Intuitive Interference in Probabilistic Reasoning

One theoretical framework which addresses students’ conceptions and reasoning processes in mathematics and science education is the intuitive rules theory. According to this theory, students’ reasoning is affected by intuitive rules when they solve a wide variety of conceptually non-related mathematical and scientific tasks that share some common external features. In this paper, we explore the cognitive processes related to the intuitive rule more A–more B and discuss issues related to overcoming its interference. We focused on the context of probability using a computerized “Probability Reasoning – Reaction Time Test.” We compared the accuracy and reaction times of responses that are in line with this intuitive rule to those that are counter-intuitive among high-school students. We also studied the effect of the level of mathematics instruction on participants’ responses. The results indicate that correct responses in line with the intuitive rule are more accurate and shorter than correct, counter-intuitive ones. Regarding the level of mathematics instruction, the only significant difference was in the percentage of correct responses to the counter-intuitive condition. Students with a high level of mathematics instruction had significantly more correct responses. These findings could contribute to designing innovative ways of assisting students in overcoming the interference of the intuitive rules.     

Intuitive Interference in Geometry: An Eye‐Tracking Study

Interference of irrelevant salient variables may cause difficulties for students. This study focused on eye tracking during the comparison of perimeters task, in which area is the interfering irrelevant salient variable. There were three trial types: congruent (larger area—larger perimeter), incongruent inverse (larger area—smaller perimeter), and incongruent equal (larger area—equal perimeter). Behavioral findings corroborated previous studies: congruent trials yielded higher accuracy and a shorter reaction time than did incongruent trials. Surprisingly, the area saliency could not be revealed in fixation location and duration measurements in incongruent inverse trials nor in the heat maps for incongruent inverse or incongruent equal trials, suggesting that such processing does not require overt attention; measures of attention shift from one geometrical shape to another were higher for incongruent equal trials, as was pupil dilation, suggesting that greater effort is associated with solving incongruent equal trials.     

Complexity of Shapes and Quantitative Reasoning in Geometry

ABSTRACT— We explored the effects of task‐related factors on reasoning processes in geometry focusing on a comparison‐of‐perimeters task in which the irrelevant feature area interferes with the reasoning process. We studied the effects of congruity, salience, and complexity on participants’ accuracy of responses and reaction times. The study shows that incongruent trials yield lower accuracy and longer reaction times than congruent ones. In addition, increasing the salience of the irrelevant feature results in a lower rate of success. Previous research suggested that a high cognitive load leads to higher interference in the incongruent condition. The current study shows that, in this condition, increasing the level of complexity of shapes associated with a higher cognitive load decreases the accuracy of responses and increases reaction times. Our findings can aid educators to select appropriate tasks and examples in their classrooms, taking into account how the level of difficulty is affected by complexity, salience, and congruity.     

The role of inhibition in overcoming arithmetic natural number bias in the Chinese context: Evidence from behavioral and ERP experiments

The natural number bias (NNB) in arithmetic operations refers to the application of natural number properties to reasoning about rational numbers. Previous studies found the NNB interferes with students’ problem-solving. However, few studies have examined it in the Chinese context or the underlying mechanism by which it can be overcome. Addressing these gaps, in Experiments 1a (n = 31) and 1b (n = 30), we found that Chinese students demonstrate the NNB despite linguistic differences between Chinese and western languages. Experiment 2 (n = 38) adopted a negative priming paradigm and found that inhibitory control was necessary to overcome the NNB. Experiment 3 (n = 34) employed the event-related potential technique; we observed increased P2 amplitude when students solved congruent problems, and increased N2 and decreased P3 amplitude when they solved incongruent problems. These results indicated that the NNB is rooted in intuitive thinking, and overcoming this bias relies on inhibition.     

Samples of stimuli,responses, and reinforcers: Effect of incongruent sample type,serial position,and mode of presentation

In two matching-to-sample experiments, pigeons’ performance with samples of stimuli (red and green), number of responses (1 and 20), and reinforcers (food and no food) was assessed. Samples of red, 20 responses, and food were associated with the red comparison stimulus, and samples of green, 1 response, and no food were associated with the green comparison stimulus. On interference trials, three sample types were presented on each trial, and two of the samples (congruent) were associated with the correct comparison and the third sample (incongruent), with the incorrect comparison. Performance on interference trials was compared with that on control trials in which either two (Experiment 1) or three (Experiment 2) congruent samples were presented. It was found that presentation of an incongruent sample reduced matching accuracy markedly, and about equally, whether samples were presented successively or in compound. Although the type of sample that was incongruent was without effect, matching accuracy declined strongly as the recency of the incongruent sample increased. Serial position of the incongruent sample also influenced the shape of the retention function on interference trials. Presentation of the incongruent sample either first or second resulted in accuracy decreasing across the retention interval, whereas presentation of the incongruent sample last in the input sequence resulted in increasing accuracy across the retention interval. The theoretical implications of the findings are considered.     

高中生数学核心素养发展水平分析与教学建议--2019年高考(天津卷)数学(理工类)考试评价

以2019年高考天津卷数学(理工)试卷实测数据为依据,分析考生从高中入学到高中毕业的数学素养发展情况,着重对数学运算、直观想象和逻辑推理素养的发展水平进行分析与评价,从而得出基于学生数学核心素养发展的教学改进策略。     

Cognitive conflict and intuitive rules

This paper is a part of an extensive project on the role of intuitive rules in science and mathematics education. First, we described the effects of two intuitive rules ‐‐ ‘Everything comes to an end’ and ‘Everything can be divided’ ‐‐ on seventh to twelfth grade students’ responses to successive division tasks related to mathematical and physical objects. Then, we studied the effect of an intervention, which provided students with two contradictory statements, one in line with students’ intuitive response, the other contradicting it, on their responses to various successive division tasks. It was found that this conflict‐based intervention did not improve students’ ability to differentiate between successive division processes related to mathematical objects and those related to material ones. These results reconfirmed that intuitive rules are stable and resistant to change. Finally, this paper raised the need for additional research related to the relationship between intuitive rules and formal knowledge.     

Event‐Related Potential Evidence for Persistence of an Intuitive Misconception About Electricity

Event‐related potentials are used to test the hypothesis that an intuitive misconception persists in the mind even after the acquisition of scientific knowledge. We investigated the temporal dynamics of neural mechanisms in solving a scientific problem involving a common misconception. It showed that the increased P2 component was elicited by the stimulus congruent with the misconception. The increased N2 component was elicited by the stimulus incongruent with the misconception. Additionally, the enhanced late positive potential component was evoked by the stimulus involving the conflict between the misconception and scientific knowledge. These results suggest that an intuitive misconception might still survive after the acquisition of scientific knowledge. It could be invoked automatically to evaluate the plausibility of incoming information. Subsequently, scientific knowledge is explicitly retrieved to monitor the conflict and inhibit the misconception. This study might have values in revealing the neural correlates of knowledge representation and scientific reasoning in education.     

The Role of Intuitive Rules in Science and Mathematics Education

During the last two decades many researchers in mathematics and science education have studied students’ conceptions and ways of reasoning in mathematics and science. Most of this research is content‐specific. It was found that students hold alternative ideas that are not always compatible with those accepted in science. It was suggested that in the process of learning science or mathematics, students should restructure their specific conceptions to make them conform to currently accepted scientific ideas. In our work in mathematics and science education it became apparent that some of the alternative conceptions in science and mathematics are based on the same intuitive rules. We have so far identified two such rules: “More of A, more of B”, and “Subdivision processes can always be repeated”. The first rule is reflected in subjects’ responses to many tasks, including all classical Piagetian conservation tasks (conservation of number, area, weight, volume, matter, etc.) in all tasks related to intensive quantities (density, temperature, concentration, etc.) and in all tasks related to infinite quantities. The second rule is observed in students’, preservice and inservice teachers’ responses to tasks related to successive division of material and geometrical objects and in seriation tasks. In this paper, we describe and discuss these rules and their relevance to science and mathematics education.     

Use of context in the word recognition process by adults with a significant history of reading difficulties

We examined whether university students who report a significant history of reading difficulties (RD; n = 24) differed from university students with no history of reading difficulties (NRD; n = 31) in how sentence context affects word recognition. Experiment 1 found no differences in how congruent sentence primes or syntactic manipulations of the sentence primes affected the performance of the two groups. However, only the RD group displayed a significant inhibition effect when the target word was preceded by an incongruent sentence prime. Experiment 2 found that the groups differed in how meaning frequency of the target word and context strength of the sentence prime affected word recognition latencies. The results suggest that the RD participants’ performance is context-sensitive and better explained by interactive models of language processing than by modular models.     

Automatic Activation of Adolescents’ Peer‐Relational Schemas: Evidence From Priming With Facial Identity

This study provides experimental evidence for automatic, relationship‐specific social information processing in 13‐year‐old adolescents. Photographs of participants’ liked, disliked, and unknown peers were used as primes in an affective priming task with happy and angry facial expression probes and in a hypothetical vignette task. For the affective priming, reaction times were faster for congruent than for incongruent prime–probe pairs when the prime visibility was high and the prime–probe stimulus onset asynchrony was long. In the vignette task, participants attributed more hostility toward the protagonist, experienced more anger, and were more likely to retaliate when the disliked peer served as a prime. It is concluded that peer‐relational schemas and related affect are activated automatically upon perception of a peer.     

The predictive power of intuitive rules: A critical analysis of the impact of `more A–more B' and `same A–same B'

In the international community of mathematics and science educators the intuitive rules theory developed by the Israeli researchers Tirosh and Stavy receives much attention. According to this theory, students' responses to a variety of mathematical and scientific tasks can be explained in terms of their application of some common intuitive rules. Two major intuitive rules are manifested in comparison tasks: ‘More A—more B’ and ‘Same A—same B’. In this paper, we address two important questions for which the existing literature on intuitive rules does not provide a convincing research-based answer: (1) are the reasoning processes of students who respond in line with a given intuitive rule actually affected by that rule or by essentially other misconceptions (leading to the same answer), and (2) are individual students consistent in their choice of one of the intuitive rules when confronted with different, conceptually unrelated tasks? A test consisting of five comparison problems from different mathematical subdomains was administered collectively to 172 Flemish students from Grades 10 to 12. An analysis of students' written calculations and justifications suggested that the students were considerably less affected by the intuitive rules than their multiple-choice answers actually suggested. Instead, essentially different misconceptions and errors were found. With respect to the issue of individual consistency, we found that students who made many errors did not answer systematically in line with one of the two intuitive rules.     

PIAGETIAN COGNITIVE LEVEL AND THE TENDENCY TO USE INTUITIVE RULES WHEN SOLVING COMPARISON TASKS

According to the intuitive rules theory, students are affected by a small number of intuitive rules when solving a wide variety of science and mathematics tasks. The current study considers the relationship between students’ Piagetian cognitive levels and their tendency to answer in line with intuitive rules when solving comparison tasks. The findings indicate that the tendency to answer according to the intuitive rules varies with cognitive level. Surprisingly, a higher rate of incorrect responses according to the rule same A–same B was found for the higher cognitive level. Further findings and implications for science and mathematics education are discussed.     

高中数学学困生学习兴趣培养方法探赜

高中阶段的数学学习对学生的记忆能力、逻辑推理能力和空间想象能力要求较高,很容易出现学困生。教师应根据学困生形成的原因,指导学困生正确学习,善于发现学困生的闪光点,创设情境辅助教学,重视分层教学,激发数学学困生的学习兴趣,使他们顺利实现转化,提高数学素养。     

First‐year secondary school mathematics students’ conceptions of mathematical proofs and proving

The aim of this study is to investigate students’ conceptions about proof in mathematics and mathematics teaching. A five‐point Likert‐type questionnaire was administered in order to gather data. The sample of the study included 33 first‐year secondary school mathematics students (at the same time student teachers). The data collected were analysed and interpreted using the methods of qualitative and quantitative analysis. The results have revealed that the students think that mathematical proof has an important place in mathematics and mathematics education. The students’ studying methods for exams based on imitative reasoning which can be described as a type of reasoning built on copying proof, for example, by looking at a textbook or course notes proof or through remembering a proof algorithm. Moreover, they addressed to the differences between mathematics taught in high school and university as the main cause of their difficulties in proof and proving.     

Categories in conflict: Combating the application of an intuitive conception of inheritance with category construction

The idea that characteristics acquired by an organism during its lifetime can be inherited by offspring and result in evolution is a substantial impediment to student understanding of evolution. In the current study, we performed a preliminary examination of how acquiring physical changes in a question prompt may differentially cue intuitive and scientific justifications of inheritance and evolution and how this varies based on how student learned the concept. Middle school students in a suburban northeastern district (N = 314) either learned about evolutionary change with a category construction task (with different levels of feedback support) or completed a worksheet. Three days later students responded to two free response scenarios (one where a physical change is acquired). Responses were coded based on student justifications for either science accuracy or intuitive nature. Specific reasons were coded by justification type with high inter-rater agreement (k > 0.93). Results showed that students were more likely to apply intuitive reasoning when a physical change was acquired (50%) than if the change was behavioral in nature (16%). Additionally, students who completed the category construction task provided significantly more scientifically accurate justifications about inheritance (M = 1.12) than control students (M = 0.47), and significantly less intuitive justifications (M = 0.67) than control (M = 1.13). Finally, category construction produced the most scientific reasoning when feedback was provided. Taken together, these results suggest that intuitive reasoning is differentially applied based on physical organismal changes, intuitive reasoning is less frequent when learning via category construction, and the category construction task is more effective for this population with the inclusion of feedback.     

Patterns of informal reasoning in the context of socioscientific decision making

The purpose of this study is to contribute to a theoretical knowledge base through research by examining factors salient to science education reform and practice in the context of socioscientific issues. The study explores how individuals negotiate and resolve genetic engineering dilemmas. A qualitative approach was used to examine patterns of informal reasoning and the role of morality in these processes. Thirty college students participated individually in two semistructured interviews designed to explore their informal reasoning in response to six genetic engineering scenarios. Students demonstrated evidence of rationalistic, emotive, and intuitive forms of informal reasoning. Rationalistic informal reasoning described reason‐based considerations; emotive informal reasoning described care‐based considerations; and intuitive reasoning described considerations based on immediate reactions to the context of a scenario. Participants frequently relied on combinations of these reasoning patterns as they worked to resolve individual socioscientific scenarios. Most of the participants appreciated at least some of the moral implications of their decisions, and these considerations were typically interwoven within an overall pattern of informal reasoning. These results highlight the need to ensure that science classrooms are environments in which intuition and emotion in addition to reason are valued. Implications and recommendations for future research are discussed. © 2004 Wiley Periodicals, Inc. J Res Sci Teach 42: 112–138, 2005     

Dyscalculia and Typical Math Achievement Are Associated With Individual Differences in Number-Specific Executive Function

Deficits in numerical magnitude perception characterize the mathematics learning disability developmental dyscalculia (DD), but recent studies suggest the relation stems from inhibitory control demands from incongruent visual cues in the nonsymbolic number comparison task. This study investigated the relation among magnitude perception during differing congruency conditions, executive function, and mathematics achievement measured longitudinally in children (n = 448) from ages 4 to 13. This relation was investigated across achievement groups and as it related to mathematics across the full range of achievement. Only performance on incongruent trials related to achievement. Findings indicate that executive function in a numerical context, beyond magnitude perception or executive function in a non-numerical context, relates to DD and mathematics across a wide range of achievement.     

Consistent occupational preferences and academic adjustment

This study explored the differences in academic adjustment variables between students who made congruent, incongruent, and undecided college major choices. The academic adjustment variables were operationally defined by the College Inventory of Academic Adjustment. Congruent, incongruent, and undecided college major choice groups were defined operationally using the Vocational Preference Inventory. The analysis of variance revealed the main effect of groups (congruent, incongruent, and undecided) to be significant for two academic adjustment variables. The test for the main effect of sex was found to be significant for one adjustment variable. The test for interaction was found to be significant for one variable. The findings suggest that students in the congruent female and male groups tend to be more academically adjusted than students in the undecided male group.