Ami Mamolo

• PhD
• Professor (Associate) at University of Ontario Institute of Technology

This article extends the notion of 'knowledge at the mathematical horizon' or 'horizon knowledge' introduced by Ball and colleagues as a part of teachers' subject matter knowledge. Our focus is on teachers' mathematical knowledge beyond the school curriculum, that is, on mathematics learnt during undergraduate college or university studies. We expl...

This case study examines the salient features of two individuals’ reasoning when confronted with a task concerning the cardinality and associated cardinal number of equinumerous infinite sets. The APOS Theory was used as a framework to interpret their efforts to resolve the “infinite balls paradox” and one of its variants. These cases shed new ligh...

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,...

This research is part of a broader research program that explores teacher educators’ mathematical knowledge. We examine the experiences, perceptions, and needs of prospective teachers as they navigate a complex set of new and interweaving ideas for how to teach mathematics with socially relevant and responsible connections. In doing so, we draw on...

We report on a group theory activity in which learners explored dihedral symmetries through a tangible geometric model. This approach has historical roots in the work of Felix Klein’s Erlangen Program and his Elementary Mathematics from an Advanced Standpoint. We situate our study with respect to this history as well as current educational research...

This paper examines preservice teachers’ experiences when they engage with tasks using dynamic technological applets to analyse data pertaining to societal issues. We examine two vignettes that discuss preservice teachers’ interaction with dynamic visual representations of data related to plastic pollution and food supply. We analyse this data with...

Teachers who commit themselves to change embark upon a difficult and sometimes lonely journey of self-doubt. In this chapter, we meet Nora, a Canadian educated middle school teacher who was hired as part of a school-wide initiative to introduce curricular change in an elite South American international school. Nora’s attempts to introduce socially...

The integration of coding in K-12 mathematics education is occurring throughout jurisdictions around the world. Best practices for applying coding practices in teaching mathematics are gaining research attention, and teacher education programmes are increasingly offering professional development to support this initiative. This paper discusses stra...

We present a case of a female Canadian pre-service teacher learning to code to support her future mathematics teaching. Learning to code for teaching is fraught with experiences of uncertainty, intimidation and overwhelmedness that are influenced by negative stereotypes, a lack of prior exposure to coding, and the vastness of the perceived learning...

This volume contains the papers presented at the International Conference Building on the Past to Prepare for the Future held from August 8-13, 2022, in King’s College, Cambridge, UK. It was the 16th conference organised by The Mathematics Education for the Future Project - an international educational and philanthropic project founded in 1986 and...

In Ontario, a mandatory high-stakes standardized literacy test called the OSSLT is administered in the tenth grade. With notable failure rates and acknowledged test anxiety, students are in need of better test preparatory methods. In this article, we examine some of the challenges and analyze ways that online multimedia learning tools can be design...

This chapter explores how competing influences from K-12 schooling and teacher education, respectively, can be examined through the lens of actor-oriented transfer (AOT). A form of scripted role-play is used to evoke an experience of contingency in the form of an unexpected mathematical approach. Of particular interest are the possible tensions and...

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...

In this commentary chapter, we draw on ideas from Baldinger and Murray; Cuoco; Wasserman and Galarza; and Zazkis and Marmur to articulate our views on the importance of mathematical structure and its relevance in secondary mathematics teachers’ disciplinary knowledge. In particular, we organize our discussion around two related questions—about the...

This paper explores how maker pedagogies helped middle school students develop transferable competencies, such as creativity, problem solving, self‐directed learning and citizenship skills. We offer an in‐depth look at how purposeful or critical making related to bullying awareness can help struggling students develop positive attitudes related to...

We present a story of a teacher educator’s response to a situation of contingency and describe how her experience enhanced her personal mathematical knowledge and influenced her teaching. In our analysis, we attend to different levels of awareness that support a teacher educator’s work and illuminate the qualities of a teacher educator’s knowledge,...

The recent change in teacher education in Ontario, moving from a single year to a two-year program, has offered us an opportunity to rethink and redesign our Kindergarten – Grade 12 (K-12)teacher education programs. A major shift has been happening within and outside of education due to a renewed focus on different mathematical ways of thinking, in...

This paper investigates pre-service secondary teachers’ perceptions of learning and teaching mathematics through extended explorations that are contextualized in issues of social importance. The study is situated within a research program concerned with mathematical knowledge used in, and useful for, teaching, and how such knowledge may be fostered...

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, v...

This chapter presents tasks and task structures for incorporating socially relevant mathematical explorations in secondary school learning. We introduce and develop Social Justice Context Problems, highlighting issues in food affordability, fairness, and bullying, with connections to Canadian curricula. The structure for our tasks is discussed, and...

Our study focused on building pre-service teachers’ capacity in maker pedagogies.

By focusing on a particular alteration of the comparative likelihood task, this study contributes to research on teachers' understanding of probability. Our novel task presented prospective teachers with multinomial, contextualized sequences and asked them to identify which was least likely. Results demonstrate that determinants of representativene...

This paper presents a case study of a mathematics teacher educator, Leanne, and her story of trying to support the development of two pre-service elementary school teachers with recognized learning disabilities. We analyze data through a lens of mathematical knowledge for teaching, focusing in particular on concerns and tensions about (i) maintaini...

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.

In this article, we develop a theoretical model for restructuring mathematical tasks, usually considered advanced, with a network of spatial visual representations designed to support geometric reasoning for learners of disparate ages, stages, strengths, and preparation. Through our geometric reworking of the well-known “open box problem”, we sough...

In this article, we explore notions of risk as perceived or experienced by individuals involved in mathematical education. We present this exploration in the form of vignettes, each illustrating a form of risk: a parent's reaction to classroom "propaganda"; a teacher trying to do justice by her students; a teacher confronted by his administration;...

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 pre...

Re-imagining pre-service teacher education in Ontario

This paper presents a new twist on a familiar paradox, linking seemingly disparate ideas under one roof. Hilbert's 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...

This article examines pre-service secondary school teachers’ responses to a learning situation that presented a student's 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 influe...

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...

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 theor...

In this article, we explore the responses of a group of undergraduate mathematics students to tasks that deal with areas, perimeters, volumes, and derivatives. The tasks challenge the conventional representations of formulas that students are used to from their schooling. Our analysis attends to the specific mathematical ideas and ways of reasoning...

Investigating rates of change in volume without calculation leads to an enriched sense of the optimization process and encourages reflection and connection among different approaches.

This article focuses on the development and problematization of a task designed to foster spatial visual sense in prospective and practicing elementary and middle school teachers. We describe and analyse the cyclical stages of developing, testing, and modifying several “task drafts” related to ideas around dilation and proportion. Challenged by par...

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 consist...

A three-dimensional model and geometry software can help develop students' spatial reasoning and visualization skills.

Many characteristics describe the work of a mathematician. These characteristics just as readily apply to the work of "professional" mathematicians (e.g. people who "do math" as a career, researching and publishing in the field) as they do to "amateur" mathematicians (e.g. people who "do math" (without funding), be it students, teachers, or teacher...

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 k...

This study examines undergraduate students' emerging conceptions of infinity as manifested in their engagement with geometric tasks. Students' attempts to reduce the level of abstraction of infinity and properties of infinite quantities are described. Their arguments revealed they perceive infinity as an ongoing process, rather than a completed one...

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 imagina...

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...