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 5. What was new and what was the same?

I don't think my mathematics department was so different to some of the most inspirational department I had encountered and heard about in my early days of teaching.

Some of the subtle skills of teaching we wanted had been obliterated by our managerialist, ofsted centred culture but because our need and desire for them became obvious, we could spark them back into life with the help of retired teachers who came in and reminded us what to do.  But, as I've said before, they were only part of the story.  There were many types of teaching going on, the majority of which would not have appeared different to mathematics lessons in other schools.

My students matured in a way I liked.  Instead of having to sit down and shut up until they were permitted to express themselves, they were encouraged to express themselves until they learned how to control themselves.  A student passing test at level 6 might suddenly realise that they did not properly understand place value.  If I didn't have the time to work with them one-to-one I would show them the relevant part of mymaths and let them explore it for themselves.  Having done that they might do the same mymaths work again the next lesson, and again until the penny finally completely and irrevocably dropped.  20 years ago a teacher might have done the same with a text book, so my thinking is not so different.

So teachers were freer and students were freer and yet all the disciplines and demands in place previously were still in place.  We quite simply had better tools at our disposal to facilitate what we wanted to achieve.  I have chosen to teach in a school where I had a big, flexible classroom with plenty of space and ICT. It would be in appropriate to claim or recommend that it was just the ICT which mattered.  I had an interactive whiteboard at one end of the room and ordinary whiteboards at the other.  This helped me teach different groups different things.

I don't want to talk too much about exactly what I did.  Each teacher did it differently and that was a big part of the point and I'm sure if another school used the same structure for thinking they would be different again.  I'm not trying to get people to copy the detail of the way I or any of my teachers taught so I won't say any more about it in this part of the blog.  Hopefully I've said enough to stimulate others to think about their situations which is my main aim.

The yin and yang of maths education - Part 4. Harnessing Technology.

 
By the time I was ready to apply to be a head of department, I had a new tool at my disposal.

Mymaths (mymaths.co.uk) provided online interactive lessons on every secondary mathematics topic to tie into the UK National Curriculum.  Although the accompanying homeworks were not yet in place at that stage there were already booster packs which could be used as homeworks.  These gave me detailed information on the progress of each students and, in doing so, empowered me to require that each students work through the curriculum at their own pace.  The quality of the interactive lessons was much higher than previous online tools due both to the high level of mathematical and pedagogical insight of the designers and also to the interactive visual tools they had incorporated.

I had a very clear vision that I wanted use this technology, combined with individual examples of key topics incorporated into lesson starters, to take a great deal of the strain of teaching the National Curriculum, leaving the teachers at my school with substantially more freedom to decide how best to use significant portions of lesson time.

The 'synthesis of paradigms' strategy was probably most evident in students homework, where I set them an online homework each week which was automatically marked and therefore had more energy to spend on the detailed marking of personal investigations, also set as homeworks.  In this second type of homework students were required to think for themselves, independently interpreting and analysing pure and applied mathematical situations.

Classes probably didn't look so different as you might expect.  We did some applied tasks and some rich investigations.  Some lessons the students worked primarily with high quality texts.  Students were explicitly taught to explain their thinking to each other (see or example the previous section of this blog on how the Chinese do it) and their skills in listening to, explaining to and supporting each other were explicitly cherished.  We were dealing with many challenges at the time, such as mixed ability classes (due to the school being small and working towards closure) and having many students with challenging behaviour and specific learning difficulties and it was my perception that the framework I had allowed us more time to focus on using teaching methodologies which responded the needs of our students rather than on our need to deliver a National Curriculum which contained more material than many classes could cover.

By focusing on teaching rather than on the content to be covered a substantial proportion of the time, in our different ways we covered many elements of that National Curriculum in powerfully connected contexts.  Thus they were relatively easy for students to understand during starter questions and during those online homeworks.  Those individual questions and the computer based teaching wouldn't have worked without the rich earth of experience in which they were taking root.

This structure for organising teaching and learning also allowed me and other staff to focus on some of the more challenging pedagogical strategies which are often neglected in the pursuit of curriculum content coverage.  In experimenting in this way I found that I experienced substantial personal growth as a teacher and my student clearly benefitted from this.

I felt I was delivering on Barcelona.  The two sides of maths education were empowering each other:

Personal investigation of the world is much more effective if  the personal investigation has mastered the use of many key tools and the universal vocabulary which will help them analyse and interpret what they see.

It is easier to learn and more deeply understand can personally make connections in a curriculum if you have spent time considering the situations where the maths it involves is generated and you have experienced the need for that maths.

Verification of my claim for the success of this strategy is limited.  Our results were very good despite the considerable challenges we faced but there were only four externally examined year groups and they were small.  But I think I can validate my claim in the theory of education and in the intuition of experienced teachers.  Much of the content based curriculum was being discovered in context and then taught in abstract.  Students needed core techniques in order to be able to think mathematically and mathematise situation.  To me it just all made perfect sense.  And I think that, as we move from Web2.0 and 3.0 to Web4.0 it will make easy and fluent sense to many others too. I'll continue to work on this validation in my future posts in this section.

I wrote up my efforts to synthesise the paradigms of teaching when the school where I was working closed in 2008 and these can be found here, under my previous name (Rebecca Teasdale).  Sadly there is a £3 charge to access this article.   http://www.atm.org.uk/journal/archive/mt210.html

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.