Why Maths Education isn't Working for All Pupils
Children giving up on maths
Steve Chinn
This article was published in the NASEN magazine ‘Special’ in January 2010
I lecture on learning difficulties in maths around the UK and around the world. Lately I have been asking the teachers a question, ‘At what age are there enough pupils for you to notice, in your school, who are giving up on maths?’ The responses are so very similar wherever I am in the world, ‘Seven or eight’. Some teachers say six years. This worrying information raises several questions, such as ‘Why is this?’ and ‘How do these pupils survive another eight or nine years of compulsory maths?’
Why children give up on maths
I have asked some teachers to rank my speculations on why pupils might be giving upon maths. The items ranked as most influential were:
- having to answer questions quickly,
- memorising facts and
- mental arithmetic.
Although these responses were not focusing on pupils with special needs, there can be no doubt that pupils with special needs will feature heavily in the ‘giving up’ group. This will be in part down to the fact that the three key contributors to de-motivation in maths are three problems that impact especially heavily on pupils with special needs.
Some learning characteristics of children with special needs
Children with special needs have more problems with aspects of mathematical long-term memory, notably retrieving basic facts from memory. They are slower to retrieve and process information. Also, they often have poor short term and working memories, both of which are key pre-requisite skills for mental arithmetic and indeed mathematics in general.
These three deficits combine to make a powerful obstacle to learning maths. Unfortunately it is hard to find comments from policy makers that acknowledge this fact. Instead they tend to say ‘more mental arithmetic’ and ‘children will learn the times table facts at an earlier age’. And then, to make sure learning really is a problem there is within the culture of arithmetic the expectation that it has to be done quickly.
Mathematics memory
Maths is often taught as an exercise in memory... ‘This is what you do’. For example, for division by a fraction, ‘Turn upside down and multiply’. Apart from the quaint image this generates, it is a highly attractive solution to a difficult process in maths because it works and is easy to remember.
In Bloom’s taxonomy of learning, recall and retrieval is the lowest level of intellectual activity. An informal survey of the Irish equivalent to GCSE maths found that the majority of questions on exam papers were at this level. The same was true of the national Grade 12 assessment in South Africa in 2008. I suspect that this is another universal trend. By taking this ‘remember and retrieve’ approach with all children we make the assumption that it is effective for all children. Not surprisingly, children who do perform badly in maths do not remember facts and processes as effectively as their better performing peers. This problem is exacerbated by their lack of effective compensatory strategies. Their strategy for overcoming the deficit is counting and counting is a low level cognitive skill with many disadvantages. Better performing peers use linking strategies which are based on understanding numbers, operations and how they relate. Lower performing pupils seem far less able to employ these significantly useful strategies. We disadvantage our special pupils by not being aware of their reliance on counting and then not placing more emphasis on developing strategies that take them beyond counting. The attitude, even if not deliberate, of keeping them at the lowest cognitive level is patronising.
I am sure that success in maths motivates and that failure de-motivates. One of the issues with learning basic facts is that pupils cannot avoid knowing that they have failed. A consequence is a sense of helplessness. In the 2007 DFCS document ‘Getting back on Track’ a 13 year old girl is quoted, ‘I don’t get stuck in other subjects – only maths. When I’m doing English I can always get on with my work, If I’m not sure about a spelling, I can just have a go and still get my work done. But I can’t do that in maths. If I’m stuck I can’t do anything but wait for help.’
Some students who have better memories can survive maths for quite a while simply by remembering. Sadly this will not last forever and then they begin to fail. So a better memory can actually hide a potential maths LD.
Slow processing
This is the second example of a clash between the learning characteristics of the student and the demands of the subject. I am unconvinced that doing maths quickly is a necessary demand. I heard that some Australian pupils are encouraged to play the ‘gunfighter game’. Two pupils face each other, pretending to be gunfighters. The teacher says a maths fact, for example, ‘Six times seven’ and the pupils who draws his (pretend) gun and shouts out the right answer wins the gunfight. Not much differentiation there.
One consequence of having to do a challenging task quickly is increased anxiety. Increased anxiety has a negative impact on working memory and so the pupil has less ability to perform the task. Students use working memory to process information mentally.
There is an interaction with memory for facts. The pupils who can retrieve facts from memory tend to be much quicker than those who use counting.
If pupils can learn linking strategies such as ‘doubles plus one’ they will improve their number sense, but they will be slower. Thus they may still suffer despite developing an understanding of number.
Short-term and working memory.
I should first explain what these memories are by giving an example. If you ask a pupil to repeat a series of single digits, said at one second intervals, they will be using their short term memory. Most adults can manage about 7 digits. Many pupils with special needs will struggle to recall three. The working memory task involves digits again, but now the pupil has to repeat them in reverse order. This means they have to hold the sequence in their short-term memory and reverse them. It is a harder task and scores will be lower.
These two memories have a significant impact on learning. For example, many research papers equate poor working memory with poor arithmetic skills. Indeed it seems quite logical that a good working memory is a pre-requisite skill for mental arithmetic.
Starting any lesson with mental arithmetic tasks, unless there is very careful and empathetic differentiation, will create anxiety and failure for many pupils with special needs. I think that every teacher should know the short-term memory and working memory capacities of their pupils. Of course, for most pupils, the memory capacity will increase as they get older. Unfortunately, this is not so for every pupil.
Understanding to learn
Students with low achievement levels in maths often have only one strategy for accessing facts... counting. Counting is an inefficient, often inaccurate and a very slow strategy particularly for working out multiplication facts such as 6 x 7. If learners can’t count backwards (and many cannot) then they can’t subtract. So the double whammy is: can’t remember, bad at counting.
We need to reassure pupils that rapid retrieval is great, but not achievable for every fact for everyone. Pupils should be encouraged to learn accessible facts such as those involving 1, 2, 5 and 10. They can then relate these to other facts and thus help to build a sense of numbers, operations and how they inter-relate. The level of success will, obviously, depend on the pupil, but then introducing something that can lead to success is better than practising over and over again something which does not.
Insecure learners look for consistency and patterns. Consistency is very reassuring. Patterns help memory. Of course, patterns and consistency help all learners.
Outcomes
There is enough evidence around to confirm the importance of mathematics in everyday life and in employment. Much of what is needed post-school is about number sense and estimating. Memorising without understanding will not develop number sense nor will it give lasting skills. Our memories are designed to forget what we do not practise. If students and adults avoid maths their skills will become less and less. Avoiding maths when you are seven is not a good thing!
Dyscalculia and Mathematics Learning Difficulties.
This paper was published in Dyslexia Review, the journal of Dyslexia Action’s Dyslexia Guild in Nov 2011.
Dyslexia is a specific learning difficulty that affects English. Dyscalculia is a specific learning difficulty that affects mathematics. Both are significant factors for pupils and students because English and mathematics are the two core subjects in the school curriculum.
There has been an awareness of dyslexia for over 100 years. Over the past forty years there has been extensive research into dyslexia such that our level of understanding of its aetiology, manifestations and educational intervention is reassuring. Our understanding of dyscalculia lags far behind.
It seems that around 5% of the population may be dyscalculic, but if we believe that there is a spectrum of difficulties, as there is with many learning difficulties, then some 25% of the population have difficulties with learning mathematics.
So, there will be some people at a very low level of mathematics skills and understanding who struggle to have any concept of what a number means. This lack of skill with even basic arithmetic will have an enormous impact on their lives. Students can also experience serious consequences with a less severe level of maths difficulty, for example, the person who cannot push a Grade D GCSE up to a Grade C. This may prevent them from obtaining a place at University, even though their A level grades in other subjects are good.
People may say that they don’t need maths, but they do.
I believe that the relatively recent research into effective teaching for dyslexic pupils has influenced mainstream teaching. When Ministers of Education are asked how they will improve reading they have an answer that is soundly researched, ‘More phonics’. When it comes to maths they slip back to the Victorian era and say, ‘More learning times tables’, ‘More mental arithmetic’ or ‘Make it harder’.
There is a problem with these reactions, particularly for students with specific learning difficulties. After 24 years of teaching dyslexic students and some thirty years studying maths learning difficulties, lecturing and listening to teachers from around the world, I know that those interventions are pretty much as inappropriate as they could be for children and adults with maths learning difficulties. Until that is acknowledged we will continue to have 25% of the population displaying learning difficulties in maths, a situation that has existed for decades.
Currently maths is compulsory until age 16 (but in reality beyond that). It has been suggested that problems could be solved if it was compulsory for all up to 18 years. Let’s do the maths…. 5 years to 18 years is 13 years of 40 week long school years. Some 520 weeks are used to teach students maths. Maybe it’s not the time that is the problem…..
When I ask teachers, across the UK and in many other countries, ‘At what age are enough children in a class giving up on maths for you to notice?’ the most frequently occurring answer is ‘Seven years old’. I find that depressing and sad. It would be helpful to know why this is so. I also ask teachers what percentage of children at age 10 or 11 years ‘do not know their times tables’. I rarely get an answer below 50%. I find that telling, but not in that knee-jerk reaction way of some traditionalists, ‘Well, give them more practice then.’ It is far more complicated than that.
Maths learning difficulties often occur alongside dyslexia. The prognosis for individual learners is, of course, individual. In some 24 years of working with significantly dyslexic students I have seen many A grades and a high percentage of grades at C and above in maths as the outcome of appropriate teaching.
Memory is a key factor in learning maths. Just how key depends on the way maths is taught. One characteristic that is common for specific learning difficulties (and many other learning difficulties) is poor memories, that is, working memory, short term memory and mathematical long term memory. There are several implications, interactions and consequences here.
I like to link theories of education and other fields of knowledge in the hope that the links will enhance my understanding of each the theories involved in the link. For example, I admire Howard Gardner’s theory of multiple intelligences. I think a similar situation applies to long-term memories. So, for example, you may have a great memory for faces, but a really poor one for numbers. Specific learning difficulties can be linked and seens, not just ‘a difficulty’, not just ‘an intelligence’, not just ‘a memory’, but to specific strengths and weaknesses.
If maths is largely taught as a matter of memorising facts and procedures, and there is credible evidence to suggest that this has been the case for many years, then students with poor long term mathematical memories will fail.
A poor short-term memory has a pervasive impact on learning. It controls how many items you can hold in your memory for a short time. For example, remembering oral instructions on how to do a calculation or copying information from a board. Working memory is a vital skill for mental arithmetic and mental arithmetic is often advocated as a way to improve maths performance. There would have to be empathetic differentiation applied in the classroom to meet the potential problems that will be created here, problems such as frustration, over-experience of failure and de-motivation.
There is a book, ‘How People Learn’, based on extensive research in the USA. It delivers its main message in three succinct Key Findings. One of them emphasises the importance of using understanding to support memory. In maths we can use that advice, we can inter-relate facts and procedures to help develop concepts. Sadly we tend to patronise learners by merely telling them what to do without explaining why.
Maths starts with counting, in ones. Most parents are comfortable with this level and can help their child. If children stay at that ‘counting in ones’ stage then they will not learn the concepts of maths. Tallies are not a developmental strategy.
Then there are the problems created by the introduction of new procedures for doing maths. Every time educators change or tweak a maths procedure, for example, the grid method for multiplication, they disenfranchise many parents.
But old beliefs can be a problem, too. If teachers insist that children have to rote learn facts and retrieve them, quickly’, from memory, they de-motivate many children.
Both these are examples of unforeseen consequences. We should by now have enough understanding of how people learn to make them not unforeseen.
Children have to understand maths in order to succeed. This is not about saying that they have to understand algebra or fractions. That may be the case eventually. What it does mean is that children have to understand maths from the very beginning, for example, understanding the relationship between 1 and 2, between 1 and 10 and 100, what ‘add’ and ‘divide’ really mean. It is our vulnerable learners who need good, conceptual teaching, not just a ‘Learn this’ approach. The clue is in the label. Maths Learning Difficulties and dyscalculia need appropriate teaching, not just ‘good’ teaching. They need an appropriate curriculum, not just a curriculum that works for the 75% who get by in maths.
© Steve Chinn 2011