Teacher Talk: Calculus – Super or Scary?

This week’s Teacher Talk comes from Spark’s KS5 Maths lead Chris, who is determined to show us that calculus is not as terrifying as you might believe…

What’s involved in A-Level Maths? Since the start to term in August, our Year 12 students have been revising and improving on the knowledge of several topics. These include Completing the Square, Surds, Coordinate Geometry, and Factorising Quadratics.

Now that this introductory period is over, we will now begin to look at the first really new topic in KS5. This is the first topic to break away from the regular GCSE curriculum: Calculus.

Calculus is an umbrella term for two key topics: Differentiation and Integration.

The History of Calculus

One popular topic in KS3 and 4 was to determine the gradient of a straight line. This would be rather easy considering that the gradient would be constant throughout all values of x on the graph.

What do you think would happen if you tried to find the gradient of a curve instead?

From the animation below, you can see that the value of the gradient (sometimes called a derivative) will alter, depending on the x-value on the graph. So instead of a constant value, we need to find a general equation for this gradient. How is this done?

Well, with differentiation of course!

We can date the first evidence of gradient dates back to Euclid (c. 300 BC) and Archimedes (c. 287 – 212 BC). However, we credit the modern development of calculus to two mathematicians. These are Isaac Newton (1643 – 1727) and Gottfried Leibniz (1646 – 1716).

Historically, it has been long debated over whether Newton or Leibniz was actually the first to “invent” calculus.

Newton vs. Leibniz

Newton had begun working on his investigations into physics and geometry in 1665 – 1666. He used calculus as the scientific description of the generation of motion as time changes. He had called these “fluxions”, rather than derivatives.

However, he did not publish his work officially.

In 1684, Leibniz had published a book explaining the concepts of calculus for differentiation, and another in 1686 for integration.

One year later, Newton had published his book Philosophiæ Naturalis Principia Mathematica (or: Mathematical Principles of Natural Philosophy). This is widely considered as one of the greatest science books of all time. The book contained Newton’s three laws of motion and Newton’s law of universal gravitation. These both help to form the foundation of classical mechanics (but this is a story for another day!).

Throughout the book, Newton uses mathematical methods included in modern-day calculus. However, the notation that we use in the 21st century was largely absent in Newton’s book.

It is actually Leibniz’s notation such as for differentiation

or for integration

that we continue to use in classrooms and lecture theatres to A-level and undergraduates throughout the world to this day.

Newton began to discredit Leibniz’s reputation. He used his newly found status and influence from his latest book. This came to a head in 1715.The Royal British society decided to settle the infamous argument once and for all.

The President of the society in 1703 was Newton. He appointed an impartial committee to decide the issue. The committee concluded in its official report that Newton was sole inventor of calculus. Newton was the anonymous author of the report.

What a coincidence, you might say! 