, p = .04, respectively). In contrast, there was no evidence for gender differences in other types of think‐aloud responses. These findings suggest that males and females differ in their use of cognitive processes that underlie reading comprehension, particularly with respect to the likelihood of retrieval of information from episodic memory.     

The Development of Size Sequencing Skills: An Empirical and Computational Analysis

We explore a long-observed phenomenon in children's cognitive development known as size seriation. It is not until children are around 7 years of age that they spontaneously use a strict ascending or descending order of magnitude to organize sets of objects differing in size. Incomplete and inaccurate ordering shown by younger children has been thought to be related to their incomplete grasp of the mathematical concept of a unit. Piaget first brought attention to children's difficulties in solving ordering and size-matching tests, but his tasks and explanations have been progressively neglected due to major theoretical shifts in scholarship on developmental cognition. A cogent alternative to his account has never emerged, leaving size seriation and related abilities as an unexplained case of discontinuity in mental growth. In this monograph, we use a new training methodology, together with computational modeling of the data to offer a new explanation of size seriation development and the emergence of related skills. We describe a connected set of touchscreen tasks that measure the abilities of 5- and 7-year-old children to (a) learn a linear size sequence of five or seven items and (b) identify unique (unit) values within those same sets, such as second biggest and middle-sized. Older children required little or no training to succeed in the sequencing tasks, whereas younger children evinced trial-and-error performance. Marked age differences were found on ordinal identification tasks using matching-to-sample and other methods. Confirming Piaget's findings, these tasks generated learning data with which to develop a computational model of the change. Using variables to represent working and long-term memory (WM and LTM), the computational model represents the information processing of the younger child in terms of a perception-action feedback loop, resulting in a heuristic for achieving a correct sequence. To explain why older children do not require training on the size task, it was hypothesized that an increase in WM to a certain threshold level provides the information-processing capacity to allow the participant to start to detect a minimum interval between each item in the selection. The probabilistic heuristic is thus thought to be replaced during a transitional stage by a serial algorithm that guarantees success. The minimum interval discovery has the effect of controlling search for the next item in a principled monotonic direction. Through a minor additional processing step, this algorithm permits relatively easy identification of ordinal values. The model was tested by simulating the perceptual learning and action selection processes thought to be taking place during trial-and-error sequencing. Error distributions were generated across each item in the sequence and these were found to correspond to the error patterns shown by 5-year-olds. The algorithm that is thought to emerge from successful learning was also tested. It simulated high levels of success on seriation and also on ordinal identification tasks, as shown by 7-year-olds. An unexpected finding from the empirical studies was that, unlike adults, the 7-year-old children showed marked difficulty when they had to compute ordinal size values in tasks that did not permit the use of the serial algorithm. For example, when required to learn a non-monotonic sequence where the ordinal values were in a fixed random order such as “second biggest, middle-sized, smallest, second smallest, biggest,” each item has to be found without reference to the “smallest difference” rule used by the algorithm. The difficulty evinced by 7-year-olds was consistent with the idea that the information in LTM is integrally tied to the search procedure itself as a search-and-stop based on a cumulative tally, as distinct from being accessed from a more permanent and atemporal store of stand-alone ordinal values in LTM. The implications of this possible constraint in understanding are discussed in terms of further developmental changes. We conclude that the seriation behavior shown by children at around 7 years represents a qualitative shift in their understanding but not in the sense that Piaget first proposed. We see the emergent algorithm as an information-reducing device, representing a default strategy for how humans come to deal with potentially complex sets of relations. We argue this with regard to counting behaviors in children and also with regard to how linear monotonic devices for resolving certain logical tasks endure into adulthood. Insofar as the monograph reprises any aspect of the Piagetian account, it is in his highlighting of an important cognitive discontinuity in logicomathematical understanding at around the age of 7, and his quest for understanding the transactions with the physical world that lead to it.     

What the Pendulum Can Tell Educators about Children’s Scientific Reasoning

Inhelder and Piaget (1958) studied schoolchildren’s understanding of a simplependulum as a means of investigating the development of the control of variablesscheme and the ceteris paribus principle central to scientific experimentation.The time-consuming nature of the individual interview technique used by Inhelderhas led to the development of a whole range of group test techniques aimed attesting the empirical validity and increasing the practical utility of Piaget’s work.The Rasch measurement techniques utilized in this study reveal that the Piagetian Reasoning Task III — Pendulum and the méthode clinique interview revealthe same underlying ability. Of particular interest to classroom teachers is theevidence that some individuals produced rather disparate performances across thetwo testing situations. The implications of the commonalities and individualdifferences in performance for interpreting children’s scientific understanding arediscussed.     

Cognitive Style and Motor Skill and Sports Performance

A pool of 116 14-18-year-old secondary school pupils who had been given the computer-presented Cognitive Styles Analysis was used to provide two sub-samples to explore the relationship between style and motor skills and sports performance. The Motor Skills sub-sample of 69 (32 males and 37 females) did a battery of motor skills tests. A factor analysis suggested four skills factors - bodily movement, interactive skills, mechanical skills and aiming. All of these except the mechanical skills showed a significant relationship to style. The Sports Performance sub-sample of 99 (46 males and 53 females) were rated on a five-point scale by their teachers on performance in rugby, soccer and cricket for the boys, and hockey, netball and tennis for the girls. Here, there was a significant effect of style for tennis but not for the team games. The findings were discussed in terms of their practical implications.     

GENDER DIFFERENCES IN THE INFLUENCE OF EARLY PERCEIVED PARENTAL SUPPORT ON STUDENT MATHEMATICS AND SCIENCE ACHIEVEMENT AND STEM CAREER ATTAINMENT

The lack of females entering STEM careers is well documented. Reasons for the gender gaps at all stages of the educational pipeline include both internal factors such as self-concept and external factors such as the influence of parents, media, and educators. Using latent growth curve analysis and nationally representative longitudinal survey data, this study compares differences in the relationship between a critical external factor (perceived early parental support), student mathematics and science achievement trajectories, and persistence in STEM career by gender. Mathematics and science trajectories were positively related to STEM career persistence for males and females. Perceived early parental support was related to growth in mathematics achievement for males but not females. There was no relationship between early perceived parental support and growth in science achievement for either males or females. These findings indicate differences in the relationship between parental support, achievement, and career persistence depending on content area and gender.     

“Is there an equal (amount of) juice?” Exploring the repeated question effect in conservation through conversation

The aim of this paper is to highlight and discuss advantages and constraints of different methods applied within the field of children's thinking studies, through the test of the repeated question hypothesis validity, using the conservation of liquid task. In our perspective, the Piagetian interview is an ecologically valid context for externalization and modification of children's thinking. We used an experimental procedure organized in standard and modified tasks, involving primary school children in Serbia. The results of quantitative and qualitative analyses show that the repeated question is not the unique cause of children's misleading in demonstrating to understand conservation. Other dimensions explain why children change their answers when they are tested by the two tasks we used, which offers an insight into the influence of research procedures on children's answers.     

CURRICULUM RESEARCH IN SWEDEN IV‐‐THE GYMNASIUM AND PRE‐UNIVERSITY EDUCATION

Fifty‐six children between the ages of two years four months and four years ten months were given tests relating to the acquisition of both ‘specific’ and ‘generic’ concepts. Several tests of concept acquisition (HAPCAT) ^{1 1}The tests may be obtained from the NFER (Guidance and Assessment Service) to whom inquiries should be addressed. were devised by the authors utilizing pictures drawn on card. A number of the children (N=29) were also given some ‘Piagetian questions’ concerning class inclusion.

The results conflict with the widely held view (following the writing of Jean Piaget) that preschool children cannot form generic concepts. Nearly all the children were able to identify the specific concepts as presented in the HAPCAT items and about half the group responded appropriately to the HAPCAT items involving an understanding of generic concepts. There was a clear and statistically significant relationship between the number of correct answers given to the HAPCAT items and the age and ability levels of the children. This was not the case, however, with the Piagetian test questions which showed no discriminatory power in these respects. It is argued that the Piagetian questions do not represent a sensitive or meaningful measure of concept acquisition at this age level whereas the HAPCAT items do appear to present pre‐school children with intelligible tasks which allow many of them to demonstrate an understanding of generic concepts involving class inclusion.