In 1784 in Germany, a seven-year old boy, Carl, was enrolled in a typical public school, one of those old schoolhouses where hundreds of poor pupils of varying ability congregate, and are taught by a disagreeable teacher whose primary tool of instruction was the liberal use of his whip: something out of a novel by Charles Dickens. The class was asked to sum all the numbers from 0 to 100, while the teacher probably settled at his desk for a nice long snooze as the pupils patiently pencilled away.

How would one usually approach this problem? Precisely the way that most of the students did:

0 + 1 + 2 + 3 + etc.

Carl, however, discomposed his teacher by announcing a few seconds later that the answer was 5050. He noticed that the numbers formed pairs of 100 i.e. 0 + 100 = 100, 1 + 99 = 100, … 49 + 51 = 100 to a total of 50 pairs. Then there was the single median number 50 which did not form a pair. Now the summing was easy: 50 pairs of 100 makes 5000, plus the median 50 makes 5050. From this simple arithmetical example, we can learn how new ways of seeing can substantially change our preconceived notions of approaching problems. Carl went on to become Carl Friedrich Gauss, one of the greatest mathematicians in history. But is this sort of revelatory insight only reserved for the geniuses of the world? I think not.

Having been a teacher and business coach for many years, I have seen similar moments of insight in my students. Thinking may start with a random thought or observation, but then one needs to grasp that thought and examine it: *What does it imply? When is it true? When is it false? How can it be tested? What significance does it have to the problem at hand? *In short, we start with a thought and we probe it, sift it through our experiences and learning until it gets converted from a mere thought to an insight. This is ** thinking**.

One of the best ways to think is through the *writing of my thought processes*. Writing helps me ground my thoughts: they are right there on the page for me to recall, elaborate, modify and hopefully to illuminate. Consider how many interesting casual observations you make in the course of each day. If you were to pick just *one* of them and spend about 15 – 20 minutes writing about it, sifting through it, and mentally working through it, you have a journal, a record of the idea. Presumably in a class of over a hundred students there must have been at least one other pupil who noted the multiple pairs that added up to 100. Carl, on the other hand, ** thought** through the observation until it yielded an insight. You doubtless have many thoughts and observations each day. Pick one, think through it, and write through it. Every day then becomes a day of discovery.

Nicely written. I have a similar article here:

http://math4allages.wordpress.com/2010/09/15/sum-first-n-positive-integers/

You may want to check it out.