What is an Investigation?

We discuss this under four headings:
  Definition      Investigations      Why Investigations?     What is mathematics?

Definition
There are at least three types of investigation. One type is essentially a library exercise where you are trying to find out why Pythagoras slaughtered 100 oxen or why Pierre de Fermat scribbled in the margin of a book. A second type of investigation is the statistical sort, where you might be trying to find the average height of 10-year-olds or the popularity of a particular politician and how to improve that popularity. And the third type of investigation is a mathematical investigation that aims to discover as much as possible about a particular mathematical question.  It is this third type that we are discussing here.

Investigations
In Problem Solving Units we talked about the problem given in 3c and 5c Stamps. The problem there is interested in the amounts of postage we can make with an unlimited number of 3c and 5c stamps. This is a one-off problem. The solution that we give there is that we can make up 3c, 5c, 6c, and everything else from 8c onwards.

In the Extension to the problem above, we look at 3c and 7c stamps and come to the conclusion that here we can get 3c, 6c, 7c, 9c, 10c and everything from 12c onwards. Once we’ve done 3c and 5c, the 3c and 7c problem is not too difficult.  Precisely the same techniques can be used.

But about now our curiosity might well have been aroused and we might start to wonder what amounts of postage we can get with 3c and nc stamps where n is any whole number. The number that seems to be of special interest here is the ‘onwards’ number. That is, the first number such that every other number from that one onwards, can be made up with sufficient of the 3c and nc stamps. To get some idea of what patterns we can find, we might set up a table. With a bit of effort we could find the ‘onwards’ numbers for various values of n using the  3c  and  5c  approach. At first these might give a confusing pattern. For n = 5 and 7, the ‘onwards’ number is 2n – 2. For n = 6 and 9 there seems to be no ‘onwards’ number at all. We present such a table  below.

(This table doesn't have to be constructed by all the groups in the class.  Some children may not want to use a table but those who do may combine with other groups and divide a set of numbers between them.  This will speed up the construction of the table and provide more data from which to see a pattern.)

Gradually we begin to realise that when n is a multiple of 3 we can only get multiples of 3. On the other hand, for all other values of n we do get an ‘onwards’ number and it does seem to be 2n – 2.

About now things start to get hard. We would very much like to justify this 2n – 2. But doing that is quite difficult. It needs a confident knowledge of algebra and is probably only possible with students who are on top of the Level 6 Algebra content.

With really able students, even at Level 6, we could perhaps investigate further. What if we change the 3 to an m? Again it is probably a good idea to experiment with a number of values of m and record the results in a table. That may lead us on to the guess that the ‘onwards’ number is (m – 1)(n – 1). But justifying this is quite a task.

The point of outlining this particular investigation is that it has a number of exit points. The original problem is probably enough for most children at Level 4. However, a development band student could go at least as far as the Extension. Then the problem could be revisited for a Level 5 class and the problem taken as far as the 2n – 2 guess. At Level 6 many students could appreciate the solution to this. But the problem is a non-trivial exercise for Level 7 and above.

One thing that we have tried to do, and will continue to try to do, is to provide problems at Levels 5 and 6 that have multiple entry points and multiple exit points. In this way we hope to provide situations that will challenge a range of students at each Level.

Why Investigations?
There are really three reasons why investigations are important. The first is that in future employment, students may have to produce information, consider different outcomes and put together a coherent report describing the field and the possibilities. Mathematical investigations provide a means to develop many of the skills that are needed for this type of ‘research’ because mathematical investigations are about developing scenarios and lines of inquiry: What if this happens? Suppose we tried that where would it lead? Combining mathematical investigations with library investigations just about covers all the skills that are required for ‘research’ problems in most employment situations.

But investigation is the lifeblood of mathematics. It is what research mathematicians spend their lives doing. Children can get some idea of the feel of mathematics and what mathematicians do, by engaging in investigations. This should be done gradually. Start first with relatively simple one-off problems and then work on harder problems and their extensions, until the students are capable of tackling investigations that may take a week to complete.

And the final advantage of such investigations is that it gives students the opportunity to practice a range of skills from all areas of the curriculum. By controlling the investigation you can to some extent control the skills that the students will use. However, you should expect them to come up with the unexpected.

What is Mathematics?
It’s not good enough to say that mathematics is what mathematicians do. Below we produce a diagram that shows the way that mathematics develops and we hope in the process that you will see how this applies to problem solving and more particularly to investigations. A more detailed discussion of this can be found in Holton, Neyland, Neyland and Thomas, Teaching Problem Solving (see References).

The bones of mathematics are in the diagram above. All mathematics starts with a problem that is either a well known one or invented by someone on the spur of the moment (or after a great deal of thought). To make any progress on a problem usually requires a lot of playing around with the ideas and making up lots of examples. We call this experimenting.

If we refer back to the postage stamp investigation, the problem was posed for us. However, it is worth noting that it was originally a problem that was posed by mathematicians. The person who solved it was Sylvester at the end of the 19^th Century. It is almost certain that he didn’t see the problem and immediately write down a complete solution. He undoubtedly spend some time experimenting in the way we did above. He would have needed to have guessed the pattern 2n – 2 and then (m – 1)        (n – 1) for the onwards numbers, somehow. To do this he would have needed data that could only have been obtained by experimentation.

But, finally, as a consequence of his experiments he would have arrived at the guess (m – 1)(n – 1). Actually mathematicians prefer to call their guesses, conjectures. Generally it is not realised that mathematicians do a lot of conjecturing, guessing. In fact it is one of their major tools. Getting a good conjecture is 90% of their battle. However, on a difficult problem they make a thousand guesses before they make the right conjecture.

So how do they know if a conjecture is false? They find a counter-example. This is an example that runs counter to their conjecture. Suppose Sylvester had conjectured 3(n – 2) – 1 for the onwards number. This works for n = 5, after all 3(5 – 2) – 1 = 3 ´ 3 – 1 = 8. But it doesn’t work for n = 7. Here 3(7 – 2) – 1 = 3 ´ 5 – 1 = 14, while the answer for 7 is 12. So n = 7 gives a counter-example to the conjecture that the onwards number is 3(n – 2) – 1.

If you find a counter-example, then you need to think again, maybe experiment again, to find a new conjecture.

But Sylvester, at least, was able to prove his conjecture. Now proofs are actually what separate mathematics from other disciplines. No other discipline proves things without any doubt. These proved things are generally called theorems. Perhaps the most famous theorem is Pythagoras’ Theorem, that the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides. As far as the stamps go the theorem would say that, given m, n with no factors in common, any amount of postage from (m – 1)(n – 1) onwards can be made.

Having produced a theorem what mathematicians do next is to try and generalise or extend their result. For instance, going from 3c and 5c to 3c and nc is a generalisation. This is because 5 is a special case of n. If we know what happens with 3 and a, we can certainly find out what happens with 3 and 5. But we can generalise 3 and n further still. After all the result for 3 and n is a special case of the result from m and n. So m and n generalises 3 and n. Summarising, then, a generalisation of a problem is a bigger problem that contains the original problem as a special case.

That leads us to extensions. These are also problems that grow out of other problems but here the link is often less strong. For instance, with 3c and 5c stamps we know that we can add any non-negative multiple of 3 to any non-negative multiple of 5. It’s possible that we might think that adding squares of 3 and 5 would be interesting. So we would look for all possible numbers of the form 3²a + 5²b. This is a similar problem but the original problem is not a special case of this problem. The squares idea is therefore an extension of the first problem.

Mathematicians working on generalisations and/or extension are going around the diagram again, starting from problem.

Now give up is an interesting feature of mathematical research. Obviously no one can solve all the problems they attempt. Death gets in the way from time to time. So sometimes people have to give up, for ever. However, giving up, for now, is a good problem solving strategy. Very often, after giving up, for a while, new ideas suddenly seem to pop up. This happened to Hamilton when he was walking with his wife along a canal in Dublin. Suddenly an important idea popped into his brain. But this giving up does come with a warning. It only works after an intensive period of activity. So don’t let your students think that they can lie in bed and have theorems jump out at them!