SHAPING THE FUTURE PHYSICS IN MATHEMATICALM · Liz Swinbank describes the approach to Maths in...

SH A P I N G T H E FU T U R EPH Y S I C S I N MAT H E M AT I C A L MO O D

2

S e r i e s E d i t o r P E T E R C A M P B E L L

The Discussion Series of bookletsOne main aim of the Institute of Physics post-16 Initiative is to promote debate and ideas about the way forward for physics education in schools and colleges, in the next decade and beyond. To pursue this aim, the Initiative is publishing a series of booklets, each addressing a matter of general and major concern about the shape of post-16 physics.

Topics in the series will span five clusters of concerns:● teaching specific topics in physics from a contemporary perspective● pedagogical issues● recruiting and supporting physics teachers● course and institutional structures● responding to long term social and industrial trends

With each booklet we organise an open, one-day conference or discussion meeting. Participants receive the manuscript of the booklet in advance, and discussion at the conference yields Discussion Points and the Loose Ends section for the final publication.We hope each booklet will be widely read, and form the basis of further discussion at local meetings aroundthe country. Written comment about any of the issues raised is invited, and should be sent to the Series Editor, at the post-16 Initiative address on the back cover.

2 Physics in Mathematical MoodMuch of the difficulty of A level physics is perceived to lie in the mathematical nature of the subject. Yet a mathematical view of the world is intrinsic to physics. This, the second booklet in the series, takes as its themes how physics can be taught in an appropriately mathematical way and how the beauty and power of mathematical reasoning can be conveyed to students.It offers both practical suggestions for immediate use and views on how we can improve on the situation in the future .

Contributions to this booklet were discussed at the ASE Annual Conference in January, 1999.

Many aspects of physics in mathematical mood are considered, including:● the mathematics students bring with them when beginning a post-16 course● links between mathematics and physics courses● how to improve problem solving and mathematical understanding● the essential mathematical functions encountered in physics● the importance of graphical skills and how they can be developed● how information technology can be used to support students.

The last sections suggest ways forward for teachers, students, exam boards, universities, QCA and government agencies funding education.

Shaping the Future2 Physics in Mathematical MoodEdited by Simon Carson

Institute of Physics PublishingBristol and Philadelphia

I N S T I T U T E O F P H Y S I C S2 P H Y S I C S I N M A T H E M A T I C A L M O O D

© Institute of Physics Post-16 Initiative 1999

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmittedin any form or by any means, electronic, mechanical,photocopying, recording or otherwise, without prior permission of the publisher.

British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from theBritish Library.

ISBN 0 7503 0622 X

Published by Institute of Physics Publishing, whollyowned by The Institute of Physics, London.

Institute of Physics PublishingDirac HouseTemple BackBristolBS1 6BE, UK

US Office: Institute of Physics PublishingThe Public Ledger Building, Suite 1035150 South Independence Mall WestPhiladelphiaPA 19106, USA

Printed in the UK by Page Brothers (Norwich) Ltd

Contributors

Jack Abramsky is Principal Subject Officer for Mathematics at the Qualifications and Curriculum Authority

George Adie lectures in Physics at the University of Kalmar, Sweden

Richard Boohan is Lecturer in Science Education at the University of Reading

Philip Britton is Head of Physics at Leeds Grammar School

Margaret Brown is Professor of Mathematics Education at King's College London

Simon Carson is Head of Science at Norton College, Malton, North Yorkshire

Graham McCauley formerly of the School of Mathematics and Statistics at the University of Birmingham

Jon Ogborn is Director of the Institute of Physics Post-16 Initiative

Andrew Raw is Head of Physics at Tring School, Hertfordshire

Helen Reynolds is Head of Physics at Gosford Hill School, Kidlington, Oxfordshire

Laurence Rogers is Lecturer in Education at the University of Leicester

Tim Sloan is a student at Leeds Grammar School

Elizabeth Swinbank is in the Science Education Group at the University of York, where she directs the Salters Horners Advanced Physics Project

with illustrations by Ralph Edney

I N S T I T U T E O F P H Y S I C S P H Y S I C S I N M A T H E M A T I C A L M O O D 3

Introduction 4

1. Pleasures of a Mathematical Kind 6

2. Physics without Mathematics at A Level: A Student’s View 9

3. Basic Problems? 11

4. A View from the Mathematics Side of the Fence 14

5. Improving Students’ Mathematical Problem Solving Ability 17

6. Maths in Salters Horners Advanced Physics 21

7. The (So-called) Elementary Functions 23

8. Wherefore dN/dt 5 –lN ? 26

9. Using Graphical Representations to Support 30Mathematical Reasoning

10. Graphical Calculators and Mathematics in Physics Teaching 33

11. Graphs in the Service of Physicists 36

12. Maps and Models – Approaches to Vectors 40

13. Free Standing Mathematics Units 44

Loose Ends 46

Some Ways Forward 48

Contents


‘Philosophy is written in this grand book. I mean the universewhich stands continually open to our gaze. But it cannot beunderstood unless one first learns to comprehend the languageand interpret the characters in which it is written. It is writtenin the language of mathematics … without which it is humanlyimpossible to understand a single word of it; without [this] oneis wandering about in a dark labyrinth.’GALILEO GALILEI, The Assayer, 1623

‘The enormous usefulness of mathematics in the natural sciencesis something bordering on the mysterious and there is no rationalexplanation for it... . The miracle of the appropriateness of thelanguage of mathematics for the formulation of the laws ofphysics is a wonderful gift which we neither understand nordeserve.’EUGENE P. WIGNER, Communications in Pure and AppliedMathematics, 1960

‘Every one of our laws is a purely mathematical statement.Why? I have not the slightest idea... . It is impossible to explainhonestly the beauties of the laws of nature in a way that peoplecan feel, without their having some deep understanding of mathematics. I am sorry, but this seems to be the case.’RICHARD P. FEYNMAN, The Character of Physical Law, 1965

Much of the difficulty of post-16 physics is perceived to lie in its mathematical nature. In the QCA discussion paper, ‘The transition from GCSE double award balancedscience to A level in the sciences: a review of findings’ (1998), many teachers identified mathematics as the principal sourceof difficulty for GCSE students moving on to study A level physics.

However, when students were asked about why they hadchosen to study physics at A level as part of the Post-16Initiative, ‘liking a subject involving mathematics’ figuredpositively. It is also true that those subjects that seem most toexcite potential future physicists, such as particle physics andcosmology, are the most mathematical of disciplines.

It is not simply that mathematics is a useful tool inphysics as it is, say, in geography or economics, but ratherit is its ‘unreasonable effectiveness’ (as Wigner has it) as alanguage for describing nature that is so remarkable.Mathematics as a tool does, of course, play its part inphysics in analysing or representing data, but it is the factthat fundamental laws can be formulated mathematicallyand that abstract reasoning can then be used to discovernew truths wherein lies some of the uniqueness and beau-ty of physics.

In considering how physics can be made attractive topost-16 students, we should keep this idea at the front ofour minds: we should not pretend that physics is not math-ematical but rather we should exploit this uniqueness andbeauty. Rather than fearing that the mathematical contentof our discipline will deter students, we should rejoice inour ability to analyse and understand the world in a quan-titative way, and use this as an enticement to attract poten-tial students.

Feynman said: ‘Even artists appreciate sunsets and theocean waves and the march of the stars across the heavens ... .As we look into these things we get an aesthetic pleasurefrom them directly on observation. There is also a rhythmand a pattern between the phenomena of nature which is notapparent to the eye, but only to the eye of analysis; and it isthese rhythms and patterns which we call Physical Laws.’Notice that surprising ‘even’! In Pleasures of a MathematicalKind, Jon Ogborn discusses the pleasures of physics in math-ematical mood in more detail.

Of course, we must be pragmatic: if we are successful inattracting more students to study our subject we must beaware of their potential difficulties and must have readystrategies that will help them to overcome their problemsand to enjoy their achievement. In Physics withoutMathematics at A Level, Tim Sloan gives us his view of whatit is like to be a student of physics. Some of the problemsthat students encounter are not to do with sophisticatedmathematics: Philip Britton considers some of these in

Simon Carson

Introduction: Mathematics and Physics


Basic Problems? and Margaret Brown gives A View from theMathematics Side of the Fence.

One way in which we can help our students is by lookingat the way we teach. In Improving Students’ Mathematical ProblemSolving Ability, Andy Raw takes ideas from research on cogni-tive development and applies them to his own teaching andLiz Swinbank describes the approach to Maths in SaltersHorners Advanced Physics.

The position taken here is that a mathematical view of the world is intrinsically a part of physics. But whatmathematics do we need to teach physics at A level? In his article, Graham McCauley describes ways in whichThe (So-called) Elementary Functions can be developed inphysical contexts. In Wherefore d N/dt52l N ?, HelenReynolds goes on to consider one of the issues raised inmore detail: that of differential equations.

Remember that mathematical understanding is not allthere is to physics: the great debate between Bohr andEinstein about the interpretation of quantum mechanicsand the nature of reality was not about the details of themathematical formalism. Faraday, probably the greatestBritish experimental natural philosopher, was not an adeptmathematician; however, he was able to develop theoreti-cal models of the mechanisms underlying electromagneticphenomena – the ideas of electromagnetic fields – thatwere of great use both to him in understanding his exper-imental results and to Maxwell in developing his fieldequations.

This is something that we should we bear in mind whenthinking about ‘mathematical’ understanding in physics: it isnot simply the ability formally to manipulate algebraic equa-tions, to substitute figures into formulae or to plot points ona graph, for example, but it is also the ability to understandthe meaning encoded in the shape of a graph, to understandthe qualitative relationships and connections between physi-cal quantities and the ability to develop models that lead toreal understanding of physical phenomena. We do well toremember Heisenberg’s difficulties with relativity:

‘Wolfgang [Pauli] asked me whether I at long last understoodEinstein’s relativity theory. I could only say that I did not reallyknow what was meant by ‘understanding’ in physics. The mathe-matical framework of relativity theory caused me no difficulties, but that did not necessarily mean that I had ‘understood’.’WERNER HEISENBERG, Physics and Beyond, 1971

Much of the qualitative feel that we have for the subject isadmittedly born of long experience with the quantitativemathematics but we should not shy away from developingthe ability to reason qualitatively in our students andRichard Boohan develops representations that might helpstudents do this in Using Graphical Representations to SupportMathematical Reasoning.

In Graphical Calculators and Mathematics in Physics Teaching,George Adie examines how technology can be used to develop mathematical understanding and Laurence Rogerscontinues the theme in Graphs in the Service of Physicists. In Mapsand Models – Approaches to Vectors, I suggest two ways ofapproaching the topic of vectors, again using technology tohelp.

It is important that physics is studied in an appropriatelymathematical way and to do so, some students amy needadditional support. In Free Standing Mathematics Units, JackAbramsky considers one way of providing this.

Without mathematics, the power of many physical argu-ments is lost. Mathematics provides a clear and succinct wayof describing how the world works and of speculating abouthow it might work. It gives rise to testable predictions aboutthe world and enables experimental results to be analysed ina way that allows comparison with theory. Mathematicsmakes physical reasoning easier – a point sometimes lost onour students! This is physics in mathematical mood.

If we accept that to be true to the subject, physics must betaught with at least some mathematical content then we haveto address the question of how the mathematical knowledgeand ability of students is to be developed. That is what thisbooklet is about.


One of the virtues of physics is that itprovides pleasures of a mathematicalkind. These are not the only pleasures itoffers – others include deft experimentingand vivid new imaginings – but they arereal and they are essential to the subject.

Simple pleasuresConsider some of the simpler such pleasures:

• Finding that very simple proportional relationships account for so many things: the stretching of a spring,the current in a wire, the acceleration of a mass, the temperature rise of a heated object, even the red-shift of distant galaxies.

• Finding that by taking thought many things can be calculated in advance, knowing just a few relationships: the movement of a ball, the level at which something will float, the distance of an image from a lens.

• Finding experimental results plotted on a graph stretching out along a straight line.

There are also of course the subtler pleasures of findingthat all these are not quite right; that linear relationshipsand simple calculations have their limits, and understandinga little of why this can be.

Pleasure in different ways of looking at thingsAt a deeper level, there is the pleasure, absolutely funda-mental to physics in mathematical mood, of seeing thatwhat look like very different ways of looking at the samething are ultimately the same. For example, that a storyabout rays being bent by a lens is in the end the same as astory about waves having their curvature changed by a lens

– that the awkward quantity 1/f in a lens equation is justthe curvature added by the lens. Or, similarly, that P 5 IV 5

I2R 5 V2/R are all the same expression. There are somekey moments in physics when this realisation was veryimportant. One was when ‘potential’ as defined by mathematical physicists was understood to be the samequantity as the ‘whatever it is’ which a voltaic cell pro-duces, which of course we now call ‘potential difference’.Another was when it was understood in quantum theorythat Schrödinger’s and Heisenberg’s ideas, seeminglytotally different, said exactly the same thing.

Related to this pleasure in how ideas connect is the pleasure of seeing the power one can gain by looking atthings in another way. Parallel circuits analysed in terms ofresistance are ugly and clumsy; parallel circuits analysed interms of conductance are simple, even obvious. The conductances just add, because the currents just add. Thereciprocal relationships R 5 1/G and G 5 1/R betweenresistance and conductance affords a switch of point ofview which simplifies and illuminates.

It is worth thinking of speed and journey time as anothersuch pair. Dualities like this turn out to be crucial in physics– one has only to think of the dualism between time and frequency domains in Fourier transforms. It is a great andimportant message: that looked at in the right way, manythings become simple, even obvious.

Another example is the great power of the logarithmicpoint of view. Quantities such as resistivity which vary overa wide range, or the size scale of objects from atoms togalaxies which defeat the imagination, suddenly makemore sense if plotted on a logarithmic (‘times’) scale. Andhow pretty it is that on ‘times’ scales reciprocal quantities(such as resistivity and conductivity) just run in oppositedirections.

Pleasure in model-makingI have so far ignored one of the most important and fundamental pleasures of all: the pleasure which drives

Jon Ogborn

1 Pleasures of a Mathematical Kind


some to do their physics mainly in the theoretical mood. Itis the pleasure of making models account for reality.

One such is the gravitational model of the motions ofplanets and satellites. Another, also well known at A level,is the kinetic theory of gases. The latter performs themagic of getting an explanation of what Robert Boylecalled ‘the spring of the air’ out of a tiny number ofassumptions. And it gives a new depth to what it means tobe hot or cold.

The strategy for such model-making is one that the laymanfinds utterly implausible: namely, writing down equationsbetween variables and conjuring something new fromthem. It is an important lesson that it can be done, andthat to do it requires imagination and daring.

An associated pleasure, the taste forwhich is not easily acquired, is that ofsimplification and idealisation. Howdare one model molecules as pointmasses which occupy no volume?How dare one suppose that moleculesbounce elastically from the walls oftheir container? And so on. It takestime to grasp that idealisation is not amatter of keeping calculations easy,but instead that it is crucial; thatadding assumptions which do notalter the main structure of the out-come even subtracts from the value ofa model. Beginners want to add detailto achieve greater realism; expertswant to remove detail to get to thebottom of things.

Models also afford, as one encoun-ters more of them, the unexpectedpleasure of finding that there are notso many of them after all. Or, rather,

that the same structures keep recurring: exponential decaymodels, oscillatory models, flux models, systems which are

gas-like. But within this is also the fascination with detailand particularity. Models may have family resemblances,but one kind of matter is not generally exactly like anoth-er. Silver is silvery, gold is gold; both shine because of thefree electrons in them, but why then the difference insheen? The study of condensed matter is full of such broadsimilarities and particular specific differences.

Pleasure in the power of your tool-kitAn example to illustrate some of these points about modelsmight be to imagine trying to model the vanishing of foamput on a river by a polluting source. Do you envisage foambubbles breaking essentially at the air surface, at the watersurface, or throughout the volume of the foam? Do youenvisage them breaking at random, or as having a definitelifetime? Clearly, different ideas will lead to different outcomes. A picture of bubbles breaking randomly at thesurface suggests a linear decrease in thickness of a foamlayer with time. The same but with bubbles breakingthroughout the volume suggests a relationship to exponentialdecay. Change the assumption to one of finite lifetime, andthe prediction might be of sudden collapse. And so on.The point is that one can bring to bear a tool-kit of knownforms of behaviour, and use analogies with them toexplore possibilities. Modelling is indeed in the imagina-tive business of inventing possible worlds. In physics, thisimagination is disciplined by insisting that predictionsabout a possible world do actually appear in reality.

Notice also that this thinking about foam soon producedsuch expressions as ‘linear’ or ‘exponential’. Model think-ing in physics is dominated by a quite small set of kinds offunctional relationships: linear, inverse proportion, quadraticterms, exponential and logarithmic forms, sinusoidalforms. That about exhausts the pop-chart of functions. Asa result, in time one comes to know them rather well,alone and in combination. Their shapes – slope, curve andintercept – become knowledge ready-to-hand. Their elegance and simplicity are another source of pleasure

•One of the most important and fundamental pleasures of physics is the pleasure of making models to account for reality

I N S T I T U T E O F P H Y S I C S

•Modelling is indeed in the imaginative business of inventing possible worlds

•Certain aspects of mathemat-ical thinking are so deeply entwined with physics that it makes no sense to think of them apart

8 P H Y S I C S I N M A T H E M A T I C A L M O O D

(with, later, the even greater pleasure of finding how theyare deeply related to one another).

Some doubtsThere will be many who will regard what I have written sofar as utterly unrecognisable as a picture of the relationshipof physics and mathematics, particularly at school level.Some, I fear, see mathematics as a regrettably difficultaspect of an otherwise potentially attractive subject. Morewill regard mathematics as somebody else’s business; theyexpect mathematics teachers to provide what is needed,ready and fully functioning when it is wanted. Others mayjust have given up, regarding mathematics as a hopelesscase so far as physics is concerned.

Caricatures these sceptical portraits certainly are. Butthey have in common a very prevalent tendency to draw aline between physics and mathematics; to split the issues intwo. I do not believe that the issues can be split in this way.Understanding radioactive decay is, all at the same time,

understanding about nuclei, about probability and aboutexponential change. The ‘mathematics’ involved is simplyan integral part of this piece of physics. Neither remainswhat it is if the other is subtracted out.

Thinking aheadThe contributions in this book look positively at what rolestheoretical – that is, mathematical – thinking can play inpost-16 physics courses, and look at what kinds of pleasureand profit this can afford to students. They regard certainaspects of mathematical thinking as being so deeplyentwined with physics that it makes no sense to think ofthem apart. Doing one is doing the other. We are notthinking here of physics as ‘having a problem’ with some-thing else – namely ‘mathematics’. We are thinking, self-ishly as physicists, about one part of doing physics. It mightbe called ‘theoretical physics’. It might be called ‘mathe-matical physics’.

At all events, it is physics in its mathematical mood.

•It’s hard to find pleasure in doing physics in a mathematical way when so much work is needed for students to improve their basic skills.

•Is the world mathematical? Why does mathematics find so natural a place in physics? Some students have an interest in philosophical issues so we should find time to discuss these questions.

•If different sorts of mathematicalunderstanding are to be valued bystudents and teachers, then they needto be built into the assessment system.

•Teachers should sometimes be stopping to say ‘Wow!’ and enthusing about what the maths does. We tend to hide how we feel about it.

D I S C U S S I O N P O I N T S


The difficulties that I have experiencedover the duration of my course can bedivided into three distinct categories:new mathematical concepts not intro-duced until A level; problems withinvolved algebra; following the logic ofmathematical derivations when aknowledge of A level mathematics isassumed.

The first problem is probably themost significant as well as being themost far-reaching. This is to do withnew mathematical tools and proce-dures such as exponentials, logarithms,the small-angle theorem and even vec-tors. Some of these, such as vectors andexponentials, form a central part of thephysics course that I am taking. Inaddition to this, there are non-essentialtechniques such as differentiation andintegration which, while not vital, canhelp in improving understanding.There have frequently been times dur-ing the course when new areas ofmathematics have popped up unex-pectedly in a lesson and caused mesome considerable confusion. As aresult of this lack of explanation, I havehad to pick up a lot of these new ideasas we went along. A good example issimple harmonic motion: we wereusing circular motion to develop anunderstanding of the SHM equations;differentiation was used to provide analternative viewpoint. While my col-leagues were able to follow both argu-ments, I spent most of the lesson tryingto understand why sin(t) had become

cos(t). I would like to see some of theseconcepts taught up front to remove themystery surrounding them. Simpleways of visualising functions are essen-tial; for example, for the exponentialfunction, the fact that the rate ofchange of a variable is proportional toits value provides a way of understand-ing the shape of the graph. Even if it isnot possible to have a mental picturefor some functions, such as logarithms,the reason for using them should be atleast made clear.

The second issue concerns the alge-braic competence required for A levelphysics. Despite the fact that I canoften get by with skills learned fromtaking GCSE mathematics, there areother times when a greater degree ofalgebraic skill is required to gain afuller understanding of the subject. Ihave found this to be a significantproblem when writing my researchand analysis project in the Nuffieldcourse, for example. In addition to this,there are times when, in exam situa-tions, a lack of experience or practicein using algebra can lead to simpleerrors being made which cost marks.The box contains one such exam ques-tion in which some algebraic manipu-lation was required. Although thismight be trivial to an A level mathe-matician, it took me some time to com-plete. While I eventually arrived at theright answer, I made a simple mistakethat could have prevented me fromdoing so.

2 Physics without Mathematics at A level : a Student’s View

Tim Sloan

I am a student at LeedsGrammar School currentlyin my upper-sixth year. My subjects are biology,chemistry, physics andFrench. I therefore have tocontend with the problemof trying to succeed atphysics A level without theadded benefit of a separatemathematics A level.

Kepler’s third law can be explained usingNewton’s law of universal gravitation. Writedown expressions for the gravitational fieldstrength due to a planet of mass M at a distance r from its centre and the centripetalacceleration experienced by a moon in orbitaround the planet with radius r. Hence show

that the constant value of is given by }4Gp

M2}

where G is the universal gravitational constant.

Here is my first attempt at the questionin the exam:

a 5

a 5

v 5

v2 5

a =

5

}GM

rT2

} 5 4p2r4

5 (What do I do now?)r 5

}T2

GM}4p2

}4p

T

2

2r2

}}

rGM}r2

}4p

T

2

2r2

}

}r

4p2r2

}T2

2pr}

T

GM}r2

v2

}r

r 3

}T 2

(This is where I went wrong.)

I N S T I T U T E O F P H Y S I C S

If I had realised, as I later did, that

divided by r was in fact

then I would have had no problemanswering the question: a trivial errorthat may seem obvious but caused mesome difficulty.

As well as these two specific mathe-matical problems, there is the moregeneral difficulty that arises whenknowledge of A level mathematics isassumed. This can lead to ‘logic gaps’from a non-A level mathematician’spoint of view when calculations areperformed in class. This can be amore subtle issue as it is often difficultto know when this is coming into play.Such logic gaps can have a consider-able detrimental affect on my under-standing of the subject matter. Wespent one lesson trying to come upwith a derivation for the equationsexplaining simple harmonic motionin pendulums. While this was notessential from an examination pointof view, it was still a useful exercise.Over the course of the lesson, I wasunable to arrive at an answer,although one of my fellow students,taking A level mathematics did. Heexplained it to us by writing his find-ings out on the board. In mid-deriva-tion, I suddenly found that some-where along the way, I had complete-

4p2r}

T24p2r2

}T2

ly lost the plot. I later found out thathe had, in one step, invoked the small-angle theorem so familiar to A levelmathematicians. This one step causedme completely to fail to understandthe entire derivation.

A more careful use of mathematicsIn general, I would like to see a morecareful use of mathematics in physicslessons. I find that it is often easier toapproach a problem using logicalphysics-style reasoning, only usingmathematics when it is no longer pos-sible to maintain a visual picture ofthe situation. A multiple-choice exer-cise that we did in class showed thisperfectly. One question involved ahollow cylindrical pipe – the objectbeing to find its volume in terms ofthe radius, the thickness, and thelength. I approached the problem byimagining the pipe as a flat, unrolledobject with dimensions length 3 thick-ness 3 circumference and this, despitebeing not entirely accurate from amathematical point of view, quicklyyielded the correct answer.Meanwhile, another colleague ofmine was answering the question byfinding the total volume of the cylin-der, pr2l, and subtracting the volumeof the empty space, p(r 2 thickness)2l.While this would have eventually ledto the answer, it was, sadly, not possi-

1 0 P H Y S I C S I N M A T H E M A T I C A L M O O D

ble to achieve this within the timeboundaries of the paper. I have alwaysbeen led to believe that this kind ofintuitive calculation is part of whatphysics is about. If this is the case,then why is this sort of physics notencouraged more in A level physicscourses? Physicists seem reluctant todo this, almost as if proper physicistswould not resort to such things whenmathematics is a much better way ofsolving problems. While I appreciatethe importance of mathematical rea-soning, shouldn’t physicists be taughtto try and gain a deeper understand-

ing of their subject? In my experience,once ‘the rate of change of N is propor-tional to the value of N’ becomes

5 2lN, some understanding is lost.

Perhaps this is due to my lack ofmathematical ability, but it seems tome that once we have downloadedour thoughts into mathematical nota-tion it is harder to see the subjectclearly, to think laterally and to comeup with the kind of inspirations thatphysics is all about.

dN}dt

•Is it necessary to do A levelmathematics alongside A level physics to under-stand the physics properly?

•The mathematical skills of students differ widely, so how do we support some while at the same time stimulating others?

•Physical ideas and the way notations express them need emphasis; notation used alone may block understanding.

•Physical reasoning can bemathematical without depending on algebraic manipulation.


P H Y S I C S I N M A T H E M A T I C A L M O O D 1 1I N S T I T U T E O F P H Y S I C S

Two physics teachers meet over a coffee.‘They can’t do maths anymore,’ saysone. The teachers go on to share theproblem. But does each really knowwhat the other is talking about?

For the first, it is problems with mathematical modellingand a lack of appreciation of calculus and differentialequations ... and only last week a student had trouble withmanipulating an algebraic expression. Interpretation offunctions and graphs isn’t easy nowadays either.

Coffee cups are rattled. There is disagreement. For theother teacher, problems are far more basic. The friendsmeet rarely and this is no time to argue. But basic does not mean trivial. These problems are of overwhelming significance. They loom large and bar understanding inalmost every area of physics. They are a daily frustrationto students and teacher. Early in many courses comes thestudy of the Young modulus (in materials teaching) andresistivity (as part of electricity). Surely many teachers willbe familiar with the annual fray, depressingly early interm, when all pretence of teaching physics is maroonedon an island of mathematical banality.

So the teacher tries to explain: ‘Standard form is usuallythe first problem’.

‘Yes, no feel for logarithmic scale.’‘Well, no, not really that ... ,’ in fact mainly incorrectly

reading the display on, and entering numbers into, a cal-culator resulting in lost (or gained) powers of ten. A lessoncan be lost in a discussion of calculator displays: 2 3 108

becomes 2 3 10EXP8, indistinguishable to some students,who then appear simply incompetent to teachers. Then,on particularly bad days, we get a display reading 208

being interpreted as 208 not 2 3 108. Worse are thoseannoyingly frequent occasions when the number of powers

of ten lost is equal to those gained (a conservation rule?)and a bewildered student takes some convincing that anumerically correct answer is wrong.

Of course, the reason that standard form is being used isbecause the numbers are large or small. It soon emergesthat a feel for how large and small these numbers are islacking. A discussion on magnitudes becomes necessary.

This ‘power of ten’ theme continues when lengths aremeasured in mm. Conversion between mm and m is prob-ably straightforward, mm2 to m2 is difficult, mm3 to m3

most unlikely to be successful. Then follows another lessonon unit conversion, brought about by the realisation thatthe meaning of ‘micro’ is a mystery.

‘Powers of ten’ behind us, the troubles for the Youngmodulus and resistivity are not yet over. Most likely thevalue given for the thickness of a wire is its diameter sincethat is what is measured. Calculating the area thenrequires the use of A 5 pr2. That this is the same as

A = }p

4d2

} is the work of another lesson. And does pr2 mean

p2 3 r2 or p 3 r2?

Introducing the Young modulusHowever, the Young modulus has often run into difficultiesbefore that. Without care, half a lesson has been devotedto get from stress over strain to Fl/Ae. First the definition:the Young modulus 5 stress/strain. Next a substitution. Theresult, the famous ‘four layer’ fraction. Now the bottom ofa denominator becomes a numerator and the bottom of anumerator becomes a denominator and so it is clear that ...

Of course it is! Numbers are put in: }13

} divided by }12

} is }23

}.

Not clear? Oh well, do it on your calculator: 0.33 dividedby 0.5 . See? It must be right! Now what were we doing?Yes, I remember, it was the Young modulus.

Only then dare we say how obvious it is that this can berearranged to give F5EAe/l ! We await the dreaded question:what does the triangle look like? Triangles used as a rote

3 Basic Problems?

Philip Britton

1 2 P H Y S I C S I N M A T H E M A T I C A L M O O D I N S T I T U T E O F P H Y S I C S

•When the number of powers of ten lost is equal to those gained, a bewildered student takes some convincing that a numerically correct answer is wrong

•Students are troubled and teachers are frustrated, as the best planned lessons turn into tutorials in basic calculation and algebraic manipulation

means of manipulating GCSE formulae become less useful as study progresses. Too often it becomes clear thatwithout this scaffolding, provided to help at GCSE, thehouse still collapses. In fact the triangle scaffolding was so good that the house of algebraic manipulation wasnever even built inside it. That makes it difficult to sort

out }GM

r2m

} 5 .

Now it may be desirable to plot a graph of F against e.Does this stir distant memories of y5mx with the slopebeing equal to EA/l ? Well, perhaps. This translation fromone set of letters to another just touches on the edge ofanother issue. What exactly do the algebraic letters meanto the students? Do they represent a value, the unknown,or do they rather represent all possible values of a quanti-ty? Do they convey the relationships between variables inthe same way that they do to us? That issue aside, why isthe graph being plotted anyway? Is it to interpolate orextrapolate or to order data and detect trends? What doesthe now overused phrase ‘direct proportion’ mean?

The coffee is cold. The friends sit in companionablesilence for a while before thinking on.

If the course started with electricity or materials, it hasbeen only a few days before any sense of the whole, of thepurpose of, and the interest in, studying physics has beenlost in this basic, detailed, but important, calculationalwork. After several days of admiring these calculationaltrees, the wood is forgotten. We have a situation where stu-dents are troubled and teachers are frustrated, since thebest planned lessons turn into tutorials in basic calculationand algebraic manipulation.

What about vectors and resolution?Yes indeed, what about them! Here was common groundat last. After all, that is why the course no longer startedwith statics and mechanics, which had simply been a dis-aster. To our second teacher, teaching vectors meant get-ting over the hurdle of drawing lines of a certain length

mv2

}r

and in a certain direction that represented a velocity. Thatwas a reasonable conceptual difficulty.

The friend sighed. The message was still not clear. Wasit not obvious that the real issue was squaring, adding androoting in the correct order when using Pythagoras’ theo-rem and remembering that sine was opposite overhypotenuse?

The two teachers live in different worlds but both sets ofproblems are very real.

What might be done about this problem with the basics?Perhaps this is not a difficulty with maths, it is a problemwith numeracy. How might physics, when in mathematicalmood, acknowledge and deal with this problem?

Whose problem is it? It is easy for teachers to blame theincreasingly ill-equipped students. If feeling supportive ofthe students, we accept that they are not to blame: it is thefault of their GCSE maths teachers. If we wish to avoidblaming a colleague, we decry the DfEE. It is their fault,not ours.

Yet the above is unimportant. We adapt yearly to takesteps more slowly, to accept that the obvious isn’t, to covermore elementary examples, to distinguish calculation fromphysics so the story line is preserved. What is important isthat in making this adaptation we must move more slowlythrough physics content, and the syllabus does not recog-nise this. So we still have to move too quickly. We get frus-trated and the students get lost (some within and some byleaving the classroom).

One step at a time ...One of the issues is accepting that the obvious isn’t. Wemust always be aware of using linked-up logic, and beingsure no step is missed. Are we careful enough to showevery step? We can be sure that the step we miss will be theone that makes the whole thing incomprehensible to onestudent. Can we address how this is to be done while notboring the able and confusing the slow?


Use of calculators ...‘Powers of ten’ problems often stem from ignorance of howcalculators work. We must invest the time to ensure thatcalculators are understood so they become useful tools andnot stumbling blocks. This issue is repeated and oftenamplified for the use of computer software packages.

What do I do?Question translation is a hidden difficulty, where a prob-lem is well understood when explained, but identifying thenature of the problem from a wordy question introduces alevel of sophistication which proves to be too much. Wemust be aware that it is possible for a question to be acomprehension exercise rather than a test of physics.

Many questions on the Young modulus and resistivity areabout powers of ten or radii and diameters rather thanmaterials, however enlightened is the mark scheme.

And whose fault is it? Not an answer but a constructivesuggestion: perhaps asking who is in a position to help thesituation rather than asking who is to blame will produceresults. When a dead body is found a murder inquiry islaunched. An ambulance is called to an injured person.The mathematical aspects of physics are injured but notdead, so where is the ambulance? In the short term thismust be us, the physics teachers interacting carefully andthoughtfully with our students. In language familiar toschool teachers, we didn’t drop the litter but we shouldpick it up and put it in the bin anyway.

•Teaching more slowly does not always help. If we wait for full mastery at every step, we never progress. Understanding grows with familiarity and use.

•Notation needs to be standardised to support weaker candidates. But for more able students seeing different representa-tions for the same thing can be very useful.

•Can more be done at GCSE level to build good mathematical foundations?

•Some schools offer bridging courses after GCSE exams. Should funding be availableto offer those starting post-16 physics courses a supplementary course in basic mathematics?

•Students’ facility with technology – calculators, computers – should not be assumed. Teaching technological skills needs space within the curriculum.


•We must invest the time to ensure that calculators are understood so they become useful tools and not stumbling blocks

•Sometimes the obvious just isn’t


The views of physics teachers and studentshave played little part in the recent skirmishes around school mathematics.Mathematics teachers perceive schoolphysics at 11–16 as having becomemerged into a largely qualitative subjectcalled science, thus forsaking its formerdependence on mathematics.

Meanwhile it is clear that A level mathematics has main-tained its proportionate share in the age-cohort only byenlarging its entries among students who are not also takingphysics, and particularly among girls. Many younger mathe-matics teachers have not themselves ever studied physics as aseparate subject, and thus owe no allegiance to it.

Part of the reason for the skirmishes about both primaryand secondary mathematics is a lack of clarity about why weare teaching the subject. The general view is that it is a collection of some bits of useful knowledge and a rather larg-er number of techniques. Of course these are important, orat least most of them are. However there is more to mathe-matics than this.

The technique trapWhen we teach mathematics as if it were only a set of techniques, we now know pretty well what happens: asmall number of pupils construct meanings and relation-ships for themselves in order to make sense of the reper-toire of techniques, while others try and learn the techniques separately and end up forgetting the interme-diate steps, confusing them with each other, and not know-ing when to apply any of them. This renders most of thepopulation paralysed with fear when presented with any-thing that looks like mathematics, with a small and rather

Margaret Brown

4 A View from the Mathematics Side of the Fence

arrogant elite who cannot appreciate everyone else’s problems.

To counteract the emphasis on techniques, an attempt wasmade to shift the balance, at both primary and secondarylevels, towards laying down an interconnected network ofmeanings and images across a broad conceptual area, andencouraging pupils to have a go at tackling investigations andproblems. The official recognition of the wide spread ofattainment in the 1982 Cockcroft Report also led, uninten-tionally, to a much increased use of individualised learning.The combination of individualisation and a preponderanceof contextual problems meant that in some cases the pendu-lum swung too far, and pupils were left dependent on primi-tive methods they had developed themselves, and with littleview of what it was important to know.

Emphasis on ‘real-life’ settingsThe GCSE examinations, the National Curriculum andSATs (national tests) generally supported the moves towardsa broader and more conceptually based curriculum. Manyitems were set in ‘real-life’ contexts and there was less em-phasis on abstract standard techniques. Investigations andproblem-solving marginally survived, but were only assessedformally in a rather stereotyped format at GCSE. Examina-tions consist of tiered papers and more straightforward sin-gle-step problems in all three tiers.

The net result of the changes, demonstrated by results ininternational tests, showed some success in tackling what hadbeen seen as the major problems of the earlier technique-rid-den curriculum. In comparison with other countries, Englishpupils now had a very positive attitude towards mathematics,good performance in problem-solving and in areas like sta-tistics and geometry. However this was accompanied by abelow average performance in core technique-heavy areaslike number and algebra.

Politicians of both parties have tried various means toswing the pendulum back. The National Curriculum large-ly maintained the previous emphases and subverted the


‘back to basics’ intentions, with thesupport of an industrial lobby particu-larly interested in ICT, oral communi-cation and group problem-solving.The many changes of structure, ratherthan content, in the NationalCurriculum had the negative effect ofdiverting national energies and pre-venting any serious attempt at devis-ing forward-looking schemes.

Structural changes in the NationalCurriculum also led to some unfortu-nate side-effects: passing examinationshas always been perceived as moreimportant than what is actuallylearned. This has led teachers tobecome very alert to devising optimumstrategies for examination success. Forexample the possibility of more easilyachieving grade C, and in some casesalso grade B, from the intermediatetier examinations at GCSE has meantthat many pupils with grade C and afew with grade B have little knowledgeof algebra or trigonometry.

Even for the higher tier, the smallnumber of marks awarded to formalalgebra, in comparison, for example,to data handling, has meant that someteachers for a while decided that it wasnot worth the investment of timerequired to master formal algebra.

However changes in GCSE examinations and NationalCurriculum tests have more recently taken place to try tocorrect some of the side-effects, which has meant thathigher tier GCSE examinations now contain more alge-bra. Some attempts are also now being made to alter theGCSE structure to avoid a straight choice between inter-

mediate and higher tier for potential grade C candidates.At primary level similar changes are taking place. A second

attempt at ‘back to basics’, strongly supported by Ofsted, hasculminated in a National Numeracy Strategy. This is likely tohave some success, not least as it assists teachers to appreci-ate what is expected and builds on strengthening of mentalarithmetic which has already been largely implemented,thanks to the changes in the national tests.

However the downside is that renewed practice of stan-dard pencil-and-paper algorithms, a curriculum which isshort on meanings and applications, and a perceived ban oncalculators in primary schools, is likely to bring us full circleat primary as well as secondary level.

Modularisation rather than modernisationAt A level, much of the available energy has gone intomodularisation rather than curriculum modernisation.Modular syllabuses are in some cases now carved up intomodules in pure mathematics, and either applied mathe-matics (actually mechanics), or statistics. Pure mathemat-ics is mainly mathematical methods (of differentiation,integration, solving trigonometric equations, etc.), ratherthan ‘big ideas’ like functions, rates of change, probabilityand risk, vector spaces. There is some evidence that mod-ular tests have reduced the range of classroom activity topractising short routine items, of an easier type than thosewhich occurred in the previous terminal examinations.

Of course there are some honourable exceptions, likethe SMP and MEI A level courses which have tried toincorporate ICT and to base themselves on the notion ofmodelling. These introduce the various mathematicalfunctions in their roles as models and require attention tothe characteristics of real-life problems. In such courses,kinematics has been regarded as an important applicationof models of continuous rates of change. Physics exampleshave also been used in probability models. Work on forces,dynamics and statics has generally been confined to lateroptions. These syllabuses have also tended to include


sented, including the translations from one mode to an-other;• understanding and ability to use fundamental mathe-matical processes, including induction, deduction, andmathematical modelling;• an overview of what can be done by computers and ad-vanced calculators, and an appreciation of how these facil-ities can be used to solve problems, both in mathematicsand in other areas of application.

Instead of this potentially exciting agenda, which I believecould re-invigorate teachers and attract new students intothe declining university degree courses in mathematics andin mathematics-based subjects, we are in danger of retreat-ing into a narrow view of the subject as only a collection oftechniques, some potentially important but unmotivated andungrounded, and others simply outmoded.

I believe there will be some advantage to physics teachersfrom the current changes, in providing students with greatertechnical facility, which may be combined with understand-ing for the minority who are motivated to make sense of itall. Nevertheless I fear that this advantage could be out-weighed by the bulk of students bringing even less insight,less confidence, and less ability to remember and apply tophysics what they have learned in mathematics lessons.

coursework and problem-solving, and have given rise tomore classroom discussion. They have been more attractiveto girls, who tend to perform better on them in comparisonto the more traditional schemes.

The future under the new A level arrangements is likely tobe little different. The proposed A and AS level cores arealmost entirely composed of (pure) mathematical techniques.While the AS level core would superficially equip a studentwith most of the mathematics needed for A level physics, thesyllabus is so full of techniques and so unmotivated by appli-cation that more average students may be more confusedthan enlightened by it. In fact the only advantages of thechanges are for mathematically able students for whom theproposed new style of S level papers is likely to provide morechallenging questions than occur in the current A level.

SummaryTo sum up, I believe that the key rationale for mathematicsteaching at the start of the 21st century is that of conveying:

• understanding of what the subject is about, in particu-lar of the big ideas which underlie it, why they have devel-oped and proved to be useful;• facility in handling each of the different ways, spatial,graphical and symbolic, in which these ideas can be repre-

•Examinations need to be better designed so that a good GCSE grade in mathematicsreally does mean a basic competence, sufficient for the student beginning A level physics.

•Science and mathematics departments in schools should cooperate more at GCSE level. Only extra funding can make the necessary time available.


•Changes in the National Curriculum may lead tounfortunate side-effects as teachers devise optimum strategies for examination success

•Average students may be more confused thanenlightened by the new AS level maths


Andrew Raw Research has shown that there are differences in how experienced physicists,‘experts’, solve problems compared withless experienced ‘novices’, including A level students tackling a topic for the first time.

It is the purpose of this article to argue that there areteaching techniques that have been shown to help students in their progression to becoming ‘experts’ which should beinvestigated.

How would you solve the following problem?An object of mass 5 kg is placed on a 40 0 smooth slope. When itstarts moving from rest it is 10 m above ground level:

a) how fast is it moving after 2 seconds?b) how fast is it moving when it hits the ground

(at the bottom of the slope)?

Did you draw a diagram to help? Were you able quickly tospot a route to the solutions? Experts tend to draw diagramslike figure 1 for the above question, including physical quan-tities like forces and all data given in the question. Novices

either do not use diagrams or draw them with the data andphysical quantities missing. Using diagrams and noting down

data is thought to help, because it focuses attention on thedata and goals of the problem and because it allows ‘brainpower’ to be used to think about the problem rather than tohold data. It is thought that the brain can assimilate infor-mation from diagrams more easily than from text.

Experts, unlike novices, can quickly deduce which princi-ple(s) to use to solve a problem – for example, part (a) of theabove question requires the resolution of the force,Newton’s second law, and a constant acceleration equation,while part (b) may be more easily solved using conservationof energy. Experts are thought to be able to spot a solutionmethod quickly because they have a database of problemtypes, compiled from the experience of solving many prob-lems, stored in their long term memory along with themethod of solving each type. Another reason why diagramsare useful is that diagnosis of the type of problem is easierbecause these are often stored in diagrammatic form.

Experts tend to ‘work forwards’ to a solution, using phys-ical principles to derive equations from which they calcu-late the unknown. Novices tend not to use physical princi-ples but instead focus on the unknown and then, sometimesfrantically, try to find an equation that includes it and vari-ables with known values.

Novice students have cognitive difficultiesNovice students have cognitive difficulties in the way inwhich they approach the solution of the problem, in addi-tion to the difficulties they have in diagnosing the type ofproblem and the method of solution. Research, mainlywith college students, has shown that they often:

• do not scan the question carefully to extract all the relevant data

• act impulsively, latching on to the first piece of data that they see

• do not take a planned approach to the solution of the problem, and sometimes resort to trial-and-error approaches

5 Improving Students’ Mathematical Problem Solving Ability – Lessons from Research

Figure 1

40°

R

W = 50 N

10 m


•Algorithms show expert methods to the students and teach them how to apply the principles of physics in a more explicit way than a worked example

•A heuristic may be more usefulfor A level physics than an algorithm since it will be applicable to many more varied types of problems

• miss data in the question that is not explicitly stated but is relevant to the problem – for example, in the above question one would need to know that ‘smooth’ meant ‘frictionless’

• find it difficult to visualise the situation in the question and thus find diagrams difficult to draw

• lack the motivation to make the cognitive effort needed to solve the problem

• find difficulty in using two or more pieces of information at once

• do not monitor the approach that they have chosen to check for suitability but carry on until they reach a dead-end

• have difficulty in putting themselves in a listener’s shoes and so cannot explain their ideas clearly.

Thus students have less chance of spotting a useful methodfor solving a problem and, even if they do, will find it moredifficult to arrive at and report a satisfactory solution.

So, in order to become more expert in their problemsolving, the students must:

• learn to diagnose the type of problem that they are faced with and the methods of solution

• improve their problem-solving techniques and thus lessen their cognitive difficulties

• change from ‘working backwards’ to ‘working forwards’.

Using algorithmsThese are a series of instructions that describe in detail thethinking needed to solve a certain type of problem. Oneexample is an algorithm for the solution of problems usingthe law of moments, which directs the students to draw adiagram, giving algebraic symbols to unknowns: choosethe point about which to take moments; decide whichforces have a moment about this point; write an equationusing the law of moments.

The algorithms show expert methods to the students and

teach them how to apply the principles of physics in a moreexplicit way than a worked example. The above algorithmis an expert method, since it involves ‘working forwards’ byusing a principle to write an equation and then to find theunknown.

Algorithms may help with some of the cognitive difficul-ties listed earlier because they may urge a student to scanthe question carefully for relevant data, to use diagramsand to show a planned approach to a question.

However, algorithms are not a miracle cure. Studentscan find them difficult to understand, particularly if theyare struggling to master the basics on which the algorithmdepends – for example, trigonometric functions, splitting aforce into components. They are also of little use unless thestudent knows when to apply them, and they can eachonly be applied to one type of problem.

Using a heuristic A heuristic may be more useful for A level physics than analgorithm since it will be applicable to many more of thevaried types of problems facing students. Such a heuristicmay, among other things, direct the student to:

• read the question carefully, noting down data given inthe question, including data implied but not explicitlystated

• make a mental picture of the situation in the questionand then draw a diagram

• make a note of the goal of the question and variables that have known values

• have a plan in mind before attempting a solution • check a solution to see if it is of the right order of

magnitude with the correct units and number of significant figures.

Heuristics have been shown to work very well with college mathematics students when reiterated over along period.


Remarkable evidence of successThe Modelling Workshop Project from Arizona StateUniversity found that traditional instruction was uniformlypoor at helping students to learn Newtonian physics. Lectures,demonstrations, problem-solving sessions and student prac-tical work aimed at demonstrating concepts were relativelyineffective – whatever the experience or style of the teacher.

When teachers used the techniques suggested by theproject, they have found large gains in their students’achievement – particularly less able students – comparedwith those taught in a more traditional way.

The main elements to the approach, common to practicalwork and problem-solving sessions, are as follows:

• Students are encouraged to work in groups and to present the results of their work, with other members ofthe class interrogating them and commenting on theirwork. Theory suggests that students will make moreprogress when working with peers than they would ontheir own, as they can discuss ideas and get help from eachother. This approach may also encourage ‘metacognition’– becoming conscious of one’s own thinking – which is believed to be crucial in developing cognitive skills. (Thisidea has been used in the CASE project which has hadgreat success in accelerating the cognitive development ofBritish adolescents, leading to greatly improved English,maths and science GCSE results.) Students become moreconscious of their thoughts when discussing their ideas inthe group, when explaining something to another member and through presenting their work to the class.• Like the CASE project, the teacher does not tell thestudents the answer, but leads them to it by skillful questioning (although, of course, there has to be tuition insome lessons). The teacher encourages metacognition byconstantly pushing the students to justify their ideas, act-ing as a ‘Socratic Inquisitor’.

In experimental work, students are encouraged to find a

mathematical model that describes the experiment. Theyare, after class discussion, encouraged to consider the vari-ables that describe the experiment, to investigate the rela-tionship between two of the variables, to find a mathemat-ical model which describes this relationship and to evalu-ate how good a description the model is.

In problem-solving sessions, students are first told tochoose a model that describes the situation in the questionhelpfully. So, a student might describe a mass movingdown a slope by using constant acceleration equations orthe principle of conservation of energy, depending onwhich would be most useful. Thus their problem solving isguided by the use of principles – ‘expert’ behaviour –rather than being led by a search for an equation. Studentsare encouraged to present solution methods to the class:this may encourage metacognitive thinking and may helpthem to store the solution methods in their memory, likeexperts. The students are also, as with the CASE project,encouraged to think of how the techniques used to solveone problem could be used to solve other types of problem.Thus any newly acquired techniques become more generalised and thus more widely applicable.

I used this method with an experiment to model a suspension bridge. A beam is suspended from newtonmeters at either end to model the suspension bridge (figure 2). A weight is suspended from the beam and is

Figure 2

•Students make more progress working with peers than they would on their own, as they can discuss ideas and get help fromeach other

•Problem solving is guided by the use of principles – ‘expert’behaviour – rather than beingled by a search for an equation


•Students are introduced to the process that is at the heart of physics – making models to describe the world– and the idea that these models have limitations

•Through listening to the presentations, teachers can see clearly the areas of difficulty for students

moved across the bridge to model the motion of a lorry. Students were invited to:

• draw a diagram and identify variables that model the apparatus effectively

• predict relationships between the variables using physicsprinciples like the principle of moments – these are mathematical models describing the apparatus

• test these predictions• give a presentation using an OHP, justifying their

predictions and experimental method and explaininghow accurate their predictions were and whether their model described the experiment effectively

• explain why their model did not describe the experiment effectively.

When I tried this approach with two groups, one con-taining students with grades A* or A in GCSE maths, the other with students mainly having grade B, I found that it challenged both sets of students and helped them to make progress although the lower

ability group did find it hard. The approach had somepositive effects:

• it is very active and forces students to think about and discuss the principles that they have been studying – they learn from mistakes and misconcep-tions when these are pointed out to them by others in the group

• it introduces students to the process that is at the heart of physics – making models to describe the world – and the idea that these models have limitations

• it guides students to consider experimental errors and the limitations of measuring instruments

• students are required to have experience of presentingideas – one of the ‘Key Skills’ mentioned by Dearing

• teachers, through listening to the presentations, can see clearly the areas of difficulty for students.

Thus, there seems to be overwhelming evidence that thisapproach is worth investigating further with A level physicsstudents in UK schools.

•As teachers, we need first to be clear about our own thinking and then communicate this to students. Heuristics and algorithms encouragea more logical, systematic way of approaching problems.

•Are the likely benefits of theseteaching methods sufficient to warrant the extra time required?

•To encourage such approaches, students need access to banks of skills-orientated questions. Recent textbooks have fewer such questions.

•Integrated Learning Systems should be designed for physics education, to diagnose weaknesses and provide appropriate questions, hints and solutions for individual students.

•Support with problem solving can help students to gain confidence.



This short article outlines one approachto tackling maths in an A level physicscourse. It is more of a description andless of a discussion document than someof the others in this booklet, but I hope itwill none the less provide a useful contribution to the discussions and willbe helpful to others involved in syllabusdevelopment.

The mathematical backgrounds and needs of studentswere among the many issues we discussed when designingthe Salters Horners Advanced Physics course. We knewthat our students and teachers would be in the same situation as those taking any A level physics course:

• some students have a strong mathematical background at GCSE whereas others do not

• some students are taking an advanced maths course alongside physics while others are not

• some of the maths covered in an A level maths course is relevant to physics and comes at an appropriate time, whereas much is not relevant and comes too soon or too late

• some of the notation and terminology used in a maths course is the same as that commonly used in physics, but some is different

• some schools and colleges provide additional maths lessons and support for those taking physics but others are unable to do so

• some students will be going on to further study (e.g. physics or engineering degree courses) that requires more advanced mathematics.

Rather than leaving the mathematical elements to chanceand hoping that they would be covered somehow, we decid-ed from the start that some maths needed to be integratedinto our physics course.

Salters Horners Advanced Physics is context led Like all science courses with the Salters brand name, thestructure and content of the Advanced Level Physics courseare driven by contexts, applications and issues, with princi-ples and techniques introduced on a ‘need to know’ basis.We decided to take the same approach to maths withinphysics – namely, to include mathematical ideas as andwhen they were needed in order to tackle the physics.

This decision has meant that we include the ‘basic’ mathsneeded in A level physics, but we do not attempt to teachthe more advanced maths (e.g. calculus) that would berequired for further study. If we had decided to do the lat-ter, we would have ended up trying to cover substantialamounts of A level maths in addition to the physics content,making the course highly overloaded. In any case, we feelthat if students are intending to go into areas that requiremore advanced maths, then they should have been advisedto take A level maths as well as physics – and if they are notfollowing such a course, then that is a problem that isbeyond our scope to tackle. (We have, however, providedthose students who are studying A level maths with oppor-tunities to apply some of the more advanced techniquesthat they will have learned.)

Building in the mathsThe choice of contexts for the teaching units was influ-enced by considering the mathematics that would be need-ed to tackle the relevant physics. This worked in two ways.First, we deliberately looked for contexts that would pro-vide opportunities to tackle some key areas of maths inphysics. For example, the use of sporting contexts enablesideas about areas and gradients of graphs to be introduced naturally when considering athletic performance, and

Elizabeth Swinbank

6 Maths in Salters Horners Advanced Physics

Further informationSalters Horners Advanced Physics AS LevelBooks 1 and 2, Heinemann Educational, 1998.GCE Salters Horners Physics AS/A Level Syllabus (pilot),Edexcel Foundation, London, 1998.


addition and resolution of vectors are highly relevant torock-climbing. Second, the mathematical as well as thephysics content of the units was designed so that there is agradual development of ideas. For example, powers of tenare used in the early units and throughout the course, base-ten logs are used in the second half of the first year, and log-arithmic graphs and exponential changes feature in the sec-ond year of the course.

Further, we decided to make the mathematical aspects ofthe course explicit in the syllabus, rather than tacitly assum-ing that they would be covered in passing. For example, the requirement that students should be able to‘determine the area of a graph by drawing and (in the caseof a straight-line graph) by calculation’ is part of the syllabus for the ‘sport’ unit in the first module of the course.

‘Maths Notes’ and other resourcesMaterial covered at GCSE, such as algebra, trigonometry,basic graph plotting and the use of calculators, is sum-marised in a set of ‘Maths Notes’, printed at the back of themain student text book. Where these items are needed, ref-erences in the main text, or on the photocopiable sheetsthat support some of the activities, direct students to the rel-evant place.

Additional maths required for A level physics (such asgradients of graphs, logarithms and radians) is developed inthe main student text and also summarised in the ‘MathsNotes’, to which there are references in later units wherethe same ideas are needed again.

We envisage the ‘Maths Notes’ being used in variousways. Particularly for students with a weak GCSE back-ground, they provide a readily accessible ‘reminder’ ofsome key ideas that students can use individually withoutthe embarrassment of always having to ask for help. Theydemonstrate to all students the conventions of notation that

are commonly used in physics, which might differ fromthose used in separate maths courses, thus helping studentsto make connections between maths and physics. Also theyprovide an on-going summary of the maths introducedduring the course, to which students can readily refer asneeded. In some schools and colleges, they might also beuseful in a separate ‘maths for physics’ course.

There are also some photocopiable sheets in the teachers’resource pack which use more advanced maths. These areintended for students who are also studying A level maths,and they provide opportunities for such students to applymore advanced maths in physical situations. For example,there is a sheet about the maximum power theorem thatuses differentiation, and sheets that take a calculusapproach to equations of motion and to the gradients andareas of motion graphs.

Evaluation of the course is taking place from September1998, when the course started as a pilot with a cohort of athousand students (the maximum allowed under QCArules). Feedback collected during the pilot will help pointthe way forward for future developments involving Saltersand other courses. Twenty of the pilot centres have there-fore been asked to provide the project team with detailedfeedback from students, teachers and technicians.

So far (October 1998), students and teachers have beenasked about their initial expectations of the course. Whenthey were asked (in a free-response question) what theyexpected to be the main advantages of the Salters’ Hornerscourse, several students mentioned the potential usefulnessof the ‘Maths Notes’.

The project team will continue to solicit feedback overthe next two years. We will then be able to evaluate in muchgreater depth the extent to which our approach to maths inphysics is helpful to students.

•Is a ‘need to know’ approach the best one, or would it be preferableto teach the relevant mathematics before it is needed in physics?

•Mathematical require-ments need to be explicitly identified within the syllabus and time made available for them to be taught.

•Could, or should, an A level physics course seek to address mathe-matical ideas that go beyond those required at the time?

D I S C U S S I O NP O I N T S


The laws of physics are traditionally expressed in mathematical terms. There seems to be no escape from thisfact, and for many physicists it is a majorpart of the charm of the subject.

When things move, we need dynamical laws that are mostnaturally expressed using differential equations. We dis-cover what the laws predict in particular circumstances byfinding solutions of these differential equations. The mostinteresting solutions are expressed in terms of special functions such as trigonometric and exponential functions.Together with the natural logarithm, these are commonlyknown as ‘the elementary functions’.

In this article I address the question of how theseremarks might be illustrated within the scope of A levelphysics, even for students who are not devoting a second A level to the study of mathematics.

I suggest an approach to the related topics of uniformcircular motion and simple harmonic motion that offers apossible way of doing so. I indicate how the necessary bitsof calculus can be developed within the context of thephysics.

Simple harmonic motionLet me first remind you how we can best understand thesimple case of a mass hanging on a spring, and then turnto the question of how to convey this to A level students.

We know that the mass bobs up and down in simple harmonic motion. This means that if the mass is pulleddown a distance b and released, then its displacementfrom equilibrium t seconds later is given by the func-tion x (t )5b cos(vt ), where v is a constant which is characteristic of the particular mass and spring. Merelyto describe accurately what simple harmonic motion

looks like we need to invoke the cosine function. For understanding the dynamical processes that give rise

to oscillations, we need to know more about the trigono-metric functions. The crucial law governing the motion isNewton’s Second Law. It tells us that the acceleration isproportional to the net applied force. The law is mostcompactly expressed by the algebraic formula F 5 ma.However this form is deceptive.

If we want to make predictions about the displacementx (t ), then we need to say explicitly how the acceleration isrelated to the function x (t ), and how to find the force F.The latter part is easy if we stay within the elastic limit ofthe spring and use Hooke’s law to write F 5 2 k x. (I use alower case for ‘law’ here, as it is on quite a different levelfrom Newton’s Laws!)

The more difficult part is to identify the acceleration asthe rate of change of the rate of change of x (t ). Despite the infelicities of the traditional notation for second derivatives, it is worth introducing it so that we can see themathematical problem posed by our bobbing mass. We arelooking for a function x (t ) with the magical property thatd2x/dt2 5 2 k x (t ). The miracle is that the trigonometricfunctions that were first encountered in quite different circumstances provide the answer.

So how might we convey this miracle to students? The first step is to make more precise sense of the rathervague phrase ‘rate of change’ using a context that is bothfamiliar, and relevant to the uses we have in mind.

We might use the scenario of a car journey. A car has a trip recorder that tells us the total distance the car has travelled, and a speedometer that presents us with an estimate of our current speed. Although the speedometeris calibrated in miles (or kilometres) per hour, we don’thave to wait for an hour to get a reading of a speed from it. A discussion of what this speed is should lead students to the idea of seeing how far we move in a second,or ten times how far we move in a tenth of a second,

Graham McCauley

7 The (So-called) Elementary Functions


•For understanding the dynamical processes that give rise to oscillations, we need to know more about the trigonometric functions

•There is no pressing need for A level physics students to learn anything about the techniques of differentiation

and so on to arrive at a value for the instantaneous speed. If we use s (t) to denote the reading of the trip recorder,

and v (t) the reading of the ideal speedometer, then theabove discussion has told us how the function v(t) is related to the function s (t). We get v (t) as the ratio of the tiny change in s to the corresponding tiny interval of time around the instant t. It can be expressed in words asthe ‘rate of change’ (with respect to the passage of time) ofdistance. To remind us that it is this precise interpretationof ‘rate of change’ that we have in mind, we use calculusterminology and say that the function v (t) is the ‘derivative’of the function s (t). The above discussion helps motivatethe use of the notation v (t)5ds(t)/dt; we need merely associ-ate the prefix d with the phrase ‘tiny difference in’.

There is no pressing need for A level physics students tolearn anything about the techniques of differentiation.These are concerned with finding expressions for thederivatives of explicit algebraic functions, and are surelythe business of a mathematics course. All that we needhere is a feel for the precision of the language, and a recog-nition of the notation.

The next step is to take a fresh look at the trigonometric functions I hope that these will have been used in a discussion ofbearings in navigation. In that context the role of thecosine and sine in resolving a velocity into componentstoward the north and east respectively should already befamiliar. This context also makes sense of the extension toall angles of any earlier definitions that might have beenrestricted to acute angles.

With these preliminaries out of the way, we turn ourattention to uniform circular motion. This is the next sim-plest motion after uniform motion in a straight line, andthe uniformly accelerated motion of projectiles. It can beintroduced by extrapolating the process of steering of a carround a corner. To formulate an algebraic description of itwe note that a particle with co-ordinates (a cos(vt), a sin(vt))moves round a circle of radius a at a constant speed.

If students have not already met radian measure forangles, then the case for it has to be made at this point.The use of radian measure means that the rate of changeof the angle vt, which appears in the co-ordinate formu-lae, determines the constant speed to be simply v 5 vawithout any further factors.

Since this motion is certainly not uniform motion in astraight line, Newton’s First Law tells us that we will needa force to bring it about. Anyone who has ever been in acar that enters a corner too fast is acutely aware of theneed for some force pointing into the corner, though theprecise direction usually comes as a bit of surprise. Sothere is a fairly good intuitive feel for it. To find out exact-ly what Newton’s Second Law tells us, we consider therates of change of the co-ordinates.

It is evident from figure 1 that the velocity vector in uniform circular motion always points along the tangent tothe circle at the current position of the particle. We havejust seen that this vector has magnitude v 5 va. Projectingit onto the co-ordinate axes gives its components as (2 va sin(vt) , va cos(vt)). This tells us that the rate ofchange of a cos(vt) is 2 va sin(vt), and the rate of changeof a sin(vt) is va cos(vt). Thus we know the time derivatives

a

vt

a cos (vt)

a sin (vt)

v = vav cos (vt)

v sin (vt)

Figure 1


of both cos(vt) and sin(vt), without doing any difficult algebraic manipulations!

The acceleration follows suit using these results, and isfound to be (2v2 a cos(vt), 2v2 a sin(vt)). The accelerationhas constant magnitude, and always points preciselytowards the centre of the circular arc on which the particle (or our car) is moving. So to keep a particle of massm moving at constant speed v on a circle of radius a, we needto provide a force mv2/a pointing directly towards the centre.

Finally comes the punch lineEach of the two Cartesian components of uniform circularmotion follows a sinusoidal pattern which is known as ‘simple harmonic motion’. The component of force neededto explain that motion is simply 2 m v2 times the compo-nent of displacement. So the functions cos(vt) and sin(vt)automatically satisfy the second order differential equationm d2x/dt2 5 2 k x for the motion of a mass m on a Hooke’slaw spring with spring constant k, so long as we choose

v2 5 k/m. We have found the solution to this second orderdifferential equation without even mentioning the idea ofintegration. This is surely a major economy if we have nofurther mathematical ambitions.

SummaryThe approach I have suggested to the trigonometric func-tions involves the ideas of differential calculus, includingthe formulation of physical laws as differential equations. Ihave tried to suggest a path through the mathematics thatis guided by some physical intuition, and that avoids call-ing on manipulative skills in calculus even at the level ofdifferentiation.

The concept of integration is not needed at any stage. Ina sense it has been replaced by the idea that a differentialequation may be solved to tease out the implications of aphysical law. Solutions are declared and their propertiesverified rather than being found by any constructivemanipulative processes.

•An understanding of physics requires a basic ‘tool-kit’ of mathematical functions.A good knowledge of the properties of these functions should be developed and their wide applicability made explicit in our teaching.

•Physical reasoning can be used to develop mathematical understanding. Such approaches are vital for those not following A level courses.


•Anyone who has ever been in a car that enters a corner too fast is acutely aware of the need for some force pointing into the corner

•We have found the solution tothis second order differential equation without even men-tioning the idea of integration


Helen Reynolds This is how the joke goes: an accountantwas sent to see the scientists working onthe Star Wars programme, developing giant laser weapons for use in space.

‘We have a problem,’ the scientists said.‘We have only reached a laser power of1010 [units unspecified] and we need toreach 1020.’

‘That’s not a problem,’ the accountantreplies. ‘You’re half way there.’

This is not a joke to my lower sixth students. They think theaccountant is right. With experience, we have become fluentin the language of mathematics and we forget any difficultywe had initially.

Consider the equation:

5 2lN

You know what this means. The rate of decay of a radioac-tive nucleus at any time is proportional to the number ofundecayed atoms at that time and the constant of propor-tionality is l, the decay constant. So why didn’t I write it inwords?

Most of my sixth form students think that the equationis concise and saves us writing a lot of words. They are notwrong. Half think the equation is easier to understandthan the words. A small number say that writing the equa-tion means you can manipulate the symbols and can ‘seewhat is happening’. I would not disagree with them, butwhy don’t all of them feel this way?

Each time I get to the point in the course where I teach

dN}dt

this I have a small crisis of confidence. I want to show whatthe power of mathematics can do. I want to show what wecan derive mathematically and how it relates to the physicsof the situation. I feel guilty about this because of the con-ventional wisdom that mathematics is difficult. So what doI do? I run around the subject for a while, playing withdice, popping popcorn, looking at graphs. Then I do themathematics. I do not imagine I am alone.

How do we arrive at the equation and what do we do when wehave it? Here is the way I do it: writing it down highlights the kindof mathematical hoops I expect students to jump through.I have tried to pick out the hoops (see margin boxes) butyou may find subtleties I have missed. This is very much aprécis, with a great deal of the ‘running around’ and expla-nation omitted; the order of ‘phases’ changes from year toyear.

Phase 1: PicturesFor homework plot by hand a graph of for values of xbetween 0 and 5. Use a calculator to work out what e21 isand so on.

Armed with this beautifully drawn graph we are going tolook at the amazing world of exponentials. Draw tangentsto the curve at x 5 1, 2 and 3 and find the gradient.

Compare it with the value of e21, e22 and e23. What doyou notice? Isn’t that amazing? What is so special aboutthe number e? It is the only number that you can do thiswith, the only number for which the gradient equals thevalue at any point.

What we are saying is that the equation of the curve

is y 5 e2x, so }dd

xy} 5 2y where }

dd

xy} is the gradient.

We also know that logarithms and exponentials areinverse functions. If we take logarithms of both sides of thefirst equation, we get ln y 5 2x , so if we plot ln y against xwe get a straight line (of gradient 21).

8 Wherefore dN/dt 5 2lN ?

Hoops – Phase 1

Use a calculator to work out:

• e21

• the symbol d

• the mathematical meaning of }dd

xy}

• logarithms, ln and log10

• taking logs• e 5 2.718 …


Phase 2: Protactinium decayLet’s collect some data and plot a graph of decay rateagainst time. We can show it is exponential by finding thegradient at different points and comparing it with thecount rate at the same time. We can also plot ln (count rate)against time and get a straight line. We should also calcu-late the ratios of successive readings.

Phase 3: Concrete blocksDice are tangible objects and that will help you to under-stand what nuclei do. If you put a white dot on one side ofa die then you will know how often that side is likely tocome up. You understand that there is a likelihood that ifyou throw it six times then you will get it coming up whiteonce. If you have 100 of them, that translates to one-sixthof them coming up white every throw. It cannot always beexact because throwing dice is a random process and thenumber of dice is not a continuous variable. If we removethose that come up white each time we throw, we can seehow the number remaining changes.

It is the same with nuclei. Trust me.I am saying that the probability per throw (or per second)

in both cases is constant. For dice it’s always one-sixth, for nuclei it is l, or what we call the decay constant. Let us get some data for 201 throws. We will do two things with these numbers. We will calculate the ratios ofsuccessive readings and also plot the graph of number of remaining dice against throw. We can then find howmany throws it takes to halve the number of dice. It’s always the same, just like the half life of a radioisotope. Why do nuclei do this? If you start with Natoms and the probability of one decaying in one secondis l how many decay in one second? And in time Dt?This gives us DN 5 2lNDt (where did the minus sign

come from?). So we get 5 2lN , and if the time interval

is small enough, }ddNt} 5 2lN .

DN}Dt

The rate of change of the thing is proportional to the thing– this means it’s exponential. Because the number is goingdown, it is exponential decay.

Phase 4: The easy mathematicsWe can write down N 5 N0 e2lt . I can show you that thisis the case by differentiating.

Watch: }ddNt} 5 2l 3 N0 e2lt , so }

ddNt} 5 2lN

Phase 5: The hard mathematics

Start with }ddNt} 5 2lN . Manipulating this E}

dNN} 5 2El dt

and so ln N 5 2lt 1 constant . At t 5 0, N 5 N0 so the con-stant is ln N0 , so that (ln N 2 ln N0) 5 2lt .

Therefore, ln 5 2lt and finally, N 5 N0 e2lt .

We can now show why the half life is a constant and workout a way of finding l from a graph.

Put N 5 N0 /2, take logs and rearrange to get T1/2 = }

lnl

2} .

Take logs of the equation for N: a graph of ln N against t isa straight line with a gradient of l.

Let me pick out two of the potentially big mathematical hoops First, I remember that a very, very long time ago I learntabout how to take logs. I know that I did because I cannow take logs. I confess that I could not remember what itwas all about so I looked it up just now on a very interest-ing website. The explanation was concise and it all cameflooding back: bases, exponents and logarithms. Whatstruck me reading the explanation was that there were twodistinct parts to it. One was terminology: in bx 5 y , b is the

N}N0

Hoops – Phase 2

• decay rate 5 }dd

xy}

Hoops – Phase 3

• D means ‘change in’• in the limit as Dt → 0 we get dt

potentially big hoop

• probability

Hoops – Phase 4

• }ddx} econst 3x 5 const 3 econst 3x


Hoops – Phase 5

• E}1x

} dx 5 ln x 1 constant


• manipulation of logs• constants of proportionality• taking logs• algebra with exponential

functions


•Another way of approaching rate of change is by popping popcorn – the rate of poppingis proportional to the number of unpopped kernels

•It’s a bit like doing a hill start in a car: all you really need is good clutch control and an ear on the engine revs

base, x the exponent. The other was what you ended updoing with the symbols on the page. I had very distinctlyremembered the latter and forgotten all of the former. Ican still take logs, though.

Second, E}1x

} dx 5 ln x 1 constant; why? Because that is

what I learnt, one of the many things that have becomemathematical tools from a toolbag I carry around in myhead. If I take a precedent from the previous paragraph,then I should be just fine if I can remember this and trustthat there are various reasons for it which I could find outif I wanted to. Of course, I am happy about using thisbecause I know I did it sometime in the past.

Getting the feelStarting with the dice, and plotting the graph, you can geta feel for the way these functions work. The shape of thegraph tells a story; the gradient ‘slows down’ as x (or t, orthe number of throws) increases. The rate of change is pro-portional to the size of the thing that is changing. Anotherway of approaching it is by popping popcorn.Qualitatively, the rate of popping is proportional to thenumber of unpopped kernels. It is quite straightforward toplot a graph while the popcorn is popping and get a feelfor the shape.

Similarly water through lock gates, and charge on acapacitor. This is certainly more accessible for a number of

students and is as essential for those who can ‘do themaths’ as for those who can’t.

Another way of looking at it all ...Looking at the picture above an analogy strikes me. Youdon’t really ‘get’ the graph without ‘knowing’ the mathe-matics and the dice only go so far to explaining things ... .It’s a bit like doing a hill start in a car. Perhaps we havebeen too worried about teaching the workings of the inter-nal combustion engine when what you really need is goodclutch control and an ear on the engine revs. And how doyou get good clutch control? Practise.

Drawing graphs of e–x orplotting data

Half life

Dice analogy or similar

Differentiating to showN = N0e–λt means

Constant ratios

Integrate to show that

dN dt

= –λN

= –λN dN dt

= –λN means N = N0e–λt dN dt

•If the story the graph tells is to be understood, a qualitative under-standing is essential of how the differential equation leads to the exponential shape of the graph.

•New mathematics is only assimilatedslowly. There is no single explanationthat works for all students.

•It is important, if difficult, to empathise with students and see the challenges of new mathematical ideas from their point of view.



One way of thinking about differentphenomena is in terms of the nature ofthe phenomena themselves – thermal,electrical, and so on. Another way ofthinking about them is in terms of theirfundamental mathematical nature: forexample, linear growth, exponential decay, oscillation.

Seen in this way, a leaky tank, radioactive decay, capacitordischarge, and so on, are all similar as they are all examplesof exponential decay. This article explores some uses ofgraphical representations which are intended to makeexplicit these kinds of fundamental mathematical similari-ties (both of model structures, and of data or model output).

A constraint modelA constraint model is one which represents fixed relationsbetween parts of a system. Ohm’s law is a simple example,which may be represented as shown in figure 1. This indi-cates that three variables are linked by one relationship –knowing the values of two of the variables, the other can

be determined. Only one representation of the equation isshown at the node (a direction of calculation is implied) –algebraic manipulation is required to find others. In writ-ing a model, e.g. on a spreadsheet, a direction of calcula-tion will be specified.

As well as ‘one-off ’ calculations, the nature of the rela-tionship in such models may be explored by changing onevariable and looking at the effect on another variable, withthe others(s) held constant. It would be thus be useful forstudents to try to think qualitatively about the nature of therelationships by choosing a graph from a ‘library’ ofgraphs which gives the best description.

In figure 2, more substantial reasoning is involved. Oneintention of these diagrams is that they are able to providesupport for reasoning about the relationships. For exam-ple, a student might be asked to predict the effect of dou-bling the potential difference on the work done. The think-ing involved would be that doubling the potential differ-ence will double the current (if the resistance remains con-stant). In the period of time, therefore, double the chargewill flow. Since both potential difference and charge aredoubled, the work done will increase by a factor of four.

In figure 2, some relationships are made explicit: forexample, R 5 V/I, V 5 W/Q, and so on (it may help to sup-

9 Using Graphical Representations to Support Mathematical Reasoning

Richard Boohan

Figure 2

Figure 1

potentialdifference

V

workW

timet

powerP

resistanceR

chargeQ

currentI

V = W/Q

R = V/I

I = Q/t

P = W/t

potential differenceV

resistanceR

currentI

R = V/I


port reasoning, if these are expressed in such diagrams innormal form, IR/V 51 etc, so that no direction of calcula-tion is implied). Others are not explicitly represented butmay be derived: for example, P 5 IV , P 5 V 2/R, and so on.

An evolutionary modelIn an evolutionary model, the values of the variableschange as the model iterates, often though not always overtime. Figure 3 represents such a model.

The convention used here is that the shaded ellipse repre-sents the increment in the independent variable. The rectangles with arrows pointing to them represent variables that ‘accumulate’ – the calculated value is addedto the existing value of the variable – an initial value (may

be zero) needs to be specified for each of these variables.Figure 3 could represent, for example, the rate of increaseof a population of bacteria which grows linearly.

Bacterial populations, however, tend to grow exponen-tially, in which the rate of growth at any instant is deter-mined by the current value of the total population. Figure4 shows how the previous diagram may be modified by theaddition of a rate constant in order to represent this.

Simplifying the representations One difficulty with such representations is that by showingsufficient detail to support the construction of an algo-rithm for solving the problem, it is not so easy to see simi-larities and differences between different models. Thus,the essence of above two models is that the first representsa quantity changing by a constant rate, while the secondrepresents a quantity changing by a rate which is deter-mined by the quantity itself.

The following ‘simplified representations’ are intendedto show more clearly the similarities and differences, andalso extend the ideas to models in which a rate is itselfaffected by another rate.

a) If the rate affects the quantity ina positive way, then this leads to linear growth. If the rate affects thequantity in a negative way, then theresult will be linear decay.

b) If the rate has a positive effect onthe quantity, then this is a positivefeedback loop leading to exponentialgrowth. If the rate has a negativeeffect on the quantity, then this is anegative feedback loop leading toexponential decay.

quantity

rate

quantity

rate

rate of increase

number added total number

rate constant

time intervalδt

total timet

number added =rate of increase × time

rate of increase =rate constant × totalnumber

rate of increase

total timet

time intervalδt

total numbernumber addednumber added =rate of increase × time

Figure 3

Figure 4 Figure 5

•Students can explore what happens to one variable as another changes

•Thinking about similarities and differences helps understanding


c) The shape of the graph will depend on the initial values of thequantity and of rate 1. For example,constant acceleration of an objectfrom rest could be represented withquantity (displacement) and rate 1(velocity) initially zero. A graph ofthe quantity against time wouldshow a maximum if, for example,rate 1 was initially positive, but rate 2had a negative effect on rate 1 (a ball being thrown in the air)

d) If the effect of rate 2 is to in-crease the quantity, then this is apositive feedback loop. The quan-tity grows without limit. If the ef-fect of rate 2 is to decrease thequantity, then this is a negativefeedback loop with a delay. Thisleads to oscillation.

These four simplified representations can be extended orcombined. For example, the acceleration of a car subjectto air resistance could be represented by modifying figure5c to show rate 2 being affected by rate 1. Projectilemotion can be represented by a combination of figure 5a(horizontal motion) and figure 5c (vertical motion).

The use of these node and link representations may helpin supporting students to reason both qualitatively andquantitatively about relationships. They may be usedeither in finished form as a map of relationships or as atool-kit for students to use in making their own represen-tations. They may be useful in helping students developproblem solving strategies.

quantity

rate 1

rate 2

•Students yearn to see a bigger picture sometimes, not simply more meaninglessnumbers popping out of an equation.

•Why not insist that every graph must have a caption which tells what the graph says, not what the graph is about

(e.g. ‘Y increases linearly with X ’, not ‘Plot of Y against X ’)?

•Sometimes we rely too heavily on algebraic reasoning and students cannotsee the wood for the trees. We need waysof conveying the relationships representedby mathematical formulae.

D I S C U S S I O N P O I N T S•Node and link representationscan help students clarify relationships

quantity

rate 1

rate 2

Figure 5 (cont.)


Graphical calculators are going to forcerapid advances and changes in physicsteaching over the next few years. Soon, students will turn up to our classeswith palmtops that are as powerful as today’s PCs complete with modems, a direct link to the web and all thephysics texts they need.

These small machines are at the vanguard of a didacticrevolution in physics. Today, for less than £100, studentscan buy a calculator which is in many respects a smallcomputer. It can handle all of the important mathematicalactivities associated with today’s physics and more. In thisarticle, I will highlight the most important area in whichcalculators have changed the character of my own teach-ing of the mathematical aspects of physics. I call this ‘realworld physics’. Experiments take just a few minutes andresults are fed directly into the calculator via an interface.Students differentiate, integrate and analyse real resultsusing numerical methods on the calculator. They can alsocompare experimental results directly with the relevantdifferential equation.

Calculators can also perform algebraic manipulation ofexpressions and algebraic differentiation and integration.

Enthralled by mathematics for too longThe physics we teach describes a mathematically createdworld where point particles move in straight lines andundergo constant acceleration, friction is negligible, fieldsare homogeneous, liquids are incompressible and gases areperfect. It is a two-dimensional soap-opera world whereeverything follows simple predictable mathematical rules.

Modern calculators and computers can handle millions

of calculations per second. We can use numerical methodsinstead of algebraic methods to analyse relationships inphysics and work directly from experimental results. Thesenumerical methods are often faster than the correspondingalgebraic methods, surprisingly easier for students tounderstand and they give honest results that are physicallymore accurate. I think we should introduce these ideasinto our teaching.

Graphical calculators have spreadsheets that can be used inthe same way as spreadsheet programs on the computerThe difference is that every student with his or her own calculator means that analysis of data can take place anywhere and anytime; in the classroom, in the laboratory,at home or even on the bus. We do not need to be restrictedby the availability of computers in the school or college.

Results can be fed into the calculator spreadsheet one byone or they can be transferred between calculators, but themost effective way of using the calculator is with experi-ments using a calculator based interface.

An example illustrates how numerical analysis methodstake over from algebra based methods and shows how theideas of differentiation can be made meaningful for stu-dents. A wagon is rolled down an inclined plane and thedistance measured every 20 ms using an ultrasound detec-tor coupled to the calculator. Figure 1 shows the experi-mental results with the regression curve and equation.

The experiment takes less than five minutes to perform.The student gets the graph directly on the calculatorscreen and can relate it to the actual motion of the wagon.Individual points can be studied and physical ideas likechange in potential energy, velocity and kinetic energy canbe calculated using co-ordinates shown on the screen.What is important here is that students are using realexperimental values to do their calculations. The physicshas not been reduced to an approximate quadratic equa-tion. Figure 2 shows how the calculator can be used todraw the tangent at a point and work out the velocity.

10 Graphical Calculators and Mathematics in Physics Teaching

George Adie

Figure 1

Figure 2

Figure 3


This is the way I want my students to understand aderivative as the slope of a tangent. It takes away all thatalgebraic hocus-pocus and makes the ideas accessible forall students.

The calculator can also calculate the experimentalvelocity at every point (figure 3) and draw a graph of veloc-ity against time. Students can work out the accelerationfrom the slope of the line.

Calculations of kinetic energy (WK) and the change inpotential energy (DWP) can be made for every point andthe graphs plotted so that students can look at how muchwork is done against friction and air resistance (figure 4).

We can now let our students analyse large amounts ofdata from physics experiments. We have inexpensive tech-nology which can be used to collect real data and we haveinexpensive calculators to do the number crunching. It isnow faster and better physics to analyse a few hundreddata points numerically than to try and fit results to analgebraic equation and work with that.

Integrals are essential to an understanding of physicsIf we treat them simply as the area under a curve and usenumerical methods, the concept of integration becomesaccessible for all of our students. This is important whenwe want to teach them about charge and current, impulseand force, work and force and many other things.

Figure 5 shows the acceleration of an accelerometer as itis moved a distance of just over a metre by hand in astraight line.

Figure 6 shows the numerical integral of the first graphand figure 7 shows the next integration.(The co-ordinatesof a marked point are given on each graph to establish ascale.) This is an obvious way of integrating that makes thereal physics of the experiment immediately understand-able for the students.

We are integrating using real experimental results andthere is no complicated algebra fogging up the physics.

It is possible to fit the original experimental points to a

fourth order polynomial using regression analysis and thenintegrate that a couple of times with the correct limits, butis that what we want? Is it physics?

The understanding of differential equations is central to aproper understanding of physicsUnfortunately, only a small fraction of differential equa-tions in physics are solvable algebraically at a level whichcan be understood by our students. This has historicallygoverned the contents of our physics courses, giving stu-dents a completely wrong picture of our world. They getthe impression that it is described by exact algebraic equa-tions and deviations from this are just rather annoying per-turbations which can be added on afterwards. With mod-ern calculators, graphs of differential equations can be dis-played on the screen and compared directly with experi-mental results. Students can investigate what happens bychanging the coefficients in the differential equation; theydo not need to study a large amount of difficult, and formany of them, incomprehensible mathematics in order tounderstand the physics.

Here is a short description of some areas where thisdirect approach to physics through the differential equa-tion can change the way in which we teach.

Consider the beautifully simple equation, x´ 5 2kxThe rate of decay of a variable is proportional to the mag-nitude of the variable. This is easy to explain from a phys-ical point of view when we talk about radioactive decay orthe discharge of a capacitor, but what then do we do? Wesolve the equation, get an exponential function, lose overhalf our students in the maths and then try and get themto understand the variables in the exponential expression.The calculator lets us work directly from the differentialequation.

Figure 8 shows the experimental results when a1mF capacitor is charged from a 9 V battery througha 33 kV resistor. The differential equation is drawn

Figure 4

Figure 5

Figure 6

Figure 7


on the same graph. It is obvious that it is a good fit.Students can change the decay constant and the initial

values and see how the graph changes. They do not needto have a deep understanding of the exponential functionin order to understand the basic physics.

Air resistance is usually ignoredWe know that a falling body is acted upon by two forces,the gravitational force and a force which depends on airresistance. This second force can be shown to be propor-tional to the velocity squared, even at quite low speeds.The differential equation that describes the motion isy ″ 5 29.81 1 Ky 2 where y is the height.

Figures 9 and 10 show how the experimental values for a falling plate change with time compared with the differential equation with K 5 0 and the correct value K 5 0.71 m-1s-2.

For the simple pendulum, u ″ 5 2gL21 sin uIn order to get a simple equation to solve, we make the approximation sin u 5 u. We do this because we are imprisoned by old fashioned algebra. We spend time teaching maths instead of physics. Why? Modern calculators allow us to show the real solution in seconds.

Figure 11 shows a comparison of the numerical solution tothe differential equation and the traditional ‘approximate’solution.

All we need to do is motivate the differential equationusing Newtonian mechanics and then let the students seewhat happens with different start conditions. They can ofcourse compare their results with experiment.

Using modern calculators means that we can be honestabout real physics. I think we should grab this opportuni-ty. It allows us to spend more time deliberating over theimportant aspects of Newton’s law and less time on thetedious mathematics.

Graphical calculators in schools offer a better, inexpensive andflexible alternative to computersWhen every student has his or her own graphical calculator,physics teaching can move into the fast lane. This is notonly true for physics, it applies equally to the other sciencesat the same time as posing a very exciting challenge tomaths teaching.

In my experience, students spend too much time on themathematical aspects of physics problems. The calculatorfrees them from this and allows them to spend more timeon understanding and explaining real physics.

Figure 8

Figure 9

Figure 10

Figure 11

•The cheapness of such tools means they will be available to all – we cannot ignore them.

•Do numerical solution techniques make some traditional mathematical techniquesobsolete?

•Is the ‘fast lane’ too fast for many teachers?Is teacher INSET readily available to enable new teaching methods such as this?



The visual impact of graphs and their relational properties make them extremely valuable for analysing and appreciating the properties of data, butmore importantly the graph containsvaluable information about the relationship between the variables represented by the data.

Why do physicists use graphs? When physicists accumu-late numerical data from practical experiments, they needto store the data in some form and this is traditionallydone in a table of results. When it comes to analysingresults, although a table has a certain amount of use forcomparing items of data and deriving further information,the graph is a far more versatile tool for this purpose.Students should be encouraged to gain experience of themany aspects of graphs by playing with them, exploringtheir properties, transforming them and linking them withstandard algebraic descriptions. To facilitate such play,computer graphing software readily comes to the aid ofstudents, providing an interactive experience through arange of visualising and analysing tools.

Comparing variables – graph shapesIn physics the YX graph provides the natural means ofcomparing two variables. With the variables plotted indirections at right angles to each other, the relationship isrepresented in two dimensional space. Although this isexcellent for summarising the connections between thevariables over a range of values, the perpendicular geom-etry of the two scales reduces the feeling for two magni-tudes varying simultaneously. On the computer this isvividly restored by the animated bar display, which, in

showing both magnitudes in one dimension only, amplifiesthe correspondence between the variables. This is illus-trated in figure 1.

This software tool for studying graphs is very useful forconveying the idea of a connection between variables. It soonbecomes apparent to the student that as the value of one vari-able is varied, the behaviour of the second variable can bepredicted from the general shape of the graph line. Theshape is a key feature which allows the general characteris-tics of the relationship to be identified. Steepness on thegraphline becomes associated with rapid changes in thedependent variable, whereas flat sections indicate slowchange. Curves and straight lines are shown to have distinc-tive properties when rates of change are studied in more detail.

Comparing variables – graphs and numbers Moving on from simple qualitative explorations using thecursors and animated bar display, a large amount of further information about the graph becomes available

Laurence Rogers

11 Graphs in the Service of Physicists

Bar display

X variable

Y

X cursor

Y cursor

X Y

Figure 1 As the X cursor is moved from left to right, the Ycursor follows the graphline and the bar display showsthe corresponding variations in the X and Y readings.


from numerical analysis. There are many software toolsfor analysing the numbers associated with a graph, but this article will focus on the use of three basic tools usingcursors.

Readings The simplest use of the cursors provides a readout of thecoordinate pair for any point on the graphline. Usually,the software constrains the vertical and horizontal cursorlines to intersect on the graphline under investigation (see figure 1) and the coordinate readings are viewed on a sidepanel.

Change The second cursor tool calculates the changes betweenthe X and Y coordinates for a pair of readings on thegraphline. This simply involves the subtraction of the respective values of the coordinates to yield Dx and Dy,but since the computer is able to process the calculationso rapidly, the values for the changes can be viewed continuously as the computer mouse is manoeuvred.

Ratio The third cursor tool calculates the ratio between the Y coordinates for a pair of readings on the graphline.This is useful for expressing the comparison in a relativemanner; one value is shown to be bigger or smaller thanthe other as a percentage or fraction.

These tools give prompt and accurate readouts and calcu-lations from the data contained in graphs. In use theirmain purpose is looking for connections between succes-sion of pairs of points on the graph. For common mathe-matical graph shapes, this usually yields a pattern of con-nections which is unique to each particular graph shape.Describing such a pattern is only a small step from describ-ing the relationship between the x and y variables. Twoexamples are given to illustrate this.

Example 1: A quadratic graph passing through the originThe shape of this graphline is a rising curve starting fromthe origin. In this example the ‘Change’ cursors are usedto calculate successive increases in y for a given increase in x.Some of the results are shown in figure 2; x is incrementedin steps (Dx) of 10 and the corresponding increases (Dy) in ybecome successively larger. Analysis shows that the differ-ence between successive values for Dy is always the same;in this example the difference is always 4 units. This is aunique characteristic of the quadratic curve. The ‘rate ofchange’ between these two variables is essentially a com-parison of how the variables grow relative to each other.The fact that the line is a rising curve affirms that the rateof change increases with x. In contrast, a curve which lev-elled out would represent a decreasing rate of change, or astraight line would represent an unvarying rate of change.Thus Dy/Dx is a key concept in describing how variablesare related to each other and to the shape of the graphline.

Example 2: A graph showing exponential decayThis curve shows the y variable diminishing as a functionof x and the shape has a number of special features; here

•Students gain experience of graphs by playing with them, exploring their properties, transforming them and linkingthem with standard algebraicdescriptions

•Describing common mathe-matical graph shapes is only a small step from describing the relationship between the x and y variables

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Figure 2 Successive readouts using ‘Change’ shows thatthe increases (Dy) in y become progressively larger, in

proportion to x


•Graphical software provides a valuable experience of‘unpacking’ a formula and recognising its physical significance

•In practice, curve fitting can beusefully limited to three generalformulae which accommodatea wide variety of phenomena in A level physics

we shall consider a key property which can be exploredusing the ‘Ratio’ cursors. The process involves identifyingtwo data points and allowing the program to calculate theratio between them. Clearly the second value is always lessthan the first, but, unlike the ‘Change’ tool which gave Dy,the reduction is here expressed as a fractional changey2 /y1; that is, y2 is expressed as a fraction of y1. This is use-ful because it turns out that for a given horizontal separa-tion Dx of the data points, the ratio y2 /y1 is always thesame, irrespective of the starting position on the curve; acertain separation for the points always yields the sameratio. This property of the exponential decay curve issometimes referred to as the ‘constant ratio property’ andit describes a wide range of natural phenomena.

The properties for the two curves described here areunique to each curve; the constant ratio property cannotbe identified for the quadratic curve; nor can the changesin y be shown to vary in proportion to x for the exponen-

tial. Similarly, other curves can be investigated for theseproperties but to no avail.

Rate of changeIn the chosen examples, care has been taken to avoid ref-erence to the particular case of ‘time’ as the independentvariable plotted on the x axis. The properties of the graphsare generic in the sense that they are independent of thephysical nature of the variables. In the foregoing discus-sion, the concept ‘rate of change’ frequently appears, andit must be emphasised that this also has been in a genericsense, although ‘rate’ might imply time dependency. Theactivities using the computer animated bar display aregood at conveying a feeling for speed of change, but it isonly in the particular case of time as the independent vari-able that this rate of change is appropriately associatedwith a physical speed of change. Frequently, the indepen-dent variable on the graph is time and the bar display pro-vides a graphic visualisation of the sequence in time of thedata represented on the graph. However, although com-mon, this has to be regarded as a special case; more gen-erally, the concept of rate of change describes changes inthe y variable relative to the corresponding changes in thex variable; it essentially describes an aspect of the relation-ship between two variables.

Comparing variables – graphs and formulae Having gained an appreciation of the properties of graphshapes through numerical exploration, it is necessary tomove on to more formal descriptions with mathematicalformulae. The previous examples have shown that thenumbers associated with the graph are useful in giving afirst hand feel for a relationship, but there are limits totheir usefulness; they tend to be clumsy when broader gen-eral descriptions are needed. To meet this need, a formu-la can be an elegant tool for describing a relationship. Theformula can be regarded as a sort of mathematical ‘short-hand’. Its power is that, despite its compactness, it can

Figure 3. For the exponential decay curve, when pairs ofvalues, each separated by the same Dx, are compared as

a ratio, the ratios are found to have the same value:y2 /y1 5 y4 /y3 5 y6 /y5

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contain a wealth of valuable information. It will be arguedhere that graphical software provides valuable experiencesof ‘unpacking’ a formula and recognising its physical sig-nificance.

Curve fitting tools are very useful for finding the most appropriate algebraic description for a set of dataUnfortunately, many software packages focus much atten-tion on fitting polynomials to the test data, and this is oflimited use in helping students gain a better understandingof the relationship. A much more useful exercise is to focuson a modest selection of simple general formulae (linear,quadratic, inverse, inverse square, exponential) andattempt to obtain the best fit of such curves with the dataconcerned.

One of the merits of this approach is the involvement ofthe student in the fitting process itself, rather than relyingentirely on automatic execution by the computer. If thesoftware permits the student to select the general formulato be fitted on a trial basis, the student’s knowledge of the-ory can be applied, a prediction made of the most suitableformula and finally a test made of that prediction.

Example 3: Fitting a graph for free fall From an experiment in which a card is dropped through atiming gate for a series of different starting heights, thegraph of velocity against height is plotted (figure 4). Thestudent can choose the type of curve for the fitting trial andit transpires that either a straight line or a parabola can becomfortably fitted to the data.

Since the straight line produces an intercept whichimplies a finite velocity when the height fallen is zero, thisoption must be incorrect. Theory demands that the graphline passes through the origin, making the parabola theappropriate choice. This example illustrates the impor-tance of involving students’ ideas about the physics in thefitting process.

In practice, curve fitting can be usefully limited to three

general formulae which accommodate a wide variety ofphenomena in A level physics:

• for linear graphs: y 5 mx 1 c

• for power law curves: y 5 axb 1 c

• for exponential curves: y 5 a ebx 1 c

If students have previously built up their understanding ofthe properties of these curves through the numericalanalyses described earlier, the formulae can then be seento embrace those properties conveniently. The significanceof the parameters m, a, b and c can be explored directly byadjusting them in the software and observing how theshapes are affected by the choice of values.

BibliographyRogers, L.T., The Computer as an aid for exploring graphs, School Science Review, 1995, 76(276), 31–39.Insight 2 Data-logging for the Science Laboratory, Longman Logotron, 1996.Understanding Insight - An interactive guide to Data-logging, Longman Logotron, 1998.

Figure 4 Both the power law curve (parabola) and thestraight line can be made to fit the data for a card in freefall, but the straight line gives a false prediction about the

velocity of the card when falling from zero height

•Computer tools can be usedto ‘make graphs speak’. Qualitative understanding,as well as quantitative, can be developed.

•These tools represent a serious challenge to many teachers. If students are tobenefit, teachers will need time and training. And many schools and collegeswill need extra funding to purchase new apparatus.

•The graphs generated by standard spreadsheet software can confuse students. They often use non-standard notation (e.g. in formulae) and often join data point to point.

D I S C U S S I O NP O I N T S 150

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Simon Carson When students first meet vectors theycan get lost in the complexities ofPythagoras and trigonometry, losingsight of the fundamental, geometricideas. To help them with the detail theyfirst need to develop intuitive ideas aboutthe meaning of what they are doing.

Very often, vectors are introduced in the context ofmechanics and the vectors involved – velocity, accelera-tion, force – are rather abstract. This abstraction presentsan additional obstacle to students when they begin tograpple with mathematical detail. So how can we developgood physical insight – a ‘feel’ for the mathematics –alongside technical detail?

Two approaches are considered here. The first begins bytaking a familiar example – a road map – and developingideas about vector addition, invariance under rotation andresolution into components in a geometric way. By initial-ly avoiding more abstract ideas about forces and motion,the formalism can be developed without too many newconcepts being introduced together. The second involvesthe use of computer modelling to enable students to seewhat a vector is, what it means to add vectors together andto find components, by making abstract ideas more con-crete.

The vector quantities that we encounter when we study mapsare displacements Motorists’ handbooks often give tables of the distancesbetween towns. What the table does not tell us is the direc-tion from one town to another. Curiously, it is possible to findthe relative directions just from the distances. Finding outhow to do this can help students to understand how vector

quantities combine, indeed what a vector quantity actually is.Consider one simple case. The distance as the crow flies

from Calais to Marseille is 879 km. The distance from Calaisto Lyon is 612 km, and the distance from Lyon to Marseilleis 270 km. Lyon should lie very nearly on a straight line fromCalais to Marseille and you should be able to see why! Thissituation approximates very closely to the sort of scalar addi-tion that students are used to – but not quite.

Marseille to Bordeaux is 498 km, and Lyon to Bordeaux is426 km. We already know that Lyon to Marseille is 270 km.These distances do not add up! The paths must be angled.

12 Maps and Models - Approaches to Vectors

Figure 1 Calais, Lyon and Marseille must lienearly on a straight line

Calais

879612

Lyon270

Marseille

Figure 2 Lyon, Marseille and Calais are NOT in a straight line

Calais

879612

270426

Lyon

Bordeaux

Marseille

498


•Using the table of distances in a road atlas, a map showingthe relative positions of towns can be built up and the geo-metric meaning of vector addition emphasised

•At a more advanced level, one might then go on to makepoints about writing the laws of physics in an invariant way

To get at a possible layout of these towns, lines propor-tional to 426 km and to 498 km can be anchored on Lyonand Marseille, and rotated until they meet. (It should beclear that the location of Bordeaux is not the only possibleone – the lines would meet just as well above and to theright of Lyon and Marseille.) From this example, studentscan begin to get an idea of how vector addition works: vec-tors are added geometrically by placing them tip to tailand the magnitude of the vector sum is not the same as thesum of the magnitudes.

Using this example, we can introduce students to anoth-er important piece of mathematics. The triangle Bordeaux–Lyon–Marseille looks as if the angle at Lyon is nearly aright angle. This can be tested, using Pythagoras’ theorem:(Bordeaux–Lyon distance/km)2 + (Lyon–Marseille distance/km)2 5 254 000

The square root of this is 504 km. It is the distance itwould be from Marseille to Bordeaux if the angle were aright angle. The actual distance, 498 km, is quite close.

By using a road atlas table of distances, a map showingthe relative positions of towns can be built up and the geo-metric meaning of vector addition emphasised. However,it is quite clear that we have no way of knowing which wayup the map should go. North might be in any direction.Figure 3 shows the map rotated so that North is, in fact,vertical.

What this tells us is that the coordinate system we usedoes not alter the relationship between vectors, nor thevectors themselves. They are geometric objects. In partic-ular, their lengths are invariant under rotations, and simi-lar conclusions can be drawn about reflections. At a moreadvanced level than we are discussing here, one mightthen want to go on to make points about writing the lawsof physics in an invariant way. A consideration of suchinvariants, even at A level, could be used to make a linkwith special relativity, where the space–time intervalbetween two events is an invariant, the ‘length’ of a fourdimensional vector.

The resolution of vectors into particular componentscan be considered, once a particular coordinate system hasbeen established. We could ask the question, ‘If we drivefrom Marseille to Bordeaux, how far towards Calais havewe got?’ One answer is that since we were not heading theright way, we have gone 498 km and have 696 km left to goin a different direction. This is the answer the driver of thecar would have to give to the not-too-pleased passengers.

Another answer is that we have gone about 200 km alongthe true direction, Marseille–Calais. That does not soundvery sensible, since at Bordeaux we are nowhere nearLyon, having gone more than 400 km West as well. Itsounds a bit more sensible if we ask how far North we havegone from Marseille. Then (if Calais is due North ofMarseille, which is nearly true) the answer is about 200 kmdue North. This is shown in figure 3.

By considering such ideas, students can begin to under-stand what a component actually is in a very concrete way,before the introduction of trigonometric techniques andbefore the resolution of more abstract vectors.

Figure 3 How far towards Calais do we get by drivingfrom Marseille to Bordeaux?

Calais

LyonBordeaux

Marseille

~200

N


So, the idea of maps can lead students towards an under-standing of what a vector actually is, of how vector addi-tion works, of the meaning of Pythagoras’ theorem, ofwhat components are and towards ideas about invarianceand symmetry. All these ideas are developed in a familiarcontext and in a very concrete way. More abstract appli-cations of the ideas can now be tackled with students con-fident in their understanding of the fundamental ideas.

Computer modelling provides another way of seeing directly whatthe mathematics means and of developing an intuitive feel for it The models described here were written using ‘Modellus’,a system developed at the New University of Lisbon. Thesystem allows you to define a mathematical model (usingthe same notation as one would on paper) and then repre-sent the model graphically, as a tabulated set of values oras an animation. Figure 4 shows the animation of a verysimple model that develops an idea of what the velocityvector means.

The software provides its own time variable, t, so thatdifference equations can be written to define two compo-nents of velocity. These are used in the animation windowto attach a vector with these components to an object, inthis case a ball. The position of the ball on the screen isgiven by the values of x and y. When the model is run, theuser is able to grab the ball with the cursor and move itaround the screen. Doing so changes the values of x and yand the software adds the velocity vector to the object. Bydragging the ball around the screen, the student experi-ences directly what the velocity vector means: it tells herwhich way she is moving her mouse and how fast. Theabstract becomes concrete and the mathematics is tied toexperience.

The ability to grab hold of objects in the animation win-dow and move them around provides a way of developinga good intuition for the mathematics. A second exampleshowing how vectors are added illustrates this. In themodel, a diagram of two vectors and their resultant is

shown, with construction lines to show how the parallelo-gram rule is used. When the model is running, the studentis able to grab hold of either of the vectors and move itaround. The resultant changes accordingly.

The power of the program is difficult to demonstrate onpaper, but when students can move vectors around and seethe result of the vector addition instantly, their intuitiveunderstanding surely has to grow. The complications ofscale drawing are handled by the software, while the stu-dent gains direct experience of what vector additionmeans.

A third example concerns resolution of vectors into com-ponents. Again, Modellus allows direct manipulation of avector and a visual resolution into components. The rele-vant window is shown in figure 6.

Again, the tip of the vector can be moved around onscreen and the components follow. The mathematicalmodel controlling this animation is a simple statement ofPythagoras’ theorem, but it could equally well have beenwritten using trigonometry. By setting up such a model forthemselves, students are able to check that they have han-dled the mathematics correctly: any errors are immediate-ly apparent on screen when the model is run.

As a final example, consider projectile motion. The par-abolic path followed by a projectile is the result of vector

Figure 5 Vector addition in Modellus

Figure 4 A Modellus model that canbe used to develop intuitive ideas

about the meaning of a velocity vector

Figure 6 Modellus can be used to show how vectors are resolved into components, and the resolution can be

animated by dragging the grey vector around on the screen


•When students can move vectors around and see the result of the vector addition instantly, their intuitive understanding surely must grow

•Students should ultimatelybe able to resolve and add vectors without the aid of the computer

changes to the velocity over a finite time step. The newvelocity is then used to compute the path over the nexttime step before the velocity is updated again. This wholeprocess may be animated as, step by step, the projectilecrosses the screen and vectors are added to the path. Theanimation window from Modellus in figure 7 shows thisvector addition pictorially.

Perhaps, then, computer modelling can help at leastsome students gain an understanding of vectors throughinteracting with animations. In this way, abstract conceptscan be made more concrete and the complex mathemat-ics given meaning in the mind of the student. Of course,computer modelling is not the solution to everyone’s prob-lems and the examples given here are no more than abeginning. We would still expect our students ultimately tobe able to resolve and add vectors without the aid of thecomputer, but the computer may provide a gentler intro-duction to some abstract ideas. Students might then be lessintimidated and more successful when they encountermore abstract ideas.

addition. A small change in the velocity of the projectile isinduced moment by moment by the acceleration due togravity. This process can be modelled by calculating

Figure 7 Animating parabolic motion and showing howvector addition leads to the projectile’s path

•Defining vectors simply as quantities that have magnitude and direction is unhelpful.The key thing is knowing how they are added.

•Beginning with displacements that can be added ‘tip to tail’ builds on students’ intuition and makes vector addition straightforward.

•The displacement vector is too directly descriptive a starting point. It is quite different from other vectors such as force.

•Notation between mathematics and physicscourses needs to be standardised. There could be more two-way cross-fertilisation between mathematics and physics.



The Qualifications and Curriculum Authority (QCA) has been developingand piloting new mathematics qualifica-tions for post-16 year olds, which willhelp to address concerns about themathematical competence of sciencestudents.

The pilot, involving all the examinantion boards, began inSeptember 1998 and will last for two years. Many studentscurrently commence A level science studies with an inad-equate background in mathematics. It is clear that anintensely mathematical subject like physics demandsmathematics that goes beyond GCSE in its scope andapplication.

Unfortunately, it is also true that many A level physics stu-dents do no formal mathematics beyond GCSE, and that agrade C pass at GCSE is often all that is required as a pre-requisite: this may not even guarantee the minimum alge-braic competence that is expected of physics students.

Bridging the gapThe new Free Standing Mathematics Units can be used tofill a gap in provision. These units were developed with theintention of increasing the numbers of post-16 year oldsparticipating in the study of mathematics; they will alsohelp to increase standards of mathematical attainment ofthis age cohort.

The units are stand alone, and each is expected to be delivered in 60 hours of class contact time. Students arerequired to use graphical calculators, spreadsheets, orother relevant software packages that would usefully support mathematical work in science subjects like physics.

A twin track approachAn important feature of the units for those wishing to usethem to support scientific study is the twin-track approach toassessment which separately assesses both principles andapplication. For each unit, students have to complete a seriesof assignments and a written examination with a 50:50weighting: candidates will have to pass both components tobe awarded an overall pass.

The examination will test the principles of mathematicsassociated with a particular unit. Portfolio evidence shouldbe used to apply the mathematics to other areas of work. So,a physics A level student studying the advanced mathematicsunit ‘Working with algebraic and graphical techniques’, orthe intermediate unit ‘Using algebra, functions and graphs’could apply the mathematics to aspects of physics. There isscope for the mathematics teacher and the physics teacher tocollaborate and work out commonly accepted goals.

Twelve units have been developed. These operate at threedifferent levels (foundation, intermediate and advanced)respectively covering the lower end of GCSE mathematics(below grade C), the upper end of GCSE mathematics(grade C and above), and mathematics that goes beyondGCSE into work of AS or A level standard.

Six of these units could have direct relevance to science students. Each student could do a unit that caters for his or herimmediate mathematical needs: to consolidate work alreadydone, but not completely understood, or to study aspects ofmathematics directly relevant to the other areas of the student’s programme of study. In this latter category, mathe-matics for A level science students could be the desired goal.

Free Standing UnitsEach unit will be separately certificated at its own level,with pass grades reported on a five-point scale. Theadvanced units will be worth the same UCAS tariff as allnew A level modules, or GNVQ units, to be introduced inSeptember 2000. The units pitched at GCSE mathemat-ics standard are smaller than a full GCSE, but compensate

Jack Abramsky

13 Free Standing Mathematics Units


by focusing on a restricted field of mathematics, so thatwhat is lost in breadth is gained in depth of study.

QCA welcomes enquiries from centres that would be interested in participating in the second year of the pilotwhich will run from September 1999 until August 2000. Itis intended that the mathematics units will become a fullpart of the national qualifications structure fromSeptember 2000.

Further InformationIf you are interested in the pilot, or would simply like further informationon the units, please contact:Dr Jack Abramsky (tel: 0171 509 5738), Dr Mike Coles (tel: 0171 509 5604), or Clare Tighe (tel: 0171 509 5581) at QCA.

•The units may help to build bridges between mathematics courses and physicscourses: portfolio work can be drawn from science contexts. Notation will have to be standardised.

•The use of new technologies is encouraged:what are the implications for training teachers?


•Each student could do a unit that caters for his or her immediate mathematical needs

•Unfortunately, many A level physics students do no formal mathematics beyond GCSE


Broadening mathematical thinkingThe articles in this booklet are largely about developingphysical intuition and a feel for mathematics that is morethan the ability to manipulate equations.

Of course students have to be able to do algebra, ofcourse they have to be able to analyse data and plotgraphs, but unless they know why they are doing this andwhy mathematics is essential to an understanding ofphysics they will not be motivated to overcome their difficulties.

A key to this has to be talking about how mathematicalexpressions communicate physical ideas with elegance,economy and power, not just practising techniques whosevery point is left unclear.

Students should begin to see why the same mathe-matics applies in different contexts. That the rate of decrease of a quantity should be proportional to the quantity is intuitive for water emptying from abath; the same reasoning can be applied to capacitordischarge.

Why the same is true for radioactive decay is less obvi-ous but developing the mathematics in a familiar contextmeans there is room to discuss the physics.

Graphs are not simply a way of presenting data: they tella story about relationships. Students need to put this storyinto words as well as symbols. Requiring that students talkabout their mathematical reasoning – not just graphical –is important in developing understanding and in revealingmisconceptions.

Philosophical questionsWhy is mathematical thinking so central to physics? Whyis mathematics so fruitful a language for the descriptionof nature? Such philosophical questions interest manyscientists, teachers and students and can enrich thephysics curriculum.

Supporting mathematical thinking in physicsTo support their learning in physics, students might berequired to take supplementary mathematics courses –perhaps QCA’s free standing mathematics units, or ASlevel mathematics. However, the danger in attaching anyadditional workload to A level physics is that some students will be deterred from choosing physics. The alternative is to build at least some of the necessary mathematics into physics courses, as physics.

How much mathematics should we teach and howquickly can we teach it? If we slow down our teaching,more able students will be bored. We should not expectmastery of mathematical techniques when they are firstencountered. Familiarity and frequent exposure to thesame ideas in new contexts gradually bring understanding.

Let us get students to think about their own thinking(metacognition) in order to develop understanding.Sometimes we will cover content very slowly but thereshould be a pay-off in improved understanding and theability to move faster later on.

Maths education, physics educationMathematics courses can provide the tools needed tounderstand physics; physics courses can provide contextsin which mathematics can be applied. However, the connections between mathematics and science depart-ments are often weak, and teachers on both sides areunaware of where and why there are differences. Studentstoo have difficulty transferring their knowledge betweensubjects. Where supplementary mathematics courses areoffered to students they do not always relate well to thesubjects they are designed to support. We need to workhard in schools and colleges to build bridges betweendepartments and subjects, to look actively for opportuni-ties to make connections, to have time to work and planwith colleagues. There are opportunities at GCSE level to

Loose Ends

Derived from a discussionmeeting at the ASE AnnualConference, January 1999


some of our formal techniques are now outdated. A care-ful and thorough evaluation of the opportunities present-ed by graphical calculators and computers is needed.Where is this technology best used? When should it not beused?

There are also practical challenges: first, ensuring thatteachers and teacher trainers are themselves properlytrained in the use of new technologies; and second, ensur-ing that appropriate equipment is available in schools andcolleges, in sufficient quantity and sited where it is needed.

Related DiscussionsIn recent years many organisations have critically exam-ined mathematics education in the UK and factors affect-ing it, each from a different perspective. We list only three.

The Engineering Council et al., Mathematics Matters in Engineering, London, 1996.Concerned with the mathematical skills of those enteringfurther and higher education in engineering and science,this short paper proposes specific actions to improve stan-dards.

Wake, G. D., and Jervis, A., Mathematics and Science Capabilities of Students with TechnologyGNVQ , UCAS, Cheltenham, 1997. The authors develop a framework to identify student capa-bilities in respect of the knowledge, skills and understand-ing underpinning a range of qualifications at level 3.

The Royal Society and The Joint Mathematical Council,Teaching and Learning Algebra pre-19, London, 1997. This detailed and thorough report was written to informrevisions of the National Curriculum and post-16 courses.It includes a comprehensive bibliography as well as recommendations.

share the teaching of some topics – speed–distance–timerelationships, the drawing and interpretation of graphs,the use of algebra - and to adopt common notations andlanguage. If there are differences – for example equationswith or without units, vectors as columns of components oras ‘arrows’ – everyone needs to understand why they exist.

Curriculum links with local industry and with universi-ties can help students experience mathematics applied toreal scientific tasks.

Sharing good practiceThere are many people with good ideas about how best tocommunicate mathematical ideas. Some of those are pre-sented in this booklet, but there are many that are nothere. Publications such as Physics Education, Physics Reviewand School Science Review regularly carry articles in this vein;for example, Peter Gill’s article in Physics Education March1999.

No two students and no two teachers are alike. There isno single approach that will work for everyone. Goodpractice never stops needing to be shared. One way isthrough meeting face-to-face in local teacher networks,something the Institute of Physics post-16 Initiative is keento support.

Information TechnologyNew technologies are often attractive to students and canbe used to support independent learning. Computer ani-mations can make the abstract more concrete. Numericalmethods can be used to model physical behaviour and tocompare models with reality. We hope to see new softwaredeveloped which will facilitate computer-captured databeing directly compared with computer models. In all, thecomputer must now be regarded as an indispensable toolin physics teaching.

But new technologies present new challenges. Perhaps


1For Examination BoardsQCA specifies the mathematical content to be builtinto A level physics syllabuses. In designing anyphysics course, Boards should both consider howthe mathematics is incorporated and also allowtime for its teaching.

Boards can encourage the development of newways of teaching and learning by designing assess-ment schemes which require students to use newtechnologies. For example, in developing theAdvancing Physics AS/A course, OCR is buildinginto the syllabus ideas about numerical modelling.

QCA also requires A level physics courses toinclude philosophical matters. The remarkable suc-cess of mathematics as a language for describingthe physical world has prompted philosophical dis-cussion and writing from a variety of scientists;their ideas could be included in a list of topics forconsideration by students. For this to be taken seri-ously by teachers and students, such discussionsneed to be assessed in some way.

Subject panels for mathematics and physicsshould meet together periodically. Their syllabusesoften use different notation and language todescribe the same ideas, differences which canprove an unnecessary stumbling block to students.

2 For QCAStudents need time to consolidate their knowledgeand understanding of mathematics. When definingthe subject criteria for physics, thought needs to begiven to how much mathematics has to be taught aspart of physics and appropriate time allowed.

Reviews of National Curriculum subject Ordersshould look for opportunities to link subject areas,to make notations standard, to identify commonmaterial and avoid duplication.

3 For UniversitiesMore dialogue is needed between universities,schools and colleges. As new ways of teaching math-ematics within physics are developed for use inschools and colleges, universities need to be aware ofthem. If we become more successful in motivatingstudents in schools and colleges then this has to bemaintained through university. There may be a priceto pay: students may have learnt fewer techniques.This price is worth paying if more students are moreinterested in and enthusiastic about physics.

More dialogue is also needed so that schools andcolleges are kept aware of how mathematics is usedin university courses.

4 For Teacher TrainingEntrants to the science teaching profession may notbe aware of the mathematical difficulties they willmeet in their students nor of what their mathemat-ics colleagues are doing. Trainee mathematicsteachers may never themselves have studied sciencecourses post-16. It is vital that trainee teachers shareideas and teaching methods wherever possible sothat they can help their future pupils make connec-tions and so that they can themselves begin to buildbridges between subjects in their own schools.

Trainee teachers need to have the time andexpert support to help them explore the opportuni-ties and challenges of new technologies.

5 For Government agencies funding educationAs the technologies available for teaching mathe-matics in physics courses change, schools and col-leges need extra funding to acquire appropriatesoftware and hardware - which is not always com-puters - and the in-service training to support itseffective use.

Funding is also needed to facilitate cooperationbetween departments, to build in curriculum linksand coordinate schemes of work.

Physics in Mathematical Mood – Some Ways Forward

The curriculum framework,those who write physics coursesand those who teach, should explore new ways of developing mathematical understanding alongside physical reasoning.

The Institute of Physics post-16 Initiative is aforward-looking initiative to revitalise physicspost-16, attract students, support teachersand influence the direction of future syllabuses.

This Discussion Series is intended to promote debate and ideas about the way forward for physics education in schools andcolleges in the next decade.

SHAPING THE FUTURE

1 MAKING PHYSICS CONNECT

2 PHYSICS IN MATHEMATICAL MOOD

If you find this booklet stimulating, please encourage others to read and discuss it.

Your views are welcomed – send them to the Series Editor through the post or the website.

The Institute of Physics post-16 Initiative team:Jon OgbornPeter CampbellPhilip BrittonSimon CarsonIngrid EbeyerIan LawrenceAveril MacdonaldMary WhitehouseCatherine Wilson

Institute of Physics post-16 InitiativeInstitute of Physics76 Portland PlaceLONDON W1N 3DH

email [email protected] http://post16.iop.org

SHAPING THE FUTURE2 PHYSICS IN

MATHEMATICAL MOOD

Including discussion points raised at an ASE Annual Conference session in January1999, the following aspects of physics inmathematical mood are considered:

● The mathematics students bring with them to start a post-16 course

● Links between mathematics and physics courses

● How to improve problem solving and mathematical understanding

● The essential mathematical functionsencountered in physics

● The importance of graphical skills andhow they can be developed

● How information technology can be used to support students

Also included are suggested ways forward for exam boards, universities, teacher training,the QCA and other government agencies.

Further copies of this booklet may be purchased from the post-16 Initiative at the address opposite.

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