Two words can mean different things for different people. And so it is also in mathematical teaching; There is both an instrumental approach, and an relationship approach. It is not about who is right or who is wrong, but rather about how we can stand back to examine what we are actually trying to explain, and come to a mutual ground of understanding we can meet together.
The instrumental method is usually easier to understand, and quicker to answers to problems, and there is less information involved. This is the more traditional method for math teachers. The relational method is often more adaptable to new problems in the real world, and thus, more intuitive, perhaps easier to remember. Once a concept is understood relationally, the student is more likely to explore other problems which stem from the original one. A student's understanding will grow beyond the scope of the original problem, to develop new problems, or new ways of getting the answer. Hence, achieving this basic understanding can be an effective goal in itself. However, with examinations, over-burdened syllabi, and other assessment difficulties, traditional teachers often shy away from relational understanding. This type of teaching and understand takes longer.
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