My definition of the concept of reflection

I think of reflection as a cognitive activity, a process of thinking. It is a mental process by which our actions, beliefs, knowledge or feelings are consciously considered and examined.

To reflect involves more than just recalling or considering something consciously. A process of reflection provides enlightenment about the actions or ideas that are being considered. A process of reflection involves a kind of “Aha! moment“ in which something is discovered or revealed. I want to illustrate this idea with an example.

Example 1. I made this example up to try to illustrate my own interpretation of the concept of reflection. The example consists of a dialogue among two mathematics teachers. One of the teachers, named Luis, has been participating in an in-service course on mathematics education. As part of the course activities, Luis has been reading some research papers. The dialogue begins when Luis comes across a colleague who asked him about the in-service course he is attending:

Luis: Hello Julio!

Julio: Hello Luis. How are you doing? How is the course you are attending?

Luis: It has been very interesting! I have just finished my homework this morning.

Julio: Yes? What does the homework consisted of?

Luis: I had to read a paper called tacit models and infinity²⁰. It is quite interesting.

Julio: Why do you think the paper is interesting?

Luis: Well, in the paper it is claimed that when we have to deal with concepts which are highly abstract or very complex, our reasoning tends to replace

20 Here Luis is referring to the paper Fischbein (2001) which is included in the bibliography.

them by substitutes which are more familiar, more accessible, more easily manipulated. These are mental models. Sometimes, mental models are used intentionally, consciously, but sometimes we are not aware of their presence and/or of their impact. Apparently these kinds of models have a considerable effect on our thinking strategies and conclusions.

Julio: Mmmm…

Luis: And you know Julio? The paper made me remember an experience I had two weeks ago.

Julio: Yes? Tell me about it.

Luis: I was in the classroom with my students. We were studying the characteristics of the graph of the function y=log

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Julio: o.k

Luis: I draw on the blackboard a graph like this one [Luis shows to Julio a piece of paper with a drawing on it. See figure 4]. Then I tried to explain them that in this region, the graph approaches the y-axis but never touches it. [Here Luis is referring to the vertical asymptote of the function]

Figure 4. Representation of the graph used by Luis to illustrate the graphical behaviour of the function y=log

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Julio: Yes

Luis: To me it was something very natural to refer to this property, but then one of my students asked how that was possible. She said that if she walked towards one of the classroom’s wall, even by using very short steps, there would be a moment when she would reach the wall. Then she asked how it was possible that the function would not touch the y-axis.

Julio: I see.

Luis: But to be honest Julio, I sort of ignored her. I though she was just a dull pupil unable to understand the mathematical idea I was trying to explain.

Nevertheless, after reading the paper about tacit models and infinity, I realised that my student was actually using a mental model. She was using a mental model where the behaviour of the asymptote was substituted for the metaphor of “walking towards a wall”. This is the reason why she found difficult to grasp the mathematical idea I was explaining. I think I should try to pay more attention to this kind of situations…

I consider the above example as a manifestation of a reflection process that Luis has experienced. He is not only recalling and consciously considering one of his teaching experiences. He also discovered an aspect within that experience that previously was not visible or perceptible. I refer to the mental model that his student probably used to try to understand the concept of asymptote.

Thus, a reflection not only consists in explicitly considering your actions, values, knowledge or feelings. A reflection also implies that an aspect of the element being considered is discovered or becomes visible.

This is what I mean by the “Aha! moment”.

The previously presented example illustrates a process of reflection which is anchored in a teaching experience. However, there are other kinds of reflections that are also important to the development of mathematics teachers. In particular, in my research I have identified three

types of teachers’ reflections: didactical reflections, mathematical reflections and extra-mathematical reflections.

A didactical reflection refers to the process of reflection in which a teacher consciously considers her teaching practice. Her values and actions associated with this practice and/or the consequences of such values and actions. The above-mentioned example 1 is an instance of this kind of reflections.

In a mathematical reflection a teacher consciously considers and revisit aspects of her own mathematical knowledge. During this type of reflection a teacher consciously examine for example, her interpretation of mathematical concepts or her way of solving mathematical tasks. Such reflections can lead to an improvement of teachers’ mathematical knowledge, since this kind of reflections can help the teachers to identify personal misconceptions or even help them to acquire new mathematical knowledge.

An extra-mathematical reflection occurs when a teacher consciously considers the role and application of mathematics in non-mathematical contexts. It can also include a consideration of the consequences of such application. For example, by means of an extra-mathematical reflection, a teacher can become aware of the undesirable and irreversible societal and economical consequences that a mathematical-based decision-making model can potentially produce in the life of the members of a particular community.