Shaheda Begum, Ian Johnson, Adam Newell, Riddhi Vyas
University of Warwick PGCE Secondary Mathematics
1. Introduction
2. History of development of topic
3. Application in real world and other curriculum areas
4. 3 investigatory tasks
5. Difficulties
6. Relevant research into teaching and learning of topic
7. Further resources
Calculus (Latin, calculus, a small stone used for counting) is a branch of
mathematics focused on limits, functions, derivatives, integrals, and
infinite series.
It has two major branches, differential calculus and integral calculus,
which are related by the fundamental theorem of calculus.
Calculus has widespread applications in science, economics, and
engineering.
Much debate around who should be accredited with the discovery of
Differentiation and/or Integration.
Key players include:
Gottfried Wilhelm Leibniz
And
Isaac Newton
When Newton and Leibniz first published their results, there was great
controversy over which mathematician (and therefore which country)
deserved credit. Newton derived his results first, but Leibniz published
first. Newton claimed Leibniz stole ideas from his unpublished notes,
which Newton had shared with a few members of the Royal Society. This
controversy divided English-speaking mathematicians from continental
mathematicians for many years, to the detriment of English mathematics.
A careful examination of the papers of Leibniz and Newton shows that
they arrived at their results independently, with Leibniz starting first with
integration and Newton with differentiation. Today, both Newton and
Leibniz are given credit for developing calculus independently.
It is Leibniz, however, who gave the new discipline its name. Since the time
of Leibniz and Newton, many mathematicians have contributed to the
continuing development of calculus.
One of the first and most complete works on finite and infinitesimal
analysis was written in 1748 by Maria Gaetana Agnesi.
Egypt
Calculating volumes and areas, the basic function of integral calculus, can be traced
back to the Moscow papyrus (c. 1820 BC), in which an Egyptian mathematician
successfully calculated the volume of a pyramidal frustum.
China
In the third century Liu Hui wrote his Nine Chapters and also Haidao suanjing (Sea Island
Mathematical Manual), which dealt with using the Pythagorean theorem (already stated
in the Nine Chapters), known in China as the Gougu theorem, to measure the size of
things. He discovered the usage of Cavalieri's principle to find an accurate formula for
the volume of a cylinder, showing a grasp of elementary concepts associated with the
differential and integral calculus.
India
The mathematician-astronomer Aryabhata in 499 used a notion of infinitesimals and
expressed an astronomical problem in the form of a basic differential equation. Manjula
and Bhāskara II expanded this thought .
Islamic
In the 11th century, when Ibn al-Haytham (known as Alhacen in Europe), an Iraqi
mathematician working in Egypt, performed an integration in order to find the volume
of a parabolic shape, and was able to generalize his result for the integrals of
polynomials up to the fourth degree. He thus came close to finding a general formula
for the integrals of polynomials, but he was not concerned with any polynomials higher
than the fourth degree.
Japan
In 17th century Japan, Japanese mathematician Kowa Seki made a helped
determine areas of figures using integrals, extending the method of
exhaustion. While these methods of finding areas were made largely
obsolete by the development of the fundamental theorems by Newton
and Leibniz, they still show that a sophisticated knowledge of
mathematics existed in 17th century Japan.
And many more……
Differentiating the equation of a curve, gives the gradient function. The
gradient function can be used to determine the gradient of the tangent
and normal to the curve. The behaviour of the curve can be analysed by
looking at the sign of the gradient function. On critical points the curve has
no gradient. The first derivative test and second derivative can be used to
distinguish among maximum or minimum or inflection point. All this
information can be used to sketch the curve and illustrate all the important
features.
A medical statistician studies a drug that has been developed to lower
blood pressure. The average reduction R in blood pressure from daily
dosage of x mg will be recorded during the experiment. He will be
interested to know the sensitivity of R to dosage x at different dosage
levels. The sensitivity of R to x is dR/ dx. This can be then used to make
inference about which dosage level works best to reduce blood pressure.
In economics the term marginal denotes the rate of change of a quantity
with respect to a variable on which it depends. For example, the cost of
production C(x) in a manufacturing operation is a function of x, the number
of units of product produced. The marginal cost production is the rate of
change of C with respect to x, so it is dC/dx.
Integrals can be used to express volume of solids, lengths of curves, areas
of surfaces, forces, work, energy, pressure, probabilities and a variety of
other quantities that are in one sense or another equivalent to areas under
graphs.
Many quantities of interest in physics, mechanics, ecology, finance and
other disciplines are described in terms of densities over regions of space,
the plane, or even the real line. To determine the total value of such a
quantity we must add up (integrate) the contributions from the various
places where the quantity is distributed.
The volume of any prism or cylinder is the base times the height. Solids can
be divided into thin “slices” by parallel planes. The volume of the solid can
be determined using the cross sectional area of each slice.
A. Introduction to differentiation
B. Differentiation dominos
C. Introduction to integration
D. Integration jigsaw
Research into students’ understanding of differentiation and integration is
still a challenge and new recommendations for improving this may not be
widely taken onboard.
Tony Orton from the Centre for Studies in Science and Mathematics
Education, School of Education, University of Leeds which appeared in the
International Journal of Mathematical Education in Science and Technology,
1986, Vol. 17, No.6, pp 659-668.
Although it may appear somewhat dated, it gives us an historical reference
point from which we can begin to have an idea of the obstacles facing
pupils in the UK when they start learning calculus. For number of decades
up to the mid-eighties, it was still debatable whether to introduce calculus
to students before age 16.
In introducing, the pupil to the concept of the derived function in
differentiation, he should know the distinction between the use of the secant
and obtaining the tangent at a particular point on the curve. For the first, he
should clearly understand that it represents an average rate of change while
the second an instantaneous one.
On his /her introduction to integration, the pupil will develop a progressive
concept of it as the process of finding the area under a curve, rather than just
being left with it as the reverse process of differentiation. As the width of the
rectangles gets smaller the numbers of them increase which give a better
approximation to the area each time. They can then go on to use trapezia to
get a better approximation to the area using the same method as before.
For both differentiation and integration, the idea of the limit was generally
neglected he argued. More create ways could be used to bridge this idea
from ‘simpler’ areas of mathematics, eg, the relationship of the circle to
polygons, finding the areas of a circle by using sectors, etc can help to build
an adequate concept of a limit.
British Society for Research into the Learning of Mathematics, June 2003,
Victor Kofi Amoah takes a more serious look at the effects of two teaching
approaches to pupils’ understanding of differential calculus. This against a
background in which the use of computers has now become second
nature.
In the situated cognition approach, pupils were taught with an emphasis on
the underlying concepts.
The unify approach, was less structured and no lecturing took place at the
beginning or during any learning activity. It stressed learning that is about
engaging the pupils in learning and keeping track of the ideas the pupils
come up with. Teaching was aimed at the ‘ mathematics making sense’ and
continuously discussing their ideas with their peers and teachers.
The results of the first approach showed that through the authentic
activities and graphing software students can improve their conceptual
understanding of differential calculus. They did not suffer from a loss of
computational skills as some educators fear and ended up with a strong
graphical understanding of the derivative.
Websites for further background and extra investigations.
MEI
http://integralmaths.org/resources/course/view.php?id=32
NRICH
http://nrich.maths.org/public/
STEMnet
http://www.stemnet.org.uk/