Children's conditional reasoning

This study stemmed from a desire to redress the distorted view of mathematics in the elementary curriculum, created by the current imbalanced emphasis on computational rules and some applications, but very little logical analysis. The study is intended to show that fifth-grade students can significantly improve their use of logical analysis through a suitable instructional unit taught under ordinary classroom conditions.Concrete teaching materials were developed, through several trials and revisions, to familiarize students with the distinction between the valid inference patterns — Modus Ponendo Ponens and Modus Tollendo Tollens (AA, DC), and the fallacious ones — Affirming the Consequent and Denying the Antecedent (AC, DA). No formal rules were taught.The experimental unit was implemented four to five times a week for 23–25 sessions, by 4 fifth-grade teachers in their ordinary classes. The teachers participated in a twelvehour pretraining workshop.A pretest/posttest treatment/no-treatment design was applied to assess resulting improvement in students' conditional reasoning ability. The sample consisted of 210 fifth graders in a suburban area, 104 in 4 experimental classes and 106 in 4 control classes.A written group test was developed, through trials and revisions. Test items are formulated with a reasonable hypothetical content. Each item includes two premises: the first a conditional sentence, and the second either its antecedent, its consequent, or the negation of one of these, thus determining the logical form: AA, DC, AC, or DA. The question following the premises is stated positively. AA and DC are answered correctly by yes or no: AC and DA by not enough clues (NEC).The test contains 32 randomly-ordered three-choice items, eight in each logical form (two of the eight in each of the four possible modes in which negation may or may not occur in the antecedent or consequent). No sentential connective other than negation and conditional appears in the premises. Test/retest reliability was 0.79.Experimental and control group pretest performance levels did not differ (=0.05). More than 78% of the answers on AA and DC, and fewer than 33.1% on AC and DA, were correct. Overall pretest mean scores were 54.3% and 53.8% for the experimental and control groups respectively.There was a significant difference (=0.01) between the experimental and control groups' posttest overall performance—74.7% and 55.4%, respectively.There was no significant change in the control group's pretest and posttest performance levels on any logical form, or for the experimental group's on AA and DC. However, on AC and DA the two groups' gain scores were found significantly different.Negation mode, unlike logical form, was not found to be independently influential in analyzing test scores, but interacted with logical form.There was a pretest/posttest increase of 3.5 in experimental group frequency (percentaged) of incorrect NEC answers (AA and DC). As NEC appeared infrequently on the pretest, this increase was interpreted as learning that NEC is an acceptable answer. Separating out this effect from the percentaged frequency of correct NEC answers (AC and DA) left a pretest/posttest average increase of 37.8. This increase was attributed to learning when NEC is correct.Teachers were excited at the beginning, frustrated in the middle, and felt competent and involved in the project at the end. They felt the teaching should be less condensed. The majority of the students reacted positively to most parts of the experimental unit. However, some thought the unit as a whole was too repetitive and boring.No correlation was found between learning logic through the experimental unit and standard school achievement as measured by the Stanford Achievement Test (SAT). High, average, and low SAT achievers of the experimental group did not differ significantly in their pretest and posttest gain scores.Results of the study call for further investigation of the value and usefulness of teaching various parts of logic as an ordinary part of the elementary mathematics curriculum.This study was supported by NSF grants Nos. GW 7659 and PES 74-018450 and by an AAUW International Fellowship. It was carried out at the University of California, Berkeley, in 1974/75 under supervision of Professor Leon Henkin.     

The curious effect of using drawings in conditional reasoning problems

College level students were given two paper and pencil conditional reasoning tests in varying order. The first test consisted of three problems presented entirely verbally. The second test was identical to the first except that two of the three problems made use of line drawings as concrete referents. Performance on the problems with drawing was significantly worse than for purely verbal presentation. In addition, interaction effects between the two forms of presentation were observed over the one-week interval between the two tests.     

传递推理研究中的滞差

本文针对一些研究者就水平滞差任务对皮亚杰理论的结构基础提出的质疑,通过对儿童传递推理能力的研究中所出现的两种水平滞差(即内容滞差和程序滞差)的分析,以及对儿童可能的两种推理(即函数性推理和操作性推理)的分析,进一步讨论认为程序滞差并非真正的滞差,它与修改后的程序是否改变任务性质有关;内容滞差的出现,源于任务内容的认知复杂性不同,还与使儿童产生误解的线索有关.     

Developmental and individual differences in conditional reasoning: effects of logic instructions and alternative antecedents

In Study 1, 10-, 13-, and 16-year-olds were assigned to conditions in which they were instructed to think logically and provided alternative antecedents to the consequents of conditional statements. Providing alternatives improved reasoning on two uncertain logical forms, but decreased logical responding on two certain forms; logic instructions improved reasoning among adolescents. Correlations among inferences and verbal ability were found primarily when task conditions created conflict between automatic and controlled inferences. In Study 2, when the cognitive demands of the logic instructions were reduced, 10-year-olds made more logically correct inferences, but only when a conditional's consequents were strongly associated with alternative antecedents. Discussion focuses on the ability to inhibit invited inferences and the role of automatically activated memories.     

A research framework for creative and imitative reasoning

This conceptual research framework addresses the problem of rote learning by characterising key aspects of the dominating imitative reasoning and the lack of creative mathematical reasoning found in empirical data. By relating reasoning to thinking processes, student competencies, and the learning milieu it explains origins and consequences of different reasoning types.

儿童比例推理研究的最新进展

儿童比例概念和比例推理能力的形成和发展是自皮亚杰以来的心理学研究焦点之一，也是学科教学中的一个难点。章简要介绍了近年来该领域的一些研究进展。     

Using analogy to improve abstract conditional reasoning in adolescents: not as easy as it looks

Abstract reasoning refers to the ability to reason logically with premises that do not allow reference to knowledge about the real world. This form of reasoning is complex and difficult, and at the same time, it is critical for understanding science and mathematics. Two studies examined the use of analogy as a method to bridge reasoning with familiar content and abstract reasoning among older adolescents. The results showed that the ability to make an appropriate analogy depends on reasoning ability. Neither of the two procedures used resulted in an improvement in abstract reasoning. However, generating an inappropriate analogy actually decreased abstract reasoning performance among students who showed greater initial levels of reasoning ability. These results highlight the problems associated with using familiar reasoning as a basis for learning abstract reasoning skills.     

A review of research on formal reasoning and science teaching

A central purpose of education is to improve students' reasoning abilities. The present review examines research in developmental psychology and science education that has attempted to assess the validity of Piaget's theory of formal thought and its relation to educational practice. Should a central objective of schools be to help students become formal thinkers? To answer this question research has focused on the following subordinate questions: (1) What role does biological maturation play in the development of formal reasoning? (2) Are Piaget's formal tasks reliable and valid? (3) Does formal reasoning constitute a unified and general mode of intellectual functioning? (4) How does the presence or absence of formal reasoning affect school achievement? (5) Can formal reasoning be taught? (6) What is the structural or functional nature of advanced reasoning? The general conclusion drawn is that although Piaget's work and that which has sprung from it leaves a number of unresolved theoretical and methodological problems, it provides an important background from which to make substantial progress toward a most significant educational objective. All our dignity lies in thought. By thought we must elevate ourselves, not by space and time which we can not fill. Let us endeavor then to think well; therein lies the principle of morality. Blaise Pascal 1623-1662.     

Informal reasoning regarding socioscientific issues: A critical review of research

Socioscientific issues encompass social dilemmas with conceptual or technological links to science. The process of resolving these issues is best characterized by informal reasoning which describes the generation and evaluation of positions in response to complex situations. This article presents a critical review of research related to informal reasoning regarding socioscientific issues. The findings reviewed address (a) socioscientific argumentation; (b) relationships between nature of science conceptualizations and socioscientific decision making; (c) the evaluation of information pertaining to socioscientific issues, including student ideas about what counts as evidence; and (d) the influence of an individual's conceptual understanding on his or her informal reasoning. This synthesis of the current state of socioscientific issue research provides a comprehensive framework from which future research can be motivated and decisions about the design and implementation of socioscientific curricula can be made. The implications for future research and classroom applications are discussed. © 2004 Wiley Periodicals, Inc. J Res Sci Teach 41: 513–536, 2004     

科学推理学习研究转向及其教育意蕴

受科学知识论和学习观演变的影响,科学推理学习研究情境从日常事务转向科学问题,研究焦点从个体内部转向整个学习境脉,科学推理的基本内涵从领域一般性转向领域特殊性,并形成了新的科学推理观。新科学推理观对改进课程、教学和评价具有重要教育价值,它强调自然科学主要使用六种类型的推理,每种类型的推理依赖于一套领域特殊的本体性知识、程序性知识和认识性知识,科学推理在不同分支学科之间存在共性。     

Effects of different forms of tutor action in a conditional reasoning task: An experimental approach to the tutorial dialogue

This study investigated how different forms of tutor action influenced novice students’ performance on the Wason selection task, and how the students perceived the situation. In the control condition, the tutor provided supportive feedback which was minimal in terms of content (CG), in contrast to help in the form of directives (IM) or questions (QM). Based on the findings of studies of tutorial dialogue in the literature, we expected to observe: lower performance for CG (H1) and better performance with QM than IM (H2), a less positive perception of the tutor’s assistance for CG (H3), and that the tutees would differentiate between the IM and QM conditions (H4). These hypotheses were confirmed, with the exception of H2, IM and QM producing similar performances.     

Proportional reasoning and the linguistic abilities required for hypothetico-deductive reasoning

The hypothesis is advanced that a necessary, though not sufficient, condition for the acquisition of proportional reasoning during adolescence is the prior internalization of key linguistic elements of argumentation, essentially those used in hypothetico-deductive reasoning. This hypothesized internalization, which does not occur in all individuals, results in some who have acquired the ability to reflect upon the correctness of self-generated answers in a hypothetico-deductive manner, and others who have not. As an initial test of the hypothesis, 46 subjects (Ss) (mean age = 21.03 years) were classified into additive, transitional, or proportional reasoning categories based upon responses to a proportions task. Group differences were found in which proportional Ss performed better than transitional Ss who in turn performed better than additive Ss on a number of items testing Ss' abilities to identify, generate, and use the linguistic elements of argumentation. Further it was found that some Ss who were successful on the linguistic items failed the proportions task, but no Ss who were successful on the proportions task failed the linguistic items. This result supports the hypothesis that the internalization of linguistic elements of argumentation is a prerequisite for proportional reasoning and by inference other advanced reasoning schemata as well. Implications for science instruction are drawn.     

Shifts in reasoning

The process of transition from a novice's state to that of an expert, in the constrained domain of decimals, is described in terms of explicit, intermediate, and transitional rules which are consistent, yet erroneous. These rules can be traced to former rules already established in earlier knowledge domains. Empirical data from children at grades 6, 7, 8 and 9 will demonstrate the evolution of an expert's knowledge through an elaborated learning path.     

Conditional reasoning with false premises facilitates the transition between familiar and abstract reasoning

Abstract reasoning is critical for science and mathematics, but is very difficult. In 3 studies, the hypothesis that alternatives generation required for conditional reasoning with false premises facilitates abstract reasoning is examined. Study 1 (n = 372) found that reasoning with false premises improved abstract reasoning in 12- to 15-year-olds. Study 2 (n = 366) found a positive effect of simply generating alternatives, but only in 19-year-olds. Study 3 (n = 92) found that 9- to 11-year-olds were able to respond logically with false premises, whereas no such ability was observed in 6- to 7-year-olds. Reasoning with false premises was found to improve reasoning with semiabstract premises in the older children. These results support the idea that alternatives generation with false premises facilitates abstract reasoning.     

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.     

Conceptual conditional discrimination inSaimiri sciureus

The present work introduced a task which superimposed a tray brightness, stimulus-response contingency on previously acquired, highly successful, one-trial oddity performance. Continuing with new one-trial oddity problems, the new contingency was that responses to the odd objects were rewarded on a white tray and responses to the nonodd objects were rewarded on the black tray. Since there is no opportunity to learn specific stimuli or stimulus patterns, successful performance may be interpreted as having a conceptual basis. All monkeys achieved criterion (90% based on 18/20) and statistically significant performances (p<.001). Discussion considered the appropriate nomenclature to describe a conceptual conditional discrimination task and the necessary evidence to justify a conceptual interpretation of conditional discrimination behavior.     

关于条件概率的几点注记

给出了条件概率的加法公式，乘法公式和—般的全概率公式。     

Children's beliefs about everyday reasoning

Two studies investigated children's metacognition about everyday reasoning, assessing how they distinguish reasoning from nonreasoning and "good" reasoning from "bad." In Study 1, 80 1st graders (6-7 years), 3rd graders (8-9 years), 5th graders (10-11 years), and adults (18+ years) evaluated scenarios where people (a) used reasoning, (b) solved problems with nonreasoning approaches, or (c) reacted appropriately but automatically to events. All age groups distinguished reasoning from type (b) nonreasoning cases, but age-related improvement occurred for type (c) cases. Study 2 tested 160 1st, 3rd, 5th graders' and adults' evaluation of good and bad reasoning processes, finding 2 developmental changes: initial improvement in discriminating thinking processes by 3rd grade, and emergence of an adult-like, process-focused (vs. outcome-focused) concept of thinking quality by 5th grade.