Everyone knows that maths is important for science, but it's not always easy to present demonstrative examples of this to students. If a high school student studies physics or chemistry, the most advances mathematics they will need is probably solving or rearranging an equation, which is a Year 8 or 9 skill.
One of my favourite examples to talk about with students is calculus. It could be considered the crown jewel in the mathematical canon, but the fact is it was developed by Newton with an eye to his studies of planetary motion. It's a great example of a mathematical development being motivated by science. OK, that much I knew, but what about some details.
In the Feynmann book I referred to yesterday, it digs a little deeper. Kepler discovered three laws about planetary motion by observation. He looked at heaps of data that was collected on the location of planets at different times and noticed that:
- planets orbit the sun in an ellipse, not a circle
- planets sweep out equal areas in equal times, rather than equal distances
(Image taken from Wikipedia.)The beautiful thing is that from this one equation, Newton could mathematically prove that orbits are elliptical and that planets trace out equal areas in equal times. This is significant because no observation or experiment is involved (except in forming the original equation, which was a hypothesis). It was through mathematics, not science, that he could show that the known facts about planetary motion were a consequence of gravity.
Newton's work on planetary motion was great science and great mathematics. As a maths teacher, I want students to appreciate the importance of mathematics, even if they're not going to make a lot of use of it in their lives. Calculus is one of mankind's greatest intellectual achievements, and many high school and university students study in in detail. It's good for them to have some idea of how it has contributed to our understanding of the universe.