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Teaching computation in the calculator era

Students should:

become good mental calculators using methods they understand.

develop a level of efficiency in some written algorithms consistent with their mathematical ability. Click here to see appropriate goals or continue reading below.

be able to use calculators well for a wide range of questions.

Not everyone thinks written algorithms should be taught at school. Click here to find out why or continue reading below.

Teaching mental computation - general principles

The excellent book by McIntosh, de Nardi and Swan (1994) provides this and more advice, as well as games and activities:

Teachers should learn about the strategies that children use and learn how to describe mental strategies to children.

Offer 15 minute sessions a few times a week, involving class sharing, some instruction, then practice with well chosen games and activities.

Class discussion is important for sharing mental methods among students. Even the weaker students have interesting methods.

Some strategies can be taught through class discussion, explanation and practice. Be wary of including rules to learn by rote (e.g. adding zeros) since they are almost invariably misused by all but the most competent.

Value creativity, exploration, efficiency and inventiveness.

Encourage mental methods before, as well as after, written computation.

Teach mental computation, don't just test it. Emphasise how answers are obtained, not just speed or the correct answer. Tests of mental arithmetic that emphasise basic facts and speed tend to increase mathematics anxiety.

 

 The subtraction and multiplication sections of this resource include class worksheets for Years 3 to 6 and Years 5 to 8 which focus on mental computation. These worksheets can be used in many of the ways described above.

Heirdsfield, Cooper and Irons (1999) summarise the key features of a good program as:
variety
individuality
emphasis on number sense
building understanding of place value and other arithmetic principles.



Goals for subtraction (and addition)

The following table lists mental and written whole number subtraction (and addition) goals for students by the completion of primary school.

Almost all children

Most children

Number combinations up to ten plus ten quickly and accurately
(e.g. 4 + 7 = 11)

Number combinations up to ten plus ten quickly and accurately

 

Automatic recall of special combinations e.g. useful doubles (15 + 15 = 30), pairs adding to hundreds (25 + 75 = 100)

Automatic recall a range of specific useful facts and benchmarks, e.g. doubles, pairs adding to hundreds (e.g. 150 + 150 = 300, 60+40 = 100)

Links between addition and subtraction facts (e.g. link 5 + 4 = 9 to 9 - 5 = 4 etc)

Links between addition and subtraction facts
Written addition of any whole numbers (moderate amount of practice). Written addition of any whole numbers quickly and accurately
(moderate amount of practice).

Written subtraction of any two digit number from any two or three digit number (extensive practice not required).

 

Written subtraction of any two numbers quickly and accurately (extensive practice not required).

Add and subtract on a calculator

Add and subtract on a calculator efficiently using the memory if needed

Mental addition and subtraction of one digit numbers and of single numbers of tens and hundreds (e.g. subtract 3 or 30 or 300)

A wide range of mental strategies for addition and subtraction, applying to a wide range of numbers.

A moderate range of mental strategies for addition and subtraction.

 

A wide range of mental strategies for addition and subtraction, applying to a wide range of numbers.


Goals for multiplication

The following table lists whole number multiplication goals for students by the completion of primary school.

Almost all children

Most children

Multiplication "tables" facts up to ten tens quickly and accurately

 

Multiplication "tables" facts up to twelve twelves quickly and accurately

 

Automatic recall of repeated addition and multiples of 25, 50, 100, 500
eg. able to count 25, 50, 75, 100, 125

Recall a range of specific useful facts and benchmarks, e.g. multiples of 25, square numbers, etc

Links between multiplication and division facts (e.g. link 5 x 4 = 20 to 20 ÷ 5 = 4 etc)

Links between multiplication and division facts

Multiplication of any whole number by a single digit (e.g. 3) on paper efficiently

Multiplication by a simple multiple of ten on paper efficiently eg. x20, x900

Written multiplication by whole numbers with two or more digits efficiently (extensive practice not required)

Multiply and divide on a calculator

Multiply and divide on a calculator efficiently using the memory if needed

Multiplication and division by 10 and 100 mentally

Multiplication and division by any power of ten (10, 100, 1000 etc) mentally

A moderate range of mental strategies for multiplication and division

 

A wide range of mental strategies

The argument against teaching written algorithms

The tables above listing the statements of goals include both written and mental computation. However, some researchers argue that students should not be taught algorithms, but should invent their own methods instead. This argument has been put most strongly by Constance Kamii (see for example, Kamii and Livingston, 1994).

The argument against teaching algorithms:
It is unnecessary. In an classroom emphasising appropriate number activities, children will invent and refine their own methods of calculation.
It is harmful. Traditional teaching of algorithms stops children thinking for themselves, which forces them to give up their own numerical thinking and make children dependent on others.
It un-teaches place value by focussing attention on the digits rather than their values.
It cannot really work. Understanding algorithms requires having both social knowledge (of how they are set out, what goes where etc) and logico-mathematical knowledge of the number principles underlying them. The social knowledge can be learned from being told, but the logico-mathematical knowledge cannot. Because it is to do with relationships between things and is not empirical knowledge observable in the world, it must be constructed in the child's mind.

Kamii and others (1989, 1994, 2000) report on the good results of classroom experiments having children invent their own methods of arithmetic, and also give many examples of the classroom activities they used. The overarching principle is to have children construct their understanding of maths themselves, rather than internalise what others present to them. What do you think?