Early Stages in Mathematical Development

Normal Development in Simple Arithmetic

There are four main operators in arithmetic: addition, subtraction, multiplication and division. Of these, the first encountered by children, and where they begin to show difficulties with procedural mathematics, are addition and subtraction.

An understanding of counting is considered to be an essential foundation to develop arithmetic skills. For example, understanding of addition may begin by using counting strategies as addition generally implies that the total set of items will increase.

Addition

Similar to counting, addition typically progresses through a number of strategies. Initially children add by counting all the numerals in a sum (e.g. for 2+3, a children would count ‘one, two, and then three, four, five’). Addition progresses to a counting on strategy where the counting begins from the second number in the sum (e.g. 2+3 a child would start counting from the two and add on ‘three, four, five’). A final counting on strategy is used when the child realises that it doesn’t matter which number you begin with, the answer is the same. This is known as commutativity. When a children understand commutativity they count on from the biggest number (e.g. 2+3 a child would count ‘start with three, add four, five (the two)’). These strategies may be used with fingers or blocks rather than rote memory. Eventually retrieval becomes the most common strategy and counting is used only as a back-up mechanism.

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Subtraction

Although subtraction is often learned around the same time as addition in the first year of formal schooling, it is often considered more difficult to teach. In particular the language that is used to describe subtraction such as ‘take away’ or ‘what is the difference between’, make it much more difficult to acquire. The main types of questions used in subtraction are:

  • Taking away -  In these questions the goal is to partition off some of a set of objects. For example, in the question ‘There are 12 apples on a tree and I pick 3 of them. How many apples are left on the tree?’ there is a set of objects of which the cardinal number is 12, 3 are removed in some way and we are left asking ‘How many are left?’
  • Comparison -  These questions ask students to compare two quantities. For example, if you place 12 red apples next to three green apples, you can compare the quantities and ask ‘what is the difference between the numbers of apples?’ Furthermore, the question can be phrased as ‘How many more red apples?’ or ‘How many fewer green apples?’
  • Complement of a set - These questions ask how many of a set do not share or have a particular attribute. The inclusion of the term ‘not’ in these kinds of subtraction questions can make them particularly difficult to interpret. An example question could be ‘There are twelve apples, how many are not red?’
  • Reduction - This is the opposite of counting on in addition; now you have to count back. In these questions, the child is encouraged to think about subtraction as decreasing the number of counting back. For example, ‘if an item in a store is $12 and it is reduced by $3, how much would it cost now?’ This can be rephrased so that the item starts at $12 and is now $3 less. So if you count back by 3 you will get the new price.
  • Inverse addition - In these questions, the answer can be calculated by working out what needs to be added to make up a particular number. For example, ‘What must be added to 3 to make 12?’ The inclusion of the word ‘add’ in these types of question can also make them difficult to interpret.

Often subtraction is described as ‘taking away’ but it is clear that the language surrounding this operation is much more complex than that. Thus difficulties can emerge due to the formulation of the problem; the actual underlying process of subtraction is the same.

Multiplication

Multiplication is taught after addition and subtraction; although the foundations for multiplication are formed through pattern recognition in Grades 1 and 2 and Foundation level it is not part of the US national curriculum until Grade 3. One reason for this is that early multiplication builds on knowledge of addition by interpreting multiplication as repeated addition. With a repeated addition structure, the multiplication sign can be interpreted as meaning ‘sets of’; e.g. 6 x 3 means 6 sets of 3, and written as 3 + 3 + 3 + 3 + 3 + 3. However, although this makes intuitive sense, multiplication has the property of commutativity (see explanation for addition); this means that 6 x 3 is equivalent to 3 x 6. If we express this multiplication sum using repeated addition, 6 + 6 + 6 does not appear the same as 3 + 3 + 3 + 3 + 3 + 3. The property of commutativity can be explored by using rectangular arrays. For example six rows of three dots when rotated 90 degrees becomes 3 rows of 6 dots. See below.

two rectangular arrays showing 6 x 3 and 3 x 6

Rectangular arrays to represent commutativity for 6 x 3 and 3 x 6

One issue with repeated addition is that it appears to be focused on sets of items, but multiplication and the use of repeated addition in multiplication can be used in other contexts. For example, if we go to the shops and buy three things that cost $6 each, essentially we want to multiply 3 x 6 to find out how much the total cost will be. This time we do not have three sets of 6 things but three times a measure of a value of goods. Repeated addition can be used in the same way but the outcome takes into account what the measure is; if it is 3 x $6, the answer will be $18; if it is 3 x 6 inches, the answer will be 18 inches. With practice, most children learn the multiplication tables from 1 x 1 to 10 x 10 and repeated addition is replaced by memory retrieval.

Although repeated addition is a good conceptual way for a child to understand multiplication, this operator also encounters some of the language issues discussed for subtraction which means that the underlying operation is not clear. For example, in the word problem ‘John has three sweets and Bill has six times more, how many sweets does Bill have?’ the connection between the words and 6 x 3 is not as clear as each boy has their own set of sweets. This question is asking about a quantity scaled by a factor. Nevertheless to find the solution, the actual multiplication remains the same and once again the language of math can lead to difficulties.

Division

Division is usually introduced in schools through the idea of sharing as this is a concept that is assumed to be familiar; importantly in division it must be equal sharing. Thus is we have 12 ÷ 4, we need to share the 12 equally between 4. This can be done with objects to show how many items would be in each of the four groups. However another way of expressing division is as the inverse of multiplication; 12 ÷ 4 can be interpreted as ‘how many fours make twelve?’ This is also known as ‘grouping’ as it involves working out how many groups of a four can be found in twelve. This connection between division and multiplication leads to another interpretation of division. In multiplication repeated addition was the most common way to interpret multiplication, thus it follows that if division can be expressed as inverse multiplication, it can also be thought of as repeated subtraction. To phrase another way ‘how many times can I take four away from twelve until there is nothing left?’ Both inverse multiplication and repeated subtraction can be represented on the number line to illustrate the steps. This is illustrated below:

12 divided by 4 shown as reverse multiplication

12 ÷ 4 using reverse multiplication

12 divided by 4 shown as repeated subtraction

12 ÷ 4 using repeated subtraction

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