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580 Pages

### PrefaceEdit

Why is this paper suitable for wiki development? Well principally it is difficult to get good data and hard facts. Everyone involved in this field agrees there are severe problems but much of what is proposed is based on anecdotal evidence like the first draft of this paper. It is hoped that a multiple author continuous review process might tighten up what starts as the principal author's collection of ideas and prejudices.

The initial ideas are specific to a UK context.

### AuthorsEdit

Dr. Martin Grayson, UK based scientific consultant, MartinY 08:52, 8 September 2006 (UTC)

### KeywordsEdit

Mathematics, CAL,Algebra

### AbstractEdit

This paper deals with the thorny issue of how to teach mathematics to the non-maths-A-level streams of university science degree courses with particular reference to a Chemistry course. (Equivalent problems exist in Physics, Engineering and Biology.) The recommendations are that there is no easy fix and that the time must be put in by both staff and students. The use of computers or of pen and paper is not especially relevant to the outcome of the teaching. Some pen and paperwork is recommended as the final exams are dominated by paper answers and fluency at this stage is required.

### IntroductionEdit

For many universities the days when admission to a Chemistry, Chemical Engineering, Materials Science or even Physics course could require A-levels in Chemistry, Physics and Mathematics are probably over for ever. The broadening out of the school curriculum has had several effects, including student entry with a more diverse educational background and has also resulted in the subject areas Chemistry, Physics and Mathematics becoming disjoint so that there is no co-requisite material between them. Typical syllabuses are available on the following web sites the Assessment and Qualifications Alliance (AQA) and EdExcel (EdExcel). This means that, for instance physics cannot have any advanced or even any very significant mathematics in it. This is to allow the subject to be studied without any of the maths which might be studied at the ages of 17 and 18. Thus physics at school has become considerably more descriptive and visual than it was 20 years ago. The same applies to a lesser extent to chemistry.

This has meant changes in the way chemistry has been taught at university from the days when lecture 1 at 9am on the first Monday morning began with particle in a box, (quantum mechanics thereof), followed by the hydrogen atom etc and its orbitals. Within a few weeks 1st and 2nd order rate equations and pH would follow quickly. (Logical as this approach apparently may have been there is anecdotal evidence that it was not understood by the majority of 1st year students forty years ago.) In a contemporary approach the chemistry being taught must be synchronised in with the learning of the non A-level maths stream students. This means complications in a modular system as there must be cross talk as the modules are delivered to make sure this synchronisation takes place.

At present this paper draws largely on one author's current and past experience, quoting data which is largely anecdotal, but supporting conclusions with hard peer reviewed literature based facts. The intention is to point out generic modes of thought, which are sometimes good and sometimes oft-repeated mistakes.

### How Much Math do We Really NeedEdit

There is an issue here as there is a point of view that the sciences have an excess of algebra and analysis in them which comes from the subject curricula being still influenced by the pre-computer age. Modern science could be practiced using spread-sheets, simulations, graphics, pre-coded linear algebra programs and statistical analysis. This might be appropriate for the right course, however we have the issue of where the expertise to write the programs and spread-sheets is to come from if we remove too much of the nitty-gritty from the course.

It is interesting to see that the French Nobel Prize winner, Pierre-Gilles de Gennes (de Gennes and Badoz,1996) expresses the opinion in a chapter entitled The Imperialism of Mathematics that in France, at least, there is too much maths in science courses, and students are encouraged to regard mathematical skills as superior to observation, manual skills, common sense and sociability. This might not be a general observation and may be a product of the formality of the French education system, which some people in the UK regard as a great asset rather than a disadvantage. Also I suspect the level of the excess mathematics is way beyond and more abstract than the basic skills required by chemistry undergraduates.

There was a lot of analytical theoretical chemistry originally worked out in the 1950s and 60s before the advent of computers which might be truely no longer worth teaching because essentially the same problems are approached with a greater degree of accuracy using techniques enabled by powerful desktop computers such as simulation and numerical Monte-Carlo and massive linear algebra.

### The Philosophy of a Course DesignEdit

There are two very different kinds of student requirements here. The non-A-Level maths student module will be essentially a remedial course and the majority of students are self-selected to be not sympathetic to maths. The students with strong mathematical ability are to a statistical extent streamed out of this class. It is therefore necessary to teach by many elementary examples and marking work to provide feedback. But there will also be students with strong mathematical ability who, because of a variety of reasons happened not to do maths A-level. It will be difficult but necessary to teach these in the same class. How do we stop this group being bored?

Having two kinds of students in the group could provide an opportunity for peer instruction (Crouch and Mazur, 2001). One would however need to get permission and put people into algebra expert / algebra novice pairings. In 1st year tutorials peer mini-lecturing is employed by some but by no means all tutors. This has not been as successful as anticipated but there is evidence also from Crouch and Mazur that PI only works well if it is supplemented by marked preparatory work.

Some universities use the mathematics department, not chemists, to deliver the A-level level maths and they teach with a large amount of contact time. In recent years there has been a move from teaching remedial maths with mathematicians to using chemists because the self-selected student group who require this teaching are not interested in mathematics from a mathematical perspective but require what they need to know in order to get a chemistry degree, and perhaps no more. (Many students have a strong vocational bias and see subjects in any way peripheral to the main degree course as obstacles to qualification, not areas of interest.) This is one reason why it is desirable to be familiar with the requirements of the professional society, the Royal Society of Chemistry (RSC) and the employers of chemistry graduates and the subsequent content of the chemistry curriculum (Cockett & Doggett, 2003). Excursions of non-vocational interest might not be appreciated by the students.

However when it comes to the issue of which textbook to use whilst there are very good beginners maths for chemistry books such as (Scott,1995) or (Tebbutt,1994) it is possible that the chemistry is better taught in embedded form in the main course. A concentration on algebra, equation manipulation skills and calculus might be better done using a purely A-level text book such as (Bostock and Chandler,2001).

An issue where there are real philosophical issues to be resolved is to what extent one attempts to teach technique i.e. what to do, and how much fundamental understanding has to be imparted. An example is the calculus of simple polynomial functions. For differentiation bring the power down and reduce the power by one can be applied as a rote formula successfully and the reverse for integration without ever doing the derivation or really understanding what is going on. In fact there are students who can successfully do integration but still ask questions like Do you put the squiggly integral sign at both sides of the equals sign? thereby showing that they have the procedure but its meaning is a total mystery. Perhaps the mechanics should be taught first but there should also be a continuous embedded attempt to get over the true meaning. It has even been suggested that knowing the mechanics of the algebraic procedure alone is enough for both the calculus and complex numbers required by chemists. (It is possible to manipulate i, the square root of -1 very successfully without at all appreciating how it is a fundamental part of a full counting system.) However understanding how complex numbers are clearly necessary in the counting system once you have the concept of areas and areas which can be negative is a great advantage. In accounting for land ownership you might owe a field to someone. In this situation you own negative area which has a complex perimeter distance.

### Implementation IssuesEdit

There is the question of whether to deliver the 1st year Maths for Chemists modules with a mixture of chalk and talk and problem workshops or with extensive use of Computer Aided Learning (CAL). There is an issue that sometimes the lack of empathy for maths goes over into a dislike of computers. It might be desirable to make computer based teaching, algebra and display systems available but not compulsory. A website is of course highly desirable as a meeting- place / bulletin board and as a reference for students away from the library but with access to a computer.

On equiring about what students actually know, as opposed to what the syllabus suggests they know, when they come to university there is some evidence that some schools have an de-facto practice of teaching less than the full curriculum and teaching a reduced set much harder. This might be an optimum tactic in a league table situation depending upon the risk of there not being enough choice on the paper to answer sufficient questions! There is some evidence of this in a quote from a recent Australian study (Stacey & MacGregor, 1999): subtle reductions in goals and the isolation of topics in the curriculum were disturbing trends.

Some efforts have been made in the UK to develop World Wide Web based materials for Maths for Chemists. (A proper bibliography of these are being collected.) Given the computer-phobic and algebra-phobic nature of some of the students this might not repay much effort compared to a traditional workshop style environment where the students are encouraged to do some of their own peer-group mentoring and use pencil and paper. A system whereby a maths workshop runs in a large room but there is a small room with a few computers running an algebra system nearby so that students can leave their paperwork and use an algebra system for a few minutes if they find that helpful is very good. However it should be possible to do the entire course using only pencil and paper. A computer algebra system being available but not compulsory is probably the most desirable solution.

Bonham et al. (2003) have conducted research suggesting that pencil and paper versus computer makes no difference: the medium itself has limited effect on student learning, so paper and chalk teachers could feel vindicated in this respect. However the authors still suspect the right program in the right context would be very positive, as has been described by Budenbender et al. (2002). Some aspects of exams: pen and paper, time pressure, artificial and stressed environment, need appropriate practice for optimum performance so too much computer use versus pen and paper can degrade examination performance.

Many universities have had experience with using computer sessions to replace tutorials. These can be very successful in terms of getting lots of questions and facts through per hour at the screen but surprisingly are not always liked because of the time pressure and lack of human contact even though there might be a fair amount of allowed collaboration going on in these sessions.

The primary author developed a card game / jigsaw style puzzle to help teach integration by parts. This was partially successful but of course could not remove the fact that it was algebra! The author always tried to relate the mathematics to things like the use of quadratic equations after the floods in Mesopotamia and the ages of dinosaurs as well as chemical problems so as to continually emphasize the usefulness rather than the abstract properties of maths. There is a wonderful old story retold in (Dehaene, 1997) about the curator at the Natural History Museum being asked about the age of the dinosaur fossil in the entrance. Question to curator - how old is this dinosaur/ It is 65 million and thirty-eight years old/ 65 million and thirty-eight, how can you be so sure about the 38 years/ well it is because when I started work here 38 years ago on my first day I asked, how old is the dinosaur, and was told 65 million years, so it is 65,000,038 years old. I cannot think of a better way to impress upon students the importance of significant digits. The classic mistake in a laboratory situation is to quote a number to the accuracy of the calculator when the experimental data is rather inaccurate. (There are however some situations, when proving computer programs, where the full accuracy of the computer must be recorded even if an experiment could only replicate 4 significant digits.)

If the mathematics teaching is given to someone too involved in advanced research to relate to people who are fresh out of school it can be hard not to go off at research related tangents and refer to concepts, even obliquely, which might not be understood until much later in the undergraduate course. There is a known problem of the expert blind spot (Nathan and Koedinger, 2000) where the student's perception of the problem and its difficulties are at variance with the teacher's (Silberman, 1981). One way of investigating this is to get the class to rank the problems in a test paper in order of difficulty. The result is not always what you as an expert would predict.

There are many things a teacher might expect to be widely known but are in fact not. Examples might be if you refer to something as being a Judgment of Solomon or a Gordian Knott situation. Biblical and classical allusions will only be understood by a small portion of the class!

One of the ways in students subsequently practice their maths for real is in the problem solving classes. Transfer of expertise from problems where the procedure is identical to a given worked example but only has different numbers (analogical transfer), to problems which require a leap of imagination is clearly not trivial. (Indeed if one sets an exam which has such a leap at the beginning of a question one runs the risk of wholesale zero scores.) Even a problem of the imitative type may be found to be difficult (Robertson, 2000). Mayer, 2004, has given a model of the mental processes required in problem solving. Success comes from making a situation model. Lack of success comes where a student concentrates on the numbers trying to grab a connection thereby making sign or multiplication / division mistakes. This grab it and hope method has even been seen used when trying to rearrange $PV = {\rm R}T.$

(Something has gone so wrong with the student's command of manipulation that what should be a relatively automatic technique has been reduced to clutching at straws for the answer.)

(Mayer, 2004) breaks up problem solving into:

• Problem translating, into an internal mental representation.
• Problem integrating, into a situation model.
• Solution Planning, find related problem and form sub-goals if necessary.
• Solution Execution, apply algorithms.

The drawback to such a structured approach is that if you understand how to do this systematic breakdown you are unlikely to have a problem! It can however direct you towards where the blockage in the problem solving process is. Successful problem solvers correlate highly with successful word puzzle solvers of missing or irrelevant information in word problems (Low and Over, 1989). (Mayer, 2004) believes a workshop aimed at solving long problems ideally needs explicitly taught and labelled sub-goals but this is likely to reduce the adventurous side of the problem solving. A peer instruction situation can be used to produce competing situation models. Deeper understanding is often facilitated by being aware of multiple representations of the same thing.

### Competence Based Methods of Mathematical TrainingEdit

The principal author has investigated formulating a new mathematics course based on competence based methods (Burke,1989) but is now not working in this area and so would like feedback on this issue from active practitioners. (Pilling, 1996) expressed dissatisfaction with Competence Based Training (CBT) in a teacher training context but this was because of the difficulties in defining and measuring the competences. In a maths situation this would however be very clear as there are definable and measurable skills such as:

• Mental arithmetic.
• Factorisation.
• Algebraic manipulation.
• Mechanical differentiation and integration skills.
• Imaginative integration skills.

Regrettably in this age of calculators mental arithmetic is still necessary to be able to factorise, rearrange and simplify expressions with any degree of facility. The author has seen students using calculators to factorize expressions such as:

$x^2 - 56 x + 1 = 0$

If your multiplication tables are weak for large numbers, (most people have much better mental arithmetic on numbers 1-5 than 6-9), you need to check 7x8 on a calculator! Similarly you would not spot that the square root of 64 was 8. Any weakness in mental arithmetic would make the following simplifications difficult

$x^2+2x-15 = (x-3)(x+5)$

$x^2-4x-12 = (x+2)(x-6)$

$3x^2-12x-36 = 3X~~~~ {\rm where} ~~~~ X ~~~~~ {\rm is }~~~~(x^2-4x-12)$

$2x^2-10x-12 = (2x+2)(x-6)$

This lack of fluency in factors means that evaluating a third plus a half (1/3 + 1/2) without a calculator becomes impossible for some students. The need for this fluency with integers needs to be fed back to schools as by university year 1 it is probably too late to be acquired.

It has been shown (Dehaene et al.,1999) that arithmetic and algebra are disjoint skills as is to some extent estimation. There is a physiological condition somewhat like dyslexia where sufferers cannot do mental arithmetic correctly, dyscalculia, but such people often have normal estimation skills, which seem not to be connected to mental arithmetic. For a scientist estimation skills could be regarded as much more important than arithmetic especially in the age of calculators. Approximation can be done on quantity representations using visuo-spatial skills, (Dehaene,1999) or by exact arithmetic to 1 decimal place, which is the more common method among professional scientists. (Butterworth, 2000) explains how a carpenter's estimation of wood requirements can be so accurate without using a calculator or conventional algorithm.

It is very useful to have a physical picture which might lead to a quantity guess such as a barrow load of coal is about 20 kilograms not 2 nanograms, or the energy of a light photon is about a chemical bond i.e. 200kJ per mole. This corresponds to yellow light, not 2 attoJoules per kilograms.

• A barrow load of coal - 20 not 2 x 10 60 kilograms.
• The Planck constant - 6.626 x 10 -34 J s. (An unbelievably small energy related quantity not 6 followed by 33 zeros.)
• The number of water molecules in a glass of wine - 2 x 10 24.
• Yellow light - 200kJ per mole.
• kT the ambient energy at room temperature - 2.5kJ per mole, (about 1/100th of a chemical bond).
• The energy by which a 1s electron is held in a xenon atom - over 1010 J per mole, over a thousand chemical bonds and in the X-ray region.
• 2 cm-1, the fine splitting in an IR spectrum - 0.024 kJ / mol, much less than an average bond energy of about 300 kJ / mol.
• The large magnetic field in a superconducting magnet NMR machine - 23 Tesla. (1 Tesla is a big magnetic field.)

In a 1st year student question to work out the mass of a calcium metal unit cell answers were given of 8 x 10 60 kilograms. Even a neutron star is not that heavy but this was regarded as a reasonable answer. In another problem students divided by the Planck constant instead of multiplying and said to the tutor my calculator is too small. as it ran over the exponent limit. Broadsheet newspapers are not immune to this problem and even on their financial pages frequently use million when billion was intended and vice versa. This implies a lack of visualization of quantity of three orders of magnitude which is regarded as a trivial mistake!

Algebra skill is related to grammar and word use (MacGregor & Price, 1999) and there may be a correlation between problems with algebra and poor writing skills or dyslexia. Dehaene has postulated that reading and writing developed too quickly and recently to be a product of evolution (Dehaene,2003). Therefore the writing systems must have evolved to take advantage of pre-evolved systems in our brains. The cultural invention has adapted to pre-existing brain mechanisms. Thus algebra piggybacks on writing and grammar skills. There is strong evidence (MacGregor and Price, 1999), that linguistic skills (meta-linguistic awareness), are strongly correlated with algebra ability. Maths requires both linguistic competence and visuo-spatial representational competence (Dehaene,1999) In terms of becoming proficient and automatic at algebra and calculus it is estimated (without publishable evidence) that about 300 hours of work are needed.

One is sometimes driven almost to despair as to how algebra and numeracy can be incalculated in student's brains. One colleague when talking about energy said "the kinetic energy is half m v squared" and was asked in all seriousness: "what is the other half equal to"? On reflection one can see the question is not at all stupid because there are many different kinds of 2 in mathematics such as the denominator from integrating a linear function, or a 2 from a doubly occupied orbital, or the 1/2! from a power series, or even just 1 thing in one room added to another in the next room. These are all subtle variations on the beauty of the use of numbers and clearly this has to be successfully put over to produce good scientists.

### A Biographical Note on AlgebraEdit

Algebra was invented and reinvented at various times from Babylon through the Hindu mathematical period about 650 CE. It seems however in the public mind to be widely erroneously thought that it was invented by the Arabs in about 1000 CE (Ifrah,2000). The idea crossed the mediterranean in the 16th century and Franciscus Vieta (1591) made the great improvement of using letters for continuous variables. Subsequently René Descartes (1637) put algebra into a recognisable modern form. Interestingly enough an early Hindu written source, author Brahmagupta, in 628 CE (Dvivedi,1902) describes algebra in words and seems to get the operation minus times minus equal to minus! I am sure for practical calculations they would have got in right and this is a misinterpretation of the 1902 translation by the current author.

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