A friend recently suggested to me that we should perhaps tell students earlier what she was told by her professor in Germany only in her third year or so.

**The most important thing you learn at University is how to learn.**

This applies to all subjects, I think. In a particular subject, you might be able to specify this a little more. You (should) learn how to think about problems, how to find enough material to understand a new problem, how to deal with lots of material, how to work to deadlines, … You learn how to persist on hard topics/problems, you learn how to think creatively, and in maths you certainly (should) learn how to think logically, how to make a proper logical argument, the importance of making the right assumptions and of knowing your assumptions, and so on. In Science I think you learn how to deal with data, what you can and can’t learn from data, etc.

Of course you also learn some facts. You can’t teach someone mathematical argument without given them some specific mathematical arguments in a specific subject. And yes, the material you learn in the first year or first few years will be relevant to (some of) the courses later on. And yes, if you want to stay in maths, you need some of the actual knowledge you have learnt in certain areas. But much much more, and no matter whether you stay in maths or go and do something else, you will need to know how to think for yourself, how to tackle problems, to be creative in your thinking, how to question things, etc.

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I would tend to agree with you: university is about learning how to learn. And I think some of the skills and techniques that we learn about constructing arguments and constructing proofs come from lectures – and not from the formal lecture notes, but from the dynamic lectures. Because the notes tend to go “Lemma X: If X, then Y; Proof. Let X be a set with the properties… algebraic statement, manipulation, QED.” But in a lecture, you will often take the time to say “why did we think of this initial algebraic representation. Why did we apply the group axiom about closure rather than about identity?” So you are seeing and (hopefully) mimicking the intellectual process, in the same way that part of learning to play tennis to the very highest standard you will watch how Roger Federer constructs his serve, in the hope of improving your own. So you are watching the construction of a mathematical argument for the same reasons – to improve how you construct your own arguments. You can get this from certain blog posts – Tim Gowers I find particularly good at indicating how he constructs arguments, but not from most articles, or even most textbooks.

However, it’s not the only part of a university education, certainly not the only part of a Cambridge education. I don’t know how true it is now, but I found that doing my first undergraduate degree left me very much in control of how I did my learning: I had to write essays (which weren’t formally graded) and sit exams (which were). But I was completely at liberty to work out how I could convert lectures and a reading list into a properly connected information set that I could use to best effect in an exam [no mathematical terms were harmed in the making of this sentence…]. I can understand that this makes it hard for some people, who may need more support, but I liked being left alone with my initiative, and also, with no forms of work which distracted me. I don’t know if this was the result of a privileged education in which I had already largely worked out how I could learn well. I can certainly say that I found doing my second undergraduate quite annoying in terms of forced participation in learning activities I already didn’t get on with. Also, I particularly disliked the fact that every piece of work you submitted was assessed – there was an immense pressure to get it right first time; in contrast, if you handed in a disaster for a weekly supervision, it didn’t really matter – you could learn what you’d done wrong and work out how (or whether) you could fix it. So I’d say that one of the things Cambridge does best is allowing you some space for submitted work that isn’t assessed, where you can try something new and fail, and still have a chance to fix it before your assessed work.* And that’s one of the important parts of learning how to learn – learning how you can go wrong and how to fix that.

Sorry for the long comment – I got quite interested in pedagogy while I was supporting CamTools, and helping academics get the most out of it in supporting their teaching.

*However, I did get a 2.i from Cambridge, and a 1st from the Open University, so it may be that while I enjoyed my learning more at Cambridge, I certainly optimised my strategy for success better at the OU.

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I meant to add a comment to this as well but didn’t get round to it. In fact, getting a 1st in a second degree may entirely be due to the fact that by then you had learnt how you learn, and indeed were more mature and had generally developed, so I don’t think we can infer tooooo much from that.

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(Coming to this rather late…)

vla22, another explanation for your different results would be that any given grade is easier to achieve at the Open University than at Cambridge.

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Reblogged this on Think for yourself and commented:

To add to my previous blog-post on “what is the most important thing we learn at University”: the fact that mathematics graduates have learnt to think (as well as learnt to think logically, and critically) and have experience in learning complex difficult new material is one of the main attractions to employers. See this Cambridge Maths Admissions leaflet for some “destinations” information and data on average salaries.

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