Well I survived marking 221 first-year exam scripts for G11FPM Foundations of Pure Mathematics (though it did take me a long time to get through them).

I then wrote a set of model answers which probably included about 4 sides of comments, and a separate feedback document on common errors and pointers for improvement (another 4 sides). Far more people are looking at the solutions than reading the separate feedback document, so it is probably good that I included some of the comments there.

I’m not going to reproduce the full set of comments here. Instead I’ll try for a very concise summary. [I failed on the “very concise” bit.]

Students were generally good at checking whether relations were reflexive, symmetric or transitive, could do long division to find recurring decimal expansions of rational numbers, could calculate very fast and accurately when multiplying permutations, and could convert permutations between two-line form and disjoint cycle form.

Students also made very good use of modular arithmetic, correctly establishing what the possible cubes are modulo 7, and then (as indicated as one possible method in the question) using this to prove that there are no integer solutions to the equation .

That was the good news!

Students did **not** do well on stating standard definitions of concepts or giving standard statements of lemmas. But some bookwork **proofs** were reproduced correctly. This suggests that, after just one semester at university, the importance of knowing (and understanding) the main definitions and statements accurately may not yet have sunk in.

The single biggest problem with definitions came with the definition of **injective** for functions. Most students wrote that this meant that every point of the domain is mapped to a unique point of the co-domain. Many of them don’t really believe that, as was shown by the functions they suggested when asked for examples with various combinations of properties (but some of them really did appear to believe it).

This was a bit disappointing, because I know this common error well from teaching 2nd-years (and 3rd-years), and I had been hoping to nip it in the bud. I had hoped that discussing graphs of functions properly might help, especially the fact that (for us) functions are never “one-to-many”. Certainly there were indications in classes (with voting using the voting packs) that the ideas were making sense. But in the end, I think the unfortunate (but standard) term “one-to-one” always ends up being thought of as the opposite of “one-to-many” instead of “many-to-one”. I’ll have to try another strategy next year. Maybe I won’t even mention the term “one-to-one”!

The big statement that very few students could remember was that of **Bézout’s Lemma**. This hadn’t come up in the previous three years, so perhaps students assumed it wouldn’t come up this year. However we did dedicate an entire workshop to proving it this year.

I won’t mention all the difficulties students had with constructing their own proofs or thinking up examples with specified combinations of properties. I will say that students need to practise these skills more. (In my feedback I have suggested that my big question sheet on Definitions, Proofs and Examples might be one place to start.) I will mention a few common problems: many students suggested that integers that were divisible by must be even, or that if was divisible by 2 then must be even (or similar). Many students claimed that if was divisible by 16 then must be divisible by 16. More generally students often claimed that if , then or (in settings where was not prime, or at least was not known to be prime).

There is more that I could say, but I think that is enough for now!

In the above, where the students wrote “every point of the domain is mapped to a unique point of the co-domain”, possibly they were trying (incorrectly) to use “unique” to mean something like “its own unique”, so that each point x in the domain gets assigned its own point in the co-domain, not shared by any other point in the domain. (This last part is actually the right idea, but it is incorrect to try to use unique in this way.)

Students did write many variants on this, some more clearly completely wrong than others. Some explicitly said that “for all x in the domain there exists a unique y in the codomain with y=f(x)”. Others said “for each x in the domain there is a unique f(x) in the codomain”.

When it came to giving examples of non-injective functions (with additional properties), some students did produce many-to-one functions, while others attempted to produce “one-to-many functions” such as .

I suppose the moral here is that you can’t always tell from what the students write when asked for the definition, what their “concept image” actually is; but this does usually become more apparent if you ask them to give examples with justification.

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