How To Write Solutions
MATH 239: Fall 2021
The learning objectives of this course include not only understanding the course material, but also
learning to write math clearly, precisely, and correctly. To that end, you should think of your handed-in
solutions not merely as evidence that you know the relevant results from class, but as formal documents
being submitted for review: the mathematical content is important, but so too is the presentation. It is not
enough to leave your reader with the general impression that you have understood the solution; you must
leave the reader with no doubts. When grading your assessments, we will only take into account what you
have written down. We cannot assign grades to what it seems like you might have meant. By the same token,
the grades you receive have nothing to do with your inherent worth nor indeed necessarily your potential
as a mathematician. They are only indicative of the solutions you handed in. In order to ensure that your
solutions accurately reflect your understanding of the material, we advise you to read this document closely.
Please do not mistake our preoccupation with good writing as pedantry. There is a long history in
mathematics of false-proofs standing for years before their untruth is eventually uncovered (see for e.g. early
attempts at a proof of what is now the Four Colour Theorem). Clear, unambiguous writing —where every
step is fully justified —is the best defense against errors in reasoning.
Topics covered:
1. Correctness of Solutions
2. Citing Results
3. Relevance of Statements
4. Notation & Nomenclature
5. Writing Quality
6. Solution Length
7. Crowdmark Submission
1 Correctness of Solutions
When writing a proof, remember that every statement should follow from earlier statements or from results
covered in the lectures and course notes. If you use a result from the lectures or course notes, please be
sure to cite it appropriately (see 2 Citing Results). Bear in mind that we will take off marks for every
incorrect statement, even if the statement turns out not to be relevant to the answer. This is in part to
ensure that you know when you have said something incorrect, and so to ensure that you do not use the
incorrect statement in your solutions to later assessments.
If you are unable to fully explain a step of an argument, mention this (by saying for e.g., “This step
requires more explanation.”) Knowing when a statement is trivial or whether it requires explanation is an
important skill for a successful mathematician; similarly, it is important that you learn to recognize when
your work is incomplete.
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2 Citing Results
You may cite any result in the course notes and lectures, as well as any result from a previous assessment
for which you have received solutions from the instructors. There are several acceptable ways in which to
cite a result. We will use the following adage to demonstrate:
Law 1.1 (Parkinson’s Law). Work expands so as to fill the time available for its completion.
Suppose that you have an assignment due in ten days, and that you wish to invoke this law. You might
do so:
• By name. “By Parkinson’s Law, it will take me ten days to complete the assignment.”
• By number. “By Law 1.1, it will take me ten days to complete the assignment.”
• By stating the result explicitly. “A law in the course notes says that work expands so as to fill
the time available. It follows that it will take me ten days to finish the assignment.”
• By stating the result implicitly. “Since the assignment is due in ten days, by a law from class it
will take me ten days to complete it.”
Note that it would not be sufficient to say “By a law from class, it will take me ten days to complete
the assignment”. A skeptical reader might not believe you; you should aim to never leave your reader in
a position where they might doubt what you are saying. If it is not immediately clear from the current
or previous few sentences why the result applies or how exactly you are using it, it is always preferable to
reiterate for the reader.
3 Relevance of Statements
You should aim to only include in your final solution what is required to answer the question at hand. We will
take points off for including statements that are not relevant to the answer. This holds even if the statements
are correct. When writing your final draft, it may be helpful to stop and ask yourself the following questions
before each sentence.
• What is the purpose of this sentence?
• Does my answer make sense without it?
Often students start out solving a problem by writing down a list of every potentially relevant piece of
information they can think of. This can be an excellent way to approach a problem, but this list does not
belong in a final draft.
4 Notation & Nomenclature
One of the skills we hope to help our students develop during this course is the ability to clearly commu-
nicate their ideas. To that end, we will take points off for any ambiguous or incorrect use of notation or
nomenclature. This holds even if we can guess what you had in mind.
E.g.: You should not say “one of the pieces of the graph” if you mean “one of the components of the graph”.
Note that component has a clear meaning here (and is defined in the course notes), whereas piece does not.
Remember: a good proof is unambiguous and clear.
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5 Writing Quality
Your answers should be understandable to anyone that has the necessary background in the subject. Note
that just because someone who has already worked out a solution to the problem at hand is able to figure
out what you had in mind does not mean that your answer is clear. We would like to emphasize that we
are not judging your English language skills but assessing the clarity of your ideas. That being said: if your
work contains many language errors, it be difficult for us to understand what you mean.
There are many ways in which to improve the quality of your writing. We give some ideas below:
• Always write a second draft. Barring knowing the material, this is the single best piece of advice
to help ensure your writing reflects your understanding. For assignments and graded tutorial problems,
you should not hand in the sheet of paper on which you do your rough work. When writing out your
final copy, it might help to ask yourself the following questions every so often:
– What is the purpose of this sentence? Does it need to be here? (see 3 Relevance of
Statements)
– Is everything I have written perfectly legible? If in doubt, erase your work or use liquid
paper and try again. Try to avoid crossing things out, as this greatly impacts how legible your
work is as a whole. As a general guideline: if your work has more than two instances of crossed-out
text, you should draft a new copy.
• Use spell- and grammar-checkers if you are typesetting your solutions.
• Always proofread your work. Any proofreading is better than no proofreading. If possible, try
to leave some time between finishing your assignment and proofreading it. You will be better able to
spot your own mistakes if your writing is not fresh in your mind.
6 Solution Length
We may give guidelines on the length of answers. The length limit will be given in terms of the number of
pages. If we say the answer has to fit on n pages, we mean both
1. it has to fit on n pages, and
2. the answer should use at most 300 × n words (where here one equation counts as a single word).
As mentioned earlier, one of the skills we hope to help our students develop during this course is the ability to
clearly communicate their ideas. A concise, clear solution is nearly always preferable to a long, equally clear
solution. There are also practical reasons for this: since this is a large class, marking is a very time-consuming
task. Since we have a limited amount of people marking the assessments, if we do not set constraints on the
length of answers returning graded work in a timely fashion quickly becomes an impossible task.
7 Crowdmark Submission
Please follow the guidelines below when submitting solutions on Crowdmark.
1. Each question has its own upload area, so complete each question on separate pages. If you have a
question with multiple parts, it’s fine to have the answers to the different parts be on the same page.
2. Make sure that each answer is clearly labelled with the question number.
3. Before submitting your solutions, please make sure that your answers are clearly legible. If we cannot
read your work, we cannot give you credit.
4. Before submitting your solutions, make sure that your answers are oriented correctly.
5. Log out of Crowdmark after submission, and log in again to check whether the upload has succeeded.
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