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# Composition, Decomposition, Relationships, Representation, Context (page 4)

By Pearson Allyn Bacon Prentice Hall
Updated on Jul 20, 2010

### Context

Our discussion of the relationship between abstract symbols and the ideas those symbols represent suggests another important relationship: the relationship between the world around us and the abstract world of mathematics. Our fifth and final powerful idea is that a student of mathematics almost always benefits when he or she can see a real-world, practical context from which mathematical abstractions can be derived. We refer to this powerful idea as context. A deep frustration that many students of mathematics have faced in the past is their inability to see the usefulness of the mathematics they were being required to learn. If and when these students asked why they needed to know some particular area of mathematics, they were often told things like, "It is a good mental discipline," or, "It will teach you how to think," or, "You will need to know this in order to move on into more advanced mathematics." These are unacceptable answers to the question. A child or student who wonders about the rationale for a mathematical topic is not typically going to be satisfied with such answers. This student is asking why she should invest the necessary effort to learn something, and typically what she is looking for is a use for the mathematics in her everyday life. If we do not know the use of the mathematics that we are teaching, then we need to revisit our own understanding of the mathematics. All of the mathematics that is a part of the elementary school curriculum have practical applications. All of it can be learned in a meaningful context. The idea that mathematics, as a study of abstract symbols and the relations between those symbols, arises from real-world contexts is a pervasive, powerful idea. We maintain that the first step in learning any area of mathematics is to first become familiar with a problem situation. (A problem situation is a situation in which we have a goal that we wish to attain and we face an obstacle. The obstacle is one for which there is no learned procedure to overcome it.) Having come face-to-face with a problem situation, the learner is then in a position to begin to see a mathematical idea as a remedy to the problem situation. Mathematical ideas or procedures have usually been developed in response to specific problems, therefore it only makes sense to learn (and teach) mathematical ideas and procedures in relation to specific problems.

Sometimes we hear of a distinction between pure mathematics and applied mathematics. Pure mathematics is mathematics that is studied and developed with no concern for any possibility of real-world applications. It is a study of mathematics for its own sake. The study of pure mathematics is undertaken in almost the same way that a person studies art: it is done for beauty, aesthetics, and for the pure pleasure of it. Applied mathematics, on the other hand, has a declared goal of making mathematics useful in some real-world application. It is developed specifically in response to a practical need. We envision the possibility of a new piece of computer equipment, but we lack the capability to manufacture it. In the process of developing a manufacturing process we may develop a new mathematical algorithm that unlocks our manufacturing capability and makes it possible for us to make the computer equipment we wanted. Applied mathematics is developed in this way, in response to a problem or a need. This distinction between pure and applied mathematics is important to our discussion because the number of people who can appreciate pure mathematics for its beauty is small. Very few people are willing to invest the time and energy necessary to understand mathematics for its own sake. On the other hand, most of us are willing to work hard to learn something that is useful to us. When a mathematical technique can make a job that we need to do go faster, then we see it as a worthwhile investment of our time and energy to learn that technique. If a mathematical way of solving a problem is shown to be more efficient and effective than a nonmathematical way, then we don't mind learning the mathematical way. There may have been a time when educators had the luxury of being concerned only with the learning of the few who loved mathematics for its own sake. If so, that time is gone. In today's world, educators must be concerned with the learning of all children. The use of meaningful contexts (real-life applications and word problems) as settings within which to learn mathematics makes it more likely that all children do learn to understand mathematics.