ESTIMATION TEACHING PROGRAMME
Textbooks contain far too little material encouraging estimation and no estimation programme has been tried on any sustained or systematic basis. I realise that some of the suggestions which follow will create more work for teachers who are already stretched by the demands of the National Curriculum, SATs, etc. However, if some of these suggestions can be implemented, I feel certain that the pupils will benefit.
The programme should start with pupils in the primary school. The size of a collection of items could be estimated and pupils should, from the very beginning, be encouraged to think in terms of `between-ness' for their estimate i.e. the `Establishing Bounds' strategy should be developed and the pupils will give an estimate range. The earliest estimates will be in the form of `How many objects on the table?'. The teacher should ask the pupil to make an estimate of some quantity. A judgment needs to be made by the teacher as to the reasonableness of the estimate. I believe a reasonable criterion could be "50% of the known answer and, therefore, if the pupil's estimate range overlaps "50% of the correct answer, his/her estimate should be considered correct. This will be a period of time during which the teacher will need to give positive feedback to the pupil without relating the `correct' answer. The pupil should be encouraged to value his/her estimate. Obviously, this task could be undertaken by a classroom assistant but if the pupil does not satisfy the "50% criterion, the teacher will need to attempt to discover the reason for the problem. Once pupils become adept at guessing a small number of objects, within the "50% criterion, larger collections of objects could be attempted.
All estimates, therefore, would be stated in the fashion of "at least X but not as much as Y". The teacher's most important role will be in discussing with the pupils their results and assessing the pupils' progress. I realise this could be time consuming for the teacher but I believe the benefits to be worth it.
The use of the calculator will be an important element in this work and activities with calculators can allow pupils to understand the nature of arithmetic operations. Once the pupils became adept at estimating answers, a calculator could be introduced and the pupils could then be encouraged to use the calculator after the estimate to find a more precise answer.
Consider a programme where pupils were never taught the formal algorithms of addition or subtraction but were encouraged to estimate the results of these operations on objects. One major problem with the algorithmic methods for addition, subtraction and multiplication is that they do not give the `important' part of the answer until the final stage of the operation. It is no wonder that pupils have difficulties with these artificial methods. The formal arithmetic algorithms may be taught but the pupil who can estimate well and use a calculator properly will have the tools for his/her arithmetical needs. However, a challenging activity for pupils might be to develop their own algorithm which gave the calculator answer. They will, of course, know the approximate value of the answer. Estimation of all numerical calculations should become second nature to the pupils and the calculator would remove the tedium from arithmetic. The reality of modern life is that when an exact answer for a numerical computation is required, a calculator usually will be close to hand.
The `Front End' method should occur naturally and pupils can be encouraged to develop the strategies of `Rounding', `Compatible Numbers', `Averaging' and `Adjusting'. All estimates should be given in the form of a range or `bounded'. The pupils will begin to realise that their Front End strategy - for addition - always gives a lower bound. Important facts about the structure of mathematics will develop as pupils use the Front End method. When the pupil is ready, other strategies could be introduced. As a pupil found the Front End strategy to be unsatisfactory, i.e. 1,945 + 1,896 would give 2,000, the strategy of Rounding could be explained. This should only happen when the pupil is desirous of a better estimate. The Compatible Numbers strategy would only be presented when a pupil expressed difficulty in manipulating existing numbers and, by this time, altering a number should not present grave difficulties to the pupil. All strategies previously mentioned could be presented but, at all times, the bounding strategy should be the one upon which one should concentrate. The pupil who persists in using the Front End strategy does not have to be left behind as their strategy does give a reasonable answer preventing most calculator errors.
`Reformulation' and `Translation' will probably not be appropriate for the primary age range but some secondary pupils will be able to utilise these processes with a little guidance from the teacher. Finally, it is important for pupils to realise when an estimate is precise enough for the purpose at hand. This will be a potential area of discussion.
An interesting activity might be for a pupil to guess how often his/her heart beats in 30 seconds and then trying to calculate the heart beats in an hour, day, or even a year. Here it may be appropriate that the pupils check their heart rate, after their first estimate, and then re-estimate the `actual' number of beats. The numbers probably are beyond many primary pupils' level of perception but this may be one way of raising that level.
Computational estimation must, however, never be developed as an algorithm nor it fall into the trap of being yet another part of the curriculum to be separated into a compartment. If this programme is to be successful, pupils must be looking for an estimated answer almost as naturally as they would pick up a pencil.
Every opportunity should be taken from the beginning to develop the habit of estimation in counting, measuring and calculating. Not only does the habit of estimating reveal errors in calculation, but it also indicates pupils' understanding of ideas of number and measurement.
The ability to perceive numbers greater than 100 may be difficult for many primary pupils but they should at least be given some opportunity for determining the order of magnitude of some large quantities and a greater awareness of thousands and millions would become available to some pupils.
Most authors agree that pupils should, initially, be given a great deal of experience of `the unit' to be used in quantitative estimation. This could be the centimetre or the decimeter but could also be the length of their shoe, width of hand, etc. An advantage can accrue from the use of non-standard units initially as pupils will understand the reason for standardisation of units. The estimating process, however, is enhanced by the use of many non-standard units. When pupils have a sense of the unit size, they can be asked to estimate the lengths of various other objects. The strategy which one is trying to impart to the pupils is that of unit iteration. Again, the strategy of Establishing Bounds will be useful and the measurement can be stated in a `bounded' fashion. It must be remembered in organising activities to ensure that the ratio of the unit size compared to the object size is not too small. When pupils become proficient at unit iteration, they will be building a firm foundation for the later use of the fractional and multiple benchmark strategies.
The main criticism that I have of many estimation lessons is that the pupil is required to estimate and then measure but often only with the standard units and only for that one lesson. A variety of units should be available and the checking could be delayed to avoid the estimate losing its value. They will also begin to realise the importance of the unit of measure in performing either an estimate or a measurement.
As the pupils improve and can cope with fractional and multiple benchmark, they can be introduced to the processes of decomposition/recomposition arising from more difficult problems. The estimation programme should include estimates of dimensions that cannot be measured such as the height of a tall building or the weight of a football stadium. The methodology remains the same and if students are interested in `solving' such a problem the working of the problem will be a valuable exercise for them. The major problem that exists is the `check' on such problems but by the time the student has reached this stage in the estimation programme such `checks' can be from a reference source if necessary.
The `errors' which the pupils make can be analyzed for corrective activities. The pupils will also require assistance in determining the level of accuracy which they need to achieve. This is a very difficult part of the programme and the teacher will need to exercise considerable care. However, the estimate must never be required to be of an inappropriate accuracy. The ability of the pupils to determine the required accuracy will be a good indication of how well they have progressed in the programme.
I have used the following technique with pupils to ascertain their confidence in their estimation ability.
The pupil is asked for an estimate which will be one number within the estimate range. Henceforth, this will be referred to as the Estimate. The pupil will then be told that his/her estimate is close enough if the correct answer is between x and y, minimum and maximum allowed values determined by the teacher. For example, if the pupil estimated 50 items in a collection and a "50% criterion was used, then, provided the actual number of the collection was between 75 and 25, their estimate was `correct'. This is opposite to the standard procedure of setting the limits upon the allowable answers based on the actual number of the collection. The pupil will indicate the level of their confidence that the answer was within the two limits on a scale of 1 (highly unlikely) to 10 (a good bet!). Pupils appear to understand and enjoy this `game'.
This technique allows the teacher to have some access to the pupil's level of confidence in their estimating ability. For large magnitude answers, the teacher is encouraged to use the Criterion Of Reasonableness and the technique shown in Figure 10.2 in the main text and the technique I have described could be adapted for pupils to determine the values of x and y - possibly through computer software; some pupils could be encouraged to make their own assessment by using a simplified graph.
It will be apparent that the programme as outlined above will place great demands on the teacher's time but I think that the pupils would gain a better understanding of estimation and this would improve their number sense.