It was perhaps inevitable that sooner or later something would appear here about that thing that occupies most of my time. While it is technically still a Letter from
As far as maths is concerned, people seem to fall into one of two categories: those who get it and those who don’t. There are quite a few people in the world who, through no fault of their own, think that they belong in the latter of those two categories when actually they belong in the former, but more of that later. I don’t mean to suggest that those who will always be somewhat confounded by numbers are a lost cause; they simply need to be taught in a very different manner to those who, for some reason another, come equipped with an innate understanding of algebra and trigonometry and similar mathematical marvels.
The difference between the two groups appears, superficially at least, to concern retention. This is perhaps best illustrated with an example. Last week, my class were learning about angles. On being told that an acute angle is any angle between zero and ninety degrees, half the class immediately found a small shelf in their brain on which to store that information and could without hesitation identify which of a page of angles were acute (actually they weren’t looking at a page of angles, we were wandering around the school looking for and identifying angles in buildings and notice boards and staircases, I just thought it would be easier to say they were looking at a page of angles, but then I didn’t want people thinking that my lessons were boring so I thought I better let you in on what we were actually doing). The other half of the class were beginning to get it by the end of the lesson but many were still confusing acute angles for obtuse angles and couldn’t really remember what the word for an angle greater than one hundred and eighty degrees was (reflex…).
That half the children should struggle to grasp the difference between an acute and an obtuse angle is not indicative of their general ability to retain information, though for some this certainly is a problem, but rather concerns their conceptual understanding of angles. In order to quickly assimilate what an acute angle is, a child must first understand that an angle is the measure of the rotation involved in moving from one initial axis to coincide with another final axis (though obviously I would not necessarily expect them to express their understanding in such terms). They must also have understood and accepted that in a complete rotation there are three hundred and sixty degrees (and what follows from this: that there are one hundred and eighty degrees in half a rotation and ninety degrees in quarter of a rotation). What this really means is that those children who immediately grasped what an acute angle is were able to do so because the shelf on which they neatly filed ‘Acute angles’ was in a room with ‘Geometry’ written on the door and part of a stack which had ‘Angles’ written above it in bold lettering.
For the children who get maths, having divulged the names given to angles of different sizes it is possible to immediately move on to calculations involving angles. For those who don’t, one must look for as many different pathways as possible to the Geometry room. When it is eventually found it will almost certainly never be the most organised of places: for some, that there are three hundred and sixty degrees in a complete rotation may always be baffling. And why shouldn’t it be? Why are there three hundred and sixty degrees in a circle? (We have the Babylonians to thank I believe, something about the number of days in a year, but I doubt that is of much comfort to anybody).
If children are not set by ability, and they often are not in primary schools, teachers are confronted by three unenviable options: a) teach two lessons simultaneously; b) teach to the highest ability in the class; c) teach to the middle ground. (The actual answer is secret option d) teach through open ended investigations, which is not always possible and is far from the easy option but it allows children to work at their own pace and potentially stretches them as far as they are able without going beyond into the land of disillusionment. I have seen few teachers go for this option, maybe because it’s secret, so I’m going to stick with the three alternatives previously identified). Planning for one Maths lesson a day is time consuming enough, to plan for two and then deliver them on top of each other is unappealing (I have seen few teachers go for this option either actually, I think I might be the only one, and I only go for it when I’m not busy with option d), obviously). Teaching to the highest ability in the class works well in other subjects but will leave those children whose conceptual understanding of Maths is weak baffled and disheartened for reasons that are hopefully explained by what I have said before. The middle ground satisfies the needs of neither group of children but it does at least come with a government seal of approval.
The demand that every child reach level four by the end of Key Stage 2 has meant that the arbitrary single mark that distinguishes a level three from a level four has become an obsession in many schools (the prevailing middle ground), a line to be crossed at any cost. That cost is quite often fewer children achieving level five; they, after all, are already on the right side of the line that matters. The government may have expected a minimum standard to improve all results equally (a rather foolish expectation if that were the case): that the normal distribution would slide further along the x-axis while maintaining its perfectly formed bell shape. But, of course, that isn’t what happened. Instead, the peak remained exactly where it was, hovering over what has become the Holy Grail of level 4, but extended up the y-axis as the curve narrowed around it (the mean remained the same, the standard deviation from that mean decreased). This meant more children getting level four (good), fewer children getting level three (also good), but also fewer children getting above level four (not so good). The not so good consequence is most ably demonstrated by the fact that it is no longer possible to get level six in Maths at the end of Key Stage 2: so few children were getting it they scrapped the paper. (If you fall into the group of people for whom Maths poses something of a problem then please accept my sincerest apologies for all the preceding talk of x- and y-axes and normal distributions and standard deviations – I hope it didn’t put you off too much and that the point, which didn’t really need those things to be made but I am a maths teacher after all, still came through).
I have no objection to National Curriculum levels; so long as they are applied appropriately. The levelling of work completed in class provides a very useful means of assessing pupils’ progress and, what is more, of ensuring that pupils are making the progress of which they are capable. But insisting that a population of individuals whose circumstances vary so considerably all achieve above a certain level is nonsensical. These are unique human beings with diverse backgrounds. Some will progress slowly, some will progress rapidly and, yes, there will be many in between who progress at a middling rate. But what advantage comes from identifying the average rate of progress? It hinders both those who progress slowly - by labelling them as underachievers - and those who would progress much faster – by considering them needless of attention. As for the middling children, what good does it do for them to hear that they are average, that they are much the same as everyone else? Surely it makes more sense to acknowledge the individual progress each child makes. And so I suggest the following: that every child is encouraged to make the best possible progress of which they are capable. And that their progress is monitored by assessing them according to the National Curriculum levels and that teachers and parents collaboratively establish whether a child is fulfilling their potential. Which leads rather nicely onto my second suggestion: that education is openly acknowledged as a partnership between teachers and parents, even if an unequal partnership.
Children spend around thirty-five hours a week in school. That leaves one hundred and thirty-three hours to fill with other things. Many of those hours are obviously spent sleeping and eating and engaged in other such necessaries, but still, for a few short hours every a child may do as they, or their parents, please. Imagine two boys, both in Year 6 and both wobbling along that divine line that separates a level three from a level four, in English say. These two boys attend the same school and keep the same teacher, overly concerned by SATs results, awake at night thinking of ways to help them fall forwards onto a level four rather than backwards onto a level three. The parents of one of the boys listen to him read every night, they talk to him about what he has been learning in school and they make sure he is in bed by seven every evening. The parents of the other boy largely ignore him; he comes home from school and sits in front of the television or plays on his Nintendo DS until someone realises he should probably go to sleep, at perhaps nine o’clock. Results day comes (this is all fantasy remember) and the boy whose parents listen to him read every night gets a level four while the other boy gets a level three. Should the teacher feel guilty because one of her pupils got a level three? (ignoring for the moment the irrationality of anyone caring so much for anything so arbitrary). I would argue that they should not, that responsibility is elsewhere found.
I have digressed somewhat from my original ramble about the nature of learning and teaching maths. But I must return to address that group of people who think they don’t get maths when really they do; or, at least, the capacity to get it is there. The blame, though perhaps blame is too strong a word - those responsible for this incongruity are maths teachers. Specifically, for not all maths teachers are at fault, maths teachers who equip those in their charge with a method that enables them to solve exam questions, rather than furnishing their pupils with an understanding of how the pieces of the mathematical puzzle neatly fit together. If taught well, a child will end up with a whole mathematical wing to their brain: with a geometry room, full of parallel lines and polygons, regular and irregular, and the stack all about angles; and an algebra room, where numbers are banned and x and y are meaningful abstractions; and the first room to begin to form in the mathematical wing, the calculation room, in which there are four oversized toolboxes, one each for addition, subtraction, multiplication and division, all overflowing, added to with every passing year to provide an ever expanding tool kit of mental and written methods to confront the innumerable and unpredictable mathematical challenges that life has waiting. And displayed prominently on the walls of every single one of these rooms, and in the many other rooms in the mathematical wing that have not been mentioned, are examples of how that particular realm of maths has relevance and applicability to the world beyond the classroom and exam papers. If taught badly, however, rather than having a mathematical wing to their brain, a child will end up with a shed instead. The walls of this shed (as well as probably the ceiling and the floor) will be covered with post-it notes, on which might be scrawled ‘a2 + b2 = c2’ or ‘Area of a triangle = ½ x base x height ’ or ‘SOHCAHTOA’. As the years pass, the post-it notes will fade, their meaning will become ever harder to recall; if, that is, their meaning was ever really clear. The problem is teachers who tell their pupils that pi is ‘just a number you have to learn’, or worse, ‘just a button on a calculator you have to push’, so that when those pupils, who some time sooner or later must escape the confines of the classroom, are presented with a problem that requires the beginnings of a conceptual understanding they soon begin to flounder.
The above is full of crude generalisations. But they are crude generalisations rooted in general truths. People can perhaps not be ever-so-neatly split into those who get maths and those who don’t. But on the spectrum of mathematical understanding, most people I know, including those I teach, tend to lean strongly towards one end or the other. And this, I believe, can and should inform teaching practices. As far as mathematical wings and sheds are concerned, most people, I would assume, have something in between, but I hope the analogy illustrates the flaws of teaching a method rather than teaching for understanding and excites in some the idea that their perceived inability to put the pieces of the mathematical jigsaw together is not the end of the story.
No comments:
Post a Comment