Awakening

The phrase “the sense that one is transcending the limitations of our normal state and gaining a more intense awareness” is exemplified by the experiences of the student, Sandra, in the article  “A Rush of Connections and Insights, a Glorious Moment of Clarity“, Davis & McGowen (2004). Sandra wrote about her insights into combinatorial problem solving:

“… sometimes, it takes a number of different approaches or methods to help students understand math. It didn’t become clear to me until we did the coin toss. And there was a rush of connections and insights. … while playing the dice game, I noticed that certain numbers were more likely to be rolled than others because there were multiple ways of adding up to those numbers. This reminded me of Pascal’s triangle, which functioned in a very similar way. I remembered that the dots in the middle of the triangle occurred more frequently than those at the periphery, much the same way that the numbers in the middle of one to twelve range appeared more frequently on a pair of dice in the dice game. Making that connection enabled me to make a prediction about how frequently certain numbers would be rolled in the dice game. I simply recalled the number of dots in each place along Pascal’s triangle and applied those numbers to the dice game. I think that connection reflected a level of comfort that I was beginning to acquire with the concept of exponents. It was a glorious moment of clarity, which inspired me to want to think even more about probability, a math topic which had always been intimidating to me as a learner. … I look forward to those rare, but beautiful opportunities to make connections across the discipline, knowing that it is those eureka moments which make it all worth while. And I can’t wait until the day when I have the privilege of seeing my students experiencing those same moments of clarity and comprehension, exhilaration and accomplishment.”