In this paper I was providing data about the state of maths abilities for all pupils across the UK and discussing some of the implications for educators. Maybe our current curriculum and pedagogy are not as effective as they could be (and not just for the pupils with special needs).
This is an edited extract.
Mathematics Teaching (MT) Issue 232 Jan. 2013
Is the population really woefully bad at maths?
This article is based primarily around data from 1783 students aged from 7 years to 15 years from over fifty schools across England, Wales, Isle of Man, Scotland and Northern Ireland, and from 766 ‘adults’ aged from 16 to 59 years old, collected to standardise a 15-minute mathematics test (from ‘More Trouble with Maths. A complete manual to identifying and diagnosing mathematical difficulties.’)
The 2010 Sheffield Report’s analysis of evidence on levels of achievement in maths showed that 22 per cent of 16 to 19-year olds in England are functionally innumerate, a problem that has been in existence for at least twenty years. In the 2011 (Porkess et al) report for the Conservative Party, ‘A world-class mathematics education for all our young people’ it states that, ‘much greater attention needs to be paid to those students (nearly half of each cohort) who currently are deemed to ‘fail’ mathematics at age 16’. In 2006, the CBI also commented on this issue with adults, using as a benchmark the ‘maths that would be demanded of an eleven year old’ and claiming that almost 50% of the working population of the UK fail to reach this standard.
Those three examples provide three descriptions of under-performance: ‘functionally innumerate’, ‘failing to achieve Grade C in GCSE’ and ‘not possessing a grasp of numeracy that would be demanded of an eleven year old’. The last description points towards the objectives listed by the National Numeracy Strategy back in 1999 when it was set up to address the problems perceived to exist in maths education in Primary schools. The NNS gave key objectives for the maths that children should know at 11 years old. These included:
At age 9 years
Know by heart all multiplication facts to 10 x 10
Multiply and divide any positive integer up to 10 000 by 10 or 100 and understand the effect.
Calculate mentally a difference such as 8006 – 2993
Carry out column addition and subtraction of positive integers less than 10 000
Carry out long multiplication of a two-digit by a two-digit integer
Carry out short multiplication and division of a three-digit by a single digit integer
At age 10 years
Reduce a fraction to its simplest form by cancelling common factors (Y5 Relate fractions to division and their decimal representation)
Understand percentage as the number of parts in every 100 and find simple percentages of whole-number quantities.
Carry out long multiplication of a three-digit integer by a two-digit integer
And at age 11 years
Use letters or symbols to represent unknown numbers or variables
Know that algebraic operations follow the same conventions and order as arithmetic operations
Convert from one metric unit to another
Refine written methods of multiplication and division of whole numbers to ensure efficiency and extend to decimals with two places.
I shall return to these Key Objectives using data from the standardisation process for my 15-minute test to place the achievement levels of 10 years old, 13 year old and 15 years old pupils and 16-19 year old students into a perspective.
The 2011 Conservative Party report clearly asserted the need to look at all learners:
‘It is essential for us to consider all young people and much greater attention needs to be paid to those students (nearly half of each cohort) who currently are deemed to ‘fail’ mathematics at age 16. We believe that it is largely the system which fails those students. We must recognise that their requirements are different from those of the top 15% who currently go on to study mathematics to a more advanced level.’
However, in their graphic presentation of ‘The Mathematics Education Circle’ there is no mention of how children learn or fail to learn mathematics.
There is now considerable evidence on how students learn maths, for example from the USA’s National Research Council, from Hattie’s meta-analysis on what works in education, from a Russian psychologist (Krutetskii in 1976) on what students need to be successful in maths, on the individual differences in children from Dowker’s (2005) work, on how children can be taught from Boaler’s (2009) best-selling book, ‘The Elephant in the Classroom’ and on how dyslexic students can be taught from my own work based on thirty years of experience of working with dyslexic students (Chinn 2017a, 2017b).
Perhaps it is time to reconsider the faiths, cultures or beliefs about mathematics and their possible influence on how maths can be taught to ‘all’. For example, the belief: expressed in the Conservative Party’s report, ‘As with any language, the fundamentals of mathematics (for example, multiplication tables and number bonds) are most easily learnt when you are young.’ I would say from many years of experience that, as Ben Goldacre says, ‘I think you will find it’s a bit more complicated than that.’
In England we devote many lessons over many weeks attempting to teach students mathematics. If we consider a school year to contain 40 weeks and we look at the pupils who attend school from age 5 to 16, then we have 11 years or 440 weeks to teach them the Key Objectives listed above. That would seem to be enough time without extending it another two years as the Conservative Party’s study group has suggested.
As I lecture to teachers across the UK and in many countries I ask the question, ‘At what age are enough children giving up on maths for you to notice?’ Sadly the modal answer is 7 years old. Sadly (again) my informal sample is now very large. De-motivation starts early, despite the belief or maybe partly because of the belief that the fundamentals are most easily learnt when you are young. And there are a lot of those 440 weeks left to endure maths when you are 7 years old.
I considered myself a successful teacher during my fourteen years in mainstream schools (evidence-based, of course). Moving to work in special education (primarily dyslexia) in 1981 revealed that I was not good enough to succeed unless I learnt much more about how children learn especially when ‘normal’ teaching had consistently failed them. I had bright, intelligent, dyslexic students who did not respond to my ‘good’ teaching.
The mantra ‘Teach the subject as it is to the child as he is’ that is often used in special education can be attributed to a number of people, for example, the pioneering USA educator, Margaret Rawson. This succinct philosophy suggests that we do not need to mess with the subject. We do not need to dilute its academic content. In fact, we should not diminish the integrity of the maths nor insult the subject by making it a collection of facts and rules to be learned rather than understood. This is what one could call the Mr Punch pedagogy, ‘That’s the way to do it.’
The second part of the mantra is that we have to understand the student, how he learns and the barriers that handicap or prevent the learning process. And then do something with that knowledge.
There are certain characteristics of dyslexic, dyscalculic and many other special needs learners that make the traditional pedagogy of teaching ineffective. Since dyslexia and dyscalculia are a spectrum and characteristics such as these are on a normal distribution, then many of those characteristics will be present in children who are not formally recognised as having special educational needs. Like in osmosis there is upward movement.
One example is a poor working memory. One of the key principles on which the National Numeracy Strategy is based is extensive exposure to mental arithmetic. Mental arithmetic requires a good working memory. Of course, teachers can teach different ways of solving mental arithmetic problems, but whichever method is used, an adequate working memory is needed. There must be reasons why children give up on maths so soon. I suspect that the emphasis on mental arithmetic is a contributing factor.
A second characteristic is long term mathematical memory, that is, long term retention and retrieval from memory of mathematical facts and procedures. When I started to work with dyslexic students it did not take me long to realise the intransigence of this problem and the strong emotions associated with it. One of the Key Objectives of the NNS at age 9 years is to ‘Know by heart all multiplication facts to 10 x 10’. There is so much wrong with that objective. The USA is now moving towards using the term ‘number combinations’ to acknowledge that many children do no retrieve this information direct from memory. Again, my informal surveys of hundreds of teachers across the UK reveal that they estimate that 50% to 70% of 10-year old pupils have not achieved this objective.
There are some problems with relying on memory as a major contributor to maths education. One is that what you fail to refresh by on-going exposure is forgotten or less well remembered. This may be one reason why maths in the adult population is poor. Adults do not have daily maths lessons. The second reason is that what you learn the first time that you meet a fact or a concept is a dominant entry to the brain, identified in research by Buswell and Judd that dates back to 1925. If you get the wrong idea first time, you are stuck with it. There is a message there for the importance of how we teach maths to young children.
This test was designed, trialled and norm-referenced over a total period of five years. The final version used was the fifth modification. A time limit of fifteen minutes was chosen primarily so as not to over-face students and adults and yet still lead to meaningful information. The data for norm-referencing the test for school age students was collected from 1783 pupils from over fifty schools across the UK, State and Independent, rural and urban. Further data was collected from 766 students and adults aged from 16 years to 59 years old.
The data from individual questions provided information about the maths that ‘would be expected of an eleven-year old pupil’, which really should read as, the ‘maths that we arbitrarily think an eleven year old pupil should be able to do.’
Although the test was standardised for ages 7 years to 59 years, my focus in this article is on the data for four ages, ten (the last year in primary), thirteen (the end of Key Stage 3), fifteen years old (the year leading to GCSE) and a post-school group ages 16 -19 years old. This gave a sample size of 631 school pupils and 281 for the 16-19 years group.
(The errors for this item were extremely varied, but the most frequently occurring error was 5067 metres, which could be explained as the consequence of a partially remembered mnemonic about moving the decimal point).
The percentages of correct answers for the 10 to 15 years old pupils showed a steady improvement by age for all items from the test. Items that featured addition and subtraction attract the highest percentages of correct answers. Division and multiplication items did not generate similar levels of success.
The data suggests that any move towards a less than straightforward example creates a significant drop in success rates. Could this suggest a cohort who can only do maths when it is the mathematical equivalent of literal?
This seemed to be the case with the two algebra questions, where over 65% of 15 years old pupils gave correct answers for Item 41 compared to 23.6% for Item 42. Similar results were found for the 16-19 year old cohort.
There are a number of issues around making maths education efficacious for more students, especially those with special needs or indeed the bottom quartile of achievers. I believe that one of the prime contributors to low success is the belief that maths is predominantly about facts and procedures. Over-emphasis on rote learning facts, methods and content will be inefficient at best and ineffective at worst for many children. Mental arithmetic which is heavily dependent on strong working memories will be discriminatory, particularly when the expectation is that answers should be provided quickly.
There is also some evidence to suggest that some topics may be introduced too soon. For example, the percentages achieving a correct answer for converting 2 ¼ hours into minutes progressed steadily from 38.2% at 9 years, 50.1% at 10 years, 70.2% at 13 years to 84.0% at 15 years. Here the most frequent error was 145 minutes (which is based on ¼ hour being ¼ x 100).
A 2000 publication by the National Research Council of the USA provided three key findings about learning. The second finding is:
To develop competence in an area of enquiry, students must:
(a) have a deep foundation of factual knowledge,
(b) understand facts and ideas in the context of a conceptual framework, and
(c) organise knowledge in ways that facilitate retrieval and application.
The requirement (a) can be debated in that we need to ask, ‘What is a ‘deep’ foundation of factual knowledge?’ For example, way back in time when I was at primary school we were required to learn all the multiplication facts up to 12 x 12. The requirement to be able to retrieve from memory 12x facts is no longer there in our twenty-first century lives. However, if children understand what the multiplication facts are and how they are constructed then they can work out 12w for any value of w. That same way of facilitating retrieval can be applied to other facts, for example, the 6x or 7x facts. Basic facts can be placed in the context of a conceptual framework and can be organised in ways that facilitate their retrieval. That may make us re-define the adjective ‘basic’. Maths is a subject that builds. We must be cognisant of its developmental nature right from the start.
There is a core conceptual framework for maths. Its formation begins as soon as children learn to count. The children who are not taught the implications of the earliest of maths will not understand maths. It is a key requisite for all learners that they understand the deceptively simple stuff, not just retrieve it from memory. A part of that developing understanding of concepts is meta-cognition. Meta-cognition is not the exclusive prerogative of the more able.
It seems that there is a need to take an approach to teaching maths that is more cognisant of how pupils learn using our knowledge of why children with special needs do not learn to inform how we teach all our children. And we need to include parents into that process, by explaining, or at least accepting, that any new version of a procedure will disenfranchise parents. This is about long-term goals, including the cultural rehabilitation of maths. It doesn’t need a revolution or radical changes, but curriculum designers may have to be educators first and mathematicians second.
Boaler, J (2009) The Elephant in the Classroom. London. Souvenir Press
Bransford et al (2000) How People Learn. Washington, DC. National Academy Press
Buswell, GT and Judd, CM (1925) Summary of educational investigations relating to arithmetic. Supplementary Educational Monographs. Chicago. University of Chicago Press.
Chinn, S (2017a) The Trouble with Maths. A Practical Guide to Helping Learners with Numeracy Difficulties. (3rd edn.) Abingdon. Routledge
Chinn, S (2017b) More Trouble with Maths. A Complete Guide to Identifying and Diagnosing Mathematical Learning Difficulties. (2nd edn). Abingdon. Routledge
Dowker, A (2005) Individual Differences in Arithmetic. Hove. Psychology Press
Hattie, J (2009) Visible Learning. A Synthesis of over 800 Meta-Analyses Rekating to Achievement. Abingdon. Routledge
Krutetskii, VA (1976) The Psychology of Mathematical Abilities in Schoolchildren. Chicago. University of Chicago Press.
Porkess R et al (2011) A world-class mathematics education for all our young people.
Rashid, S. and Brooks, G. (2010). The levels of attainment in literacy and numeracy of 13- to 19-year olds in England, 1948-2009, Literacy Today. 32, 1, September 2010, pp 13 – 24
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