Maths: a hidden ability??

manipulatives

 

Dyscalculia is a term used to describe extreme, specific difficulties in maths. Many of these difficulties may be found in a dyslexic profile.

 ‘Dyscalculia is a condition that affects the ability to acquire arithmetic skills. Dyscalculic learners may have difficulty understanding simple number concepts, lack an intuitive grasp of numbers, and have problems learning number facts and procedures. (DfES, 2001) 

In Steve Chinn’s book, he suggests that many children ‘give up’ on maths around the age of 7 and lists the following factors:

  • Having to do maths calculations quickly.
  • Learning facts and procedures (without understanding them).
  • The extremely judgemental nature of maths. An answer is right, or it is wrong.
  • The inconsistencies in early arithmetic confuse children.
  • Being asked to do tasks that are beyond the capacity of the child’s working and short-term memories.
  • The vocabulary and language of early maths is often everyday vocabulary and language, but used in a maths setting.

(Chinn, 2012).

Haylock and Cockburn (2013) feel that much can be done in the early stages of teaching mathematics and that some difficulties are created by teachers, who themselves were taught ‘by drill’. They call for teachers to have a thorough understanding of the basic mathematical concepts and principals. This will, in turn, enable them to help children construct that understanding for themselves.

Butterworth and Yeo (2004:1 ) make the point that there are many reasons for underachievement in mathematics, including ‘inappropriate teaching, behavioural problems, anxiety and missing lessons’ which makes ‘identifying a special condition difficult’.

It is difficult for researchers to make conclusions about the cognitive processes involved (Ansari and Bugden, 2015). Just as neuroscience (Magnetic Resonance Imaging, MRI and Electroencephalography, EEG) is being employed to understand literacy development, so too is it being employed to further understanding of Developmental Dyscalculia: its diagnosis, intervention and treatment (Ansari and Bugden, 2014).

As with literacy and dyslexia, multiple-deficit models of dyscalculia can explain the heterogeneity of the difficulty and the range of approaches required to compensate (Landerl, 2015). Butterworth (2003) has developed a ‘Dyscalculia Screener’ (2003), identifying dyscalculic learners between 6-14+ years. It aims to assess prerequisite skills that underlie efficiency in maths and recommends intervention strategies.

The Dyscalculia screener tests elements of numerosity (subsitising), within a time constraint. Pupils are shown collections of dots and asked to quickly and accurately report how many they see. This ability, Butterworth argues, distinguishes those with dyscalculia from others.

Chinn (2012) suggests using a range of tests and informal diagnostic activities that probe the way a child is thinking; arriving at the underlying misconception. The WRAT4 can provide a comparison between literacy and maths. The MALT assessment (2009) gives useful information about answers. Kelly, Phillips and Symes (2013) provide a useful dyscalculia checklist.

If many children ‘give up’ age 7, this resonates with early work from Buswell and Judd (1925) who found that the first time you learn a new idea, it creates a dominant learning experience. Chinn (1998) lists the following areas of potential difficulty for those with dyslcalculia, all of which might apply to a dyslexic profile:

  • Short Term Memory

If instructions or information given is too demanding for the child, they will have memory overload and will not retain the information.

  • Working Memory

Working memory is especially important in maths, as numbers have to be held in memory during mental arithmetic. There are usually steps involved too and these may also overburden working memory so that information is lost.

  • Consistency

Consistency is reassuring and it makes it possible for pupils to be relaxed and deal with new experiences, assimilating them. Vocabulary and procedures need to be consistent so that teaching and learning is effective and explained carefully.

  • Speed of Processing

Dyslexic children and those with dyscalculia are usually slow to process auditory information and to work through steps as working memory is generally weak. Having to do maths calculations quickly can create anxiety, which in turn reduces working memory capacity and also affects long term memory

  • Rote facts (Long Term Memory)

Committing basic facts to long-term memory, in particular times-table facts, is extremely challenging. Time is not put aside in the curriculum for this. It is demoralising for children who cannot master them in the way their peers can.

  • Language (vocabulary of maths)

Eg The English language does not offer consistency for two digit numbers, the numbers eleven and twelve do not fit the rest of the pattern. In fractions, the pattern of a quarter and a half does not fit with a fifth and an eighth;       maths vocabulary is often abstract and hard for the learner to retain.

Telling the time, using an analogue clock, can be particularly challenging for dyslexic children: the language, number knowledge and processing involved is all too much! Much of being a successful learner at Primary level is to do with memory functioning. Cognitive weaknesses associated with dyslexia, such as short term, working and long term memory; speed of processing, sequencing and difficulties with attention and concentration are all likely to impact on maths. The Baddeley model of working memory (2000) shows short-term memory (STM) as split between visual semantics (Visuospatial sketchpad) and the verbal STM (Phonological loop). These two areas are not directly connected but interact with the episodic buffer, which transfers information to the long-term memory.

It would appear that in dyslexia, certain information does not get readily stored in the long-term memory (LTM). Auditory information is held in the store where it rapidly decays. In dyslexia, this is thought to be impaired, as tested using Nonword Repetition (Roodenrys and Stokes, 2001). Maths auditory input can be lengthy and dyslexic learners may struggle to retain instructions and the steps required for task completion.

Chinn, (2012) suggests that visual representations of number, which create a number sense, and the use of concrete materials (numicon, dienes, Cuisenaire) helps to engage struggling learners in maths. Visualisation is a key concept when teaching maths to pupils with a dyslexic and dyslcalculic profile. Ronit Bird (2009) also suggests activities that involve concrete manipulatives and visualisation. Dominoes and playing cards can be used to develop a number sense. Maths language can be practiced and supported through a number pack, akin to the spelling and reading packs used in literacy programmes. Sharma (2016) recommends using Cuisenaire rods to support learners, these are especially good for fractions.

Chinn (1998) adds spatial awareness as a potential problem for dyslexic learners. Importantly, spatial awareness is needed for the layout of simple work on paper, such as the column method but also in drawing and interpreting graphs, charts and other symbolic information. It may cause problems in geometry, algebra and copying from the board e.g. writing down the wrong question number. It can be helpful to plot the space out with dots before beginning the task.

As suggested by Butterworth and Yeo (2004), anxiety is a huge factor in maths. ‘Number Anxiety was first introduced as a concept by Dreger and Aiken (1957). Anxiety can also be an issue in dyslexia and the acquisition of literacy skills, (Carroll et al., 2005; Carroll and Iles, 2006). Worry and anxiety are associated with the need to be prepared; fear and panic are primitive responses to a real or perceived defined threat. This will be exacerbated by having to do calculations quickly. Overall, studies suggest that attitudes to mathematics tend to deteriorate with age during childhood and adolescence (Wigfield and Meece, 1988; Ma and Kishor, 1997).

Young, Wu, and Menon (2012), in a math anxiety study, found brain activation in 7 to 9 year olds was in the right amygdala, a site previously tied to the learned fear response in adults. They also found that math anxiety was associated with reduced activity in posterior parietal and dorsolateral prefrontal cortex – regions involved in mathematical reasoning. Mathematics anxiety has been defined as:

“a feeling of tension and anxiety that interferes with the manipulation of numbers and the solving of mathematical problems in … ordinary life and academic situations” (Richardson and Suinn, 1972).

It is, of course, unclear where math’s difficulties create anxiety or vice versa, a self-perpetuating cycle. Moreover, there seems to be two aspects: affective and cognitive. Cognitive might be worry about performance and affective is the emotional response: nervousness and tension – exacerbated by exam situations (Liebert and Morris, 1967). Competence in numeracy needs to be addressed because difficulties in maths impacts negatively in life. Dyscalculia refers to severe difficulties with number and people with dyscalculia may struggle with: handling money, budgeting, time-telling (including recording times, dates and appointments), using pin numbers, remembering personal information (like date of birth) addresses and post codes, travelling and directions. Difficulties with these areas can prove incredibly stressful.

Within maths, if there is not a secure foundation, it is necessary to define the key concepts and go back, before moving forwards. There are many books available offering games and puzzles, which provide opportunities to overlearn, Bird (2011) and apps such as Doodlemaths are very successful for the same reason. Bryant, Bryant, Shin and Pfannenstiel (2015) highlight, ‘Providing a cumulative review of previous concepts’, as an example of a practice designed to deliver explicit and systematic instruction. It is important to analyse the mistakes student make in maths and look for patterns, ask questions and address misconceptions. This is all part of good practice within the ‘plan, do, review cycle’.

Whilst auditory processing is a weakness in dyslexia, there is evidence to show that the visuospatial sketchpad is not impaired and may even be a strength (see meta-analysis by Swanson, 2006). This is important information as it suggests that visuals can be used in interventions to boost confidence and improve outcomes; using areas of strength to support weaker areas, i.e. Hebb’s law: “Neurons that fire together wire together”. When taught simultaneously with the strong one, the weaker modality becomes strengthened.

There has been greater research into literacy difficulties, than into dyscalculia and difficulties with maths but with a recent emphasis on achievement within maths nationally, comes a new focus. The new national curriculum (2014) whilst more rigorous, does not address instructional practices. By applying what works within literacy, maths may become easier to access for many students.

The methods for improving literacy for pupils with Specific Learning Difficulties are well established:

  • Multi-sensory
  • Phonics based (especially early in reading i.e. KS1)
  • Systematic (structured, cumulative and sequential)
  • Include opportunities for overlearning/’little and often’
  • Teaching learning strategies is also considered important – this might also be considered ‘metacognition’.

(Rose, 2009, Singleton, 2009)

 

Apart from ‘phonics based’ all of the above criteria can be applied to mathematics. There are fewer specialist teachers for dyscalculia and fewer specialist programmes. Chinn (2017) suggests that adjustments to lessons be based on the following four principles:

 

  1. Empathetic classroom management, which is aware of the needs of pupils and the adjustments required to support learning strengths and weaknesses, such as working memory.
  2. Flexibility and a responsive, open approach, allowing for the use of a repertoire of resources and strategies. These change according to the needs of the pupil over time.
  3. Developmental methods, addressing the underlying need, while developing mathematical skills and concepts.
  4. Effective communication, which takes into account working memory weaknesses, slow processing, and different thinking and learning styles. The layout and presentation of work must have clarity.

 

Reid (2016) makes the point that the reading aspect of maths can be a real challenge for children, as dyslexia often overlaps with dyscalculia and maths difficulties. The text should be simplified and children with dyscalculia will need the vocabulary explaining more often and rephrasing. Often, it is the simple, abstract words such as ‘more’ or ‘less’, ‘before’ and ‘after’, which create ambiguity. Where there is more language, and complicated syntax e.g. in a word problem; dyscalculic children may benefit from using highlighters and drawing pictures.

Professor Mahesh Sharma (2015), suggests the following procedure:

  1. Explain the linguistic aspects of the concept.
  2. Introduce the general principle.
  3. Let students use investigations with concrete materials to discover these principles.
  4. Give many specific examples using concrete materials.
  5. Allow students to talk about their discoveries about how concepts work.

Hattie (2009) points out in his meta-analysis search, that the most successful mathematics programmes are strategy-based, involving working out an answer. He recommends a formula, encompassing all learning, to D.I.E for: Diagnose what they do/don’t know, Intervene, Evaluate (reflect). Maths is developmental; it relies on there being solid foundations in earlier stages upon which to build. Areas such as number bonds may not be secure and will prevent entry into more advanced maths. One cannot assume knowledge, it is important to personalise learning and get to know how the student is thinking.

In his 2012 book, Chinn has a chapter on ‘Cognition and Meta-cognition in Maths. Meta-cognition is becoming increasingly important as a concept in all areas of learning and is taught in many literacy programmes ‘thinking about how you are thinking’. There seems to be a worldwide consensus that flexible thinking and meta-cognition are more important than the use of formulas in maths. The use of formulas (or algorithms) is successful for many students who can remember and apply them without understanding what they are doing e.g. for dividing by a fraction: ‘turn upside down and multiply’. For students who don’t remember (dyslexic/dyscalculic), and who need the meaning to help them, this is not successful. They need to fully understand the process in order to apply it successfully the next time. The positive here is that they cannot ‘ape’ understanding. Sometimes, a small misconception is hindering them, when addressed, they take flight! Singapore, with their Singapore method that is creating an impact in the UK, overtly encourages meta-cognition.

From an American study into cognitive style (Bath, Chinn and Knox, 1986), Chinn formed a hypothesis that there are 2 distinct learning styles in maths. The team labelled them ‘grasshopper’ and ‘inchworms’. Grasshoppers are flexible and intuitive, Inchworms are formulaic; slow and steady. Chinn makes the point that often mathematics requires both approaches in the solving of one maths question. He suggests that if there is a non-judgemental approach to maths in the classroom; with emphasis on the method rather than the outcome, it will lead to a positive learning environment, where flexibility is allowed to grow.

Self-esteem is connected to being a successful and independent learner (Burden, 2005). If a learner hasn’t understood a concept, it is important to try and understand how they are thinking. Just as in literacy, the relationship between teacher and pupil is integral:

Building strong and trusting relationships between teacher and child is an essential prerequisite for accelerated learning (Brooks, 2007: 31).

In summary, dyscalculia is a relatively new area of learning difficulty. Following a new focus on Maths, with initiatives like Numbers Count, the DfE recently revealed that Maths is now the most popular A Level, which suggests that there have been improvements. However, work to ensure that all pupils leave school with the necessary skills in Maths to be successful in managing their finances and daily activities, is continuing. This can only be done by finding out how pupils are thinking, offering them visuals, concrete materials and alternative methods i.e. flexibility. In this way, it is hoped that anxiety within Maths can also be properly addressed.

 

References

Ansari, D. and Bugden, S. (2016) Probing the nature of deficits in the ‘Approximate Number System’ in Children with Developmental Dyslexia). Developmental Science, vol 19, issue 5.

Baddeley, A. D. (2000). The episodic buffer: A new component of working memory? Trends in cognitive sciences, 4 (11), 417-423.

Bath, J. B. Chinn, S. J. and Knox, D. E. (1986) The Test of Cognitive Style in Mathematics. East Aurora, NY, Slosson.

Brooks, G., (2007). What works for pupils with literacy difficulties? London: DCSF

Bryant, D. P., Bryant, B. R. Shin, M. and Pfannenstiel, K. H. (2015) Learning Disabilities: Mathematics Characteristics and Learning Exemplars in Chinn, S. International Handbook for dyscalculia and Mathematical Learning Difficulties. London and New York. Routledge.

Burden (2005) Dyslexia and Self-Concept seeking a dyslexic identity. London: Whurr.

Buswell, G.T. and Judd, C. M. (1925) Summary of educational investigations relating to arithmetic. Supplementary Educational Monographs. Chicago. University of Chicago Press.

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Butterworth, B. and Yeo, D. (2004) Dyscalculia Guidance, London: NferNelson.

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Carroll, J. M., and Iles, J. E. (2006). An assessment of anxiety levels in dyslexic students in higher education. Br. J. Educ. Psychol. 76, 651–662.

Chinn, S and Ashcroft, R (1998) Mathematics for Dyslexics: A Teaching Handbook. London: Whurr.

Chinn, Steve (2012) Maths Learning Difficulties, Dyslexia and Dyscalculia. BDA.

Chinn, S. (2017) Dyscalculia and other D’s. SpLD Central Conference.

Dfes (2001) The National Numeracy Strategy. Guidance to support learners with dyslexia and dyscalculia. London. DfES 0512/2001

Dowker, A, Sarkar, A., Looi, Chung Yen (2016) Mathematics Anxiety: What have we learned in 60 years? Frontiers in Psychology, vol 7.

Dreger, R.M. and Aiken, L.R. (1957) The identification of number anxiety in a college population. Journal of Educational Psychology, vol 48, issue 6.

Hattie, J. (2009) Visible Learning Meta Study: a synthesis of over 800 meta-analyses relating to achievement. Routledge.

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Landerl, K. (2015) How Specific is the Specific Disorder of Arithmetic Skills? In Chinn, S. International Handbook for dyscalculia and Mathematical Learning Difficulties. London and New York. Routledge. (pp115-124).

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Reid, G. (2016) Dyslexia: A Practitioner’s Handbook. Oxford: Wiley

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Swanson, H. L. (2006) Working memory and reading disabilities: Both phonological and executive processing deficits are important. In T.P. Alloway & Gathercole, S. E., (Eds). Working memory and neurodevelopmental disorders pp.59-88. Hove: Psychology Press

Wigfield, A., and Meece, J. L. (1988). Math anxiety in elementary and secondary school students. J. Educ. Psychol. 80, 210–216.

Young, C. B., Wu, S. S., and Menon, V. (2012). The neurodevelopmental basis of math anxiety. Psychol. Sci. 23, 492–501.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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