Ami Mamolo

Paradoxes as a window to infinity

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 Masters Programme. Our data are drawn from students’ engagement with two well-known paradoxes – Hilberts Grand Hotel and the Ping-Pong Ball Conundrum – before, during, and after instruction. While graduate students found the resolution of Hilberts 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.

Factors Influencing Prospective Teachers' Recommendations to Students: Horizons, Hexagons, and Heed.

This article examines pre-service secondary school teachers’ responses to a learning situation that presented a students struggle with determining the area of an irregular hexagon. Responses were analyzed in terms of participants’ evoked concept images as related to their knowledge at the mathematical horizon, with attention paid toward the influence of one on the other. Specifically, our analysis attends to common features in participants’ understanding of the mathematical task, and explores the interplay between participants’ personal solving strategies and approaches and their identified preferences when advising a student. We conclude with implications for mathematics teacher education research and pedagogy.

Unasked But Answered: Comparing the Relative Probabilities of Coin Flip Sequence Attributes

The objective of this article is to contribute to research on teachers’ probabilistic knowledge and reasoning. To meet this objective, prospective mathematics teachers were presented coin flip sequences and were asked to determine and explain which of the sequences was least likely to occur. This research suggests that certain individuals, when presented with a particular question, answer different questions instead. More specifically, we found that participants, instead of making the intended relative probability comparison, compared the relative probability of a number of particular attributes associated with coin flip sequences. Further, we interpret participants’ attempts to reduce levels of abstraction in order to reason about probability, in a relative sense. Embracing the research literature suggesting that responses reflect individuals’ understandings of the question they were asked, this article suggests potential questions that participants have not been asked but are answering. In doing so, this article suggests that participants are providing reasonable relative probability comparisons for questions that are unasked. Finally, implications for future research are also discussed.RésuméL’objectif de cet article est d’apporter une contribution à la recherche sur les connaissances des probabilités chez les enseignants. Pour atteindre cet objectif, on a présenté à des futurs enseignants des mathématiques des séquences de tirs à pile ou face, et on leur a demandé de déterminer et de justifier quelles séquences étaient les moins probables. Cette recherche montre que certaines personnes, lorsqu’on leur pose une certaine question, répondent en fait à une question différente de celle qui est posée. Plus précisément, certains participants, au lieu de faire la comparaison attendue entre les probabilités relatives des séquences présentées, comparent plutôt les probabilités relatives d’autres caractéristiques associées aux séquences de tirs à pile ou face (par exemple l’équiprobabilité, les modèles de répétition, le hasard, les alternances, les revirements et les séries). Des implications pour d’autres recherches futures sont également abordées.

Contextual Considerations in Probabilistic Situations: An Aid or a Hindrance?

We examine the responses of secondary school teachers to a probability task with an infinite sample space. Specifically, the participants were asked to comment on a potential disagreement between two students when evaluating the probability of picking a particular real number from a given interval of real numbers. Their responses were analyzed via the theoretical lens of reducing abstraction. The results show a strong dependence on a contextualized interpretation of the task, even when formal mathematical knowledge is evidenced in the responses.

How to Act? A Question of Encapsulating Infinity

This article investigates some of the specific features involved in accommodating the idea of actual infinity as it appears in set theory. It focuses on the conceptions of two individuals with sophisticated mathematics background, as manifested in their engagement with variations of a well-known paradox: the ping-pong ball conundrum. The APOS theory is used as a framework to interpret participants’ efforts to resolve the paradoxes. The cases discussed focus on how transfinite subtraction may be conceptualized, and they suggest that there is more to accommodating the idea of actual infinity than the ability to act on a completed object—rather, it is the manner in which objects are acted upon that is also significant.RésuméCet article se penche sur certains traits spécifiques qui entrent en jeu lorsqu’il s’agit d’accorder une place à l’infini tel qu’il apparaît dans la théorie des ensembles. L’article est centré sur les conceptions de deux personnes hautement qualifiées dans le domaine des mathématiques, telles que ces conceptions se manifestent dans les variations apportées à un paradoxe bien connu: celui des balles de ping-pong. La thèorie APOS est utilisèe comme cadre pour interprèter les efforts des participants lorsqu’ils tentent de rèsoudre les paradoxes. Les cas analysès sont centrès sur les façons dont la soustraction transfinie peut être conceptualisèe, et suggèrent que le concept d’infini réel implique plus qu’une simple capacité ‘d’agir’ sur un objet complété: la manière dont se produit l’action sur les objets serait également significative.

Riffs on the infinite ping-pong ball conundrum

This article presents a novel re-conceptualisation to a well-known problem – The Ping-Pong Ball Conundrum. We introduce a variant of this super-task by considering it through the lens of ‘measuring infinity’ – a conceptualisation of infinity that extrapolates measuring properties of numbers, rather than cardinal properties. This approach is consistent with a nonstandard analysis approach to infinite numbers, and gives credence to the intuitive (but otherwise normatively incorrect) resolution. We explore the mathematical motivation and consequences of this variant, as well as offer further ‘riffs’ on the infinite ball problem for consideration.

Eyes, Ears, and Expectations: Scripting as a Multi-lens Tool

Invitations to envision what might occur in a teaching and learning situation can be used as both instructional and research tools for teacher education. When presented in the form of script writing, such invitations can afford opportunities to awaken important sensitivities for effective teaching, as well as shed light on the conceptualizations, values, and rationales held by pre-service teachers. This chapter highlights how scripting tasks may provide glimpses of participants’ mathematical awareness and horizon knowledge applicable for teaching. Two contexts for scripting tasks are discussed: (i) responding to parents’ concerns for a student’s future trajectory, and (ii) engaging in an unexpected mathematical exchange. Of interest is what participants noticed or overlooked (their “eyes”) as they assigned voices to their characters, attributing to them the emotions, attitudes and types of utterances, which they were prepared to hear (their “ears”). Participants’ scripts revealed different perceptions and expectations regarding the roles and relationships of players in the classroom. What was noticed and attended to may have influenced the advice and explanations given to student-characters. Through investigation and reflection of their scripts, shifts in participants’ attention occurred, prompting new insight and understanding for engaging with and responding to students.

Reflecting Upon Different Perspectives on Specialized Advanced Mathematical Knowledge for Teaching

Teachers’ knowledge assumes a major role in practice and in the students learning and achievement. In particular, the construct of horizon knowledge or, what can be termed specialized advanced mathematical knowledge for teaching (in order to capture the overall perspectives we are dealing with within this proposal) has been the focus of attention from some researchers with different foci of attack (e.g., Carrillo, Climent, Contreras, & Muñoz-Catalán, 2013; Jakobsen, Thames, Ribeiro, & Delaney, 2012; Wasserman & Stockton, 2013; Zazkis & Mamolo, 2011). In that sense, and aiming to deepen our understanding of such a construct, the aim of this working group is to discuss and reflect upon, different theoretical perspectives, methodological approaches and analytic methods used when focusing on such specialized advanced mathematical knowledge for teaching. In particular, we consider the activities of analysing and conceptualizing situations where access and development of such teachers’ knowledge is of primary importance.

On Numbers: Concepts, Operations, and Structure

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.

Hilbert's Grand Hotel with a series twist

This paper presents a new twist on a familiar paradox, linking seemingly disparate ideas under one roof. Hilberts Grand Hotel, a paradox which addresses infinite set comparisons is adapted and extended to incorporate ideas from calculus – namely infinite series. We present and resolve several variations, and invite the reader to explore his or her own variations.