Mental computation for numeracy
Dr Shelley Dole
Senior Lecturer
Mathematics Education
The University of Queensland
Shelley Dole: Mental computation is a key component of numeracy. It is supported by estimation skills
that serve as a quick means to analyse numerical situations in our daily lives. For example, if you are
dining in a restaurant with 6 other friends and the bill is $236, you know that each person will be paying
approximately $30 each, and a little bit.
In this instance, you have used your knowledge of multiplication facts that 3 times 7 is 21 and related
this to the final total that is over $210. Estimation skills and basic fact recall have assisted you in this
real-life situation and your mental computation skills would then come into play to determine the exact
amount for each diner.
Hello, my name is Shelley Dole and I am a researcher and teacher educator from The University of
Queensland, in Brisbane. Mental computation is one of the areas of my research.
In this presentation, I will provide an overview of issues relating to mental computation, its definition and
approaches to mental computation in schools today. I will then outline approaches to developing
students’ proficiency with the basic facts of addition and multiplication and how we can bring meaning
to learning the basic facts while fostering students’ thinking and strategic knowledge.
First, what is mental computation? Mental computation has been described as the ‘ability to calculate
exact numerical answers without the aid of calculating devices or recording devices’.
This does not mean that the solution must be determined instantly or that there is simply one best way
to calculate a problem mentally. Answers are not always attained merely through mental means. Some
people keep track of long mental calculations by recording steps towards the answer. The degree to
which a solution must be exact, or a ‘good enough estimate’, will be determined by the situation. Yet
there a many myths about the concept of mental computation and how it should be taught.
If you are old enough, you might remember the days when ‘mentals’ were a major part of the
mathematics lesson. The lesson started with chanting of tables facts, for example, once times 3 is 3,
two times 3 is 6, three times 3 is 9 … and so on, and then the teacher would fire off questions, for
example, 3 x 6; 9 x 3, in rapid succession. And you were required to write the answer on a piece of
paper. Woe betide the child who couldn’t keep up or who got less than 10 out of 10 correct! Detention
in the form of writing out the tables during lunch breaks, was the norm for those children who didn’t
know their tables.
In the past, tables were the basis of the daily ‘mentals’ time and computation was expected to occur
automatically and ‘in the head’. However, mental computation does not always have to be completely
‘in the head’ and it doesn’t have to be automatic. There may be several steps involved in completing
the calculation and then checking to see if the solution makes sense.
For mental computation to be valuable, children must be supported to use a combination of mental
computation and estimation skills, but also to make decisions about the most appropriate approach
depending on the situation. One of the important messages about mental computation is that it does
not have to be performed instantly, without thought, but children should have the confidence to use
mental computation—and estimation—in real situations and to value this approach to finding numerical
solutions.
Mental computation is important because it assists estimation of numerical situations in day-to-day life;
it is portable; it is a means to check accuracy of calculator and pen and paper solutions; and it is the
most common form of computation used by adults. Thoughtful mental computation is also an excellent
way of learning about how numbers work; it promotes number sense; it is a creative and problemsolving approach to number; and it is the easiest way of doing many calculations.
Before we go any further, take some time to try to calculate the following mentally. Don’t worry, they are
two-digit additions and I will pause between each one so that you have time to perform the calculation.
After you have worked out an answer, think about the way you approached the calculation. What
numbers did you work with first? Why? What number knowledge did you bring to support you? Each
item will be displayed on the screen so you have time to consider the numbers and your strategy for
working it out.
HOW DID YOU DO IT?
95 + 53
1.
9 + 5 is 10 + 4 which is 14
2.
5 + 3 is 8
3.
140 + 8 is 148
Here is the first one: 95 plus 53.
What did you do? Did you add 9 and 5 to get 14, then add 5 and 3, and then put these two parts
together to get 148? How did you know that 9 and 5 make 14? Was this an instant fact that you know?
What other strategies did you use?
Here is the next one: 75 plus 38.
HOW DID YOU DO IT?
75 + 38
1.
7 + 3 is 10
2.
5 + 8 is 10 + 3 which is 13
3.
113
In this one, did you instantly see a connection between the 7 and the 3 and know that it made a total of
10? If you did, you might also stop to pause at the fact that you started with numbers in the tens
column, yet this is not how you would start if you were working this out with pen and paper.
The next one: 84 plus 87.
HOW DID YOU DO IT?
84 + 87
1.
8 and 8 is 16 (double)
2.
4 and 7 is 11 (because it just is)
3.
171
Was it the double 8 that drew your attention to this situation?
If the strategies I’ve outlined were not the ones that you used, don’t worry. They are just some ways of
looking at the numbers that some people notice. But for other people, there are many different ways of
looking at and working with the numbers that work just as well and they are all successful mental
computers.
Those who work with numbers a lot have a wide repertoire of mental strategies. They use number
knowledge and understanding of number relationships and they exhibit flexible thinking strategies. In
contrast, those who struggle with numbers often rely on basic counting procedures; they are not
confident to perform calculations mentally; they show little understanding of the relationships between
numbers; and they often have poor place-value knowledge. So, how do we support learners to become
successful and confident users of number?
Successful mental computers
•
•
•
Have a wide repertoire of mental strategies
Use number knowledge and understanding
of number relationships
Exhibit flexible thinking strategies
The basic building blocks for mental computation are the basic facts of addition and multiplication. So,
having students learn the facts in their ‘tables’ is very important. Chanting of tables to learn
multiplication facts was a common practice in the old days and appeared to be an effective strategy.
The drill and practice method was regarded as effective, but at what cost? How much curriculum time
was devoted in the school year to chanting tables? And what did this do to students’ confidence in
learning mathematics. Our current approach to promoting students’ knowledge of basic facts no longer
relies on chanting tables. Rather, students learn the number facts within the tables by engaging in
activities to promote number sense and a range of strategies that rely on understanding of the
relationships between numbers.
POOR MENTAL COMPUTERS
•
•
•
•
Often rely on primitive counting strategies
Are not confident to perform calculations
mentally
Show little understanding of the
relationships between numbers
Often have poor place value knowledge
Developing students’ knowledge of number bonds between single-digit numbers supports mental
computation with larger numbers—two-digit and beyond. Developing number bonds is assisted through
promoting students’ understanding of strategies for addition: counting-on, doubles, bonds to 10,
counting by 10, bridging 10, doubles plus 1 and doubles plus 2.
Counting on is a useful strategy when the number to be added is small. This strategy should be
restricted to adding on 1, 2 or 3. Adding any amount larger than that becomes inefficient and prone to
error.
ADDITION STRATEGIES






COUNTING-ON
DOUBLING
BONDS TO TEN
COUNTING BY 10
BRIDGING 10
DOUBLE PLUS 1 & DOUBLE PLUS 2
The doubling strategy is very useful for mental computation and for many students as some doubles
are known before they come to school. To promote students’ understanding of other doubles, think of
real things that ‘come in’ specific amounts. For example, a tricycle has three wheels, so two tricycles
has, double 3, 6 wheels. An insect has 6 legs, so two insects will have, double 6, 12 legs. For 7, think of
a week on a calendar. For 8, an octopus. Nine? How about the 3 x 3 dot arrangement of the Channel 9
TV symbol?
DOUBLES PLUS 1, 2
- Requires facility with doubles
5 + 6 is double 5 plus 1
To promote the doubles plus 1 strategy—which is further down our list, but I will address here as we
are talking about doubles—use counters of two different colours on grid paper to provide a visual image
for students.
This representation shows that 5 and 6 is the same as double 5 and add one more. For doubles plus 2,
use a similar representation, as shown on this slide.
This slide shows 5 plus 7. It can be seen to be the same as double 5 add 2 (5 and 5 and 2). So, one
way of thinking about 6 and 8 is to think 6 plus 6, that is, double 6 equals 12, plus 2 (14).
5+7

Is the same as double 5 plus 2
The next strategy relates to knowing the bonds to 10. A ten-frame is useful here. By placing counters in
each cell of the ten-frame, numbers that add to 10 can be visualised. In the slide, the ten-frame shows
that 6 and 4 is 10. The next slide shows that 7 and 3 is 10. Note that just two colours are used in the
ten-frame to show the two numbers that, together, add to 10.
The counting by 10 strategy is often known as the ‘counting off the decade’ strategy. Knowing what
happens to a number when 10 is added, relies on place value. What occurs when 10 is added or
subtracted to a number is not immediately obvious to students unless attention is explicitly focused.
Once established, the pattern applies to powers of 10, that is, adding 100, adding 1000 and so on.
TENS FACTS

Ten frame assists in becoming familiar with
combinations to 10
TENS FACTS

Ten frame assists in becoming familiar with
combinations to 10
The bridging 10 strategy is useful for adding numbers that are close to 10. For example, for learning 9
plus 5, the strategy is to think 10 add 4. To assist students to see why this strategy works, a double tenframe is used. In the slide you can see that 9 is represented on the top frame and the 5 is represented
on the bottom frame. Two different coloured counters are used once again. The students can see that
there is one counter missing from the top frame, so if one counter was taken from the bottom, then the
resulting addition will be 10 plus 4. The same image can help students visualise 8 plus 5 as being the
same as 10 plus 3.
BRIDGING 10

Use a double ten frame
9+4=
10+3
By introducing addition strategies, students become familiar with the number bonds for single-digit
numbers. Knowing the bonds is an important component of mental computation for larger amounts. The
approach suggested for introducing students to the strategies is through visual images and concrete
experiences. Students will be able to refer to these visual images if they have difficulty remembering a
number bond.
For young children, each strategy outlined above introduces them to a group of addition facts—number
bonds—and this approach can become part of a structured mental computation program. For older
students, less time may need to be devoted to each strategy. However, it is still important for students
to have opportunities to experience these strategies and then apply them to mental computation with
larger numbers.
As each strategy is explored, selected examples are used to emphasise the strategy. For example, the
counting-on strategy is useful when you have a large and a small number, such as 7 plus 2. Students
need to identify the larger number and then count on from that number.
If you used 2 plus 3 as the example, students would not be exactly sure of which number to begin and
which one to count on. At the same time as introducing a new strategy, and thus a new addition fact
category, students are learning ‘two facts for the price of one’. That is, they are learning the fact as
presented, but they are also then learning that facts ‘spin-around’.
For example, they are learning that 8 plus 5 is the same as 5 plus 8. In this example, we want students
to see that 5 and 8 is the same as 10 and 3, a bridging 10 fact, but that for 5 and 8, there is no need to
start with the 5 and ‘fill up’ the ten-frame, but to start with the 8. Students must also come to see the
relationship between the three numbers in any number fact. For example, 8, 5 and 13 share a special
relationship.
As students become very comfortable with strategies for addition facts, they are ready to learn the
subtraction facts. For every fact they learn, students should become aware that there are four facts that
can be generated from three numbers: two addition facts and two subtraction facts. With 8 and 5 and
13, the four facts are as follows: 8 plus 5 equals 13, 5 plus 8 equals 13, 13 take 5 is 8, 13 take 8 is 5.
For subtraction, the simplest strategy is to ‘think addition’. If students know that 8 and 5 is 13, then
when they are presented with 13 take 5, they should think of a number that, when added to 5, gives 13.
This strategy, of course, relies on automatic recall of the addition facts in order to be able to think this
way.
Learning multiplication facts takes a similar approach. In this approach, particular facts are used to lay
the foundation for learning new facts. This requires that students have automatic recall of sets of facts
before then learning new sets of facts. The multiplication facts can be grouped into four major
categories as follows: easy facts—the twos, fives, tens, ones and zeros; pattern facts—fours, nines,
square numbers; the build-up facts—threes, sixes; the remaining facts.
Multiplication Fact Grid
0
1
2
3
4
0
2
2
4
7
8
9
12
14
16
18
5
6
8
10
3
6
15
4
8
20
5
6
10
0
1
2
5
5
10
15
20
25
10
20
30
40
30
35
40
45
50
6
12
30
60
7
14
35
70
8
16
40
80
9
18
45
10
10
20
30
40
50
90
60
70
80
90
100
Of the easy facts, the twos, fives and tens are readily recalled through students’ prior experiences with
skip counting. When asked to find three fives, children should be encouraged to count on their fingers
three times to get to 15.
For a visual image, students can be referred to the face of an analogue clock. Each number on the
clock is 5 minutes, so to determine what are eight fives, students should locate the 8 on the clock face
and then count in fives to that number. The twos are doubles, and this links to the doubles that were
learnt when learning addition facts. The strategy, therefore, for 2 times 8 is to think double 8.
If we look at the multiplication grid, we can see how many facts children most likely already know even
before they have a targeted program for multiplication facts.
Multiplication Fact Grid
0
1
2
3
4
5
6
7
8
9
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
2
3
4
5
6
7
8
9
10
2
0
2
4
6
8
10
12
14
16
18
20
3
0
3
6
15
4
0
4
8
20
5
0
5
10
6
0
6
12
30
60
7
0
7
14
35
70
8
0
8
16
40
80
9
0
9
18
45
10
0
10
20
15
30
20
40
25
50
10
30
40
30
35
40
45
50
90
60
70
80
90
100
For multiplying by 1, just like adding zero, students need to understand what happens. To give meaning
to a situation that is multiplied by 1, think of real-life situations where the number is multiplied by itself.
One example is queuing at the tuckshop. There are 8 students in the line, which is single file. That
means there is 8 times 1 in the line. Tell students the number in the line and ask them to tell you how
many all together. Although this sounds a bit silly, it is a good way to show that one row, or line of a
certain number of things has an easily determined total.
For multiplying by zero, ask students what happens if they have five groups of nothing. How many do
they have all together?
By just looking at the easy facts, students have covered a substantial number of the 121 multiplication
facts that they need to learn, as can be seen by the grid.
The next group of facts is the pattern facts. For the fours facts, the strategy is to double the doubles. So
if students know that 2 times 7 is 14, they should be able to determine 4 times 7 by thinking double 7
(14), doubled (28).
Patterns - Fours
 Double
the doubles
4
x 7 is the same as…
2
x 7 doubled
For the nines facts, there is a two-step thinking strategy. First, students think tens. For 9 times 7, for
example, students should think that 10 times 7 is 70, so 9 times 7 will be less than 70. Second,
students need to know the pattern of the nines, that when you multiply by 9, the result is two digits that
add to 9, as seen in the slide.
Pattern Facts - Nines

STEP 1

Think 10s
1 x 9 ~ 1 x 10 = 10
2 x 9 ~ 2 x 10 = 20
3 x 9 ~ 3 x 10 = 40
3 x 9 ~ 3 x 10 = 40
So, following through the above example, 9 times 7 is a number less than 70. Thinking of two numbers
in the sixties that add to 9, it must be 63. Therefore, 9 times 7 is 63.
Patterns Facts - Nines

Step 2



Know the
pattern





1x9=9
2 x 9 = 18
3 x 9 = 27
4 x 9 = 36
5 x 9 = 45
6 x 9 = 54
7 x 9 = 63
0+9=9
1+ 8 = 9
2+7=9
3 +6 = 9
4+5=9
5+4=9
6+3=9
The square numbers are the third set of facts in the patterns facts group. Some students don’t realise
that square numbers are called ‘square numbers’ because they make a square when constructed. It is
important that students have experiences constructing square numbers using counters to come to this
realisation. This helps them to visualise square numbers and to recognise the sequence of the square
number pattern: 1, 4, 9, 16, 25, 36, and so on.
PATTERNS - SQUARES

Some students don’t realise that square
numbers are called squares because they
can be made into squares…
1x1

2x2
3x3
When we look at the number of multiplication facts now covered using these strategies, it can be seen
that there aren’t many more left to learn. This means that, as a new strategy is introduced, you select
only the facts that have not been previously learnt using a different strategy.
Multiplication Fact Grid
0
1
2
3
4
5
6
7
8
9
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
2
3
4
5
6
7
8
9
10
2
0
2
4
6
8
10
12
14
16
18
20
3
0
3
6
9
12
15
27
30
4
0
4
8
12
16
20
24
28
32
36
40
5
0
5
10
15
20
25
30
35
40
45
50
6
0
6
12
24
30
36
54
60
7
0
7
14
28
35
63
70
8
0
8
16
32
40
64
72
80
9
0
9
18
27
36
45
54
63
72
81
90
10
0
10
20
30
40
50
60
70
80
90
100
49
10
The build-up facts are the threes and sixes. Have a look at the grid to find out which particular facts are
to be targeted. For the threes, there are only three facts: 3 times 6; 3 times 7; and 3 times 8. For the
sixes, there are only two facts: 6 times 7; 6 times 8. For the threes facts, the strategy is to think
doubles, plus one more group. For 3 times 6, think double 6 (12) plus one more group of 6 (18). Do the
same for the other threes. Practise until these are automatic.
Build up - Threes



3x8
is the same as…
2 x 8 + 1 more group of 8
For the sixes, the strategy is to double the threes, which highlights the importance of automatic recall of
the threes in order to use this strategy. For 6 times 7, think 3 times 7 (21, doubled (42). For 6 times 8,
think 3 times 8 (24), doubled (48). Then practise these until automatic.
Build up facts - sixes

6x8

Is the same as…

3 x 8 doubled
If we look at the multiplication grid now, we can see that there are two facts remaining. But on closer
inspection, the two facts are actually one: 7 times 8 and its spin-around, 8 times 7. For this one, the
strategy is that there is no strategy and students should be encouraged to devise their own strategy.
Or, as it is the last fact, this may be a special fact that is memorised as an individual item that has no
strategy.
Multiplication Fact Grid
0
1
2
3
4
5
6
7
8
9
0
0
0
0
0
0
0
0
0
0
0
10
0
1
0
1
2
3
4
5
6
7
8
9
10
2
0
2
4
6
8
10
12
14
16
18
20
3
0
3
6
9
12
15
18
21
24
27
30
4
0
4
8
12
16
20
24
28
32
36
40
5
0
5
10
15
20
25
30
35
40
45
50
6
0
6
12
18
24
30
36
42
48
54
60
7
0
7
14
21
28
35
42
49
63
70
8
0
8
16
24
32
40
48
64
72
80
9
0
9
18
27
36
45
54
63
72
81
90
10
0
10
20
30
40
50
60
70
80
90
100
Like learning subtraction facts, learning division is to think multiplication. Through their number work,
students should be seeing relationships between number triples: 4, 8, 32, and use this to think division:
32 divided by what number is 8?
Through a structured mental computation program, students are provided with strategies for thinking
about numbers and expanding their repertoire of approaches for working with numbers.
They are developing flexible ways of approaching computation and building their understanding of how
numbers work. With automatic recall of the 121 basic addition and multiplication facts, students are in a
strong position to be successful mental computers and through knowing strategies, they are building
flexible ways of exploring numbers and number relationships.
Often mental computation consists of merely drill and practice and games of competition. A structured,
dedicated mental computation program follows four steps: strategy, thinking, practice and, finally, to get
to automatic.
Structured Mentals Program




Strategy
Thinking
Discussion
Practice
The goal for basic facts is automatic recall.
The goal for mental computation is thinking and
taking time to DO THE MATHS.