 sideshow:The quadradtic equations proves what you are going against! you say that you have to remember facts. NO! I don't know my times tables that well either, but I do know HOW TO WORK THEM OUT... so I can work out that it has to be something to do with 7 and 8 as the answers...
If you don't know your times tables well, where did the 7 and 8 come from? My point was that if you want to answer that question it really would take a long time if you had no ideas about the factors of 56. The more fluent you are at times tables the easier it is, and I find it hard to imagine somebody progressing to maths at degree level without at least the level of times table knowledge to think "7 times 8" when they see 56. sideshow: the only way to do that by rote is to remember the x = -b - or + square
root etc.
There's no point attacking the strawman position of "learning only through memorising lists of facts". The Chinese system is not so extreme as to leave students incapable of mathematical thought beyond the use of formulas. Far from it. sideshow: To be honest I learnt how to do simultaneous equations aged 6
without help of my teachers doing puzzle books... no one got me to
remember stuff. Thinking skills. If you don't think that there are real
thinking skills Andrew, then you are just plain wrong, there are
explicit thinking skills in my subject that you can relate to all thing
that you learn.
I was very careful to say no generic thinking skills. Every subject has its ways of thinking and nobody is endorsing the idea that learning can be completely passive. The point is, however, that there is no way to simply separate subject knowledge and thinking, and just teach thinking without teaching knowledge, including factual knowledge. Nor can we simply attack an education system for its emphasis on memorising factual knowledge if we have no reason to believe that it doesn't produce effective thinkers. sideshow: There is no point in remembering everything,
understanding how to work it out is more important and we need to
encourage unique ways in doing this to encourage innovation and
creativity. To do this students need to discuss their work... think
about their thnking. metacognition.
The point is that the ability to understand how to work something out is largely dependent on knwoledge. As for "innovation and creativity" these always conjure up mental images of inventors and artists but if we stop and think about it for a moment it is neither clear that these can be taught, that they don't depend on existing knowledge, nor that innovation is always for the best. Even in science there are things that have been known for thousands of years. We really don't need to expect students to be constantly reinventing the wheel. More generally, it is far from clear that all classroom discussion is beneficial, students often spread their misconceptions. As for metacognition, it is one of those jargon terms that obscures matters. It can mean the small body of study skills that are among the few generic skills that can be taught, but it can also be used to refer to nonsense like identifying learning styles.
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