An ending and a beginning

Classes ended today, but fieldwork continues. While waiting for my ride, I began to realize that I need to have a working definition of mathematics, something I can hold in mind and use to frame my planning and teaching.

Thus: number / form / pattern. These are the objects of study in mathematics.

Numbers are used to distinguish, to arrange, to count, to measure.

Forms describe shapes or structures and the way parts fit together into a whole.


Patterns describe the way things repeat and change.



Numbers, forms, and patterns pair up to create other objects. A tessellation is a repeating pattern of geometric forms. An arithmetic sequence is a pattern of numbers. Numbers characterize polynomials and polyhedra, two different types of forms.

In addition to this triad of objects, my definition of mathematics involves four ways of thinking about the objects. These come in two pairs.

Abstraction is the process of isolating some features of interest from a domain, which could be mathematical or non-mathematical, for the purpose of reasoning about them, making discoveries, and drawing conclusions. Generalization is the reverse process: the application of of the results of reasoning about mathematical objects back to the original domain, or to another domain that shares the features of interest.

Following Polya, I see two types of mathematical reasoning: plausible reasoning and demonstrative reasoning. The two complement each other: the first is used to discover and to conjecture, the second to verify. Both are necessary for proofs and problem-solving.

This, then, is my working definition. I intend to use it not to define the field to others (though it does provide a ready answer) but rather to provide a structure or framework to the way I present mathematics in my teaching.

I'm not completely satisfied with this definition. Other definitions split out "change" as a separate object of study, but I've left it as an aspect of patterns. The idea of "correspondence" is central to mathematics but I'm not sure how it fits into this definition. I'm also not sure how "understanding" - a goal of education - could be defined in the terms I've used. For me, one key part of understanding is the ability to make connections, something that's only partly captured by the process of generalization.

But it's a beginning, both for my thinking about mathematics and for this blog.