An attempt to understand students' understanding of basic algebra

This paper reports the results obtained with a group of 24 14-year-old students when presented with a set of algebra tasks by the Leeds Modelling System, LMS. These same students were given a comparable paper-and-pencil test and detailed interviews some four months later. The latter studies uncovered several kinds of student misunderstandings that LMS had not detected. Some students had profound misunderstandings of algebraic notation: Others used strategies such as substituting numbers for variables until the equation balanced. Additionally, it appears that the student errors fall into several distinct classes: namely, manipulative, parsing, clerical, and “random.” LMS and its rule database have been enhanced as the result of this experiment, and LMS is now able to diagnose the majority of the errors encountered in this experiment. Finally, the paper gives a process-oriented explanation for student errors, and re-examines related work in cognitive modelling in the light of the types of student errors reported in this experiment. Misgeneralization is a mechanism suggested to explain some of the mal-rules noted in this study.     

Effects of spelling errors on the perception of writers

The authors investigated the perceived relationship between spelling errors and cognitive abilities in a series of 3 experiments. Specifically, they examined whether college students' ratings of an author's intellectual ability, logical ability, and writing ability were affected by the presence of spelling errors. In the 1st experiment, the presence of 4 spelling errors in a short essay did not significantly affect the ratings. The spelling ability of college students, as measured by a standard oral dictation spelling test, was moderately conelated with a brief test of intelligence. In a 2nd experiment, college students rated the author of a short essay as having lower ability when there was a large number of spelling errors. The effect was more pronounced on the ratings of writing ability than it was on the ratings of logical ability or intellectual ability. This finding was replicated in a 3rd experiment, in which the essay contained misspellings actually made by writers. The results suggest that spelling errors can affect how people perceive writers, particularly when there are many spelling errors. College students appear to attribute spelling errors more to writing ability than they do to general cognitive abilities such as intelligence and logical ability.     

Personality Correlates of Confidence in One's Decisions

ABSTRACT Taylor and Brown (1988) hypothesized that certain illusions, such as unrealistic optimism, are predictably associated with positive affect, social skills, and intellectual functioning. To test this hypothesis, we obtained data from 162 students who first filled out a questionnaire containing dispositional measures of affect, social skills, and approaches to problem solving; in a second session they completed three tasks requiring difficult decisions, reporting their confidence in each decision. Accuracy of judgments was found to vary considerably from task to task, but confidence ratings showed a consistent pattern of individual differences. This result lent support to Taylor and Brown's hypothesis, as did other features of the data, most notably several small but significant Pearson r s between confidence ratings and dispositional variables. Removing the effects of accuracy from these r s reduced their magnitude very little; it yielded partial r s interpretable as evidence that illusory confidence is associated with personality traits which, in this case, load saliently on factors labeled ( a ) affective, and ( b ) cognitive/social.     

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.     

The hypercorrection effect in younger and older adults

ABSTRACT

The hypercorrection effect, which refers to the finding that errors committed with high confidence are more likely to be corrected than are low confidence errors, has been replicated many times, and with both young adults and children. In the present study, we contrasted older with younger adults. Participants answered general-information questions, made confidence ratings about their answers, were given corrective feedback, and then were retested on questions that they had gotten wrong. While younger adults showed the hypercorrection effect, older adults, despite higher overall accuracy on the general-information questions and excellent basic metacognitive ability, showed a diminished hypercorrection effect. Indeed, the correspondence between their confidence in their errors and the probability of correction was not significantly greater than zero, showing, for the first time, that a particular participant population is selectively impaired on this error correction task. These results potentially offer leverage both on the mechanisms underlying the hypercorrection effect and on reasons for older adults’ memory impairments, as well as on memory functions that are spared.     

Updating misconceptions: Effects of age and confidence

Young adults are more likely to correct an initial higher confidence error than a lower confidence error (Butterfield & Metcalfe, 2001). This hypercorrection effect has never been investigated among older adults, although features of the standard paradigm (free recall, metacognitive judgments) and prior evidence of age-related error resolution deficits (see Clare & Jones, 2008) suggest that they may not show this effect. In Study 1, we used free recall and a 7-point confidence scale; in Study 2, we used multiple-choice questions, and participants indicated how many alternatives they had narrowed their options down to prior to answering. In both studies, younger and older adults showed a hypercorrection effect, and this effect was equivalent between groups in Study 2 when free recall and explicit confidence ratings were not required. These results are consistent with our previous work (Cyr & Anderson, 2012) showing that older adults can successfully resolve learning errors when the learning context provides sufficient support.     

Social influences on remembering: Intellectual,interpersonal and intergroup components

When people recall an event collaboratively we may expect the product to be influenced by the combined cognitive resources of the group, interpersonal acquaintance of the members, and social competition engendered by salient intergroup considerations. Using undergraduate students and serving police officers as subjects, a range of experimental conditions was established which varied on the three factors of cognitive resources of the participants (Cognitive Resources), interpersonal acquaintance of the participants (Interpersonal Acquaintance) and professional salience (Professional Salience) of the recall material. Participants answered a questionnaire concerning a police interrogation they had witnessed, and rated how much confidence they had in their answers. Multiple regression revealed statistically significant associations between (i) Cognitive Resources and number of correct answers, (ii) Interpersonal Acquaintance and confidence for implicational errors, and (iii) Professional Salience and (a) number of implicational errors and (b) confidence in confusional errors. The theoretical implications for our understanding of memory as a social process are discussed, and the practical implications for courtroom testimony briefly described.     

Encouraging Intellectual Growth: Senior College Student Profiles

Researchers and educational writers provide many views on what characteristics and skills the college graduate should possess (P. D. Gardner, 1998). Given the rapidly changing nature of knowledge in all disciplines and the subjectivity of grading systems, we argue the best indicator of a successful college education is a student's ability to think complexly about difficult problems. In this paper we examine students' levels of intellectual development to inform our understanding of how college experiences affect students' abilities to think in complex ways about difficult problems. Using both qualitative and quantitative research techniques, we examine and compare the experiences of 19 senior college students who were rated at opposite ends of the Perry scheme of intellectual development. The 19 students were purposefully selected to represent the high and low ends of the Perry scale from an overall sample of 27 senior students at a large eastern university. Ten students from the original sample of 27 demonstrated the most advanced cognitive development with Perry scale ratings of 5–7. Nine students had Perry ratings below 4; this was the lowest rating represented in the sample. We examine the similarities and differences between these two sets of students' academic profiles, and their comments about their curricular experiences, and how the Perry scale reflects on these results. We then discuss how these similarities and differences relate to students' development of the desirable characteristics of a college graduate, implications for faculty and administrators, and the relationship of these results to prior studies using the Perry scheme.     

Prior knowledge is more predictive of error correction than subjective confidence

Previous research has demonstrated that, when given feedback, participants are more likely to correct confidently-held errors, as compared with errors held with lower levels of confidence, a finding termed the hypercorrection effect. Accounts of hypercorrection suggest that confidence modifies attention to feedback; alternatively, hypercorrection may reflect prior domain knowledge, with confidence ratings simply correlated with this prior knowledge. In the present experiments, we attempted to adjudicate among these explanations of the hypercorrection effect. In Experiments 1a and 1b, participants answered general knowledge questions, rated their confidence, and received feedback either immediately after rating their confidence or after a delay of several minutes. Although memory for confidence judgments should have been poorer at a delay, the hypercorrection effect was equivalent for both feedback timings. Experiment 2 showed that hypercorrection remained unchanged even when the delay to feedback was increased. In addition, measures of recall for prior confidence judgments showed that memory for confidence was indeed poorer after a delay. Experiment 3 directly compared estimates of domain knowledge with confidence ratings, showing that such prior knowledge was related to error correction, whereas the unique role of confidence was small. Overall, our results suggest that prior knowledge likely plays a primary role in error correction, while confidence may play a small role or merely serve as a proxy for prior knowledge.     

A Developmental Analysis of Performance Ratings

In this study, we examined the performance appraisal process from a developmental perspective. Components of cognitive structure, measured in terms of differentiation and hierarchic integration, were obtained from 45 American students representing three different academic levels at three times during a 15-week semester. These students' performance evaluations (ratings) of their psychology instructors were collected during the first and final sessions. The results indicated that although both differentiation and hierarchic integration seemed to change systematically, even over a short period of time, there was no corresponding change in the halo inherent in the raters' ratings. We present possible explanations for the failure of developmental changes in cognitive structure to be reflected in the halo inherent in the performance ratings, as would be predicted by Schneier's (1977) cognitive compatibility hypothesis. Increased emphasis on developmental aspects of the performance appraisal process is recommended.     

Response selection strategies and realism of confidence judgments

Two studies were conducted to examine how response selection strategy is related to confidence ratings and to performance on general knowledge questions. In both studies subjects were asked to answer 80 general knowledge questions and to rate their confidence in the correctness of the answer selected. A pilot study, in which subjects thought aloud while answering general knowledge questions, was carried out to identify different response selection strategies. In the first study, 40 subjects were asked to indicate which of four strategies (immediate recognition, inference, intuition, or guessing) they used for selecting an answer. In Study 2, think aloud reports from 20 subjects were coded into the same four strategies. The distribution of strategies differed between the studies, but there were very similar relations among strategy, confidence, and correctness of answer in the two studies. Response selection strategy was related to correctness of answer when confidence was partialed out. More specifically, immediate recognition was associated with higher proportion correct than with the other strategies. It was also found that ratings of how difficult the knowledge questions were to fellow students of the subjects were on a much more realistic level than the confidence ratings were. It is concluded that people could improve their confidence judgments by taking into account (a) how difficult a question is to other people, and (b) the response selection strategy used for answering the question.     

Studies of Diagnosis and Remediation with High School Algebra Students

Despite extensive discussion in the literature about the diagnosis and subsequent remediation of students' errors, few studies have compared the effects of different styles of error-based remediation. Swan (1983) found that a conflict approach (pointing out errors made by students and demonstrating their consequences) was more effective than simple reteaching, but Bunderson & Olsen (1983) found no difference between error-specific remediation and reteaching. More studies are needed in order to understand the factors which lead to successful remediation. The three studies discussed in this article compared error-specific or model-based remediation (MBR) with reteaching in algebra. MBR bases its remediation on the model inferred for an individual student before reteaching the correct procedure. Reteaching simply shows students the correct procedure without addressing specific errors. The results show that MBR and reteaching are both more effective than no tutoring; however, MBR is not clearly more effective than reteaching. The results are discussed in terms of stability of errors, their relevance to educational practice and to intelligent tutoring systems (ITS). Although the studies were carried out using human tutors, the results suggest that for the purpose of remediation in the algebra domain, when taught procedurally, “classical” computer-assisted instruction (CAI) would be as effective as an ITS.     

Confidence and Uncertainty in Judgmental Forecasting: Differential Effects of Scenario Presentation

Multiple scenario analysis is used by many organizations for long-term planning, in part because the development of more than one scenario of the future is thought to prevent overconfidence in any one specific forecast. The present study attempts to clarify some of the questions concerning the cognitive effects of scenario presentation by distinguishing between confidence and subjective uncertainty. It examined the impact of different types of scenario information on confidence ratings, credible intervals, and features of forecast series. Subjects (N=186) read either one, two, or no scenario(s) describing potential changes in the level of a stimulus variable, then made forecasts of that variable over the next fifty years. Reading any type of scenario information increased confidence in forecasts; there was no reduction in confidence due to presentation of multiple conflicting scenarios, compared to presentation of only one scenario. Differences in number and in type of scenarios presented did, however, affect subjective uncertainty.     

Does calling it “Morgan's way” reduce student learning? Evaluating the effect of person-presentation during comparison and discussion of worked examples in mathematics classrooms

Mathematics textbooks sometimes present worked examples as being generated by particular fictitious students (i.e., person-presentation). However, there are indicators that person-presentation of worked examples may harm generalization of the presented strategies to new problems. In the context of comparing and discussing worked examples during extended classroom instruction, the current study compared the impact of person-presentation to strategy labels on students' posttest accuracy and ratings of strategy generalizability. Five algebra teachers and their 168 students used worked examples either presented using fictitious students or with a strategy label during a multiweek unit on equation solving, with teachers randomly assigned to condition. All students compared and discussed the worked examples. In this context, we found no effect of condition on student accuracy at posttest, nor on their ratings of the generalizability of the presented strategies. We discuss why previously found negative effects of person-presentation may not have extended to this context.     

Is probability matching smart? Associations between probabilistic choices and cognitive ability

In three experiments involving over 1,500 university students (n = 1,557) and two different probabilistic choice tasks, we found that the utility-maximizing strategy of choosing the most probable alternative was not the majority response. In a story problem version of a probabilistic choice task in which participants chose from among five different strategies,the maximizing response and the probability-matching response were each selected by a similar number of students (roughly 35% of the sample selected each). In a more continuous, or trial-by-trial, task, the utility-maximizing response was chosen by only one half as many students asthe probability-matching response. More important, in both versions of the task, the participants preferring the utility-maximizing response were significantly higher in cognitive ability than were the participants showing a probability-matching tendency. Critiques of the traditional interpretation of probability matching as nonoptimal may well help explain why some humans are drawn to the nonmaximizing behavior of probability matching, but the traditional heuristics and biases interpretation can most easily accommodate the finding that participants high in computational ability are more likely to carry out the rule-based cognitive procedures that lead to maximizing behavior.     

Parental Cognitive Errors Mediate Parental Psychopathology and Ratings of Child Inattention

We investigate the Depression‐Distortion Hypothesis in a sample of 199 school‐aged children with ADHD‐Predominantly Inattentive presentation (ADHD‐I) by examining relations and cross‐sectional mediational pathways between parental characteristics (i.e., levels of parental depressive and ADHD symptoms) and parental ratings of child problem behavior (inattention, sluggish cognitive tempo, and functional impairment) via parental cognitive errors. Results demonstrated a positive association between parental factors and parental ratings of inattention, as well as a mediational pathway between parental depressive and ADHD symptoms and parental ratings of inattention via parental cognitive errors. Specifically, higher levels of parental depressive and ADHD symptoms predicted higher levels of cognitive errors, which in turn predicted higher parental ratings of inattention. Findings provide evidence for core tenets of the Depression‐Distortion Hypothesis, which state that parents with high rates of psychopathology hold negative schemas for their child's behavior and subsequently, report their child's behavior as more severe.     

Psychopathologie de la vie quotidienne : validation d’un nouveau questionnaire de défaillances cognitives

In our daily lives, we make a number of errors and blunders for all our activities. These are the cognitive failures of everyday life, which are conventionally measured by the Cognitive Failure Questionnaire (CFQ). Literature has demonstrated not only that this tool has many limitations especially regarding its factor structure which is not identified by consensus but also that the origin and nature of cognitive failures are not interviewed because they are considered in terms of executive dysfunction. Also, this study will be to describe the various stages of validation structure of a new assessment tool, the Cognitive Failures Questionnaire Daily (questionnaire de défaillances cognitives quotidiennes [QDCQ]) and expose the different levels of analysis of our methodological approach that we have allowed to get it. Therefore, it was administered to 237 all comers people of all ages and proposed them to assess the frequency of cognitive failures over the last 6 months. As a result of is a first step in item analysis using indices of normality, of difficulty and discrimination and a second at different scales, we removed twenty-five items out of fifty-two which the statistical power was not sufficient to be kept. We then obtained a new version of QDCQ, which comprises twenty-seven items that apprehend the most cognitive failure's construct. The results of several confirmatory factor analyzes identify a structure with three factors (attention, language and memory) as best fitting our data. That is consistent with the general literature on this construct (0145 and 0240), and it tends us to prefer the hypothesis that the occurrence of cognitive failures would be the result of a momentary disruption of specific cognitive processes (probably executive processes which it remains to be defined) that influenced in different spheres of activities. The questionnaire opens new perspectives both in terms of clinical practice, particularly in that it could participate in early detection of several behaviours involved in various forms of pathological cognitive aging such as Alzheimer's disease or Parkinson's. Moreover, conceptually, it could also lead to clarify the relationship that the construct of cognitive failures has with executive functions.     

The Prediction of Negative Events Associated with Anxiety and Dietary Restraint: A Test of the Content Specificity Hypothesis

The prediction of future negative events was studied in a sample of undergraduate students varying on measures of anxiety and dietary restraint. Cognitive theories of psychopathology hypothesize that the emotional concerns of an individual determine which types of information are most often cognitively processed with a biased interpretation. This prediction has generally been called the content specificity hypothesis. Based upon this aspect of cognitive theory, we hypothesized that a high level of trait anxiety would be associated with higher ratings of the probability of future physically threatening events. In addition, it was hypothesized that a high level of dietary restraint would be associated with lower ratings of the probability of future occurrences of overeating. The results of the study supported these hypotheses. These prediction biases were found to be more pronounced in women, as compared to men. The findings were interpreted as supportive of the content specificity hypothesis.     

A year’s memories: The calendar effect in autobiographical recall

When asked to recall autobiographical events from the past year, students tend to recall more incidents from the beginning and the end of school terms than from other periods. We investigated thiscalendar effect in Experiment 1 by comparing free recall at schools with different academic calendars. The event distributions tracked the individual calendars, helping to eliminate the possibility that the calendar effect is due to seasonal, nonschool factors, such as holidays. In Experiments 2–4, we checked explanations based on the ideas that events at term boundaries are more important or distinctive than others, that events are incorrectly dated too near the boundaries, and that boundaries serve as implicit cues for recall. These experiments revealed no evidence that importance or errors in dating could explain the effect. Manipulating cues, however, did change the size of the effect, implicating retrieval from very long-term memory as the effect’s source. We suggest that when people have to search episodic memory, they consider their own calendar rhythms (such as a student’s academic schedule) and let the temporal structure of their personal context guide their search.