In article <email@example.com>, firstname.lastname@example.org (K.Ann Renninger) wrote:
> RESEARCH SUMMARY: REPOST > > LEARNING AND MATHEMATICS > > PAPERT (1993) > > Quotes and Comments: > > Discussion can be used to bring an issue to > mind and can then be followed by direct instruction; It can also > be used following instruction to enable students to consolidate > their "real" understanding of what has been presented. Discussion > can also be used in evaluating how students have synthesized their > understanding of a new skill in combination with prior skills.
I have found it to be the case that the best way to test my understanding of something is to discuss it, or even better try teaching it to somebody else. It is this _discussion_ of mathematics that makes me want to be a teacher as opposed to a mathematician who only gets to "do" math. What are other ways in which we can test whether or not a student has learned something?
It seems to me that the method used most often now is to give written examinations. Exams however tend to test a student's ability to DO similar (if not essentially identical) problems. It was rare in my education that I was forced to discuss concepts in math class.
> o Papert argues for a constructionist philosophy that will promote > teaching "in such a way as to produce the most learning for the > least teaching" (p. 139). He contrasts this view with that of > instructionism, in which "the route to better learning must be the > improvement of instruction" (p. 139). According to Papert, > constructionism is tied to mathetics in the sense that children > can often learn without the benefit of schooling, and, if given > the incentive to learn it independently, will even learn a subject > better than they would have learned it in school.
This concept of the most learning for the least teaching interests me. What came to my mind was a picture of a kid who's been labelled a "difficult student" or "slow learner", yet who has learned an unbelievable amount of information about an out-of-school hobby (cars, dinosaurs, whatever).
> It should be noted, however, that instituting discussions or > moving toward more student-directed classes requires that teachers > be ready to take up questions to which they do not know the > answers, come prepared with a variety of resources, and in > general, be well-versed in the content that they are teaching.
This description of how teachers might need to adapt gets at a completely different concept of what a teacher is. It echoes the calls of teachers as co-learners, guides, and friends rather than as simple purveyors of knowledge.
Eric Sasson The Geometry Forum email@example.com