Recently I have discovered by chance a book that many of you may already know – Thinking Mathematically” by J.Mason, L.Burton and K.Stacey.


As it is usually the case, one thing leads to another (click after click :-)), and from the introduction of the book I learned about a video that records a “lecture” by George Polya “Let us teach guessing”. It is long, old (almost archaic, from 1966!)  but brilliant and definitely worth seeing.

[youtuber vimeo=’′]


Returning to the book – in the introduction I noticed a curious passage about the pedagogical benefits of reflection:

“Activity produces experience, but as Immanuel Kant might have said,

A succession of experiences does not add up to an experience of that succession.

Something more is required. Pólya called it the stage of looking back. We chose to call it the Review phase, which is more precise than the term reflecting which we also use. Reflection has a multiplicity of meanings. Jim Wilson once said that this stage was the most talked about and the least used of Pólya’s four stages. Most educationalists are agreed that some sort of drawing back from immersion in activity is necessary in order to learn from experience. After all,

One thing we do not seem to learn from experience is that we do not often learn from experience alone.”

I find it interesting (and shall use this passage with my students) that among the criteria of the exploration there is also one (Criterion D) on reflection.