Early Stages in Mathematical Development

Expected curriculum attainment.

Addition, subtraction, multiplication and division are all expected learning outcomes in the new US curriculum. Addition and subtraction are learned during Kindergarten and 1st Grade. To begin with children should use fingers and other objects to represent addition and subtraction up to 10. They are expected to know addition and subtraction up to 5. Then in Grade 1, the language of arithmetic is outlined so that children learn how to represent different problems.

Furthermore, children are expected to know about the property of commutativity and understand the relationship between addition and subtraction. There is a consolidation of simple addition and subtraction through Grades 2 to 4. The foundations of multiplication are introduced in Grade 2 particularly through the use of rectangular arrays. This is extended in Grade 3 when multiplication facts are learned and division is introduced, and consolidated in Grade 4.

Mathematical difficulties and simple arithmetic

Children with a mature understanding of simple arithmetic will usually start using memory recall of simple arithmetic facts when calculating problems (especially for multiplications), having understood the principles underlying operations. On the other hand, children with mathematical difficulties often find it hard to reach this level of fluency. They tend to use immature strategies even at later learning stages. For example, continuing to use ‘counting all’ to do addition when typical developers have moved onto ‘counting on’. The use of these immature strategies leads to two problems. First they often make errors because the immature procedures are more laborious than advanced strategies. For example, when using ‘counting all’ they may count wrongly and thus get the wrong answer. And secondly, because these immature strategies use more resources they often forget the problem before they have worked out the answer. This means that a correct answer cannot easily be stored for specific facts.

As an illustration, if a child wants to calculate the answer to 3+4, if they ‘count all’, they may sometimes get the answer 7, but also get answers such as 6 and 8 due to counting errors. As ‘counting all’ takes a long time to do, the child may also forget that 3+4 was the original question. Thus problems with both these elements mean that it is much more difficult to memorise the arithmetic fact that 3+4=7.

These problems could occur due to a poor understanding of counting or underlying memory deficits. Furthermore, difficulties with the language of arithmetic may lead to a lack of understanding of what is expected.

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