> >"Conceptually based instruction" is used here to mean the kind of > >teaching that fosters understanding of the reasoning and logic > >behind mathematical ideas. Place-value concepts include grouping > >issues (10s, 1s, etc.) and the use of units for writing and using > >numbers. The authors maintain that while in text-based instruction > >these ideas may not be transitted to or understood by the students, > >in conceptually based instruction the focus is on learning why we > >use place value and the ability to transfer this knowledge to > >mathematical achievement. > > > I honestly think that it would make more sense if more clasases were > taught in this fashion.I am a firm believer that by teaching students > why they are doing things your life and theirs is made a lot easier. > I know from a student who I currently tutor in mathematics that when > taught in school the equations for doing things she can sometimes do > the problems but, when I explain to her the reasoning behind why she > is > doing certain operations or procedures it becomes so much clearer and > makes it possiblt to transfer this knowledge into other further > studies in mathematics.
I cannot resist jumping into the fray here -- we are agreeing with ourselves too much.
Let me first count myself to be on the side of the good. I am a strong believer in teaching for understanding. The basis of my support, I think, is that the learner can only build on her/his understanding. If the student's experience of mathematics classes is that one must memorize a bunch of facts and algorithms and catch-phrases, certainly s/he will abandon that part of learning at the first opportunity, and write her/himself off as incapable of learning math and subjects that rely on math -- a major, lifetime limitation.
Yet ... My 16-year-old son has shown me that we can err in our zeal to teach "understanding." He, along with many other students, has made up his mind that school is a coercive system. Like prisoners, or slaves, one learns to stay out of trouble without entirely giving up one's dignity by doing the minimum needed to compy with the demands of the system. (Sorry to paint such a grim picture -- he'll pull out of this eventually.)
Therefore, when we refuse to tell him what tricks he must perform to win the right to be left alone for awhile, and instead insist that he proclaim he sees the light, that he do something as nebulous as "state this in your own words" he feels angry that we're really becoming sadistic in our coercion. We're no longer content with regimenting his behavior, we're after his mind now.
Here's my present thinking about all this: my goal of liberal education (in the traditional sense of that term) is a radical one. It is based on the premise that people want to learn and that this desire must be the driving force of learning.
There are cases, like my son's, where we -- the systemm, the grown-ups, the people running the educational show -- fail to demonstrate or convince the student that our aim is to empower that student to take whatever role the student may desire in society (I'm throwing to the winds qualifications against empowering psychopaths). Instead we demonstrate our insistence that the student conform to her/his current role as a student, with the promise of future assignments in society being as good or better -- a rather bleak picture.
Okay, so far so good. Here's the controversial bit. If we cannot create an education that is liberating for a student, then perhaps we ought, in some cases, bow to the demand that we keep our demands simple, clear, and fairly easy to comply with.
This is a retreat for me from my former insistence on "understanding." But it is a tactical retreat. I do not yield on the larger goal of encouraging individuals to take charge and take responsibility for their own lives, and their own learning.
We can do some coaxing of children to learn what we know they need. But as they mature I believe we are gradually forced to a different position -- of showing how we can help, but waiting to be asked for that kind of help.
at work: | at home: | Anthony D. Thrall, PhD | Etak -- The Digital Map Company | Tony Thrall 1430 O'Brien Drive | 2131 Ashton Avenue Menlo Park, CA 94025 | Menlo Park, CA 94025-6501 | 415/328-3825 x 1300 | 415/854-6449