Comments by W. W. Sawyer, in his wonderful, little book, Mathematician’s Delight:

Earlier we considered the argument, ‘Twice two must be four, because we cannot imagine it otherwise.’ This argument brings out clearly the connexion between reason and imagination: reason is in fact neither more nor less than

an experiment carried out in the imagination.

People often make mistakes when they reason about things they have never seen. Imagination does not always give us the correct answer. We can only argue correctly about things of which we have experience or which are reasonably like the things we know well. If our reasoning leads us to an untrue conclusion, we must revise the picture in our minds, and learn to imagine things as they are.

When we find ourselves unable to reason (as one often does when presented with, say, a problem in algebra) it is because our imagination is not touched. One can begin to reason only when a clear picture has been formed in the imagination. Bad teaching is teaching which presents an endless procession of meaningless signs, words and rules, and fails to arouse the imagination.

I’ve been thinking about this quote as I prepare workshops for the homeschooling convention this spring: How can I spark the imagination of my workshop parents, and how can I help them spark the imagination of their students?

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Imagination comes first because clearly without imagination, what we know is worthless!

Many of the greatest inventors were driven by wildly imaginative minds and not just mathematics or even knowledge in general.

Knowledge is just detail; the method one uses to calculate something is only important when they need to calculate it, and will always have its place in a text book or Wikipedia or wherever.

In my opinion, many people fear, and hate mathematics because it offers little to an imaginative mind; mostly endless ugly equations and countless monotonic methods.

During my school days, the best math teachers, always taught like it was art class and the worst art teachers always taught like it was maths class.

I think that was a great idea, for the teacher to teach math like art. Also i think it was better for the student’s in her class that was thinking math is boring.