This article throws light upon the eight major groups involved in the process of reasoning development of a child. The groups are: 1. Grouping I: Additive Composition of Classes 2. Grouping II: The Secondary Addition of Classes 3. The Bi-Univocal Multiplication of Classes 4. Co-Univocal Multiplication of Classes 5. Addition of Asymmetrical Relations 6. Addition of Symmetrical Relations and Other Details.
Grouping I: Additive Composition of Classes:
It describes the organisation of classes among a set of nested classes. By nested classes, we mean that each class is included in the next larger class until we come to the largest class which includes all the numbers of the set. (Fig. 9.2).
Here, B is the result of A + A’; C, of 5 + B; D is the additive composition of C and C’. The inverse relationship may also be followed subtracting C from D to get C; B’ from C to get B; A’ from B to get A. In such a grouping, there may be working a number of elements as identity elements.
On the one hand, the union of any class with itself, is equivalent to the class itself, that is, A + A is equal to A. If zero is added to any class even then the group will remain y unchanged. Such sort of organisation—does not fulfill the conditions required for the formation of a group. The elements which lie in vicinity to each other can only be combined together—only A and A’ can be combined together.
Likewise, A and B can be combined. Piaget in his research has thoroughly studied this additive grouping of classes. The child’s understanding of all the implications of this organisation of classes, develops only gradually.
Reasoning Development: Group # 2.
Grouping II: The Secondary Addition of Classes:
“Whereas grouping I covers the relationships of inclusion of one class into the superordinate class that contains it. Grouping II is concerned with the relationship between alternative divisions of the same class.” For example, mammals as a whole make a class which can be divided into dogs and non-dogs. Piaget describes this algebraically as A1 (dogs) plus A’1 (non-dogs) = B (mammals).
The class of mammals can also be divided as cats and non-cats. There is an equivalence between the sum of two classes: A1 + A1‘ = B and A2 +A’2 = B. Each sum represents the class of complete mammals, (see Fig. 9.3). These equivalences have been called vicariences by Piaget.
It is through reasoning, which develops along with maturity that the mind searches out such vicariences, and develops logical organisations of mutual relationship.
Reasoning Development: Group # 3.
Grouping III: The Bi-Univocal Multiplication of Classes:
Piaget describes four groupings that involve classes. The first two involve addition of classes; the second two multiplication of the classes. The first two groupings may be red [A] and non-red [A’]. At the same time they may be separated into square [B] and non-square [B’]. In each case, the set is total—A + A’ or B + B’ is total. The relationship between A versus A’ and B versus B’ is different from the relationship between A1 (dogs) versus A1‘ (non-dog); and A2 (cats) versus A2‘ (non-cats) described in the above section; because A1 (dog) and A2 (cats) are mutually exclusive, while A and B are not. We can thus multiply (A – A’).
Here, A B represent objects which are both red and square; AB’ correspond to objects which are red but not square. A’ B’ describe the objects which are neither red nor square. “Thus, by multiplying a dichotomy, we can divide the total set into four sub-classes, A + A’ is equal to the total set; B + B’ is also equal to the same total.
Considering red (A) and non-red (A’), and square (B) and non-square (B’) as elements of two separate classes, we can see that the separation (A versus A’) is connected to the separation (AB versus A’ B versus A’ B’) by a relationship of multiplication by (B versus B”).
The separation can be multiplied further by (C versus C) and we can obtain (ABC versus A’ BC versus AB C versus A’B’C versus ABC versus A’BC versus AB’C versus A’B’C). In this way, a whole series of separation can be arranged in a sequence, (see Fig. 9.4).
These multiplications are reversible; we can remove the classification (B versus B’) from the fourfold classification (AB versus AB’ versus versus A’B’), and we shall have the original separation (A versus A’). Thus, this whole set of multiplications is the example of an organisation, where there are consistent relationships among the elements.
Piaget defines such an organisation as a grouping. It is, also, a concrete example to show how we reason out relationships among elements of the same grouping, for further development of our cognitive ability, and for resolving a problem which we may be facing. These relationships are logical relationships, the relationships between one division and a finer division.
Reasoning Development: Group # 4.
Grouping IV: Co-Univocal Multiplication of Classes:
The grouping III differs from grouping IV, in the sense, that while in grouping III, the multiplication of red versus non-red by square versus non-square leads to a division containing all four classes; red square versus red non-square versus square non-red versus non-square, non-red.
It is not always the case in grouping IV, where the multiplication of two attributes does not result in all the four classes. For example, if we multiply the division (dogs versus non-dogs) by (cat versus not-cats), we find there are no animals which are both cat and dog.
For Piaget, grouping IV represents a logical completion of the grouping of multiplication. This grouping includes cases where each class in a classification corresponds to differing numbers of classes in another. We are reproducing a matrix here as given by Piaget to make the pattern of grouping clear.
This is grouping IV, based on co-univocal multiplication of classes. Here, classification K1 has the classes A, B, C, included in it. The classification on K2 includes the class A2 A2‘ and B2‘. When these are multiplied together, we get the only classes as shown in the triangle on the lower left-hand side of the matrix. There is correspondence between the numbers of categories in the two classes, that is, K1 and K2.
To realise the classification concretely, we should take A1 as the class of sons of a particular person P. B1 is the class of grandsons of that person. And C1 is the class of great-grandsons of P. This is the K1 division of the male members of the family tree. Another classification is A2, A2‘ and B2‘.
A2 is a class of brothers; and A’2 is a class of first cousin of people in A2. It is implied that A2+ A’2 = B2, the class of males who are children of the same grandfather. Class A’2 consists of these males who are grandchildren of the same person; but who are not members of the same class A2.
They are remainder of the class C2who are not in B2. The class C2 is made up of great-grand-children of the same person. And, when these two classifications K1 and K2 are multiplied together, we find that all those in A1 who are sons of the same person, are all brothers. People, making B1 may be either brothers (A2) or first cousins (A’2) or second cousins, (see Fig. 9.5).
Such sort of classification may, no doubt, appear quite arbitrary. Nonetheless, Piaget considers it to be a logically necessary grouping for the cognitive development of the child.
Reasoning Development: Group # 5.
Grouping V: Addition of Asymmetrical Relations:
Grouping V to VIII are all concerned with relationships, rather than with classes as we have seen to be the case in groupings I through IV. There is one-to-one correspondence between the four groupings of classes with the four groupings of relationships—grouping V corresponds with grouping I.
For understanding the logical significance of these groupings, we shall have to consider the different kinds of relationships and the names that have been given to them in logic and mathematics. A has relationship with B, and the same is shown through the notation AgB.
The terms symmetric and asymmetric are used to show relations. The relation of sibling is symmetric; for example, if A is a sister or brother of B, then B is also a sister or brother of A.
This relationship may also be shown as a relationship of equality. If A is equal to B, B is also equal to A. A relationship between an object and its element is called the reflexive relationship. When we say A and B are the sons of the same father, we are showing the reflexive relationship.
Unequal relationship can never be reflexive, though symmetrical it may be. When we say A is the son of the same father as B, then the relationship is reflexive.
“Grouping V describes relationship that are asymmetrical between two or more individuals, objects or classes. The analogy between grouping V and grouping I, stems from the fact that the sizes of the classes is a nested set, described in grouping I form, an asymmetrical series in which each member is asymmetrically related to other members of the series”.
The number of members in class A is ii the lowest. The class B has greater J number of members. Then the class C has still greater number of members, s The next is class D which has the i greatest number of members in the ; existing four classes (see Fig. 9.6).
The classes may be arranged from any dimension—weight, size, and may be asymmetrically arranged. This asymmetrical arrangement may also be read as ‘C, is smaller than ‘D’; ‘S’ is smaller than ‘C; ‘A’ is smaller than ‘B’.
A series of objects may, thus, be arranged in order of size. New elements may also be fitted into such a series. Arranging objects in such a series, is a cognitive operation which needs the knowledge of asymmetrical relationship existing between the objects. Such an operation forms a grouping of the same type, as it happens in case of the nested sequence of classes in Grouping I.
Movement is possible both from A and B, from B to C, and so on, in an onward way, as well as in an inverse way from D to C, from C to B, and so on, as each one is related to its next in a nested sequence. In such a sequence, A is less than B, and, B is greater than A; relationship between each adjacent duo, is also of the same nature.
Piaget has pointed out the difference between the inverse relationship that exists in grouping I and that in grouping V. Grouping in I happens to be based on the operations of subtraction or addition-class B is reduced to class A when some of the members of the former are taken out, reducing the class into A.
This relationship is based on negation. But relationship in grouping V, is reciprocal, when we say A is less than B, reciprocally, B is greater than A. “These two different kinds of inverses ‘negation’ and ‘reciprocation’, are important for some of Piaget’s conceptualizations about the thinking of older children in the stage of formal operations.
At that time, the child is capable of understanding an organised set of relationships in which both kinds of inverses exist within the same organisation. The combination of the two makes it possible to define a special group called the “4-groups.”
For the development of a true number system, conceptualization of the relationship between grouping I and grouping V, must have been the basis. For example, the cardinal number 7, is a label for any set containing seven members. 7 describes a class that is included in 10 in grouping I.
“That 7 is less than 10, is established when the two sets are juxtaposed, and cardinals in the first are exhausted at 7 while in the second set there are still more cardinals up to 10. Both groupings I and V are involved when numbers are used ordinally as points on a line to form an asymmetric series.
For the establishment of a unit, we move from an ordered series to a number system of integers. A set of five is transformed into a set of six, when one unit is added to the former set. This operation may be repeated over and over again. The numbers form a true group rather than a mere grouping, on the patterns of groupings I and V, the classes are given cardinal numbers.
Reasoning Development: Group # 6.
Grouping VI: Addition of Symmetrical Relations:
Preliminary grouping of equalities, illustrates cases of symmetrical relationships. Grouping VI, also, illustrates relationship within a genealogical tree. If “B” has the same grandfather as “5”, we can build up a hierarchical set as was illustrated in grouping IV.
“These relationships can be combined; thus, if X is also a brother to F; and 7 is a first cousin of Z, then X is a first cousin of Z. Thus, there are logically necessary relations among these different kinship terms and altogether they constitute an organised system of relationship—which Piaget describes as Grouping VI.”
Reasoning Development: Group # 7.
Grouping VII: Bi-Univocal Multiplication of Relations:
“This grouping is quite important as it describes the kinds of relationships that can exist when objects are ordered asymmetrically with respect to two attributes at the same time” (see Fig. 9.7).
In the Fig. 9.7, there are a series of rectangles; the two attributes are the width and the height—in respect to which the rectangles have been ordered asymmetrically. One dimension of looking at the relationships of these rectangles, is horizontal-B1 is wider than B2, B2 is wider than B3 and so on.
It is also possible to rearrange the rectangles in such a way that the widest one is in the first column, the next widest in the second column, the third widest in the third column, and so on.
It is not always the case that the width decreases evenly from column to column. Likewise, the rectangles can be arranged in an asymmetrical sequence from the dimension of height. The elements in the first row are taller than those in the second; the second are taller than those in the third, and so on down to the shortest.
When the two sets of relations are multiplied together, “We get a matrix in which there is decreasing width as one move from left to right across the table and decreasing heights as one move from the top down through the table.
If such sorts of relationships are appreciated by a child then, he or she can understand the problems of invariance. Understanding such a relationship, the child can realise that the amount of liquid will remain invariant, when it is changed from a wider pot to a narrower one, as the latter is much taller in height than the former.
Piaget, in a very important experiment, asked a child to arrange a number of dolls in a series, in order of size and then, to arrange cones in a series, in the same sequence so that each doll will have its due place in order of size, vis-a-vis each cone, again, arranged in order of size.
“This represents the multiplication of two asymmetrical series” where “the diagonal cells in the matrix are occupied. The tallest doll (A1) has the tallest cone (B1); the next tallest doll has the next tallest cone, and so on.”…..no complete matrix but rather a series of diagonal cells running from the upper left to the lower right.”
Piaget contends, that putting two kinds of elements in correspondence with each other, when each is arranged in series of its own individual numbers, requires the understanding of the possibility of multiplying one asymmetrical series with another.
In another research, the child is shown a matrix arranged in this fashion with certain cells missing. The child is asked to fill in the blanks looking from both the dimensions, horizontal as well as vertical, as to what would fit in that particular cell. The correct response by the child, verifies that he understands the essentials of this kind of multiplication of asymmetrical relationship.
Reasoning Development: Group # 8.
Grouping VIII: Co-Univocal Multiplication of Relations:
Co-univocal multiplication of relations is to grouping VII what grouping IV is to grouping III. A1 is the grandfather of A3—this is one dimensional relationship. A1, A2 and A3 make an asymmetrical series. Grouping VIII is concerned with the multiplication of such relationships, when we say B2 is the brother of B1, B3 is the first cousin of B1 etc.—we are referring to some other dimension. This kind of relationship can also be predictably multiplied together.
If A is the father of B, and B is a brother of C, this implies that A is the father of C also. The other dimension is, if A is the father of B, and B is a first cousin of C, then A is an uncle of C; combination of these two kinds of symmetrical and asymmetrical relationship, forms a logical organisation which Piaget describes as Grouping VIII.