What is Mathematics?
There are many answers to this question. The academician, the curriculum designer, the student, the engineer, the politician... are all likely to have different notions about the fundamental nature of mathematics which colour their opinions on what should be taught in school mathematics, how it should be presented, and what the ultimate purpose of the exercise is. Such notions about the nature of a thing are called 'conceptual systems'.
A conceptual system is an operationally useful metaphorical structure that we use to help us make sense of some aspect of the world. There is no such thing as a universally correct conceptual view of anything. But it may be that some perspectives are more appropriate in particular settings than others. In this section I advance the notion that present school mathematics reflects an inappropriate perspective of what mathematics is.
Conceptual systems can be broadly classified as being either objectivistic or experiential in nature. Of these, objectivism represents the traditional and experientialism represents contemporary positions of cognitive science.
According to an objectivistic perspective it is self evident that the characteristics of mathematical concepts are independent of any understanding being. A line, for example, is ethereal and timeless; a transcendental thing. Mathematical entities and processes have their own existence. The mind is only a mirror of nature, and thought is correct only to the extent that it mirrors the logic of the external world. A person holding an objectivistic view might make statements such as:
Mathematical laws are universal and have remained unchanged since the beginning of time. Euclid, Euler and all the others discovered and put into words things that were already there.
Reasoning and thinking involves manipulation, according to established rules, of symbols which are abstract representations of the real external world.
Thought is atomistic and algorithmic; it can be broken down into 'building blocks' which are combined and manipulated by formal logical rules.
By contrast, the experiential view sees concepts as a product of individual and collective human experience. Thought does not exist without a thinker. Concepts have existence only in the mind. A person holding an experiential view might make statements such as:
Mathematics is a product of Man. Its laws were invented to help make sense of the universe, and are descriptive, not prescriptive.
Reasoning and thinking are imaginative in that those concepts which are not directly grounded in experience employ metaphor and mental imagery.
Thought and concepts have gestalt properties. They have structures that go beyond putting together building blocks.
Thus, an objectivist might answer our question "What is mathematics?" something like this:
Mathematics is a rigorously defined corpus of knowledge, intimately connected to our cultural history. It is logical, consistent and beautiful. Mathematical laws govern the universe. Mathematics is inherently sequential and most of the fundamentally important discoveries have already been made.
An experientialist might counter:
Mathematics is a dynamically defined discipline used by Man to study and come to terms with the world about him. Though Pure Mathematics has led to useful discoveries, much of the impetus has come from the physical sciences. In the last couple of decades there has been an explosion in new mathematical knowledge.
Thus, an objectivist curriculum design stipulates that:
We must pass traditional mathematical knowledge on to our heirs intact.
One learns mathematics by practicing logical reasoning on traditional problems.
The essentials of a mathematics curriculum is pre-defined. Any changes should be limited to scope-and-sequence of topics and possibly some additions as time allows.
... while the experientialist curriculum designer would state:
There are no sacred cows in mathematics.
People learn by interacting with reality.
We must develop new course content and approaches in which students learn to use contemporary tools to apply contemporary mathematics to contemporary problems.
I believe present school mathematics curricula reflect objectivistic perspectives of mathematics. Jim Swift (who needs no introduction in BC mathematics education circles), in a private communication once observed:
...if mathematics is a collection of techniques for rearranging symbolic expressions, and the mathematics curriculum consists of achieving mastery in those techniques as long as we deal with the purest form of history, we can allow most of our students the luxury of pretending that they are in the 17 century and that nothing they do in the math classroom will ever have any real connection with they way they live.
Here is another comment:
There is something odd about the way we teach mathematics in our schools. We make little or no provision for students to play an active and generative role in learning mathematics and we teach mathematics as if we expect that students will never have occasion to invent new mathematics.
We don't teach language that way. If we did, we would never require students to write an original piece of prose or poetry. We would simply require them to recognize, appreciate, and memorize the great pieces of language of the past- literary equivalents of the Pythagorean Theorem and the Law of Cosines.
Most mathematics instruction is a kind of satire on the nature of mathematical thinking and the process of creating mathematics.
Sunburst Communications
Introduction to The Geometric Supposer.