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# Mathematics (Bhutan Certificate of Secondary Examination)

The first question that comes in my mind is “What is mathematics?” It might
seem strange, probably spent several years being learn and taught mathematics.
However, for all the times schools devoted to the teaching of mathematics very little, if
any, is spent trying to convey just what the subject is about. Instead, the focus is on
learning and applying various procedures to solve mathematics problems that is a bit
like explaining soccer by saying it’s a series of maneuvers executed to get the ball into
the goal. Both accurately describe various key features. However, they miss what’s and
the why of the big picture. If all we want to learn new mathematical techniques to apply
in different circumstances, then we can probably get by without knowing what
mathematics is really about.
One thing I realize is that a lot of school mathematics dates back to medieval
times, 17th century at the very latest. Virtually nothing from the last 300 years has found
its way into the classroom. Yet the world we live in has changed dramatically in the last
ten years, let alone the last 300 years. Most of the changes in mathematics over the
centuries were just expansion. But in the 19th century, there was a major change in the
nature of mathematics. First, it became much more abstract. Second, the primary focus
shifted from calculation and following procedures to one of analyzing relationships. The
changing emphasis wasn’t arbitrary, it came about through the increasing complexity of
what became the world we are familiar with. Procedures and computation did not go
away which are still important, but in today’s world, they are not enough. We need
understanding. In our education system, the change of emphasis in mathematics usually
comes when the students transition from high school to university. However, the basics
of mathematics begins from high schools and it lacks to emphasize the mathematics to
real life situations. Among many topics taught in mathematics, I realize that the calculus
is one of the most important and prominent topics that the mathematics teachers can
relate with real world situations−experimentally real, visually and graphically.
Calculus is one of the greatest achievements of human intellect and
demonstrates the power to illuminate the most fundamental problems in mathematics,
physical sciences, biological sciences, and engineering. Calculus can reduce
complicated problems to simple rules and procedures by using symbols and notations.
vi
However, use of symbols and notations might lead to losing the original pictures of the
problems. Despite its importance, the teaching of introductory calculus always
emphasizes manipulation of algebraic notations and rote learning. Students memorize
algebraic procedural steps rather develop conceptual understanding. Most students learn
the how instead of the why of calculus due to extensive use of algebraic symbols and
notations. The real meanings of symbols and notations learned in the classroom are not
interpreted explicitly in the context of real world situations.
To address this issue, I have designed a contextual and graphing activities
based on the learning cycle approach to enhance students’ conceptual understanding of
the fundamentals of calculus and the relationship between differentiation and
integration. Experimentally real activities for students were developed to convey the
concepts of the fundamentals of calculus realistically and then represented in the form
of graphs. The activities given in this book were thoroughly researched, implemented
and the results were also successful. The activities are very simple and used locally
available materials in the Bhutanese context. However, the result of the research study is
not mentioned in this book, only discussions and impact of the research study is
presented at the end.
This is book can be exclusively used by the mathematics teachers teaching
introductory calculus in higher secondary schools. I hope that the mathematics teachers
would make best use of this book to help students understand calculus better.
If readers come across any mistakes, your suggestions will be gracefully
acknowledged for the further improvement of the book.