Kamii, C., & DeVries, R. (1976). Piaget, Children, and Number: Applying Piaget’s theory to the Teaching of Elementary Number. Washington, D.C: National Association for the Education of Young Children. ISBN: 0-912674-49-0
(Guest Post by Alprata Ahuja)
Chapter 2 – Principles of Teaching (pp. 11-25)
The principles discussed in this chapter are implications of Piaget’s theory to teaching particularly to teaching of numbers.
1. “Teach number concepts when they are useful and meaningful to the child”
This principle is derived from Piaget’s idea of Constructivism. The child should be taught numbers when they feel the need of using numbers and contextually understand the importance of numbers like when they are playing games, counting who has more books, counting the presents they have received. Thus, the idea of timetable or allocating specific time for doing Mathematics for the children between four and six years is not an effective idea.
2. “Use language that elicits logical quantification and the comparison of groups”
Children should be encouraged to do logical quantification instead of just numerical quantification. Children should be provided with situations and contexts where children use logical quantification. For eg – contexts which can be explored are “Bring enough cups for everybody at your table, Do we have enough for everybody to have one?, Do we have too many cups?, Do we have more?” (pp. 13). The children can use one to one correspondence, getting a handful or can even use counting to provide answer to the presented situation or context. It is important to understand the just numerical quantification is not enough and child should not be told directly to count as child may not yet have understand the underlying idea of quantification and its linkage to counting. Children should be given opportunity to figure out and establish the relationship.
3. “Encourage children to make sets with movable objects”
To help child understand the quantification of group of objects; exercises or tasks of comparison of sets forms a good mode of exposure.
Comparison of sets can be done either by comparison of two already available group of objects which has possibility of three answers – same, more or less. A more engaging task, helping child understand quantification, is to tell child to make group of objects having same number as already made group. This helps child to think more as in child starts “from 0, then takes 1, then one more and so on” (pp. 14) and child reasons out when to stop. Thus making equal groups of objects help child to understand the idea of quantification deeply.
Some of ideas which do not promote logical quantification are exercises given in books where two sets have to matched, Cuisenaire rods[i], Seriated rods[ii].
4. “Getting children to verify answer among themselves”
Children should be encouraged to give feedback to each other. Conflicting answers and ideas should be discussed in classrooms. A conflicting idea presents a great opportunity to children to be able to re-think their ideas, revise their understanding and justify their thoughts to others and to oneself also.
Teacher should also act as mentor in these situations instead of giving leading hints which points to correct answers. This will promote the wrong idea that teacher is in position of authority and knowledge can be handed from teachers to students only. Thus students should be made to discuss and verify answers among themselves.
5. “Figure out how children are thinking”
For every right and wrong answer given by the child, there is a reasoning behind it, which a teacher should explore. It is not necessary to correct every wrong answer but to understand the child’s reasoning and help the child to correct that reasoning. For eg- sometimes child forgets to count oneself when told to bring enough objects for everyone in the group then a mere hint would help child to include himself or herself in the group.
Also, sometimes reasoning behind correct answer is faulty too so it is equally important to understand the reasoning leading to correct answer.
6. “Encourage children in general way to put all kinds of objects, events and actions into relationships”
It is imperative for a child to put all “things and events into relationships and structures them in his or her life” (pp. 22) to be able to form comprehensive understanding of concepts. A child doing the above is ready to construct number concepts.
It is important to note that “the ability to coordinate the elements in a hierarchical system requires thought which is both mobile and well structured” (pp. 22) [iii].
The implications of Piaget’s theory can be explored by creating class room situations where child can learn numbers by own’s construction.
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[i] Cuisenaire rods have biggest drawback as it connects length with numbers. It has fundamental flaw that length is continuous and numbers are discreet.
[ii] Seriated rods form step wise pattern which provides a leading hint to children to arrange rods of 1 unit then 2 unit and so on, but does not lead to understanding of numbers.
[iii] Refer to ‘The early growth of logic in the child – Inhedler and Piaget (1964)’ to understand further.