Introduction for 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. 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 (a
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 math in art, dance, and music.
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.
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
above three categories. 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.
Beauty in Mathematics
G.
H. Hardy
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 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.
1. A prime number is a
positive integer greater than 1 whose 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.
