Monthly Archives: June 2015

A controversial maths question

The following question which appeared on an Edexcel GCSE maths paper (for readers outside of the UK, this is a test that would be taken at the end of high school, at the age of 15 or 16) has gone viral, with many students complaining about its difficulty.

There are n sweets in a bag. 6 of the sweets are orange. The rest of the sweets are yellow.

Hannah takes at random a sweet from the bag. She eats the sweet.

Hannah then takes at random another sweet from the bag. She eats the sweet.

The probability that Hannah eats two orange sweets is 1/3.

Show that n2n − 90 = 0.

I sympathise with the students complaining about this question’s difficulty. I don’t think it is an easy question for GCSE students. I found it easy to answer, but I’m studying for a Maths degree.

However, there are different kinds of difficulty. Sometimes a question is difficult because it involves the application of knowledge which is complex and/or less prominently featured in the syllabus, which makes acquiring this knowledge and keeping it in your head more difficult, and also makes it harder to realise when this knowledge needs to be applied. This was not one of those questions. The knowledge required to complete this question was very basic stuff which I expect most of the students complaining about it already knew. As far as I can tell, the following mathematical knowledge is required to answer this question:

  • If m of some n objects have a certain property then the probability that one of these n objects, picked at random, will have that certain property is m/n.
  • The combined probability of two events with probabilities x and y is xy (as long as the events are independent of each other, although you could get away with not understanding that part for this question).
  • Basic algebraic manipulation techniques, so that you can see that the equation (6/n)(5/(n − 1)) = 1/3
  • can be rearranged into n2n − 90 = 0. All this takes is knowing how to multiply the two fractions together on the left-hand side (new numerator is the product of the old numerators, new denominator is the product of the old denominators), then taking the reciprocals of both sides and bringing the 30 on the left-hand side over to the right-hand side (there are different ways you could describe this process).

Certainly for some of the students, the knowledge was what was at issue here, but I think the main challenge in this question was the successful application of this knowledge. Most of the students knew the three things listed above, but they failed to see that this knowledge could be used to answer the question.

Unfortunately for the students, failure to apply knowledge successfully is in many ways a much more serious failure than failure to possess the appropriate knowledge. If you fail due to lack of knowledge, there is an obvious step you can take to prevent further failure in the same situation: acquire the knowledge that you lack, by having somebody or something teach you it. Crucially, when you successfully learn something you not only end up knowing it, but you also know that you know it.

If, on the other hand, your problem is that you failed to apply your knowledge successfully, then it is much less clear what the next step you should take is. And, also, you never know whether you are capable of applying your knowledge in every situation where it might be useful, because there are usually a whole lot of different situations where it might be useful and it is impossible for you to be familiar with them all. This is why cramming for tests is not a good idea. Cramming may be the most efficient way to obtain the knowledge you need for a test (I don’t know whether this is actually true, but I don’t think it’s impossible) but it certainly won’t help you with applying your knowledge.

There is one relatively straightforward way to become better at applying your knowledge. If you remember how you applied a certain piece of knowledge in a previous situation, then, if the exact same situation occurs again, you won’t have any trouble, because you will just apply the knowledge in the same way again. In a similar way, the more similar the new situation is to the old one which you remember how to deal with, the more likely you are to be able to successfully apply your knowledge.

But, in a way, this is a trick. I am about to get a bit esoteric here, but bear with me. Let’s say your mind has two “departments” which work together to solve a problem. One of them is the Department of Knowledge-Recall, or the DKR for short, and the other is the Department of Knowledge-Application, or the DKA for short. I think that when you carry out the strategy above, what is happening in your mind is that the DKA is pre-emptively passing on the tough part of the work to the DKR for them to deal with. If you remember how you used a certain piece of knowledge (let’s call it X) to deal with a previous situation, then that memory, in itself, has become knowledge (let’s call it Y). The DKR has worked so that Y is easily recalled. When the situation comes up again, the DKA’s task is really easy. It’s looking at the most prominent pieces of knowledge that the DKR has made you recall, and it notices that Y has a kind of direct link to this situation. If the DKR hadn’t worked to make Y a piece of easily-recalled knowledge for you, the DKA would have to do more work, sifting through the knowledge that the DKR has made you recall, possibly asking the DKR for more, inspecting them more closely for any connection to the situation at hand.

There’s a good chance the above paragraph made no sense. But basically, I’m trying to say that familiarising yourself with the situations where you need to apply your knowledge works because it is a process of acquiring specialised knowledge about where you need to apply your knowledge—it is not truly applying your knowledge. But probably the more important point to make is that this process is inefficient. It would be much simpler if you could simply apply your knowledge to unfamiliar situations straight away. And it’s often impossible to familiarise yourself with every conceivable situation, because the range of conceivable situations is so vast.

Students taking tests try to carry out the familiarisation process by doing past papers. This is often effective because the range of questions that can be on a paper is often quite limited, so it really is possible to familiarise yourself with every situation where you might need to apply your knowledge. But this isn’t a good thing!

As I argued (somewhat incoherently) above, the skill of being able to apply your knowledge is only separate from the skill of being able to recall your knowledge if it includes the sub-skill of being able to apply your knowledge to unfamiliar situations. That particular sub-skill is one which is hard to improve. Arguably, this sub-skill is what we mean by “intelligence” (the word “intelligence” is used to refer to a lot of different things, but this might be the most prototypical thing referred to as “intelligence”). It’s certainly possible to get better at this sub-skill, but I don’t know if it is possible to get better through conscious effort.

But intelligence (by which I mean, the ability to apply your knowledge to unfamiliar situations) is often the main thing the tests that students take are trying to assess. After all, there are few vocations, if any, that a person cannot do better at by being more intelligent. You don’t always need intelligence to get to an acceptable level, sure, but intelligence always helps. The purpose of the GCSEs is not just to find out which students are more or less intelligent—they are also supposed to increase the amount of people who have useful knowledge—but it is one their main purposes.

That’s why I don’t think this question was unfair, as some students have been saying. Yes, it was quite different from anything that was on the past papers. But that was the whole point, to test people’s ability to apply their knowledge in unfamiliar situations. It is natural to be disappointed if you couldn’t answer the question and to complain about your disappointment, but saying that it was unfair is a step too far. It did what it was meant to do.

I think there is possibly a genuine problem here, though. If questions like this which strongly tested intelligence (as defined above) were usual on the GCSE papers, you wouldn’t expect this one to become such a topic of interest. Perhaps the GCSEs have suffered in the past from a lack of questions like this, which has affected students’ expectations of what these tests should be like. I should be clear that I have no idea whether this is true. I took my GCSEs in 2011, but I don’t remember what the questions were like in this respect.

PS: I’ve seen some people saying “this is easy, the answer is 10”. These people are making fools of themselves, because the answer is not 10, in fact that doesn’t even make any sense. The question is asking for a proof. (“Show that…”) It seems these people have just seen the quadratic equation at the end and assumed that the question was “Solve this equation” without actually reading it. Perhaps this is evidence that the question really isn’t easy. Or maybe these people just aren’t thinking about it very much.

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A summary of Cannibals and Kings by Marvin Harris, chapter 2

Cannibals and Kings is an anthropology book by Marvin Harris aimed at a popular audience. Since I’m trying to read this for education rather than entertainment, I’m not reviewing it in the usual way, but instead I’m trying to understand what it is arguing for. Hopefully, this will be part of a series of posts where I summarise each chapter; I’m starting with chapter 2 because the first one is a general overview. I wrote a brief introduction to Marvin Harris and Cannibals and Kings in this post on Tumblr.

Somebody not familiar with the topic might be inclined to think of agriculture as an invention, like the steam engine or the light bulb. The question of the origin of agriculture would not seem well-posed to such a person. They would say that until the first farmer societies appeared 12,000 years ago in the Middle East, the idea of agriculture had simply never occured to anybody, and for this reason every society was a forager society. But once the first person had the idea (who happened to be in the Middle East, 12,000 years ago), people saw that the farmer lifestyle would be better for them, and therefore they adopted it. Neighbouring societies came into contact with the first farmer societies, came to the same realisation, and became agricultural themselves. Other societies came up with the idea independently too (but later than those Middle-Eastern pioneers), and passed the idea on to their own neighbours. In this way most of the societies in the world became agricultural, except for a few in places like Siberia where the environment made agriculture impractical, and in places like Australia where the societies were too isolated to be sufficiently exposed to the idea and happened not to come up with it themselves.

There are a lot of problems with this explanation. One of them is the assumption that the idea of agriculture was sufficiently unlikely to occur to people in forager societies that the first farmer societies only appeared 12,000 years ago. Humans are thought to have reached behavioral modernity around 40,000 years ago, so that’s 28,000 years where the idea of agriculture, if it occured to anybody, occured only to people who were unable to get it across to others, or to people in those places like Siberia where agriculture was impractical. Was the idea of agriculture really so inaccessible? Anthropologists have found that people in modern forager societies often have extensive knowledge of the natural world in which they live. Presumably, prehistoric foragers were the same. In particular, the mechanics of plant growth would probably not have been a mystery to them. And if they knew how plants grew, it doesn’t seem like a massive leap for them to have the idea of planting seeds and encouraging their growth in order to eat the plant once it was fully grown.

But the most fundamental problem with this explanation is the idea that the farmer lifestyle would be attractive to foragers. This is far from obvious. In fact, it appears that, at least in its primitive stages, farmer societies were in many respects less conducive to general well-being than forager societies. There are lots of points that could be made here, so let’s just focus on one metric by which forager societies have an advantage over farmer societies (at least the primitive ones): the amount of leisure time available, as opposed to time spent obtaining food. The modern San foragers of the Kalahari desert spend about three hours per adult per day hunting and gathering and have a diet rich in animal and plant protein. Modern Javanese peasants (as of 1977), on the other hand, spend about six hours a day working their farms and get much less protein for their efforts. Even modern workers still spend about four and a half hours a day earning the wages they need to obtain their food (assuming a 40-hour week), although they have access to an enormous range of different kinds of food, so the comparison is less straightforward. And, let’s not forget, the Bushmen live on the edge of the Kalahari desert, not one of the most hospitable environments in the world. Most prehistoric foragers would have lived in environments where food was easier to access.

That’s the empirical side of the argument. It is also possible to explain why it makes sense that the early farmer societies would have had a worse standard of living than forager societies. The reason is that there is a crucial difference in the nature of the means of production in forager and farmer societies. Foragers depend on the amount of resources present in the surrounding environment. It is impossible for them to increase the amount of food produced per unit area, because the more they hunt and gather, the more scarce the animals and plants that they sought become. On the other hand, farmers can increase the amount of food produced per unit area by planting more crops per unit area; there are limits on the amount of crops that can be grown per unit area too, but the limits are sufficiently high that the maximum amount of food that can be produced per unit area in a farmer society is much higher, and the early farmers would not have needed to reach it. Harris refers to increase in the amount of food produced per unit area as “intensification of production” (it’s a concept that re-occurs and is important throughout the book). The difference between forager and farmer societies can therefore be briefly summarised as this: foragers cannot intensify production, but farmers can.

Since foragers cannot intensify production, forager societies are motivated to maintain a constant population density. If the amount of foragers within a given area increases, the foragers in the area have to eat less on average. However, if the amount of farmers within a given area increases, the farmers in the area don’t have to eat less on average, because they have the option of increasing the amount of food produced in the area and thereby cancelling out the effect of the increased population density. As a result, farmers are not motivated to maintain a constant population density, and, in general, the population density in a farmer society tends to increase over time. But the intensification of production that farmers must carry out in order to accomodate the increase in population density necessitates an increase in workload.

One last question remains. How do forager societies maintain their constant population density? In the second chapter of the book, which is called “Murders in Eden”, Harris talks about some of the methods that they probably used. As the name of the chapter suggests, some of these might turn you off from the somewhat idyllic picture of forager life that has been painted so far.

Foragers had no access to effective methods of contraception. They had access to methods of abortion, and some of them were probably quite effective, but they tended to also be very effective at killing or seriously injuring the mother, so abortion was not an attractive option. But there was another way of getting rid of unwanted children which carried zero risk of harm to the mother: infanticide. Infanticide is, of course, morally objectionable to most people in my society, and probably yours as well, if you’re reading this. But other societies, both historical and modern, it is not; in these societies infants are considered to be non-persons, just as many people in my society consider foetuses to be non-persons. I think Harris would agree that a society’s nature generally determines its moral system, not the other way round. The necessity of infanticide in forager societies would cause these societies to define infants as non-persons. There would, therefore, be no reason for foragers to refrain from carrying out infanticide for moral reasons. The only reason infanticide would be disfavoured to some degree would be so that the effort of pregnancy, and, especially, the extra food that the mother would need to eat, would not be wasted. But the benefits of keeping the population constant would have outweighed these costs.

Another, less morally questionable method that was used was late weaning. After birth, women ovulate only after their body has built up enough fat reserves to allow the next baby to get enough food, at least according to Harris (I don’t know if this is an accepted fact or not). If they breastfeed, they expend a significant amount of calories feeding the baby, so that their fat reserves build up more slowly, and ovulation is delayed. (The same mechanics are behind the fact that menarche occurs earlier among more well-nourished populations.) In farmer societies, people usually consume enough carbohydrates that it is impossible to delay ovulation for more than a year or two, even if the baby is not weaned for the whole of this time. But foragers’ diets are much less carbohydrate-rich, so they can delay ovulation for much longer. And as long as ovulation has not occurred, impregnation is impossible. Studies of Bushmen women have found that, by putting off weaning, they often avoid getting pregnant for four years or more. That means that, within in the approximately 26-year-long span in which they are fertile (between the ages of 16 and 42, or close to those ages), there is only time for five or six pregnancies. Accounting for the effects of miscarriage and infant mortality, this might result in only three or four children who survive puberty on average, given no infanticide. In order to maintain constant population this number needs to be cut down to two, and this would have to be done via infanticide, but the average woman would only need to kill one or two babies during her lifetime. It still probably doesn’t sound like an ideal way to live to anybody reading this, but it’s not quite as bad as you might have thought. Again, though, it is important to understand that while this would be a moral cost for us, it would not have been a moral cost for foragers in which infanticide was an accepted practice.

Now that we have established that the naïve description of the origin of agriculture given above is incorrect, the next question to answer is this. In general, farming was not an attractive option for foragers. What were the particular circumstances in the Fertile Crescent 12,000 years ago, and in the other areas where agriculture originated independently, that made farming an attractive option in these particular times and places? This is the subject of the next chapter, “The Origin of Agriculture”, which I’ll write about in the next post.