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# Composition, Decomposition, Relationships, Representation, Context (page 3)

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

### Representation

Our discussion of the powerful idea of relationships has led us directly into our fourth powerful idea: representation. Often when mathematics students are in the process of trying to communicate their ideas they need to let a written symbol or drawing stand for an idea. The concept of letting something tangible or written stand for an abstract idea is an important component of mathematics. We use the term represent when we are letting something tangible stand for an abstract idea. We might say, "This little circle represents five votes for SUVs as the favored method of transportation in our survey." Or you might hear a student say, ''I'm letting this ¤ represent a pig in the problem about pigs and chickens." When the term represent is used in this way it is being used as a verb, an action, a process. The word representation is the noun form of the same idea. The symbols themselves are referred to as representations. Used more broadly, the term representation can refer to both the process of representing and the product of that process. The idea of representation permeates any meaningful study of mathematics. A study of mathematics includes work with creating representations as well as work with learning the standard mathematical representations. One of the most significant insights that a student of mathematics can have is the understanding that the symbols we deal with in mathematics always represent ideas.

When we perform operations (such as addition, subtraction, multiplication, and division) with the written symbols we call numbers, we are actually performing operations on the ideas of numbers that those symbols represent. That is, there is a difference between the written symbols and the ideas behind those symbols. Sometimes the term numeral is used in reference to the written symbol, and the term number is used in reference to the concept of a given quantity. Each number (the abstract idea) can be represented in a variety of ways. The number of things in a dozen, for example, can be represented by the numeral 12, or the word dozen, or the word twelve, or with tally marks, or in a variety of other ways. A successful student of mathematics works to understand the ideas behind all the symbols used in mathematics. Then, the mathematician is able to manipulate the symbols as representations of increasingly abstract ideas. The symbols themselves do not mean anything, but to a person who has applied herself to understand the ideas, the symbols represent meaning. The manipulation of symbols becomes an economical and efficient way to manipulate complex ideas.

A student who struggles with mathematics often fans to perceive that the symbols relate to abstract ideas. He attempts to learn rules for manipulation of the symbols themselves, but the rules appear to be arbitrary. Without an understanding of the ideas behind the symbols, the student has no choice but to try to memorize all the (apparently arbitrary) rules of symbol manipulation. The difficulty of memorizing all the rules of mathematics (without understanding the ideas) becomes so burdensome that the student generally reaches a point where he gives up trying. From that point on, further mathematics study is fruitless. Meanwhile, this student observes the success of his classmate who seems to enjoy and do well with mathematics. The struggling student does not realize that the successful student is dealing with abstract concepts and ideas rather than with rules for symbol manipulation. The struggling student concludes that he must just be poor at mathematics while his classmate is good at mathematics.

In reality these two students are experiencing two completely different perceptions of what mathematics really is. To the failing student, mathematics deals only with arbitrary rules of symbol manipulation. To the successful student, mathematics deals with ideas that make sense and with the symbols that represent those ideas. The failing student fails because of memory overload; the successful student succeeds because of understanding. Often the difference is not so much due to differing ability between the two students, but with differing perceptions of the nature of mathematics. Chances are the successful student would fail too if he tried to memorize all the mathematical rules and procedures without understanding the concepts. The notion of representation in mathematics may be the key that unlocks understanding for many students who have given up on mathematics.

Many of the ideas of mathematics are very complex, and the representation of those ideas in symbolic form gives us an abbreviation and a shorthand method of dealing with those complex ideas. This is a very good thing when it comes to the efficiency of manipulating abbreviations for complex ideas. However, it makes the learning of the symbol system especially challenging. The first time a learner encounters a particular symbolic representation she may not understand the complex idea that the symbol represents. With further work and study, the student constructs some vague, fuzzy idea of the concept behind the symbol. With even more study and practice, the student reconstructs her understanding and deepens it further. More study and more practice lead to even deeper insights into the meaning of the complex idea behind the symbol. The process of learning mathematics is a process of coming to deeper and deeper understandings of the ideas that the symbols represent. This is why it is never enough in mathematics to simply learn the symbols and the rules that govern their use. The symbols are only representations for complex ideas, and it takes time and effort to fully explore those complex ideas.

### 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.