Here's a diagram which I offer as a tool for thinking. All diagrams are partial. They are only in 2-dimensions after all.
In the traditional paradigm, the maths to be learned in a lesson is selected from the mathematics which has been proved and written down.
In the constructivist paradigm, however, the focus is instead on nurturing the way that students mathematise the world around them. Because the focus is on the thinking processes to be developed, the maths to be learned is not clearly specified. Hence it becomes irrelevant as to whether it is mathematics which has been previously proven or not. In my opinion and experience substantial amounts of mathemics is learned during these lessons, but that mathematics is most powerfully learned when it is backed up with some traditional teaching at another time. There is also a job to be done to fill in the gaps of core mathematical techniques which have not been learned.
But - oh - the magic of invention. When students have aha moments. Which a child expresses something incredible that you've never imagined before. The intellectual journeys you take as a teacher. Please excuse me while I get a little misty eyed. I can understand why some teachers who discover the constructivist paradigms for themselves can become rather messianic about it. I wouldn't want to teach without it being part of my teaching. It inspires me. My students inspire me.
Is "D" accounting for the math yet to enter the mind, to be proven and written down or does it only represent everything inside rectangle?
ReplyDeleteYou have constructionist arrow coming from outside the rectangle D. How can we construct what we do not know if based on what is not clearly specified, or even acknowledged as being Math? How does the rectangle prove anything about the circle when it is in the constructed limitations of four sides?
Yes you spotted the obvious missing dimension straight away Brad. This diagram allows no 'degree of awareness'. Things are either known or unknown. Gone are the insights into relevant contextual experience, partial awarenesses, conscious experience, what is 'agreed' and so on. We could create a 3-dimension diagram but I don't want to do that at the minute.
ReplyDeleteYes, D is the mathematics which has not yet been discovered. The rectangle is 'all mathematics'. D is only the bit outside the circles not the whole rectangle.
The arrow going from D to the mathematics in the mind of the individual is just there to show that when students are constructing their worlds for themselves they may discover things which are not part of established mathematics.
Thanks for your comment. Please do continue to probe and that goes for other readers too. I really value your comments.
Are you defining the role of the teacher in this model as "the guide at the side" rather than "the sage on the stage"? (borrowed from Philosophy for Children)
ReplyDeleteKev
Excellent point Kev. Is the traditional route just 'sage on the stage' or would it also incorporate, say, working from text books? or working with online interactive teaching programs.
ReplyDeleteI see the traditional route as being teaching in which the mathematical vocabulary and techniques to be learned are defined before the lesson. So it incorporates methods other than sage on the stage (I love that phrase).
Is that sufficient clarification? Please do probe more if you feel you'd like to.
In drawing a diagram showing these relationships I would first draw a circle that represents accumulated math knowledge. Within that can be constructed the rectangle representing the “proved and written down” parts we embrace. Mathematics in the mind of the individual is that area that is outside the rectangle to the circle boundary. The potential of mathematics is in the opening degrees of awareness that are inherent in the concentric nature of the circle moving out beyond the boundaries of our presently defined circle. The constructed rectangle of knowledge (box) may or may not increase with the enlarging of the circle boundary. That has to do with the paradigm used, and willingness to consider beyond the boundaries of imagination, thus intuition and insight.
ReplyDeleteIntersecting circles are a construction different than the single circle of the yin/yang image where dual circles are a reflective division within the Whole that creates multiplication of same. Both the Venn and the yin/yang are constructed ideas held within the quadrilateral within the circle that through waves of infinite harmonic concentrics move into and out from the single circle image of the Whole.
The first fold of the circle reveals the quadrilateral that holds fundamental “proved and written down” knowledge. Ref. “OneFoldCircle”
Do you think you could draw your diagram Brad? Could you try to post it as a comment here? If you cant then mail it to me and I'll post it?
ReplyDeleteI accept your point about the opening of degrees of awareness but have deliberately left it out of this diagram because this diagram is meant to support and provoke thinking about other issues. I think there is definitely room for other diagrams which do things that mine does not and warmly invite participants in this blog to attempt to draw some.