How many triangles are there in this diagram? How many are there in this? If I call the first one a 2-stack triangle and the second a 4-stack, how many would you find in a 100-stack? Problems like these seem standard fare for puzzle pages but that’s not how things are done in a classroom, surely? Many would deny it; it all depends on what generation of students are approaching it.
The Victorian mathematics classroom, where mass education began, was either concerned with the functional task of teaching the third R (to workers) or the classical duty of teaching Euclid (to gentlemen). Following the reforms of the early to mid-20th century, an older generation recognises this arithmetic and geometry and would add algebra, trigonometry and perhaps calculus to the mix. This has formed an image so fixed in everyone’s mind that that must be how maths teaching has always been and, more importantly, that is how it should always be. A younger generation will merely add the ‘new’ technologies in place of the blackboard.
One of the myths of progressive education is that it was realised by well-meaning liberals pushing an ethical agenda to spare children from a grinding routine and bring all social classes and abilities together into one learning community. In fact, conservatives were complicit in abolishing grammar schools motivated by the population explosion after the second war which made building three schools less appealing economically than building one. Furthermore, technological demands had raised the stakes on not properly training future workers. Proportionally more investment went into Mathematics education than ever after Sputnik was launched and the West realised it was losing the space race, and Harold Wilson’s White Heat of technology only stoked the fire.
There was a race to find reforms that would generate a modern schooling for a modern workforce and for a short while, there was evidence that they took seriously the idea that engagement of the child might produce a more motivated therefore more productive student. New curricula like the School Mathematics Project (SMP) were introduced in the 60s. Initially they avoided dry routine exercises that taught a fixed model to follow and attempted deeper explanations and more realistic applications in scenarios that students could easily convince themselves would be relevant when they left school. In the less optimistic decade to follow, the inner-city challenges of maths teacher shortage and large populations of migrant students with limited English, the Secondary Mathematics Individualised Learning Experiment (SMILE) was born. Here, a classroom of thirty pupils would be given thirty different routes through a large bank of resources and, at their own speed, develop a mathematical knowledge from a cluster of well thought out activities. The students were completely differentiated but not entirely self-directed: the activities were linked by the teacher in advance.
That is not to say the vulgar materialist motives of reforms did not have a progressive outcome. Certainly, it gave a lot of space for teachers to allow many of the parameters of learning to be wrest from their control. When I started teaching about twenty-five years ago the orthodoxy in mathematics education centred on Constructivism which demanded the student construct their own understanding of mathematical concepts rather than have them transmitted by the higher authority that is the teacher. Constructivism was not a new idea. Every mainstream teacher for the last half-century would have some exposure to Piaget and even Vygotsky. What was new was the degree to which it informed classroom practice. Mathematics educators encouraged students to engage in problem solving, identifying inherent patterns and discover the mathematical principles that had until recently been delivered ready-made to a passive student body at the chalk face.
The protest that part of the role of a teacher is to protect their student from reinventing the wheel and expose them to results that have been developed over the centuries completely misses the point. It is vital that the student be given the opportunity to reinvent that wheel as learning cannot be effective if its most active component is copying. “Maths is not a spectator sport” is the cliché amongst professionals. I could tell my students that the sum of consecutive numbers is a triangle number or the sum of consecutive odd numbers is a square number perhaps in this form . But how much better it would be that they come by the same results in their investigation of pictures of triangles. As technology advances in the classroom this process could only accelerate. Or so we thought.
For all the fuss made of learning interactively with computers, learning in general has taken an increasingly conservative turn in the last couple of decades. Teachers came to find that the parameters that were wrest from their control did not devolve to the students but were firmly in the hands of a new bloated tranche of management who were directing state pronouncements on what constituted good practice. Today, many professionals are feeling the wind from the East which excels in international comparisons. They are exhorted to carry out Mastery programmes where all pupils work on the same material simultaneously. Differentiation of pupils would seem to be a thing of the past.
So what is the solution to the problem?
In a 2-stack there are (1+2) up 1-triangles, 1 1-down triangle and 1 up 2-triangle which makes 5. In a 4-stack there are (1+2+3+4) up 1-triangles, (1+2+3) down 1-triangles, (1+2+3) up 2-triangles, 1 down 2-triangle, (1+2) up 3-triangles and 1 up 4-triangle which makes 27. In a 100-stack there are 256,275 triangles in total. I think.
Oh, to the teaching problem?
Clearly the mainstream is in no hurry to relinquish its micromanagement of students’ progress through a programme of “mathematical competence for tomorrow’s productive workplace”, especially as it must not be seen to be falling behind in the “global race”. It obviously knows what is best for its citizens and it would not want to stand by and watch them reinvent the wheel.