Exploring Students’ Metacognitive Knowledge: The Case of Integral Calculus

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Online mathematics teaching and learning during the COVID-19 pandemic: The perspective of lecturers and students

The global spread of the novel coronavirus, Covid-19, reached Norway at the end of February 2020. MatRIC, Centre for Research, Innovation and Coordination of Mathematics Teaching conducted a national survey in Norwegian higher education institutions (HEIs) during June-July 2020 to explore lecturers’ and students’ experiences of online mathematics teaching and learning and to enable sharing of solutions to the challenges encountered. One hundred and twenty-seven students and eighteen lecturers participated in this survey. In this presentation, we will share some of the findings of the survey in relation to the following two themes: challenges of learning and teaching mathematics online, and the psychological impact of lockdown on student learning and lecturer teaching. The study findings show that many students missed the social contact, being physically present at the university, and face to face interaction with their lecturers. Additionally, several students experienced a degree of anxiety through the lockdown period to the extent that they perceived their learning was negatively affected. Lecturers took a number of steps to ensure lines of communication remained open. Our findings show that simple actions by the lecturer to open channels of communication can be very effective.publishedVersio

Switching to Fully Online Teaching and Learning of Mathematics: The Case of Norwegian Mathematics Lecturers and University Students During the Covid‑19 Pandemic

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Exploring engineering students’ engagement with proof without words: the case of calculus

Previous studies have reported that many engineering students struggle to develop a conceptual understanding of mathematics in university calculus courses. Engaging with mathematical proofs is one of the approaches to developing a conceptual understanding of mathematics; however, it is not always easy to use standard proofs for this purpose. In the present study, we focus on a type of mathematical proof known as Proof Without Words (PWW). A PWW typically consists of pictures or diagrams that help readers understand why a mathematical statement is true without providing verbal justifications. This study investigates how engineering students engage with PWW tasks related to calculus and also how they perceive its usefulness for teaching and learning calculus. Twenty undergraduate engineering students participated in semi-structured group interviews (in groups of two students), engaging with three PWW tasks related to calculus. Afterwards, students’ perceptions of PWW tasks were explored using several open-ended questions. The findings indicate that many students engaged well with this type of activity. Furthermore, all students perceived that PWW tasks could positively impact their mathematical understanding, and many believed they could be used for teaching and learning calculus. Furthermore, several students highlighted that PWWs could also help with comprehending standards proofs and make mathematics learning more enjoyable

A transition to online teaching and learning of mathematics in Norwegian higher education institutions: the perspectives of lecturers and students

This paper reports a study of university lecturers’ and students’ experiences of teaching and learning mathematics following the abrupt requirement to switch to online teaching in 2020. A goal of the study is to share experiences that could be useful to improve the teaching and learning of mathematics in online settings. The qualitative research described is a phenomenological study and draws on interviews with ten mathematics lecturers and six undergraduate students who were enrolled in at least one university mathematical course during the lockdown in 2020. The interview data were analysed using a thematic approach. This paper reports findings regarding perceptions of lecturers and students about the challenges and benefits of online teaching and learning of mathematics, how the transition to online education has influenced assessment and sharing useful approaches for teaching and learning mathematics in online settings.publishedVersio

Advancing engineering students’ conceptual understanding through puzzle-based learning: a case study with exact differential equations

Current views on the teaching of differential equations (DEs) are shifting towards the use of graphical and numerical methods. Motivated by recent research suggesting that puzzle-based learning (PzBL) can improve the teaching and learning of STEM subjects and by the lack of relevant studies for DEs, we designed two tasks—sophism and paradox—to explore undergraduate engineering students’ conceptual understanding of a classical topic—exact DEs—and to analyse the process of meaning-making during collaborative learning in small groups. One hundred and thirty-five undergraduate engineering students from a public university in Iran participated. In response to recent research signalling the tendency of the students to procedural learning of DEs, we analyse how the students in our study engaged in small group work on puzzle tasks, gaining a more conceptual understanding of exact DEs and acknowledging the efficiency of PzBL in their responses to a questionnaire and in interviews.publishedVersionPaid Open Acces

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

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

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

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

A Study on K5 students’ mathematical problem solving based on Revised Bloom Taxonomy and psychological factors contribute to it

In this study, Revised Bloom’s Taxonomy (RBT) was used as a touchstone for obtaining a profile of K5 students’ mathematical problem solving in different cognitive process and levels of knowledge. In addition, the relationship between students’ mathematical problem solving and psychological factors (i.e. Mathematics Anxiety, Mathematics Attitude, Mathematics Attention, Working Memory Capacity and Cognitive Style) has been discussed through the lens of RBT. A total 212 K5 girls (aged 11-12 years old) were tested on (1) K5 Mathematics questions based on RBT, (2) Digit Span Backwards Test (DBT), (3) Cognitive style (FD/FI) test, (4) Mathematics Anxiety Rating Scale, (5) Modified Fennema-Sherman Attitude Scales, (6) Mathematics Attention Test. Data of this research was analyzed by MANOVA repeated measure, General Linear models and graphs error bars from SPSS (Statistical Package for the Social Sciences) software. Obtained results indicate that students’ have serious difficulties in solving Metacognitive knowledge problems and those concern to complex cognitive process. Moreover, psychological factors in question could predict Mathematical problem solving in different cognitive process and levels of knowledge. Overall, these findings could help to provide some practical implications for adapting problem solving skills and effective teaching/learning.En este estudio, Taxonomía revisada de Bloom (TRB) fue utilizado como una piedra de toque para la obtención de un perfil del problema matemático de los estudiantes de K5' de problemas en diferentes procesos cognitivos y niveles de conocimiento. Además, la relación entre el problema matemático de los estudiantes de problemas y factores psicológicos (por ejemplo ansiedad Matemáticas, Matemáticas Actitud, Matemáticas Atención, capacidad de memoria de trabajo y el estilo cognitivo) se ha discutido a través del lente de la TRB. Un total de 212 niñas K5 (entre 11-12 años de edad) fueron probados en (1) preguntas K5 Matemáticas basado en la RBT, (2), el estilo cognitivo (FD/FI) Prueba Digit Span Prueba revés (DBT) (3), (4) Matemáticas Anxiety Rating Scale, (5) Modificado Fennema-Sherman Attitude Scales, (6) Prueba de Matemáticas de Atención. Los datos de esta investigación se analizó mediante MANOVA repite medida, modelos lineales generales y las barras de error gráficos de SPSS (Statistical Package for Social Sciences) de software. Los resultados obtenidos indican que los estudiantes tienen serias dificultades para resolver problemas de conocimiento metacognitivas y las preocupaciones de proceso cognitivo complejo. Por otra parte, los factores psicológicos en cuestión podrían predecir la resolución de problemas matemáticos en diferentes procesos cognitivos y niveles de conocimiento. En general, estos resultados podrían ayudar a proporcionar algunas implicaciones prácticas

Exploring students’ metacognition in relation to an integral-area evaluation task

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