Diagrams as sketches

This article puts forward the notion of ??evolving diagram?? as an important case of mathematical diagram. An evolving diagram combines, through a dynamic graphical enrichment, the representation of an object and the representation of a piece of reasoning based on the representation of that object. Evolving diagrams can be illustrated in particular with category-theoretic diagrams (hereafter ??diagrams*??) in the context of ??sketch theory,?? a branch of modern category theory. It is argued that sketch theory provides a diagrammatic* theory of diagrams*, that it helps to overcome the rivalry between set theory and category theory as a general semantical framework, and that it suggests a more flexible understanding of the opposition between formal proofs and diagrammatic reasoning. Thus, the aim of the paper is twofold. First, it claims that diagrams* provide a clear example of evolving diagrams, and shed light on them as a general phenomenon. Second, in return, it uses sketches, understood as evolving diagrams, to show how diagrams* in general should be re-evaluated positively.     

The Observation of Incremental Validity Does Not Always Mean Unique Contribution to Prediction

Statistical analyses require proper interpretation. Misinterpretation leads to a lack of understanding of the relationships among variables. Worse, it can lead to misunderstanding of these relationships, which sometimes lead researchers and practitioners to infer the presence of a source of variance that is not present. This is especially true in regression where increased predictiveness from an additional variable may be due to either common or specific variance. In many instances, erroneous interpretation leads to erroneous attribution of the source of the improved prediction. Three examples are provided and methods for detecting specific variance are suggested.     

Facilitating children's understanding of misinterpretation: explanatory efforts and improvements in perspective taking

The authors investigated children's understanding of how mistaken beliefs can arise through misinterpretation of ambiguous information. Children (N = 91), aged 4 to 5 years, were given pre- and posttests on their ability to infer a puppet's interpretation of a restricted-view drawing after the puppet had been led to an erroneous expectation about the drawing's identity. Before the posttest, the children received either self-explanation training or other-explanation training in which they explained the source of their own or a puppet's misinterpretations of drawings; a control group received no training. The children who received training improved from pre- to posttest, and those who had practiced explaining misinterpretations by referring to previously viewed pictures or to features of a target picture showed the greatest improvement. These results indicate that learning to explain misinterpretations can help children recognize situations in which misinterpretations are likely to occur.     

And so on . . . : reasoning with infinite diagrams

This paper presents examples of infinite diagrams (as well as infinite limits of finite diagrams) whose use is more or less essential for understanding and accepting various proofs in higher mathematics. The significance of these is discussed with respect to the thesis that every proof can be formalized, and a ??pre?? form of this thesis that every proof can be presented in everyday statements-only form.     

The Nature and Origin of Rational Errors in Arithmetic Thinking: Induction from Examples and Prior Knowledge

Students systematically and deliberately apply rule-based but erroneous algorithms to solving unfamiliar arithmetic problems. These algorithms result in erroneous solutions termed rational errors. Computationally, students' erroneous algorithms can be represented by perturbations or bugs in otherwise correct arithmetic algorithms (Brown & VanLehn, 1980; Langley & Ohilson, 1984; VanLehn, 1983, 1986, 1990; Young S O'Sheo, 1981). Bugs are useful for describing how rational errors occur but bugs are not sufficient for explaining their origin. A possible explanation for this is that rational errors are the result of incorrect induction from examples. This prediction is termed the “induction hypothesis” (VanLehn, 1986). The purpose of the present study was to: (a) expand on post formulations of the induction hypothesis, and (b) use a new methodology to test the induction hypothesis more carefully than has been done previously. The first step involved teaching participants a new number system called NewAbacus, a written modification of the abacus system. The second step consisted of dividing them into different groups, where each individual received an example of only one port of the NewAbacus addition algorithm. During the third and final step, participants were instructed to solve both familiar and unfamiliar types of addition problems in NewAbacus. The induction hypothesis was supported by using both empirical and computational investigations.     

Can students evaluate their understanding of cause-and-effect relations? The effects of diagram completion on monitoring accuracy

For effective self-regulated study of expository texts, it is crucial that learners can accurately monitor their understanding of cause-and-effect relations. This study aimed to improve adolescents' monitoring accuracy using a diagram completion task. Participants read six texts, predicted performance, selected texts for restudy, and were tested for comprehension. Three groups were compared, in which learners either completed causal diagrams immediately after reading, completed them after a delay, or received no-diagram control instructions. Accuracy of predictions of performance was highest for learning of causal relations following delayed diagram completion. Completing delayed diagrams focused learners specifically on their learning of causal relations, so this task did not improve monitoring of learning of factual information. When selecting texts for restudy, the participants followed their predictions of performance to the same degree, regardless of monitoring accuracy. Fine-grained analyses also showed that, when completing delayed diagrams, learners based judgments on diagnostic cues that indicated actual understanding of connections between events in the text. Most important, delayed diagram completion can improve adolescents' ability to monitor their learning of cause-and-effect relations.     

The effects of self‐explaining when learning with text or diagrams

Self‐explaining is an effective metacognitive strategy that can help learners develop deeper understanding of the material they study. This experiment explored if the format of material (i.e., text or diagrams) influences the self‐explanation effect. Twenty subjects were presented with information about the human circulatory system and prompted to self‐explain; 10 received this information in text and 10 in diagrams. Results showed that students given diagrams performed significantly better on post‐tests than students given text. Diagrams students also generated significantly more self‐explanations that text students. Furthermore, the benefits of self‐explaining were much greater in the diagrams condition. To discover why diagrams can promote the self‐explanation effect, results are interpreted with reference to the multiple differences in the semantic, cognitive and affective properties of the texts and diagrams studied.     

Predicting Support for Arabs' Autonomy from Social Dominance: The Role of Identity Complexity and Dehumanization

The present study extends previous research on the influence of social dominance orientation (SDO) in international affairs by investigating the role of sociocognitive processes such as social identity complexity (SIC) and intergroup dehumanization in explaining the relationship between SDO and support for Arabs' autonomy. An Italian heterogeneous sample (N = 123), in terms of gender, age, political orientation, religiosity, and income level, was considered. We expected that those low on SDO would be more likely to support Arabs' autonomy, would have higher SIC, and would be less likely to dehumanize Arabs. We also expected that SIC and dehumanization would sequentially mediate the relationship between SDO and support for Arabs' autonomy. Results revealed that Italians low on SDO have higher SIC (i.e., a more inclusive social identity) which endorses lower dehumanization of Arabs. This in turn explained the influence of SDO on support for Arabs' autonomy. These findings integrate the literature about SDO and SIC and enhance the understanding of the sociocognitive processes underlying people's support for international egalitarian movements.     

Eliciting information and cues to deceit through sketching in interpreter‐based interviews

As interviewees typically say less when an interpreter is present, we examined whether this was caused by interpreters not interpreting everything interviewees says or by interviewees providing less information. We further examined (a) the effect of a model drawing on providing information and (b) the diagnostic value of total details and the proportion of complications as cues to deceit. Hispanic, Russian, and South Korean participants were interviewed by native interviewers or by a British interviewer through an interpreter. Truth tellers discussed a trip they had made; liars fabricated a story. Participants received no instruction (condition 1) or were instructed to sketch while narrating without (condition 2) or with (condition 3) being given examples of detailed sketches. Interviewees said less when an interpreter was present because they provided less information. Truth tellers gave more details and, particularly, obtained a higher proportion of complications than liars. The sketching manipulation had no effect.     

Representational Translation With Concrete Models in Organic Chemistry

In representation-rich domains such as organic chemistry, students must be facile and accurate when translating between different 2D representations, such as diagrams. We hypothesized that translating between organic chemistry diagrams would be more accurate when concrete models were used because difficult mental processes could be augmented by external actions on the models. In three studies, the task was to translate between different diagrams of molecules with or without a model. The model groups outperformed control groups, and students who received and used models outperformed those who received but did not use models. Uses of the model suggested that participants were performing external actions to support or replace difficult mental spatial processes. Spatial ability was a much weaker predictor of performance than model use. Results suggest that concrete molecular models can be an effective learning tool but some students need direct instruction to be able to take advantage of models.     

The use of diagrams in analogical problem solving

In four experiments, we examined the impact of perceptual properties on the effectiveness of diagrams in analogical problem solving, using variants of convergence diagrams as source analogues for the radiation problem. Static diagrams representing the initial problematic state (one large line directed at a target) and the final state for a convergence solution (multiple converging lines) were not accessed spontaneously but were often used successfully once a hint to consider the diagram had been provided. The inaccessibility of static diagrams was not alleviated by adding additional diagrams to represent intermediate states (Experiment 1), but spontaneous access was improved by augmenting static diagrams with a verbal statement of the convergence principle (Experiment 3). Spontaneous retrieval and noticing were increased markedly by animating displays representing converging forces and thereby encouraging encoding of the lines as indicating motion toward a target (Experiments 3 and 4). However, neither static nor animated diagrams were effective when the arrows were reversed to imply divergence rather than convergence (Experiment 2). The results indicate that when animation encourages the interpretation of a diagram as a helpful source analogue, it can greatly enhance analogical transfer.     

物体受力分析正误样例组合的学习效果

分别采用正误样例组合、有无错误标记的正误样例组合和有标记的正误样例配对组合设计方法,以物体受力分析正误样例为实验材料,以初中三年级学生为被试进行3项实验,考察了被试物体受力分析图正误样例组合的学习效果。结果表明,正误样例组合的学习成绩显著优于正确样例组合的学习成绩;有标记的正误样例组合的学习成绩显著优于无标记样例组合的学习成绩;有标记正误样例配对组合的学习成绩明显优于正确样例组合的学习成绩。     

The Role of Task Experience and Prior Knowledge for Detecting Invalid Augmented Feedback while Learning a Motor Skill

Two hypotheses were investigated. These were generated from results reported by Buekers, Magill, and Sneyers (1994) and an uncertainty account of those and other effects on skill learning of erroneous knowledge of results (KR). The first hypothesis proposes that if experienced performers have developed the capability of detecting and correcting errors, then they should not be influenced by erroneous KR in the same way as novices. The second proposes that if information about the invalid feedback is given to subjects prior to the beginning of the practice trials, then the capability for assessing feedback can be accelerated such that subjects who receive this advance information should not be misled by the erroneous KR. The first hypothesis was investigated in Experiment 1 by having subjects practise an anticipation timing task for 450 trials. One group received correct KR on all trials, and another group received no KR. Half of the subjects in each group were unexpectedly switched to erroneous KR after 400 trials. Results indicated that the erroneous KR influenced both groups during acquisition and on a retention test one minute later. However, on a retention test given one week later, only the group that had practised with no KR before being switched to erroneous KR continued to show the negative influence of the invalid KR. The second hypothesis was examined in Experiment 2 by telling subjects prior to beginning practice trials that it was possible that the KR they received would be erroneous. Results showed that this intervention strategy was effective on the no-KR retention tests only for subjects who had no-KR trials alternated with trials on which they received the erroneous KR. The results of these experiments provide evidence that uncertainty about the validity of sensory feedback increases the likelihood that subjects will be misled by invalid augmented feedback. Moreover, this uncertainty can be overcome by increased experience or by providing advance knowledge about the nature of the invalid feedback.     

Context and structure: The nature of students' knowledge about three spatial diagram representations

The authors investigated whether college students possess abstract rules concerning the applicability conditions for three spatial diagrams that are important tools for thinking—matrices, networks, and hierarchies. A total of 127 students were asked to select which type of diagram would be best for organising the information in each of several short scenarios. The scenarios were written using three different story contexts: (a) neutral, presenting a real-life situation but not cueing a particular representation; (b) abstract, presenting only variable names and relations; and (c) incongruent, in which the context and informational structure cued different representations. The results indicated above-chance performance on the abstract scenarios, as well as comparable performance on the abstract and neutral context scenarios. In a follow-up study in which eight students thought out loud while selecting diagrams for the abstract scenarios, there were almost no references to concrete examples. The results of these studies suggest that students possess abstract rules concerning the applicability conditions for matrices, networks, and hierarchies.     

Mistakes on display: Incorrect examples refine equation solving and algebraic feature knowledge

Although findings from cognitive science have suggested learning benefits of confronting errors (Metcalfe, 2017), they are not often capitalized on in many mathematics classrooms (Tulis, 2013). The current study assessed the effects of examples focused on either common mathematical misconceptions and errors or correct concepts and procedures on algebraic feature knowledge and solving quadratic equations. Middle school algebra students (N = 206) were randomly assigned to four conditions. Two errorful conditions either displayed errors and asked students to explain or displayed correct solutions and primed students to reflect on potential errors by problem type. A correct example condition and problem-solving control group were also included. Studying and explaining common errors displayed in incorrect examples improved equation-solving ability. An aptitude-by-treatment interaction revealed that learners with limited understandings of algebraic features demonstrated greater benefits. Theoretical implications about using examples to promote learning from errors are considered in addition to suggestions for educational practice.     

Patterns of Response Times and Response Choices to Science Questions: The Influence of Relative Processing Time

We report on five experiments investigating response choices and response times to simple science questions that evoke student “misconceptions,” and we construct a simple model to explain the patterns of response choices. Physics students were asked to compare a physical quantity represented by the slope, such as speed, on simple physics graphs. We found that response times of incorrect answers, resulting from comparing heights, were faster than response times of correct answers comparing slopes. This result alone might be explained by the fact that height was typically processed faster than slope for this kind of task, which we confirmed in a separate experiment. However, we hypothesize that the difference in response time is an indicator of the cause (rather than the result) of the response choice. To support this, we found that imposing a 3‐s delay in responding increased the number of students comparing slopes (answering correctly) on the task. Additionally a significant proportion of students recognized the correct written rule (compare slope), but on the graph task they incorrectly compared heights. Finally, training either with repetitive examples or providing a general rule both improved scores, but only repetitive examples had a large effect on response times, thus providing evidence of dual paths or processes to a solution. Considering models of heuristics, information accumulation models, and models relevant to the Stroop effect, we construct a simple relative processing time model that could be viewed as a kind of fluency heuristic. The results suggest that misconception‐like patterns of answers to some science questions commonly found on tests may be explained in part by automatic processes that involve the relative processing time of considered dimensions and a priority to answer quickly.     

Can People Identify “Deceptive” or “Misleading” Items that Tend to Produce Mostly Wrong Answers?

In many domains, two‐alternative forced‐choice questions produce more correct responses than wrong responses across participants. However, some items, dubbed “deceptive” or “misleading”, produce mostly wrong answers. These items yield poor calibration and poor resolution because the dominant, erroneous response tends to be endorsed with great confidence, even greater than that of the correct response. In addition, for deceptive items, group discussion amplifies rather than mitigates error while enhancing confidence in the erroneous response. Can participants identify deceptive items when they are warned about their existence? It is argued that people's ability to discriminate between deceptive and non‐deceptive items is poor when the erroneous responses are based on the same process assumed to underlie correct responses. Indeed, participants failed to discriminate between deceptive and non‐deceptive perceptual items when they were warned that some of the items (Experiment 1) or exactly half of the items (Experiment 2) were deceptive. A similar failure was observed for general‐knowledge questions (Experiment 3) except when participants were informed about the correct answer (Experiment 4). Possibly, for these tasks, people cannot escape the dangers lurking in deceptive items. In contrast, the results suggest that participants can identify deceptive problems for which the wrong answer stems from reliance on a fast, intuitive process that differs from the analytic mode that is likely to yield correct answers (Experiment 5). The practical and theoretical implications of the results were discussed. Copyright © 2017 John Wiley & Sons, Ltd.     

Modeling Corroborative Evidence: Inference to the Best Explanation as Counter–Rebuttal

Corroborative evidence has a dual function in argument. Primarily, it functions to provide direct evidence supporting the main conclusion. But it also has a secondary, bolstering function which increases the probative value of some other piece of evidence in the argument. This paper argues that the bolstering effect of corroborative evidence is legitimate, and can be explained as counter–rebuttal achieved through inference to the best explanation. A model (argument diagram) of corroborative evidence, representing its structure and operation as a schematic pattern of defeasible argument is also supplied. In addition to explaining the operation and theoretical foundation of corroborative evidence, the model facilitates the correct analysis and guides the evaluation (assessment and critique) of corroborative evidence as it occurs in argument.