*Ever wondered what it means to know and do mathematics? Stellenbosch University lecturers Zurab Janelidze and Sophie Marques share their conclusions after months of debate and discussion.*

This article is the result of numerous discussions between the authors around the question: What lies at the core of mathematical comprehension? We arrived at the following answer: *The art of distinguishing*. In this article we will try to explain some intuition behind this answer. Disclaimer: the complexity of the art of distinguishing cannot be settled in a single column, so what we write below is only scratching the surface.

As a starting point, the art of distinguishing can be thought of as the ability to decompose a phenomenon or concept being comprehended into more basic contrasting constituents. We believe that application of this ability leads to an improved comprehension of the phenomenon or concept, which in turn can be applied to, for example, solve a problem, find a work-around, or simply find a way to new ideas and new perspectives that may positively shift one’s mental state.

Applying the very same art to the comprehension of this art itself, we would like to distinguish between two processes of distinguishing: *subconscious distinguishing* and *conscious distinguishing*. As the term suggests, subconscious distinguishing is when distinguishing occurs by itself in our subconscious mind, without much deliberation or effort from our conscious self. We might not even be aware of the process, but we will be able to apply its result in our actions or thoughts. In contrast, conscious distinguishing is when the process of distinguishing requires our conscious deliberation or effort. Note that in reality, it is not merely a dichotomy between the two, but rather a complex mix.

Let us illustrate the contrast between subconscious and conscious distinguishing on an example. Consider the following image:

At the very moment you look at the image, subconscious distinguishing offers you two objects, the circle and the square, as the basic constituents of the image. Now look at the following image:

Subconscious distinguishing may offer here more than two constituents, especially if you have not seen the previous image (try showing the second image, without showing the first one, to someone else and ask them what they recognise to be the constituents of the image). With some effort of understanding the structure of the second image, one may discover that it is composed of two basic shapes – the circle and the square. Arriving at this conclusion will be a result of applying a certain amount of conscious distinguishing.

Actually, applying some conscious distinguishing, we may come to see that the example above, in addition to showing us the contrast between subconscious and conscious distinguishing (which was the original aim), also reveals two other features of distinguishing:

- The whole is more than the collection of distinguished parts. For instance, when describing the first image above, the two objects that arise from distinguishing (the circle and the square) do not describe the image fully – in addition to these two objects, the image consists of the white space surrounding the objects. Also, the two different images above have the same constituents (circles and squares), but the two images differ from each other: the distinguished parts thus do not determine the whole uniquely. (One may argue that this may be fixed by incorporating further constituents, such as the number of circles and squares that appear in each image, as well as their relative positions. But then, what about the colour? We can have an image that looks the same as one of the above, but whose lines are green instead of black. Of course, we can incorporate the colour as well as a constituent, but… will this process ever stop?).
- One distinguishing experience may condition another. Seeing the image of the separate square and circle may well have an influence on identifying the constituents of the second image to be a square and a circle.

We propose the following names for the two features above. The first, we will call *gestaltism*, in reference to the concept of “gestalt” in psychology (see e.g. [George Mather, *Foundations of Sensation and Perception*, Routledge (Taylor & Francis Group), London and New York, 2016]). The second, we call *conditioning*.

It is worth remarking that gestaltism and conditioning both have something to say when it comes to approaches in pedagogy. When trying to teach some material to a student, it is not sufficient to teach each of the distinguished constituents separately, since those constituents, when considered in isolation from each other, may not be sufficient to recover the whole. In fact, one could argue that true learning occurs when the student themself is able to carry out the necessary distinguishing for comprehending the material, rather than when the result of distinguishing is offered by the teacher. On the other hand, adapting the material to a simpler scenario where the student can themself succeed in pursuing a distinguishing process, may lead to conditioning that could help the student to carry out the more complex distinguishing.

An example of conditioning in mathematics is when simple examples of a mathematical phenomenon or concept help one unpack it in more complex and abstract scenarios. This and other types of conditioning are used not only in learning mathematics at school or university, but also by professional mathematicians in their mathematical research.

In mathematics we can also give a good illustration of gestaltism: different concrete examples of an abstract concept do not determine the abstract concept. Indeed, reaching abstract thought requires something more than just observation of its concrete instances. The engineering that takes place to manufacture abstract thoughts is a true marvel, which seems to be a joint work of subconscious and conscious distinguishing.

And now that we are back to subconscious and conscious distinguishing, let us touch on a phenomenon that intimately ties the two together, and which is actually a central phenomenon to the art of distinguishing. It is when conscious distinguishing empowers subconscious distinguishing. This phenomenon can be described as the phenomenon of *true learning* – the learning as a result of which knowledge is acquired predominantly in the form of skills rather than in the form of memorised facts/procedures. The process of true learning can be pictured as a cycle, where conscious distinguishing gradually gets transported into subconscious distinguishing, which in turn enables a further-reaching conscious distinguishing to occur.

Mathematical comprehension is a good example where true learning is a must. One may argue why this is so. It could be that because of the abstractness of mathematics compared to the concreteness of the world we comprehend using our five senses, one has limited initial subconscious distinguishing to employ. Hence effort must be placed in conscious distinguishing to grow subconscious distinguishing to a point where one can begin to acquire the minimum level of fluency that we have when applying distinguishing in comprehending the real world around us.

If true learning is a must in mathematics, and if true learning requires effort accompanying any conscious distinguishing, it seems to follow that learning mathematics ought to be difficult. This need not be bad news at all, however, since it also suggests that (true) learning of mathematics may well be a path to acquiring the art of distinguishing to a point where the difficulties in other areas of life may be overcome with less effort.

One other conclusion from the possibility that the art of distinguishing indeed lies at the core of mathematical comprehension, which we as mathematicians passionate in teaching mathematics were happy to draw, is that mathematics is accessible to all. Yes, for some, at start, it may be easier than for others, due to an initial surplus of subconscious distinguishing. However, the good news is that the subconscious distinguishing can be upgraded through conscious distinguishing, although a correct approach to teaching and learning, ensuring that true learning is taking place, plays a crucial role in this.

### Dr. Sophie Marques

Researcher and lecturer

Department of Mathematical Sciences

STELLENBOSCH UNIVERSITY, SOUTH AFRICA

### Prof. Zurab Janelidze

Professor

Department of Mathematical Sciences

STELLENBOSCH UNIVERSITY, SOUTH AFRICA