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to Hypotenuse ContentsThis is a report of the lecture by Professor
SK Houston. It is written in the first person.
INTRODUCTION
Everyone who teaches A-level Mathematics cannot but be aware of the
requirement to teach "The Application of Mathematics".
Section 3.11 of the SCAA Common Core syllabus has the following:
The Application of Mathematics
Understanding the process of mathematical modelling with reference to one or more application areas:
Then in Section 4.2 we find the Assessment Objectives.
Mathematical Skills and Understanding
Candidates should be able to demonstrate that they can:
(iii) evaluate mathematical models, including an appreciation of the
assumptions made, and interpret, justify and present the results from a
mathematical analysis in a form relevant to the original problem.
This is one of the compulsory assessment objectives. We also have the
optional ones:
(iv) read and comprehend a mathematical argument or an example of the
application of mathematics,
(v) appreciate how to use appropriate technology, such as computers
and calculators, as a mathematical tool, and have an awareness of its limitations.
I shall return to the these later, but for now I want to concentrate
on modelling.
MATHEMATICAL MODELLING
It is my thesis that Mathematical Modelling is a way of life, for everybody.
We all do it, but usually we do it unconsciously, without being fully aware
of what we are doing. Mathematical Modelling is certainly the way of life
of a professional applied mathematician. Now I know that only a very small
percentage of A-level students will become professional mathematicians,
but many more become scientists or engineers or economists and they make
use of mathematical modelling in their work.
Modelling is about simplifying, structuring and solving; it is about
convincing and justifying; it is about describing and predicting. Mathematical
Modelling is when we bring mathematics into it, and it could be argued
that this happens every time. If you get bored with this, why don't you
try to think of a model which does not involve anything that could
be construed as mathematics!
There are various ways of approaching the teaching of modelling and
they should all be used because this contributes in different ways to a
pupil's understanding.
First of all pupils have to be fairly fluent at mathematics itself,
otherwise they get bogged down in their mathematical solution and cannot
complete their problem. Algebra, geometry, trigonometry, calculus, probability,
statistics, numerical methods - all of these are useful at one time or
another, not necessarily all at the same time.
Then pupils should have opportunities to do two things:
What I propose to do next is to work through some situations which are
amenable to modelling and which could be used for illustration or for engagement.
I am borrowing heavily from two books published recently by Collins
Education in their "Discovering Advanced Mathematics" series.
The books are
and I think they are very well written and presented . When studying
a model or engaging in modelling, it is important to be aware of the stage
in the modelling process you are at. This encourages a structured
approach to the problem. It also, heightens awareness - we know
when we are making simplifications, we know when we are validating
the model, and so on. We make assumptions much of time when we are doing
applied mathematics and we need always to be aware of these and even to
articulate them. Then there is no doubt that we know what the assumptions
are and we can take them into account later on. The books I have just mentioned
identify very clearly the different stages in each modelling case study.
Mechanics is, of course, a rich field for modelling and Ted Graham gives
a case study in almost every chapter. Pure Mathematics is not just quite
so rich. Nevertheless there are about 10 case studies in Bob Francis's
book.
Let's have a look at some of these.
[Professor Houston went on to discuss a number of the case studies from
these books.]
COMPREHENSION TESTS
There are other ways of getting students acquainted with the process
of modelling. One of these is to use comprehension exercises. You will
remember that objective (iv) - one of the optional objectives is
read and comprehend a mathematical argument or an example of the application
of mathematics.
Some examining boards have already taken this idea on board. For example
the MEI Structured Mathematics A-level Scheme incorporates both pure and
applied comprehension tests. This is a modular A-level and module 9665/3
- Pure Maths 3 - incorporates a 1 hour test. Students are given an unseen
article to read and are asked questions about it. Also module 9665/22 -
Modelling in Mechanics involves a project and a 75 minute comprehension
test. For this test candidates are given an article on an application of
mathematics about 6 weeks before the test. They are to study it thoroughly
and then they answer unseen questions on it in the test.
Of course all of these ideas were pioneered here in Northern Ireland
about a decade ago through our Mode 2 Further Mathematics course which
was examined for 5 years from 1987 to 1991.
In my view the aims of a comprehension test are
(1) To encourage students to read, with understanding, a mathematical article
(2) To provide students with an opportunity to demonstrate their understanding of general mathematical processes, both pure and applied
(3) To encourage students to develop their skills of communicating mathematics - reading, asking, answering and writing
(4) To demonstrate to students that mathematics is a living subject
and is used in contemporary situations.
While the objectives of an applied comprehension test are
(1) Explain all statements such as "It can be shown that ..." or "It follows from the above that ...." in the article.
(2) Identify and explain all mathematical modelling assumptions made in the article.
(3) Make constructive criticisms of assumptions made, mathematical analysis and calculations carried out, inferences and deductions made, processes carried out.
(4) Locate any inconsistencies or incorrect deductions made in the article.
(5) Locate and correct any mathematical or typographical errors in the article.
(6) Have some wider background knowledge of the situation described in the article.
(7) Generalize the ideas or apply the ideas to a different situation.
If you are not entering pupils for these examinations then old papers
can be used as "end of term" activities. (There is a collection
of comprehension tests in my book with John Berry - Mathematical
Modelling - published by Edward Arnold, 1995).