We need to set maths in context, argues Tom Roper. There are times when it comes in very useful
You're out skiing, off the piste, you shout to a friend and the echoes bounce around the rock faces and the steep slopes holding fresh snow. Suddenly it happens, the low growl as the snow on the high slopes starts to move - an avalanche. Can you outrun it down the slope to safety?
Fearsome though an avalanche is, with its tremendous destructive power, and as quickly as it appears to move, the answer to the question is yes - provided you see it in time.
The next question is, why?
Thinking of you as a particle sliding down a perfectly smooth slope inclined at angle , your acceleration down the slope is g sin , where g is the acceleration due to gravity. That is just you, nothing else is happening to you. However, the avalanche is picking up more and more snow as it makes its way down the slope. Put simply, its mass is increasing all the time. Therefore it does not gain velocity quite as fast as you do. In fact the acceleration is only a third of yours. So, if you see it in time, you should be able to out run it.
The same maths handles both situations, you and the avalanche. Newton stated in his second law of motion that the resultant force, F, is proportional to the rate of change of momentum, a form of words often neglected in favour of the more popular but limiting F = ma. Thus for the force, Fs, causing the avalanche to accelerate down the mountain, But for you, the particle, the mass remains constant. The rate of change of the mass, is therefore zero. Hence, for the resultant force acting on the particle, Fp, we have, The extra term, due to the fact that the mass of the avalanche is changing, makes all the difference but the mathematics is basically the same.
Maths is remarkable in that it is not uncommon for the same maths to apply to apparently completely different situations. Thus the motion of a boomerang through the air, a bowl moving across a bowling green and a gyroscope are all the same from the point of view of maths.
The almost uncanny success of mathematics to model the real world that is all around us is one of the main reasons for studying the subject. But to make use of that success we also need to be able to "read" the maths that results. The calculation can be as accurate and as sophisticated as we can make it, but if we cannot read and interpret the outcomes for non-mathematicians then we have cut ourselves off from the rest of the population. Maths will remain the subject of routine computation instead of the subject that tells a story and predicts the future.
With the avalanche, we can take some comfort in the fact it is possible to outrun it. However, it is difficult to see how we might predict something that could happen in the future from this example. To show the capacity of maths to predict particular behaviours, let us consider how fast you can walk and enlist the help of the team from the 1970s television comedy Monty Python's Flying Circus, in particular their Ministry of Silly Walks.
The question raised is: just how fast can you walk? We model walking as being similar to the John Cleese walk at the beginning of the Ministry of Silly Walks sketch. Walking consists of planting one foot firmly on the ground and allowing the body to rotate in a circular arc about the foot as centre. The rigid leg is the radius of the circle. The other leg swings forward in its turn, the foot to be planted equally firmly. Each stride repeats the process.
If the length of the leg is l and g is the acceleration due to gravity, then treating the body as a particle moving in a vertical circle of radius l, about the foot as centre, the maximum walking speed v is given by: For an adult, a leg length of about 0.9 m is reasonable and gravity is about 10 metres per second per second, which gives a maximum value of v of 3 metres per sec. But this is the least interesting part of the answer. Any result in mathematics needs to be read like a piece of text and interpreted. A wealth of meaning is packed tight into these few symbols. Shakespeare himself could not have made them more pregnant with meaning.
The result tells us that the maximum walking speed depends upon the length
* of the leg. Thus when a parent grasps their small child firmly by the hand and walks quickly off to catch the bus, the child has to run. Its little legs simply are not long enough to enable it to walk as fast as the parent. But there is more to be had from this assortment of symbols.
The result clearly depends on the size of g. For most practical purposes g is constant over the surface of this planet. However, mankind has walked where the value of g is less than what it is on Earth. On the Moon, the value of g is about one-sixth what it is on Earth, and hence the maximum speed of walking is much slower. This means that if you tried to walk on the surface of the Moon using a similar method to that you use on Earth, you would end up running. Visualise the video footage of the men who walked upon the Moon. You will recall that they did not walk, but in fact bounced along - the lunar form of running!
There is much talk of going to Mars. Unmanned expeditions are being sent to explore the famous Red Planet in a search for life within the solar system but outside the confines of Earth. Some day mankind will go there. How fast will they be able to walk? What will the video footage of that first mission show?
The Cockcroft Report in 1981 talked of maths as a means of unambiguous communication and gave this as the major reason for teaching the subject. Yet in very few syllabuses, at any level, are young mathematicians asked to 'read' their mathematics and interpret it within a given context. The original national curriculum in mathematics of 1989 did not refer to the maths used in other subjects or contexts. Successive versions, including the latest one, have perpetuated the myth that maths has nothing to do with anything but itself. Calculation and manipulation are all-important, meaning in context is of little concern.
In the new national curriculum, other subjects do refer back to the maths curriculum. This serves only to make the omission of forward references all the more glaring. Let us take one simple example, in biology. The maths curriculum deals with surface area and volume of regular solids. So what about a cube? For a cube of size one unit, the surface area is six units, the volume one unit. The surface area to volume ratio is high, 6 : 1. In a small mammal, such as a mouse, the surface area to volume ratio is high. This means it has little space for the food which gives it its energy, but lots of surface area through which it can lose energy in the form of heat. Hence small mammals are either furry or eat a great deal, or both.
We can now predict the behaviour of large flocks of penguins in the Antarctic night. They must huddle together to cut down their collective surface area and hence reduce heat loss. But those at the edges of the huddle will lose heat more quickly and must be kept warm. Hence the flock must exhibit a spiral motion as those at the edges make their way into the centre and those at the centre make their way out towards the edges. David Attenborough has made a film showing this in the Antarctic. The prediction is correct. Powerful stuff.
Perhaps one day the fact that small mammals are either furry, or eat a great deal, or both, will become a theorem to be established in the maths classroom and investigated in the biology laboratory. If you can read maths, it can tell you quite wonderful things about other subjects and the world that is all about us. Perhaps those who write the curriculum could give us all a little help by putting some hints into it as to where we can find our subject being used and how.
Tom Roper is senior lecturer in mathematics education at the School of Education, University of Leeds