Re: Strange mathematics trends in Singapore
- From: daevida@xxxxxxxxxxxxxxx
- Date: 17 Oct 2005 19:12:26 -0700
The guys at the Ministry of Education forgot one important point: the
PSLE is for primary school kids, not mathematics majors. They are
supposed to be introduced to the discipline of mathematics, not
convoluted permutations of mathematical logic. Before they run in the
marathon, they need to walk uprightly first.
Which brings me to my pet bitch - model math. Someone has likened it
to be like a lake 10 miles wide, but only one centimetre deep. All
those diagrams and "simplification" when straight forward algebra will
solve the problem all the time - model math does NOT solve all
problems. And that's why when the kids make it to secondary one, they
have to redo all the math problems, this time using algebra. So who's
the clown behind model math? Another mercenary trying to milk the
parents of hard earned money for assessment books when traditional math
books should have sufficed.
Der Zählmeister wrote:
> Chameleon wrote:
> > Mathematics education in Singapore seems to have adopted a strange
> > syllabus. Ungeometrical untrue sketches designed to stretch the truth
> > are posed to our students.
> >
> > What is the philosophy behind such mathematics ideas? Believe the
> > numbers and the words but don't believe what you see in the skecth?
> >
> > Maybe the teachers just don't know how to use a ruler's dimensions
> > correctly.
> >
> > What is their philosophy behind this form of teaching? Bend the young
> > minds?
>
> ~~~~~~~~~~~~~~~~~~~~
>
> [refer to the diagram in
> http://www.channelnewsasia.com/stories/singaporelocalnews/view/173118/1/.html
> ]
>
> the intended method of "short-cut" solution, i believe, was for pupils
> to observe that the white area just below area B has exactly the same
> area as B itself. if you imagine B to be moved to this white area, you
> see that A + B + C forms one big triangle, whose area can be quickly
> worked out to be
> ½ × 15 cm × 6 cm = 45 cm²
>
> then (if one takes the given areas of A and B at face value) the area
> of C can be quickly worked out by subtracting the areas of A and B from
> the above total thus:
> area of C = (45 - 4 - 10) cm² = 31 cm² (supposedly)
>
> ~~~~~~~~~~~~~~~~~~~~
>
> the intention was good: to get pupils to realise that
>
> (1) maths is not just about numbers and calculation, but also about
> concepts and relationships between concepts, the ability to make
> observations, and the ability to link the observations to one's
> knowledge to solve problems.
>
> (2) by the clever exploitation of deeper concepts such as "same area",
> "movement", "problem equivalence", problems can be simplified to
> yield easier solutions. i.e. "short-cuts"
>
> the pupils who are able to make the observations and apply the concepts
> will be able to find an efficient solution path (such as the one
> presented above, which takes only 2 calculation steps). those who
> can't make the observations and apply the concepts will get bogged down
> by tedious long-winded calculations that may lead nowhere.
>
> if the intention was to "bend the young minds", then it was to bend it
> for the better. however, ... (read the next part)
>
> ~~~~~~~~~~~~~~~~~~~~
>
> the knowledge required in properly setting a primary school maths
> exam question like this is actually more than the knowledge required
> in primary school maths syllabus itself. there is a subtlety
> involved (it's not just about being able to use a ruler, although
> trying to draw an accurate scale model of the problem might have
> uncovered the error), and the examiners should have been conciously
> aware of this subtlety. apparently, in this case, the examiners were
> oblivious to it when they set and reviewed the question.
>
> the subtlety is the involvement of 'square roots' if one were to try
> to reify the problem onto a flat piece of paper.
>
> Let x cm be the base of triangle A. Since the height:base
> ratio is 6:15, the height of the triangle A is (6/15) x cm.
> And since the purported area of A is 4 cm², we have
> ½ . x . (6/15) x = 4
> x² = 20
> x = ¬/ 20 (it's about 4.47)
>
> Let y cm be the base of triangle B, by a similar reasoning
> we have
> ½ . y . (6/15) y = 10
> y² = 50
> y = ¬/ 50 (it's about 7.07)
>
> If you look at the diagram, the total of bases should tally to
> the base of the big triangle i.e.
> x + y should= 15
> but by calculation (or other ways of reasoning) we find that
> x + y not= 15
>
> ~~~~~~~~~~~~~~~~~~~~
>
> actually, we don't have to go into all this analysis "on paper". the
> are now computer software available (e.g. Geometric Sketchpad,
> Mathcad, ... etc.) that can do all the modelling easily. but someone
> who is reasonably well-trained in mathematics to be an examiner for
> a national high-stakes examination ought to be able to at least realise
> that square roots may be involved, and the proposed diagram may not fit
> all the given conditions.
>
> what the examiners need to realise is that subtleties and possible
> complications may exist even though it's "just a primary school maths
> problem". they should be able to spot such subtleties when they arise,
> and use the necessary tools (ruler, calculator, software, ... most
> importantly their own brains) to deal with it. this incident serves
> as an object lesson to illustrate my point.
>
> ~~~~~~~~~~~~~~~~~~~~
>
> The Count, Singapore
.
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- From: Der Zählmeister
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