Focused and Diffuse: Your Companions For Mathematical Thinking

Whenever you approach a piece of text, whether it is a novel, essay or if you are really unlucky, a textbook article, you are approaching that piece of text with a mind engrossed in whatever captivating content it was previously attending to. That is why it can be useful, as a mode of organisation, to quickly glance through the text to prime your mind for what it is about to perform.

The brain is on one side unbelievably capable and on the other frustratingly fickle. Plans made in one state of mind can be destroyed in another. I want to stop myself from eating swiss chocolate cake but then hey, did you see how nice that looked?. Since the beginning of the 21st Century, neuroscientists, hooking people up to complicated machines that are able to map brain activity, have made substantial advances in understanding the two types of neural networks that the brain seems to switch between. These two states are highly attentive and the resting . For the purpose of the discussion lets transform this scientific jargon into conversational colloquialism, and call these two states the focused mode and the diffuse mode.

For mathematical thinking, the ability to rationally ponder, sequentially dissect and analytically investigate a problem, the focused mode in a necessary companion. This mode of the brain, masked inside the frontal lobes sitting right behind your forehead, requires paying attention. It is the flashlight of the brain, it’s targeting system; highlighting the signal and discriminating the noise.

So, thats it, like your teacher always said “Pay attention and you’ll be fine”. But wait, are we to ignore the other state, the second mode? Does it play no part in our climbing of mathematical mountains? Should we leave it at the gate? Not exactly. Focus is all good and well, it allows us to concentrate, to think, to zone in on an issue, but by the very naturing of zoning in we forget the things that may have left out. Can you zone in on a problem whilst at the very same time keep in mind why it is that you are doing it? Where you are going with it? Where it fits in? I doubt it. This is why we need the second mode, we need to take a rest. The diffuse mode, the power to stop, to take stock, to step back in order to make sure you are going forward, gives us insight into a problem, helping us to see the wider picture.