Discovering mathematics

What sort of beliefs about the nature of mathematics should undergird the curriculum and instruction when we live in a world that requires mathematical proficiency to live a productive life?

Discovering mathematics

Those readers who have ever been close to a mountain range, or better still have adventurous enough to trek the majestic highlands of Pakistan or elsewhere, would know well the capacity of the mountains to hold them in awe. Since I have had the good fortune to trek on the Baltoro glacier to the base of the K2, I can vouch for this feeling.

The majesty and awe that I experienced on the mountains was not too different from the feeling I had when learning mathematics and its application to physics as a graduate student. In this article my concern is not mountains but mathematics education. But the simile sounds too useful to ignore.

Let me begin by briefly describing some beliefs about the nature of mathematics before getting back to the mountains. There are mainly two ways of looking at mathematics: mathematics as superhuman, abstract, ideal, infallible, and eternal and mathematics as a human creation. There is a large number of mathematicians and mathematical physicists who subscribe to the former.

Mathematician Reuben Hersh labeled the former as the mainstream and the latter as the humanist conception of mathematics. The very choice of words to describe these orientations toward the nature of mathematical knowledge suggests that for most mathematicians mathematical objects exist in an a priori realm. Humans cannot create mathematics under this description. They can only discover it.

This point may be illustrated metaphorically by considering a mountain range as a scene, climbers as agents, and climbing as an act. Climbers do not invent or construct the peaks that they scale. The job of climbers may be described as finding the best possible routes and using them to reach the summits. Once a particular summit has been scaled, the beaten paths come on record, and become available for future generations of climbers. These climbers may also discover new, more efficient, and safer paths to reach the summit. Yet, they, and those before or after them, cannot change the shape of the mountain.

The mathematician does not act to create, but to only discover. Discover, not invent or construct, is the verb that describes his act.

Like a mountain range and its many summits, mathematics comes across in the mainstream description as an a priori. Just like mountain range has various features, the math world is also populated with mathematical forms. On this view, the mathematician’s job, i.e. doing mathematics, involves gaining access to the pre-existing world of mathematics. Since the world of mathematical forms is supposed to precede and exist independently of the mathematician, it constrains what can or cannot be done as mathematics.

The idea of mathematics as conveyed in this description imagines mathematician as someone who has somehow been trained to access the preexisting realm of mathematical forms and discover mathematical objects, much like our climber who is trained to navigate peaks in a mountain range.

These aspects of mathematical acts are also exemplified in the perspectives that undergird theoretical physics and other mathematical sciences. Consider, for example, the work of theoretical physicists. Most of them believe in a preexisting mathematical language in which the universe is written. As a student of physics in my early career, I was also initiated into such a conception of mathematics. All great equations in physics were spoken about as discoveries. The list of these discoveries was long: Newton discovered laws of motion; Hamilton discovered quaternions; Maxwell discovered laws of electromagnetism; Einstein discovered relativity, Erwin Schrödinger discovered wave mechanics, etc. Discover was the verb that described the main acts of the physicists in this discourse.

But, does the verb discover not restrict what can be done as mathematics by completely containing the doing of mathematics in a preexisting mathematical universe? Indeed one cannot discover a mathematical object that does not exist, even though we only find out after the fact that it did indeed exist. Doing mathematics on the stage so set is like finding one’s feet, and ultimately, one’s way in this mathematical equivalent of the mountain range.

The mathematician does not act to create, but to only discover. Discover, not invent or construct, is the verb that describes his act. In addition to discover, this description of mathematics can also accommodate other verbs such as, reading, following, discovering, comprehending. Each one of theses verbs falls within the realm of what mathematicians and the students of mathematics can do. Notice, these verbs do not typically include construct. Like the climbers do not create the peaks and only discover them and the paths leading to them, the mathematicians and students of mathematics also do not create mathematical objects.

Does such specification of the nature of mathematics imply an elitist approach to knowledge? Probably! Typically, the great mathematicians who subscribed to this view did not like to teach mathematics to all students. They searched for the gifted students. It is thus that G.H. Hardy, the famous English mathematician, discovered Ramanujan, the great Indian mathematics genius. Hardy did not train or create a Ramanujan but could only be fortunate enough to discover him. Likewise the 19th century American mathematician Benjamin Pierce, who is often called father of pure mathematics in America, was also contemptuous of teaching anyone but the most gifted students.

The acts of the ‘unrepentant elitist’ as a teacher are entirely consistent with a belief in mathematics as preexisting. For the believer in such a view of mathematics, reforming mathematics education can mean getting rid of those without mathematical powers, not inculcating them in those who did not possess them. It may mean institutionalising low expectations about the potential of mathematical achievements by all students. It may also imply strict sorting practices in schools and higher education institutions.

Unsurprisingly, some mathematicians who subscribe to these views are horrified when they are told that schoolteachers expect all students to do well in mathematics. Slogans like Mathematical power for all appear to them as incomprehensible gibberish.

Some reforms in school mathematics, however, have involved moving away from such conception of mathematics to a more humanist conception of mathematics. The manifestation of this in the realm of mathematic education is such learning philosophies as constructivism. The reform texts define the nature of mathematics as being produced as a result of purposive human engagement with their lived realities. When the belief shifts from mainstream to the humanist nature of mathematics, the wanderer in the math world becomes a creator of mathematical landscapes.

What sort of beliefs about the nature of mathematics should undergird the curriculum and instruction when we live in a world in which mathematical proficiency is a prerequisite for a productive life? Mathematics education is no longer about finding students gifted in mathematics and turning them into mathematicians. When the goal of school mathematics is to produce a numerate citizenry, the educators will need to expect all students to become knowledgeable and skilled in mathematics. Yet their beliefs about the nature of mathematical knowledge may come in the way of achieving the goals of mathematics education.

It is possible for teachers to think of themselves as great teachers, without fully recognizing that their beliefs about knowledge and ways of knowing are restricting the reach of their instruction to a handful of their students. The least we can do is to make teachers aware of their own beliefs about the nature of knowledge and ways in which they influence their teaching.

Discovering mathematics