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Like I said yesterday, I’m reading through Seymour Papert’s Mindstorms: Children, Computers, and Powerful Ideas right now. It is full of potent ideas about education that are reverberating in my brain as I read it. Here’s another quote from the chapter titled “Mathophobia: The Fear of Learning”:
Our children grow up in a culture permeated with the idea that there are “smart people” and “dumb people.” The social construction of the individual is as a bundle of aptitudes. There are people who are “good at math” and people who “can’t do math.” Everything is set up for children to attribute their first unsuccessful or unpleasant learning experiences to their own disabilities. As a result, children perceive failure as relegating them either to the group of “dumb people” or, more often, to a group of people “dumb at x” (where, as we have pointed, x often equals mathematics). Within this framework children will define themselves in terms of their limitations, and this definition will be consolidated and reinforced throughout their lives. Only rarely does some exceptional event lead people to reorganize their intellectual self-image in such a way as to open up new perspectives on what is learnable.
Haven’t all of us who teach seen this among the people in our classes? The culture in which our students grow up unnaturally, and incorrectly, breaks people into “good at math” or “bad at math”, and students who don’t have consistent, lifelong success will put themselves in the second camp, never to break out unless some “exceptional event” takes place. Surely each person has real limitations — I, for example, will never be on the roster of an NFL team, no matter how much I believe in myself — but when you see what students are capable of doing when put into a rich intellectual environment that provides them with challenges and support to meet them, you can’t help but wonder how many of those “limitations” are self-inflicted and therefore illusory.
It seems to me that we teachers are in the business of crafting and delivering “exceptional events” in Papert’s sense.
Casting Out Nines 
I was one of those people who grew up thinking I was dumb at math, but my grades in math don’t bear that out. I realized I wasn’t dumb at math when I was reading Understanding by Design, and Wiggins and McTighe described a math problem on the NY Regents test that a lot of students missed because they didn’t know it was asking for application of the Pythagorean Theorem. I looked at the problem, and I remember realizing (before I got to the reveal that the Pythagorean Theorem would solve it) that I knew how to solve it. I drew a diagram right in the margin of the book, solved, and discovered I was right. In a sense, I’m mad that I told myself math was something I couldn’t do for so long, but I was also happy to discover it wasn’t true.
Speaking of reverberations, I just finished Papert’s “The Children’s Machine” not too long ago, and one chapter in particular got me thinking about abstract vs. concrete reasoning. Papert says (and I hope I’m not mangling this, as I’m going from memory here) that we traditionally hold abstract reasoning up as the more advanced of the two, but I thought he made a strong case for concrete being just as cognitively demanding, if not more so.
I was planning to blog about it when I have a chance to re-read the specific section and digest a little further (had to return the book to the library and didn’t think to scan the pages for later reference); I’ll drop you a link when I do.
Please do. The Childrens’ Machine is on my reading list — our college library didn’t have it though.
I’m a teaching artist who integrates the ‘abstract’ world of math with the ‘concrete’ world of dance. Art/dance/music/etc. are also things people think they are either ‘good’ or ‘bad’ at which leads to them missing out on the process of making, discovering, and then, many times, understanding, maybe even enjoying. Same with math. It’s really just cultural baggage we all carry that dictates we have to be ‘good’ at something in order for it to be important, in order for us to be important, blah, blah, blah. More here: http://mathinyourfeet.blogspot.com/2011/01/making-things.html and although this does not specifically speak to math, other posts on my blog make the connection. If you’re interested, read through it and see that the process of what I call ‘facilitated making’ can be applied to pretty much anything you need to ‘teach’, including math.
Well, in this country I think that’s true. It’s not true in Asian countries, where being good at math is assumed to be the result of hard work. This really is a cultural thing.
Also, see Carol Dweck about fixed vs growth mindset. This is similar to what she talks about.
Professor Dweck of Stanford directly addresses this issue with her work on the “Grow Mindset” and the software program “Brainology” at http://www.brainology.us. The cultural problem is partially solvable.
“many of those “limitations” are self-inflicted and therefore illusory.”
Just because the limitations are self-inflicted does not make them illusory. Once a student has internalized that they are bad at something, they often fail to learn the basics of the skill, making it very much harder for them to learn the next steps. The limitations are real, even if self-inflicted. Until the initial barrier to learning is removed, the limitations will not go away. It is often beyond the ability of a classroom teacher or tutor to remove deep-seated fears that are blocking learning. Even professional psychotherapists have a rather dismal track record.
What I mean by that is that the origin of the limitation didn’t have to happen and, if the perceived limitation hasn’t become too firmly entrenched, the person can work his or her way out of it — so it’s not a real limitation at all. An intellectual limitation that can be overcome is not a limitation, it is a mental block.
Example: I have a perceived limitation that I cannot make a ponytail my daughters’ hair. (Ask my wife.) If at some point in the future I have an “exceptional event” that shows me that I indeed can do my kids’ hair, then I would have to look back and say that the thing I thought was a limitation was all in my mind. Unless overcoming the limitation involved developing a mental or physical capacity that was simply not available to me at the time; for instance my 2-year old son can’t do hair, and that’s a true limitation.
Obviously, and this is a big point in Papert’s book, if you take a perceived limitation and reinforce it for a sufficient amount of time, then it becomes insurmountable without a truly exceptional event. That’s the state of a lot of kids learning math today — and a lot of adults who say they “can’t do math” — and for all practical purposes the limitation is real, because the amount of work needed to wash off all the layers of cultural and personal reinforcement of that statement is gigantic.
My daughter was placed in the dumb at math category at an early age. But once she was lucky enough to find a teacher who had been labeled that as well but excelled anyway. He helped her overcome this tag and make math easier. It is all in how it is presented to some students.
Robert, I read Mindstorms many years ago. You have inspired me to order a copy from Amazon.
This also led me to SCRATCH – a programming language developed for kids by MIT on a similar principle to Papert’s constructivist philosophy. It looks ideal even for young children – very intuitive (even for a 6 year old.)
Very much a latecomer to this discussion, and even more an outsider to the topic, but I saw a quote recently (darn, I wish I could remember it precisely–along with the quotee…MAYBE Picasso??) that was something like this: I do things that I can’t to–in order to learn to do them better.
For what it’s worth,