Theme(s): Numeracy
Page 1
Writing numbers in figures and mental calculation in the
early years
Author(s):
Johansson, B.S., Uppsala University, Sweden
Publisher:
Scandinavian Journal of Educational Research, Vol. 49, No. 1, pp 3-25 (2005)
[Original title: Numeral writing skill and elementary arithmetic mental
calculation]
Writing numbers in pre-school years does it help?
This Swedish study explored how writing numerals in their pre-school years (pre-7 years
in Sweden) influenced children’s later performance in arithmetic. Specifically the
research set out to investigate:
 the effects of early numeral learning on mental calculation strategies; and
 the effects of poorly formed or reversed numerals – whether such errors have a
negative impact on the development of mental calculation procedures.
The findings and discussion shed light on what young children may naturally learn to do
for themselves in relation to arithmetic problem solving and how they do it. The study
found that acquiring the skill of writing numerals early on was linked to the development
of children's ability to solve number problems; the greater the number of digits they could
write correctly, the more arithmetic problems they could solve in their first year of school
(aged 7 in Sweden). Even where children wrote some digits backwards, this positive
correlation still held. The findings suggested that children just starting school who were
used to writing numerals at an early stage went beyond counting to recalling number
facts from memory and used strategies to simplify calculation. The study’s author
concluded that learning to sequence number-words and acquiring numeral writing skills
provided the ground for efficient and effective methods of solving number problems
based on thinking of numbers as members of a sequence of numerals.
The study lends support to the view that pre-school children find numerals easy to learn,
visualise and memorise and this makes them easy to use in mental calculations.
Note: The text of this digest uses the author’s terms for numeracy strategies: the English
equivalent is put in brackets afterwards.
Keywords:
Sweden; Key stage 1; Primary schools; Pupils; Numeracy; Arithemetic; Learning
strategies; Problem solving; Teaching methods
Page 2
Contents
How do early number skills affect children’s numeracy learning?
(page 3)
How might children’s numeracy thinking develop?
(page 4)
How were the children tested?
(page 5)
What strategies did the children use to tackle the arithmetic problems?(page 6)
How was the study designed?
(page 7)
Implications
(page 8)
Where can I find out more?
(page 9)
How do early number skills affect children’s numeracy
learning?
The researcher’s findings showed that:
 the more digits they could write correctly, the more arithmetic problems the
children could solve in their first year of school;
 writing a digit back to front did not seem to make any difference to the child’s
problem-solving abilities;
 omitting digits was linked to low mathematics performance;
 as children became better at manipulating number sequences mentally, they
began to move away from counting methods to ordinal number methods;
 in no case did children use formal (algorithmic) methods such as place value
(putting numbers in tens and units columns, adding and carrying etc); and
 children who were skilled in number sequencing showed an ability to retrieve
number facts from memory.
In contrast to counting-all methods, when children use ordinal number methods they
think of the number as the numeral in the number sequence; for example, four is 4, the
number after 3 and before 5.
Children who were skilled in putting number-words in order and who could use numerals
to represent numbers tended to use calculation strategies which involved ‘jumping’ from
one number to another mentally. Children who were weak at sequencing number-words
and had difficulties in representing the numbers with numerals tended to solve arithmetic
problems with ‘counting-all’ strategies or were unable to solve the problems.
The researcher concluded that an ability to represent numbers by numerals is an
important prerequisite for being able to form a mental number sequence. Once they
acquire this skill, children can imagine the number sequence as a series of numerals,
that is, they can represent the numbers one, two, three, four… as 1, 2, 3, 4,….
Page 3
How might children’s numeracy thinking develop?
The author suggested that children’s arithmetic problem-solving goes through a threestage development process:
 initially, numbers are represented by objects or images of objects which only
allow for counting one-by-one solution strategies (click to page 5);


as children become better at sequencing number-words and using numerals to
represent numbers, they are more able to represent the numbers in a mental
sequence; and
writing numerals appears to help children form a mental number sequence more
or less as soon as they learn to do so. The children may now be finding
regularities among number sequences that can form the basis of new, more
efficient strategies for solving arithmetic problems.
Drawing on the evidence from the tests and interviews the author suggests that as
children get better at sequencing numbers in their heads, they move towards new and
more efficient strategies for solving arithmetic problems. For example, when tackling the
problem “17+6=?”….. they break 6 into 3 and 3, then counting on from 17 to ”18,19, 20”;
then on again to “21, 22, 23.” Counting-on in this way, from the first number in a
calculation to familiar staging posts such as multiples of ten seems to serve as a
transition stage in the formation of quantity value strategies. A more developed approach
to tackling the same problem is to decompose (or partition) using tens (click to page 5).
This means breaking the 6 into 3 and 3 as before, then adding 3 to 17 to make 20 (a
multiple of ten), then 3 to 20 to make 23, without counting. A further evolutionary step in
the development of quantity value methods is represented by the child using immediate
recall (click to page 5) to say 17+6=23”.
Page 4
How were the children tested?
These examples show the range of tests presented to the children over the three
studies:
 Numeral writing skills:
Single-digit writing: “Would you like to write the digits you know?”
Writing 2-, 3- and 4-digit numbers: The child was asked to write spoken numbers
in numerals, including 18, 54, 106 and 2089.

Number-word sequence
Counting forward (counting on): The child was asked to count from 1 until told to
stop (maximum 45).
Counting down (counting backward): The child was asked to count, for example,
from 7 to 5, 17 to 11, and 22 to 14.
Doubles tests were used as a measure of an advanced understanding of the
number-word sequence.
These included questions such as 6+6, 13+13 and 18+18.

Arithmetic tests
Understanding of digits by going from numerals to cardinal numbers and the
reverse: The children were given a numeral on paper with a written instruction “I
would like you to draw as many tallies as the numeral says” An example of the
numerals used is the series 3, 5, 7, 14. For the reverse test the children were
presented with a series of tallies covering a similar series of numerals and given
the written instruction to “Draw the numeral that tells how many tallies there are”.
Addition and subtraction
The child was given a series of number problems in order of difficulty:
Addition example: 1+4, 4+5, 6+9, 5+16, 27+14;
Subtraction example: 4-3, 8-3, 14-5, 23-15, 43-19.
The questions were phrased as word problems such as: ”Imagine you have 3
balls and then you get 4 more. How many do you have altogether?”, rather than
the more abstract form 3 + 4 =?
Page 5
What strategies did the children use to tackle the
arithmetic problems?
These fell into a number of groups including:
 Counting:
o Concrete counting-all: The child counted fingers out one by one
separately for both parts of the sum then counted all the fingers to
determine the result;
o Sequence counting-all: The child started with “1” and counted to the value
of the first part of the sum. The child then continued counting on the
required number of steps for the second part of the sum; and
o Sequence counting-on: The child began with the value of the first part of
the sum then counted on the number of steps for the second part of the
sum.

Decomposition (Partition):
The child first manipulated the given numbers to create easier number
combinations; for example, 5+4=?: 5 and 5 make 10, one less gives 9. Another
example is 16+8=?: 16 and 4 gives 20, adding 4 makes 24.

Immediate recall:
The child relied on facts recalled from memory and gave an answer within 3
seconds.

Subtraction solution procedures:
These involved counting up or down procedures. For example, “7-2=?” The child
began with 2 and counted on to 7 stating the number of steps used to do this.
Alternatively the child began with 7 and counted down the required number of
steps to reach 2.
Page 6
How was the study designed?
The researcher conducted three studies designed to:
 analyse the relation between writing numerals and arithmetic performance (study
1);
 investigate whether the writing of reversed digits was related to a high or a low
level of arithmetic performance, whether children understood what the numerals

represented and whether writing numerals helped children retain a better
memory of digits (study 2); and
test whether writing numerals, together with the number-word sequence skill is
related to children’s use of ordinal number strategies (study 3).
Altogether, 650 children aged between 4.8 and 7.8 years were involved in the
experiments. Study 1 involved children in the first year of school as well as pre-school
children. The other studies involved children in pre-school classes. The children
completed tests during individual interviews which involved writing the digits 0-9, writing
2, 3 and 4 digit numbers, counting forward and backward, doubling numbers and
addition and subtraction tests. The test sessions in the last study were videoed. The
data was collected and scored by teacher trainees for the first experiment and by trained
research assistants for the second and third experiments. Inter-rater agreement was
high. After the children had solved the arithmetic problems they were asked to explain
how they had solved the problem. The results from the tests were analysed statistically.
All the data from the three experiments showed the same basic relation between number
writing and arithmetic performance.
Page 6
Implications
This study was conducted in Sweden where the educational system and children start
school at age 7 years. Nevertheless the writer of the digest believes that there are some
implications for practice in schools in England.
For teachers and leaders in primary schools:
 The evidence suggests that children acquire and develop in a natural way certain
approaches to number and that writing numerals may be a good indicator of
number skills. Would it be helpful for teachers to find out more about what the
children are already able to do when they enter school and to build on this
appropriately? Would finding out about their numeral knowledge, including their
skills in writing numerals, be helpful in this?
 A finding of the study is the importance of mental number sequencing in the
development of the child’s problem solving skills. Would working with a colleague
to assess children’s number sequencing skills and to plan strategies for
responding to problems be helpful in taking forward your knowledge of how this
aspect of numeracy learning unfolds?
For parents and teachers in pre-school settings:
 The study provides evidence that the acquisition of number-word sequence skills,
followed by early numeral knowledge, are useful preliminary steps to being able
to solve arithmetic problems effectively and efficiently. As a first step could you
as a parent encourage and stimulate your child’s interest in numbers through
counting games, nursery rhymes and play activities such as skipping? (There are
many websites that feature children’s counting games.)
 How could you, as a parent support your child’s interest in number sequences
and in numerals at home? For example, you might find an opportunity to do this
by encouraging your child to look at the clock on the micro-wave oven and count
down as the timer display changes. Perhaps you could stimulate more actively

this kind of spontaneous interest in numbers, thus advancing your child’s
numeral knowledge?
The Foundation Stage places a lot of emphasis on children learning how to use
words for numbers, on calculating using number words and on manipulating
cubes, etc. Would it, given the evidence in this study, be helpful to include a
more explicit use of numerals, for example, by asking children to record numbers
of objects as numerals as well as tallies?
Page 7
Where can I find out more?
A National Teacher Research Panel study described teacher-led action research aimed
at constructing a programme of multi-sensory teaching activities to develop mental
arithmetic capability in children from Nursery through KS1: A multi-sensory approach to
teaching mental arithmetic by Romy Tacon and Tony Wing. A summary of the study can
be found at:
http://www.standards.dfes.gov.uk/ntrp/lib/pdf/TaconandWing.pdf
A TRIPS digest that explores the relationship between children’s mental arithmetic
strategies and formal methods introduced in year 5 numeracy classes can be accessed
at:
http://www.standards.dfes.gov.uk/research/themes/thinkingskills/ThuOct101531302002/
A website which provides enjoyable games in which children practice mental arithmetic
including guess a number, subtraction and multiplication sums for young children:
http://www.teachingideas.co.uk/maths/contents2.htm
For young children (age 4-6) starting to count, this website provides ideas for parents
who want to help children practice arithmetic:
http://education.guardian.co.uk/netclass/schools/mathematics/links/0,5607,98408,00.htm
l
Mental calculation strategies for addition and subtraction are explained in this article:
http://www.m-a.org.uk/docs/library/2076.pdf
Readers might like to visit this website. Its aim is to begin the child’s arithmetic at a
young age by using real objects:
http://www.shirleys-preschool-activities.com/preschool-math-activities.html
This program is for young children; it covers the numbers 0 through 9 and improving
mental arithmetic:
http://www.bluechillies.com/list.html?k=arithmetic
This website aims to help parents practise mental arithmetic with their children:
http://www.alite.co.uk/groups/parents/continued.htm