Dr. Timo de Wolff
Institute of Mathematics
Texas A&M University www.math.tamu.edu/~dewolff/
Doing exercises (that is: 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, once you understood (in your mind), how to solve it. It is done after 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.
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First, work on your exercise on scratch paper. Once you obtained a solution, present it on a new, extra sheet of paper.
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Write readably. Be aware: If the grader cannot read your solution, then he cannot regard the exercise as solved. Thus, you will not get credit for the problem. Using
LaTeX is optimal and also a good training for you, but not a must, of course. If you write by hand, then take your time (in particular, if you have a bad handwriting, like the author of this text). Put formulas in extra lines, if it simplifies the reading.
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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, then use a ruler to cancel the mistake and restate the part. In my personal opinion one should avoid
Wite-Out or similar products; it never looks nice. Keep your solution free of blurs, dirt, dog-ears, etc.. Staple together your solution sheets.
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Leave enough space for comments.
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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.
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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.
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Similarly, a solution is not a novel. We are doing math. Hence, for your solutions
sagen l¨aßt, l¨aßt 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 are convinced 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 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 you are unsure about 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.
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Be honest about what you do not know and cannot solve. Do not write a wild collection of everything that 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 afterwards that it is just hot air. If you do not have a solution, then just leave out the exercise instead of handing in wild guesses. Do not feel bad about this.
Normally, a math homework in a 300-level course or higher has a 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, then state as clearly and precisely as possible, what you have achieved and at which point you do not know, how to go on. Do not 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.”