How do we teach non-university mathematics to university chemists?

===How do we teach non-university mathematics to university chemists?===
===Preface===
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. 
===Authors===
Dr. Martin Grayson, UK based scientific consultant, MartinY 08:52, 8 September 2006 (UTC)
===Keywords===
Mathematics, CAL,Algebra
===Abstract===
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. 
===Introduction===
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 <I>disjoint</I> so that there is no
<I>co-requisite</I> 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 <I>particle in a box</I>, (quantum mechanics thereof), followed
by the hydrogen atom <I>etc</I> 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 <I>facts</I>. 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 Need===
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 Design===
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
<I>peer instruction</I> (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>i.e.</I> what to do, and
how much fundamental understanding has to be imparted. An example is
the calculus of simple polynomial functions. For differentiation
<I>bring the power down and reduce the power by one</I> 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 <I>Do you put the squiggly
integral sign at both sides of the equals sign?</I> 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>i</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 <I>own</I> negative area which has a
complex perimeter distance.
=== Implementation Issues===
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 <I>optimum </I>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): <I>subtle reductions in goals and the isolation of
topics in the curriculum were disturbing trends. </I>
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 <I>et
al.</I> (2003) have conducted research suggesting that pencil and
paper versus computer makes no difference: <I>the medium itself has
limited effect on student learning</I>, 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 <I>et al.</I> (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 <I>card game / jigsaw</I> 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 - <I>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> 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 <I>must</I> 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 <I>expert
blind spot</I> (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 <I>Judgment of
Solomon</I> or a <I>Gordian Knott</I> 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 <I>situation model</I>. Lack of
success comes where a student concentrates on the <B>numbers</B>
trying to grab a connection thereby making sign or multiplication /
division mistakes. This <I>grab it and hope</I>
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: 
<UL>
	<LI>
	Problem translating, into an internal mental representation.
	<LI>
	Problem integrating, into a situation model. 
	<LI>
	Solution Planning, find related problem and form sub-goals if
	necessary. 
	<LI>
	Solution Execution, apply algorithms. 
</UL>
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 <I>sub-goals</I> 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 Training===
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 <I>competences</I>. In
a maths situation this would however be very clear as there are
definable and measurable skills such as: 
<UL>
	<LI>Mental arithmetic. 
	<LI>Factorisation.
	<LI>Algebraic manipulation. 
	<LI>Mechanical differentiation and integration skills. 
	<LI>Imaginative integration skills. 
</UL>
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 <I>et al.</I>,1999) that arithmetic and
algebra are disjoint skills as is to some extent <I>estimation</I>.
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>i.e.</I> 200kJ per mole. This corresponds to yellow light,
not 2 attoJoules per kilograms. 
<UL>
	<LI>A barrow load of coal - 20 not 2 x 10 <SUP>60</SUP> kilograms. 
	<LI>The Planck constant - 6.626 x 10 <SUP>-34</SUP> J s. (An unbelievably
	small energy related quantity not 6 followed by 33 zeros.) 
	<LI>The number of water molecules in a glass of wine - 2 x 10 <SUP>24</SUP>.
	<LI>Yellow light - 200kJ per mole. 
	<LI>k<I>T</I> the ambient energy at room temperature - 2.5kJ per mole, (about
	1/100th of a chemical bond). 
	<LI>The energy by which a 1s electron is held in a xenon atom - over 10<SUP>10</SUP>
	J per mole, over a thousand chemical bonds and in the X-ray region. 
	<LI>2 cm<SUP>-1</SUP>, the fine splitting in an IR spectrum - 0.024 kJ /
	mol, much less than an average bond energy of about 300 kJ / mol. 
	<LI>The
	large magnetic field in a superconducting magnet NMR machine - 23
	Tesla. (1 Tesla is a big magnetic field.) 
</UL>
In a 1st
year student question to work out the mass of a calcium metal unit
cell answers were given of 8 x 10 <SUP>60</SUP> 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 <I>my calculator is too
small</I>. 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
<I>evolved</I> 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 <I>m v
</I>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 Algebra===
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 <I>using letters for continuous variables</I>.
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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