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Can anyone learn maths to PHD level?
just wondering if everyone has the ability to learn maths
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Hmm, you're really asking two questions here and I'd like to answer them in reverse order.
Yes, everyone has the ability to learn maths, but having taught the subject, it depends on the age of the learner and the type of maths involved. Age makes a difference as we often come across learners who were simply not able to take in the complexities of maths whilst at school either because they weren't interested or had poor teachers. On the other hand, older learners can often study the basics of arithmetic quite easily whereas they find great difficulty with such things as trigonometry and calculus.
As for learning maths to PhD level, you need to appreciate that at this level, you only study one particular and often obscure area of maths and not the subject as a whole. The area chosen is usually something that you have a particular interest or ability in and sometimes something you'll have touched upon whilst taking your first maths degree. Consequently, the person that "examines" your work and marks it, often has specialist knowledge in this area themselves. Mine for example was in J-spectral factorization (don't ask!) and had to reviewed by an expert in another university before they awarded me my PhD.
Yes, everyone has the ability to learn maths, but having taught the subject, it depends on the age of the learner and the type of maths involved. Age makes a difference as we often come across learners who were simply not able to take in the complexities of maths whilst at school either because they weren't interested or had poor teachers. On the other hand, older learners can often study the basics of arithmetic quite easily whereas they find great difficulty with such things as trigonometry and calculus.
As for learning maths to PhD level, you need to appreciate that at this level, you only study one particular and often obscure area of maths and not the subject as a whole. The area chosen is usually something that you have a particular interest or ability in and sometimes something you'll have touched upon whilst taking your first maths degree. Consequently, the person that "examines" your work and marks it, often has specialist knowledge in this area themselves. Mine for example was in J-spectral factorization (don't ask!) and had to reviewed by an expert in another university before they awarded me my PhD.
i'm already off to a bad start because i didn't realise until you told me that i had, indeed asked 2 questions!
i am someone who disliked Maths at school and scraped through his GCSE Maths exam with a C but as i go through life i'm thinking surely Maths can be quite interesting because it's incorporated in our lives completely.
i am someone who disliked Maths at school and scraped through his GCSE Maths exam with a C but as i go through life i'm thinking surely Maths can be quite interesting because it's incorporated in our lives completely.
with my humble A-Level standard (before they where dumbed down), I think yes anyone of average intelligence can learn maths, that is to say that processes and techniques can be explained and learnt. However to reach the top level it's a bit like sport, you can learn all the techniques and be perfect but you need tat l;ittle something extra to be any good, that little bit of natural flair that can see where others are blind.
Maths is a curious thing... It is generally pretty logical and straightforward, but there are a whole host of different ways of approaching it.
I feel lucky that I generally understood the way that my teachers explained maths (from primary school to college) but in every lesson I had friends who just needed it explaining in a slightly different way before they twigged.
When I leant maths at school / college, I was always given an example, you have this data, from this, in order to get this bit of useful data you must use this process because... this taught me the basic principles very well.
When at university studying engineering, we were taught that if you start with this 'data', apply this 'process' to it, you'll get this 'data'....... I did not like this style because there was no apparent sense to it. It was just changing one set of data into another with no aparent reasoning. Luckily, my good grounding in the basic principles meant that I could work through it all and do the work. Also, a lot of self teaching, finding the real world applications helped enourmously!
7 years ago, my girlfriend squirmed at the thought of any maths; now she has me - her expert teacher, I can explain things in a way that is clear to her and she says (usually after three different explanations) 'oh, well thats obvious, why didn't you say that the first time?'.
Anyway, sorry to waffle.... I think that anyone can be taught maths to a very high level if they have the right teacher, but to teach yourself maths I think you have to have a bit more intelligence and determination!
I feel lucky that I generally understood the way that my teachers explained maths (from primary school to college) but in every lesson I had friends who just needed it explaining in a slightly different way before they twigged.
When I leant maths at school / college, I was always given an example, you have this data, from this, in order to get this bit of useful data you must use this process because... this taught me the basic principles very well.
When at university studying engineering, we were taught that if you start with this 'data', apply this 'process' to it, you'll get this 'data'....... I did not like this style because there was no apparent sense to it. It was just changing one set of data into another with no aparent reasoning. Luckily, my good grounding in the basic principles meant that I could work through it all and do the work. Also, a lot of self teaching, finding the real world applications helped enourmously!
7 years ago, my girlfriend squirmed at the thought of any maths; now she has me - her expert teacher, I can explain things in a way that is clear to her and she says (usually after three different explanations) 'oh, well thats obvious, why didn't you say that the first time?'.
Anyway, sorry to waffle.... I think that anyone can be taught maths to a very high level if they have the right teacher, but to teach yourself maths I think you have to have a bit more intelligence and determination!
When you reach university level the nature of learning maths changes from using maths to do sums or work stuff out to a much more abstract study of how numbers and maths actually work. Once a new technique or method for manipulating maths is found it can be hundreds of years before any practical use is found. But learning maths as a set of numerate tools to work stuff out is unbelievably useful, you can quickly work out if loans are good or a horrible rip off and ignore all the razzle dazzle, just work the numbers yourself. Also the logical methods help with any science and computing.
Do it!
Do it!
Yes do it! I took me about 39 years to realise that it doesn't matter whether you're "good" at something, it matters whether you enjoy it. I was "good" at maths AND I enjoyed it, so I ended up doing a degree in it. Lucky me. It's full of beautiful things that are like magic. Lots of graduate mathematicians stop after their degree and never open another maths book. I still do. But I was "no good" at music and so I stopped when I was about 6. Lots of people unfortunately are told when they are about 6 or 16 that they are "no good" at maths. More recently I've taken up music. Actually I'm not very good - lots of my bandmates are miles better - but music is also full of beautiful magical things, and it doesn't matter how quickly you learn them. With maths there are lots of things that nobody understands yet. But everyone with (say) a degree in maths knows and understand hundreds of things that the very finest minds could not grasp a thousand years ago. In a thousand years will our top mathematicians be fools? On the other hand, if you can see that something has to be true in maths, nobody can ever change that. Today's magical mathematical fact: 4-4/3+4/5-4/7+4/9-4/11+... for ever = pi! That's in the magical category, I think. (And was known a long time ago - long before the simple proof using calculus could have been found). Warning: maths is hard. But it's hard for everyone - it's just a matter of discovering what you find hard enough to be interesting but not so hard as to be discouraging. There's plenty to choose from.
Sorry for such a long rant but I feel passionate about this.
Sorry for such a long rant but I feel passionate about this.
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