As I said, small village school, with a full range of ability and achievements. My wife, similar vintage, had the same experience in her school in Borehamwood, with the same mix of children. So it worked with small and large groups across the ability spectrum. Could it be down to a better teacher, with faith in her methods and understanding?
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There may be an element of that in there. I can't possibly say. It isn't going to be simple to pin down why some people do well but others badly with some methods. It could be down to the teaching method, or even just plain cognitive differences of some sort which we don't really know.
Except that the visual methods such as you describe do require pencil and paper, and so are unsuitable for many contexts. To my mind the more graphic approaches are more confusing to children who are capable at a very early age of grasping procedures and implementing them. It is patronising, IMHO, to think that you have to explain everything graphically.
They do, but so does the specified method the Education Secretary is going to force upon every school-leaver. For the avoidance of doubt, I don't mean something like 800 x 1200 as that is not long multiplication, but something that cannot simply be worked out with the 12 times tables like 76 x 123, for example.
All multiplication methods require an understanding of place value if not taught simply through rote learning. Some of the graphical methods only require this understanding and are easier to understand and remember for those who cannot visualise place values, eg. the box method of multiplication, where each digit (with its place value) is split along the sides of a large triangle and cross-multiplied.
I don't think it is patronising to have to explain things graphically. Graphical perception is a perfectly reasonable way of learning and understanding, as long as it is done correctly, and for some people, far easier than other approaches. The government keeps talking about inclusive education. That should not just be about being inclusive in terms of behavioural issues, but more importantly recognising the different learning styles and catering for them. I cannot speak for other subjects but mathematics is ultimately about empowering the population with the ability of (independent) thinking, the ability of being analytical, the ability of solving a problem with one's own method. It does not matter what method one uses, as long as it is
a correct one and gives the right solution.
Rote learning has its place when a necessary skill is not easy to explain and the mechanism behind it tricky to understand. You will never convince some pupils at a young age (even up to 16 in top ability sets) that long multiplication works because shifting some digits gradually and padding zeros at the end is because of place value, because they cannot "see" it. In the days when manual calculation was widespread and common practice in the workplace (remember log tables?), this was a vital skill therefore it was a good idea if taught through that approach, but the world has moved on. Like the example you gave earlier of square-rooting by hand, you no longer remember it because after a period of disuse, since you don't understand why it works (I make a brave assumption here), it is difficult to reconstruct the method for yourself. This is simply a side-effect of rote learning. Same will likely happen (and has probably already happened with many) with long division for example.
But having said that, the level of maths required - differentiating x^n and sketching a graph - was first year 'A' level back in my day (1980s), and the level of economics required is basically an understanding of what production and consumption are. It seems extraordinary to me that final year economics degree students wouldn't understand those topics and hence be able to answer the question. I would guess from that that you are correct and the problem is the students didn't know how to apply their knowledge in different situations. (Of course, without knowing what was actually taught on the course and what was said by the lecturers to the students, it's impossible to tell whose fault the problem is).
Final year economics students don't always have many of the necessary mathematical skills if our ones from a few years back (BSc Economics with Mathematics - not sure if they still run that course) were anything to go by. You will be surprised how many cannot differentiate, how many cannot work with continuously compounded interest rate, to the extent that some simply had to be told to remember the formula, and even then some still struggle if a change of subject is required before using that formula (or alternatively solving the equation after substitution), all of which stuff from the first year of A-Level or earlier. Some cannot read log tables properly, nor use a calculator competently when log is involved.
This also applied to a large proportion of those doing BSc Management, or even some on MSc Financial Management courses.
