This is going to be true of maths and science right up until you are doing it 'properly'. I do explain this to all my students. But they have to start someone and they need the basic skills to get there. You cannot have just the creative and innovative and thinking skills. You need the basic tools as well. Both need to be taught, and realistically speaking it is a lot easier to 'teach' the procedures, basic skills and knowledge at an earlier age than it is to churn out ready made right thinking mathematicans at a young age. It is not a perfect system but having been involved at most levels (teaching young children, secondary school teacher, undergraduate teaching, done my PhD. etc) I can see the problems that everyone has at each stage. We are all going to complain that those sent to us do not have the right skills (I spend enough time moaning that my Year 7s and Year 12s are not coming up prepared) but in reality what would you change about that?

My selling point on Further Maths is just that. It gives you a taster, a little insight at some of the basic skills you will pick up and run with at a higher level. It is not the pinnacle of mathematics and nor should it be. That is what university is for! The next level of education. The original post here is talking about UK Secondary Education, not preparing that tiny handful of students that are motivated and intelligent enough to be able to do things that their peers will take a few years more to grasp.

Doing mathematics at a suitable level, let us say postgrad, will of course give you a skewed view. I have taught last year some Further Maths students, one of which went off to Cambridge. However, at the same age, in the same year group, the same cohort, we were teaching students GCSE Maths for the third time, and some of those still fail to gain any grade at all. Any given year group in secondary school has a massive range of ability, that is quite apart from the quality of teaching, social skills, home support or any other factors. This is what the curriculum has to deal with and it is all too easy for those not doing it to say that everyone should come out in tip top condition ready to roll for that subject for the rest of their life. This is before you consider they have some 9 other subjects or so in which we could discuss the same thing.

The context of this original post is in UK education. You do not learn about norms in any spaces and certainly not Banach fixed point theorem. That would be covered at university.

There is a wide range of material to be covered up to age 16 and up to age 18. Students have a very wide range of abilities and it is absurd to think that everyone at that age could possibly learn some high level maths. My Further Maths students would be among the brightest and they get a taste of some of the material that would be covered in the first year or so at university.

It is simply not possible to cover all the material required in maths, especially if you are looking at any on particular subject at degree level and demanding all the prerequisites be covered before they enroll on your course.

A Level Mathematics (and Further Mathematics... and for those really really dedicated students something like Additional Further Mathematics) gives them a good grounding in a range of mathematics that would be required at university for maths, science, engineering etc. Learning about differential equations, some graph theory, some mechanics, some statistics, vectors, matrices etc. gives a pretty good preparation for most courses.

You can consider any level of mathematics and talk about it being "beginner" mathematics in terms of what is 'left' to learn. Bear in mind that in the UK here those Further Maths students would be up to age 18, how does this compare with, say, the USA and its education system? I do not know in detail what mathematics they would have before they whizzed off to their next level of education.

So do you want students, who are just learning about calculus in terms of differential equations, integration, differentiation etc. to cover all of fields, rings, vector spaces etc? What do you take OFF the curriculum in order to make time and room for that?

In principle not as hard. The trouble is the very people you might recruit from ARE the same people you need for the other shortage subjects. There are plenty of maths and physics graduates that could teach computing... but they all tend to be siphoned off at university by industry, academia or just the lure of big money. Teaching is not an attractive option for them.

The new computing course at GCSE is definitely a step in the right direction. Done correctly it can really open up a good skills base and at least teach the kids to differentiate between learning what menu item to select in a particular program as opposed to some basic idea of coding. There are many schools starting that this year that could lead to an improvement in the situation over the next few years. The trouble will always be getting specialist teachers... in any subject. We still have a shortage of (good) maths and science teachers ... and to a lesser extant in other areas. So getting expertise in such a field is even more difficult for the non expert who is doing the hiring. IT/Computing etc is a minefield of terminology and confusion about qualifications for those that do not understand the area.

At what age would you like to teach rings and fields? Before or after they have mastered the art of simultaneous equations?

Ah... I must be doing something else with my day then whilst I thought I was teaching mathematics. Come and sit in some of my Further Maths lessons and tell me that is not mathematics

You dont want to know how many emails I have sent fred@fred.com 's way

Sometimes it is very much needed for clarity

http://www.barking-dagenham-sc...

My main PC is supposed to be an outdated dinosaur that should have been upgraded and replaced every two years several times over - if you listen to the mainstream hype.

In actual fact my main computer was ordered in March 2005 for just under £1000 (I built it myself, the parts were ordered). This has just surprised me actually. Although I have swapped out the heart of it more recently it means my PC is now 'ten years old' and is still going strong and beating hands down the hundreds of machines I have looked at and had to fix for other people over the years. I regularly have to explain to others that no, you don't need to buy a new laptop, your one is just over a year or two old and is running slow because of how you have looked after it, not because all of a sudden the technology inside has deprecated and become obsolete.

Since then I still have the same case, the same RAM (OK ok .. it was swapped out after failure by Crucial with their excellent warranty), the same power supply, same original hard disk (now the third oldest set in the machine but still there), same DVD Write, same floppy and card reader. In actual fact the original hard disk STILL has the same Windows XP install on it, even though it has suffered a complete motherboard change (along with RAM, Graphics card) due to a motherboard problem a few years ago. Worth noting XP coped with the change and carried on as well!

So yes I have a 'new' motherboard but this was one bought second hand off eBay that is almost as old as my original one. I have a 'new' graphics card but again this is many generations old.

And does my machine work? All day every day. I use it heavily for work (education, programming, office, file related, graphics related etc) and for gaming (always been heavily into gaming). Do I feel the need to have the very latest games at the absolute top resolution? not quite because I am not willing to pay the premium to have the rig to run that as soon as the game comes out. Give it a year or two and all of a sudden this games crash in price and in the ability for the latest computers to handle them. Having said that my machine has still not had much of an upgrade and I have little problem running the vast majority of every day titles.

My bottle necks are

RAM (in the sense of the ability to have lots of programs working on the fly with no delay switching to them) Storage (always an issue no matter the upgrade, I now have a few TB of storage spread among my hard disks I have accumulated, that doesn't count external or cloud backups. Never enough.)

For work I have been given an iPad. I barely use it. At all. For anything. It is a pain in the backside for connectivity to my normal workflow and does not have the day to day items I need/want to use in work. It is excellent for connection in the sense of simple sharing/browsing online/using web features but that could be achieved by any handheld device. The biggest drawback for me with that, given the way I now work, is the coping with the file system storage and sharing structure. Not the processor, screen or RAM.

My phone gets daily use for all sorts of things. The biggest problem I have with it is simple storage. Given the price of smartphones today (especially on contract) why they cant all come with at least 100Gb of storage on a couple of cards is beyond me.

These problems are exactly the same ones I had with my first machine. A 386 DX (yes ...my dad paid a LOT of extra money for the co-processor for my FORTRAN and to upgrade the hard disk from 20Mb to 40Mb). Then there were a lot of bottle necks: processor, co-processor, graphics, memory, highmem, RAM, storage, connectivity.

But the main ones I see across all devices for the last twenty plus years I have been using computers? RAM (to a lesser extent today on PCs) and storage.

I teach maths and have taught various subjects for over 20 years. Since I have worked in mainstream schools the last 5 years or so I have come to use the electronic whiteboard more and more. I have no problem teaching off the top of my head with a pencil and paper (or a stick and some sand) but all my planned material revolves around the use and functionality of the IWB (and other devices such as using mobile devices - ipads in our case - to access electronic material).

It is so much more useful in the long run. The students can have an electronic copy of the lesson (before and after), you dont waste time "rubbing stuff out", you can come back to items whenever you please and you can reuse material so easily from lesson to lesson, year to year.

That is before you consider the inbuilt (and other) functionality the electronic whiteboard offers. In our case we have built in transformation tools, on screen rulers, protractors, compasses, graphing facilities etc. Then when you branch out and use apps, websites etc the amount of interactivity and usefulness increases massively.

This never replaces good old fashioned teaching and knowledge and methods, but as an added bonus it can be massive.

However, at the end of the day, you cannot do proper mathematics without resorting to proper work on paper, with a pen or pencil and a bit of brainpower and a lot of patience. That will never be replaced electronically.

What sort of summary were you expecting to be 'written in stone'?
These ideas, fleshed out in detail, in science and mathematics, may grow to be so much more than a cliched one liner:
http://en.wikipedia.org/wiki/E...
compare with
http://en.wikipedia.org/wiki/G...

In what sense are these not science?

Maybe you are being confused over the terminology:

http://en.wikipedia.org/wiki/E...

or more simply

http://www.notjustatheory.com/

Why is that? They have a valid model of a given universe with given parameters to a certain degree of "fineness". Why is that NOT modelling the universe? It is a model.. a mathematical model. (hmmm... interesting... one reply while logged in and one not)

I would contend that she does use algebra but she merely does not recognise it as such

Algebra essentially is "finding the missing thing" and it can be as simple as working out what your change is when you go shopping (as well as a million and one other examples).

People think that because they dont sit down with a pencil and paper and scribble some funny symbols they are not doing maths when in actual fact they are using those skills (maybe innate skills) all day, every day.

It doesnt actually matter how many planets or brown dwarfs you think we have missed

There are limits (for very good and well checked reasons) on how much ordinary (baryonic) matter there can actually be

We may have understimated the numbers of extrasolar planets or similar but that still wont account for the vast majority of the missing matter. In any case such calculations have been well looked at for a long period of time and screwed down pretty tight (this is what I did for my PhD almost 15 years ago. Even then it was pretty clear that brown dwarfs were not the be all and end all of accounting for dark matter within galaxies).

Regarding "move beyond the assumption that if we cant see it it isnt there"...surely that is the whole point of dark matter/dark energy. We are confident that 'something' is there, but we cant 'see' it, hence our insistence on using the term 'dark'.

It doesnt actually matter how many Brown Dwarfs we have missed

There are limits on how much "ordinary" (Baryonic) matter there can be, regardless of how much we actually have down on our named list here. So no matter how much we have underestimated the number of Brown Dwarfs (and we have done a pretty good job on estimating those numbers, that is what I was doing for my PhD pretty much 15 years ago and even then it was getting obvious that Brown Dwarfs or similar was not the answer) the fact remains that they cannot account for any significant proportion of "dark matter"

As regards "if we cant see it it isnt there" surely astrophysics actually assumes the opposite. Namely that there definately is something there but we cant "see" it. Hence the term dark.