How to Present Homework Solutions

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Dr. Timo de Wolff
Institute of Mathematics
Texas A&M University
www.math.tamu.edu/~dewolff/
How to Present Homework Solutions
Doing exercises (normally handed to you as homework) is an essential part of
understanding mathematics. Although reading and learning the material is the first step,
it does not suffice. Understanding math is similar to learning to bike or to swim – you
cannot just read about it; at some point you have to do it!
The presentation of your solutions is an important part of any homework. An exercise is
not already done, if you understood (in your mind), how to solve it. It is done when you
have communicated your solution in a clear, clean and convincing way to the reader.
Here is a loose collection of suggestions, how one should solve exercises and how to
present their solution.1
• First, work on your exercise on scratch paper. Once you obtained a solution,
present it on a new, extra sheet of paper.
• Write readably. Be aware: If the grader cannot read your solution, then he cannot
regard the exercise as solved and you won’t get credit. Using LaTeX is optimal and
also a good training for you, but not a must, of course. If you write by hand, take
your time (in particular, if you have a bad handwriting as the author of this text).
Put formulas in extra lines, if it simplifies the reading.
• Work cleanly. Your solutions should look in a way such that you would also hand it
to your tax authority (or a similar institution) and sleep well afterwards. Handing
in disorganized homeworks is disrespectful to your grader. They are a pain to
grade. Hence, if you make a mistake when writing down your solution, use a ruler
to cancel the mistake and restate the part (in my opinion one should not use
Tipp-Ex or similar products – it never looks nice). Keep your solution free of blurs,
dirt, dog-ears, etc.. Staple together your solution sheets.
• Leave enough space for comments.
• A solution is not just a collection of formulas. An exercise is solved in complete
sentences adjusted with formulas (more of one or the other depending on the
particular exercise). In the first sentence of every exercise, you explain, what you
are about to show. The last sentence of an exercise concludes, why the exercise is
solved. An ideal solution can be read as if it was a prose text from the beginning to
1
Thanks to Chris O’Neill for his various helpful comments.
1
the end and is immediately clear to the grader after the first reading. It is not the
graders job to guess, how your formulas can be puzzled together in order to form a
solution out of them.
• Similarly, a solution is not a novel. We are doing math. Hence, for your solutions
(and any mathematical writing) keep Wittgenstein in mind: “Was sich überhaupt
sagen läßt, läßt sich klar sagen.”2 . This means in particular:
– Think carefully, what you want to say, before you start to formulate it.
– Avoid unnecessary variables / constants / formulas. Formulas are meant to
make a solution (or statement or proof) clearer, not to confuse the reader. If
you use variables and constants, then make sure that they are well defined.
– Use short, precise sentences. Avoid any unnecessary figurative language.
Cancel out everything that is not necessary for your argument (unless you
think that it helps the reader to understand, what you mean).
– Make sure that each step of your argumentation follows fluently from the
previous ones. The parts of a proof should fit together as the parts of a gearing
mechanism or the code lines in a software. Make sure, not to culminate three
steps in one. The reader has to be able to follow your line of argumentation.
– Never try to hide parts, about which you are unsure, in vagueness. Every good
grader will interpret vague speech as a lack of understanding.
– When you are done with writing down your exercise, re-read it and make sure
that every single sentence makes sense. When you are done with writing down
your entire homework, re-read everything again in order to make sure that you
understand every single one of your own solutions. If you figure out that
something is unclear, then reformulate your solution.
• Be honest about what you don’t know and cannot solve. Don’t write a wild
collection of everything you have learned and every rough thought you have and try
to sell it as “the” solution. Nothing is more annoying for a grader than to think 10
minutes about a one page solution of an exercise and to figure out then that it is
just hot air. If you do not have a solution, then just leave out the exercise instead
of handing in weird stuff. Don’t feel bad about this – normally, a math homework
has a level of difficulty such that the instructor by far does not expect everyone to
be able to solve everything.
If you have an idea or a partial solution, state as clearly and precisely as possible,
what you have achieved and at which point you don’t know, how to go on. Don’t
hesitate to hand it partial solutions or ideas, if they are stated as partial solutions
or ideas clearly.
2
“What can be stated at all, can be stated clearly.”
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