Issue Number 290
September 30, 2020

This free

All back issues of the newsletter and subscription information are available online. A number of the newsletters are available in Spanish on the AGATE website mentioned above. Dave Moursund’s book,

I am currently writing a book tentatively titled

Introduction
to ICTing and Mathing

Across the History Curriculum

What Is Mathematics? Part 1

Across the History Curriculum

What Is Mathematics? Part 1

David Moursund

Professor
Emeritus, College of Education

University
of Oregon

“God [nature] created the natural
numbers. All the rest is the work of man [humans].” (Leopold
Kronecker; German mathematician and logician; 1823-1891.) [Content
in square brackets added by David Moursund.]

Introduction

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 Numbers

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.

Expanding the
Definition of *Number*

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 a*zero*
-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.

History of
Mathematics

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.

A
Vertically-structured Discipline

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.)

Mathematics and
Problem Solving

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 have written extensively about the general topic of problem solving, as well as teaching/learning math problem solving. Almost all of these documents are available free on the IAE website. (Moursund, 2020, 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. (Moursund, 9/13/2020, link.)

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.)

Mathematics and
Measurement

“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.)

Final Remarks

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.

References

Over the years, I have taught and written extensively about math
education and problem solving. Most of what I have written on
these topics is available on the IAE website. The *IAE-pedia*
currently has 291 entries. (IAE Main Page, 2020, link;*
IAE-pedia *Most Popular Entries, 2020, link.)
Nearly half of the 20 most popular entries are about math
education. The IAE website also contains a more extensive list of
my math education writings. (Moursund, 2020, link.)

Author

**David Moursund** is an
Emeritus Professor of Education at the University of Oregon, and
editor of the *IAE Newsletter*. His
professional career includes founding the International Society
for Technology in Education (ISTE) in 1979, serving as ISTE’s
executive officer for 19 years, and establishing ISTE’s flagship
publication, *Learning and Leading with
Technology* (now published by ISTE as *Empowered
Learner*). He was the major professor or co-major
professor for 82 doctoral students. He has presented hundreds of
professional talks and workshops. He has authored or coauthored
more than 60 academic books and hundreds of articles. Many of
these books are available free online (IAE Books, 2020, link.)

Moursund founded Information Age Education (IAE) in 2007. IAE
provides free online educational materials via its *IAE-pedia*, *IAE
Newsletter*, *IAE Blog*, and IAE
books. Information Age Education is now fully integrated
into the 501(c)(3) non-profit corporation, Advancement of
Globally Appropriate Technology and Education (AGATE) that was
established in 2016. David Moursund is the Chief Executive
Officer of IAE and AGATE (IAE, 2020, link;
AGATE, 2020, link.)

Email: moursund@uoregon.edu

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