The man who thinks backwards

G.K. Chesterton’s essay “The man who thinks backwards” is a useful allegory for the contrast between the Intellectual Mathematics teaching philosophy and its more prominent counterparts. I begin by quoting it at length.

The man who thinks backwards is a very powerful person to-day: indeed, if he is not omnipotent, he is at least omnipresent. It is he who writes nearly all the learned books and articles, especially of the scientific or skeptical sort ...

Thinking backwards is not quite easy to define abstractedly; and, perhaps, the simplest method is to take some object, as plain as possible, and from it illustrate the two modes of thought: ... I will take a poker and think about it; first forwards and then backwards; and so, perhaps, show what I mean.

The sage desiring to think well and wisely about a poker will begin somewhat as follows: Among the live creatures that crawl about this star the queerest is the thing called Man. This plucked and plumeless bird, comic and forlorn, is the butt of all the philosophies. He is the only naked animal; and this quality, once, it is said, his glory, is now his shame. He has to go outside himself for everything that he wants. He might almost be considered as an absent-minded person who had gone bathing and left his clothes everywhere, so that he has hung his hat upon the beaver and his coat upon the sheep. ... This external need of his ... has lit in his hand ... the red flower called Fire. Fire, the most magic and startling of all material things, is a thing known only to man and the expression of his sublime externalism. ... But there is about this generous and rejoicing thing an alien and awful quality: the quality of torture. Its presence is life; its touch is death. Therefore, it is always necessary to have an intermediary between ourselves and this dreadful deity; to have a priest to intercede for us with the god of life and death; to send an ambassador to the fire. That priest is the poker. Made of a material more merciless and warlike than the other instruments of domesticity, hammered on the anvil and born itself in the flame, the poker is strong enough to enter the burning fiery furnace, and, like the holy children, not be consumed. In this heroic service it is often battered and twisted, but is the more honourable for it, like any other soldier who has been under fire.

Now all this may sound very fanciful and mystical, but it is the right view of pokers, and no one who takes it will ever go in for any wrong view of pokers ... He who has thus gone back to the beginning, and seen everything as quaint and new, will always see things in their right order, the one depending on the other in degree of purpose and importance ...

This is thinking forwards. Now our modern discussions ... are all entangled in an opposite train of thought, which runs as follows: A modern intellectual comes in and sees a poker. He is a positivist; he will not begin with any dogmas about the nature of man, or any day-dreams about the mystery of fire. He will begin with what he can see, the poker; and the first thing he sees about the poker is that it is crooked. He says, “Poor poker; it’s crooked.” Then he asks how it came to be crooked; and is told that there is a thing in the world (with which his temperament has hitherto left him unacquainted)—a thing called fire. He points out, very kindly and clearly, how silly it is of people, if they want a straight poker, to put it into a chemical combustion which will very probably heat and warp it. “Let us abolish fire,” he says, “and then we shall have perfectly straight pokers. Why should you want a fire at all?” They explain to him that a creature called Man wants a fire, because he has no fur or feathers. He gazes dreamily at the embers for a few seconds, and then shakes his head. “I doubt if such an animal is worth preserving,” he says. “He must eventually go under in the cosmic struggle when pitted against well-armoured and warmly protected species, who have wings and trunks and spires and scales and horns and shaggy hair. If Man cannot live without these luxuries, you had better abolish Man.”

Indeed, the world is full of curriculum designers and textbook authors who want to abolish fire in order to have straight pokers.

The Intellectual Mathematics teaching philosophy is based on thinking forwards. It is based on understanding and respecting the purpose of things, rather than letting some dictatorial know-it-all uproot all natural order because it straightens out the world according to his narrow field of vision.

In forward-thinking Intellectual Mathematics you introduce things only when their natural place and purpose can be understood. In backwards-thinking traditional mathematics teaching you hubristically and senselessly pontificate on the properties of objects without any understanding of what the natural purpose and motivation of these objects are in the first place.