In this unit, we explain what is currently known about the development of basic mathematical skills and the difficulties experienced by pupils with dyscalculia. The unit covers the following areas:
- How the mathematical skills of number concepts, counting and arithmetic typically develop
- Definitions of dyscalculia
- The difficulties in identifying pupils with dyscalculia in light of the wide variety of skills that can be affected
- The relationship between dyscalculia and dyslexia
- Working memory and dyscalculia
- The effects of maths anxiety
- Supporting pupils with maths difficulties
This mind map shows the five principles of effective counting, as described in your PDF. Select each principle to find out more.
This principle is based on only assigning one count word to each individual counted object or item. It involves three processes:
- Itemising, where the pupil singles out each item that they want to count.
- Tagging, where they correlate words with each of these items.
- Partitioning, where they distinguish items that they have already counted from those that still need to be counted.
The process of itemising is the first to emerge in children, while tagging and partitioning are often supported by gesturing, particularly in children between the ages of four and five, who will touch or point at objects to count.
To count, children initially need a stable sequence of words that can be used in a fixed, repeatable order. Understanding counting as a form of progression is an inherent feature of initially learning to count, which is further supported by an early capacity to grasp serial order.
Learning stable order can come before learning traditional sequencing. By the age of 2 ??, children tend to use stable sequences of words as tags in counting routines, but may at first, for example, count 1, 2, 4, 9. So a child who counts 1, 2, 3, 4 for one set of objects and 2, 1, 3, 4 for another has not grasped the stable order principle. However, a child who consistently counts 2, 1, 3, 4 has ??? without yet understanding traditional sequencing.
The cardinal number is the number a child reaches at the conclusion of a counting sequence. This count word represents the total property of the sequence, and tells the child how many objects have been counted overall.
It is not simply about ???naming??? the last object in a sequence, but knowing how many items have been counted. A child who recounts a collection when asked how many items they have counted has not grasped the principle. It is also not just a case of repeating the last number they said in a sequence; for example, a child counting five items starting with 3 may say they have counted 7 items.
To understand this principle, children must move past needing tangible, movable objects to count.
The principle states that the one-to-one, stable order and cardinality approaches can be used on any collection of objects, regardless of whether they are tangible.
It focuses on children grasping that they can count non-physical things such as sounds, imaginary objects or even counting words.
To understand this principle, children must know that objects can be counted in any order; it doesn???t matter about sequencing, just as long as everything is counted once and only once. So, for example, a child that understands this principle could start in the middle of a row of objects and count to the end, and then from the beginning to the middle.
Children generally start to show at least some understanding of this principle when they are about three years old, although it is not very strongly established at this point. A study by Arthur Baroody, published in 1993, found that some 4-year-olds did not realise that counting the same row of objects twice from different sides would result in the same outcome (Wiese, 2003).
- View sources
Gelman, R and Gallistel, CR (1986), The child's understanding of number
Wiese, H (2003), Numbers, language, and the human mind
Arithmetic: basic counting to early addition
Early addition can be seen as an extension of counting. This resource shows how children progress from initially learning basic forms of counting, and the ways in which this can influence the development of their future application of maths skills.
Select each circle to find out more.
First stage development
- Single digits, objects, and counting on fingers
- Count-on strategy
- Automatic retrieval
- Min strategy
Single digits, objects, and counting on fingers
Single digit addition is the first approach to arithmetic that children are taught in school. Younger pupils might use visual aids such as regrouping objects and counting them, or counting on their fingers.
The effort and concentration involved in this early method requires some complex formative skills, even for simple sums such as 2+3.?? Return
When using this approach, a pupil will start with the first number in a sum and count on an amount equal to the second number to work out the answer. So for example in the sum 4+3, they would start with 4 and then count on 5, 6, 7.
Counting on uses the cardinality principle, and will generally be used in the early development of counting skills, subsequent to using visual aids and objects.?? Return
Frequent calculations of number bonds such as 3+6 are likely to encourage pupils to remember these so they can retrieve them automatically. Pupils are expected to remember number bonds and effortlessly, automatically retrieve these as their arithmetic skills develop.
As such, pupils who struggle with the early stages of addition may subsequently experience difficulties when this level of retrieval is required. If they are prone to making errors, they may as a result store incorrect answers in their long-term memory.?? Return
This is a more developed version of counting on, whereby the pupil starts with the larger of the two numbers and counts on an amount equal to the smaller number to reach their answer. So for example, they would solve 3+6 by starting with 6 and counting on 7, 8, 9.
Pupils using the min strategy must understand the commutativity principle ??? that changing the order of numbers in a maths problem will not change the result.?? Return
Identifying a specific condition such as dyscalculia can be difficult, given the many explanations for poor performance on maths activities. There are also a wide variety of skills involved in learning mathematics, such as those presented in this mind map.
Early maths skills
- Retrieving arithmetic facts
- Understanding arithmetical operations and laws
- Knowing procedures in multi-digit tasks
- Solving novel word problems
Dyscalculia and dyslexia
Dyscalculia has a high prevalence among people with dyslexia. However, pupils with dyslexia who do not have dyscalculia often perform as well in maths as their peers, with around 10% being among the higher-achieving pupils.
60% dyslexia and dyscalculia
40% dyslexia only
Feelings of anxiety related to maths are common and can develop in many pupils. This can hamper their learning, although there is no evidence that dyscalculia is caused by anxiety. Select each stage on the progress diagram to find out more about the development of maths anxiety.
- Most commonly comes to the fore in adolescence
- Pupils avoid situations which require maths skills
- Progress in maths is hindered