Areas of insecurity
What on earth does this mean? For instance, is our knowledge about area “secure”? Is anyone’s?
Could students following our primary mathematics courses explain area? We asked them. They might, we warned, know something without being able to articulate it fully. The health warning proved to be appropriate. Indeed, in discussing their responses, our own words did not come easily.
Recourse to standard texts or dictionaries did not particularly help. Derek Haylock’s excellent Mathematics Explained for Primary Teachers describes area as “a measure of the amount of two-dimension al space inside a boundary”. Several students also mentioned boundaries. But what of the surface area of a sphere or a torus?
We discussed how the “amount of space” was measured. Conventionally square units are used. We can make sense of this even for irregular shapes, circles and so forth. So measuring area by repeating squares seems unexceptionable for plane surfaces. What about curved surfaces?
A cone is easily dealt with. Imagine it spread flat, into a sector of a circle. But the troublesome sphere cannot be spread flat. The history of attempts by geographers to offer plausible projections of the globe vividly testifies to this fact. We could cover the sphere with little squares to measure its surface area. But these squares do not fit on exactly. If you try to press them down, they will crinkle. Of course, the smaller the squares, the nearer we come to being able to fit them snugly to the surface of the sphere. We can approach, but never quite reach this satisfactory limit.
This is intriguing, but it is becoming very abstract. Beyond level 8? How many angels can stand on a pin?
“Two-dimens ional” is an idea which most of us felt we understood, but would not care to have to explain in words. Interesting little pitfalls await those discussing the “difference” between 2-D and 3-D shapes with primary age children. 2-D shapes are “flat” while 3-D shapes have “depth”, we may find ourselves suggesting. Does this imply that we can hold up examples of each, and compare? We have seen this done. Thin plastic shapes,or shapes cut from paper are offered to represent the 2-D shapes. You may be lucky enough (or unlucky, depending on your point of view) to have a child kindly pointing out that your supposed triangle is actually a triangular prism, albeit a very thin one.
All 2-D shapes are the surfaces or parts of the surfaces of 3-D objects. Whereas some surfaces of 3-D objects are not 2-D shapes. Even if we can explain the idea of “two-dimensional”, there are problems about building this into an account of area. The very idea of a curved surface seems, at least to the non-mathematician, to possess a three-dimensional flavour. Yet we talk about the areas of both the flat and the curved surfaces of cones and cylinders.
Of course sectors of circles may be transformed into the curved surfaces of cones, and rectangles into the curved surfaces of cylinders. The fact that these actions are reversible may be claimed to make all the difference. It may be said that the curved surfaces of cones or cylinders can be assigned areas because of the possibility of flat versions of the said surfaces.
Children, and even initial teacher training students could be forgiven for feeling a little puzzled about all this. After all, the curved surface of a cylinder cannot be curved unless a third dimension is invoked. It is supposed to make perfectly good sense to speak of the area of this surface when it is a curved surface. And in any case, to repeat, the sphere and the torus lack flat versions of their surfaces.
Note that primary pupils will find our language difficult. “Space” might suggest volume or capacity. The Pan Dictionary of Mathematics talks of the extent of a plane figure or surface. “Extent” is better than “space” in that it does not hint at volume. Nevertheless we would not be keen to use such a term with primary children since they would have to learn it simply as a piece of mathematical jargon. It does not figure in their ordinary language. Our objection here is not to technical terms as such. A precisely defined expression is often better than a struggle with inadequate everyday phrases. However, in this instance the use of “extent” is unlikely to increase pupils’ understanding of area.
Other area “explanation s” include the thought that it relates to the amount of paint required to cover it. This nicely avoids problems about curved and plane surfaces. The account is intuitively fruitful, despite the fact that it is circular (logically!). We are implying that the paint is applied evenly. “Evenly” could not be defined without saying something like “the same quantity of paint for a given area”.
Sometimes we know more than we can tell. This is just as well. There’s a memorable moment in an early Open University video where a child, attempting to convey the idea of area, moves her hand horizontally to and fro. This captures part of her conception. However, we would have liked to ask her about spheres . . .
In many ways, the fact that the notion of area fairly bristles with conceptual and linguistic challenges does not bother us at all. It is after all a mathematical abstraction. The application of area to certain curved surfaces represents an enrichment of its standard meaning for plane surfaces. This is perfectly legitimate. Mathematicians do likewise for the term “number” when calling -2, or even #195; -1 numbers.
Now we have proposals for specified knowledge within a national curriculum for initial teacher training. Imagine a written test about area being set for students.
Would this not inevitably miss the points we have just explored? Would it not sideline many difficult but crucial questions concerning the nature of subject knowledge required to support good teaching?
These questions do not admit of easy answers. Nor are the answers necessarily the same across different subjects and levels of abstraction.
To pretend that they were would woefully distort the whole enterprise of teaching.
Andrew Davis and Maria Goulding are lecturers in maths education at Durham University