# Study Skills in Mathematics – Set Work

### Do all of it, do it honestly and learn from the feedback. Study is not a spectator sport.

IDENTIFY YOURSELF! For all work you hand in, whether for marks or not. Follow your universities instructions about identifying yourself properly (usually via student id number). We need to be able to attribute marks after all.

If there are no university instructions, put your full name(s), student id number(s) and date on your work. Remember, common surnames can cause your marks to be wrongly attributed. Your marker may print it out so your electronic submission details will be lost when printed. If you put your name in a header, make sure it is still within the print area. It is usually better to put your name(s) etc. in the document body.

Do NOT rely on your lecturer knowing who you are when communicating with them (submission of work or otherwise). So don’t use just your first name, still less nickname or street name. They will not know who ‘Asbo’ and ‘Gazza’ are, they will not bother to find out and so you’ll get no marks. Do not abbreviate your name, but state it exactly as you did when you registered. CAPITALISE your surname, give your first name in full. Lecturers hate having to track down whether “A Fotheringay” is “Alan Turnstile-Fotheringay” or “Alice Fotheringay-Beast-Major” … give them a break!

If possible, check your marks have been correctly entered into your records and query this promptly if need be (not after the module has ended and the admin staff have entered them into a central system – changing them then may not be allowed, and in any case it is very annoying for them).

For any sort of assignment make sure you fully understand the question before you attempt it. What is required of you/your group? Analyse the question carefully, noting any restrictions on the length of your answer, how many marks the assignment is worth and how they will be distributed between the tasks required or various aspects of the assignment e.g. execution of the project, conclusions, presentation etc. If in doubt, ask.

Make sure you add it to your diary so that it doesn’t come as a nasty shock! Remember, deadlines give you the LAST date you can submit, not the time you should do it … or even the day before!

For arts and humanities, a great deal is said about essay writing since it forms such a major part of the learning process. Indeed the ‘act of writing’ helps develop ideas and a viewpoint in the first place. What’s more, learning to ‘argue your case’ from a variety of alternative viewpoints is pivotal to developing a sophisticated understanding of the content there.
This is far less true in STEM subjects, where there is often only one correct answer and your viewpoint or personal opinion is irrelevant (even if supported by evidence). Many would dispute this, but it is still true that content alone is never sufficient; one has to show how one topic follows from another in a way that brings out the internal logic of the subject. In fact, this structure is even more important in STEM, where assumptions need to be stated and built on, usually mathematically. Indeed mathematics, not writing, is normally used to develop ideas here, with writing being used to organise and communicate the subject in a coherent way. Writing is certainly not relegated to a mere reporting role, but is still an essential part of your learning. So if you have chosen e.g. maths or engineering at university to avoid writing, you are in for a nasty shock!

For all academic study, developing sophisticated writing style is certainly not necessary (or even desirable since it often raises the alarm that you are seeking to impress rather than communicate – some academics would do well to remember this too!). Just say what you need to briefly and clearly in ordinary plain English – do not adopt a style. Here’s a real example: students wrote “It was anonymously agreed …” which (from the context) should have been “It was unanimously agreed …” but why not just say “We all agreed …”?

Writing mathematics correctly is a task that requires no less attention to detail than actually doing the mathematics in the first place. A good guide can be found in F. Vavaldi’s book. For more general writing see the Thinking Wriitng pages at QMUL

For most essays it is assumed you will use a word processor, so that the final submission is the result of several sessions of developing and editing your drafts. Work directly onto the screen from your notes. Some dyslexic students are enthusiastic about these new versions of Word which have grammar and spell checkers running while you type. Alternatively see dyslexic.com for information on textHelp! and Inspiration.

Whatever you use, be very careful to proof read your work and/or get someone else to read it.

### Group work

If the assignment is group work, your marker will also be marking how well your group has worked. Being able to set up and work in an effective group is a great way to learn new content, assimilate new ideas and is also an essential employment skill. It’s good to talk! What’s more, the person who is able to explain things to the group gains most!

If your group doesn’t perform, that’s YOUR problem, not the lecturers, so don’t expect them to sort things out for you. Instead you can seek advice. Basically it boils down to everyone being professional; set an agenda for meetings and agree a set of minutes on who is to do what by when. At the end, everyone should ‘own’ the submission. That means that everyone should be capable of presenting it if the assignment is to give a talk (my PC usually chooses a person at random to do this on the day), and everyone should be able to answer questions about the WHOLE submission (not just their bit).

Remember a study or assignment group is not a friendship group; work with new people from different cultures as much as you can to widen your horizons! Do not work with people using your native language if the assignment is not required in that language. (Some time ago I had several Greeks in my class; my rules were that every group needed a Greek and no Greek could work with another Greek!).

### Précis or summary work

View an example of a precis

### Projects and essays

Start early, allowing time to give your supervisor (or at least let a friend) a draft to read, before re-writes/edits and final submission. Final year projects are typically 70 pages long and this seems daunting. To get started, make a list of all your objectives and all possible resources. Avoid the temptation to write a book, and do not wait until you have read everything on the subject – you will never get started of you do! Instead delimit the essay/project scope clearly so that it is not too big, but make a note of everything you have omitted in case you need it later (e.g. in writing Recommendations for Further Work).
Having decided the scope of your essay, draw a diagram of the subtopics and how they are connected together logically. Redraw this “mind map” several times until you are satisfied that it is complete and that you understand the connections fully. Only then should you start writing the ideas as a narrative.

Breaking the topic into such subtopics and writing them up before fully researching the others is often helpful; it helps you focus your ideas and gives you something to show for your efforts early on. Do not include material which is not relevant, simply because you find it interesting or think you need padding. Meet the required specification of the work in terms of subject and page or word limits – if you don’t you may get zero marks! Remember that garbage will still be recognised as garbage no matter how beautifully presented – there is no substitute for proper content.

The nature of a final year project has much in common with postgraduate research work, where responsibility for the management, content and content of the work passes to the student. You might therefore find the Postgraduate Research section helpful.

Another good source of advice is that given by Martins Bruveris in his blog.

PROJECT PLANNING: In planning your project (or even essay) you will find it useful to

• Make a precise statement of your problem
• Draw up a scheduling chart (Gantt chart – see image below)
• Outline your ideas using spray diagrams, category charts, flow diagrams etc.
• Identify resource and time constraints, clear connections between ideas and information flow, to identify the data you need for application of chosen statistical techniques; you will certainly need to discuss experiment or questionnaire design with a statistician before spending time collecting, analysing and reporting data and drawing conclusions from it,
• Specify any preliminary experiments or pilot studies you need
• Pinpoint areas of concern where you need to focus your attention, requirements such as equipment, training, software etc.

For projects and postgraduate work these charts and ideas should form an important part of your dissertation/thesis plan, and might even be included as an appendix in the final report. You will need to discuss these charts with your supervisor and update them regularly in the light of your agreed objectives and time plans. As a very rough guide, you could start with the schedule in LINK, mapping the 3 years of the PhD study discussed there into the time you have available.

At some universities, final year maths students may have to submit a draft plan when they give their project talks about half way through the project. The following examples demonstrate good planning and serious engagement with the projects!

Draft Plan 1

Draft Plan 2

ORIGINALITY: Students often overestimate or underestimate what is required of them for a major report, such as the final year project dissertation. Undergraduates do sometime make important discoveries … but more usually the final year project is required to be solid, honest and display some originality such as applying a known technique to a new situation. Keep a workmanlike approach (e.g. regular hours, set objectives, phone off the hook, “do not disturb” note on door) but be prepared for that “inspirational moment” and act on it immediately (keep a pen and paper by the bed).

INDEPENDENCE: Your supervisor will probably define what needs to be done quite closely at first, for example by setting model problems for you to work on. You will be expected to stand on your own feet more and more as time goes on, not only in the execution of set tasks, but also in generating and acting on your own ideas.

### Keep the audience and purpose of your work in mind

Purpose: is the requirement for a technical report, or does it need a more discursive (or even historical) setting that would require some sort of literature survey? Why would anyone want to read your report if you do not say why it is relevant in some way? This relevance could be that it fits into or extends some broader mathematical framework, or has some applications in science or social science. Do not assume your reader will know this – give it prominence in the introduction section.

Once you have the above clear in your mind, then and only then start to make short notes of all the points you need to cover. Decide which are main points and which are subsidiary. Understand how the main points are logically connected. This prioritisation and synthesis of ideas and approaches is very important for your learning and understanding. Do not attempt to recycle lecture notes or to cover everything in the hope that some of what you say will be to the point; you will only get marks for discrimination and hard thinking about the subject. Copying, even if you edit the material afterwards, is NEVER acceptable.

Departmental or University regulations apply to the contents of dissertations and theses (ask your supervisor). Note that a satisfactory final year project may be an absolute requirement; you will not get a degree without it! Also note that it is usually worth a large part of your degree mark, so you should be spending a commensurate amount of time on the project. The marking scheme for the project can be found on the syllabus.

Your essay/project should flow logically, so a clear and ordered structure is vital. This will avoid repetition and your reader will not then feel lost but rather, know why it is that they are reading the content at that point and not elsewhere. At all costs, avoid steam of consciousness writing where you put down everything that might be relevant/everything you know in the hope that your reader will be able to make sense of it. Readers will not bother … and why should they?
In general, longer work such as dissertations/theses should have section headings below (N.B. see your thesis/dissertation/project report requirements guide for the exact format).
For shorter essays, formal division into sections may not be needed, since this can be done by the use of paragraphs, reflecting the sections below.

Before you start you should write at least an initial plan and add some notes about what will be in each section. Note that the ORDER of writing the sections is rarely the order in which it is finally presented. Usually the Introduction is written near the end when you know what it is you are introducing! I would start somewhere in the middle with the section you know best. Then make notes, a plan and a draft before discussing with your friends or supervisor. Then rewrite to get something in the bag that’s worth reading. You will have time to polish it up and integrate it better with the rest of the work later. Then do other sections the same way … and finally write the Introduction, Conclusions and Recommendations and Abstract.

• Start Roman numbering
• Title/your name/ your year, the module/lecturer for the work and the date of submission. Make sure your title is correct, informative and short.
• Acknowledgements – mention any people who have helped you with the content of work you are submitting (not your lifestyle, beliefs etc.). This should include your supervisor, and might include other lecturers, academic advisors, technical support staff and possibly people who have provided or permitted the use of data (e.g. placement employers). NOBODY ELSE! A dedication is the place to mention parents/spouses or friends but remember ‘less is more’ and a simple phrase like “To my parents” will have far more impact than a long list of everyone you know. In any case, do keep it sensible: your examiners may not be impressed by such statements as “To Bob Dylan, whose songs kept me sane during this work!” unless they are Dylan fans themselves! Who? I hear you ask … look him up on Google and join the 1960’s.
• Consider adding a Nomenclature defining all non-standard terms and symbols used.
• Abstract – stating the main area of work covered and main results obtained. This should be concise, often not be more than half a page of type-written text (approx. 300 words) and is usually written last. Do not make the common mistake of ‘leading into’ the Abstract by including background material which properly belongs in the Introduction. Simply say what you have done, nothing else. Start your abstract with “This {report/dissertation/thesis} {comprises/considers/addresses/analyses/discusses/etc.} …” where you should choose the appropriate word in the braces.
• End Roman/start ordinary numbering. Use decimal notation and a larger font for the 1. Main Heading and 1. 1 Sub-heading.
• Introduction – usually a statement of the problem, why the problem is important and a review of previous work describing its strengths/generality and weaknesses/limitations. You should make clear where your work fits into existing knowledge and what gaps you have filled.
• Method – describes the methodology you are using. This might be an overview of your experimental design, data collection methodology or an overview of the main mathematical or statistical theories you will be using.
• Other sections – as needed, e.g. examples which fit into your main thrust. You should also include ideas which did not work out here, specifying (if possible) the reasons for their failure, and any unresolved issues or anomalies.
• Results – describes the results obtained and any limitations on their accuracy and generality. This section will usually contain analysis of the results/data and interpretation. An analysis of the sensitivity of the results to changes in parameters should also be considered.
• Conclusions – at least half a page of conclusions is needed. The conclusions must stress the main results only and require clear thinking. They are important and you cannot expect the reader to draw his/her own conclusions – if you are unable or can’t be bothered to draw conclusions why should you expect your reader to do so? Make sure your conclusions are objectively supported by your results, and not influenced by what you “wanted” them to be! Be self-critical and absolutely honest. Do not include new material in the conclusion.

### References

Keep a careful note of exact references you have made, otherwise it will be a lot of extra work finding the articles again when you write it up. You can record your references using either the British Standard or Harvard methods, see for example this guide produced by Anglia Ruskin University.

Journal articles may be referenced thus:
Greenhow M. “All you wanted to know about Calculus, but were afraid to ask” J. Interesting Results, 8, no. 4, (1983), pp 1-13.

Articles in books may be referenced thus:
Greenhow M. “Another boring paper about mathematics” in A collection of essays in mathematics by some people who ought to know better, Ch 6, ed. M, Thatcher, C.U.P., 1993.

Books, CD-Roms etc may be referenced thus:
Greenhow M. A self-study guide to newt racing, J. Wiley & Sons, 1979.

Web sites should be referenced by the full URL of the site (at the top starting with http://…) and the date when it was accessed.

In the journal reference above, bold, indicates volume number, or you can write Vol 8. The standard abbreviation for the journal can usually be found inside the front cover – if in doubt use the full title. When you write the references up, put them in alphabetical order by authors surname. Finally when you proof read your project, tick each reference mentioned in the text to make sure they are all listed, and that you haven’t listed any which are not referred to. In the text, you should make references as explicit and informative as possible; do not say at the end “The books I used were …” but rather, ” … as shown by Greenhow (1983), but later questioned and extended by Smith et al (1984)”. In the reference to Smith et al , full details of all the authors should be stated e.g. Smith P., Jones M. & Aardvark S. P.

• You could also include a Bibliography which contains material not explicitly referred to in your text e.g. background or further reading.
• Appendices A1, A2, A3 etc. – these may or may not be needed. They should include highly technical detail, such as tables of results, computer code or proofs of results which are not essential to the flow of the main write-up. You should also consider including a list of the software you used for various parts of the work and/or your logbook (see 4.10) – both are useful to other students reading your thesis. You should also include programs or data on a CD as an appendix with written instructions on what the disc contains and how to run it.

Number all the pages, including the appendices, and run the spell checker again.

### Writing style

For all academic writing, use in ordinary plain formal English – no slang! Remember your reader may not speak English as a first language, and will be unable to understand slang terms and their connotations. Avoid emotional or emotive language (no matter how strongly you feel), personal opinions (you are not a world leader in your field yet, so nobody cares what you think – at least not in a formal submission!), unsupported conclusions (see above) and jokes!

You can get better marks by avoiding common mistakes.

• Learn the difference between there/their/they’re, to/too/two, its/it’s.
• Learn how to use apostrophes e.g. Students’ Union, but do not use them to abbreviate it is, they are, will not, etc. which would then sound too informal. Do not use apostophies for plurals e.g. supplies not supply’s. If in doubt about a plural, consult a dictionary.
• Hyphens should be used for compound adjectives such as five-carbon molecule, free-surface effect, etc. Some nouns are also hyphenated e.g. cross-section.
• Do not start sentences with AND or BUT. Use moreover, however, nonetheless etc.

A comprehensive list of these issues (and much else) can be found at the Grammar slammer web site

Using bullet points is a good idea. Make sure that each point can be read as the continuation of the preceding sentences as in the following example. “The main points to remember when swimming are that you should: avoid breathing when underwater, keep the legs moving, staying fairly flat in the water.” Here the first two points complete the sentence grammatically, but the third should have started with “stay …”.

For protracted projects, such as the final year dissertation, keep your supervisor informed of your progress all the time by submitting thesis plans and draft chapters (not first drafts). Note that the abstract, introduction and conclusions are usually the last to be written after you have got the whole problem in your mind. You should aim to submit a short summary of what you have achieved, and a plan for what you will do next, roughly half way through the project. Near the end of the project, your supervisor will need to see a draft of the complete thesis in order to judge the overall development of the story line. Keep a logbook of what you have done (failures as well as successes, and how long each topic took to do) – this will make writing up far easier and provide evidence of your work. If you do stop work for a while (e.g. to concentrate on exams), make sure that you document where you are and what to do next – don’t rely on your memory.

### Using Tables and Diagrams

Do not use words when a table or diagram would show your ideas better, and refer to them in the text, e.g. “Figure 2 shows the current trend, from which we see … “. Equations should be read as part of the text and properly punctuated and numbered e.g.

Be aware that figures can be highly misleading, see e.g. “How to lie with statistics” D Huff, Penguin 1991. You need to choose the most appropriate form of diagram for your data, or perhaps show the data plotted in two separate ways to stress different points.

Pictogram; even though these are visually attractive they are often too ambiguous for scientific reports e.g

Classification/Consequence trees have the main class/idea at the top and sub-classes/consequences below e.g.

Spray diagram or skills tree have the main idea/highest-level skill in the centre and related ideas/more basic skills all round it e.g.

Flow Chart

Bar chart, comparative bar chart and component (or stacked) bar chart

Pie chart

Scatter diagram

Other charts include:

• Histogram; these are like the bar charts above but stress the area of each column which needs to be accounted for when displaying unevenly grouped data by varying the column widths and heights. The area scale should be stated in the caption and a square depicting 1 (or 10, 100, …) measured units should be drawn on the figure. Data with no upper (or lower) limit such as >1000kg should not be placed on the histogram, but should be stated in the caption.
• Linear graph; be very careful if you use different scales for the x and y axis as this will distort the graph, so that e.g. a slope of 1 will not look like 45 degrees and this will convey a misleading impression of either weak or strong dependence of the y variable on the x variable. Similar distortions occur when using the origin not at (0,0) or if the data ranges are not contiguous; both cases must be clearly marked on the axes with a break symbol or dotted line.

• Log graphs never have data at the origin and cannot be used for negative data.
• On a log-log graph, a datum point of (1,1) will be plotted at (0,0) since log(1) = 0,
• x & y data between 0 & 1 will be plotted in negative quadrants (since log(p) is negative if p is between 0 and 1),
• Data will be greatly compressed for large x (e.g. 10^6 is plotted at 6 and a million times larger 10^12 is plotted at 12), whereas data will be greatly expanded for small x (e.g. 10^-6 is plotted at -6 and a million times smaller 10^-12 is plotted at -12). Thus log to base 10 scales are often useful for depicting data covering a wide range such as the diameter of a molecule to the size of the Milky Way (-12 to 22 on a log base 10 scale when measured in metres).
• Moreover if y = kx^m where k and m need to be determined from the data, then a log-log graph of Y=log(y) against X=log(x) will give the straight line Y= mX + C where the constant C=log(k) and the slope m can be measured from the plot.
• Label the figure and each axes with the variable name and units, and mark the scales clearly. Finally all tables, diagrams and figures must have a self-explanatory caption so that they can be understood without reading the text by someone broadly familiar with the subject.

It is usually much better to include figures straight from other software (e.g. Excel, Mathematica, etc.) or drawing packages in the document via the clipboard; this makes managing and printing the document much easier. If you do need hand drawings, photos etc. you should scan them in and include them as imported pictures in the word processed report. Pasting them in (physically) and photocopying does not give such a good result.

This is an example of a bad bar chart; it has no title, no scale or axes titles and no legend.

This is an example of a good bar chart; it has titles, scales and a legend.

Always use S.I. units and standard form e.g.$1.2 \times 10^{-5}J$ or $3ms ^{-2}$ (not metres per sec per sec or m per sec squared etc.).

Learn the following notation:

G giga $10 ^{9}$

M mega $10 ^{6}$

k Kilo $10 ^{3}$

d deci $10 ^{-1}$

c centi $10 ^{-2}$

m milli $10 ^{-3}$

$\mu$ micro $10 ^{-6}$

As an aside, have a look at this web site on sizes of different objects: my daughter described this as the coolest thing ever – I agree!

For word processed reports use 10 or 12 point in a plain font such as Times New Roman. Titles and section headings should be larger e.g. 16 and 14 point. Dissertations should be one and a half or double spaced with a left margin of at least 40 mm (to allow for binding) and a right margin of about 20 mm. A professional effect can be achieved by using headers (with title and date) and footers (with author and “Page # of 6” where # is the page number input by the word processor), as in the top and bottom of this page. Cross-referencing to other sections, pages or equations can often only be done at the end; in the meantime, mark them with e.g. “see section ++” and use the EDIT, FIND command to pick these up in the final edit. Make regular backups of all word processed work.

### Before you submit your work

You must allow time to proof read your work. This should be done a day or two after you have finished, not straight afterwards. Then you will read it with fresh eyes and read what you have actually written, not what you think you wrote!

Firstly go through your words and cross out everything you do not actually need. This will make it much briefer and clearer. Then simplfy your English as much as possible by keeping your sentences short, clear, concise and (above all) correct (i.e. not phrases, but including a verb in the correct tense).

For example “I personally expect this mathematics to make more sense, however its structure and also complexity have gotten difficult for myself.” is horrible! It should be “I expect this mathematics to make more sense but its structure and complexity have become difficult for me.” (Note the word ‘gotten’ went out in the Middle Ages – except in America where they still seem to like it!).

Avoid the word ‘would’ unless there is a condition attached; for example “I would say …” makes no sense unless qualified, as in “I would say I could write much better than this, given some training.”

Do not adopt what you think is a mature style since your purpose is to communicate, not impress. Just say what you mean in ordinary plain English. Then you should do everything you can to order your material, diagrams, tables, paragraphs and sentences clearly to make it as easy as possible for the reader. Make sure that diagrams and tables have a number and a caption. If a report is required, use separate sections each with a section header (and possibly sub-sections and subheaders). If an essay, use paragraphs to indicate that you have moved on to a new topic or idea. Make sure you have proper pagination and that you do in fact submit all the content (I have seen submissions with whole sections missing or sudden jumps from e.g. page 13 to 27).

Avoid slang and jargon: for example, ‘out of my comfort zone’, ‘at this moment in time’ or any use of the word ‘massive’ (apart from a descriptor of a physical body like the Earth) have no place in academic English and make it difficult for non-native English readers. Check you have correctly CITED your references in the text itself, and included full references at the end in the References section. Finally, make sure it has a TITLE and your FULL NAME (and student number if required). Your marker needs to know who to award the marks to!

Remember: no waffle, no padding, no unsupported claims!

### Problem sheets

Make a real effort to do all the assignments and worksheets for every module you are taking in your study time and do them on time. It is pointless to wait for the answer handouts and then think “I could have done that!” because in real life or the exam you will not be able to. Note that exam questions are often similar to, and sometimes easier than, those on the problem sheets. If you fall behind with problem sheets, lectures will become (even more) mysterious and far less useful to you; think about planning better by consulting the Time Management section of this guide and get some help from your study advisors if the problem persists or gets worse. Don’t ignore it!

When you are doing a worksheet, FOCUS on it exclusively! No phone or television should be on to distract you. Don’t attempt to do an entire sheet of problems at one sitting! Instead set yourself a clear goal and time limit for doing problems e.g. problems 1-3 in 20 minutes. When 20 minutes is up, either finish them off if you are ‘on a roll’, or stop and write down what it is that you don’t follow. Take these questions to a seminar or tutorial … or even friends; only then you should go back to the problems that you haven’t tackled (again as a 20-minute blitz).

For many topics, a good solution will start with a fairly accurate diagram. Define the problem clearly. For complicated problems you may need several goes around a MODELLING CYCLE as follows: a) simplify the situation to produce a model, b) solve, c) interpret the results to see if they are sensible and adequate for your purposes: if so, stop, else d) refine the model, e) return to loop at b).

Does your solution contain some English sentences explaining clearly what you are doing and the strategy for achieving it? Refer to any theorems/results you use. Check your solution(s). I know it’s stating the obvious, but does your solution work? Substitute your values/solutions back into equations, differentiate indefinite integrals, check inverse matrices by evaluating AA-1, multiply out partial fractions, check boundary and initial conditions are satisfied for O.D.E.s, etc.

Is your answer dimensionally correct when expressed in terms of M(ass), L(ength) & T(ime), left in an exact form if possible e.g. ln(2) or if not exact, is it quoted to an appropriate level of accuracy (with an error estimate if possible) and in standard form using S.I. units (e.g. 3.2 x 105 ms-1)? Is it reasonable, both in terms of size and sign? For example, were you expecting the energy to be negative? If your answer is a formula, can you understand it in the limit of small or large x? Does the solution reflect the symmetry of the problem, if any?

For multi-choice questions, read and consider ALL the choices before deciding. If you can’t decide, make an informed guess (except in diagnostic tests of course!).

Before you get started with a problem, can you:

• specify precisely what the question is asking? Often this involves drawing a diagram, flow chart etc.
• define the dimensions of all the terms in formulae and equations? If not, stop and get help.
• simplify the problem further before attempting a solution? For physical non-dimensionalisation, so that only the important parameters are left, is often very useful. This depends on the nature of the problem and can often be hard, but is vital to your understanding of the true nature of the problem and well worth getting help with. Non-dimensionalisation can also dramatically help when seeking to write numerically stable programs and avoiding overflow problems.
• see any symmetry in the problem e.g. what happens if x -> -x or x and y are exchanged?

If you do get stuck, can you

• identify what mathematics is involved from your notes and books?
• make any estimates and see how sensitive the results are to your input assumptions?
• do sample calculations with numbers to see any sort of pattern?
• solve a simpler problem, perhaps by fixing one of the variables, or looking at what happens if z is real rather than complex?
• look at any sort of limiting case e.g. as x gets small or large in comparison to another parameter or variable?
• use a symbolic manipulator (e.g. Wolfram alpha or Geogebra) to solve the problem and then consider why the solution is as it is? (N.B. this is not a substiitute for doing the problem yourself!)
• plot a graph of the functions involved to see what’s going on? Pay attention to any special features like vertical asymptotes, turning points and large x behaviour.
• do the problem backwards from the answer to see what you can learn?
• get any help from your friends? Can you explain to them exactly what the problem is, and why you are getting stuck? This often results in a solution. If not and there is a group of you, you could start by “brainstorming” which involves writing down uncritically any idea anyone has for 5 minutes, and then examining these ideas for relevance and order of importance for the problem.
• If you still cannot do the problem, take a break, sleep on it or ask your lecturer. Remember that you will still have learned something in the attempts you have made and this will make you more receptive to the correct method or solution when you see it.

A useful and readable book is “How to solve it” G Polya, Penguin.

### Computing

Many assignments involve computing and/or word-processing. It will be a real advantage to you if you get your own PC and some final-year projects are not feasible without your own machine.

When programming make sure you really understand the structure of the problem and the flow of data before you attempt any actual coding. If you cannot draw an accurate flow diagram for the problem, stop and seek help. Do not attempt to start coding in the hope that something will become clear later – it will not – and any code you do write will be impossible to debug.

• Divide the problem into modules, write subroutines or external function subroutines for them and check each individually and thoroughly before linking into the main program. Always check each module with a problem you can solve analytically if this is at all possible e.g. for a root finding routine, will it give 1,-2 and 5 for the polynomial (x-1)(x+2)(x-5) or does the numerical integral of sin(x) work out before you attempt evaluation of functions you cannot integrate analytically? Always alter any numerical variables, such as number of iterations or integration range subdivision, to make sure your program works and your answers have converged.
• Is your code flexible? For example, can you easily alter the function to be integrated (leaving most of the program unaltered) and does it give the correct answer if the limits of integration are negative, or the top limit is less than the bottom limit?
• The main program should only control the data flow by calling these subroutines and should not contain any calculations. Check as above for subroutines.
• Write a subroutine for input; if input is to be from the screen, make sure the user knows what to input by writing statements to the screen such as “Input the number of iterations you require (an integer between 1 and 100)”. For complicated input, the chances of typing it all in correctly are slim, so read it from a previously-edited input file.
• Write a subroutine for output, with all the input data repeated first and clearly labelled, and then the results in a table with headers, or labels. If only a small amount of data is to be output, you can display this on the screen; large amounts of data should be written to a file (for later editing, printing or pasting into a report), but in interactive running, it is helpful to output some data to the screen so you know what is happening e.g. messages like “Starting iteration number …”.
• Use plenty of comments so that you and others can understand your program. Do this as you go along, not as an afterthought at the end.
• If your program gives unexpected results, write out most or all of the values of the variables to a file for inspection. This could be big, so don’t print it out and do only a few test cases/iterations; delete it afterwards.

Backup everything you do regularly on your University Network or to a cloud based service. If you use memory sticks be aware that they are easily lost!. Remember that PCs, especially notebooks PCs sometimes fail and regularly get stolen. Neither will be accepted as a valid excuse for non-submission.

Remember to use the spell-checker when word processing!. You should aim to write your essay directly onto the screen from your notes rather than writing it out in full beforehand. Get up early! All PCs are free at 8 am and most free at 9 am. Inability to find a PC is not an excuse for non-submission of work.

You might want/need to produce a Web page of your document, or put it on the Web for downloading. Here is a guide to creating a simple HTML page.

The temptation to download large quantities of material from Web sites etc. must be resisted. You will be discovered and get zero marks for it – or worse! Reference carefully any material, such as diagrams, you do use, including full Web site URL and date accessed if used.

### Finally

Hand in your work on time, and to the correct place. Don’t wait until it is perfect … it never will be!

Do not cheat in any way; it is very easy for your to spot work which is not your own and, in the case of copying from another student, both original and copied work may get zero marks, so don’t let others copy your work. If you do copy, any feedback you get will be of little benefit to you to improve your own understanding and writing style. Copying large sections out of a book, CD-ROM or Web site is equally pointless (whether you have referenced and/or edited the material or not) and will get you very low or zero marks. You need to read, digest and analyse the material, and then write it in your own words, preferably synthesised from several sources. There is no substitute for hard thinking about what you write, and copying everything down in the hope that some of it will be relevant will result in poor marks.

Quoting material from books or other sources without referencing them is plagiarism! This will normally result in you being awarded zero marks and disciplinary action being taken against you. If you are in doubt about what is acceptable and what is not, discuss it with the module lecturer, your tutor, your supervisor or the Undergraduate Course Director. You can learn more about understanding and, especially, avoiding ” unintended” plagiarism here

Look at these examples to see how (and how not!) to apply these rules in technical report writing, especially mathematics.

You should also look at E. Hunsicker, Plagiarism in Maths, Loughborough University course notes for MAC200 Mathematics Report, 2011 for more mathematically- advanced examples.

About the Study Skills for Mathematics pages

These pages were originally created by Martin Greenhow.

Study Skills in Mathematics – Introduction

Study Skills in Mathematics – Time Management

Study Skills in Mathematics – Lectures

Study Skills in Mathematics – Reading

Study Skills in Mathematics – Set Work

Study Skills in Mathematics – Experiments

Study Skills in Mathematics -Presentations

Study Skills in Mathematics – Revision

Study Skills in Mathematics – Exams