An arithmetic programme for the subnormal pupil

J. R. Locking.
Senior Clinical Psychologist, Monyhull Hospital, Birmingham. 1966

Introduction

This paper gives a short account of a programme which presents the teacher of the subnormal with carefully formulated objectives in arithmetic teaching. It provides the teacher with precise knowledge of what she should be trying to teach at each stage of her teaching programme. The programme is based on the work of logicians who have attempted to 'reduce arithmetic to logic'. This is a legitimate application of logical principles and it enables us to develop a programme for the teacher which is guided by logic. Developing a programme on logical principles must however not be confused with any attempt to get the pupil to apply logic to his work and behaviour.
This of course would be a ludicrous undertaking.
It is important to define each objective or target in teaching arithmetic in very precise terms. Suppose we wish the pupil to perform additions in hundreds, tens and units. A teacher should not be satisfied if the pupil can do this operation by mere mechanical operations. The pupils behaviour should indicate that he knows what he is doing. If he is to add up 13 and 18 the steps should be as follows: Eight ones and three ones make one ten and one one, write down the one, carry one ten. One ten and another one ten and a further ten make three tens. So the answer is three tens and one one. The pupil is required to behave in a certain sequential manner, i.e. he should give a sequence of responses. In order to achieve this we shall have first to teach one component of this fairly complex behaviour and then another component leading to a larger fraction of the final desired behaviour untill at last the objective represented by the addition of 18 and 13 by a meaningful process has been achieved.
This paper is concerned with the stages which are required in order to achieve the final aim of adding up in hundreds, tens and units and making this into an arithmetical operation that is meaningful, ( i.e. the pupil knows what he is doing), rather than a mere routine operation. The main stages will be called

(a) Nominal
(b) Ordinal
(c) Interval
(d) Ratio

(in view of a parallel with Stanley Smith Stevens's theory of measurement (Stevens 1951)
Within these broad divisions the aim will be to produce very small fine steps which follow a proper sequence and which should make it possible to teach subjects who would probably otherwise not be able to understand arithmetical processes. These finer steps have been derived with the help of the work of Willard Van Orman Quine, (a famous American logician well known for his work in the field of symbolic logic.)

It will be the task of the teacher to assess on which stage of the sequence the pupil is functioning. As in Steven's system the sequence is regarded as cumulative. In the present context this means that the pupil is regarded as functioning at e.g. the Interval stage only if he can also perform the tasks of all previous stages, i.e. Ordinal and Nominal. Such an assessment will help the teacher to decide where to start with her teaching, i.e. whether to begin at a higher or lower level.

First Stage--Nominal One
The final aim of this stage in teaching should be to enable the pupil to answer questions of these types:-
(a) How many are there?
(b) Give me the same number of chairs as there are dolls.
(c) Give me 5, 8, or 10 dolls.
In this stage the pupil should be taught to determine the numerical equality of two sets e.g. two blocks and two dolls. He uses at this time the method of one-to-one correspondence and can therefore determine how many there are in any given group.
The general idea of the techniques used in this stage can perhaps be best conveyed by an example.
Show the pupil a set of three dolls' chairs. Then show him a set of three dolls. Ask him to seat each doll on a chair and see that every chair is occupied. He must therefore state that there are as many dolls as there are chairs. (If he is given more dolls or fewer dolls he must state that they are not the same number.) Then, when the teacher tells him that there are three chairs he must conclude that there are three dolls. Such a method is more effective than to simply tell the pupil that e.g. there are three blocks here, there are four blocks here etc. and later have the pupil try to give the correct answers, confirming or correcting as necessary. This would leave the abstraction required (seeing what is the common property of three blocks, three dolls, three chairs etc.) up to the pupil, who may be incapable of such a feat. Instead, the pupil must learn that the name of a group of items ( ' 4 ' or ' 7 ' or ' 3 ' etc) is found by seeing whether the group can be put into one-to-one correspondence with a standard group, when the number of things in the latter group is stated for the pupil by the teacher.
The details of the sequence used in this stage are as follows:-
Situation A (Dolls and Chairs I)
1. The pupil is presented with a set of e.g. one doll and a set of e.g. one toy chair. In this case the one-to-one correspondence is already set up by the teacher because the doll should be sitting on the chair. The pupil must reply appropriately to a question like "Is this doll sitting in a chair?"
2. Increase the size of the sets gradually until sets of any reasonable size may be handled. The questions should now be "Is every doll sitting in a chair?"
3. Then ask the converse type of question "Does every chair have a doll sitting in it?" (Point out chairs which have no doll sitting in them, if any).
4. In the next step the cues "Is every doll sitting in a chair?" and "Does every chair have a doll sitting in it?" should no longer be used. They are now withdrawn and the question is now changed to: "Are there the same number of dolls as there are chairs?" If then the teacher tells the pupil there are 10 dolls or 5 dolls then the pupil must conclude that there are 10 or 5 chairs. (Note that this is the first time that figures have been used).
5. The pupil is next asked to set up the one-to-one correspondence by himself so as to respond appropriately to the question "Give me the same number of chairs as there are dolls", "Give me the same number of dolls as the chairs I have here".
At this point the pupil can establish numerical equality between two sets where the one-to-one relationship involved is that describable by the term 'sitting in'. This however applies only to this one situation. Many other situations will have now to be provided so that the definable relations between corresponding members of the two sets are made broader and broader with the result that the number of situations where the pupil can determine equality of two sets is increased. This will help to make the concept of each individual number broader and broader.
Situation B. ( Dolls and Chairs II )
The teacher should now talk about this situation in terms of the relation of possession - "this doll has this chair". This is a step forward which follows naturally in view of the fact that this relation includes the previous relation of 'sitting in'. It is however a slightly more general relation and hence more useful in determining the equality of sets e.g. a doll may 'have' a chair and yet not be sitting in it.
Situation C. (Party)
Using the same relation 'has', the teacher should now get the pupil to imagine a party situation. For this photos of men and girls can be used . The pupil should observe or ensure that there is one boy for each girl and one girl for each boy. Questions should follow the pattern of situation A.
Note: Since in steps 1-4 of situation A for physical reasons the teacher does not present the pupil with situations where the definable relation involved is not of the one-to-one functional type it is not necessary for the pupil to determine whether or not the relation is one-to-one in order to arrive at the correct answer. In fact he will probably not do so. For example it is natural for just one doll to sit in a chair. In step 5 of situation A the pupil's performance takes place within similar limitations. In situation B and subsequently however the pupil must ensure that the relation is the one-to-one functional one in order to give the right answer since there is a possibility that it might not be. There is a chance that the teacher might present him with such instances or that he may set up such a situation e.g. give one doll more than one chair etc.
Situation D. (Written and spoken number names)
The aim is now to assign spoken and written symbols to objects. The pupil has now to work with a collection of 1 inch square cards, each of which bears a written numeral. The teacher presents the pupil with a row of items and she will now try to induce a pupil to relate each object to one and just one number name, (matching). Although we can describe this as 'counting ', at this stage the process involved is of a very mechanical type, it does not necessarily imply that the pupil will know, for example, that since the numeral 5 is said after three the number five is more than three.
Situation E. (Spoken number names)
In this situation the help provided by written numerals is eliminated. When this has been accomplished the pupil should be able to count without the cue of the written number. In the last phase of this stage the pupil should be able to appreciate the relationship of equality and non-equality between numbers. He should for example be able to say whether any given two numbers are the same or different. He should also be able to "Tell me a number that is different from 3", and so on.
Note: In accordance with Stevens (1951), the arithmetical system may be regarded as a kind of scale by which the numerousness of a collection is measured. The usefulness of that scale to the pupil is determined by the quality of teaching and the variety of situations employed in stage I. If the pupil could determine only the equality of, say, sets of dolls and chairs he would only be capable of applying his knowledge to very limited purposes, no matter how developed were his skills in Stages 2, 3, and so on.
Second Stage--Ordinal I
The teaching target of this stage is the pupil's ability to answer questions such as "Which is more (less) 3 things or 5 things?", "Tell me a number which is less than 7" and so on. This can be achieved in exercises by the following sequence:-
Situation A. Numbers larger than 1. Ist example
1. The pupil is presented with a set consisting of e.g. a girl's photo and a man's photo. He should be able to answer appropriately such questions as "Is this girl a person?", "Is this man a person?" He must then be able to say that "every girl is a person", i.e. he must recognise that the class of girls is included in the class of people.
2 The pupil must then answer correctly the question "What do we get if we take the girls away?"-- the answer should be "the man". (The term 'taking away' here refers to a very concrete operation or relation)
3 At this step in the sequence the pupil should say how many men, (and therefore people) there are by using techniques developed in the previous stage I, (Nominal)
4. Now the pupil should be encouraged to assemble his results. This can be summarised as the knowledge "If you take some away from these people, (girls and men), then there will be one left".
5. The pupil is now told "So you know now that a man and a girl, ( 2 people), are more than one". He can then determine how many things there are here, (again by the method of the first stage), and he knows now that "two is more than one".
Situation B. Numbers larger than 1. 2nd example.
The process is now repeated for another number of things which is more than one, and such that the removal of a sub- class yields a set having one thing, e.g. the set consisting of one girls photo and two men's photos and so on.
Situation C. Numbers more than 2
Repeat procedure A and B for numbers more than 2, more than 3, etc
Third Stage - Interval I
The aim of this stage should be that the pupil is able to answer questions like "How much more is 5 than 2?","What is 7 and 3 more?" etc This is essentially a different way of defining individual numbers. In stage I individual numbers are regarded as isolated classes e.g. '4 ' refers to the class of all groups having just 4 members - 4 blocks, 4 dolls etc. whereas in stage 3 one number is defined in terms of another e.g. 5 is 2 more than 3. Since we have now two independent ways of defining a number we have consequently two ways of identifying a set. This new way enables the pupil to predict the identity of a set, which he can check by the method of stage one, e.g. if he has three things before him and then adds two more he can predict that there will be five, then check the result by actually counting.
The teaching target for this stage can be achieved as follows:-
Unit Increments
A. Sets of Two and One Ist example
1. Pupil is presented with a set of, say, two dolls. The pupil must first simply state that "This (one of the dolls) is a doll".
2. The question is now asked by the teacher "When we takes this away, do we get this doll left over?". (As in the previous stage this 'taking away' refers to a very concrete operation). If the pupil answers appropriately the teacher should proceed to the next step
3. Questions should be answered "How many things are here?" (i.e. left over). This is of course work which relates to Stage One.
4. Summarising. the pupil should now answer the question "If you take a doll away, will there be just one doll left?"
5. Lastly when the pupil has done this he can now be told "So there were two things to begin with".
B. Sets of Two and One 2nd example
The process must now be repeated with a different example of the class of 'two' e.g. a doll and a chair. This has the effect of making the idea of the operation "Take one away from two" as broad as possible, having as members the simpler and more concrete operations "take this chair away", " take this doll away" etc
C. Sets of Three and Two
The same procedure is followed for three and two, then four and three etc Concurrently with learning how to subtract one from any number the teacher can go on to teach how to add one. At Stage One the pupil might regard the various number classes as isolated and unconnected with each other. On this stage however he learns that any number can be transformed to another class of number by including extra items. First he has to learn that some particular example of the number '1' e.g. 1 block, can be converted to one belonging to the class '2' by including (adding) some particular extra item. Next he learns that some particular set belonging to the class '1' (e.g 1 block) can be transformed to one of belonging to the class '2' by including any extra item. Then he must learn that all examples of the number class '1' (1 block, one doll, one chair etc) can be transformed to a set belonging to the class '2' by including any extra item in the set. A suggested way of doing this is as follows:-
Situation A. (Sets of one and two)
1. The pupil is presented with one block. The pupil should be able to say "This is one block".
2. He must then be able to answer the question "If we put another block on what will be here?" The answer should be "This block and this block".
3. The next question is "How many things is this?"
4. Summarising, the pupil should now answer appropriately the question "If you add on the block how many things will there be altogether?"
5. The teacher should then help the pupil to summarise as:- "So one and one more is two" This kind of statement can only represent a very concrete process at this stage of the programme. The teacher must make sure that the extra item put on to the starting group varies, being sometimes a block, sometimes a pencil, sometimes something else. Then the process should be repeated with a different example of the number class 1, (the starting set), e.g. one matchbox. This has the effect of making the idea of the operation 'add one on to one' as broad as possible, having as members the simpler and more concrete operations 'add one block onto a pen', 'add one pencil on to a pencil' and so on.
Situation B. (Sets of two and three)
The same procedure is carried out with sets belonging to two which are to be converted to three, then with three converted to four. This has the effect of making the idea of the operation 'Add one on' as broad as possible. When subsequently all this is repeated for additions of two, three and four etc. the pupil should have a very broad and accurate idea of the operation 'add on'.
Binary Increments
This section deals with the addition and subtraction of two. The method follows similar lines to that used for the addition and subtraction of 1 ( unit increments)
Ternary Increments
This is the addition and subtraction of three. The procedure should be similar to that used above. It does not seem necessary to deal with increments 'greater' than three.
Note. The existence of two independent ways of defining a number as discussed above is very useful to the pupil. This makes the counting of an array of objects an addition-like process and less liable to error. Suppose the pupil has got up to four in his counting of an array. He knows that the next item added on will make five, since he has learnt that the successor of four is five (i.e. 4 + 1 =5). It is true that the pupil will usually be able to give "the next number after four" at the first stage of this scheme. However this is simply a rote association of one number name with another and it does not mean that a pupil knows that "five is one more than four". In the present example the pupil, presented with four things, (e.g. four blocks), is expected to be able to answer the question "What number is one more than this?" This method of solving the problem "What number is one more than four?" is part way between a completely competent 'in the head' solution and a complete inability to solve it even when allowed to manipulate actual objects. It is possible to describe other mixed 'concrete' and 'abstract' methods by the use of special test materials developed for this purpose.
Fourth Stage- Ratio I
The aim of this stage is to understand the times tables. Numerals are used in this stage in a way different to any way so far encountered, i.e. to present the number of times an amount is added to zero, or the number of increments needed to produce a number starting from zero. For example 3 x 4 = 12. This should be understood as a statement that if one starts with 0, and then adds 4 lots of 3 or adds on three 4 times then the number obtained will be 12. Questions appropriate to this stage are such as:-
(a) "How many twos must you add on to nothing to get six?"
(b) "What is nothing and three twos more?"
The teacher should begin first with single additions of an amount, then progress to double additions and so on.
In this phase of the 4th stage the pupil is expected to answer questions such as
(a) "How many twos must you add on to 0 to get two?"
(b) "What is 0 and one more two?"
The following sequence for question (b) will illustrate the method of teaching.
Situation a (Zero, One and One)
1. The pupil is shown a tray on which he can add or subtract blocks as in the previous stages. This time it is empty. He must state "There are no things here".
2. The question is now asked "What will there be on the tray if you add on one lot of one block?"
3. The pupil must then reply to the question "How many things is this?"
4. Summarising, the pupil must answer appropriately to the question "If you add on one lot of one block (or if you add on one block once), "How many things will there be altogether?"
5. Lastly the pupil says, following the teacher,"Nothing and one more lot of one makes one".
The teacher repeats all this with other items beside the blocks, e.g. pens, toy chairs, dolls, coins etc.
Situation b (Zero, Two and One)
The same procedure is followed here as above except that now the single additions are of two things. In the same way the teacher handles zero, one and two, then zero, two and two and so on.
Here the teacher takes the pupil through all the preceding steps, this time with double additions. She starts with zero, one, and two, then goes onto zero, two and two and so on.
Fifth Stage (Carrying)
Whereas in the previous stage such expressions as "10 x 1" were meaningful to the pupil, (who interpreted this statement rather simply and concretely as one lot of 10), in the present fifth stage the pupil must learn to appreciate the meaning of such expressions as "1 x 10 x 1". When taking concretely this means "one lot of 10 additions of one on to 0". The pupil is then able to grasp the significance of the carrying operation. In this case numerals are being using in yet a new way, i.e. to refer to the number of groups of operations. This sounds more complicated than it is. Let us suppose, for example, the addition of 15 and 28. Ideally the process runs as follows:-
(a) 5 ones and 8 ones are 13 ones. (This is the stage called 'Interval 3' below).
(b) 13 ones are 10 ones and 3 ones. (This is also 'Interval 3') As we have already the pupil can, by now organise a set of items in different ways. For example, 12 can be conceived in Stage I (Nominal), simply as 12 objects. At Stage IV ( Ratio I) he can organise it in a more sophisticated manner into a certain number of sub-groups.e.g. four groups of three. Now at Stage V the pupil can impose an even more complex degree of organisation on a collection of objects. He could for example in the above case further organise four lot of three into two lots of two lots of three.
Thus,
12 (first stage) = four lots of three = two lots of two lots of three (fourth stage) (fifth stage)
In the performance of the above two place addition of 15 and 28 the next step would, therefore, be as follows:-
(c) 10 ones are (10 ones) x I, i.e. 1 x 10 x I (This is the fifth stage)
The numeral I is the one which is carried into the next column
The pupils is able at stage "Interval 3" to link together the various numerals in any particular column, to see that they are parts of a total collection, and to obtain the total by performing the appropriate operation. As well as this however the pupil must be able to link together the totals of the separate columns and combine all these into a single all- embracing total. It is the carrying operation which plays a crucial role in the calculation of this 'grand total' and which forms the substance of the 5th stage.
This 5th stage is the last stage in what might be regarded as the backbone of the programme. It consists of the development of number ideas through five stages of one scale. The different stages represent various levels of sophistication of the pupil's characterisation of a collection of objects. The higher the level the greater is the magnitude of a collection which can be grasped adequately and accurately. For example, if the pupil is still at Stage One he will not easily remember the size of a large number of objects or estimate them very well. Being able, at a higher stage, to use a more sophisticated method of specifying the number will make it much easier for him, e.g. by seeing the collection as one of 'four lots of 6 ', instead of seeing them as simply '24'. Similarly the pupil' estimation of the length of a table might be grossly inaccurate if he can only think in terms of just inches or just feet. But if he can combine these two systems then his estimate may not be far wrong.
The following diagram indicates the development of the system described in the forgoing pages and an additional two branches which represent further development. In the centre is the backbone discussed in these pages indicating the five stages Nominal 1, Ordinal 1, Interval 1, Ratio 1 and Stage 5. This can be regarded as a scale representing the various possible levels of sophistication of a pupil's ideas of the size of a collection. The two branches to the left and right can be regarded in a similar fashion. The first branch to the left of the diagram represents the possible levels of sophistication of the pupil's ideas of the size of an interval. At the first stage of this branch each interval is a class independent of any other e.g. the interval represented by the numeral three is a class having the exemplars 4 to 7, 7 to 10, 0 to 3 etc. In this case the individual intervals are simply given names and, therefore, it represents the nominal stage of this first branch. It is identical with the stage called 'Interval 1'. In the next step one interval is seen to be greater than another e.g. the interval from 4 to 7 (3) is greater than the interval from 2 to 3 (1), and so on. This therefore represents the ordinal stage of this branch .
More directly important from the point of view of the ultimate objective is the second branch on the right-hand side starting from the stage called 'Ratio 1'. This, the second branch, may be regarded as a scale which develops the idea of the size of a ratio. Whilst the nominal stage of this branch is identical with the ratio stage of the backbone, the next stage (ordinal 3), is a further development. Here the pupil must respond appropriately to questions such as:-
1. "Which is more/less, 3 twos or 5 twos?"
2. "What must you do to 2 twos to get 5 twos?", (where the answers required is "add-on some more twos"
3. "What number of twos is more/less than 2 twos?"
This is followed by the interval stage ('Interval 3'), which requires appropriate answers to questions like:-
1. "By how many threes is 5 threes more than 3 threes?"
2. "What must you do to 1 three to get 3 threes?", (where the answer required is "add-on 2 more threes"
3. "What is 2 twos and 3 twos more? How many twos will there be altogether?"
Concluding Remarks
The following points are worthy of note: 1. The questions offered as being appropriate to various stages will at times be very complex. As mentioned in the introduction the verbal expression might not be a valid indication of, and means of teaching, the relevant knowledge. The questions can be broken down into parts. It has been necessary to make the meaning as precise as possible by using this rather complicated phrasing but in actual practise the question can be rephrased and repeated as often as one would like.
2. Connected with this is the point that a pupil may use concrete material, (blocks etc), to begin with as an aid in answering the questions. Later of course the teacher should try to get him to do the required tasks 'in his head'.
3. Such 'in the head' performance may be facilitated by employing a particular kind of spatio-temporal symbolism where spatio-temporal relations between numerals symbolise arithmetical ones between numbers, e.g. the number '5 ' is uttered, in counting after the numeral '2'. This can represent the fact that 5 is more than 2. Similarly, to answer appropriately to the question "What is the number which is one more than three?" the pupil may give the correct answer by giving the numeral which comes just after three, (or is the next number after three). A final example can be taken from the ordinal stage of the first branch which has its first (nominal) stage coincident with the interval stage of the backbone. At this stage the pupil can judge the relative magnitude of two intervals. In spatio-symbolic terms, using an actual numeral line, or imagining this 'in his head', he can answer questions such as: "Is '6' closer to '1' or to '8'?" (Here the terms under single quotes represent numerals or number symbols.)
A similar procedure is appropriate for the teaching of linear measurement, volume, weight, money etc. At the present time a detailed programme for the teaching of what might be called 'elementary economic behaviour' is being developed. This starts where the present programme leaves off, i.e. at the point were the pupil's judgment of the number of discrete items in a collection, (the measurement in terms of hundreds, tens and units), of what Stevens calls 'numerosity') is at an appropriately high stage of development. The aim thereafter is first to develop the pupil's judgement or measurement of other attributes of collections which are relevant to the determination of the price of such collections, e.g. weight, length, type of article, then to develop along similar lines the attribute called 'economic worth' (which is measured in terms of pounds, shillings and pence), and finally to develop the appropriate dependence of this latter attribute upon the others, e.g. the price of a set of articles depends partly on their number.
References
1. Stevens, S.S. (1951) Handbook of Experimental Pychology (1st chapter) New York: Wiley.
2. Quine, W. V. O. (1961) Mathematical Logic, Cambridge, Harvard University Press

Note 1
As is well known, basically multiplication consists of repeated additions, and division of repeated subtractions. So the skills, and the understanding, of multiplication and division, come after those of addition and subtraction.
So, for example, in multiplying 3 by 4 we are asking what is the result if, from zero, we add one three, and then add on another, (a second) one three, and then another, (a third), one three, and finally a fourth one three? (Or the other way round, adding three lots of four) In division, an example would be dividing 8 by two. We subtract two from 8 and get 6, (one subtraction), subtract two from 6 and get 4, (two subtactions), subtract two from 4 and get 2, (three subtractions), and finally subtract two from 2 and get zero, (four subtractions). So 8 divided by two = 4
This last points up the fact that in both multipication and division, at the ratio stage of understanding, we need the concept of a real zero.
Note 2
These processes and operations can be done with actual concrete items, e.g. blocks or beads, to represent operations with abstract classes of classes. It is useful when doing this to place the group of items in the group to be added on to, on a small piece of paper, to make it clear just what this group is.
(This use of counters might be compared with the child's use of toys, e.g. dolls and toy soldiers to 'work out' phantasies.)
With practice these operations can be done internally, in imagination, 'in the head', using visuo-motor images.
Here we can use our special form to represent various combinations of 'out there', actual, real performances with counters, and 'in the head' operations.
Then again we might use the visuo-motor imagery of the numeral line, (a more abstract idea), and get the pupil to perform arithmetic operations with the aid of this, firstly with the line actually present, and later with the pupil using internal visuo-motor images.
This numeral line, in the early stages, is straight, and simply represents relations of 'more than', 'so much more than', etc. but later can be bent on itself to represent place symbolism, or even arranged in a helix etc.
This number line might be applied to fine motor processes, but it is sometimes better to embody the concept in big, gross motor processes, as in our experimental teaching with Lucy K., at A.H.S., in which we placed written numerals on the large main hall staircase, and transformed the stairs into the number line.
Also the special ruler
See also the material on conservation of discrete quantity, of countable items

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