Reflective practice is a commonly used term in mathematics education, often without careful definition, implying a contemplative reviewing of learning and/or teaching in mathematics in order to approve, evaluate, or improve practice. A feedback loop is often suggested in which reflective practice feeds back into the design or initiation of practice providing possibilities for improved practice. More precise definitions often draw on Dewey, who wrote:

Active, persistent and careful consideration of any belief or supposed form of knowledge in the light of the grounds that support it and the further conclusions to which it tends constitutes reflective thought (1933, p. 9)

… reflective thinking, in distinction to other operations to which we apply the name of thought, involves (1) a state of doubt, hesitation, perplexity, mental difficulty, in which thinking originates, and (2) an act of searching, hunting, inquiring, to find material that will resolve the doubt and dispose of the perplexity (p. 12).

… Demand for the solution of a perplexity is the steadying and guiding factor in the entire process of reflection. (p. 14)

Rather than a perspective just of contemplative thought, Dewey emphasizes the important element of action in reflection and the goal of an action outcome. This has led to a linking of reflective practice with so-called action research which is research conducted by practitioners into aspects of (their own) professional practice. Stephen Kemmis a leading proponent of action research spoke of reflection as “meta-thinking,” thinking about thinking. He wrote:

We do not pause to reflect in a vacuum. We pause to reflect because some issue arises which demands that we stop and take stock or consider before we act. … We are inclined to see reflection as something quiet and personal. My argument here is that reflection is action-oriented, social and political. Its product is praxis (informed committed action) the most eloquent and socially significant form of human action. (Kemmis 1985, p. 141)

Kemmis conceptualized action research with reference to a critically reflective spiral in action research of plan, act and observe, and reflect (Kemmis and McTaggart 1981; Carr and Kemmis 1986), and other scholars have adapted this subsequently (e.g., McNiff 1988) (Fig. 1).

Reflective Practitioner in Mathematics Education, Fig. 1
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Action-reflection cycle (McNiff (1988), pp 44, Fig 3.7)

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More recent scholars relate ideas of reflection, seminally, to the work of Donald Schön who has written about the reflective practitioner in professions generally and in education particularly (Schön 1983, 1987). Schön relates reflection to knowing and describes knowing-in-action and reflection-in-action. With reference to Dewey, he writes about learning by doing, the importance of action in the process of learning, and relates doing and learning through a reflective process.

Our knowing is ordinarily tacit, implicit in our patterns of action and in our feel for the stuff with which we are dealing. It seems right to say that our knowing is in our action (1983, p. 49).

Schön refers to knowing-in-action as “the sorts of know-how we reveal in our intelligent action – publicly observable, physical performances like riding a bicycle and private operations like instant analysis of a balance sheet” (1997, p. 25). He claims a subtle distinction between knowing-in-action and reflection-in-action. The latter he links to moments of surprise in action: “We may reflect on action, thinking back on what we have done in order to discover how our knowing-in-action may have contributed to an unexpected outcome” (p. 26). “Alternatively,” he says, “we may reflect in the midst of action without interrupting it … our thinking serves to reshape what we are doing while we are doing it” (p. 26). Schön distinguishes reflection-on-action and reflection-in-action. The first involves looking back on an action and reviewing its provenance and outcomes with the possibility then of modifying future action; the second is especially powerful, allowing the person acting to recognize a moment in the action, possibly with surprise, and to act, there and then, differently. John Mason has taken up this idea in his discipline of noticing: we notice, in the moment, something of which we are aware, possibly have reflected on in the past and our noticing afford us the opportunity to act differently, to modify our actions in the process of acting (Mason 2002).

Michael Eraut (1995) has criticized Schön’s theory of reflection-in-action where it applies to teachers in classrooms. He points out that Schön presents little empirical evidence of reflection-in-action, especially where teaching is concerned. The word action itself has different meanings for different professions. In teaching, action usually refers to action in the classroom where teachers operate under pressure. Eraut argues that time constraints in teaching limit the scope for reflection-in-action. He argues that there is too little time for considered reflection as part of the teaching act, especially where teachers are responding to or interacting with students. Where a teacher is walking around a classroom of children quietly working on their own, reflection-in-action is more possible but already begins to resemble time out of action. Thus Eraut suggests that, in teaching, most reflection is reflection-on-action, or reflection-for-action. He suggests that Schön is primarily concerned with reflection-for-action, reflection whose purpose is to affect action in current practice.

In mathematics education research into teaching practices in mathematics classrooms, Jaworski (1998) has worked with the theoretical ideas of Schön, Mason, and Eraut to characterize observed mathematics teaching and the thinking, action, and development of the observed teachers. The research was undertaken as part of a project, the Mathematics Teachers’ Enquiry (MTE) Project, in which participating teachers engaged in forms of action research into their own teaching. Jaworski claims that the three prepositions highlighted in the above discussion, on, in, and for, “all pertain to the thinking of teachers at different points in their research” (p. 9) and provides examples from observations of teaching and conversations with teachers. To some degree, all the teachers observed engaged in action research in the sense that they explored aspects of their own practice in reflective cycles. However, rather than the theorized systematicity of action research (e.g., McNiff 1988), Jaworski described the cyclic process of growth of knowledge for these teachers as evolutionary, as “lurching” from time to time, opportunity to opportunity, as teachers grappled with the heavy demands of being a teacher and sought nevertheless to reflect on and in their practice. As Eraut suggested, the nature of teaching in classrooms is demanding and complex for the teacher, as is the ongoing life in a school and the range of tasks a teacher is required to undertake. Teachers’ reflection on their practice, evidenced by reports at project meetings and observations of teacher educator researchers, led to noticing in the moment in classrooms, reflection-in-action, and concomitant changes in action resulting from such noticing.

A question that arises in considering reflective practice in mathematics education concerns what difference it makes (to reflective practice) that it is being used in relation to mathematics and to the learning and teaching of mathematics. Although in the mathematics education literature there are many references to the reflection of practitioners, there is a singular lack of relating reflective practice directly to mathematics. We see writings by mathematics educators referring, for example, to mathematics teachers who are reflective practitioners, reflecting on their practice of teaching mathematics; however, the mathematics is rarely addressed per se. We read about specific approaches to teaching mathematics and to engagement in reflective practice, for example, the identification of “critical incidents,” or the use of a “lesson study approach.” To a great extent, the same kinds of practices and issues might be reported if the writers were talking about science or history teaching. There is also a dearth of research in which mathematics students are seen as reflective practitioners.