Introduction to ICTing and Mathing Across the History Curriculum What Is Mathematics? Part 1

When I began writing this newsletter, I thought it would be easy to answer the question, What is mathematics?. After all, I have a doctorate in mathematics and have had many years of experience in teaching a wide range of students, doing math  research, and writing in this field. But, I quickly found that it is not easy to provide an answer, or even pieces of an answer, that will be understandable and satisfactory to a very broad range of people.

Each of you, my readers, has had years of math coursework during your formal schooling. Each of you routinely uses math in your everyday life. Each of you has your own understanding and answer to the What is mathematics? question. I hope you will enjoy and benefit from my answers given in this and the next IAE Newsletter. I certainly have enjoyed and benefitted from writing these two newsletters!

The natural (counting) numbers 1, 2, 3,… are the foundation of mathematics, and a good start in answering the What is mathematics? question. Mathematics is a human endeavor. Here on earth, humans have created all of mathematics beyond the natural numbers. They have been at this task for many thousands of years.

Of course, it is not possible to pinpoint when humans and/or prehumans first had oral language adequate to name and make use of the counting numbers. But we do have some archeological evidence.

Lebombo bone is a baboon fibula with incised markings discovered in the Lebombo Mountains located between South Africa and Swaziland. The bone is between 44,230 and 43,000 years old, according to two dozen radiocarbon datings. According to The Universal Book of Mathematics the Lebombo bone’s 29 notches suggest “it may have been used as a lunar phase counter, in which case African women may have been the first mathematicians, because keeping track of menstrual cycles requires a lunar calendar.” But the bone is clearly broken at one end, so the 29 notches can only be a minimum number. (Wikipedia, 2020, link.)

I think that the natural numbers are a good place to start my answer to the What is mathematics? question. The first of the natural numbers is named one in the English language, and has different names in other languages. It is represented by the symbol 1 in many different parts of the world. It may seem somewhat silly, but I looked up 1 on the Web.

1 (one, also called unit, and unity) is a number, and a numerical digit used to represent that number in numerals. It represents a single entity, the unit of counting or measurement. For example, a line segment of unit length is a line segment of length 1. (Wikipedia, 2020, link.)

Next, we need the idea of addition, which is the process or skill of calculating the total of two or more numbers or amounts. The symbol + is widely used to designate this process. It is a major step forward in a young child’s math education to develop an understanding of oneness and addition. With these human-created ideas, mathematicians can define the natural numbers. Here is a definition:

The number 1 is a natural number. If the letter A designates a natural number, then A + 1 is a natural number.

This is a rather simple idea, but is stated in the vocabulary of mathematics. Thus, because 1 is a natural number, 1 + 1 (which is named 2) is a natural number. Because 2 is a natural number, 2 + 1 (which is named 3) is a natural number. The sequence goes on and on. It has an uncountable (an infinite) number of entries.

Why do we need such a definition? A partial answer to the What is mathematics? question is to understand that math is built on a solid foundation of definitions, theorems, and proofs beginning with the natural numbers.

Here is an example. You know that 2 + 2 = 4. Are you sure? How does a mathematician prove this, starting from the definition of a natural number?

Theorem: 2 + 2 = 4.

Proof: From the definition, 2 = 1 + 1. Thus, 2 + 2 = 2 + 1 + 1. From the definition of 3, 2 + 1 is 3. So now we have 3 + 1, which (by definition) is 4.

This simple example represents a huge amount of progress in developing the discipline of mathematics. It also illustrates the idea of counting on as a way to do addition. To add two integers, start with one (preferable the larger) and do a simultaneous process of adding by one and counting up to the second quantity. (First Grade Frame of Mind, 2020, link.)

Of course, we do not begin the math education of students by using such definitions, theorems, and proofs. Instead we begin by having students learn about ideas such as integer numbers, counting, addition and subtraction, and so on. Math educators agree that a major goal in math education is for students to develop number sense:

“…an intuitive understanding of numbers, their magnitude, relationships, and how they are affected by operations.” Other definitions of number sense emphasize an ability to work outside of the traditionally taught algorithms, e.g., “a well-organized conceptual framework of number information that enables a person to understand numbers and number.” (Wikipedia, 2020, link.)

A similar type of statement holds true for more advanced math. In summary, math is a discipline of study built on the natural numbers. Informal and formal math education help people to develop math sense. A combination of math content knowledge, math problem-solving skills. and number/math sense help people to solve a very broad range of problems across a broad range of disciplines of study.

The creation and widespread use of integers was a great human achievement. But, an expansion of what we mean by a number was needed.

For example, suppose I have one apple. I cut it into two equal-sized pieces. I need some vocabulary to describe the result. I create a name, such as one-half, meaning one of two (equal) parts. That is, I divide the number 1 by the number 2 and agree (define) that the result is also a number. Wow! I have created a new type of number. As a mathematician, I then generalize what I have done.

Definition: A fraction is a number produced by dividing one integer number by a larger integer number. The words numerator and denominator are used to name the two parts of such a fraction.

Now students face the challenge of not only learning how to do arithmetic on integers, but also on fractions. There is much to learn and understand. For example, 4 + ½ is a number. We name the result four and one-half and write it as 4½. Rational numbers is a name given to the totality of numbers produced by the addition, subtraction, multiplication, and division of elements of this (expanded) collection of numbers.

This expansion of the natural numbers into the rational number system occurred before the invention of negative numbers and the number zero. Both of these major developments in mathematics came much later in the history of mathematics.

Zero’s origins most likely date back to the “fertile crescent” of ancient Mesopotamia. Sumerian scribes used spaces to denote absences in number columns as early as 4,000 years ago, but the first recorded use of azero -like symbol dates to sometime around the third century B.C. in ancient Babylon. (History Staff, August 22, 2018, link.)

During the 7th century AD, negative numbers were used in India to represent debts. The Indian mathematician Brahmagupta, in Brahma-Sphuta-Siddhanta (written c. AD 630), discussed the use of negative numbers to produce the general form quadratic formula that remains in use today. He … gave rules regarding operations involving negative numbers and zero, such as “A debt cut off from nothingness becomes a credit; a credit cut off from nothingness becomes a debt.” He called positive numbers “fortunes”, zero “a cipher”, and negative numbers “debts”. (Wikipedia, 2020, link.)

What about the number π (pi)? Its history goes back nearly 4,000 years. I find it interesting that some people were working to find the circumference and area of a circle that long ago. But, it is much more important that π has the same value throughout the world, in the past, now, and in the future.

This same observation applies to the accumulated math research results. In essence, once a mathematical theorem is carefully stated and proved, it becomes a known fact that is the same throughout the world and on into the future. Contrast this with facts in history and many other disciplines.

Math has a long history. When the Sumerians developed written language in about 3,500 BC, the first schools they established taught reading, writing, arithmetic, and history. (Rank, n.d., link.) Back in those days, Sumerian students were faced by arithmetic in a base 60 number system. (Remember, we still have 60 seconds in a minute and 60 minutes in an hour.) Probably while you were in school you memorized the multiplication facts up to 10 times 10, or even 12 times 12. I wonder if any Sumerian students memorized the multiplication facts up to 60 times 60.

From these early days on to the present, the learning and teaching of math has been a challenge to many students and teachers. Quoting Plato from about 2,500 years ago:

There still remain three studies suitable for freemen. [Slaves did not get to go to school.] Calculation in arithmetic is one of them; the measurement of length, surface, and depth is the second; and the third has to do with the revolutions of the stars in reference to one another.… (Plato; Athenian philosopher during the Classical period in Ancient Greece, 428/427 or 424/423-348/347 BC.) [Comment in square brackets added by David Moursund.]

This assertion by Plato moves us to the idea that measurement is an important area of study, and that it can be considered part of math. This topic is discussed later in this newsletter. Moreover, he seems to be suggesting that the astronomical observation that the stars seem to revolve about the earth might well be studied through the use of math.

Plato was not trying to answer the question, What is mathematics?. However, Plato named three major types of math-related problems relevant to students of his time. Aha! Math is about problem solving. I will return to that topic later in this document.

Math is a vertically-structured discipline of study, and the totality of known mathematics is growing steadily. By vertically structured, I mean that new results, understanding, and uses of math are built on the past results, understanding, and uses.

One way to measure the growth in mathematics is to examine the number of articles being published in math journals. In a 2019 article, Edward Dunne estimated that well over a hundred thousand math research articles were published in 2017.

Mathematical Reviews has been indexing and reviewing the research literature in mathematics since 1940. We have collected a considerable amount of information about this corpus over the years. As of this writing, the database contains roughly 3.6 million items and profiles for over 900,000 authors.

Counting the number of items indexed by Mathematical Reviews per year from 1985 to 2017, the number of new articles per year is well modelled by exponential growth at a rate of about 3% percent per year. Counting just journal articles, the rate is about 3.6%. That rate has a doubling time of just over 19 years. (Dunne, February 2019, link.)

Here is a definition of mathematics from George Polya:

“Mathematics consists of content and know-how. What is know-how in mathematics? The ability to solve problems.” (George Polya; Hungarian math researcher and educator; 1887-1985.) [Bold added for emphasis.] (Polya, circa 1969, link.)

I believe Polya’s content and know-how statement is applicable to any discipline of study. Each academic discipline or area of study can be defined by a combination of general things such as:

• The types of problems, tasks, and activities it addresses. (In the remainder of this document I will use the term problem to mean both problems and tasks.)
• Its accumulated accomplishments such as results, achievements, products, performances, scope, power, uses, impact on the societies of the world, and so on, and its methods of preserving and passing on this accumulation to current and future generations.
• Its history, culture, and language, including notation and special vocabulary.
• Its methods of teaching, learning, assessment; its lower-order and higher-order knowledge and skills; and its critical thinking and understanding. What it does to preserve and sustain its work and pass it on to future generations.
• Its tools, methodologies, and types of evidence and arguments used in solving problems, accomplishing tasks, and recording and sharing accumulated results.
• The knowledge and skills that separate and distinguish among: a) a novice; b) a person who has a personally useful level of competence; c) a reasonably competent person, employable in the discipline; d) an expert; and e) a world-class expert.

Notice the emphasis on solving problems, accomplishing tasks, producing products, doing performances, accumulating knowledge and skills, and sharing knowledge and skills. Because these ideas about problem solving cut across all disciplines of study, there can be significant interdisciplinary transfer of learning as students study various disciplines. However, this tends not to occur unless explicit instruction on problem solving is routinely provided to students. Moreover, this observation helps us to examine the various discipline areas that are taught as requirements and electives in our traditional precollege educational systems.

Take reading and writing as an example. Reading provides access to much of the accumulated knowledge of the human race. We can make use of such accumulated information to solve problems and answer questions. Writing is an aid to our brains as we attempt to solve problems. Reading and writing have proven to be such powerful aids to the human brain that students are required to study and practice using these skills through years and years of schooling.

Speaking specifically about the discipline of mathematics, reading allows people to access accumulated results in mathematics, and writing is an aid to the brain in using (applying) math knowledge and skills to solve or help to solve a wide variety of problems. Imagine the challenge of mentally doing multidigit multiplication and division. A few people have mastered this task, but not very many.

But, here is an aha! moment. For many centuries, people have worked on developing mechanical aids to doing arithmetic. The UNIVAC computer, developed in 1951, was the world’s first commercially available electronic digital computer; it could perform a thousand calculations per second. In those days, that was considered to be a major achievement. Today’s (2020) fastest super computer is more than 400 thousand billion times that fast! For those of you who like large numbers, this one computer has the computational power equal to 50,000 of the first commercially available computers for each person on earth! Clearly math is much more than being skilled at pencil-and-paper arithmetic calculations. (McKay, 6/22/2020, link; History, n.d., link.)

“To measure is to know. If you cannot measure it, you cannot improve it.” (Lord Kelvin; William Thomson; Scots-Irish mathematical physicist and engineer; 1824-1907.)

Being able to count is a useful skill. It is closely related to being able to measure or quantify various things. Measurement is a major source of math problems that have driven the development of mathematics over the centuries. Consider the statements:

• I am 24 years old.
• My dog weighs 24 pounds.
• The next town is 24 miles down the road.
• That rectangle has an area of 24 square inches.
• I have $24 in my wallet.

The number 24 is the same in each of these sentences. But each sentence makes use of a different unit of measure. In each case, the unit of measure is essential to the meaning of the sentence. Hmm. I imagine that very early on people understood the concept of a year. (But, early on they thought of it as being 365 days in length. A more precise measurement is 365.2422 days. That explains why we have leap years and leap centuries.) But, what about a pound, a mile, a square inch, or a dollar? Astronomy does not help us there.

I like to think of math as being divided into two major contents: pure math and applied math. My process of doing this is overly simplistic, but it seems to me to be a good way of thinking of math and its applications. When I do an addition or multiplication of two natural numbers, I am doing pure math. When I attach units to the numbers being added, I am doing applied math. I can add 24 pounds to 24 pounds and get 48 pounds. But, it makes no sense to add 24 pounds to 24 square inches. The use of units adds meaning (sense) to the numbers being added and is an aid to detecting errors in doing/using math.

The history of measurement is full of stories of attempts to develop widely accepted definitions of many different units of measure. In a local kingdom, the king could state that a foot was the length of his own foot. But, this means the length named foot varies over time and location. The history and science of weights and measures is interesting, long, and important. (History World, n.d., link.) Through creation of the metric system, the world has made considerable progress in developing precise definitions for commonly used units of measure, and considerable progress has been made in worldwide acceptance of these definitions. (Wikipedia, 2020, link.) However, instead of the metric system, those of us living in the United States still use major components of the British Common System that was adopted when the United States was established.

The history of measurement is also replete with examples of progress in mathematics related to attempting to solve applied math problems. Consider the math used in doing surveying. A king might want to tax his land owners based on the area of the land they own. How does one find the area of a circular plot of land, or a plot where one edge is circular?

Another problem that interested mathematicians quite early on is finding the length of one of the sides of a right triangle (a triangle containing a 90 degree angle), when the lengths of the other two sides are known. Sometimes a person becomes known for solving such problems. More than 2,400 years ago Pythagoras proved that, for a right triangle, if A and B are the lengths of the two shorter sides, and C is the length of the longer side, then A2 + B2 = C2. (This problem was solved by others much earlier, but perhaps Pythagoras had a better press agent.)

I hope that I have provided you with a good start on increasing your insights into the What is mathematics? question. In brief summary, I have touched on:

• The natural numbers as a starting point in the development of mathematics.
• Math definitions, theorems, and proofs.
• Number sense and math sense as key aspects of learning and using mathematics.
• Problem solving.
• Measurement as a driver of math use and the development of new math problems.

The next newsletter delves into a number of other aspects of mathematics including the language of mathematics, math roles in the creation of College and University Computer Science Departments, math modeling, more about math education, and a final touch of math humor.