Exploring Students’ Learning of Integral Calculus using Revised Bloom’s Taxonomy

<p>Integral calculus is one of the topics involved in mathematical courses both at secondary and tertiary level with several applications in different disciplines. It is part of gateway mathematical courses at universities for many majors and important for the development of the science. Several studies had been undertaken for exploring students’ learning of integral calculus, both at the secondary and tertiary level, using a variety of frameworks (e.g., Action-Process-Object-Schema (APOS) theory (Dubinsky, 1991). However, students’ learning of integral calculus has not been explored in terms of metacognitive experiences and skills, and the number of studies which have explored metacognitive strategies in relation to the students’ learning of integral calculus is limited. Therefore, this study used Revised Bloom’s Taxonomy (RBT) (Anderson et al., 2001), Efklides’s metacognition framework (Efklides, 2008), and an adaptation of VisA (Visualization and Accuracy) instrument (Jacobse & Harskamp, 2012) for exploring students’ learning of integral calculus. A multiple case study approach was used to explore students’ learning of the integral-area relationships and the Fundamental Theorem of Calculus in relation to the RBT’s factual, conceptual, and procedural knowledge, and the facets of metacognition including metacognitive knowledge, experiences, and skills. The study sample comprised of nine first year university and eight Year 13 students who participated in individual semi-structured interviews answering nine integral calculus questions and 24 questions related to the RBT’s metacognitive knowledge. Integral calculus questions were designed to address different aspects of RBT’s knowledge dimension and activate RBT-related cognitive processes. A think aloud protocol and VisA instrument were also used during answering integral calculus questions for gathering information about students’ metacognitive experiences and skills. Ten undergraduate mathematics lecturers and five Year 13 mathematics teachers were also interviewed in relation to the teaching and learning of integral calculus to explore students’ difficulties in the topic. The entire teaching of integral calculus in a first year university course and a Year 13 classroom were video recorded and observed to obtain a better understanding of the teaching and learning of integral calculus in the context of the study. The study findings in terms of the RBT’s factual knowledge show several students had difficulty with notational aspects of the Fundamental Theorem of Calculus (FTC) (e.g., Thompson, 1994) whereas this issue was not dominant for the definite integral. In relation to the RBT’s conceptual and procedural knowledge for both topics, conceptual knowledge was less developed in students’ minds in comparison to procedural knowledge (e.g., students had not developed a geometric interpretation of the FTC, whereas they were able to solve integral questions using the FTC). The obtained results were consistent with previous studies for these three types of knowledge. The study contributes to the current literature by sharing students’ metacognitive knowledge, experiences and skills in relation to integral calculus. The findings highlight some student learning, monitoring, and problem-solving strategies in these topics. A comparison between University and Year 13 students’ results showed students across this transition had different factual, conceptual, procedural, and metacognitive knowledge in these topics. For instance, University students in the sample use online resources more often than Year 13 students, are more interested in justifications behind the formulas, and have more accurate pre and post-judgments of their ability to solve integral questions. The information obtained using questions based on RBT and the metacognition framework indicates that these two together may be very useful for exploring students’ mathematical learning in different topics.</p>

Exploring Students’ Learning of Integral Calculus using Revised Bloom’s Taxonomy

<p>Integral calculus is one of the topics involved in mathematical courses both at secondary and tertiary level with several applications in different disciplines. It is part of gateway mathematical courses at universities for many majors and important for the development of the science. Several studies had been undertaken for exploring students’ learning of integral calculus, both at the secondary and tertiary level, using a variety of frameworks (e.g., Action-Process-Object-Schema (APOS) theory (Dubinsky, 1991). However, students’ learning of integral calculus has not been explored in terms of metacognitive experiences and skills, and the number of studies which have explored metacognitive strategies in relation to the students’ learning of integral calculus is limited. Therefore, this study used Revised Bloom’s Taxonomy (RBT) (Anderson et al., 2001), Efklides’s metacognition framework (Efklides, 2008), and an adaptation of VisA (Visualization and Accuracy) instrument (Jacobse & Harskamp, 2012) for exploring students’ learning of integral calculus. A multiple case study approach was used to explore students’ learning of the integral-area relationships and the Fundamental Theorem of Calculus in relation to the RBT’s factual, conceptual, and procedural knowledge, and the facets of metacognition including metacognitive knowledge, experiences, and skills. The study sample comprised of nine first year university and eight Year 13 students who participated in individual semi-structured interviews answering nine integral calculus questions and 24 questions related to the RBT’s metacognitive knowledge. Integral calculus questions were designed to address different aspects of RBT’s knowledge dimension and activate RBT-related cognitive processes. A think aloud protocol and VisA instrument were also used during answering integral calculus questions for gathering information about students’ metacognitive experiences and skills. Ten undergraduate mathematics lecturers and five Year 13 mathematics teachers were also interviewed in relation to the teaching and learning of integral calculus to explore students’ difficulties in the topic. The entire teaching of integral calculus in a first year university course and a Year 13 classroom were video recorded and observed to obtain a better understanding of the teaching and learning of integral calculus in the context of the study. The study findings in terms of the RBT’s factual knowledge show several students had difficulty with notational aspects of the Fundamental Theorem of Calculus (FTC) (e.g., Thompson, 1994) whereas this issue was not dominant for the definite integral. In relation to the RBT’s conceptual and procedural knowledge for both topics, conceptual knowledge was less developed in students’ minds in comparison to procedural knowledge (e.g., students had not developed a geometric interpretation of the FTC, whereas they were able to solve integral questions using the FTC). The obtained results were consistent with previous studies for these three types of knowledge. The study contributes to the current literature by sharing students’ metacognitive knowledge, experiences and skills in relation to integral calculus. The findings highlight some student learning, monitoring, and problem-solving strategies in these topics. A comparison between University and Year 13 students’ results showed students across this transition had different factual, conceptual, procedural, and metacognitive knowledge in these topics. For instance, University students in the sample use online resources more often than Year 13 students, are more interested in justifications behind the formulas, and have more accurate pre and post-judgments of their ability to solve integral questions. The information obtained using questions based on RBT and the metacognition framework indicates that these two together may be very useful for exploring students’ mathematical learning in different topics.</p>

Investigating Students' Metacognitive Experiences: Insights From the English as a Foreign Language Learners' Writing Metacognitive Experiences Questionnaire (EFLLWMEQ)

While research on metacognitive knowledge and metacognitive strategies in second language (L2) writing has proliferated, little attention has been paid to metacognitive experiences in learning to write. This study contributes a novel 6-point Likert scale questionnaire, EFL Learners' Writing Metacognitive Experiences Questionnaire (EFLLWMEQ), and reports insights into learners' metacognitive experiences gathered from its use. The questionnaire was designed to investigate, first, the nature of students' metacognitive experiences when they learn to write in English as a foreign language (EFL) and, secondly, the relationship between students' metacognitive experiences and their writing performance. To this end, the questionnaire was developed and validated with two independent samples of 340 and 540 Chinese undergraduates whose metacognitive experiences were measured as they learned to write in EFL. Data were subjected to exploratory factor analysis (EFA) and confirmatory factor analysis (CFA), respectively. Findings of EFA and CFA revealed a four-factor structure of students' metacognitive experiences of EFL writing: Metacognitive estimates, metacognitive feelings, online task-specific metacognitive knowledge, and online task-specific metacognitive strategies. Results showed that students' metacognitive experiences had positive correlations with their EFL writing test scores. Importantly, the CFA results from the sample of 540 students supported the four-factor correlated model with the best model fit, confirming the validity and reliability of the EFLLWMEQ. This study has theoretical and pedagogical implications for how learners' metacognitive experiences can be managed in learning to write, particularly in EFL classrooms.

Student’s Perspective and Teachers’ Metacognition: Applications of Eye-Tracking in Education and Scientific Research in Schools

This Perspective article discusses the possible contributions of eye-tracking (ET) to the field of Educational Neuroscience based on an application of this tool at schools. We sought to explore the teachers’ view of ET videos recorded while students solved mathematical problems. More than 90% of the teachers could predict with great accuracy whether the students had answered the questions correctly or not based solely on the information provided by the ET videos. Almost all participants tried to translate the students’ thoughts to understand the strategy used by the children. Our results highlight the relevance of qualitative analysis to identify the gaze strategies used by students. We propose that ET allows teachers to gain critical feedback about students’ behavior during problem-solving. Most previous studies tend to emphasize the benefits of ET applications to explore learners’ cognition. Our findings point that this system can also be useful to investigate teachers’ cognition by providing metacognitive experiences.

Μεταγνωστικές εμπειρίες στην προοπτική μνήμη νεότερων και μεγαλύτερων σε ηλικία ατόμων: παίζει ρόλο η απερισκεψία;

The present study aimed to examine metacognitive experiences in situations with respect to perspective memory (PM). The metacognitive experiences studied were blank in mind, “tip of the tongue”, mind wandering, and mind blanking. Young, middle-aged and elderly people of both sexes participated in this study. We also studied affect, cognitive failures and mindlessness as personality traits, which were also used as correlation factors of metacognitive experiences. The results of the current study showed that personality traits are related to metacognitive experience of blank in the mind, but not with performance in the PM task. Moreover, the results showed that there is a decline in task performance in older people, but no effect on metacognitive experiences. There were also high correlations between the metacognitive experiences of mind wandering, blank in mind and mind blanking, but there were no correlations between them and the “tip of the tongue” experience. Finally, regarding the correlation of affect with the task performance and metacognitive experiences, it was found that neutral affect correlates positively with mind wandering and blank in mind and negatively with mind blanking self-reports, but it does not necessarily correlate with lower task performance.

A Nonspecific Approach to Metacognitive Experiences: The Experimental Study of Control Sense and Confidence in Giving a Correct Answer

There are specific and nonspecific approaches to explaining diversity of metacognitive experiences in psychological testing. The study is concerned with verification of the effects of a nonspecific approach, according to which there is a nonspecific signal about the success of task performance behind metacognitive experiences. Unidirectional changes in experiencing confidence and control sense were expected to be revealed. The findings showed that correct answers were given with more confidence than wrong ones, confident correct answers were given faster than unconfident correct ones, straightforward cases were handled with a high level of confidence and more sense of control if long time was given to think them over. The study failed in demonstrating unidirectional character of control sense and confidence in other conditions, which can be explained by insufficient sensitivity of the technique chosen to measure control sense.

From Insight to Aha Experience: a New Theoretical Framework for Investigation Insightful Problem Solving

The article is dedicated to the insight phenomenon, namely the explanations of the rela-tionship between cognitive and affective components of insight that different theories suggest. The affective component of insight includes the subjective suddenness, certainty, and pleasure with which the solution comes to mind (Aha! experience). The cognitive component of insight is the assumed mechanisms contributing to the generation of the solution. In the first part of the article, we review classical approaches to explain the insight phenomenon's nature: special processes account and business-as-usual account. In the first approach, two mecha-nisms can be suggested: unconscious associative processing and representational change (restructuring). According to the business-as-usual account, insight solutions do not differ from routine ones and are implemented as a step-by-step movement to the goal state. In the second part of the article, we show that all classical theories of insight problem solving suggest the direct relation between Aha! experience and cognitive mechanisms of insight. Then, we analyze accumulated empirical data that stand for or against this notion. Based on this analysis, we conclude that there is no unequivocal evidence for the direct rela-tion between Aha! experience and the unconscious processing or restructuring. We propose that Aha! experience is an anomaly for the studies of insight problem-solving. The third part of the article aims to review possible ways to resolve the problem of Aha! experience. One way is to consider that the relation between affective and cognitive compo-nents of insight is indirect. We review studies of metacognition where the non-specificity of metacognitive experiences was proposed. According to this idea, metacognitive experiences reflect changes in cognitive processes dynamics (such as the information processing fluency) but do not carry specific information about their source. In the fourth part of the article, the attempts to apply the idea of non-specificity to the Aha! experience are reviewed. We analyze the empirical data supporting the hypothesis of indirect relation between Aha! experience and cognitive mechanisms of insight problem-solving. Based on it, we predict an increase in the number of studies developing this idea.

From metacognitive beliefs to strategy selection: does fake performance feedback influence cognitive offloading?

The ubiquitous availability of technological aids requires individuals to constantly decide between either externalizing cognitive processes into these aids (i.e. cognitive offloading) or relying on their own internal cognitive resources. With the present research, we investigated the influence of metacognitive beliefs on individuals’ offloading behavior in an experimental setup (N = 159). We manipulated participants’ metacognitive beliefs about their memory abilities by providing fake performance feedback: below-average feedback, above-average feedback, or no feedback (control-group). We then measured offloading behavior, using a pattern copying task in which participants copied a color pattern from a model window into a workspace window. While solving this task, participants could rely either more on an internal memory strategy or more on an offloading strategy. Fake performance feedback affected the participants’ metacognitive evaluations about their memory abilities (below-group < control-group < above-group). Although fake performance feedback did not affect actual offloading behavior, the participants receiving below-average performance feedback reported that they had relied more on an offloading strategy than those participants receiving above-average performance feedback. Furthermore, the participants in the below-group reported lower general memory abilities than the other groups at the end of the experiment. We conclude that while fake performance feedback strongly influenced metacognitive beliefs, this did not transfer into a change of strategy selection, thus not influencing offloading behavior. We propose to consider not only metacognitive beliefs but also metacognitive experiences as potential determinants of cognitive offloading.