One may be aware of oneself as time.
This cryptic quotation, taken from Gattegno's Science of Education, has remained with me. I currently take it to mean that we are formed by the way we use ourselves, and the energy available to us, over time.
In this post, I will attempt to describe how time is a crucial element in solving problems, and learning how to solve problems.
Try this problem, taken from the book Five Hundred Mathematical Challenges:
This problem took me around 30 minutes. I tried 7 different approaches and made a completely fresh start 3 times before arriving at a solution.
We do what we can in order to prepare for insight, but we must wait for a moment of grace. Knowing that solving problems takes time is a simple but important awareness.
It might be useful, perhaps essential, for teachers to work with students on remaining with problems for longer and longer periods. If time isn't taken, elusive solutions will not be reached, and perhaps more importantly, awarenesses regarding what to do when we get stuck will not be developed.
We are formed by the way we use ourselves, over time.
Here is a second problem you might like to try:
I worked on this for two days and one sleep. I was stuck for long periods. There were glimmers of hope, a number of what seemed like good ideas, followed by dead ends. There were long periods of doubt, and eventually I gave in and looked at the answers.
Sometimes the moment of grace occurs, and sometimes it doesn't.
Having the answers at hand is sometimes just too tempting. For this reason, I consider it preferable that answers are withheld from students for as long as is suitable. Only through not knowing for long periods of time do we bring everything we have to bear on a problem; only through time do we learn about the dynamics of being stuck.
Solutions are most informative having worked on a problem for a long time and exhausted the energy available.
Even though I did not reach a solution to the Problem 58, it was a rich learning experience, in terms of dynamics and content. Upon reading the solution, I created this set of questions that might help you access the mathematics involved in this problem if you became stuck, like me: