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Mathematics can be used as a tool to question and critique society and, in doing so, give us more information about the world around us and how it operates. This however, is not a common perspective that is conveyed to students during their undergraduate mathematics coursework. This paper contributes to the understanding of how undergraduate mathematics students question and critique society via mathematical modeling tasks. In two courses at two universities, 27 mathematics majors and secondary preservice teachers engaged in the modeling process situated in authentic contexts to learn specific concepts and make mathematical connections across domains and disciplines. Both courses culminated in a final project in which students created and investigated solutions to their own modeling tasks. In this paper, we describe how our courses (1) centered justice as pedagogy, (2) were environments for student agency and exploration, and (3) explicitly demonstrated how mathematics and social justice are intertwined. Drawing on frameworks of mathematical modeling for social justice, we present and analyze student-created tasks to showcase how they utilized mathematics as a tool to question and critique the world around them. [ABSTRACT FROM AUTHOR]

In this paper, we, a group of graduate students in mathematics education, discuss some of the metacognitive benefits of the non-traditional teaching methods we observed employed by one of our professors. This professor's methods challenge the common belief that well-managed class time is key for positive learning outcomes. Instead, he orients his teaching to share the exploration and sense-making phases of doing mathematics. The goal of his teaching is to share the idea that learning mathematics is a process of "refining our mathematical thinking". We argue that this approach to teaching helps students see that mathematics is a human endeavor, appreciate the cycles of learning mathematics and the importance of struggle throughout the process, and gain some insight on what it means to be an expert in mathematics. We encourage teachers and professors to move away from obsessing about structured, well-organized class time, and to instead move towards having flexible and risk-oriented class time. [ABSTRACT FROM AUTHOR]

Although much creativity research has suggested that creativity is inuenced by cultural and social factors, these have been minimally explored in the context of mathematics and mathematics learning. This problematically limits who is seen as mathematically creative and who can enter the discipline of mathematics. This paper proposes a framework of creativity that is based in what it means to know or do mathematics and accepts that creativity is something that can be nurtured in all students. Prominent mathematical epistemologies held since the beginning of the twentieth century in the Western mathematics tradition have different implications for promoting creativity in the mathematics classroom, with fallibilist and social constructivist perspectives arguably being most conducive for conceiving of creativity as a type of action for all students. Thus, this paper proposes a framework of creative mathematical action that is based in these epistemologies and explains key aspects of the framework by drawing connections between it and research in the field of creativity. [ABSTRACT FROM AUTHOR]

Mathematics has often been referred to as an art. For some it is “the purest of the arts”, where the mathematicians’ art is “asking simple and elegant questions about our imaginary creations, and crafting satisfying and beautiful explanations”. Yet with classroom time given primarily to “covering the curriculum”, testing, and practicing problem-solving procedures, students’ opportunities to appreciate the aesthetic dimension of mathematics are often limited. To promote a responsive environment in an effort to enable students to become artists of their own mathematics experience, I consider in this paper two facets of the mathematics classroom. Content-wise I make the argument that students need to see problem-clarifying strategies in conjunction with problem-solving techniques, as the former are essential for making progress when engaging a mathematics problem where an explicit solution is not apparent. The other facet aiming to promote student agency is providing them opportunities to work with their own practical/professional concerns as students so as to become more creative and productive artists of their mathematics experience. [ABSTRACT FROM AUTHOR]

When we teach mathematics, we strive to teach students to think like mathematicians. In this paper we discuss one particular mathematical habit of mind that students do not naturally display. More specifically our study of voting patterns in data collected from classroom voting questions indicates that the undergraduate students who were in the classes using these questions did not understand the significance of counterexamples to statements, or lacked the ability to construct them, or both. Searching for counterexamples to disprove statements is a natural habit of mind for professional mathematicians. In this paper we give examples, and make some recommendations. We believe that if our students get used to routinely seeking out counterexamples, as they play with various mathematical ideas, they may also end up enjoying their mathematical experiences more. [ABSTRACT FROM AUTHOR]

Good mathematics stands the test of time. As culture changes, we often ask different questions, bringing new perspectives, but modern mathematics stands on ancient discoveries. Isaac Newton’s discovery of calculus (along with Leibniz) may seem old but is predated by Archimedes’ findings. Current mathematics students should be familiar with parabolas and simple curves; in our introductory calculus courses, we teach them to compute the areas under such curves. Our modern approach derives its roots from Newton’s work; however, we have filled in many of the gaps in the pursuit of mathematical rigor. What many students may not know is that Archimedes solved the area problem for parabolas long before the use of algebraic expressions became mainstream. Archimedes used the geometry of the ancient Greeks, which gave him a vastly different perspective. In this paper, we provide both Archimedes’ and Newton’s proofs involving the quadrature of the parabola, trying to remain true to their original texts as much as feasible. [ABSTRACT FROM AUTHOR]

This article describes methods and approaches for incorporating creative projects in undergraduate mathematics courses for students of engineering and computer science in an urban community college. The topics and the grading rubrics of the projects go way beyond standard homework questions and contain elements of finding own project, incorporating historical background, inventing own questions and exercises, or demonstrating experiments to illustrate some aspects of the project. After analyzing challenges and outcomes of these projects, I identified several skills which help students be successful, including the skills of creativity. These skills are writing, oral presentation, math skills, and collaboration skills. I present ways to support students in evaluating these skills precisely and im- proving them over the time of the course. Then I present several examples of student projects, including students' solutions, and insight into how the students demonstrated creative approaches to solving the problems. The paper concludes with suggestions for implementing the projects and an assessment of the projects from the students and instructor perspectives. [ABSTRACT FROM AUTHOR]

This paper discusses the design and implementation of mathematical departmen- tal publications. We argue that these publications foster students' creativity and written communication skills. [ABSTRACT FROM AUTHOR]

Teachers and students of mathematics often view history of mathematics as just mathematics as they know it, but in another form. This view is based on a misunderstanding of the nature of history of mathematics and the kind of knowledge it attempts to acquire. Unfortunately, it can also lead to a deep sense of disappointment with the history of mathematics itself, and, ultimately, a misunderstanding of the historical nature of mathematics. This kind of misunderstanding and the disappointment following from it-both raised to the level of resentment-run through the paper "A Critique of the Modern Consensus in the Historiography of Mathematics." My review of that paper, sent to me blind, became a response to it. In particular, this essay attempts to clarify the nature of the historical discipline and to show that author of the Critique ends up, in effect, wanting and not wanting history at the same time. [ABSTRACT FROM AUTHOR]

Mathematics employs a formal language where symbols, verbs, and nouns serve to express terms, concepts, and rules that concatenate to definitions, problem-solving procedures, and proofs. Taken together these constitute the expository language of mathematics found in journals, textbooks, and demonstrations. As the object of these mathematical communications is to inform and perhaps inspire, there are epistemological and ethical considerations that deserve examination. For in keeping with the commitment to an aesthetic of concision promulgated by tradition, the formal presentation language and style of mathematics, while valuable in furthering the body of knowledge, provides only the conclusion of an inquiry, completely excluding the language of investigation that informed the many steps of decision making involved in the process, so that little if any insight into the creative process is made available. Here we explore this problem of communication in the context of mathematics presented to students early in their education as well as at university level. The argument is made that the language of investigation -- the heuristic actions instrumental for their formulation -- ought to accompany the language of formal demonstration so as to provide a communication that is in the best interests of all students and members of the mathematics profession. [ABSTRACT FROM AUTHOR]

Erlwanger [18] shook the mathematics education world when he introduced Benny, a student who successfully worked through a behavioristic curriculum. Erlwanger showed how far removed Benny's understanding of mathematics was from expectations. Erlwanger's legacy is the basis for this comparative case study which explores students' actions in the modern, in-class computer-centered emporium classroom. Many striking similarities are found between Pearson's My- MathLabs (MML) and Benny's Individually Prescribed Instruction curriculum. In this case study we meet Barbara, a student who succeeds in MML but shows little understanding of mathematical concepts and demonstrates that the legacy of Benny is his continued appearance in our current students. However, what differentiates Benny and Barbara is more than time; it's the inequities resulting from imposing a pedagogy with well-known problematic characteristics to developmental mathematics students. Most of these developmental students are members of marginalized communities. As such, the social justice and ethical implications of using such a course structure are explored. [ABSTRACT FROM AUTHOR]

The history of contemporary mathematical education is the history of a struggle against computers and IT. As a result specially selected simplified math problems are used while teaching. Just as it was a hundred years ago, contemporary students are forced to memorize a lot of rules and theorems in order to solve math problems. But we know that today they can get the same results using simple computer calculations. Information technologies can (and in this paper we argue that they should) change the traditional methods of solving mathematical problems. Here we share some simple problems that helped engineering students learn the basics of mathematics and computer science and even enjoy the learning process. In particular we point out that the ability to visualize solutions is very important in most contexts, and modern mathematical software packages offer users convenient and simple tools of visualization and even animation. Including them on our pedagogical team, we can significantly increase our students' understanding of the basic concepts and theorems of mathematics. [ABSTRACT FROM AUTHOR]

This paper recounts an experiment by a mathematics professor who primarily teaches mathematics majors. The main question explored is whether the ordering of the questions makes a difference as to how students perform in a test. More specifically we focus here on the following research questions: (1) Does arranging a math test with easy-to-hard items versus hard-to-easy items impact student performance? and (2) If so, does item order impact male and female mathematics majors and non-majors in unique ways? We examine data collected over multiple semesters with several di erent classes. We find that for most of the mathematics students who were examined, the ordering of the questions on a test did not impact performance. However, female majors performed better on classroom exams when the test was arranged with the more difficult questions presented first. Readers who are interested in teaching mathematics, educational psychology, or gender issues in the classroom may find our results intriguing. [ABSTRACT FROM AUTHOR]

Ducci’s Four-Number Game is a classic mathematical puzzle appealing to a wide audience for its procedural simplicity, mathematical richness, and aesthetic values. This article first describes a few activities appropriate for school students and mathematics teachers to make sense of the intriguing behavior of the game. Then, using a mobile simulation, we delve into the lengths of the Four-Number Game and the corresponding probability distribution. The Ducci number game is playful, engaging, and full of mathematical surprises. [ABSTRACT FROM AUTHOR]

After six months of working in aerospace, I was tasked to brief, via PowerPoint in teleconference, important customers on the resolution of a seven-month problem that had its roots in coordinate confusion. This article gives my preparation techniques for this PowerPoint presentation and lessons learned from observing other presentations. For people interested in the bare essentials and as an easy guide for students, I've included an outlined checklist at the end of this article. [ABSTRACT FROM AUTHOR]