The relationship between philosophical and mathematical inquiry
Andrew Blair examines the relationship between the two types of inquiry with reference to a new book that brings together articles on philosophical inquiry.
In the mathematics classroom, teachers are cocooned from philosophical questions. We teach the concepts and applications of the discipline, but do not consider epistemological and ethical issues. Those issues include our assumptions and beliefs about the nature of mathematics and the implications of its use in social, economic and political contexts. By broadening students' horizons beyond the utilitarian uses of mathematics, we can develop a reflective and critical citizenry that understands the subject's power and limitations.
This is the argument of the authors of a new book about enacting philosophical inquiry through dialogue. While philosophical inquiry complements the learning of mathematics, it stands outside the normal content of school lessons: "An understanding of the role of mathematics and the use of its formal methods in society are not to be found in mathematics" (Kennedy, 161). To promote a philosophical perspective, the teacher has three choices: design stand-alone lessons, plan space in normal lessons, or take advantage of spontaneous opportunities.
The first choice involves importing a model of philosophical inquiry into the mathematics classroom. The teacher designs a stand-alone lesson and generates dialogue with a diagram, picture, or text related to the subject.
In one example, the teacher used the format of Lipman's Philosophy for Children (P4C) to study a text about the possibility of an absolutely perfect cube (Daniel, 147-148). The text and accompanying discussion plans were designed to support, but stand separately from, the students' normal lessons. Through creating a philosophical community of inquiry for an hour a week, the class was stimulated to explore key concepts from different perspectives.
In another example, Martens' Five-finger model is employed to generate a comparative inquiry into pairs of pictures. The pictures illustrate a mathematical concept in two contexts: as part of the natural world and embedded in a human-made artefact or building (Marsal, 126-127). Students were encouraged to think in five distinct ways (the 'fingers') during dialogue.
(2) Create space in a normal lessons
Alternatively, teachers can create spaces during mathematics lessons for philosophical dialogue. When modelling complex open problems, students can be encouraged to examine the assumptions and implications in the solutions they offer.
In one example, 11-year-old students in Cyprus tackled the issue of a water shortage on the island (English, 66-70). Groups developed mathematical models for the cost of importing water from other countries before evaluating the social and environmental consequences of their proposals.
In another example, the teacher of a sixth grade class provided space at the start of a modelling activity for philosophical dialogue (Meerwaldt et al., 110-112). Before students considered questions about the scene of a robbery (including estimating the height of the thief from a footprint), the teacher posed a question about the ethics of stealing. Researchers postulated that, by so doing, students would learn to communicate more effectively, think more analytically, reflect more critically, and offer more creative solutions.
(3) Spontaneous opportunities
Finally, teachers can take advantage of a student's question to launch a philosophical dialogue during mathematical inquiry.
For example, during a dialogic inquiry about the relative sizes of the infinite sets of natural numbers and the even natural numbers that had got bogged down with conflicting hypotheses, a student asked "Is infinity a number?" (Kennedy, 169-170). The teacher took the opportunity to launch a timely interlude on the nature of infinity. Students developed a deeper understanding of the concept and, armed with a greater appreciation of the hypotheses, they were able to re-commence the mathematical inquiry at a higher level.
However, in my experience of using Inquiry Maths prompts, a philosophical interlude can sometimes be a distraction from the development of mathematical lines of inquiry. Rather than address a question immediately, the teacher might find a delay until later in the inquiry less disruptive. Moreover, in its use of regulatory cards, the Inquiry Maths model has a meta-mathematical means of communicating about obstacles to inquiry. Rather than wait for a spontaneous opportunity, participants use the cards to take a deliberate step outside the inquiry process.
While we might imagine that philosophical inquiry has a greater affinity with mathematical inquiry than with traditional methods, it turns out not necessarily to be the case. Indeed, Oliverio, in an article published elsewhere, argues that mathematical inquiry can be just as trapped in the "matrix of the discipline" as a lesson that requires repetition and recitation: "If not generated by a genuine doubt, the inquiry which develops, brilliant and interesting as it may be, can be less fruitful in terms of a real understanding of the topic than expected" (10). Only philosophical dialogue, Oliverio continues, can give rise to genuine doubt and only genuine doubt can drive true inquiry.
The problem with the contention that genuine inquiry in the mathematics classroom is philosophical relates to the role of the teacher. In philosophical inquiry, the role is "anchored in rigorous and critical questioning" (Daniel, 155) that requires students to refine, reformulate or critique their own or others' contributions. The skilled teacher moves students from pluralism, in which differences of opinion are accepted and tolerated, to semi-critical dialogue in which students question statements from their peers, and then to fully critical exchanges in which those questions improve and transform the group's philosophical judgements.
While teachers of mathematics should aspire to foster communities of inquiry in which critical dialogue is the norm, the content of discussion, after the presentation of the initial stimulus, remains outside the realm of the teacher in philosophical inquiry. The development of the form of dialogue alone improves judgements in philosophical inquiry "because there is no authority external to the community of inquiry that can correct limitations or short-sightedness" (Kennedy, 166).
This is not the case in mathematical inquiry. The teacher acts as a representative of a mathematical community that has established disciplinary norms of activity and communication, correcting misconceptions and contributing a new concept or procedure as the inquiry develops. In philosophical inquiry, the teacher is a facilitator of critical dialogue; in mathematical inquiry, the teacher is a facilitator of mathematical dialogue and a participant in developing an understanding of disciplinary content.
The two types of inquiry are not mutually exclusive. Mathematical inquiry can include spaces for philosophical dialogue. However, content "cannot take second place to the inquiry processes being nurtured"; rather, concepts and processes can be "developed concomitantly in ways that stimulate and challenge students' thinking about, and beyond, the content" (English, 62).
Inquiry Maths and philosophical inquiry
With regards to Inquiry Maths, philosophical dialogue about concepts linked to a prompt can enrich the inquiry. In the evaluation phase the teacher could initiate a dialogue in which the students compared their attitudes towards and assumptions about mathematics before and after the inquiry.
However, other questions addressed by philosophical inquiry are more difficult to incorporate into the Inquiry Maths model. Ethical and moral questions about the use of mathematics in models of the world would require a different sort of prompt to the ones on the website.