Traditional Views of Mathematics

Most adults will acknowledge that mathematics is an important subject, but few understand what the discipline is about. For many, mathematics is a collection of rules to be mastered, arithmetic computations, mysterious algebraic equations, and geometric proofs. This perception is in stark contrast to a view of mathematics that involves making sense of mathematical objects such as data, form, change, or patterns. A substantial number of adults are almost proud to proclaim, "I was never any good at mathematics." How has this debilitating perspective of mathematics as a collection of arcane procedures and rules become so prevalent in our society? The best answer can be found in the traditional approaches to teaching mathematics. Traditional teaching, still the predominant instructional pattern, typically begins with an explanation of whatever idea is on the current page of the text followed by showing children how to do the assigned exercises. Even with a hands-on activity, the traditional teacher is guiding students, telling them exactly how to use the materials in a prescribed manner. The focus of the lesson is primarily on getting answers. Students rely on the teacher to determine if their answers are correct. Children emerge from these experiences with a view that mathematics is a series of arbitrary rules, handed down by the teacher, who in turn got them from some very smart source.

This follow-the-rules, computation-dominated, answer-oriented view of mathematics is a gross distortion of what mathematics is really about. It cannot be very exciting. A few children are good at learning rules and thrive on the ensuing good grades. But these are not necessarily the best thinkers in the room. The traditional system rewards the learning of rules but offers little opportunity actually to do mathematics.

Mathematics as a Science of Pattern and Order

Mathematics is the science of pattern and order. This wonderfully simple description of mathematics is found in the thought-provoking publication Everybody Counts (MSEB, 1989; see also Schoenfeld, 1992). This definition challenges the popular social view of mathematics as a discipline dominated by computation and rules without reasons. Science is a process of figuring things out or making sense of things. It begins with problem-based situations. Although you may never have thought of it in quite this way, mathematics is a science of things that have a pattern of regularity and logical order. Finding and exploring this regularity or order and then making sense of it is what doing mathematics is all about.

Even the youngest schoolchildren can and should be involved in the science of pattern and order. Have you ever noticed that 6+7 is the same as 5+8 and 4+9? What is the pattern? What are the relationships? When two odd numbers are multiplied, the result is also odd, but if the same numbers are added or subtracted, the result is even. There is a logic behind simple results such as these, an order, a pattern.

Consider the study of algebra. One can learn to graph the equation of a parabola by simply following rules and plotting points. Now calculators are readily available to do that with a speed and precision we could never hope to achieve. But understanding why certain forms of equations always produce parabolic graphs involves a search for patterns in the way numbers behave. Discovering what types of real-world relationships are represented by parabolic graphs (for example, a pendulum swing related to the length of the pendulum) is even more interesting and scientific--and infinitely more valuable--than the ability to plot the curve when someone else provides the equation.

And pattern is not just in numbers and equations but also in everything around us. The world is full of pattern and order: in nature, in art, in buildings, in music. Pattern and order are found in commerce, science, medicine, manufacturing, and sociology. Mathematics discovers this order, makes sense of it, and uses it in a multitude of fascinating ways, improving our lives and expanding our knowledge. School must begin to help children with this process of discovery.