Program
Monday 12 June 2023 (live stream: StreamYard and YouTube)
| 9:30—10:00 |
Welcome & Coffee |
| 10:00—10:45 |
Colin Foster (Loughborough University)
Teaching problem solving in school mathematics |
| 10:45—11:15 |
Coffee Break |
| 11:15—12:00 |
Brendan Larvor (University of Hertfordshire)
Every classroom is multicultural |
| 12:00—14:00 |
Lunch |
| 14:00—14:45 |
Fenner Tanswell (TU Berlin)
Proof identity, essential ideas, and translations between algebra and diagram |
| 14:45—15:00 |
Coffee Break |
| 15:00—15:45 |
Keith Weber (Rutgers University)
The permissibility of diagrams in mathematical proofs |
| 15:45—16:15 |
Coffee Break |
| 16:15—17:00 |
Dirk Schlimm (McGill University)
Learning and understanding mathematical notations |
Tuesday 13 June 2023 (live stream: StreamYard and YouTube)
| 9:30—10:00 |
Welcome & Coffee |
| 10:00—10:45 |
Karen François (Vrije Universiteit Brussel)
The interaction between philosophy of mathematical practices and mathematics education—concept, clarification and examplar |
| 10:45—11:15 |
Coffee Break |
| 11:15—12:00 |
Gila Hanna (University of Toronto)
Interactive theorem provers in mathematical practice and teaching |
| 12:00—14:00 |
Lunch |
| 14:00—14:45 |
Ellen Lehet (Lees-McRae College)
Mathematical explanation in the classroom |
| 14:45—15:00 |
Coffee Break |
| 15:00—15:45 |
Silvia De Toffoli (IUSS Pavia School for Advanced Studies)
What mathematical explanation cannot be |
| 15:45—16:15 |
Coffee Break |
| 16:15—17:00 |
Nathalie Sinclair (Simon Fraser University)
Aesthetic considerations in mathematics education (research) |
Wednesday 14 June 2023 (live stream: StreamYard and YouTube)
| 9:00—9:30 |
Welcome & Coffee |
| 9:30—10:15 |
Matthew Inglis (Loughborough University)
The relationship between the philosophy of mathematical practice and mathematics education |
| 10:15—11:00 |
Yacin Hamami (ETH Zürich)
Should learning be an object of study for the philosophy of mathematics? |
| 11:00—11:30 |
Coffee Break |
| 11:30—12:15 |
Valeria Giardino (Institut Jean Nicod, CNRS)
From research to the class: on the collaboration between cognitive science and philosophy to support mathematical education |
| 12:15—14:00 |
Lunch |
Abstracts
Silvia De Toffoli (IUSS Pavia School for Advanced Studies): What mathematical explanation cannot be
Recent works in the philosophy of mathematics and mathematical education have argued for the thesis that mathematical explanation should be cashed out in terms of understanding. To be sure, the details of the various proposals are different – but they share a common core: providing an account that characterizes mathematical explanation in terms of the particular type of understanding it generates. In this talk, I explore some reasons to think that this idea is untenable. Rather than analyzing a unified phenomenon, I will suggest that the literature is conflating two different (albeit related) phenomena under the same label, a technical and a pre-theoretical one. The former is a type of scientific explanation in which both the explanandum and the explanans are mathematical. The latter is a broader phenomenon in which explaining is taken to indicate a process of clarification that answers a why or how question.
Colin Foster (Loughborough University): Teaching problem solving in school mathematics
In school mathematics curricula across many parts of the world, the practice of ‘problem
solving’ has a high profile. Teachers are exhorted to prepare students for the modern world
by equipping them with the skills needed to devise creative solutions to novel (i.e. ‘unseen’)
mathematical problems. However, ‘problem solving’ has multiple definitions in the
literature and it is unclear what exactly is meant by ‘teaching it’. In this talk, I will outline
what I have called ‘the fundamental problem of teaching problem solving’; i.e., showing
students how to solve the problem kills the problem and degrades it into an exercise,
whereas leaving students to struggle to discover their own solutions feels like not ‘teaching’.
At Loughborough University, I am currently leading a team designing a complete, free set of
teaching resources for lower secondary mathematics (the LUMEN Curriculum – see
https://www.lboro.ac.uk/services/lumen/curriculum/). As part of this, we are promising the
teaching of ‘problem solving’, and I will outline our emerging framework for what this might
look like. I would be very grateful for comment and critique on any of this.
Karen François (Vrije Universiteit Brussel): The interaction between philosophy of mathematical practices and mathematics education : concept
clarification and exemplar
During this presentation I’ll clarify the concept of mathematics education by giving a brief overview
of the different domains including (civic) statistics and (big) data education; and how it differs from
philosophy of mathematics education (Francois, 2007). In a second move I’ll introduce the
forerunners of philosophy of mathematics education focussing on (chronological order) D’Ambrosio
(1985), Bishop (1988), Ernest (1991), and Hersh (1997). I finally will introduce the ongoing research
with Joachim Frans on ‘Explanatory Proofs in Mathematics Classroom Practices’ as an exemplar of a
fruitful interaction between philosophy of mathematical practices and mathematics education (Frans& Francois 2023).
Bishop, Alan. J. [1988] (1997). Mathematical Enculturation, A Cultural Perspective on Mathematics
Education. Mathematics Education Library, Vol. 6, Kluwer Academic Publishers, Dordrecht.
D’Ambrosio, Ubiratan [1985] (1997). Ethnomathematics and its Place in the History and Pedagogy of
Mathematics. In Powell, Arthur B.; Frankenstein, Marilyn (eds.) (1997). Ethnomathematics. Challenging Eurocentrism in Mathematics Education. State University of New York Press,
Albany, pp. 13-24.
Ernest, Paul [1991] (2003). The Philosophy of Mathematics Education, Studies in Mathematics Education, RoutledgeFalmer, London.
François, Karen (2007). The Untouchable and Frightening Status of Mathematics. In Karen François &
Jean Paul Van Bendegem (eds.) Philosophical Dimensions in Mathematics Education. Springer, New York, pp. 13-39.
Frans, Joachim & François, Karen (2023 in press). Explanatory Proofs in Mathematics Classroom
Practices: An Investigation in Empirical Philosophy of Mathematics. Proceedings of the 13th
Congress of the European Society for Research in Mathematics Education (CERME 13),
Budapest, Hungary, July 10-14, 2023.
Hersh, Reuben (1997). What is Mathematics, Really? Jonathan Cape, London.
Valeria Giardino (Institut Jean Nicod, CNRS): From research to the class: on the collaboration between cognitive science and philosophy to support mathematical education
In the past years, many cognitive studies have looked into the cognitive “building blocks” of mathematical reasoning, which have been described as the cognitive foundations of mathematics or the core knowledge systems for arithmetic and geometry (Dehaene 1997; Butterworth 1999; Kinzler & Spelke 2007, Carey 2009). At the same time, some interest has grown in the philosophy community towards the practice of mathematics, by reviving topics such as the use of visualizations, diagrams and notations in mathematics (for a survey, see Giardino 2017 and Carter 2019,). In this latter framework, I discussed in previous work the role of cognitive tools in mathematics, by stressing the way in which they are acted upon and manipulated by the experts, and I proposed to think in terms of their “representational affordances” (Giardino 2018). In my view, the work with cognitive artifacts constitutes another building block for mathematical reasoning, which up to now has attracted only in part the attention of cognitive scientists. In my talk, I will discuss the possibility of a collaboration between philosophy and cognitive science on the topic of cognitive artifacts to the aim of bringing to useful suggestions for mathematical education.
Yacin Hamami (ETH Zürich): Should learning be an object of study for the philosophy of mathematics?
Learning, conceived as the acquisition of knowledge, skills, and understanding, is obviously an epistemological notion. And yet, it has received almost no attention in the philosophy of mathematics. This situation may change with the emergence of the philosophy of mathematical practice. In this talk, I will discuss a number of works in this latter field where learning appears to play a significant role. Given that the expertise on this issue is on the side of mathematics educators, this put them in a privileged position to contribute to these philosophical developments. I will then make a few suggestions on how to concretely foster this interaction.
Gila Hanna (University of Toronto): Interactive theorem provers in mathematical practice and teaching
As interactive theorem provers become more user-friendly and gain wider acceptance in mathematical practice, it is important to explore their use as an additional instructional tool. I will discuss current research on using the interactive theorem prover Lean in teaching undergraduate students to construct and understand mathematical proofs.
Matthew Inglis (Loughborough University): The relationship between the philosophy of mathematical practice and mathematics education
In this talk I discuss how we do, and how we should, conceive of the relationship between the disciplines of mathematics education (ME) and the philosophy of mathematical practice (PMP). To do this I first review some literature on interdisciplinarity, and then give a personal history of the development of mathematical cognition as a research field. Using this as one model of how interdisciplinary fields emerge, I discuss some possible options for the future of the PMP/ME relationship.
Brendan Larvor (University of Hertfordshire): Every classroom is multicultural
Ten years ago, the UK Arts and Humanities Research Council funded the Mathematical Cultures Network. This was a series of three meetings, the last of which was on education. In this talk, I will briefly recount some of the contributions that made a difference to the way I think about culture and education. I will present and argue for a general model of culture and describe some of the things I have tried to do with it.
Ellen Lehet (Lees-McRae College): Mathematical explanation in the classroom
Explanation is an inherent part of education. When we introduce students to a new concept, we provide some form of explanation to help familiarize them with the concept and situate it within their conceptual framework. When we assess students, we often ask them to not only provide an answer to a question but also to explain their answer. Explanation has also received increasing attention within the philosophy of mathematics. In particular, questions about intra-mathematical explanation—explanation within mathematics—have been considered. However, the question of how philosophical conceptions of intra-mathematical explanation relate to mathematics education has received little attention. In this presentation, I take the stance that philosophical theories of intra-mathematical explanation should be applicable to mathematics education and that results from mathematics education should be considered when evaluating philosophical theories. I will consider how some prominent features of philosophical theories of explanation apply in the classroom. In doing so, I will share some methods that I have tried in the classroom to provide students with explanations, encourage students to present their own explanations, and emphasize the value of explanation. I will comment on the successes and failures of these methods and will consider what we can learn about the relation of explanation and education from these attempts.
Dirk Schlimm (McGill University): Learning and understanding mathematical notations
The use of notations is an important aspect of mathematical practice, but in order to be used, a notation must first of all be learned and understood. This is an area where philosophy of mathematical practice and mathematics education intersect. In this talk, I argue, on the one hand, that the systematic study of notations together with the cognitive and material requirements for their use can inform mathematics education and, on the other hand, that the study of what works and what doesn’t work in mathematics education can inform us about the efficacy of notational design features. Examples from arithmetic and logic will be used to illustrate these claims.
Nathalie Sinclair (Simon Fraser University): Aesthetic considerations in mathematics education (research)
Many scholars have argued that aesthetics play a significant role in the practice of mathematics. I have been interested in how these arguments matter in the context of mathematics education. In this talk, I will survey some of the different ways in which the concept of aesthetics has been used, both historically in mathematics-related research, but also in more contemporary philosophical scholarship. I will present an inclusive materialist view of aesthetics and describe examples of how it has been used in mathematics education research.
Fenner Tanswell (TU Berlin): Proof identity, essential ideas, and translations between algebra and diagram
In this talk I will discuss what it means for two proofs to be “the same” in terms of sharing essential ideas. Looking at many proofs Sangwin (2023) collected of the summation of the first n odd positive integers, we explored whether identifying the essential ideas of various proofs could generate new proof ideas, in particular whether algebraic proofs will have diagrammatic equivalents and vice versa. We found that it is possible to identify and generalise several essential ideas, but that translation nonetheless did not match the proofs one-to-one. Form and content cannot be easily disentangled, and translation is a creative act. I will discuss this as having both philosophical and educational implications, and the broader benefits of collaboration between maths education and philosophy of maths. (This is joint work with Chris Sangwin.)
Keith Weber (Rutgers University): The permissibility of diagrams in mathematical proofs
I will present studies in which I investigate what types of diagrams that undergraduates and mathematicians believe are acceptable in proofs in real analysis. I will show how Larvor’s (2017, Synthese) distinction between metrical and non-metrical inferences can account for both undergraduates’ and mathematicians’ judgments. I will argue that this illustrates ways in which the philosophy of mathematical practice (PMP) can inform research in mathematics education (RME) and vice versa. In the PMP to RME direction, Larvor’s analysis provided a useful lens in which to interpret my data. In the RME to PMP direction, my data provided empirical support for some speculations in Larvor’s paper while adding nuance to others.
Organizers
Acknowledgement and Support
The workshop is funded by the Centre for Mathematical Cognition at Loughborough University. It is (non-financially) sponsored by the Association for the Philosophy of Mathematical Practice and the Chair Diversity of Mathematical Research Cultures and Practices at Universität Hamburg.
