In the 1980s, there was an English city school I used to visit. It was a junior high school in which the children stayed until the age of 13. The school was in one of the city's most deprived areas, if not the most deprived. In this school there was a group of children who were unusually good at arithmetic, or what is now called maths.
All their times-tables were automatic, as were all their unit number computations (one unit number +, -, x or ÷ with another unit number). In larger sums, their grasp of layout and carrying-over was solid in all four arithmetical processes, as was their grasp of place value, at least up to the tens of thousands. They could operate decimals, including correctly placing the point in decimal multiplications and divisions. They could find lowest common denominators in vulgar fractions and then use it for adding and subtracting them. They could convert decimals and the most commonly occurring fractions into percentages and the other way round. They could interpret graphs as well as bar charts and pie charts.
Moreover, the children seemed to understand what they were doing and why. I say 'seemed' because you can never know what understanding anybody has of anything as that would require the contemplation of the operation of their mind, something humanity has yet to master. But you can make a stab at it. In this case, for example, you could throw trickily expressed arithmetical problems at these children and they would be quick to spot the arithmetical sum or sums required to solve them. There was variability within the group, but the generality of the matter is as I've described. It was all as unusual then as it is now, if perhaps a little less so.
Other than their arithmetical prowess, there was nothing to distinguish these children from the other children in the school, either in their backgrounds, their gender or their performances in other subjects. The only thing they had in common was that they all had the same maths teacher, a Mr Ward. Intrigued, a colleague and I decided to have a look at what Mr Ward was doing. Mr Ward gave his agreement without hesitation.
It would be possible to go into great detail about this but there isn't the space for that here. In any case, my notes are long since gone. Instead, I would like to attend to four general aspects of Mr Ward's teaching:
First, the tasks required of his children were clear. They knew where each task began and ended and therefore where the next task began. Everyone was gainfully occupied at all times. Consequently, there was no slack time and no open-ended drudgery in Mr Ward's classes.
Second, Mr Ward's pupils were not always required to discover everything for themselves. He had himself discovered that it is sometimes alright to tell children things. For example, 8 x 7 is either two 8s + two 8s + two 8s + one 8, or it is the result of looking up a number square, or it is just 56.
The third aspect of Mr Ward's classes was to with a problem that, then as now, plagues British education (yes, Scottish education too). That is, low expectations: the more disadvantaged the children are, the lower the expectations for their educational outcomes. If ever you were struggling to explain a self-fulfilling prophecy, that should do it. Mr Ward seemed to have no expectations for his pupils' outcomes as that might normally be understood. Rather, he behaved as if he expected that it was his responsibility to create the circumstances in which it would be as easy as possible for his pupils to learn what he was aiming for them to learn. Thus, the responsibility for outcomes was his, not theirs.
The fourth aspect of Mr Ward's classes is also to do with another fundamental problem in British education. That is, the routine confusion between content and method. Content means what is to be learned. Method is the means by which it is transmitted. Lots of children, especially in the matter of primary, master lengthy and error-prone methods that are more difficult to master than the thing – or content – they are intended to transmit.
As it happens, at the time my colleague and I were looking at Mr Ward, we were, elsewhere, also looking at the mixing-up of content and method in primary schools. What we were finding, though quite informally, was that if you picked a pupil at random and asked their teacher what they were right then aiming to teach that pupil, we would more likely than not be told a method of teaching them something. For example, when asked what she was aiming to teach a child right at the minute, one teacher said it was to rummage through a bag full of sight words written on cards and match them to words written on a separate sheet of paper. That is a method of teaching. The content or purpose was to acquire a sight vocabulary (words recognised on sight without having to add its component sounds together). I make no criticism of this method, but that is not the point.
A classic example of the confusion between content and method is the decomposition process in subtraction sums. This is a complex and time- consuming process of finding an answer to simple subtraction sums (such as 3,291- 496). It is error prone because it has so many steps. Even I regularly come up with wrong answers when using it, and I am fluent in arithmetic. Decomposition and other long-winded processes were intended to help children 'understand'. All this has achieved for many is to give them the idea that maths (or arithmetic) is a difficult slog, the result of which is often the arrival at a wrong answer. Mr Ward paid no homage to such elaborate methods. His teaching was direct, simple and content-focused.
I think about Mr Ward every time a new and catchy fad for getting children to learn arithmetic comes along. They usually last for two or three years before fading away having left no evidence of any impact other than on their disciples. I also think about Mr Ward whenever I hear wishful thinking like 'Successful Learners' or 'Effective Contributors' masquerading as outcomes. Consequently, I think about Mr Ward quite a lot.