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weebecka:

When you were working with these practical and extended taskes, were they clearly structured so that students knew where they were going, what they were doing and when they had achieved the desired outcomes, or were they not - so that students had to puzzle out where to start, create or research appropriate mathematical strategies and so that some of them took the investigations well beyond the syllabus?  i.e. if you look at the Bowland CPD 1 were you being true to it in keeping the maths hazy or were you stripping out that bit, which is so easily done.  

Most of the time, it was left completely open. There were the odd pointer given, but nothing to make the objective obvious.

weebecka:

I lost it long ago.  And strange as it seems the world didn't fall in and reality didn't stop.  It got better.

I found maths at Cambridge frankly a bit dull, so in my second year I decided to study the history and philsophy of science while also studying an embryonic science (experimental psychology).  So I discovered my reality, which is that knowledge is socially and contextually constructed, and I Iooked at the philosophies of the likes of Popper, Kuhn and Lakatos (who had a direct influence on the older generation of ATM) to understand how this happens in practice.

And I'm afraid I didn't find a difference between science and mathematics scentless_apprentice.  It's not that science is perfect and mathematics is fallible.  They are both fallible. 

I've got no problem with Popper and Kuhn and their idea that science is fallible. If it wasn't, we'd still be thinking the universe revolved around the sun.

But I feel you're making an error in comparing science and Mathematics in this regard. Science certainly is a social construct - indeed, it is usually driven by political and military influences and different societies have dealt with it in different ways. Mathematics though, certainly isn't. Whilst it is something that develops and changes - the mistake Euclid made with parallel lines and the fallout from that, for example - it is not a social construct in the sense that whoever, wherever and whatever you are, there are fundamental truths in Mathematics which can't be proved or disproved - as Godel showed us - thus they are neither fallible or fallible.

weebecka:

I have what is called a pragmatic non-formalist view of mathematics.  By that I mean that I see all 'truth' as being contextual. 

So you're a postmodernist then. That's fine. No problem with that. But is it fair to teach students who don't have the cognitive maturity to deal with such ideas in this way? For most people their world needs concrete principles in which to move forward with their lives - children especially. If we're going to say that the Mathematics they're studying - and I'm not talking about Peano Logic or anything crazy like that, just the syllabus and the connections between concepts they're expected to know by the age of 16 -  has some order of 'haze' around it, then can you imagine the results?

"Yes mate the quote might be 2 grand for the extension, but it depends on how you look at it."

"The prescription is for what I think are 100mg tablets, but I'm not sure of my calculations"

"The driver hit the pedestrian at 30mph. At least, not in a relative sense, because obviously that depends on how fast the pedestrian was travelling".

Now you could argue that I'm being facetious, and yes, I probably am. But in the context of the world we move in the higher order concepts you're talking about only have a very minute influence on the reality of our children. They have to know what works, and why it works. Not that according to Popper's idea of falsification or Kuhn's 'paradigm shifts' that someone might come along and completely disprove it.

Because - and tell me if I'm wrong here - the mathematical concepts that have been taught in the programme of study in the National Curriculum work (2+2 = 4, the sine of an angle is equal to the ratio between it's opposite side and the hypotenuse, a + a = 2a, etc...), and have worked for thousands of years, no matter what shift there's been from Pythagorean, Aristotlean, Newtonian or Einsteinian scientific bases, and I'd hope that Popper and Kuhn would agree with that.

scentless_apprentice:

If we're going to say that the Mathematics they're studying - and I'm not talking about Peano Logic or anything crazy like that, just the syllabus and the connections between concepts they're expected to know by the age of 16 -  has some order of 'haze' around it, then can you imagine the results?

Yes.

Because I've done it.

The easist way to understand how I did it was through creating a culture in which there is a regular demand for reification (that is making abstract examples real or visualisable.

This is one of the reasons I chose to work at MMU.  They work extensively with RME (realisable mathematics education) which underpins the Dutch education scheme based around the Freudenthal institute.  They have the same aims but different methods to some that I've developed myself which has led to some fascinating discussions.

So - how do I begin to explain what I mean.

Well - you could take I starter I use a lot with year 7. I write two things on the board (they could be quantities, algebraic expressions, pictures) and tell them to sum, difference, product and quotient them.  1 mark for each correct answer if they can picture/explain how they got it.  Half a mark if they get it right but have used a mathematical trick they can't explain.  But, crucially, a whole mark if they get very close but can't quite get there but can explain what they did. Also a whole mark if it's just undoable and they can explain why.

One of the first hurdles is - are their 5 potential marks or 6?  What does difference actually mean?  We lean into the confusion here to become aware of the 'haziness' around language.  Then we get on with trying to picture our maths.  And the students begin to invent, to be creative.  The come up with new and original ways of picturing and explaining things, which they have to teach to each other.  And so the beginnings of an inventive culture in which students are constructing their own mathematical worlds (questioning and personalising the mathematics of others rather than just accepting it) begins.

Obviously there's more to it than that.

By the way of course I don't talk about non-formalism or post-modernism with my student - the focus is - if you can't picture it or fully explain it don't accept it yet (unless it's just before a key exam in which case do).

weebecka:

The exam results were great, thanks.

Reminiscent of the definition of blackwhite from George Orwell's 1984:

"this word has two mutually contradictory meanings. Applied to an opponent, it means the habit of impudently claiming that black is white, in contradiction of the plain facts. Applied to a Party member, it means a loyal willingness to say that black is white when Party discipline demands this. But it means also the ability to believe that black is white, and more, to know that black is white, and to forget that one has ever believed the contrary. This demands a continuous alteration of the past, made possible by the system of thought which really embraces all the rest, and which is known in Newspeak as doublethink."

weebecka:

They work extensively with RME (realisable mathematics education) which underpins the Dutch education scheme based around the Freudenthal institute.  They have the same aims but different methods to some that I've developed myself which has led to some fascinating discussions.

Before the conversation travels too far up its own posterior, have you ever actually taught anyone who has been to a school in Holland? Or Finland? Or any part of Scandinavia? ( in case you are wondering I know Holland is not part of Scandinavia - but it is a direct competitor in that part of the world ).

 I have taught students from most countries of the world, and these much vaunted mathematical educational systems are a crock of ... well ... you can guess.

Scandinavian maths education is rubbish. Figuratively, metaphorically, literally take whatever ally you like. The Dutch mathematical education system follows it closely behind and the Finnish system - despite wonderful reports to the contrary - is following the same map.

Did any of your fascinating discussions actually get round to teaching some Dutch kids? Or perhaps even Dutch kids that were not part of some tightly specified control group?

The only countries of the world that have produced students who actually have some real knowledge and skills in maths, and have produced it on a regular basis, are South Korea and China. Sure you get the odd kid from any other country with great technical skills in maths but that is the exception rather than the rule. South Korea and China have kids with excellent technical maths skills as the norm.

The other thing I have also noticed is that the kids with the greatest self-belief in mathematics but with the weakest technical knowledge ( I am not talking about ability here - whatever that may be ) tend to be from ( excluding my favourite Northern European region ) the UK with its wonderful National Curriculum.

The last part has always troubled me. As a product of the English educational system, my natural inclination is to uphold it as the best in the world, yet hard evidence in the form of students in front of me tells me it isn't. Something is going wrong, something seriously worrying.

Would someone care to explain ( although the way this thread has developed does explain a lot )?

weebecka:

Scentless_Apprentice and Betamale thank you so much for your posts.

POS = Program of study and I'm talking about the national 2007/8 versions.

Now it won't surprise either of you to know that I'm in a rather different place to you two on what I believe maths education should be and it's clear that to you that seems to be a murky, hazy, dangerous place that just doesn't make sense?

I'm glad you see it that way because if you thought you knew my views exactly you would, I'm sure, be wrong.  It's easier to explain if I start from where you are and explain why I'm no longer there and where I went.

I'm not doing this because I'm trying to convert you or prove you wrong or anything like that.  Heaven forbid!  Just so that we can communicate more clearly and you can know exactly where I'm coming from when I say unusual things and you can feel more comfortable about knowing which of them you might find relevant and which you just don't.

Is that okay?  Shall I carry on?

If so essentially there are three separate journeys which I have taken away from where your views on maths educations lie - a philsophical one, an experiential one and one which proactively explored and exploited the possibilities of ICT in maths educations.

If you want me to start, pick one of these three topics.  I'll talk about what happened and you can criticise/ask questions until were done and then we'll change topic.

 

All I can picture when reading post is a child of the 60s in a back of camper van havving just put a spliff down and readjusted their floral headband.

Its good you have your thoughts but I don't feel they should be shared with any new teachers or as part of CPD.

Deliver me kids in year 7 who can manipluate numbers, not use their fingers as soon as they have to multiply anything beyond 4 x 5, get me kids who can learn rather than ones who want to play in a sand pit and pupils who are not *@!!*@ at basic numeracy and believe every lesson should be about entertaiment an I will listen....unfortunately everything you advocate is at the heart of primary education and as a result having weak students after 6 years of exploring.

• Numeracy is not maths
• Rote learn numbers and basic operations
• Explore

Not the other way round and trying to justify it with trendy words and obscure references makes it even less credibl.

Make your own subject, sit in a cornfield talking about your feelings and how to explore....fine.....just don't call it maths or suggest people should who are teaching or coming into teaching