Hierarchial relationships among formal cognitive structures and their relationship to principled moral reasoning

Measurements of formal reasoning and principled moral reasoning ability were obtained from a sample of 99 tenth grade students. Specific modes of formal reasoning (proportional reasoning, controlling variables, probabilistic, correlational, and combinatorial reasoning) were first examined. Findings support the notion of hierarchical relationships which exist among those variables. Further results from factor analysis provide evidence that the variables represent specific cognitive structures that are interdependent with each other and precede operations in development. Finally, significant relationships were found to exist between the different modes of formal reasoning and principled moral reasoning. Combinatorial and correlational reasoning were found to significantly account for 22% of the variance in principled moral reasoning. Theoretical and educational implications are discussed.     

What do tests of “formal” reasoning actually measure?

Tests of formal operational reasoning derived from Piagetian theory have been found to be effective predictors of academic achievement. Yet Piaget's theory regarding the underlying nature of formal operations and their employment in specific contexts has run into considerable empirical difficulty. The primary purpose of this study was to present the core of an alternative theory of the nature of advanced scientific reasoning. That theory, referred to as the multiple-hypothesis theory, argues that tests of formal operational reasoning actually measure the extent to which persons have acquired the ability to initiate reasoning with more than one specific antecedent condition, or if they are unable to imagine more than one antecedent condition, they are aware that more than one is possible; therefore conclusions that are drawn are tempered by this possibility. As a test of this multiple-hypothesis theory of advanced reasoning and the contrasting Piagetian theory of formal operations, a sample of 922 college students were first classified as concrete operational, transitional, or formal operational, based upon responses to standard Piagetian measures of formal operational reasoning. They were then administered seven logic tasks. Actual response patterns to the tasks were analyzed and found to be similar to predicted response patterns derived from the multiple-hypothesis theory and were different from those predicted by Piagetian theory. Therefore, support was obtained for the multiple-hypothesis theory. The terms intuitive and reflective were suggested to replace the terms concrete operational and formal operational to refer to persons at varying levels of intellectual development.     

The rate of acquisition of formal operational schemata in adolescence: A secondary analysis

A theoretical model of cognitive development is applied to the study of the acquisition of formal operational schemata by adolescents. The model predicts that the proportion of adolescents who have not yet acquired the ability to perform a a specific Piagetian-like task is an exponentially decreasing function of age. The model has been used to analyze the data of two large-scale studies performed in the United States and in Israel. The functional dependence upon age was found to be the same in both countries for tasks which are used to assess the following formal operations: proportional reasoning, probabilistic reasoning, correlations, and combinatorial analysis. Different functional dependence was found for tasks assessing conservation, control of variables, and prepositional logic. These results give support to the “unity” hypothesis of cognitive development, that is, the hypothesis that the various schemata of formal thought appear simultaneously.     

How Swedish pupils,aged 12‐15 years,understand light and its properties‡

The reasoning patterns used by a sample of Western Australian secondary school students aged 13‐16 were investigated with regard to the following reasoning modes: proportional reasoning, controlling variables, probabilistic reasoning, correlational reasoning, and combinatorial reasoning.

There was a wide range in students’ reasoning abilities at all year levels. Large percentages of students did not use formal operational reasoning patterns when they attempted to solve problems assessing their ability to use each of the five reasoning modes. Commonly used, but incorrect reasoning patterns were identified for each reasoning mode.

The students’ ability to use formal reasoning patterns was found to be an important factor in determining student achievement in lower secondary science, in their selection of year 11 science subjects, and their achievement in these subjects.

The results of the study indicate that it is important for teachers to be aware of the reasoning patterns of their students and the cognitive demands of course content, so that they can optimally match the content and their teaching strategies with the abilities of their students. Further research is needed to establish the nature of instruction which might best facilitate cognitive growth.     

Development and validation of a scientific (formal) reasoning test for college students

We present a multiple-choice test, the Montana State University Formal Reasoning Test (FORT), to assess college students' scientific reasoning ability. The test defines scientific reasoning to be equivalent to formal operational reasoning. It contains 20 questions divided evenly among five types of problems: control of variables, hypothesis testing, correlational reasoning, proportional reasoning, and probability. The test development process included the drafting and psychometric analysis of 23 instruments related to formal operational reasoning. These instruments were administered to almost 10,000 students enrolled in introductory science courses at American universities. Questions with high discrimination were identified and assembled into an instrument that was intended to measure the reasoning ability of students across the entire spectrum of abilities in college science courses. We present four types of validity evidence for the FORT. (a) The test has a one-dimensional psychometric structure consistent with its design. (b) Test scores in an introductory biology course had an empirical reliability of 0.82. (c) Student interviews confirmed responses to the FORT were accurate indications of student thinking. (d) A regression analysis of student learning in an introductory biology course showed that scores on the FORT predicted how well students learned one of the most challenging concepts in biology, natural selection.     

Teaching by Keller Plan‐‐Comment on a Course in Spain

The use of the 'Cognitive Acceleration through Science Education' (CASE) intervention programme was explored in Malawi to see if the critical period for cognitive transition from concrete operations to formal operations at 12-14 years of age actually exists. The older pupils (16-17 years of age) in secondary school classes in Malawi made similar cognitive gains to younger English pupils after being taught CASE for 2 years. This was taken to suggest that the critical period does not exist. However, in Malawi, girls, who are on average a year younger than boys in the same class, and older boys, had lower academic achievement than younger boys. Explanations for this age and gender interaction are explored.     

Students’ socio-scientific reasoning on controversies from the viewpoint of education for sustainable development

In this article, we study third-year university students’ reasoning about three controversial socio-scientific issues from the viewpoint of education for sustainable development: local issues (the reintroduction of bears in the Pyrenees in France, wolves in the Mercantour) and a global one (global warming). We used the theoretical frameworks of social representations and of socio-scientific reasoning. Students’ reasoning varies according to the issues, in particular because of their emotional proximity with the issues and their socio-cultural origin. About this kind of issues, it seems pertinent to integrate into the operations of socio-scientific reasoning not only the consideration of values, but also the analysis of the modes of governance and the place given to politics.

Modelling activities integrating construction and simulation supported explanatory and evaluative reasoning

ABSTRACT

Model-based instruction offers numerous benefits to students, including increased content knowledge and critical thinking. This study explored the differences in the knowledge outcomes and reasoning processes employed by undergraduate students in an introductory biology lab as they constructed, revised, and simulated a computational model of a prokaryotic gene regulatory system. We analysed post-lesson conceptual models (n?=?335) and audio-recorded small group discussions (n?=?10) of students that either constructed and revised a computational model and then investigated a pre-constructed computational model or only investigated a pre-constructed computational model to determine the impact of constructing and revising a computational model. No significant differences were found between the mental models of students who constructed and revised a computational model and those who only simulated a pre-constructed computational model. However, this study demonstrated that constructing and revising a model before simulating it offered additional benefits to students by fostering deeper cognitive reasoning processes. Constructing and revising a model elicited more explanatory and evaluative reasoning and prompted students to discuss the underlying mechanisms of the biological system.     

Translation of algebraic equations and its relation to formal operational reasoning

A large proportion of science major college students are unable to translate even simple sentences into algebraic equations. Given the following sentence: There are six times as many students (S) as professors (P) at this university, most students write the following equation: 6S = P, referred to as the reversal error. In order to overcome the reversal error students need to operate in a hypothetico-deductive manner, i.e., performing a hypothetical operation that makes the group of professors six times larger than it really is (S = 6P). The objective of this study is to investigate the relation between student ability to translate sentences into equations, equations into sentences, and student performance in the following variables: formal operational reasoning, proportional reasoning, and introductory freshmen-level chemistry course. The results obtained show that as the student ability to translate sentences into equations and equations into sentences increases, their mean scores in Chemistry I, formal operational, and proportional reasoning increases. This study has found support for the hypothesis that students who lack formal operational reasoning skills (hypothetico-deductive reasoning) may experience more problems in the translation of algebraic equations.     

Models and processes for solving arithmetic word problems proposed by Spanish mathematics textbooks / Modelos y procesos de resolución de problemas aritméticos verbales propuestos por los libros de texto de matemáticas españoles

Abstract

Textbooks are frequently used as a resource to teach children how to solve word problems in mathematics classes. The solving models explicitly proposed by textbooks, and the kind of problems they propose, can therefore have a decisive influence on the way in which students learn to solve them. This paper analyses the extent to which the problem solving models proposed by the books include reasoning as one of the steps needed to solve a problem. To do this, the processes and steps articulated in the models proposed in the books by Anaya, SM and Santillana are categorized according to data, reasoning, choice of operation, strategies, execution of operations, expression of results, checking and inventing. The results indicate that reasoning processes are hardly ever found in the models, especially in lower year courses. It is concluded that mathematical textbooks provide incomplete solving models that fail to promote reasoning.     

Factors in the development of proportional reasoning strategies by concrete operational college students

This study was designed as a test for two neo-Piagetian theories. More specifically, this research examined the relationships between the development of proportional reasoning strategies and three cognitive variables from Pascual-Leone's and Case's neo-Piagetian theories. A priori hypotheses linked the number of problems students worked until they induced a proportional reasoning strategy to the variables of M-space, degree of field dependence, and short-term storage space. The subjects consisted of students enrolled in Physical Science I, a science course for nonscience majors at the University of Southern Mississippi. Of the 34 subjects in the study, 23 were classified as concrete operational on the basis of eight ratio tasks. Problems corresponding to five developmental levels of proportional reasoning (according to Piagetian and neo-Piagetian theory), were presented by a microcomputer to the 23 subjects who had been classified as concrete operational. After a maximum of 6 hours of treatment, 17 of the 23 subjects had induced ratio schemata at the upper formal level (IIIB), while the remaining subjects used lower formal level (IIIA) schemata. The data analyses showed that neither M-space and degree of field-dependence, either alone or in combination, nor short-term storage predicted the number of problems students need to do until they induce an appropriate problem-solving strategy. However, there were significant differences in the short-term storage space of those subjects who mastered ratio problems at the highest level and those who did not. Also, the subjects' degree of field-dependence was not a predictor of either the ability to transfer problem-solving strategies to a new setting or the reuse of inappropriate strategies. The results of this study also suggest that short-term storage space is a variable with high correlations to a number of aspects of learning such as transfer and choice of strategy after feedback.     

Problem-solving analysis: A piagetian study

Fifty-nine second-year medical students were asked to solve 12 Piagetian formal operational tasks. The purpose was to describe the formal logical characteristics of this medical student sample (59 of a total 65 possible) in terms of their abilities to solve problems in four formal logical schemata-combinatorial logic, probabilistic reasoning, propositional logic, and proportional reasoning. These tasks were presented as videotape demonstrations or in written form, depending on whether or not equipment manipulation was required, and were scored using conventional, prespecified scoring criteria. The results of this study show approximately 96% of the sample function at the transitional (Piaget's 3A level) stage of formal operations on all tasks and approximately 4% function at the full formal (Piaget's 3B level) stage of formal operations on all tasks. This sample demonstrates formal level thinking to a much greater degree than other samples reported in the literature to date and suggests these students are adequately prepared and developed to meet the challenge of their training (i.e., medical problem solving).     

Pedagogical reasoning: the foundation of the professional knowledge of teaching

ABSTRACT

Much of the debate about that which comprises teachers’ professional knowledge has been important in the academic literature but does not necessarily reflect the reality of how they think as they construct the knowledge that underpins their practice. Typically, teachers are not encouraged to spend time talking about teaching in ways that are theoretically robust, or to unpack their teaching in order to show others what they know, how and why. Because they are busy ‘doing teaching’ they are not commonly afforded opportunities to ‘unpack’ their practice to explore and articulate the reasoning underpinning what they do. This paper argues that the essence of teachers’ professional knowledge is bound up in the teaching procedures they employ and that knowledge is accessible and demonstrable through the pedagogical reasoning that underpins their decision-making, actions and intents; all of which come to the fore when their pedagogical reasoning is examined. If teaching is to be more highly valued, it is important to more closely examine the nature of teachers’ pedagogical reasoning as it offers a window into the complex and sophisticated knowledge of practice that influences what they do, how and why.     

Cardboard Cities,Real Mathematics: Employing Quantitative Literacy to Study Gentrification in NYC

Abstract

Gentrification is a complex issue with very real repercussions for urban youth. Games make learning fun and provide a way for students to explore complex systems through recursive play. We describe a board game designed to help students explore ratios and proportional reasoning while simultaneously engaging the causes and effects of gentrification. Situated within a multidisciplinary first-semester learning community course, the game provides a model for later experiential learning opportunities as students are asked to study the effects of gentrification in one New York City neighborhood.     

The four-card problem resolved? Formal operational reasoning and reasoning to a contradiction

To test the hypothesis that adolescents classified as formal operational, based upon use of proportional reasoning on the “Pouring Water Task” (Lawson, Karplus, & Adi, 1978) have acquired the mental structures necessary to comprehend hypothetico-deductive arguments of a pattern referred to as “reasoning to a contradiction,” while adolescents classified as concrete operational, based upon use of additive reasoning on the same task have not, a sample of 100 high school students were administered the task and three versions of a problem requiring use of reasoning to a contradiction before, immediately after, and one month after brief verbal instruction in use of that reasoning pattern. Results were generally supportive of the hypothesis as most of the concrete students failed the immediate and delayed posttest problems (62 and 80%, respectively) while most of the formal students succeeded (80 and 71%, respectively). Group differences were significant (p < .001) in both cases. These results suggest that, contrary to those who have argued that content plays a substantial role in logical performance, a general hypothetico-deductive reasoning competence exists in some adolescents and is applicable across a wide variety of task domains. Science instruction which aims to teach this competence is recommended.     

First-graders’ knowledge of multiplicative reasoning before formal instruction in this domain

Our study investigated children’s knowledge of multiplicative reasoning (multiplication and division) at the end of Grade 1, just before the start of formal instruction on multiplicative reasoning in Grade 2. A large sample of children (N = 1176) was assessed in a relatively formal test setting, using an online test with 28 multiplicative problems of different types. On average, the children correctly answered more than half (58%) of the problems, including several bare number problems. This indicates that before formal instruction on multiplicative reasoning, children already have a considerable amount of knowledge in this domain, which teachers can build on when teaching them formal multiplication and division. Using analysis of variance and cross-classified multilevel regression analysis, we identified several predictors of children’s pre-instructional multiplicative knowledge. With respect to the characteristics of the multiplicative problems, we found that the problems were easiest to solve when they included a picture involving countable objects, and when the multiplicative situation was of the equal groups semantic structure (e.g., 3 boxes of 4 cookies). Regarding student characteristics, pre-instructional multiplicative knowledge was higher for children with higher-educated parents. Finally, the mathematics textbook used in school appeared to have influenced children’s pre-instructional multiplicative knowledge.     

Reasoning strategies of students in solving chemistry problems as a function of developmental level,functional M‐capacity and disembedding ability

Achievement in science depends among other factors on hypothetico‐deductive reasoning ability, that is, developmental level of the students. Recent research indicates that the developmental level of students should be studied along with individual difference variables, such as Pascual‐Leone's M‐capacity (information processing) and Witkin's Cognitive Style (disembedding ability). The purpose of this study is to investigate reasoning strategies of students in solving chemistry problems as a function of developmental level, functional M‐capacity and disembedding ability. A sample of 109 freshman students were administered tests of formal operational reasoning, functional M‐capacity, disembedding ability and chemistry problems (limiting reagent, mole, gas laws). Results obtained show that students who scored higher on cognitive predictor variables not only have a better chance of solving chemistry problems, but also demonstrated greater understanding and used reasoning strategies indicative of explicit problem‐solving procedures based on the hypothetico‐deductive method, manipulation of essential information and sensitivity to misleading information. It was also observed that students who score higher on cognitive predictor variables tend to anticipate important aspects of the problem situation by constructing general figurative and operative models, leading to a greater understanding. Students scoring low on cognitive predictor variables tended to circumvent cognitively more demanding strategies and adopt others that helped them to overcome the constraints of formal reasoning, information processing and disembedding ability.     

Improving spatial thinking skills among undergraduate geology students through short online training exercises

ABSTRACT

Spatial reasoning is a critical skill in the Geosciences. Using a randomized control study with 592 undergraduate students enrolled in introductory and advanced Geology courses, our data indicates that regular, short interventions throughout an academic semester improve students’ spatial thinking skills significantly with a moderate to large effect size when compared to an instruction-as-usual control group. We found evidence for additional gains in students who participated also in hands-on training interventions. We found even larger training effects on students who achieved correct scores of >50% on the practice modules, with gains of almost three-quarter of a standard deviation relative to their baseline scores. Male and female students display significant differences in spatial skills, with males outperforming females. Training resulted in similar magnitudes of improvement in both genders, so we see no evidence that the interventions closed the gender gap. Using the initial performance as a baseline, poor performers’ spatial skills appear to improve through practice at the same rate as their peers. We argue that 15.4% of students improve their spatial skills through participation in the training towards a threshold that appears to be critical for participation in STEM careers. On a reflection survey, half of the students reported that they felt their spatial thinking skills improved through their participation, and over a third found the training beneficial for their overall learning in Geology or other classes. We hypothesise that formal training opportunities for spatial reasoning could increase the potential pool of students who successfully enter and succeed in Geoscience careers.     

evolving research of mathematics teacher educators: the case of non-standard issues in solving standard problems

This study describes the learning of researchers who engage in mathematics teacher education as an integral part of their practice. As teacher educators working with teachers on the subject of proportional reasoning, the authors reflected on teachers solutions to a standard problem and analyzed answers that would conventionally be considered incorrect. This exploration showed that some incorrect answers made sense, were based on problem situation analysis, and brought attention to the fact that conventional formal answers were given without much deliberation on their meaning in the situation. This insight prompted a second research phase in which teachers discussed and explained alternative solutions, and developed deeper analysis of problem situation in solutions that had been correct in the first place. The importance of making teachers aware of the nature of alternative solutions was further exhibited in a third research phase in which teachers evaluated childrens answers to the same problem. The pedagogical insight that emerged stressed the importance of making teachers aware of the tension between an almost automatic application of a mathematical model, and of analyzing problem situation during problem solving. In addition, the researchers developed better understanding of the mathematical challenge associated with the proportional reasoning problem, a stronger awareness of the role of sensitivity to their learners (the teachers), and of the role of reflection.