21 October 2013 11:07 PM

kengreen
Posts: 437
Joined: 15 April 2013

no James,
Heaviside advocated mathematical rigour  PERIOD. No doubt that is what you meant to say?
Ken Green



22 October 2013 01:48 AM

jarathoon
Posts: 1061
Joined: 05 September 2004

Originally posted by: kengreen
Heaviside advocated mathematical rigour  PERIOD. No doubt that is what you meant to say?
Just after the quote I gave above he says,
"There is a time for all things: for shouting, for gentle speaking, for silence: for the washing of pots and the writing of books. Let now the pots go black, and set to work. It is hard to make a beginning, but it must be done."
The word PERIOD meaning without doubt and without exception is too strong in this context.
In the construction of a new mathematical deduction there are many phases and not all require equal rigour (especially for intuitive mathematicians):
 the creative ideas
 the first design attempt
 the failure the new design, and a new design after that.
 ideas gradually evolving and the flashes of inspiration
 the first build with scaffolding
 the reordering of the build programme
 the mistakes found and fixed
 the removal of scaffolding
 the presentation of a rigorous deduction to the world in the language of mathematics, devoid of all evidence as to how it came about.
There are probably a large number of routes to get to the presentation of a rigorous deduction. Not everyone works within an academic institution with lots of clever colleagues to help them thrash out and clarify their ideas in confidence and in private first.
Most people just want to be remembered for that last step and throw away the working...but aren't all the nonrigorous working stages of interest as well, especially to aspiring and otherwise isolated students who want to find out how it is done!
Should we try to convince generation after generation of young students that getting to that final polished deductive framework or argument is ALL that matters, and ALL that can be presented  without giving them a clue about how to get on the road to achieving the same feats of endeavour themselves?
Should we only give books to young students that have all evidence of how one might actually, in practice, go about building up a brand new deductive framework or argument expunged or purged from them; because such a nebulous and unclear creative process lacks the necessary rigour to talk about or to present openly and honestly in public?
James Arathoon

James Arathoon



22 October 2013 02:15 AM

kengreen
Posts: 437
Joined: 15 April 2013

James,
About what, in Hades, are you talking?
once again exhort you to take account of the instruction given by Winston Churchill: "take this away and bring it back to me on one side of one piece of paper."
Ken Green



22 October 2013 11:05 AM

jarathoon
Posts: 1061
Joined: 05 September 2004

Teaching mathematics and finding new mathematics for yourself (evn if this maths was already first found hundreds of years ago) is not the same as presenting polished mathematics in book form.
SMP maths was too polished and abstract in form for teaching most children. In my opinion it gave the wrong impression about what actually process of doing maths and finding new mathematical solutions is all about for nonmathematical geniuses.
There is no deductive theory of how to derive the deductive form. If there was we wouldn't be having this discussion.
James Arathoon

James Arathoon



22 October 2013 11:55 AM

jarathoon
Posts: 1061
Joined: 05 September 2004

In the language of letters there is now a distinction between english, engilish literature and creative writing.
In analogy, the language of numbers could consist of: mathematics, mathematical literature (study of great mathematical constructions of the past) and creative mathematics (heuristics, creative problem solving, finding old solutions without knowing all mathematical techniques and tools needed first).
James Arathoon

James Arathoon



22 October 2013 11:57 AM

Zuiko
Posts: 521
Joined: 14 September 2010

Originally posted by: jarathoonThere is no deductive theory of how to derive the deductive form.
I'm with Ken on this.
That sentence isn't even English.
http://www.plainenglish.co.uk/



22 October 2013 01:14 PM

jarathoon
Posts: 1061
Joined: 05 September 2004

Sorry for the unclear english, I do my best.
Mathematics is the language of deductive argument, in it that allows a stated theorem to be proved or illuminating mathematical identities to be explored.
I am just pointing out that there is no metamathematical language that allows you to deduce the form a mathematical deduction must take when proving a particular theorem or finding a new mathematical identity.
In that sense the process of discovering a mathematical proof is not itself mathematical i.e. a deductive process. It is fundamentally a creative process, just as creative writing is a creative process.
In fact I suspect that the creative writing process itself, especially poems and songs, is more amenable to mathematical description, than the mathematical writing process.
James Arathoon

James Arathoon



22 October 2013 05:14 PM

Zuiko
Posts: 521
Joined: 14 September 2010

Originally posted by: jarathoon
There is no metamathematical language that allows you to deduce the form a mathematical deduction must take when proving a particular theorem or finding a new mathematical identity.
that's cleared that up then...
You no doubt have some excellent points to make. Maybe you could translate them into English for those of us that do not understand whatever language that sentence is written in?



22 October 2013 06:44 PM

kengreen
Posts: 437
Joined: 15 April 2013

James,
When I first joined the technical publications section of the BBC engineering training department I remember well the advice given to me by my new boss: "if ever you find yourself unable to explain with ease, then you can be sure that you do not have any idea of that that which you are preaching. Go away and do your homework!"
Later, as an editor, it was my sad duty to help one of my writers to find an alternative occupation  he could never get beyond two paragraphs at the most before diving into realms of matHhematical hieroglyphics. I maintained, and do so to this day that, if I could not fathom his meaning, then there would be little point in publishing his document.
One of my first tasks on receipt of a new offering was to check through quickly for inconsistencies  I quickly learned that the best tool for hiding such was a bucketful of words mixed with a shovel full of commas.
I urge you to consider that your posts would be much more interesting if you ceased trying to convince yourself.
Ken Green
posts



23 October 2013 10:33 AM

jarathoon
Posts: 1061
Joined: 05 September 2004

Originally posted by: Zuiko
Originally posted by: jarathoon
There is no metamathematical language that allows you to deduce the form a mathematical deduction must take when proving a particular theorem or finding a new mathematical identity.
that's cleared that up then...
You no doubt have some excellent points to make. Maybe you could translate them into English for those of us that do not understand whatever language that sentence is written in?
Sorry I have tried to put it in my own words and it doesn't seem to have worked...
http://en.wikipedia.org/wiki/Metamathematics
Also if your interested try reading "The Unknowable" by Gregory J. Chaitin
If you want to engineer computers that write new software or new mathematics you have to at least be aware of the subjects of Metamathematics and heuristics and how new mathematics is created in the first place.
This is not just esoteric philosophy, it is something that could possibly in future times give members of this institution a good living.
James Arathoon

James Arathoon



23 October 2013 02:03 PM

kengreen
Posts: 437
Joined: 15 April 2013

I'm sorry James but I still have not the slightest idea of what you are writing. You seem to have a language all your own which uses English words which are not in any sequence which informs me.
Sorry, dear boy, but what you have convinced me is that you don't even know the nature of mathematics!
Ken Green



23 October 2013 03:05 PM

jarathoon
Posts: 1061
Joined: 05 September 2004

Originally posted by: kengreen
I'm sorry James but I still have not the slightest idea of what you are writing. You seem to have a language all your own which uses English words which are not in any sequence which informs me.
Sorry, dear boy, but what you have convinced me is that you don't even know the nature of mathematics!
Ken Green
This is a thread about mathematics education and how it was organised in the past in the form of SMP modern maths. I have put forward a view on how mathematics education might be better done as we move into the future. At least you will understand that much.
Other, much more influential people have different and perhaps even more radical views than me, in regards to the future of mathematics education.
e.g. the Wolfram brothers, Stephen and Conrad
http://www.computerbasedmath.org/
"computerbasedmath.org is a project to build a completely new math curriculum with computerbased computation at its heart  alongside a campaign to refocus math education away from historical handcalculating techniques and toward relevant and conceptually interesting topics."
If your personal preference is for mathematics education to revert to what was done prior to the 1960's; by mostly working through Euclid's three books on Geometry, in various ways, then put your view precise view forward and so we can discuss its merits and demerits relative to SMP modern maths.
You could also add debates on the nature of parallel postulate and what Euclid's three lost books on Porisms might have contained (as extracurricular activities for advanced students) if you like.
James Arathoon

James Arathoon



23 October 2013 03:58 PM

jarathoon
Posts: 1061
Joined: 05 September 2004

Ken,
Read the IET education brief
http://www.theiet.org/factfile...on/edubriefpage.cfm
Read the IET Policy submission
"S956  Consultation on reform of the National Curriculum in England"
Contact the members of the Education and Skills Policy Panel
http://www.theiet.org/policy/p...ion/members/index.cfm
Please try to articulate a clear and positive view point of your own, with help from the available internet sources you find to support your case, to show me where I am going wrong.
James Arathoon

James Arathoon



23 October 2013 05:20 PM

kengreen
Posts: 437
Joined: 15 April 2013

James,
You just can't get it. Can you? I cannot possibly show you where you are going wrong because, as far as I can see, you are not even going. You are indeed making a deal of noise stamping up and down on the spot but I'll be dammed if I have the faintest idea exactly what you're trying to do.
As I've said before, there is no point in even learning how to build a house unless you first learn how to lay foundations; prior to that, you must learn how to lay drains. Even then, before you can start to build you might learn something about bricks and mortar and trowels and acquire the skills required to bring all this together and so lay bricks in an orderly and useful manner.
You can quote me all references you can find, even those that are posted on IET forums/websites, not to attempt to teach  let alone improve such teaching methods  is quite useless if you address your learning to the family dog; an essential ingredient is the power of communication which brings us into the realm of language.
Your major error is in regarding mathematics as a language; as such it is pure gobbledygook and completely beyond understanding without some specialist rigourous training. Mathematics is a tool invented by man to help when his powers of language prove insufficient. There is absolutely no point in trying to teach, or trying to improve the teaching of, mathematics unless you start with the foundations and there is no better foundation than the relentless logic of Euclid. In my mind, the world has not been advanced by the advent of educational "experts"; is not even helped by the crass ignorance of many of those who are paid to educate our children. Following the end of WW!! we had a surge of energetic reform driven by wellwishers who were determined to make easier that which was already easy but for the confusion of teachers. It was decreed that children should be given the choice of what they wished to learn and it should have come as no surprise that they elected to learn nothing at all. Those children went on to become the next generation of teachers and so on to the utter chaos that is today the highly prized ashes of what was once an education system.
I have personally encountered university graduates who are not even competent in such simple tasks as addition and subtracTtion; who were not at all aware of the interelation between addition, subtraction, multiplication and division, but most of them were pleased to boast that they did not need tables; one indeed could not use a telephone directory because of his shaky grasp of the alphabet?
James, beware of being the one who makes the most noise.
Ken Green



23 October 2013 07:52 PM

jarathoon
Posts: 1061
Joined: 05 September 2004

Originally posted by: kengreen
"As I've said before, there is no point in even learning how to build a house unless you first learn how to lay foundations; prior to that, you must learn how to lay drains. Even then, before you can start to build you might learn something about bricks and mortar and trowels and acquire the skills required to bring all this together and so lay bricks in an orderly and useful manner."
Ok I like analogies. My attempt at decoding your analogy is
foundations = mathematical foundations
bricks = numbers
mortar = mathematical operations / operators
trowels = pencils
land = paper
buildings = mathematical proofs
drains = learning how to spot mistakes and when to discard a one approach in problem solving for another (perhaps?)
The only trouble I have with this is who decides what is orderly and what is useful and what form the orderly is considered useful?
"Your major error is in regarding mathematics as a language; as such it is pure gobbledygook and completely beyond understanding without some specialist rigourous training."
I do regard mathematics as a language or perhaps even (if I am pushed to it) a collection of related languages.
"There is absolutely no point in trying to teach, or trying to improve the teaching of, mathematics unless you start with the foundations and there is no better foundation than the relentless logic of Euclid."
What the foundations of mathematics as taught at school should actually include is far from clear to me.
The oldest mathematical text I have is
"The Young Mathematicians Guide, Being a PLAIN and EASY Introduction to the Mathematicks (in five parts)" by John Ward I have the 9th Edition published in 1752, and the first edition was in 1707.
The parts are:
Part I: Arithmetick
Part II: Algebra
Part III: Geometry
Part IV: Conic Sections
Part V: Arithmetick of Infinites
The twelfth Edition is here
In Praecognita he writes
"The Business of Mathematicks, in all its Parts, both Theory and Practice, is only to search out and determine the true Quantity; either of Matter, Space or Motion, according as Occasion requires.
By Quantity of Matter is here meant the Magnitude or Bigness of any Visible thing, whole Length, Breath, and Thickness, may be measured or estimated.
By Quantity of Space is meant the Distance of one thing from another.
And by the quantity of Motion is meant the Swiftness of anything moving from one Place to another.
The Consideration of these, according as they may be proposed, are the Subjects of the Mathematicks, but chiefly that of Matter.
Now the Consideration of Matter, with respect to it's Quantity, Form and Position, which may either be Natural, Accidental or Designed, will admit of infinite Varieties: ut all the Varieties that are known, or indeed possible to be conceived, are wholly comprized under the Consideration of these Two, Magnitude and Number, which are the proper Subjects of Geometry, Arithmetick, and Algebra. All other Parts of the Mathematics being only the Branches of these three Sciences, or rather their Application to Particular Cases."
He goes on with the idea of comparing like with like, with introducing the line, the surface and the solid.
I agree, that we should avoid devising mathematics syllabuses which are overly academic and too widely divorced from the real world. I like the idea that pure numbers arise out of the real world by counting, or calculating the ratio between two objects of similar kind, etc..
James Arathoon

James Arathoon



23 October 2013 08:55 PM

kengreen
Posts: 437
Joined: 15 April 2013

James,
You are beyond belief! In fact, you fall flat on your face with the first mathematics guide whose sins can be forgiven seeing the date of its publication. Surely anyone  at least today  realises that algebra deals with the rules under which our mathematics were derived? To put algebra in second place to arithmetic is ludicrous? I'm well aware that arithmetic is taught first and it has been so since Adam was a lance corporal but I put that down to a naive belief in practicalities  speaking for myself, I have always found algebra to be the most interesting and the most rewarding of mathematical studies.
The idea of "pure numbers" most certainly could not have arisen out of the real world. It is not so long since counting was achieved by cutting notches in sticks or by sliding beads along wires? Counting as we know it was not possible until (I believe it was) an Indian mathematician invented the symbol for zero? indeed, there is a case for believing that the mathematics of areas was pioneered by humble ploughmen who brought into our language the units of length, such as "rods, poles and perches" ?
As for your final paragraph I disagree most violently with your statement that we should avoid devising... ...mathematics syllabuses ... . It is my belief that we should bury our egos and get on with living using tried and trustworthy techniques.in fact, I believe in giving my children an education that has meaning.
Ken Green



23 October 2013 09:58 PM

jarathoon
Posts: 1061
Joined: 05 September 2004

Originally posted by: kengreen
The idea of "pure numbers" most certainly could not have arisen out of the real world. It is not so long since counting was achieved by cutting notches in sticks or by sliding beads along wires? Counting as we know it was not possible until (I believe it was) an Indian mathematician invented the symbol for zero?
Complete Hog wash.
The implicit zero was there ebven if the symbol wasn't; no notches on the stick; no beads slid across the wire.
" indeed, there is a case for believing that the mathematics of areas was pioneered by humble ploughmen who brought into our language the units of length, such as "rods, poles and perches" ?"
Complete Hog wash again.
The first recorded use of the mathematics of areas was in Ancient Egypt. The use was bureaucratic, in regards to raising taxes in proportion to the area of land cultivated by farmers along the Nile. Regular surveys of farm land areas were needed because the annual Nile floods, led to changes in the land areas individual farmers could cultivate.
Apparently Babbage was strongly inflenced by John Ward's Young Mathematicians Guide.
It was certainly published early enough for Adam Smith who wrote the "The Wealth of Nations" to see.
I'm just trying to point out that link between the real world and mathematics can be less clear in modern texts...
http://www.ams.org/notices/200601/feamccrory.pdf p 24
"An Example: Defining Fractions
Three of the mathematicians' textbooks and the Milgram book define a fraction as a point on the number line with particular characteristics. One includes both a part/whole definition and a number
line definition. One uses only a part/whole definition.
Among the other sixteen books, the most common definition is similar to that in Billstein, et al (2003):
[N]umbers of the form a/b are solutions to equations of the form bx = a.
This set, denoted Q, is the set of rational numbers and is defined as follows:
Q = {a/b  a and b are integers and b ? 0} (p. 266) "
Modern Maths was about dumping this sort of stuff on kids at the earliest possible opportunity...with the predictable result that it put vast numbers of children off mathematics for life in the 1970's and 1980's.
James Arathoon

James Arathoon



24 October 2013 03:16 AM

kengreen
Posts: 437
Joined: 15 April 2013

I quote:
"An Example: Defining Fractions
Three of the mathematicians' textbooks and the Milgram book define a fraction as a point on the number line with particular characteristics. One includes both a part/whole definition and a number
line definition. One uses only a part/whole definition.
Among the other sixteen books, the most common definition is similar to that in Billstein, et al (2003):
[N]umbers of the form a/b are solutions to equations of the form bx = a.
This set, denoted Q, is the set of rational numbers and is defined as follows:
Q = {a/b  a and b are integers and b ? 0} (p. 266) "
Modern Maths was about dumping this sort of stuff on kids at the earliest possible opportunity...with the predictable result that it put vast numbers of children off mathematics for life in the 1970's and 1980's."
I thought that my wife had some of the most mobile of goal posts but you make her look like an amateur!
I really think that you should consult a Doctor  of medicine.
It's quite entertaining to argue with you but you grow tedious.
Ken Green



24 October 2013 03:56 AM

jencam
Posts: 608
Joined: 06 May 2007

Originally posted by: jarathoon
This is a thread about mathematics education and how it was organised in the past in the form of SMP modern maths. I have put forward a view on how mathematics education might be better done as we move into the future. At least you will understand that much.
Other, much more influential people have different and perhaps even more radical views than me, in regards to the future of mathematics education.
I have many gripes about professors in pure mathematics who can prove that 2=3 being the driving force in curriculum reform at primary school level but I also agree with my son that allowing middle aged grammar school educated people who do not develop software to dictate the curriculum is equally undesirable. His feelings towards SMP modern maths are mixed. The good points are that there are certain topics relating to software and digital electronics that should be reintroduced. Technology has changed since the silence of 1975. Computers are now part of mainstream life and it's not uncommon to find kids of primary school age writing software. SQL databases are based on set theory and they happen to be part of the KS2 ICT curriculum. A 9 year old today would be able to relate set theory to databases whereas most 9 year olds of the 1970s would find set theory abstract and impossible to relate to anything they had encountered in the real world. Even simple algorithms like sorting an array of numbers by size could be taught to primary school kids because they are familiar with sorting functions in spreadsheets. The bad points are that SMP modern maths introduced many abstract concepts far too early before students were proficient in arithmetic and basic algebra, and it completely went against the chronological development of maths through the centuries. In his opinion certain levels of abstraction should have been reserved for 'serious' mathematicians after they had completed A Levels.



24 October 2013 09:45 AM

jarathoon
Posts: 1061
Joined: 05 September 2004

Originally posted by: jencam
Originally posted by: jarathoon
This is a thread about mathematics education and how it was organised in the past in the form of SMP modern maths. I have put forward a view on how mathematics education might be better done as we move into the future. At least you will understand that much.
Other, much more influential people have different and perhaps even more radical views than me, in regards to the future of mathematics education.
I have many gripes about professors in pure mathematics who can prove that 2=3 being the driving force in curriculum reform at primary school level but I also agree with my son that allowing middle aged grammar school educated people who do not develop software to dictate the curriculum is equally undesirable. His feelings towards SMP modern maths are mixed. The good points are that there are certain topics relating to software and digital electronics that should be reintroduced. Technology has changed since the silence of 1975. Computers are now part of mainstream life and it's not uncommon to find kids of primary school age writing software. SQL databases are based on set theory and they happen to be part of the KS2 ICT curriculum. A 9 year old today would be able to relate set theory to databases whereas most 9 year olds of the 1970s would find set theory abstract and impossible to relate to anything they had encountered in the real world. Even simple algorithms like sorting an array of numbers by size could be taught to primary school kids because they are familiar with sorting functions in spreadsheets. The bad points are that SMP modern maths introduced many abstract concepts far too early before students were proficient in arithmetic and basic algebra, and it completely went against the chronological development of maths through the centuries. In his opinion certain levels of abstraction should have been reserved for 'serious' mathematicians after they had completed A Levels.
I agree. I am more critical of the style and notation of the modern maths curriculum, than the actual mathematical content.
A maths education should cover both the literary and nonliterary skills. You can be completely illiterate and still engage in oral and pictorial maths. A more narrative like experience. Telling a story, imagining a journey. Arriving at the destination; an answer.
The equals sign is a relatively modern literary invention. And the rigorous version of it is even newer. With the equals sign now comes a lot of other baggage in terms of rigor. So in the example I gave above we end up with two equals signs, one for the actual definition, and one for the definition of the set of numbers to which it applies.
Why not complain about this definition and ask: Is the ratio of two rational numbers, strictly speaking a rational number according to this definition? That is (a/b)/(c/d). Well no, according to this definition it is not. However it does become a rational number when simplified to (e/f). (e/f) = (a/b)/(c/d) = ad/cd Therefore strictly speaking according to the above definition we have proved that the set of rational numbers also contains numbers of the form (a/b)/(c/d), even though strictly speaking according to the definition it doesn't
If you a very logical person like me this sort of thing can be very very infuriating!
Gödel should have saved us from all this numskullery from the false prophets of rigor, but mathematicians have tried their best to ignore him ever since he came up with his incompleteness theorem.
What Ken complains about is students not being able calculating things in their heads and carry out simple logical and deductive sequences in oral conversation and written conversation. If everything becomes suffixed by the "Set" this or the "Set" that, right from the outset, the nonliterary and intuitive mathematics, without pen and paper in hand, becomes much harder.

James Arathoon


