The yin and yang of maths education - Part 6. What are these paradigms?

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.

The yin and yang of maths education - Part 3. How to make enemies of natural befellows.

I wondered about the tendency of some to attack the paradigm of maths education which is not their favourite.  Why do it?

I think there is a natural human tendency to universalise conclusions.  When you find something which is good it is natural for you to want everyone to share your experience.  It's easy not to notice that others may also have found something else which is equally exciting and fulfilling.  I think this is more true for the young and less true for those who have experienced the limitations of thinking in this way many times before.

But there is another force which has made enemies of these natural bedfellows.  It is that the organisational forces which are required to make maths education function in a school have needed to be configured predominantly one way or the other.  Schools have either had programs of study, testing and teaching which are organised according to the mathematical techniques and vocabulary to be learned (with perhaps specific times allowed for constructivist activities) or the curriculum has been child thinking centred (with perhaps specific times allowed for exam preparation).  Those teachers who have been in departments configured for teaching methodologies which have not suited them have suffered and so they have complained.  If they have not been heard then they have come to resent a status quo which prevents them being the teacher they are capable of being.

Let's imagine some parallels:
Say schools had to choose either to teach French or Spanish.  There would be great debate as to which was better and such debate inevitably includes consideration of which is worse.  Now suppose schools were required to teach both French and Spanish.  After the initial strains of reorganisation do you not think that concerns associated with which language is worse would simply disappear?  Do you not think that teachers would instead look at the synergies between the two languages and at ways of teaching which would enhance students' progress with both?

There is another parallel in society as a whole.  I have long appreciated the benefits of the philosophy of liberal freedom.  I have also understood socialist values.  And in my appreciation of both positions I have come to understand that society has had to configure itself either one way or the other.  Either you have centrally organised socialism or you have liberal freedoms with much less central organisation.  But that has changed, hasn't it?  For me the most obvious joy of the 21st century is that ICT has given us the power to develop much more complex and versatile administrative infrastructures which can more easily and effectively integrate social and liberal ideals.  Isn't socialism much more powerful when it is powered by the financial contributions to society of those who free to generate substantial wealth?  Aren't we actually more free if we know that should we become vulnerable in an aspect of our life there will be support available to us?

Broadband is a magic bullet we can use to eliminate that jar between many pairs of traditionally opposing paradigms.  By 2006, when I was preparing to become a head of department, I knew that I had new and very powerful tool at my disposal in my question to cherish both the teaching and learning of traditional mathematics vocabulary and techniques and the development my students as creators and appliers of mathematics.