Search

Creativity is crucial for doing mathematics, yet many United States students may not have opportunities to experience it in their courses. Moreover, the literature base on views of mathematical creativity lacks the student perspective. To explore the connections between views of and feeling creative, we examine differences in views of creativity between students who felt creative and did not feel creative in an interventional Calculus I course. We conducted semi-structured interviews with 37 undergraduate students taking a creativity-based Calculus I course across the United States, for their views on creativity and whether they felt creative in the course. Approximately three quarters felt creative (n = 27), while one quarter of students (n = 10) did not. Using qualitative coding, we found that students who did not feel creative were more likely to view creativity as including "understanding" and "applications." In contrast, students who felt creative were more likely to view creativity as originality and actions and attitudes they could take. We recommend instructors take actions focusing on "originality" and "actions" and "attitudes" to help foster students' creativity. Finally, we discuss how all ten students who did not feel creative came from groups that have been historically marginalized in mathematics.

As students learn to problem solve in authentic situations, they must also develop metacognitive tools to manage and regulate their problem-solving process. To foster process-focused metacognition utilized by mathematical thinkers and problem solvers, inquiry-based learning classroom practices and an adapted version of portfolio problems were implemented in a content course for pre-service elementary teachers. In this article, we describe how the combination of a process-focused (instead of product-focused) classroom culture and explicit reflection on student thinking supported students' process-focused metacognition while problem-solving. The problems, portfolio structure, and student interview reflections are shared.

As calculus instruction moves further from lecture-based instruction toward student-centered pedagogy, support services must align with the changing goals for the course. This paper describes the work of a department as they collectively design and redesign a co-requisite Calculus I course, in an effort to better promote students' academic and social integration into active, student-centered calculus classrooms. We highlight the inclusion of more cognitively demanding and group-worthy tasks, re-negotiation of support for procedures and fundamental concepts, and promotion of collaborative participation. Our discussion is supplemented with reflections from faculty, graduate student instructors, and students.

Many studies provide insights into students' conceptions of various linear algebra topics and difficulties they face with multiple modes of thinking needed for conceptualization. While it is important to understand students' initial conceptions, students' transfer of learning of these conceptions to subsequent courses can provide additional information to structure meaningful curricular materials. This study explores physics students' transfer of learning of eigenvalues and eigenvectors from prerequisite experiences to quantum mechanics. Data analysis focused on three task-based interviews with undergraduate students, observations of physics courses, and students' course artifacts. Existing studies on students' conceptions of linear algebra topics indicate the necessity of developing flexible shifts between different modes of thinking in order to grasp linear algebra. This study's participants, who had initial learning experiences of linear algebra, were also observed to struggle with such shifts prior to quantum courses. It seems that various contexts in quantum courses, and explicit instructional methods, provided opportunities for students to enhance this initial learning of eigenvalues and eigenvectors. In particular, the explicit reasoning of one of the quantum courses' instructors concerning the choice of certain representations during problem solving in class, seemed to facilitate students' construction of similarities, thus providing evidence for actor-oriented transfer. Results of this study align with goals for recently developed instructional materials and interventions that emphasize opportunities for students to inquire and connect multiple modes of thinking.

Challenging mathematics problems were posed as a substantial part of homework in an elective combinatorics course. The intent was to encourage undergraduate students' exploration as well as develop their technical writing skills. The bulk of students' grades on these problems rested on a three-page reflective write-up using a rubric on mathematical creativity, the Creativity-in-Progress Rubric on Proving. Through student reflections, this structure led to a self-reported boost in students' proving confidence and led them to view themselves as practicing mathematicians. Descriptions of the problems, how students used the rubric as their guide, and grading structure are detailed in this article.

A growing body of mathematics education research points to the importance of fostering students' mathematical creativity in undergraduate mathematics courses. However, there are not many research-based instructional practices that aim to accomplish this task. Our research group has been working to address this issue and created a formative assessment tool, the "Creativity-in-Progress Rubric (CPR) on Proving". This tool is developed to help students evaluate their own proving process, as well as provide instructors insight into students' self-evaluation and process. In this paper, we provide a brief description of the CPR on Proving and explain its implementation in two courses at different institutions. Additionally, we share several students' usages of the CPR and provide some practical implementation suggestions.

Although many policy documents include equity as part of mathematics education standards and principles, researchers continue to explore means by which equity might be supported in classrooms and at the institutional level. Teaching practices that include opportunities for students to engage in active learning have been proposed to address equity. In this paper, through aligning some characteristics of inquiry put forth by Cook, Murphy and Fukawa-Connelly with Gutiérrez's dimensions of equity, we theoretically explore the ways in which active learning teaching practices that focus on inquiry could support equity in the classroom.

"WeBWorK" CLASS combines the best features of online homework and digital whiteboards to create a system that provides instructors with information on students' problem-solving processes, as well as their final answer. The system's utilization of tools that can sort students based on correct and incorrect homework and quiz answers and its ability to display students' solutions to problems helps instructors make timely formative assessments of both in and out of classroom work. In this paper, we share how these tools allow us to engage students in the problem-solving process and provide meaningful formative assessment.

This study explores in-service high school mathematics teachers' conception of various forms of complex numbers and ways in which they transition between different representations of these forms. One 90-min interview was conducted with three high school mathematics teachers after they completed three professional development sessions, each 4 h, on complex numbers. Results indicate that, in general, these teachers did not necessarily have a dual conception of complex numbers. However, they demonstrated varying conceptions with different forms of complex numbers. Teachers worked at an operational level with the exponential form of complex numbers, but there was no evidence to indicate that they had a structural conception of this form. On the other hand, two teachers were very comfortable with the Cartesian form and exhibited a process/object duality by translating between different representations of this form. These results indicate that high school teachers need more opportunities to help them develop a dual conception of each form ("multiple duals"), which in turn can result in developing a dual conception of complex numbers. An interesting phenomenon that we found was that teachers who taught courses such as geometry and international baccalaureate were able to draw from their teaching experiences as they attempted the interview tasks. This particular observation may suggest that teachers' teaching assignments coupled with appropriate professional development activities could facilitate their understanding of these concepts.

The flipped classroom model is being used more frequently in undergraduate mathematics courses. As with any new teaching model, in-depth investigations of both various implementation styles and how the new model improves student learning are needed. Currently, many practitioners have been sharing their implementations of this model. However, there has not yet been an investigation of the various implementations of the model to discern general trends in this movement. With this research goal in mind, we conducted a study exploring various implementations of the flipped classroom model by interviewing 19 faculty members who experienced using this model at 14 different institutes. Results indicate that participants had similar motivations for implementation; however, subsequent implementations were different. In addition, we share participants' perspectives on (a) student learning of pre-requisite, procedural and conceptual knowledge, and (b) how this particular model promotes such knowledge developments. Finally, we provide suggestions for future implementations and research regarding this particular teaching model.

Creativity is crucial for doing mathematics, yet many United States students may not have opportunities to experience it in their courses. Moreover, the literature base on views of mathematical creativity lacks the student perspective. To explore the connections between views of and feeling creative, we examine differences in views of creativity between students who felt creative and did not feel creative in an interventional Calculus I course. We conducted semi-structured interviews with 37 undergraduate students taking a creativity-based Calculus I course across the United States, for their views on creativity and whether they felt creative in the course. Approximately three quarters felt creative (n = 27), while one quarter of students (n = 10) did not. Using qualitative coding, we found that students who did not feel creative were more likely to view creativity as including understanding and applications. In contrast, students who felt creative were more likely to view creativity as originality and actions and attitudes they could take. We recommend instructors take actions focusing on originality and actions and attitudes to help foster students' creativity. Finally, we discuss how all ten students who did not feel creative came from groups that have been historically marginalized in mathematics. [ABSTRACT FROM AUTHOR]

As students learn to problem solve in authentic situations, they must also develop metacognitive tools to manage and regulate their problem-solving process. To foster process-focused metacognition utilized by mathematical thinkers and problem solvers, inquiry-based learning classroom practices and an adapted version of portfolio problems were implemented in a content course for pre-service elementary teachers. In this article, we describe how the combination of a process-focused (instead of product-focused) classroom culture and explicit reflection on student thinking supported students' process-focused metacognition while problem-solving. The problems, portfolio structure, and student interview reflections are shared. [ABSTRACT FROM AUTHOR]

As calculus instruction moves further from lecture-based instruction toward student-centered pedagogy, support services must align with the changing goals for the course. This paper describes the work of a department as they collectively design and redesign a co-requisite Calculus I course, in an effort to better promote students' academic and social integration into active, student-centered calculus classrooms. We highlight the inclusion of more cognitively demanding and group-worthy tasks, re-negotiation of support for procedures and fundamental concepts, and promotion of collaborative participation. Our discussion is supplemented with reflections from faculty, graduate student instructors, and students. [ABSTRACT FROM AUTHOR]

A growing body of mathematics education research points to the importance of fostering students’ mathematical creativity in undergraduate mathematics courses. However, there are not many research-based instructional practices that aim to accomplish this task. Our research group has been working to address this issue and created a formative assessment tool, theCreativity-in-Progress Rubric(CPR)on Proving. This tool is developed to help students evaluate their own proving process, as well as provide instructors insight into students’ self-evaluation and process. In this paper, we provide a brief description of the CPR on Proving and explain its implementation in two courses at different institutions. Additionally, we share several students’ usages of the CPR and provide some practical implementation suggestions. [ABSTRACT FROM PUBLISHER]

Piaget's theory and ideas open the doors for the education community in general, mathematics educators in particular, to understand individual cognition and explore the ways of investigating it. This has helped us to develop our understanding of how individuals might think. More recent research suggests that individual cognitive development can continue throughout adulthood, expanding on Piaget's theory and ideas. Scholars who investigate this development describe a form of thinking that goes beyond formal operations as postformal thinking. In this discussion paper, we use three constructs from postformal thinking frameworks to describe what postformal thinking is and what its implications might be if it is applied in mathematics education. [ABSTRACT FROM AUTHOR]

WeBWorKCLASS combines the best features of online homework and digital whiteboards to create a system that provides instructors with information on students’ problem-solving processes, as well as their final answer. The system’s utilization of tools that can sort students based on correct and incorrect homework and quiz answers and its ability to display students’ solutions to problems helps instructors make timely formative assessments of both in and out of classroom work. In this paper, we share how these tools allow us to engage students in the problem-solving process and provide meaningful formative assessment. [ABSTRACT FROM PUBLISHER]