Work will only be of value when man [sic] gives as much as is the limit of possibility… Working like a man means that a man feels what he is doing and thinks why and for what he does it, how he is doing it now, how it had to be done yesterday and how today, how we would have to do it tomorrow, and how it is generally best to get it done - whether there is a better way. If man works rightly he will succeed in doing better and better work. But when a two-brained creature works, there is no difference between its work yesterday, today, or tomorrow… If you can make, say, a cigarette like a man, you already know how to make a carpet. All the necessary apparatus is given to man for doing everything. Every man can do whatever others can do. If one man can, everyone can. Genius, talent, is all nonsense. The secret is simple - to do things like a man. Whoever can think and do things like a man, can at once do a thing as well as another who has been doing it all his life but not like a man. What had to be learned by this one in ten years, the other learns in two or three days, and he then does it better than the one who spent his life doing it. I have met people who, before learning, worked all their lives not like men, but when they had learned, they could easily do their finest work as well as the roughest work they had never even seen before. The secret is small and very easy - one must learn to work like a man. And that is when a man does a thing and at the same time thanks about what he is doing and studies how the work should be done, and while doing forgets it all - his grandmother and grandfather and his dinner…
From notes on a talk given by G.I. Gurdjieff in 1923, from the book 'Gurdjieff's Early Talks 1914-1931'
This passage from Gurdjieff has lots in common with comments made by Caleb Gattegno (founder of the ATM) about learning mathematics. Gattegno talked about children being able to learn in two years what generally takes five years in school. I suspect he was not making this remark glibly, that he was talking from experience, and that he knew something important about how people learn.
I have some recent personal experience of learning something surprisingly efficiently. In studying for my driving theory test, I set myself to remember everything while reading the highway code, and only needed to read it once to retain most of the information. I also learned the basics of driving in an hour or two, again deciding to apply myself as attentively as possible to what I was experiencing.
I increasingly believe that what Gattegno and Gurdjieff knew is that in order to learn efficiently, we must strive to become conscious of what we are doing, how we are doing it, how we are using ourselves, why we are doing it, and how we might do it better. Underneath this lies intention, and the application of initiative.
It is becoming increasingly clear to me - in recent lessons on shape with primary children, and the lesson I describe below - that learning mathematics can be considered as a series of exercises in becoming conscious, more aware, of what we are doing. Only by learners becoming more conscious of what they are doing can efficient learning take place, and it is this that I find myself trying to educate.
I had to start by prompting them to chose a value for the perimeter. K then chose a perimeter of 10,000 and K chose a perimeter of 5. I left them to get on with it. When I returned, K had written 2000 x 3000 = 6000000 and J had written 1 x 1.5 = 1.5 .
I asked them how they knew this was the largest area. K had tried 1900 x 3100 and J had tried 0.5 x 2, and found them both to be smaller. J said there were not any other possibilities. Now, their failure to go beyond these few examples could be due to the design of the task, but I believe it was also in part due to the students' state of mind/emotion/body. We have been doing some particularly difficult problems lately (most notably for homework), and I think this may have had an effect on their willingness to explore this problem.
I had a strong sense that they were not working on this problem 'consciously', (and) that they were not applying initiative. I stopped them and read out the Gurdjieff passage above. There was a marked difference in their work from this point. Here is what they did next (click to view):
K initially wrote down 10^2 x 10 = 1000 and then circled (a). What is required here, as is often the case when solving problems, is a 'sideways movement' from the first impression. After some time, he wrote a(20-a), presumably trying to adopt the algebraic approach we had discussed for the rectangle problem, but not being sure quite how to apply to this problem. After some inaction, J realised that he could use trial and error to solve the problem as given.
I asked them to complete this problem, and the ones below (taken from NRICH) for homework, and to try to use their initiative as much as possible. It will be interesting to see what they do.