The traditional media have been making a bit of a “do” (the American English usage of the word, which connotes a commotion and derives from the term “ado”, not the Oriental usage of the word, which connotes a “way” or path of thought and action) about how maths should be taught. [Link] Evidently teaching kids inductively – from specific examples to general principles – has been indicated in a recent study by folks at Ohio State U to be less effective than teaching deductively – from general principles to specific examples.[Link]
I had been rather concerned at first glance that the study had been conducted by mathematicians since the general view of mathematicians in my experience has been that any maths done, used, or taught by other than a mathematician under the purest of circumstances is at best mediocre and usually a profound and pustulant evil. This is not to say that what mathematicians have to teach in their preferred “theorem-proof” formalism is not valuable, but if it were not for applied mathematicians of various persuasions, including physicists, we should still be at the scientific and technical level of 500 BCE classical Greece but with modern mathematics. Or at least the parts of it that are not beholding to instrumentality like the four color proofs and the like.
The effort however was performed by the Cognitive Science Center at Ohio State U, which while it brings up other concerns, namely that any discipline that has to put science in its name isn’t science, at least alleviates concerns that the effort was framed in the “theorem-proof” view. Rather it would seem that the view is a bit more mundane and applied.
This result is not really surprising. In some sense we may anticipate the result from noting that the way applied maths are taught in the sciences is rather less inductive than it is in the engineering disciplines. This is one of the reasons why scientists can often adapt to engineering problems rather more easily than engineers can adapt to science problems.
Of course, this does not address why mathematicians are not also adaptable to science problems than scientists to maths problems. The reason for this has to track back to one of my earlier blots [Link] where I talked about how one learns a new discipline. I have since evolved that explaination to what I call the Four T’s:
- Thoughts – the ideas associated with the discipline;
- Terms – the words used to express those ideas and their application;
- Tools – the means by which the discipline is practiced; and
- Techniques – how the other three T’s are combined to practice the discipline.
The reason why mathematicians can’t always adapt to the sciences and engineering discipline is because their first two T’s are different.
But what this study has cemented in place is that for many humans the ability to generalize is limited. In effect, if one teaches maths using word problems dealing with trains, the students may be incapable to generalizing to any other moving object. Not only does this demonstrate much of what is wrong with our current educational system, but it sadly also explains what is wrong with much of human society in general. As vaunted as our intelligence is, it is still limited and we are sadly lacking in capability to deal with reality.
1. Induction = The act or process of reasoning from a part to a whole, from particulars to generals, or from the individual to the universal; also, the result or inference so reached. [1913 Webster]