Once, teaching MA407, *Modern Methods in Differential Equations*, “as is my wont”, just as I was going to tell the world that locally integrable functions generate regular distributions, a student looked at me with infinite sadness and asked: Why are we doing this? Her sadness communicated itself to me, becoming my sadness in the process, but I controlled it “admirably” and said: Legitimately asked! Let us continue “with the matter at hand”; I promise we will come back to your question later.

Now is the “later”, three years or so later in fact. Truth is, I did go around my “workplace” (see how *unofficial* we have become in the meantime!) asking people what they thought, but somehow the opportunity to consider possible answers in class never arose. So let us see, and enough of quotation marks.

Answers to the question depend on its interpretation. It could be taken to be similar in spirit to:

Q1: Why do I have to jump into a swimming pool full of cockroaches?

To which a reasonable answer might be:

A1: You volunteered for it and it is the only way to get your $60 000 prize.

In other words… but I will leave it to you to put in other words.

The question could also be construed to be of the following kind:

Q2: Why do I have to learn how many AMs there are in the Northern Ireland Assembly?

This is answered by

A2: Because passing the Life in the UK Test is a necessary condition for being able to submit your request for naturalisation as a British citizen.

I.e. the whole endeavour is a useless bureaucratic formality, but what can we do? (The answer is 108 if you were wondering.)

Finally, and subtly different from Q2/A2, is this version:

Q3: Why do we have to learn the Law of Arms in England, Wales, Northern Ireland, and Scotland?

A3: Because you want to show that you have progressed beyond the elementary level for which you only have to know blazoning, emblazoning, hatching, tricking, (simple) marshalling, cadency, the achievement and its component parts (I kid you not, and I have no idea what any of that means. Well, tricking maybe).

Observe that no-one in her right mind will ask

Q4: Why do I have to learn histopathology, or passé composé, or double tongueing?

The answer here is obvious:

A4: Because you want to be a competent doctor, or speak French like a native, or play fast staccato passages on the flute.

You will not ask Q4, because you clearly already want to be a competent doctor, speak French like a native or play fast staccato passages on the flute, and you know what it takes to achieve those goals.

Similarly, if you wanted to be a good mathematician, you would not ask the question that was originally asked of me all these years ago, as by fourth year you would have a clear idea what that implies.

But that is unfair: many students come to do a mathematics-based degree aiming not to become professional mathematicians, but instead, for example, to work as school teachers of mathematics, for whom regular and singular distributions are equally useless. Why should they be made to sit through a course such as *Modern Methods in Differential Equations*?

What do you think?

Do you think that at play here is simply a more or less unpleasant and senseless jumping through arbitrary hoops, so that the material can/should be forgotten with great relief the day after the exam? If not, why not?

*To be continued…*

In defence of the student: motivation is key here…not just the motivation the student has for enrolling in a Mathematics degree…but also the motivation for individual classes which make up the degree which has to be impressed on the student by the teaching staff. Do we just say to students at the start of their degree, “all that you will learn in the next 4 years is essential…you may not understand why but trust us”? Do we provide a general motivation for each class, why each topic is important.

Your analogies are interesting

1: Histopathology: as a student starts a medical degree they learn early on that there are some cells/viruses/bacteria that cause humans to become unwell. Then when they come across histopathology they see that here is a way to actually see these nasty wee creatures, the motivation for them to study this area is clear.

2: passé composé: most/all students who start a French degree will probably have been to France and will have been immersed in the language…and will have realised that speaking in only the present tense is a little limiting. If the student ever wants to converse in French it is clear that the past tense is necessary.

3: double-tonguing: students of woodwind instruments will generally have heard and seen some of the best musicians, on many occasions. They have been able to see them perform to large audiences (usually receiving the audiences praise afterwards). Not all flautists reach these heights but it’s clear that if they want to achieve this level then double-tonguing is essential.

So..back to Maths. Do we provide students with the understanding of what is essential? Maybe the road to enlightenment and to the highest levels of mathematics is so long that some idea of the final goal is hard to convey. If there was a country where everyone spoke mathematics then maybe that would help. Or if large audiences flocked to see mathematicians prove uniqueness of a particularly difficult differential equation…

It’s not all our fault, or the fault of a very complicated subject area, and maybe a large part of the problem is that students start a mathematics degree without a clear understanding about what the degree will provide and enable you to do…but it’s it’s not the only factor.

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