USE OF MATHEMATICAL EXPRESSIONS ‐ CHALLENGES IN ECONOMICS CLASSROOMS Ashita Raveendran, Lecturer, DESSH, NCERT, New Delhi Mathematics definition as a ‘Science of order’ (White head (1929)) points on to its features‐ generalisation, simplification, representation, communication, and application and determines its wide applications in economics and other social sciences. It has been used everywhere in the world, for many centuries as a universal language. Mathematics acting as a language parsimoniously and prolifically helps in meeting the purpose of economics. Economics is often concerned with variables, which are measurable and involve study of relationships between them that can be expressed by means of functions and curves for a convenient understanding of the underlying concepts. This necessitates as well as facilitates the use of mathematics. The paper discusses the relevance of mathematics in Economics illustrating certain ideas and theorems in economics using mathematical expressions. With various illustrative examples, the paper states how the basic relationships in economics theory can be put in the form of functions and equations there by learning economics becomes easy, smooth and systematic. Mathematical approach is rightly considered and a quick mode of transportation but it is essential to accustom oneself with certain basic mathematics and their applications while driving lessons to elucidate the problems of economic theory. The new textbooks of class XII, Introductory Microeconomics and Introductory Macroeconomics, have incorporated the recent development in the theory aided by a mathematical treatment of the issues involved. The paper discusses on the difficulties encountered by the students while applying mathematical expressions in learning economic theories. Students face a variety of challenges including low level of confidence with respect to mathematical ability and limited fluency in algebraic notations. They also have limited understanding of the usefulness of mathematical principles in economics argument and have difficulties in recognising the mathematical representation of a problem within economic theory. Students anxieties and difficulties are widely acknowledged, but teachers anxieties are not often discussed. Anxious teachers and students do not contribute to a satisfying or productive educational atmosphere (Fort, 1995).Another problem is that a teacher may be too understanding of student’s difficulties or other negative altitude. Teachers may be guided on dealing with the situation when a student clearly does not get it. Analysing the possible causes of difficulties the paper opens discussion on the remedial measures to be carried out. The definition of Mathematics as a ‘science of order’ (Whitehead (1929))
points to its features: generalisation, simplification, representation,
communication and application and determines its wide applications in
natural sciences, economics, social studies and other fields. It has been
used everywhere in the world, for many centuries as a universal
language. Mathematics, acting as a language parsimoniously and
prolifically, helps in meeting the purpose of economics. In doing so it
provides concrete form to economic laws and relationships, and makes
them more precise and practical. The formal mathematical expression of
economic ideas enables us to give loose economic intuitions a coherent
and logical meaning and in derivation of certain results, which would
either be impossible through verbal logic or would involve clumsy,
complex and circular process.
The paper intends to:
z Set forth, illustrate and discuss how the use of mathematics helps
in the expression of economic theories
z Diagnose the difficulties encountered by the students while
applying mathematical expressions in learning economics
z Analyse the possible causes of difficulties and open discussion on
the remedial measures to be carried out.
Use of Mathematics in Economics.
Ruling out the question of whether mathematics has any business in the
subject I simply assert with J Willard Gibbs that mathematics is a
language, it is a language that sometimes makes things clearer than do
other languages, and that sometimes helps to discover things. Economics
are often concerned with variables, which are measurable and involve
relationships between them that can be expressed by means of functions
and curves for a convenient understanding of the underlying concepts.
This necessitates as well as facilitates the use of mathematical
techniques.
Let me set forth a few illustrations in order to convince that it is, in fact,
worthwhile to employ mathematical tools in economics.
(i)
(ii)
(iii)
(iv)
(v)
Quantity depends on price or consumption depends upon level
o income can be mathematically expressed in the form of a
functional relationship Q = f(p), C= f(y) , read as ‘quantity is a
function of price’ and ‘consumption is a function of income’.
Economics is the science of choice making and involve the use
of mathematics ranging from geometry to calculus. The
consumers equilibrium is studied in the context of given income
and prices and involves solving of simultaneous equations for
unknowns. Equilibrium itself is a mathematical concept derived
from static and dynamics.
Consumer equilibrium and equilibrium of the firm involve
decisions at the margin and mathematically turns out to be the
first derivative of the relevant function.
The change in price on account of the change in output
depends upon elasticity of demand and supply and ‘elasticity’
is, in fact, a mathematical concept.
The maximisation and minimisation of variables like profit,
cost, revenue, utility etc obviously involves mathematical
expression and can be solved with the help of differential
calculus, linear programming or theory of games.
Now let me proceed to discuss the mathematical treatment that might be
given to some of the microeconomic topics at an introductory level. This
will enable us to see that the basic relationships in Microeconomics and
Macroeconomics can be put in the form of functions and equations
thereby making it easy, smooth and systematic.
We come across many variables while dealing with economic theories.
Equations can be constructed by relating a number of variables to one
another. It is the application of the relevant mathematical operations to
these equations we seek to derive a set of conclusions.
Let’s take the equation of the linear demand curve given as Q=a-bP.
We can derive a set of conclusions from this equation, which states the
relationship between quantity demanded and price.
(i)
The slope of this demand curve is –b which shows that the
change in demand per unit
Change in the price i.e.
Δq
=− b
Δp
Let us suppose that the change in quantity demanded in and
change in price is ΔP . Then the equation becomes
q − Δq = a − b( p − Δp)
− Δq = a − bp + bΔp − q
Δq = − a + bp − bΔp + q
Δq = −a + bp − bΔp + a − bpa
(as q = a − bp )
Δq = −bΔp
Δq
= −b
Δp
The negative sign of the slope clearly shows that the demand curve
is sloping downwards and the demand function show demand as a
monotonic decreasing function of the price of the commodity. This
proves the law of demand.
Now let us discuss the elasticity of demand of a linear demand
curve. It doesn’t have a constant elasticity. As we move down the
demand curve, ∆P/∆Q may change, and the price and quantity will
always change. Therefore price elasticity of demand must be
measured at a particular point on the demand curve and will
generally change as we move along the curve.
Applying to the formulae for elasticity of demand we obtain the
elasticity of a linear demand curve at various levels of price.
Δq p
.
Δp q
p
= −b
q
Ed =
∴ Ed = −
(as
Δq
= −b)
Δp
bp
(as q = a − bp )
a − bp
Now to find the elasticity at various levels of prices, lets take price
at three levels namely
p = 0, p =
a
a
and p = .
2b
b
When p=0
b.0
= 0 ( Substituting 0 in placeof p )
a − b.0
a
when p =
2b
a
b
2b = a × 2 = 1
Ed =
a 2 a
a −b
2b
a
when p =
b
a
b
b = 1 =∞
Ed =
a 0
a −b
b
Ed = −
This helps to precisely state that the price elasticity of demand is
different at different points on the linear curve.
We have seen that the attempt to set down in mathematical form
the law of demand and its various aspects, has led to important
advances in the understanding of that concept and its various
possible meanings. It follows that there can be no doubt that
mathematical methods are possible in economics and that it can
be used to express the economics relations with high degree of
clarity and rigor.
Diagnosing difficulties in using mathematical expressions at higher
secondary level.
Mathematical approach in Economics is considered to be a quick,
precise and elegant medium of communication. But all the same,
there is variety of challenges faced by students as well as teachers
while using mathematics at the introductory level. These may be
described as:
z Low level of confidence with respect to mathematical ability
z Limited fluency in algebraic notations
z Limited understanding of the usefulness of mathematical
principles in economic argumention
z Difficulties in recognising the mathematical representations
of a problems within economic theories
A research study was undertaken by the Department of Education
in Social Sciences and Humanities with the primary task of
identifying difficulties encountered by students especially of the
humanities group, so that it may provide some suggestions for
required changes in development of curricular materials.
Preliminary mathematics diagnostic test was done at five different
schools using questions requiring an objective response which
were designed to provoke possible misunderstandings and errors.
The errors committed by students reveal a variety of difficulties
regarding specific mathematical knowledge.
Why do such errors occur?
• Most students might have forgotten what they have learned
during their schooling
• The basics might not be clear for eg, in arithmetic, adding or
subtracting negative numbers
• ‘Math anxiety’ is evident among the students mainly
because of the inauspicious manner in which the subject is
often presented to the students.
Proposals for remediation
• Equip students as well as teachers with required
mathematical skills
• Make them understand the relevance of using mathematical
expressions and how these provides a scientific approach to
the subject
• Enable them to gain confidence by helping them attain
success. Success paves the way for confidence
• Enable the students to enhance their mathematical thinking
processes. This will give them confidence in clarifying,
formulating and solving mathematical problems
Conclusion:
Before I conclude let me go back to Gibbs statement of treating
mathematics as a language to assert that it’s not the only language
with which the economic theories can be expressed with. Students
should be aware of the different representations of the same
concept- graphical, numerical and verbal. With or without
mathematics, I would stress proper understanding of basic
concepts and theorems to be important. Let me end taking sides
with Prof. Samuelson, mathematics is neither necessary nor a
sufficient condition. It can help.
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