1.2 How is mathematics at university different?

In school, mathematics is typically concerned with carrying out calculations and solving numerical or algebraic problems. You probably learned to apply processes to solve standard types of exercises. For example, consider this question from the 2024 exam for SQA Advanced Higher Mathematics (the standard course studied in the final year of secondary school in Scotland).

Exercise 1.1 ([7]).

x

  1. (a)
    ​

    Express −2x⁢(x+1)\dfrac{-2}{x(x+1)} in partial fractions.

  2. (b)
    ​

    Use integration by parts to find ∫x⁢e3⁢x⁢𝑑x\displaystyle\int xe^{3x}\;dx.

  3. (c)
    ​

    Using your answers to (a) and (b), solve

    d⁢yd⁢x−2⁢yx⁢(x+1)=x3⁢e3⁢x(x+1)2.\frac{dy}{dx}-\frac{2y}{x(x+1)}=\frac{x^{3}e^{3x}}{(x+1)^{2}}.

You might have spent lots of time in school practising this type of question, particularly parts (a) and (b). Your teacher may have have given you a list of steps to memorise and your job was to carry these out reliably. Notice that the questions essentially tell you what to do and sometimes even how to do it.

At university, your lecturers and tutors are much less likely to give you step-by-step processes to remember, and you are much less likely to be guided through how to solve a problem. To illustrate these points, a ‘university version’ of Exercise 1.1 might look as follows.

Exercise 1.2.

Solve

d⁢yd⁢x−2⁢yx⁢(x+1)=x3⁢e3⁢x(x+1)2.\frac{dy}{dx}-\frac{2y}{x(x+1)}=\frac{x^{3}e^{3x}}{(x+1)^{2}}.

Although answering this version might involve the same steps as answering Exercise 1.1, it is usually your job to figure out how to apply the techniques that you have learned. It is expected that you can carry out calculations and computations reliably.

There may be other ways you could go about solving this problem that do not follow the steps in Exercise 1.1. At university, finding your own approach to solving problems is usually allowed (and sometimes even encouraged), but you should take care to understand why your solution works and explain it clearly to convince your lecturers and tutors that you understand.

In summary, you could think of mathematics at school as being about ‘getting the right answer’. Mathematics at university is about much more: it is about understanding difficult ideas, solving problems and making logical arguments. Of course, your lecturers and tutors are here to help you. But they are not here to tell you precisely how to do things – that is the challenge and part of the fun of studying mathematics at university.