What is Mathematics?

Mathematics is an old, broad, and deep discipline (field of study). People working to improve math education need to understand "What is Mathematics?"

A Tidbit of History

Mathematics as a formal area of teaching and learning was developed about 5,000 years ago by the Sumerians. They did this at the same time as they developed reading and writing. However, the roots of mathematics go back much more than 5,000 years.

Throughout their history, humans have faced the need to measure and communicate about time, quantity, and distance. The Ishango Bone is a bone tool handle approximately 20,000 years old.

Figure 1

The picture given below shows Sumerian clay tokens whose use began about 11,000 years ago (seehttp://www.sumerian.org/tokens.htm). Such clay tokens were a predecessor to reading, writing, and mathematics.

Figure 2

The development of reading, writing, and formal mathematics 5,000 years ago allowed the codification of math knowledge, formal instruction in mathematics, and began a steady accumulation of mathematical knowledge.

Mathematics as a Discipline

A discipline (an organized, formal field of study) such as mathematics tends to be defined by the types of problems it addresses, the methods it uses to address these problems, and the results it has achieved. One way to organize this set of information is to divide it into the following three categories (of course, they overlap each other):

1. Mathematics as a human endeavor. For example, consider the math of measurement of time such as years, seasons, months, weeks, days, and so on. Or, consider the measurement of distance, and the different systems of distance measurement that developed throughout the world. Or, think about . There is a rich history of human development of mathematics and mathematical uses in our modern society.

2. Mathematics as a discipline. You are familiar with lots of academic disciplines such as archeology, biology, chemistry, economics, history, psychology, sociology, and so on. Mathematics is a broad and deep discipline that is continuing to grow in breadth and depth. Nowadays, a Ph.D. research dissertation in mathematics is typically narrowly focused on definitions, theorems, and proofs related to a single problem in a narrow subfield in mathematics.

3. Mathematics as an interdisciplinary language and tool. Like reading and writing, math is an important component of learning and "doing" (using one's knowledge) in each academic discipline. Mathematics is such a useful language and tool that it is considered one of the "basics" in our formal educational system.

To a large extent, students and many of their teachers tend to define mathematics in terms of what they learn in math courses, and these courses tend to focus on #3. The instructional and assessment focus tends to be on basic skills and on solving relatively simple problems using these basic skills. As the three-component discussion given above indicates, this is only part of mathematics.

Even within the third component, it is not clear what should be emphasized in curriculum, instruction, and assessment. The issue of basic skills versus higher-order skills is particularly important in math education. How much of the math education time should be spent in helping students gain a high level of accuracy and automaticity in basic computational and procedural skills? How much time should be spent on higher-order skills such as problem posing, problem representation, solving complex problems, and transferring math knowledge and skills to problems in non-math disciplines?

Beauty in Mathematics

Relatively few K-12 teachers study enough mathematics so that they understand and appreciate the breadth, depth, complexity, and beauty of the discipline. Mathematicians often talk about the beauty of a particular proof or mathematical result. Do you remember any of your K-12 math teachers ever talking about the beauty of mathematics?

G. H. Hardy   was one of the world's leading mathematicians in the first half of the 20th century. In his book "A Mathematician's Apology" he elaborates at length on differences between pure and applied mathematics. He discusses two examples of (beautiful) pure math problems. These are problems that some middle school and high school students might well solve, but are quite different than the types of mathematics addressed in our current K-12 curriculum. Both of these problems were solved more than 2,000 years ago and are representative of what mathematicians do.

1. A rational number is one that can be expressed as a fraction of two integers. Prove that the square root of 2 is not a rational number. Note that the square root of 2 arises in a natural manner as one uses land-surveying and carpentering techniques.

2. A prime number is a positive integer greater than 1 who’s only positive integer divisors are itself and 1. Prove that there are an infinite number of prime numbers. In recent years, very large prime numbers have emerged as being quite useful in encryption of electronic messages.

Problem Solving

The following diagram can be used to discuss representing and solving applied math problems at the K-12 level. This diagram is especially useful in discussions of the current K-12 mathematics curriculum.

Figure 3

The six steps illustrated are 1) Problem posing; 2) Mathematical modeling; 3) Using a computational or algorithmic procedure to solve a computational or algorithmic math problem; 4) Mathematical "unmodeling"; 5) Thinking about the results to see if the Clearly-defined Problem has been solved,; and 6) Thinking about whether the original Problem Situation has been resolved. Steps 5 and 6 also involve thinking about related problems and problem situations that one might want to address or that are created by the process or attempting to solve the original Clearly-defined Problem or resolve the original Problem Situation.

Final Remarks

Here are four very important points that emerge from consideration of the diagram in Figure 3 and earlier material presented in this section:

1. Mathematics is an aid to representing and attempting to resolve problem situations in all disciplines. It is an interdisciplinary tool and language.

2. Computers and calculators are exceedingly fast, accurate, and capable at doing Step 3.

3. Our current K-12 math curriculum spends the majority of its time teaching students to do Step 3 using the mental and physical tools (such as pencil and paper) that have been used for hundreds of year. We can think of this as teaching students to compete with machines, rather than to work with machines.

4. Our current mathematics education system at the PreK-12 levels is unbalanced between lower-order knowledge and skills (with way too much emphasis on Step #3 in the diagram) and higher-order knowledge and skills (all of the other steps in the diagram). It is weak in mathematics as a human endeavor and as a discipline of study.

There are three powerful change agents that will eventually facilitate and force major changes in our math education system.

Brain Science, which is being greatly aided by brain scanning equipment and computer mapping and modeling of brain activities, is adding significantly to our understanding of how the brain learns math and uses its mathematical knowledge and skills.

Computer and Information Technology is providing powerful aids to many different research areas (such as Brain Science), to the teaching of math (for example, through the use of highly Interactive Intelligent Computer-Assisted Learning, perhaps delivered over the Internet), to the content of math (for example, Computational Mathematics), and to representing and automating the "procedures" part of doing math.

The steady growth of the totality of mathematical knowledge and its applications to representing and helping to solving problems in all academic disciplines.

Source: http://pages.uoregon.edu/moursund/Math/mathematics.htm

Survival mathematics

It ma y be helpful to distinguish three categories of mathematics. First, there is survival mathematics: that is, the mathematics that we need in order to go about our daily business and make good use of our leisure time. Some people refer to this as 'the basics' or 'the core curriculum'; but this seems to imply that these needs are the same for everybody, which is clearly not true. City dwellers use different mathematics from those who live in a village; a lawyer's mathematical needs are different from those of a housewife (though neither would admit to 'using mathematics' in their work); if your hobby is photography, you want different mathematics from a person who plays football. Survival mathematics is a reflection of our personal life-style.

And yet it has certain common features for all of us. First, we almost always have to use it in a situation that requires an immediate response: paying a bus fare, deciding where a tree is going to fall, estimating the date for the completion of a contract, getting each dish in the oven at the right time, choosing the right camera exposure, positioning oneself to intercept an attack by the opposing forwards. Second, it is rarely carried out with paper and pencil (or even with a pocket calculator). Third, one is hardly aware that one is using mathematics at all. And this means that survival mathematics has little to do with formal mathematical instruction. The very process of taking a problem out of a textbook in a lesson called mathematics, and writing the answer in an exercise book in one's own time, makes it largely an irrelevance as far as survival mathematics is concerned. This does not mean that mathematics teachers cannot help children to acquire the mathematics they need. But it is an illusion to suppose that this can be left to mathematics

teachers alone. Other teachers, parents, elder brothers and sisters all have a part to play. In this sense, every teacher must expect to be a teacher of mathematics. For the most part, survival mathematics comes like most other survival skills—such as crossing the road, reading a ma p or telling the time—by experiment, using the experience of any older person who happens to be on hand to help.

Mathematics for use

Next, much of the mathematics in the school curriculum is mathematics for use. This extends from quite simple skills, such as decimal arithmetic, up to advanced topics such as the use of differential calculus to find maximum and minimum values. It describes all the mathematics that some people need in order to do their work successfully, over and above what we have already described as survival mathematics. T h e difficulty with most of the mathematics in this category is that it is job-specific; only a minority of people will ever use any particular piece of mathematics. For example, engineers and navigators obviously need to know some trigonometry, a subject that is of no use whatsoever for pharmacists and bank employees. Economists need to understand statistics, but not electricians. And, of course, few children at school can be sure what work they will do in later life. This presents us with a curricular problem: should we try to teach every mathematical topic that might be needed later by some member H o w important is learning mathematics? Of the class? Since amongst thirty or forty children we ma y find a wide variety of career possibilities, this would be a sure recipe for an overloaded curriculum. Or should we restrict ourselves to some general topics—such as proportion, the properties of some common geometrical figures, and substitution in formulae—with which man y of the pupils will need an acquaintance? If we adopt this latter course, we ma y find ourselves left with rather a small mathematics curriculum—especially since, as some recent research carried out in England has shown, 1 most employees use much less mathematics in their jobs than is commonly believed. And a corollary of this policy would be the need to incorporate more mathematics into subsequent specialized vocational training than we do at present. Of course, mathematics is also an essential tool for the scientist, and this has often been used to justify the inclusion of particular mathematical topics in the curriculum. Certainly it is desirable to have an interdisciplinary perspective in designing any mathematics curriculum. But this is an argument which can easily be carried too far. The usual assumption is that pupils should first learn the mathematics, and then apply it in the science lesson. However, if this means that they are expected to learn it in an abstract form, divorced from the context which could give it meaning for the pupils and before they have the requisite conceptual background, they ma y well fail to master it; and the failure in mathematics can lead to frustration in the science lesson as well. Much science teaching in schools is too dependent on mathematical skills; for man y pupils these can get in the way of learning the science. We also need to recognize that mathematics for use is something which changes with time. An obvious example is calculation with logarithms, which until quite recently was an essential accomplishment for anyone who had to carry out complicated calculations. Nowadays, when every person who has the need to do such calculations is likely to have access to a pocket calculator, this skill has become almost obsolete.

Mathematicians' mathematics

There are some who would claim that so far there has been nothing in this article about mathematics. 'Real' mathematics, they would argue, is about definitions, proof and abstract structures. Most curricula contain something of this kind of mathematics: for example prime numbers, geometrical theorems, and sets. We might call it mathematicians' mathematics. It would be wrong to imagine that a hard line should be drawn between this and the mathematics referred to previously. These is certainly a place for logical reasoning in teaching mathematics from a utilitarian point of view; for much of the power of mathematics lies in the connection between facts, so that a little remembered knowledge can produce a large amount of derived knowledge. If mathematics is worth its place in the curriculum, it should certainly be learnt in such a way as to bring out these relations. There are also other aspects of mathematicians' mathematics. Think of the pleasure which man y people get from solving mathematical puzzles and playing games with a mathematical structure; or the sense of personal achievement which can result from investigating number patterns (for example, that the answers to the sums i+2+i, 1+2+3+2+1 , 1+2+3+4+3+2+1 , etc., are respectively 4, 9, 16, etc., the successive square numbers), with their possibilities for making conjectures about the way in which the pattern continues, and even of trying to explain why . The n there are parts of mathematics which can only be described as 'delightful'. Quite small children enjoy trying to count as far as they can, by extending the numeration system (c . . . eighty . . . ninety . . . ninety-nine . . . twenty (!).. . twenty-one (!!) . . .'); and man y people can appreciate the fact that any map , however complicated, can be colored with just four crayons, or that in the figure shown here (Fig. 1) the three points marked with dots lie in a straight line wherever the six points marked with small rings are taken on the circle—though establishing these facts calls for advanced mathematics. It is not difficult to argue that all children should have a chance to experience this kind of mathematics, although by itself it could hardly justify the central place in the school curriculum that the subject currently enjoys. There is also the argument that learning this kind of mathematics 'teaches one to think'. But the evidence for this assertion is unconvincing. It is of the nature of mathematics that its field of operation is very narrow, but that within this field one is tied by a strict logical regime; the forms of reasoning encountered in mathematics are rarely applicable in a wider context. Certainly learning mathematics gives practice in analyzing the meaning of statements, marshaling evidence, discarding what is irrelevant, and so on; but so does learning a language, or studying a novel, or making sense of a political situation. Mathematics ma y exhibits the purest form of reasoning, but from an educational point of view this can be regarded as its weakness as well as its strength.

Source: http://unesdoc.unesco.org/images/0005/000524/052474eo.pdf