For a long time, a side interest of mine has been how people think when they are doing mathematics. Two difficulties in investigating this general question are (i) that it is quite hard to examine one’s own thought processes reliably (since very often what one remembers of these processes after solving a problem is a very tidied up version of what actually happened) and (ii) that in any case I am just one mathematician with my own particular style and my own little bag of tricks.
I would therefore be grateful to anyone who was prepared to spend about 90 seconds contributing to a little experiment. What you have to do is solve a simple equation that appears just after the fold and pay close attention to your thought processes as you do so. Once you have done that, you can look at further instructions about how to record the result of your participation.
I stress once again that the whole thing should be quick and easy. But please don’t look at the equation until you are ready to start thinking about it and remembering the sequence of your thoughts, since otherwise there is a danger that the tidying-up process will take over and it will be impossible to get a reliable result.
The equation is this.
Now there are at least three approaches to this equation.
1. The blindingly obvious approach: cancel from both sides and then solve the resulting linear equation.
2. Rewrite the left-hand side as and then see that has to be , and in fact must therefore be .
3. Rewrite the equation as and factorize the left-hand side as , exploiting the fact that it is a difference of two squares.
Now once you’ve thought of 1, anything else seems silly, and once you’ve thought of 2, 3 seems a bit silly. But there’s no disgrace in having silly thoughts — it’s an important part of doing mathematics — and the whole point of this experiment is to get some idea of the extent to which we have them.
To record what happened to you, I have a poll. I have represented the various possibilities by numbers or sequences of numbers. Each number corresponds to the numbers above, where I listed the approaches, and a sequence of numbers such as 31 would indicate that your first thought was to use 3 but then you abandoned that thought after noticing 1. I have listed only decreasing sequences, since my expectation is that people are unlikely to abandon an approach for another approach that is (as I see it anyway) strictly sillier. But if you have had an interesting experience that is not covered by the options in the poll, then there is always the possibility to leave a comment.
Please try hard to remember exactly what happened. If you very briefly thought of doing 2 but almost immediately realized that you could do 1 and that 1 was better, then your answer should be 21 rather than 1, even if you didn’t at any stage feel a commitment to 2. It’s your fleeting thoughts that interest me.