What is Functional Mathematics? A Personal View Professor Huw Williams Senior Engineering Specialist, Jaguar Research Presented at Ninth East Midlands Network June 10th 2008

Page 3 Overview The Engineering Societies: SARTOR What is Mathematics? What do we mean by Functional Mathematics? Mathematics in the Automotive Industry Case Study: Target Tracking Summary

Page 4 The Engineering Societies: SARTOR The Engineering Societies published a guide called SARTOR which stands for Standards and Routes TO Registration (NB SARTOR has been replaced by UK-SPEC) Chartered Engineers are expected to be able to “use mathematics as a tool to solve complex problems” Incorporated Engineers are expected to be able to “use mathematics as a tool for problem solving” Engineering Technicians are expected to learn mathematics “within the context of real engineering applications, integrating theory with current industrial practice and design requirements” IMA Working Group on SARTOR3: “Engineering mathematics is not simply teaching of pure mathematics techniques but requires its own motivation and skills focussed towards engineering”

Page 5 What is Mathematics? From the CTK Exchange Website The study of the measurement, properties, and relationships of quantities, using numbers and symbols. – American Heritage Dictionary The science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations. – Webster’s Collegiate Dictionary Mathematics is a language. – Josiah Willard Gibbs To those who do not know Mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty of nature.... If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in. – Richard Feynman

Page 6 What is Mathematics? From the CTK Exchange Website Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. – Bertrand Russell All science requires Mathematics. The knowledge of mathematical things is almost innate in us... This is the easiest of sciences, a fact which is obvious in that no one’s brain rejects it; for laymen and people who are utterly illiterate know how to count and reckon. – Roger Bacon We do not teach arithmetic in Kindergarten. A knowledge of mathematics is not necessary to a four year old. Any premature instruction arrests development – Philosophy of Froebel Kindergarten

Page 7 What is Mathematics? From the CTK Exchange Website Results of Poll 8234 Votes Cast Respondents asked to approve Multiple votes possible

Page 8 What is Mathematics? In Pythagoras’ day, music was considered to be a mathematical science and it still has some association with mathematics In the spirit of the quotations already given I offer the following definition Mathematics is the music that nature sings to us

Page 9 What do we mean by Functional Mathematics? How many of us can identify this tune?

Page 10 What do we mean by Functional Mathematics? Now that all is revealed… The notation makes more sense Music is sound not notation Sheet Music is the Engineer’s term for abstract mathematics

Page 11 What do we mean by Functional Mathematics? As John Ball commented in his David Crighton Medal lecture “everyone owns mathematics” Painting and decorating is a good source of examples Which do you prefer? – A tin of paint covers 3 square metres. How many tins of paint do you need to cover a wall that is 2 metres high and 4 metres wide? – Solve 3X = 8, rounding your answer up to the nearest integer. There are even applications of pure mathematics (if that is not an oxymoron) in wallpaper patterns

Page 12 What do we mean by Functional Mathematics? In the light of what I have said I would like to offer the following definitions Abstract Mathematics is the distillation of mathematical ideas to their essence. It is expressed in formal notation and is practised by people called mathematicians Functional Mathematics is the use of mathematical ideas in human society. It is expressed in natural language, diagrams and formal notation and is practised by everyone Using the musical analogy, Functional Mathematics is the lyrics and the tune and Abstract Mathematics is the sheet music

Page 13 Mathematics in the Automotive Industry This is a slide from an in-house training course for engineers The course shows how to use mathematics to design better products

Page 14 Case Study: Target Tracking Crash Sensing This is an extract from a schools’ talk of mine Current crash sensors work after the event

Page 15 Case Study: Target Tracking Crash Sensing The two distance sensors detect the target Notation chosen to connect with the abstract

Page 16 Case Study: Target Tracking Crash Sensing We have now moved into the abstract Second non-functional solution rejected

Page 17 Case Study: Target Tracking Crash Sensing Standard notation used

Page 18 Case Study: Target Tracking Crash Sensing Trigonometric theorem applied

Page 19 Case Study: Target Tracking Crash Sensing Geometric construction example to reinforce learning

Page 20 Summary There is a tendency to overvalue abstraction when teaching mathematics Most people are more comfortable with mathematics in a functional context If we are sufficiently creative we can make the link between the abstract and the functional We need a store of good worked examples with teachers notes