Mathematics Education: Taking a Clue
From the Recent Technological Revolution
Do We All Think Similarly?
Do we all have equal abilities for mathematics? Of course not. This is common knowledge. What's interesting is that apparently even people in the same profession, more relevantly, even mathematicians have their brains organized differently. Henry Poincaré says:
It is impossible to study the works of the great mathematicians, or even those of the lesser, without noticing and distinguishing two opposite tendencies, or rather two entirely different kinds of minds. The one sort are above all preoccupied with logic; to read their works, one is tempted to believe they have advanced only step by step, after the manner of a Vauban who pushes on his trenches against the place besieged, leaving nothing to chance. The other sort are guided by intuition and at the first stroke make quick but sometimes precarious conquests, like bold cavalrymen of the advance guard.
The method is not imposed by the matter treated.
Let me then state for the record that different minds function differently, and, therefore, subjecting all students to the same instructional approach can't be good for everyone.
Here is a support from the educational literature [McKillip, p 29]:
Some children will quickly abandon concrete materials as they do symbolic exercises; other children will need to use concrete materials along with symbolic exercises. Because children may be at different stages of development they may need to perform mathematical operations in quite different ways.
The first sentence in the passage states a fact which, as I surmise, was observed in practice. In the second sentence, the authors provide their explanation for the fact just stated. The differences in thinking among children are explained by differences in developmental stages. I accept the fact but have reservations about the explanation. Especially because a little later on the same page the authors write:
Those children who does not grasp the process on an abstract level continue to make errors of almost the same kinds right up through the grades despite massive amounts of practice.
From the NCTM Standards
...Inherent in this document is a consensus that all students need to learn more, and often different, mathematics and that instruction in mathematics must be significantly revised.
- W.D.McKillip et al, Mathematics Instruction: EarlyChildhood, Silver Burdett Co, 1980
- H.Poincaré, Intuition and Logic in Mathematics, in Mathematics Teacher 62 (1969) 205-212.
- F.J.Swetz, From Five Fingers To Infinity, Open Court, 1996, Thrid Printing.