:Home
A
Seminar
On
Application
and Modeling on Mathematics
By:
- Uttam Bahadur Tamang
At
the primitive time, mathematics was originated from country by using stone and
by cutting notches in a piece of sticks or by tying knots in a string. During
those days of the distant past, as the primitive human started to settle in
colonies and live as social creatures , which gave rise to the need of
primitive country and surveying. Now
mathematics has lead to the development of various subject, vocation and
technology. Mathematics is an exact science which is still playing an important
role in various field so that mathematics as an application and modeling.
An
application is a body of knowledge and techniques which has been found to have
use in a particular range of problems. Frequently the techniques have developed
through work on these problems.
In
some cases the situation will be so novel that there is no obvious choice of
mathematics to use. Thus two people studying the same situation may bring to
bear on it different mathematical knowledge and produce different models. In
other cases the position will be less open and the task may be more that of
adjusting a known model to fit the new situation. An application may claim a
place in the curriculum in their own right as a way of illuminating the
situation to which they apply and as a medium for teaching to motivate and
stimulate. But once it has been decided to include in the syllabus a particular
application then large body of contain must be taught. Actually the process of
impressing a real life problem into mathematical form is called mathematical modeling.
A real life problem may be related to a physicist, engineer, sociologist,
medicine and other streams of life. Even if the house wife also has the
opportunity to enjoy with mathematical beauty.
As
with problem solving we need a more precise definition. Is there a set of
modeling skills? Can they be taught? If so how? How for are these skills of
general application, what evidence is there that, if learn in one context, they
can transferable to others? Is transfer facilitated by work on a variety of
models which use different pure mathematics? How is modeling to be predicted? This
must be considered, if modeling is not predicted it will not form a significant
part of the teaching of the course. In an actual situation the success of the
model in the correctness of the prediction to which it leads is often the
crucial test. It is not easy to provide this acid test in the school context.
Again, for a number of the situation which might be considered by a pupil at
school there may be an effective model which is however, unknown to the
students. If the students derive a different, less satisfactory model in such a
case should his marks be therefore reduced? As is a process we are trying to
teach, then it should be the students’ ability as a modeler that we should try
to assess, attempting to judge how successful he will be when faced with a new
situation. We know how to teach applications, we have little experience of
teaching modeling.
Key issues of Modeling are dealt with, among which
are the following:
Epistemology and the relationships between mathematics and the "rest
of the world"; the meaning of mathematical modeling and its process
components; the respect in which the distinction between pure mathematics and
applications of mathematics make sense
Authenticity and Goals dealing with modeling and applications in mathematics
teaching; appropriate balance between modeling activities and other
mathematical activities; the role that authentic problem situations play
in modeling and applications activities
Modeling Competencies: Characterizing how a student's modeling competency can be
characterized; identifiable sub-competencies, and the ways they constitute a
general modeling competency; developing competency over time
Mathematical Competencies: Identifying the most important mathematical competencies that
students should acquire, and how modeling and applications activities can
contribute toward building up these competencies; the meaning of
"Mathematical Literacy" in relation to modeling
Implementation and Practice: The role of modeling and applications in everyday mathematics
teaching; major impediments and obstacles; advancing the use of modeling
examples in everyday classrooms; documenting successful implementation of modeling
in mathematics teaching
Assessment and Evaluation: Assessment modes that capture the essential components of modeling
competency; modes available for modeling and applications courses and
curricula; appropriate strategies to implement new assessment and evaluation
modes in practice
The contributing authors are eminent members of
the mathematics education community. Modeling and Applications in Mathematics Education will
be of special interest to mathematics educators, teacher, researchers,
education administrators, curriculum developers and student.
We
can argued that since must people who use mathematics use an applications and
do not have to begin from scratch to create a model, then it is applications
that we should teach. In the words school efforts should be concentrated on
imparting an appropriate body of knowledge and developing skills in the
standard topic of mathematics, or statistics, or both if possible. Teacher in
higher education generally, we seem to favour this approach, particularly those
who see mathematics as a service subject. It might also be observed that must
of us who now press for the teaching of modeling were ourselves brought of an
application and have discovered modeling at a later stages.
The
opposite extreme, to tried to teach modeling and not to attempt to teach
anybody of knowledge of one of the standard applications, if probably equally
unrealistic many of the situation which lend themselves to modeling are
inevitably ones where there is a standard model or application that works and
it will be perverse to insist that no knowledge of that application should be
taught, although it would be possible to say that no knowledge of specific
parts of the application should be expected in any examination. If therefore
the optimum should is some where between the extremes, some knowledge one or
more application but with some modeling finding the optimum blend of the two
ingredients involves, like other questions about the course assessing the
balance of advantage (Hersee,1984). The optimum balance may well be different
for different groups of students, possibly reflecting their future career
intentions, but since all students in a school are likely to be taught in the
same class the same blend will necessarily be offered to all. What is certain
is that to include modeling in any significant way requires a substantial
allocation of time and this time can only found by reducing the content taught
with in the applications, thus inevitably reducing what teachers in higher
education may expect the students to have covered. If it is desired to include
more than one application in the course, the time available for each
application is limited. Experience shows that it is difficult to build of
underlying concepts and understanding if as a result the work is hurried or if
the content of each application is severely curtailed it is all to easy for any
application to become simply a set of standard to be use in standard situation.
In recent years this has happened with statistics and with electricity in a
level course. Ironically, this time pressures coupled with teachers’ insecurity
in teaching ‘new’ application have tended to produce an effect which is
contrary to the aim underlying the inclusion of several applications.
The use of mathematical modeling has
increased in recent years specially because of increasing the power of digital
computer and new computing models. Now the handling of lengthy computational
work is not discouraging the use of mathematical modeling. It is a system and
it involves the following steps:
1. Understanding
the problem: In the first of the modeling, the given real-life problems must be
understood clearly.
2. Formulating
the model: - To formulate the problems, the variables should be identified as
well as classified as dependent and independent variables.
3. Finding
the solution:- In this stage the user have to apply the mathematical knowledge
to solve the given problems which has already being expressed in terms of
equations and in-equations.
4. Evaluation
of Model:- The results obtained by solving the model equations are compared
with the known fact of the given real problem.
“That aim has a modeling sprit, to
show the usefulness and applicability of mathematics and encourage students to
use it.”
References
Blum, W. (1996): Anwendungsbezüge im Mathematikunterricht
– Trends and Perspektiven. – In: Kadunz, G. et al. (eds): Trends and Perspektiven. Schriftenreihe Didaktik
der Mathe-matik, Vol. 23. – Wien:
Hölder-Pichler-Tempsky, p. 15 – 38.
Flick, U. (1999): Qualitative Forschung. Theorie, Methoden, Anwendung in Psychologie und
Sozialwissenschaften. – Reinbek: Rowohlt.
Grigutsch, S. (1996): Mathematische Weltbilder von Schülern.
Struktur, Entwicklung, Einflußfaktoren – Duisburg: Unpublished doctoral
thesis at the Gerhard-Mercator-Universität – Gesamthochschule Duisburg.
Kaiser-Meßmer, G. (1996): Anwendungen im Mathematikunterricht. Vol. 1–
Theoretische Konzeptionen. – Bad Salzdetfurth: Franzbecker.
Kaiser, G. (1996): Realitätsbezüge im Mathematikunterricht – Ein Überblick über die aktuelle und historische Diskussion.
– Graumann, G. et al. (eds): Materialien für einen realitätsbe-zogenen
Mathematikunterricht. – Bad
Pandit, R.P. (2064). Recent trends in mathematics education.
Kathmandu: Indira Pandit.
Upadhyay, H.P. et al. (2010). Trends in mathematics Education, Kathmandu: Balalika Education
Publication.
