Wednesday, March 08, 2017

Mathematics in education and ability grouping

Exploring geometry through play

Time to focus on mathematics in education

Recently I had a discussion with some young teachers about the teaching of mathematics in schools - the teachers taught in the middle school area. It didn't go to well! They have to do what's expected of them - and that this was  sadly influenced by what the secondary school maths teachers wanted students to have covered! Change requires leadership and a whole school approach.

Why does maths take such a dominant place in the day's programme?
Me - the classroom teacher!

I have always felt that how schools arrange their maths programme ( and to a slightly lesser degree their literacy programme) reflects the philosophy or teaching beliefs of the school. If there is a dedicated ( usually hour long) maths period with students working at teacher, or textbook, tasks in ability groups, then little has changed over the decades.

As Professor Jo Boaler sees ability grouping as 'The Elephant in the Classroom'.

My students maths work
The alternative of developing maths understanding and an appreciation of the power of maths is through  teaching maths through activities and investigations preferably integrated with the classes current inquiry study (ies). Years ago, as a classroom teacher, I did just this I spent the whole year exploring maths and, where possible, integrating it with our studies. My main aim was for
the students to enjoy maths and to develop positive attitudes towards the subjects.  For my own security I made sure students basic computation and number facts were learnt - where possible through fun activities,for example, using 100 squares colouring in patterns made by the various times tables. If I did resort to 'traditional maths' I made it clear to the class the difference between 'real' and 'practice' maths.

Once our minds are changed it all falls into place.

Maths everywhere
Once such a discovery approach is accepted there are plenty of resources to make use of.  Maths needs to be seen as an area to explore in its own right as well as  an integral part of science and environment inquiries.  There are so many exciting things to explore : the history of number; number use in different cultures; the invention of zero; maths in sports , in the arts, in food technology and music etc .Students need to 'see' maths everywhere - as a way of exploring and interpreting their environment.

At the very least one group a day could be involved in such maths activities - or one day a week - or a two or three week maths investigation. Whatever experience should become before the use of abstract symbols. Students who dislike maths are those who do not understand adult imposed symbol systems; for such students schooling has 'turned off' their innate ability to use maths.
Triangular numbers

Why has maths pride of place?

The question to ask is why do assign so much time to maths - is this a something left over from industrial aged 3Rs schooling? Do all students need to know maths ( particularly at the senior school level) they might never use?  Would maths be more enjoyable learn in collaborative mixed ability groups rather than working silently as individuals? With the appropriate approach and resources all maths can be introduced in a way that develops both maths power and understanding.

Don't take my word for it. Below are resources to support such an active and often aesthetic approach to maths.

First  from Professor of Maths Jo Boaler

Jo Boaler writes, ‘far too many students hate maths. As a result adults all over the world fear maths and avoid it at all costs…. It’s the subject that can make them feel both helpless and 
Professor Jo Boaler
stupid….Maths more than any subject has the power to crush children’s confidence.It’s time to remedy this situation and her book provides the knowledge and the details of the real maths children should be learning.
The ‘elephant in the room’ is the belief held by teachers that some people can do maths and some can’t and so, to help ‘slow’ learners, children are grouped by ability. As a result the maths that is taught, she writes, has little in common with real maths. When ‘real’ maths is taught in context many more children are successful.
Learning to love maths and moving away from ability grouping by Professor Jo Boaler.

Another blog about Jo Boaler

Jo Boaler makes two main points – maths can be a fun activity for all students but to achieve this needs the removal of an approach based on ability grouping.  The one in five currently failing in our schools, (notwithstanding the effects of poverty) see themselves as failures, as defined by numeracy and literacy, and the premise of this book that  this is, in good part, to the result of the use of ability grouping. Jo Boaler’s book reports on the depressing research to back her position on ability grouping

Learning to love maths.

The Place of maths - thoughts by Guy Claxton.

Science /tech maths art study
Mathematics ( leaving literacy aside ) seems to have claimed an unassailable place, as of right, in the daily timetable. But, according to Guy Claxton, 'its warrant is under scrutiny'. As the core purpose of education is being contested, he writes, 'powerful new models of teaching and learning are being proposed, and mathematics is in need of a new rational'.

He provides three challenges about the place of mathematics.

First it is not clear that much of mathematics is as directly useful as it has traditionally claimed to be. Many people, he writes, lead happy and successful lives with only basic maths. 
Science and maths

The second point Claxton makes is that there is no evidence to the claim that maths provides valuable generic training of the mind ( an argument once used for Latin). To achieve 'transfer' teachers have to highlight where and why maths is used in other problem solving situations. 

The third point made is that there is no reason that maths should retain its central role in the school day. The worldview that sees mathematics as so central to learning is archaic.Teachers need to be alert to see the maths potential in any learning experience including exploring the maths potential of the natural and man made environment.

  • The common practice of cross-grouping cutting maths off from the rest of the class programme
  • The resentment by children of the cross-grouping
  • The heavy emphasis on grouping in the first place
  • The way children in the top group receive a better deal than children in the other groups, thus making ability grouping a self-fulfilling placement of children
  • The way grouping and cross-grouping impede relating maths to real life applications
  • The teaching becoming routine because of a lack of attention to problem solving
  • A sense of teachers not being sufficiently on top of things to be able to provide cohesion – not being able to go backward (to concepts taken) and forward (to concepts to be taken) in mathematical references
  • Strategies being used in heavy-handed manner
  • The lack of integration of numeracy with curriculum math
  • A severe drop in lively discussion – time pressures, you see
  • The use of unmediated, downloaded teaching units
  • The need for more ancillary aide help (the recent  review office criticism of the use of teacher aides can be interpreted as providing support for the government policy of cutting back on funding for them)
  • National standards'
Views on maths teaching from Dan Murphy ex Maths Adviser and school principal

Introduction to our maths approach 
Dan dressed up for last day

Background to the Numeracy programme.

The Numeracy Project was initially intended as a Teacher Development Programme. For that purpose it is excellent. It makes teachers aware of the various stages and strategies that children use and develop. If it stopped there it would be fine - in my opinion!

As with many other mathematics initiatives in New Zealand, they begin well but there is always a group of people who capture it and turn it into a structured step-by-step, stage-by -stage, fragmented programme! That is where it fails. Mathematics easily lends itself to being broken into fragmented steps of learning, thereby making it easy to measure! This suits the scientific management types who like to measure every step of progress and graph it. It also makes placing children into ability groups very easy.

Learning is fragmented. Teachers plan lots of minute behavioural objectives (now called ‘Learning Intentions’ thanks to Hattie and his ilk). Children have to master each step to progress. 

The slow group (the donkeys) are always way behind in the programme and by the time they leave primary school they have missed half of the mathematics curriculum. They have been labelled slow because they don’t fit the model of children parroting repetitious recall of maths equations. In fact, they are usually very capable, they have been incorrectly judged and put on the scrapheap forever.

This approach is what I describe as the ‘jigsaw’ approach. The children never get to see the big picture. They never make connections of mathematical ideas. For example, the link between fractions, decimals, percentages, ratios, division etc are never made, because each bit is taught in isolation (1/2 = 0.5 = 50% etc). This is made worse by Learning Intentions because learning is confined to the identified objective, which the child has parroted at the start of the lesson. The opportunity to go beyond that is missed by the child and worse - by the teacher!

At  Our School

Teachers hated planning the meaningless lessons that were prescribed in a linear fashion in the numeracy booklets. It was a chore, not an exciting learning opportunity, to teach this stuff.
Our maths results were going down and parents, teachers, and Board of Trustees were becoming concerned.
This was contrary to the exciting, creative learning we were attempting in other areas of the curriculum. Why keep doing this?

Well we didn’t!

We moved into a problem solving approach with contextual mathematics lessons. This is being further developed right now with resources from the Maths Task Centre in Melbourne, where children are taught to work like a mathematician. Read paper by Charles Lovett

Mathematics is in context, it solves real problems and children learn the skills as part of the problem solving process.. Check out this set of wonderful tasks.

The teachers are being made aware of: fixed mindset psychologyCarol Dweck;  recent brain science that shows problem solving lights up the brain as opposed to doing pages of repetitive tasks where the brain does not light up!;   what real mathematics academics are saying - but being largely ignored.

Dan's references:

Cg      Practical maths tasks - brilliant ( from Australia) Recommended by Dan

And there is more if you aren't convinced!

Read about two New Zealand mathematicians teaching maths through crafts. Fascinating

How much maths should be taught in our schools?
Exploring triangular numbers

Maths professor urges teaches to rethink maths ( Professor Jo Boaler)

Want kids to learn maths stop teaching it.

What is important to learn in mathematics

Maths education. Harvard University

A brilliant paper on maths education by Charles Lovett and Doug Clark ( Australia)

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