Oh, you know all the words, and you sung all the notes,

But you never quite learned the song she sang. –Mike Heron

At Cambridge Math Learning we are developing a system for teaching mathematics to adult learners with math anxiety. We plan to develop it in the context of corporate training sessions, and eventually market it to individuals who need to learn mathematics for career advancement, helping their children with homework, or other reasons. Thus, we need to span main two learning environments: training and education.

We define training as the delivery of knowledge needed to enable an individual to perform a specific type of task, and education as knowledge delivered for its own sake. Training is often delivered in a "just in time" framework, where the knowledge that the worker learns will be used immediately. This is economically efficient because research shows that learning left unused is soon forgotten. This seems to be somewhat less true of education than of training, since the goal of education is to teach high-level principles, while training focuses on facts and procedures. If the reader thinks back to their school days, he or she will find that it is general methods of thought rather than specific facts and procedures that are most clearly remembered.

In recent years, both education and training have been increasingly dominated by the methodology of instructional design, a field which attempts to make instruction into a science. Although there are different theories of learning which can underpin instructional design, including cognitivism and constructivism, behaviorism is the oldest and most pervasive theory of learning used. The basic idea is to divide the material to be learned into small chunks. Each chunk is identified with one or more behavioral objectives, sometimes called performance objectives. These performance objectives must be testable. Once the student has achieved pre-defined "mastery" of these objectives, they are deemed to have learned the material.

It is easy to see why this methodology has become so popular. It promises to make learning efficient, quantifiable and replicable. We recognize that it has been effective in many spheres. And yet, something seems missing. By breaking up learning into bits, the participant may learn the words without learning the song. Creativity can be stifled, and material learned in this way is often soon forgotten.

As an example, suppose an adult with no musical training wants to learn to play an instrument, say a piano. In the traditional approach, the person is first taught how to play scales, which they practice over and over. Then they are shown how to play chords, which they practice over and over. Next they are shown how to play very simple tunes. This is a basic behaviorist approach. Many people have learned to play instruments this way, but many more have become bored and discouraged. We might even say they develop "music anxiety". However, holistic approaches to learning music do exist. For example, Paul Winter teaches a workshop in which non-musicians are put into small groups, each with a different instrument. The facilitators show people how to make noise come out of the instrument, and leave them on their own. After a few hours, as if by magic, the random noises begin to develop coherence. Through childlike play, the adults have tapped into their innate, long-buried musical talent. They are enjoying making music. At some point they can seek more formal instruction.

While you may agree that a behaviorist approach to learning an art is not ideal, you may say that mathematics is not an art, and that its mastery consists of learning step-by-step procedures, making it ideal for a behaviorist approach. This would be wrong. Almost all mathematicians (and almost no non-mathematicians) consider math to be an art. This view has been well advanced in G. H. Hardy's book A Mathematician's Apology. Even though our goal is not to make learners appreciate mathematics as an art, we think we can teach it as a skill that can be enjoyed rather than drudgery that must be endured. Also by focusing on main ideas of mathematics rather than minutia, we hope to provide learners with a foundation by which they can learn or relearn mathematics as they need it.

Let's consider a mathematical example. An even number, as you know, is a whole number that is divisible by 2. Suppose you are given a large number and asked whether it was even. You could use a calculator to divide the number by 2, of course, but there is a much faster way, which most everyone knows: just look at the last digit of the number. It is even if the last digit is even (0, 2, 4, 6, or 8); otherwise it is odd. So you can tell that 254953321650 is even faster than I can enter it into my calculator, even if my calculator will hold such a large number. There are similar, very quick rules that will determine whether a number is divisible by 3, 4, 5, 6, 8, 9, or 10.

The rule for determining whether a number is divisible by 3 is to add the digits. If the sum of the digits is divisible by 3, so is the number; otherwise not. For example, if you are asked whether 237910068 is divisible by 3, you could say 2 + 3 + 7 + 9 + 1 + 0 + 0 + 6 + 8 = 36. Since 36 is evenly divisible by 3, so is 237910068.

One could establish behavioral objectives to test whether the student has learned these divisibility rules, but that would be missing the point. The divisibility rules in themselves are of small value, other than that students find them interesting. What is important is that the student understands why these rules are true. To do this they must develop mathematical styles of thinking. Some of the mathematical ideas include the ability to search for patterns, a fairly deep understanding of the meaning of a the digits in a multi-digit numeral, prime numbers and their importance in factoring, and the distributive law. If students want to go on and look for a divisibility rule for 7, and understanding of modular arithmetic and algebraic notation will come into the mix.

At Cambridge Math Learning we recognize that many students come to us with very limited goals. Perhaps they need to be able to use basic statistical formulas, for example to determine the mean and variance of a distribution. We will teach them what they want, but we will teach them more; we plan to teach the song as well as the words. In this way they will gain long-term retention of the information, or the ability to reconstruct it.

(c) Peter Ash, Cambridge Math Learning