Another great insight from Seymour Papert, via The Daily Papert blog. I put it up on my Posterous blog this morning but I thought it could go here too:
Many children who have trouble understanding mathematics also have a hopelessly deficient model of what mathematical understanding is like. Particularly bad are models which expect understanding to come in a flash, all at once, ready made. This binary model is expressed by the fact that the child will admit the existence of only two states of knowledge often expressed by “I get it” and “I don’t get it.” They lack—and even resist—a model of understanding something through a process of additions, refinements, debugging and so on. These children’s way of thinking about learning is clearly disastrously antithetical to learning any concept that cannot be acquired in one bite.
(Papert, S. (1971) Teaching Children Thinking. In Contemporary Issues in Technology and Teacher Education, 5(3/4), 353-365.)
And on the higher education end of the spectrum, all of the things that really matter are those things that take patience, time, and persistence to acquire. But these are the very things excluded by this binary notion of understanding in which many children are immersed and to which most college freshmen are completely habituated.
This also makes a good argument for insisting that students — particularly those in the STEM disciplines, but I would argue anybody — should learn computer programming as part of their studies. You cannot learn to program without engaging in the non-binary notion of understanding Papert is describing. Papert knows a thing or two about that subject.
(By the way, I must give Gary Stager many thanks for running The Daily Papert. It is a great resource. The month of March is ridiculously busy for me but once it’s over I am going on a massive Papert reading spree.)
Casting Out Nines 
You know, this is very true, particularly, as you say, of STEM classes. I have seen it time and time again in my technology work when people FEEL as if they either get it or don’t rather than look at understanding as a process. That may be where being from an English/literature background comes in handy for me because answers vary, writing is a process, understanding is often gradual. And the educational system we’re creating will only make students MORE uncomfortable with learning as a process. There is a lot educators could learn about teaching from instructional technology. I know I was surprised to discover instructional design used by educational technologists or really any technology instructors is backward design like you’d see in Wiggins and McTighe.
Good points, Dana. I think we’re seeing an overall tendency towards instantaneity all across the board — in writing, the STEM disciplines, etc. — and any sort of activity that can be described as a discipline is needed to counteract it. Computer programming, essay writing, you name it, they all develop the right habits of mind.
It occurred to me that many students already have a model for this and follow it almost religiously, and that model is athletics. Students happily invest many hours each week working in the weight room to build their strength and don’t give up the first time they can’t do 10 reps at a given weight. So I wonder what educators can learn from what coaches are doing?
Absolutely right about athletics. I know I’ve used that analogy with my students, and they do relate to it. It’s still hard for them to make a shift in their thinking. What’s sad is that I see colleagues tell themselves “I’m not good with technology” or something similar as a sort of cop out. I’m hoping that in my new position next year I will make some headway in that regard.
The key might be figuring out why people develop that model.
A hot candidate, imho, is the process of obtaining knowledge. Not ability, raw knowledge. Nowadays, you google or go to Wikipedia and get sufficient information for almost everything up to a certain degree of specialisation. Very easy, and certainly instantaneous.
And since knowledge and ability are muddled concepts, too, people might transfer that intuition. In general, media tend to give us short reward intervals, so we expect them everywhere.
This Papert quotation is so on-point and cuts to the heart of what’s missing from so much US mathematics instruction. I’m sharing it with my distribution list and the folks I work with in Detroit. Thanks for bringing it to my attention.