The Current State of Affairs in the
Teaching of Mathematics
Mathematics is the most important
tool ever invented by man. The history of Mathematics and its impact on human
society and culture is the best testimony to this fact. For, it is the
refinements of Mathematical ideas which provides the essential turning points
in the history of mankind, giving rise not only to the enhancement of the
existing tools but also the development of new ones in different spheres of
human activities, such as architecture, astronomy, engineering, science,
physics, chemistry, biology, geology, agriculture, economics, technology,… The
manifestation of this can be seen or witnessed in the finest monuments ever
built, invented, or created by man, such as in the Pyramids of Giza, in Egypt,
in the Palaces of Alhambra in Granada, in Moorish Spain, in the majestic
buildings and monuments of the city of Isfahan, in Iran, in the music scores of
Johann Sebastian Bach (1685-1750), Wolfgang Amadeus Mozart (1756-1791), Ludwig
von Beethoven (1170-1827), in the development and refinements of micro electronic
on the silicon chip, in the construction of Particles Accelerator in Geneva,
Switzerland.
Mathematics acts, so to speak, as a
magnifying glass, enabling man to visualize and penetrate into the structure of
atoms, to provide theoretical description in the behaviour of subatomic
particles in Quantum and Nuclear Physics. It enables man to explore the space,
in the Mathematics of Johannes Kepler (1571-1630), in the Mechanics of the
universal gravitation force and the laws of motion of Sir Isaac Newton
(1642-7127), in the infinitesimal Calculus of Gottfried Leibniz (1648-1716) and
Sir Isaac Newton, giving rise to the Mathematical description of the universe,
its birth, its geometry, its topology. It provides theoretical description of
the emergence of matter and anti-matter and their transformation back into
energy in the Theoretical Physics and Astrophysics, in the equations of Albert
Einstein (1879-1955) of the Special and General Relativity, enabling man to
harness, regardless of its devastating consequences, the atomic energy.
Mathematics enables man to probe into the origin and chemistry of life, here on
earth, or anywhere in the universe, to discover the molecular structure of DNA,
providing the foundation for genetic engineering in the production of crops, in
finding cures for diseases.
Arithmetic starts with the set N
of Natural
or Counting numbers, {1,2,3,4,…}. Counting is not unique to man’s ability.
Other species can count. That is to say, they have developed the ability to
differentiate between 1 or 2 or 3, … collection or set of objects in their
habitat, in their struggle for survival, in their adaptation to their
environment in the Process of Natural Selection.
In the history of Mathematics, the
emergence of nothing or zero (0) is relatively recent. In the Roman Numerals
zero does not exist. It does not manifest itself. Hence in the Roman
arithmetic, it is quite cumbersome to count, add, multiply and provide
numerical representation, especially for very large or very small numbers. This
limitation of Arithmetic dissipates and disappears, the moment 0 (zero), the
concept of nothing prevails in human culture, giving rise, not only to the
arithmetic in the decimal system, in base 10, with ten Arabic symbols or
numerals, but also, to other number bases, such as the binary system, base 2,
with two symbols, 0, and 1, which provides the foundation for the digital
systems in the micro computers, and the Hex Decimal, base 16, which provides a
concise representation of 4 bits in the only language that the machines can
understand, i.e., a sequence of 0s and 1s.
In the Algebra of solving equations,
the set W of Whole numbers, {0, 1, 2, 3,...}, are
extended to the set I of Integers, i.e., positive and the
negative whole numbers, which in turn are extended to the field of Rational
numbers. The Mathematical structure of the field of Rational numbers remains
incomplete until the introduction of irrational numbers, leading to the
Algebra of the Real numbers. The Real numbers are in turn extended to
the field
of Complex
numbers.
Mathematics even enables and
facilitates man to formally define the grammar of natural languages, an insight
into which give rise to 3rd and 4th Generation of
Computer Programming Languages (GL), such as C in the development of UNIX
Operating System at Bell Lab, C++, Java, COBOL in the further development and
refinement of Operating Systems and application software in software
engineering, Prolog (Logic Programming) and its application to Medical and
Hardware Diagnostics, Lisp and its application to the field of Artificial
Intelligence (AI), in the robotics and automation of processes of production. They
provide the bases for the design and the development of Relational Databases,
Computer Aided Design and Manufacture, the design of Micro Processors in VLSI,
refinement of Man Machine Interfaces, such as graphical user interfaces, Speech
Recognition and Image Processing. These are only some of the fine examples of
the application of 3 GL and 4 GL programming languages in software
engineering.
What I am really trying to say is
this. Without good and in depth knowledge of Mathematics, one is not equipped
and is not well prepared for the challenges of every day life. One is unable to
make a positive contribution to the welfare of human race and the advancement
of the society.
In the human biology, Mathematics
starts at the very moment that egg and sperm unite. i.e. , the moment the human
egg is fertilized and embryo is formed, whether in the womb of a female during
the artistic act and expression in the sexual reproduction, or in the artificial insemination in a test tube, in a lab. One
and one unite; hence 1 + 1 gives rise to 1. Cell division starts, 1, 2, 4, 8,
16, 32,… In the discrete Mathematics of powers of 2; in the functional
relationship or mapping of Whole numbers, {0, 1, 2,...}, into its proper
subset, {1, 2, 4, 8, 16, 32,…, 2^(p – 1)}, closed under the operation of
multiplication of the modular arithmetic of p (p assumed to be a prime number),
after approximately 9 month, this process of cell division, comes relatively to its end, culminating in
the birth of human kind, a male or a female. This act of sexual reproduction is
to go on, so to speak, forever, until the doomed day.
In modern astronomy, in cosmology,
Mathematics starts with the birth of the universe, i.e., it starts with the act
of creation at the moment of the big bang, about 14 billion years ago, and so
it is estimated. The energy released, transforms into matter, subatomic
particles. An electron and a proton unite, forming the simplest atomic
structure, H, an atom of Hydrogen.
Hydrogen burns in the furnace of a star, say, in the Sun in our solar
system, in the Milky Galaxy, in the nuclear fusion to form Helium. An electron
and a proton unite, giving rise to an atom of Hydrogen. Hydrogen fuses with
Hydrogen to form Helium. This nuclear process goes on for millions of years,
culminating in the formation of the organic chemistry of amino acids, the basic
constituents of proteins, leading to the development of life on the planet
earth or perhaps on some other planets elsewhere in the universe.
Mathematics begins with a set of
definitions or axioms. The concept of set, i.e., a well-defined collection
of objects, with its Mathematical operations, i.e. the algebra of sets, forms
the foundation of Mathematics. Abstract algebra is the study of the set of
objects together with certain underlying binary operations, satisfying a set of
axioms, i.e., groups, rings, fields, and vector
spaces.
To start with, the set N
of Natural
numbers is closed under the operations of addition (+) and multiplication (x).
That is to say that, the addition and multiplication of two Natural
numbers is a Natural number, i.e., they are binary operations. The
set I
of Integers
is a group
under the binary operation of addition (+),
but not under multiplication (x).
The set Q of rational numbers is an ordered
field with respect to the underlying binary operations of addition (+) and multiplication (x). Naturally this also applies to the
set R
of Real
numbers, which is a proper subfield of the set C
of Complex
numbers. However there are no order relations between the members of the field
of Complex numbers.
In the study of Groups in the Abstract
Algebra, the set P of Prime numbers, {2, 3, 5, 7,…}, play a
crucial role in the algebra of groups of finite sets, i.e., in the
modular arithmetic of the prime numbers. The finite
groups play an important role in other scientific fields, such as in
the study of atomic structures, in the classification of the structure of
crystals, in cryptography. Prime numbers
play also an important role in many spheres of human scientific endeavour, such
as encryption of data, sent over a wire, sent through World Wide Web.
The algebra of functions under the
operations of addition and multiplication over the domain and the co-domain,
say, of the set of Integers, Rational, Real, and Complex
numbers, with an underlying topology or a metric or a distance,
forms the foundation of Mathematical Analysis. It underlies
the theoretical development of the limits, the study of convergence of
sequences, infinite series, the continuity of functions, derivatives, and the
theory of integration, point-wise, and uniform convergence of functions in the Real
and Complex Analysis. It forms the foundation for the study of linear
functions or operators under the domain and co-domain of norm or topological vector
spaces in the linear algebra, such as the study of Matrices, differential, and
integral operators over the norm vector spaces. It provides the foundation for
the study of general topological vector spaces of functions, of probability
spaces, which forms the Mathematical foundation of Quantum Physics, the
philosophical and theoretical discussion of the origin of the universe, its
past and its future, its coexistence with possibly other universes, in our
perception and understanding of the space and time, not only in the flat or
Euclidean geometry, but also in Non-Euclidean topological of the spaces and
time.
Mathematics is easy to master. Why is
it then that our youngsters not only struggle with the subject but also develop
a sense of inability, a sense of insecurity, learning to dislike the most
important subject in their educational career?
Why is it that given our finest educational establishments, our
universities, which attract the finest students in the subject, they still fail
in their endeavour to return to the society a large number of able
Mathematicians? How tragic it is when one learns or hears that in an act of
desperation, a student of Mathematics has committed suicide, often because of avoiding
failure, at some finest universities around the glob.
In my humble opinion, the matter is
quite simple. To teach any subject, for that matter, it requires two sets of skills:
a)
Expertise in the subject matter;
b)
The ability to transfer knowledge to the learners. This
requires a good insight into and understanding of social relations, of
sociology of education, of the child developmental psychology.
These two sets of knowledge and skills
not only are necessary, but also sufficient for teaching any subject.
Mathematics happens to be the most
important subject in the curriculum and the most difficult one to teach.
Mathematical concepts are developed, as I have endeavoured to highlight here,
at the early age. Therefore if the teaching of Mathematics is inadequately at
the very outset, it leads to the development of misconceptions in the learners.
Hence they become more difficult to diagnose and rectify at the later age.
In my opinion, the current shortcomings
in the teaching of Mathematics and the current state of affairs in Mathematical
education arise from this:
Those who teach Mathematics at
schools, in spite of their sincere efforts and their hard work, are not really Mathematicians,
and therefore they lack the first prerequisite, i.e., having the expertise in
the subject matter. Those who teach Mathematics to the students of Mathematics
at universities, i.e., Mathematicians, with the exception of some, lack
teaching skills, i.e., unable to transfer effectively their knowledge of the
subject to their students. Moreover, there is no continuity in the teaching of
the subject from nursery schools to primary ones, from primary schools to
secondary ones, and from secondary schools to universities.
To solve the problem, naturally it
requires not only sincere efforts and willingness on the part of the educational
specialists and educational establishments, in a collective fashion, but also
enough allocation of resources by the state.
The budget allocated these days to
education in the advanced industrial countries often forms a major component of
their Gross National Product (GNP). Yet, I regret to say that a great deal of
it goes to waste. In the final analysis, this is due to the crises of
leadership or management which manifest itself at all levels of social
organisations, in all spheres of social relations of production, at all levels
of micro and macroeconomics.
It is my contention here to establish
that there is a solution to the current state of affairs, at least with regards
to the teaching of Mathematics.
For all my sins, since September 15,
1971, I have lived primarily in the United Kingdom, reading and teaching, among
other subjects, Mathematics to all age groups and abilities. Prior to September
1971, I primarily lived in Iran, doing the same.
In the United Kingdom, I have tried
in my own limited capacity, to provide at least some solutions to the problems
associated with the teaching of Mathematics, and I have done so, at various
stages of my academic career, be it at schools, FE colleges, universities, or
industry. Alas, in my personal
experience, British people, with the exception of course of some individuals and
academia, for some historical reasons, have shown very little interest in what
I may have to offer.
In conclusion, the reason I was inclined
to write this short essay is simply this. I am, metaphorically speaking,
sitting on a gold mine. In order to exploit it, I need the support and collaboration
of many individuals, academia, academic, as well as non-academic organisations.
Therefore I am writing to invite, so to speak, all those who are concerned and
wishing to do something, even though it may be limited in its outcome in
providing some solutions to the problems associated with the teaching of
Mathematics.
Shabpar, Majid
Tehran, 16th April 2012
CC:
Professor David Larman, University
College London
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