Composition
The first and most fundamental concepts of mathematics are the idea of counting and the more developed idea of using a system to represent numbers. Dehaene (1997) presents the intriguing notion that several species of birds and mammals can distinguish between a single item and quantities of items. Further, there is some evidence that these birds and mammals might be able to distinguish between two objects and three objects or even between three objects and four objects. Dehaene's hypothesis, which is thoroughly developed and documented, is that certain structures within the brain seem to be specifically devoted to these primitive mathematical abilities. If this is true, then what exactly do these brain structures do? What is involved at the most fundamental level in this thing we refer to as counting?
To begin our exploration of this idea, consider two (imaginary) individuals, one who has the ability to count and one who does not. As these two individuals look out on the scenery of their lives, what do they see, and how does what they see differ? On one level we can say that what they see is exactly identical. They are both receiving the same sensory signals. What they do with those signals that they receive from their senses can differ, however. The non-counting individual probably perceives that there are multiple leaves on a tree, for instance, but beyond this perception he is unable to proceed. The counting individual is able to identify a single leaf as a unit that is repeated, he is able to momentarily focus attention on each leaf once and only once, he is able to apply a counting word to each leaf in a sequence, he is able to use the exact right sequential word for each successive leaf, and after he has focused momentarily on each and every leaf he is able to report a final "count" to tell us how many leaves there are. Furthermore, he understands that he could repeat the process using a different order of leaves but the same sequence of word-labels, and if he has been careful he will come to the same ending word each time he does this. All of this can take place without any physical evidence that he is doing anything at all; that is, it can all take place in his mind. For both individuals the scene is a single scene, but the counting individual can perform this complex analysis on the scene to provide us with some abstract information about what the scene contains. It is unlikely that he would ever do this, however, unless there were some pressing need for that information. Mathematics, while abstract, rarely makes its presence known except in response to a particular goal a person wants to achieve.
One of the most interesting parts of this counting process is the very beginning. How does one begin to count? In order for our imaginary counting person to even begin to use his ability he must first decide on an object to count. He must take what he sees as a single scene and see within it some repeated objects that he considers to be units to be counted. That is, he must mentally compose what he will consider to be units. In the example given, the countable objects were, of course, leaves. But did he have any other choices? What else could he have counted? Trees? Blades of grass? Clouds? Rocks? Any of these are possibilities. It is all a matter of choices made by the counting individual. How about clusters of leaves? What if the counter determines that each branch has, invariably, six leaves? Then, when he counts branches he can determine the total number of leaves without having to count each one. The idea of counting things in groups and then counting the number of groups is an extremely powerful idea in the development of a counting system (or a number system). In the case of counting clusters, the counter has simply composed a different unit. He has considered a branch rather than a single leaf to be his countable unit. The concept of composing various counting units is a theme that we will return to again and again.
Composition occurs in geometry as well. A key element of spatial reasoning is the ability to see a shape and mentally manipulate it. But what constitutes a shape? Is it the overall outline of the shape? Is that overall outline composed of parts? Do the parts have characteristics that can be analyzed and/or found in common? Do the parts have names? Does the overall shape have a name? Most important, can the parts and the whole be seen in more than one way? The ability to compose and manipulate units of shape is one example of the importance of composition in a geometric context.
Decomposition
Once we have composed a countable unit we can go in an entirely different direction. Aside from simply counting the number of objects of that unit, we could also decide to partition, or decompose, one of those units. There is an entire world of mathematics found in decomposing units. A unit can be a geometric shape, and decomposing it into parts can begin a traditional study of fractions. A unit can be a group of objects, and a consideration of a portion of the group involves a different use of fractions. If our unit is readily divided into ten equal pieces, then we can introduce the idea of decimal numbers. If we decompose a unit into two different parts and then we study the relative sizes of those parts (Are the parts equal in size? Is one of the parts twice as large as the other?), then we introduce a study of ratio. If we can readily decompose our unit into one hundred equal parts, then we can work with percents. All kinds of measurement activities and understandings rely on decomposing as well. Rarely can we accomplish the precision that we need using whole numbers of measurement units exclusively. Precision in measurement, then, relies on the idea of decomposing units into meaningful parts.
A slightly different consideration of parts and wholes brings us to the heart of addition and subtraction. When we add two quantities together we are taking two parts and composing them into a single whole. When we take a single whole (group) and break it apart into two or more smaller groups we are doing a certain kind of subtraction. When we consider addition and subtraction in this light we can begin to see them as two sides of a coin rather than as two distinctly different operations and concepts. Addition and subtraction are not the only basic operations that have their roots in decomposing. Division as well can be best understood in this light. What is division if it is not taking a whole and dividing it into smaller parts? If the whole is a dozen objects, then we can use decomposing to answer questions about smaller groups into which our dozen objects can be broken.
Decomposing, then, opens up a wide range of mathematical topics for study. Decomposing is one of our powerful ideas. On one hand we can use the idea of composing as the basis for counting, and on the other hand we can use the idea of decomposing as the basis for analysis. If composing is the key idea behind counting, decomposing is the key idea behind fractions and all of the things we can do with fractions.
Relationships
Another powerful idea emerges when we begin to consider the concept of relationships in mathematics. We can study relationships between numbers or between sets of numbers. We can also study relationships between shapes and between parts of shapes. We mentioned ratios in our introduction to decomposing; the examination and analysis of ratios leads us to the powerful idea of mathematical relationships. While it is true that the concept of a relationship between numbers is completely abstract, it is, nonetheless, one of the more important topics in mathematics. An ability to see relationships as something tangible, in spite of their abstract nature, will make it possible for a student of mathematics to progress into higher levels of mathematical study. As with composing and decomposing, the concept of relationship is found in numeric as well as geometric areas of mathematics. This is part of why we consider it to be a powerful idea.
Consider the set of numbers that would be used to represent pounds of bananas and the cost of the bananas. Tthe table below presents a possible set of such numbers. An examination of the relationship between pounds of bananas and cost of the bananas reveals a consistent pattern. Multiplying the number of pounds by $0.15 yields the total cost of the bananas. Conversely, dividing the cost by $0.15 gives us the number of pounds of bananas. We say that the bananas are $0.15 per pound. The term "per" indicates a relationship. Knowing this relationship we can compute the cost for any quantity of bananas, and we can compute the weight of any given purchase. Knowing a relationship between quantities gives us a lot of power. Studying relationships is a crucial part of a study of meaningful and useful mathematics.
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.
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.
May Workbooks are Here!
Get Outside! 10 Playful Activities
