Liberal Arts Inspired Mathematics: A Report OR How to bring cultural and humanistic aspects of mathematics to the classroom as effective teaching and learning tools

This is the report of a project on ways of teaching university-level mathematics in a humanistic way. The main part of the project recounted here involved a journey to the United States during the fall term of 2012 to visit several liberal arts colleges in order to study and discuss mathematics teaching. Several themes that came up during my conversations at these colleges are discussed in the text: the invisibility of mathematics in everyday life, the role of calculus in American mathematics curricula, the is algebra necessary?\u27\u27 discussion, teaching mathematics as a language, the transfer problem in learning, and the relationship between humanistic mathematics and mathematics as taught in liberal arts contexts

Implications of Foundational Crisis in Mathematics: A Case Study in Interdisciplinary Legal Research

As a result of a sequence of so-called foundational crises, mathematicians have come to realize that foundational inquiries are difficult and perhaps never ending. Accounts of the last of these crises have appeared with increasing frequency in the legal literature, and one piece of this Article examines these invocations with a critical eye. The other piece introduces a framework for thinking about law as a discipline. On the one hand, the disciplinary framework helps explain how esoteric mathematical topics made their way into the legal literature. On the other hand, the mathematics can be used to examine some aspects of interdisciplinary legal research

Washington University Record, January 11, 2002

https://digitalcommons.wustl.edu/record/1918/thumbnail.jp

Which Approaches Do Students Prefer? Analyzing the Mathematical Problem Solving Behavior of Mathematically Gifted Students

This study analyzed the mathematical problem solving behavior of mathematically gifted students. It focused on a specific fourth step of Polya's (1945) problem solving process, namely, looking back to find alternative approaches to solve the same problem. Specifically, this study explored problem solving using many different approaches. It examined the relationships between students' past mathematical experiences and the number of approaches and the kind of mathematics topics they used to solve three non-standard mathematics problems. It also analyzed the aesthetic of students' approaches from the perspective of expert mathematicians and the aesthetic of these experts' preferred approaches from the perspective of the students. Fifty-four students from a specialized high school were selected to participate in this study that began with the analysis of their past mathematical experiences by means of a preliminary survey. Nine of the 54 students took a test requiring them to solve three non-standard mathematics problems using many different approaches. A panel of three research mathematicians was consulted to evaluate the mathematical aesthetic of those approaches. Then, these nine students were interviewed. Also, all 54 students took a second survey to support inferences made while observing the problem solving behavior of the nine students. This study showed that students generally were not familiar with the practice of looking back. Indeed, students generally chose to supply only one workable, yet mechanistic approach as long as they obtained a correct answer to the problem. The findings of this study suggested that, to some extent, students' past mathematical experiences were connected with the number of approaches they used when solving non-standard mathematics problems. In particular, the findings revealed that students' most recent exposure of their then-AP Calculus course played an important role in their decisions on selecting approaches for solution. In addition, the findings showed that students' problem solving approaches were considered to be the least "beautiful" by the panel of experts and were often associated with standard approaches taught by secondary school mathematics teachers. The findings confirmed the results of previous studies that there is no direct connection between the experts' and students' views of "beauty" in mathematics

Chains of life: Turing, Lebensform, and the emergence of Wittgenstein’s later style

This essay accounts for the notion of Lebensform by assigning it a logical role in Wittgenstein’s later philosophy. Wittgenstein’s additions of the notion to his manuscripts of the PI occurred during the initial drafting of the book 1936-7, after he abandoned his effort to revise The Brown Book. It is argued that this constituted a substantive step forward in his attitude toward the notion of simplicity as it figures within the notion of logical analysis. Next, a reconstruction of his later remarks on Lebensformen is offered which factors in his reading of Alan Turing’s “On computable numbers, with an application to the Entscheidungsproblem“ (1936/7), as well as his discussions with Turing 1937-1939. An interpretation of the five occurrences of Lebensform in the PI is then given in terms of a logical “regression” to Lebensform as a fundamental notion. This regression characterizes Wittgenstein’s mature answer to the question, “What is the nature of the logical?