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Useful explanations in solutions

I always provide students with solutions to the homework exercises. One simple but effective tactic to improve student understanding is to include helpful asides in square brackets. These are bits of information that I would not expect in a handed in solution but give students insight into what I am thinking while answering the question. For example, “[We can use Method A here but Method B is actually quicker. If you like, use Method A and compare.]” or “[We can’t use the Ratio Test directly as some elements of the series are zero.]”.

If could also be information that expands on the answer to show something extra: “[If we take the imaginary parts rather than the real parts, then we get another equation we can use.]”.

This is a very simple technique but appreciated by students as it allows them to see what is actually required for the final answer.

Some subjects require more explanation than others. For example, when teaching analysis I dispensed with the square brackets and had two sections for each question in my model solutions. One was labelled Thoughts and the other Written Solution. In the Thoughts section I discussed some ideas for answering the problem including explanations of why some ideas won’t work but usually the section contained working that would be discarded in writing up. For example, in analysis, many proofs involve finding an N (or similar!) given an \epsilon >0. The usual procedure is to find the N (dependent on \epsilon ) by some calculation but when writing up we give the required N in the first line and show that this satisfies our required condition. This is problematic for students as without the initial working they often can’t see where the N comes from. This method shows where it comes from and shows that at university level we don’t have to obey the dictum of the school teacher: Show your working! Instead we show that our answer works.

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