Paradoxes, ambiguity, infinity
This paper illustrates how mathematical symbols can have different, but related, meanings depending on the context in which they are used. In other words, it illustrates how mathematical symbols are polysemous. In particular, it explores how even basic symbols, such as ‘+’ and ‘1’, may carry with them meaning in ‘new’ contexts that is inconsistent with their use in ‘familiar’ contexts. This article illustrates that knowledge of mathematics includes learning a meaning of a symbol, learning more than one meaning, and learning how to choose the contextually supported meaning of that symbol. http://sigmaa.maa.org/rume/crume2012/RUME_Home/RUME_Conference_Papers_files/RUME_XV_Conference_Papers.pdf
Mathematics as well as mathematics education research has long progressed beyond the study of number. Nevertheless, numbers and understanding numbers by learners, continue to fascinate researchers and bring new insights about these fundamental notions of mathematics.
This study examines approaches to infinity of two groups of university students with different mathematical background: undergraduate students in Liberal Arts Programmes and graduate students in a Mathematics Education Master's Programme. Our data are drawn from students’ engagement with two well-known paradoxes – Hilbert's Grand Hotel and the Ping-Pong Ball Conundrum – before, during, and after instruction. While graduate students found the resolution of Hilbert's Grand Hotel paradox unproblematic, responses of students in both groups to the Ping-Pong Ball Conundrum were surprisingly similar. Consistent with prior research, the work of participants in our study revealed that they perceive infinity as an ongoing process, rather than a completed one, and fail to notice conflicting ideas. Our contribution is in describing specific challenging features of these paradoxes that might influence students’ understanding of infinity, as well as the persuasive factors in students’ reasoning, that have not been unveiled by other means.
This story is a playful retelling of ideas related to infinity. Presented as a historical fiction, the story reflects the thinking of research participants who addressed the ping pong ball conundrum, and where indicated, the individuals who contributed to modern formal understandings of infinity. This story offers a way of engaging with questions, controversies, ideas, and beliefs related to infinity. The characters in the story are confronted with a situation that challenges the notion of an ‘objective’ truth. Through their musings, April and her friends stumble upon the contextually-dependent nature of mathematical truth and open the door to further conversation.
This paper is the first installment of a study which seeks to identify the necessary and sufficient features of accommodating the idea of actual infinity. University mathematics majors' and graduates' engagement with the Ping-Pong Ball Conundrum is used as a means to this end. This paper focuses on one of the necessary features: the leap of imagination required to conceive of actual infinity, as well as its associated challenges. Introduction The concept of infinity has a distinctive quality which rouses the imagination, provoking controversy, and challenging fundamental ideas intuited as truth. In meeting these challenges and controversy an individual is invited to think in often new and complex ways—to engage in ―advanced mathematical thinking.‖ The term ‗advanced mathematical thinking' carries with it many descriptions. Although there is no agreement on the definition, many of the characteristics describing advanced mathematical thinking are exemplified in the concept of infinity. One working description suggests advanced mathematical thinking (AMT) involves abstract, deductive thought (Tall 1991, 1992), and includes ―proving in a logical manner based on definitions‖ (Tall, 1991, p. 20). Alternatively, ideas that exercise advanced mathematical thinking may be considered as ones that are not ―entirely accessible to the five senses‖ (Edwards, Dubinsky, & McDonald, 2005, p. 18), and lack ―an intuitive bases founded on experience‖ (Tall, 1992, p. 495). The abstract and intangible nature of actual infinity epitomises both of these descriptions. This paper presents research from part of a broader investigation which aims to identify the necessary and sufficient features involved in accommodating the idea of actual infinity. It focuses on the ‗leap of imagination' required to conceive of mathematical infinity, as well as on the challenges university mathematics students and graduates faced in making such a leap.
This article explores instances of symbol polysemy within mathematics as it manifests in different areas within the mathematics register. In particular, it illustrates how even basic symbols, such as ‘+’ and ‘1’, may carry with them meaning in ‘new’ contexts that is inconsistent with their use in ‘familiar’ contexts. This article illustrates that knowledge of mathematics includes learning a meaning of a symbol, learning more than one meaning, and learning how to choose the contextually supported meaning of that symbol.