No place for thinking in math class?

I can't remember how I was taught maths when I was in primary school, but I vividly remember the maths that I was taught from the age of 11 onwards.  It was called SMP maths, and was developed by a group of researchers in the 1960s who worked out of Southampton University.  I recently found out that SMP was a response to the call for a reform in mathematics teaching, following the launch of Sputnik by the USSR.  It's kind of incredible to me now that the maths I was learning in the 1970s was a response to the Cold War and Space Race!

When I was 11 I had no idea what SMP was.  At school we called it "Stupid Maths Problems" and I do remember having to do problem after problem after problem.  SMP was abandoned in the UK in the 1980s when the National Curriculum was introduced, however before this schools were free to set their own curricula and buy the resources they wanted to support that.  Therefore at school I didn't learn about algebra, trigonometry or geometry, instead I learned about things like graph theory, non-cartesian co-ordinate systems, vectors and non-decimal number systems (I remember binary - I can't remember much about any of the others).  The aim of SMP was to improve the mathematics curriculum taught in the UK, but now it has been criticised as putting a whole generation off mathematics by trying to dive into abstraction too early.  SMP did not really fit in with the exam system either.  I remember sitting my maths O'level at the age of 16 and having to take my shoelace out of my shoe and use it to make a sort of scale model of the problem I was trying to solve that involved the circumference of a circle. However, reading Jo Boaler's book Mathematical Mindsets, I realise that in many ways I was fortunate as I was not drilled in maths facts.  Jo, who was at school in the UK at a similar time to myself, writes that her school was focused on the whole child, and as such she also didn't have to memorise tables of addition, subtraction and multiplication facts.  Instead she learned number sense, which is the deep understanding of numbers and the ways they relate to each other.

As mentioned in a previous post, Jo argues that maths facts are held in the working memory section of the brain, and that under pressure (for example during timed tests) the working memory becomes blocked, causing anxiety.  This is what puts students off maths, and Jo claims, is leading the the maths crisis that is currently being faced in both the UK and the USA.  Instead of memorising facts, Jo writes that we should be offering conceptual activities that help students understand numbers.  There is brain research that shows that the left side of the brain handles facts and technical information and that the right side handles visual and spatial information - and the learning of maths is optimised when both parts of the brain work together to develop new brain pathways.  Jo writes,

The more we emphasize memorization to students, the less willing they become to think about numbers and their relations and to use and develop number sense.

Jo goes on to compare the learning of maths with the learning of English.  In order to understand novels or poetry students need to know the meaning of many words, yet we are not teaching the fast memorization and recall of hundreds of words when we teach English - instead we take words and use them in many different situations - talking, reading and writing.

Getting back to the SMP, and indeed to the Addison Wesley mathematics text books that I was presented with when I became a homeroom teacher in Grades 5 and 6, the thing I remember the most were pages and pages of practice questions and worksheets that had to be completed every day.   I was interested to read that Jo states that we do not need to practice methods over and over again - what we really need is to reinforce ideas by using them in new ways.  Jo writes about repetitive practice:

We do not need students to take a single method and practice it over and over again.  That is not mathematics; it does not give students the knowledge of ideas, concepts, and relationships that make up expert mathematics performance ... The oversimplification of mathematics and the practice of methods through isolated simplified procedures is part of the reason we have widespread failure in the United States and the United Kingdom.  It is part of the reason that students do not develop mathematical mindsets; they do not see their role as thinking and sense making; rather they see it as taking methods and repeating them.  Students are led to think there is no place for thinking in math class.

I was also interested to read the PISA results that show the lowest scoring maths students in the world are those who use memorisation, whereas the highest scoring students are those who think about the big ideas and the connections between them.  Clearly what students need is to be given interesting situations and encouragement to make sense of them.  In this way, Jo points out, "they will see mathematics different, as not a closed, fixed body of knowledge but an open landscape that they can explore, asking questions and thinking about relationships."

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