What is Dyscalculia?
There is no national or international agreement about dyscalculia; there is a range of existing definitions. The UK government defines it like this:
“Dyscalculia is a condition that affects the ability to acquire arithmetical skills. Dyscalculic learners may have difficulty understanding simple number concepts, lack an intuitive grasp of numbers, and have problems learning number facts and procedures. Even if they produce a correct answer or use a correct method, they may do so mechanically and without confidence."
Professor Mahesh Sharma, an expert in the field from Cambridge College, USA, describes dyscalculia like this: Dyscalculia is an individual's difficulty in conceptualizing numbers, number relationships, outcomes of numerical operations and estimation - what to expect as an outcome of an operation.
It is not known how many people experience dyscalculia. Professor Sharma estimates it at 4%, and this figure is supported by Dr Bjorn Adler, a Danish neuropsychologist who takes a particular interest in the subject.
As with dyslexia, dyscalculia is identified when a person experiences most of the items on a list of indicators, all the time. It is important to note that difficulty with Maths as such is extremely common; what distinguishes dyscalculia is a fundamental difficulty with classic concepts of number, time and space.
The indicators of dyscalculia include difficulties with the following, starting with the most significant:
- Understanding the concept of positive whole numbers as indicators of order and size
- Instinctive judgement of length of time, keeping track of time and planning time schedules
- Remembering simple arithmetic facts; employing these to generalise using the place value system
- Carrying out everyday financial transactions, such as giving change and handling a bank account
- Following sequential directions - sequencing (including reading numbers out of sequence, substitutions, reversals, omissions and doing operations backwards), organizing detailed information, remembering specific facts and formulas for completing their mathematical calculations.
Of the above indicators, 2 and 5 could be indicators of dyslexia. There is also overlap between dyspraxia and dyscalculia, and the following are strong indicators of dyscalculia and/or dyspraxia:
- Acquiring spatial orientation/space organisation / direction, easily disoriented (including left/right orientation), trouble reading maps, and grappling with mechanical processes.
- Learning musical concepts, following directions in sports that demand sequencing or rules, and keeping track of scores and players during games such as cards and board games.
Professor Sharma says that dyscalculia can be quantitative, which is a difficulty in counting and calculating; or qualitative, which is a difficulty in the conceptualizing of mathematics processes and spatial sense; or mixed, which is the inability to integrate quantity and space.
In order to succeed with Maths, Professor Sharma says that these skills are needed: following sequential directions, spatial orientation/space organization, pattern recognition, visualization, estimation, inductive and deductive thinking. Dyscalculic people struggle to develop those skills.
There is much overlap between dyslexia and dyscalculia in terms of arithmetical and mathematical calculation, because they both involve difficulty with short-term memory, sequencing and dealing with symbols. However some dyslexic people are very good at Maths; this is because they can often use three-dimensional thinking to see the structure of a problem in a way which linear thinkers cannot. They can be ‘grasshoppers’ (Bath and Knox 1984): quick creative thinkers with strong visual mathematical abilities; however these frequently go unrecognised, both by the student and their tutors.
To confuse the picture further, there appears to be a group of people who are skilled at deductive analytical thinking, and thus very good at algebra yet very weak with remembering ordinary number bonds and carrying out number calculations. These people usually show dyslexic/dyscalculic tendencies, but they can be excellent mathematicians.
This diagram is in the process of development by Jan Robertson of De Montfort University, Leicester, UK. It is important to note that types B, E and I are rare, as dyscalculia so often overlaps with other types of neurodiversity:
Green = potential positive aspects; Red = potential problem areas
area |
description |
A |
Numerical techniques, arithmetic, sequencing, linear thinking Algebra, geometry, reasoning, intuitive ideas, thinking ‘out the box’ |
B |
Understanding number concepts Reasoning, algebra, responding to straightforward graded examples |
C |
Drawing, geometry, paper and pencil numerical calculations Algebra, reasoning, oral discussion of Mathematical problems |
D |
Algebra, reasoning, analysing, linear thinking, arithmetic Visual and spatial awareness, intuitive ideas, ‘grasshopper thinking |
E |
Reasoning, understanding numbers and using quantitative methods Responding to discussion relating symbols to words, writing down techniques, practical scenarios |
F |
Algebra, arithmetic, geometry, estimation Responding to oral discussion, a slow graded accurate approach |
G |
Integrating numbers with visual ideas, reasoning, moving from concrete to abstract Practical, concrete problem solving, especially with visual component |
H |
Reasoning, deductive thinking, especially on paper, algebra Responding to oral discussion, remembering arithmetic techniques |
I |
Analytical reasoning, algebra, higher Mathematical ideas Practical, concrete problem solving; simple arithmetic |
Dyscalculia Spectrum
Jan Robertson, from her experience as a mathematics tutor, proposes a dyscalculia spectrum which looks like this (the things in the boxes are what people have difficulty with):
Extreme Ordering and comparing whole numbers. Judging time and direction |
Serious Everyday tasks involving time and money computations and judgements, including with a calculator |
Moderate Understanding the concept of decimal numbers and fractions, and how they are written. Slightly more abstract concepts such as area, volume, weight. |
Mild Understanding the concepts of minus numbers and algebraic variables; comparisons of size |
Many people have difficulties with mathematical activity for a variety of reasons. This may not be the result of dyscalculia. Jan Robertson describes three modes of mathematical activity:
Intuitive mode
Everyday maths-type activities: concrete, specific, immediate problem-solving, familiar context, instinctive response, rough estimation
Tool-box mode
Numerical operations, symbolic representation, learned numerical and algebraic techniques, translation of mathematical language
Abstract mode
Creative activity, decision-making, discovery, deduction, reasoning with understood symbols.
Those who identify with ‘extreme’ and ‘serious’ dyscalculia are unable to access any of the three modes. ‘Moderate’ dyscalculia means being ‘stuck’ in intuitive mode, and ‘mild’ dyscalculia means being stuck in toolbox mode.
The definition and description of dyscalculia are not easy to agree on, as the two views above demonstrate. There is information about the subject on the Dyscalculia and Dyslexia Interest Group website, with a useful set of web links provided here:
http://ddig.lboro.ac.uk/pages/dyscalculia_web_links.html
As a mainstream lecturer, how can I help a student with mathmatical difficulties?
- Try to find out where the student is, conceptually, until you reach their level of understanding. Start the learning process gradually, controlled by their pace, with carefully graded examples
- Avoid correcting trivial copying mistakes and calculating errors
- Throw away the rule book. Students deserve explanation and justification. It inspires Mathematical thinking
- Use calculators - they are a useful tool. It helps to focus all energy on the concept you are trying to put over, not on some complicated and unnecessary calculation procedure
- Root new techniques in visual, concrete, intuitive scenarios, wherever you can
- Build on student’s strengths. This type of thinker often excels at creative, divergent, right-brain thinking. Give them ‘open’ questions- students are likely to enjoy them and they greatly help with the concept anyway
- Look at how non-dyscalculic, mathematically-intuitive thinkers tackle things, and gently encourage thinking in this way, where you can.
There are two articles about dyscalculia and mathematical difficulties linked from this page: http://www.brainhe.com/resources/
BathJB and Knox DE (1984) Two styles of performing mathematics. In Bath JB & Knox DE (eds) Dyslexia: Research and its Application to the Adolescent Bath: Better Books.