A brief guide to the teaching of Differentiation and Integration

Shaheda Begum, Ian Johnson, Adam Newell, Riddhi VyasUniversity of Warwick PGCE Secondary Mathematics

Contents

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

1. Calculus

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.

2. History of the development of the topic

Much debate around who should be accredited with the discovery of Differentiation and/or Integration.

Key players include:Gottfried Wilhelm Leibniz

And

Isaac Newton

Controversy…

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 Newton claimed Leibniz stole ideas from his unpublished notes, which Newton had shared with a few members of the Royal Society.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.

Therefore…

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.

Claims from outside of Europe…

EgyptCalculating 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. ChinaIn 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.

Claims from outside of Europe…

IndiaThe 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 .  IslamicIn 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. 

Claims from outside of Europe…

JapanIn 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…… 

3. Applications of Differentiation

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.

Medical Statistics

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.

Economics

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.

Applications of Integration

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.

Solids of Revolution

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.

4. Tasks….

A. Introduction to differentiation B. Differentiation dominos C. Introduction to integration D. Integration jigsaw   

5. Difficulties and misconceptions

6. Research Into the Teaching and Learning Into Differentiation and Integration

Research into students’ understanding of differentiation and integration is still a challenge and new recommendations for improving this may not be widely taken onboard.

Introducing Calculus: An Accumulation of Teaching Ideas?

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.

Summary…

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.

Findings…

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.

Situated Cognition and Conceptual Understanding of Elementary Calculus

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.

Different approaches…

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.

Results….

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.

Further resources…

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/