Investigations: The State of the Art

Investigations: The State of the ArtAuthor(s): John EvansSource: Mathematics in School, Vol. 16, No. 1 (Jan., 1987), pp. 27-30Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30214166 .

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INVESTIGATONS the

State of the art

by John Evans North East Wales Institute of Higher Education

Background For many teachers the first real mention of investigations and problem solving came in the Cockcroft Report, where they were specified as essential elements of mathematics teaching at all levels. In saying this, however, Cockcroft was simply reflecting the views of many mathematics educators, particularly members of both the MA and ATM, who had been advocating a more open-ended, investigative approach to mathematics teaching for some years.

The Americans, incidentally, were slightly ahead of us in time and rather more extreme in direction. Whereas Cockcroft just talked of essential elements, the American National Council of Teachers of Mathematics in their document "An Agenda for Action" in 1980 recommended that "problem solving must be the focus of school mathematics in the 1980s." Further, although this was explicitly their No. 1 recommendation, the rest of their recommend- ations were written around problem solving (the Americans do not make a distinction between investigations and problem solving in the way that some of us in the UK do _

see next section). Thus, for example, they suggest that the phrase "basic skills" should include problem-solving skills.

Back in the UK, in 1985, we had the HMI document "Mathematics 5-16" in which the thread of investigative mathematics runs right through, and problem solving and investigative work are made explicit as points 9 and 10 of their "Principles for Classroom Approaches". Not exactly the focus for school mathematics but nevertheless quite definitely there!

These ideas have even reached the examination boards at 16 +: the Joint Matriculation Board Syllabus A will include an optional investigation from 1986; the GCSE National Criteria state that mathematics courses must include prac- tical and investigational work by 1991.

Problem Solving or Investigations? All of the documents mentioned above have used the two terms "investigations" and "problem solving". So, what is the difference?

Cockcroft makes a distinction, although he doesn't say what it is. HMI say that there is no clear distinction but suggest that, broadly speaking:

*Problem Solving is a convergent activity where pupils have to reach a solution to a defined problem;

eInvestigations are more divergent activities where pupils are encouraged to think of alternative strategies, to consider what would happen if a particular line of action were pursued, or to see whether certain changes would make any difference to the outcome.

More informally, we could say:

Problem: An end without a beginning. Investigation: A beginning without an end.

Mathematics in School, January 1987

If there is a distinction, then it very rapidly becomes blurred. Most problems can become investigations by changing the numbers of conditions. Most investigations, no matter how simple the starting point, reach a stage where the pupil doesn't know what to do next - then he has a problem!

In my view, any distinction only relates to the beginning of an activity. For example, consider a quite well-known "investigation": Frogs.

R)(R)(R)(

Fig. 1 Frogs.

The object is to exchange the counters so that the blue ones move to where the red ones were and the red ones move to where the blue ones were. The usual rules are:

0 A counter may slide into an empty space next to it. SA counter may hop over one counter of the opposite colour. ONo "backward" moves are allowed.

It seems to me that for most pupils this starts off as a problem - they can't do it! Only when the problem is solved can it become an investigation. At that stage pupils can ask many questions which will lead to further math- ematically rich situations. What happens if there are 4 on each side? 5-a-side? How many moves? Can you generalise? Can you prove your result? What happens if there are different numbers of counters on each side? How many hops? How many slides? What happens if the rules change? Can we use more than two colours?

Frogs is a particularly fruitful investigation, but does illustrate the point well that we can restrict a problem to the basic starting point, but in doing so we may be missing the opportunity to develop some fascinating mathematics.

What are we Trying to Achieve? With Frogs, as with most investigations, it isn't the result obtained which is important; rather it is the way in which the pupils go about the investigation that matters.

If we, as teachers, tell our pupils to try 4-a-side, 5-a-side ..., draw a table, and show them the rule linking the number of counters with the number of moves, then those pupils will have gained nothing. On the other hand, if we don't give them any help at all, especially when they are inexperienced at investigating, then very few of them will get anywhere.

What we need to do is to develop in our pupils the strategies which will enable them to extract the mathematics

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Table 1 Specific strategies and general qualities.

Specific Strategies Draw a diagram Work systematically Draw a table Look for pattern Make predictions Make and test hypotheses Generalise Look for proof Choose simpler numbers List all possibilities Trial and error Work backwards Estimate Collect data Organise and present data Interpret data

General Qualities Discuss Work cooperatively Work independently Communicate using mathematics Define and comprehend problems Formulate key questions Experiment Explore Develop a sense for "blind alleys" Transfer skills: "Have I seen one

like this before"? Draw on background knowledge to

apply mathematics Use imagination and flexibility

of mind Reflect

from situations for themselves. Here, though, is where we hit a big snag. What are these strategies? They are in fact, very difficult to pin down. My own list is presented in Table 1.

Broadly speaking they fall into two categories: specific strategies and general qualities. Specific strategies, I think, we can teach: we can provide investigations which illustrate the use of these strategies and explicitly develop them with our pupils. The general qualities I'm not so sure about teaching in the same way, although I have no doubt that we can provide an environment which will stimulate their development.

Some of the specific strategies may benefit from further illustration. For example, looking for proof need not be highly algebraic. Suppose your pupils are investigating a row of cubes. They can look around the cubes but must not touch and can see that with one cube five faces are visible (the sixth is hidden by being in contact with the table), with two cubes eight faces are visible and with three cubes eleven faces are visible (Figure 2).

Fig. 2 Row of cubes.

A pupil who then goes on to state that for, say, 50 cubes then 152 faces are visible because there are three sets of faces, each set equal to the number of cubes in the row, plus one extra face at each end, is providing a perfectly satis- factory proof. The algebraic thinking has been carried out, even if it is not written down in algebraic terms.

Trial and error here does not mean "random guesses". For example, suppose you are asking your pupils to insert the numbers 1-9 so that each side of the triangle shown in Figure 3 adds up to the same total.

It is a perfectly reasonable approach to begin by putting the numbers in the circles at random to see what happens. It is what the pupil does next that is crucial. Many pupils, particularly younger ones, will abandon their trial if it does not give a correct solution and will start again from scratch. This is what I call "trial/trial". However, if the pupil adjusts his trial by systematically making use of the amount by which his first trial is in error, then he will "home in" on a correct solution by means of a respectable mathematical technique.

28

Fig. 3 Insert the numbers 1-9 in the circles so that each side adds up to the same total.

The last three entries under the heading of specific strategies - collect data, organise and present data and interpret data - relate in particular to the area sometimes known as "real problem solving". If you ask, for example, "Is it better to travel by bus or buy a bike?" then all sorts of considerations need to be taken into account, such as the cost of fares compared with buying and maintaining a bike, distances to be travelled, enjoyment, weather conditions, etc. Be prepared though for very short answers as from the girl who said "I'm frightened of the traffic so I'd travel by bus." For her, there was no more to be said - any suggestion that she might work out the financial break-even point would become academic, and therefore would defeat the object of the exercise. Nevertheless, problems of this type, where the pupils are required to take real factors into consideration, are an extremely important part of their mathematical education.

But what do we make of the list of general qualities? At first sight they seem a bit vague, to say the least. Again, perhaps, a few would benefit from further explanation.

Working cooperatively means a lot more than pupils checking answers and helping each other out at times of difficulty. Working cooperatively means that groups of pupils should actually develop their insight into the mathematical situation together, expanding and developing their own and each other's ideas. In that way a group of pupils can usually progress much further than any one group member would have been able to on his own. There are times, though, when it is appropriate for a pupil to break away from working with a group in order to develop an idea which is important to him but not to the others, or to try to understand an idea which he has not grasped but the others have. This is one aspect of working independently. The other main aspect is independence from the teacher: with experience, pupils should develop the confidence to explore their own ideas rather than require the teacher to be the directing force.

It is difficult to define what we mean by developing a sense for blind alleys, but we as teachers have all developed one to some extent. When we explore mathematical situations and ask our own questions, more often than not we can tell at an early stage of exploration whether or not we are going to get anywhere. Partly this is due to our mathematical knowledge and partly to the experience we have developed through being in similar situations before. There comes a time with many pupils when they have had some experience with investigations that they become very good at asking their own questions and really need to develop a feeling for which ones are worth pursuing and which are not. (At the same time I am well aware that it is all too easy to miss many mathematical gems by dismissing a line of enquiry too soon.)

Possibly the most important quality that we need to develop in our pupils is the ability - or desire - to reflect on what they have done. They are usually very solution- orientated (our fault?) and once they have an answer they




will resist looking back at the strategy they have used -

they see it as much more important to rush on to something else. However, if strategies are to be developed for future application to new situations, this will be achieved much more successfully if they are made explicit and this can only be done by reflection.

Table 1, therefore, provides a checklist of what I feel we are trying to achieve with investigational mathematics. However, the items on the list are not clearcut - indeed, to a sceptic they would seem very wishy-washy. Nevertheless, they do constitute a very powerful set of investigative tools, and are far more closely related to the heart of mathematics than is the mainly skill-based syllabus so common in schools.

The Teacher's Role

The role of the teacher of investigative mathematics is quite different from the traditional role of the teacher. Tradition- ally, the teacher is the fount of all knowledge. S/He takes total responsibility for the content of the lesson: the topic to be taught, the teaching method used, and the exercises set.

In investigative mathematics, however, the teacher does not control the work in the same way, so let us look at exactly what his role is.

Firstly, the teacher initiates the investigation, but then stands back and lets the pupils get on with it. It is very tempting to try to head off difficulties which the pupils will come up against, but it is a temptation which must be resisted. Far better to wait until the difficulties arise natur- ally and use them as a focus for discussion.

The stimulation of discussion is in fact an important aspect of the teacher's role - the pupils will need to experience the fact that discussion does not just mean comparing answers or helping each other out when they are stuck. Suppose, for example, you have asked your pupils "How many different triangles are there on a 9-pin board?" The question will soon arise whether the two triangles shown in Figure 4 are different.

Fig. 4 Are these triangles different?

Don't answer this question. Rather, allow the pupils to decide for themselves and accept the consequences of that decision. It may be necessary for you to come back to that group later and ask whether they would like to reconsider their decision, and possibly guide them towards an alternative viewpoint, but this should certainly not be done until they have perceived the need for themselves.

It is always difficult to know how much to guide pupils, and there is certainly no definitive answer to this. My own view is that in the early days of investigative work there should be plenty of guidance of all pupils, but with a gradual "weaning off" process. This weaning off process, however, will have to be on an individual pupil basis.

After a fairly short time some pupils will be able to explore quite successfully independently of the teacher while some other, less confident, pupils will lose heart


1 x9=9 2x9=18 3 x 9 = 27 4 x 9 = 36

9 x 9 = 81

0 + 9 = 9

1+8=9 2+ 7=9 3+6=9

8 + 1 =9

Fig. 5 The 9-times table.

unless they are guided much more. My experience is that there is little correlation between the confidence to explore and ability at "regular" mathematics. Indeed, some of those pupils who resist investigative work most strenuously are those who have previously been very successful in mathematics - they are the ones who are quick to pick up skills from blackboard explanations and can apply them to routine exercises but lack confidence in independent thought.

A useful part of the teacher's role can be to coordinate ideas. Generally, you are mobile while your pupils are stationary, so you are in a good position to say to a group, "That group over there are looking at the same thing as you but have a different answer. Why don't you work together for a while?" or, to a whole class, "Simon's come up with a good idea. Would anybody like to follow it up with him?"

It can turn out to be a very exciting experience if you respond to pupils' unexpected comments or questions and use them as a basis for exploration. On one occasion recently a colleague showed his class the patterns in the nine-times table (Figure 5). He intended to bring out two figures: (i) the units decrease by 1 as the tens increase by 1, and (ii) the digit sum is always 9.

At this point, however, Nicola put up her hand and said "Please sir, the next one doesn't add up to 9", to which he replied, "That's interesting, Nicola, when is the next time it doesn't add up to 9?" In fact he had no idea what the answer to that question was, and his planned lesson was abandoned while everyone searched for occasions when the sum of the digits was 18, 27, etc. A fascinating pattern was revealed which was totally new to him.

What is important about this story is not just that those pupils investigated a mathematical situation. Even more importantly than that, they were investigating a question to which their teacher clearly did not know the answer - and had never even realised a fact which was quite obvious to Nicola.

This brings us to the most radically different aspect of the teacher's role, which is to be a fellow-traveller. No longer will you always be the fount of all knowledge. You must be able to answer "I don't know". Not too often (you have to retain some credibility!) but often enough for your pupils to develop a feeling that they are exploring a genuinely worthwhile question.

Many teachers find this role very threatening. They feel that it undermines their authority if their pupils see that they don't know all the answers. In fact the reverse is true: respect rapidly grows if pupils feel that they are being allowed to make a significant contribution to their own education. Nevertheless, I suspect that this threat which is felt by teachers is one of the greatest barriers to the introduction of investigative mathematics.

Investigations in the Curriculum? The mathematics curriculum is already overcrowded.

Are we really suggesting that we put even more into it? In the long term, that isn't the real question. Because in

the long term what we are working towards is the develop-

29



ment of mathematical thinking, with content mainly in- cluded only insofar as it is helpful for the development of that thinking.

We are going to have to look very seriously at the question of how much of the present contents is of any real value. Do we still want, for example, equivalent fractions in Junior 2/3, or ratio in 2nd-year Secondary, or simultaneous equa- tions at all for the majority of pupils? There is a great deal of content, especially in Secondary schools, that could easily be thrown out. Then many of the earlier, essential, topics - which are covered much too soon for most pupils anyway - could be moved on by a year or two. That would leave plenty of space for the development of the thought processes we are talking about here.

In the short term, though, we are a long way from doing that. In the short term we must make space for investigational work and accept that we may not "get through the syllabus". It is vital, though, that we do not feel under time- pressure - that we do not rush, or push pupils towards particular results in order to short-cut the thinking process.

Consider the "insight" curve shown in Figure 6. An essential element of investigational work is the early "sort- ing out" stage, and pupils must be allowed the time they need to go through it. I once had a long talk with a parent trying to explain what investigations were all about. His parting comment to me was "You can waste an awful lot of time thinking - why don't you just tell them?" We must not give in to that sort of pressure.

Insight

Time Fig. 6 Insight curve.

When introducing investigations into a school curriculum, my suggestion is to start in a small way. Set aside, say 1-1 hours once per fortnight - half a morning for a Primary school or one double lesson for a Secondary school. Start with a series of "one-offs" - those which you like and which you think your pupils will enjoy.

After a while you will probably begin to feel that your collection ought to be structured in some way. However, it is not often appropriate to structure investigations in terms of difficulty because so much depends on the pupils' past experience of investigational work. Frogs, for example, could be an early investigation or one which comes much later. The difference would lie in the amount of guidance you gave your pupils and the distance they took their investigation.

Eventually, though, you will probably develop a set of investigations which you are happy with and will present them in a particular order. I suggest, however, that this order will not be based on level of difficulty, but on much more general factors, such as the balance between those which are basically spatial and those which are numerical, and between those which require equipment and those which use only pencil and paper.

30

Assessing Investigational Work If everyone is doing something different, how do we mark it? At the moment there are no clear answers to that question. Some of the questions we need to address our- selves to urgently, though, are:

- How do we assess such qualities as independence, creativity, insight, perseverence?

- Do we really want to assess? - If we encourage pupils to work cooperatively in

groups, is it right that we mark them individually? - Do we assess their work relative to each other, or to

themselves? - Do we assess the effort they've put in?

A large part of what we assess will be the pupils' written work, so here are some suggestions for what I should like to encourage them to write:

- Description of the starting point. - Where did you go from there? - What were your thought processes? - Any conjectures, correct or not. - If a line of investigation is abandoned, try to say why. - In that case, how did you decide what to do next? - Any thoughts or feelings as they occur

e.g. frustrated bored excited determined confused.

Thus, what I feel ought to be written is not just a cleaned- up presentation of results only, but a description of how the investigation developed.

It takes a long time and a lot of persuasion before pupils will write in this way. Nevertheless, I believe that this capturing of the thought processes is a very effective way of making them explicit, and also makes pupils reflect on the processes they have used.

Eventually, we may decide that we need to work out individual markschemes for each investigation, based on the specific strategies and general qualities mentioned earlier. However, most of us are a long way from being able to do that, so here is a fairly crude markscheme which goes some way towards being adequate, at least until we came up with something better:

Content 5 Presentation 5 Understanding 5 Perseverance 5

Total 20

By Content, what I mean basically is "Is this a reasonable piece of work for this pupil, given the time that was spent on it?" Presentation is much more than just neatness; rather it is an assessment of the extent to which the pupil has written his report according to the guidelines laid down above. Understanding is difficult to give criteria for on paper, yet in practice is much less difficult to assess: it usually is quite clear from a pupil's write-up whether he understands what he did or whether he is quoting some meaningless (for him) statement which he has obtained from someone else in the group. Finally, Perseverance does not mean "Did the pupil go away and do a lot of homework?" Pupils often find themselves with several avenues to explore and many will try one, then as soon as it gets difficult try another, and another, without even really getting anywhere. The mark for perseverance is intended to be a measure of whether a pupil explored a particular line of investigation as far as he was really able - possibly never going on to any other aspects of the investigation. Thus we are marking depth rather than quantity.




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Investigations: The State of the Art

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