tag:blogger.com,1999:blog-12692155561365621.post7355651608830125133..comments2024-03-28T07:45:39.017-04:00Comments on The Map is Not the Territory: Two Sides of the Same CoinMalkehttp://www.blogger.com/profile/09927560751422131935noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-12692155561365621.post-6353402363319935362013-12-12T19:46:47.989-05:002013-12-12T19:46:47.989-05:00Hi Bryan -- great thoughts. What if "fixed&q...Hi Bryan -- great thoughts. What if "fixed" meant working with something that has already been made, something tangible, and "flexible" meant that the learner had some impact on how that something is to be formed? What then? <br /><br />Also, honestly, I get a little itchy when I think of all knowing/understanding as lodged in the mind. Everything I speak of above is something that children can touch, hear, do, see or do with their own bodies -- which is where children do much of their thinking. There's a really interesting study about children's gestures in math learning that concludes "children think and learn through their bodies." I'm not denying mental constructs, but I believe there to be a much larger role of the body in the knowing/learning process *especially* for children.<br /><br />I suppose I have some bias against the fixed side of things -- I am known for being pretty open minded to a wide range of 'answers' within children's choreography and even in the math groups I run at my daughter's school. So, in that way I am definitely, as you say, "open to individual ways of knowing (flexible)" and I really like that phrase "the outcome the observer will be content with." That's definitely something to think about a little more. What I am trying to do here, though, is to pin down an answer to what I saw as an inverse to Papert's body knowledge (and also make sense of the difference between identifying attributes vs. using an inventory of attributes to make something. I'm still not completely sure I have it or, even, if it's really that necessary to figure out. <br /><br />Now I'm at the risk of rambling. Thanks again!Malkehttps://www.blogger.com/profile/09927560751422131935noreply@blogger.comtag:blogger.com,1999:blog-12692155561365621.post-39242740856862568342013-12-12T18:46:46.975-05:002013-12-12T18:46:46.975-05:00Your post, as usual, has me thinking quite a bit. ...Your post, as usual, has me thinking quite a bit. I find some of the distinctions here between fixed and flexible (for me, at least) are bound up in a particular view on knowledge....one which gives an ontological status or existence to "mathematical ideas." I prefer to think of mathematical ideas (or relationships, objects, etc.) as existing ONLY in the mind. Because of that, some of your statements are problematic for me. For instance, you write: <br /><br />"Illustrating math ideas using the body means there is a predetermined goal for the activity and that the outcome needs to look a specific way."<br /><br />This is only a statement that an observer, or knower, could make. It basically says that a child is moving in a way YOU recognize as matching your own mathematical knowing. So, to me, a "fixed" objective means that a teacher/person wants to recognize their own thinking in the actions of another.<br /><br />In contrast, you wrote about "flexible":<br /><br />"In making, having a large inventory of ideas/things/skills from which to choose and create your own novel ideas (like a dance step) is an open-ended investigation."<br /><br />To me, this sounds very much like how Piaget described the functioning of the mind. We build schema that are basically ways of acting/thinking that lead to viable outcomes in our experience. When we are confronted with a novel situation, we try to assimilate that experience into an existing conceptual structure. If that does not produce a viable or expected outcome, we are forced to make an accommodation (essentially, learning). But, that does not mean the accommodation will be one that the observer would identify as "correct" or matching their own way of knowing. <br /><br />So, to wrap up this little rambling, I'm reading the distinction between fixed and flexible to essentially refer to the outcome the observer will be content with. Must they require one that matches their own way of knowing (fixed) or is it open to individual ways of knowing (flexible).<br /><br />Thanks, Malke!Bryanhttp://www.doingmathematics.comnoreply@blogger.com