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

By J.E. Schwartz
Pearson Allyn Bacon Prentice Hall
Updated on Jul 20, 2010

#### Banana Costs and Weights

 Pounds of bananas 4 5 8 10 Cost of bananas \$0.60 \$0.75 \$1.20 \$1.50

The concept of a relationship between numbers or between sets of numbers is an important part of the study of ratio, proportion, percent, scale, measurement, statistics, probability, functions, and more. Each of these areas of mathematics will become comfortable for the student who takes the time and devotes the energy to gain an understanding of relationships between numbers.

We find the concept of relationship in our study of geometry as well. As we will show in our geometry chapter, there is a level of development of geometric thinking where learners begin to perceive relationships between various categories of shapes. The classic example of such a relationship is the relationship between squares and rectangles. This same relationship can be seen between an equilateral triangle and triangles in general as well as between a rhombus and a parallelogram. In general, this is the relationship known as class inclusion. An object can be a member of a class of objects that is contained within a larger class of objects. Another area of geometry where relationship is a central concern is in the study of the parts of a triangle and how they relate to each other. We all remember the Pythagorean relationship: a2 + b2 = c2. This is just one example of many relationships that mathematicians study between various parts of a triangle. To further illustrate the importance of relationships in geometry, consider the concept of parallel lines. Two lines are parallel because of their relationship to each other. The same can be said of perpendicular lines. The measurement of angles depends on the relationship between two rays.

Another way in which a study of relationships is important is in the study of relationships between and among various ways a concept can be represented. The price of bananas shown in table above can be represented in a formula, p = 0.15w if we identify p as a variable for the price and w as a variable for the weight. A third way to represent the same information is in the form of a graph. Students of mathematics need to become comfortable looking for the relationships between different representations.

### 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.

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