The
Numeracy
Continuum
Aspect 1:
Counting sequences and
numeral identification
This is an interactive Adobe
Acrobat PDF 9 document.
It is best viewed on screen as it
contains embedded video and
an interactive menu.
The Numeracy
Continuum
Aspect 1:
Counting
sequences
and numeral
identification
Introduction
Challenges in the ‘teens’
Introduction
Counting begins with the task of reproducing a sequence of number words starting from one. Over
time, children’s familiarity with the sequence of counting words develops to the point where they
are able to identify the number word before or after any number without needing to start the count
from one. That is, the process of oral counting starts as a fixed recitation but moves to a ‘breakable
chain’ of number words.
Students can use counting as the basis of a range of strategies, including various kinds of additions
and subtractions. Within these strategies, counting refers to more than producing the forward
sequence of number words, sometimes called rote counting. If you ask a student what number
comes after nine, he or she will often initially count from one to find the answer. To be able to use
the strategy of counting-on, students need to be able to know the sequence of number words well
enough to continue counting from any number. That is, you cannot count on from seven if you do
not know the number word that follows seven.
Forward number word sequences
Backward number word sequences
Zero and the counting numbers
Numeral identification
References
Acknowledgements
This student counts
from one to find the
next number word in
a counting sequence.
>
2
The Numeracy
Continuum
Aspect 1:
Counting
sequences
and numeral
identification
Challenges in the ‘teens’
When children learn to say the counting words, it is quite common for them to experience some
problems with the ‘teens’. The teen number words often sound very similar to other number words.
For example, ‘thirteen’ sounds similar to ‘thirty’, ‘fourteen’ to ‘forty’, ‘fifteen’ to ‘fifty’ and so on.
Although it can be surprising to hear the response ‘ninety-one’ when you ask a child what number
comes after ‘nineteen’, this is a logical answer if the child treats the number words ‘nineteen’ and
‘ninety’ as being the same.
Introduction
Challenges in the ‘teens’
Forward number word sequences
Backward number word sequences
Zero and the counting numbers
Numeral identification
References
Acknowledgements
This problem of understanding the naming of numbers between ten and twenty also includes the two
number words eleven and twelve that occur before the ‘teens’. The old English or Germanic origins of
eleven and twelve suggest that they originally meant “one left over” and “two left over”, respectively.
However, the etymology of the pre-teen number words eleven and twelve does little to support a
child’s search for consistent structure in the counting word sequences. As well as reversals in the
‘teens’ where the smaller value is named before the larger value (e.g. the four comes before the ten
in fourteen) students need to interpret two different modifications of “ten”; ‘teen’ and ‘ty’, neither of
which clearly says “ten”. Since the ten and ones structure of the ‘teens’ is not transparent in English,
more learning time must be devoted to developing an understanding that the numbers from 11 to 19
express one group of ten and a particular number of ones.
Knowledge of a student’s use of the sequence of number words is a powerful indicator of the
efficiency of addition and subtraction methods. Wright (1989) distinguished between the development
of the forward sequence of number words and the backward sequence of number words, describing
five levels in each as well as an emergent level. The levels are summarised in the following tables.
>
3
The Numeracy
Continuum
Aspect 1:
Counting
sequences
and numeral
identification
Introduction
Challenges in the ‘teens’
Forward number word sequences
Backward number word sequences
Zero and the counting numbers
Numeral identification
References
Acknowledgements
Forward number word sequences
L0 Emergent
L1 Initial (10)
L2 Intermediate (10)
L3 Facile (10)
L4 Facile (30)
L5 Facile (100)
Cannot count
to 10
Counts to 10 but
cannot give the
number after.
Counts to 10 and gives
the number after, but
counts from one.
Counts to 10 and
gives the number
after.
Counts to 30 and
gives the number
after.
Counts to 100 and
gives the number
after.
Level 0: Emergent
The emergent level characterises children who have not yet developed knowledge of the sequence
of number words from one to ten. They may know some of the counting words in correct sequence
but cannot consistently count from one to ten.
Level 1: Initial (10)
At the level of the initial sequence up to ten the forward sequence of numbers from one to ten can
be produced but not the word for a number immediately after one in this range.
Level 2: Intermediate (10)
At the intermediate number word sequence to ten the number word just after a given number word
can be produced within this range but only by dropping back to count from one when doing so.
Level 3: Facile (10)
At the facile number word sequence to ten the number word just after a given number word can be
produced within this range without dropping back to count from one.
Level 4: Facile (30)
When students are facile with number word sequences to thirty the forward sequence of number
words from one to thirty can be produced and the number word just after a given number word
can be identified within this range without dropping back.
Level 5: Facile (100)
When students are facile with number word sequences to one hundred, the forward number word
sequences can be produced in the range from one to one hundred (or beyond) and the number
word just after a given number word can be identified within this range without dropping back.
>
4
The Numeracy
Continuum
Aspect 1:
Counting
sequences
and numeral
identification
Introduction
Challenges in the ‘teens’
Forward number word sequences
Backward number word sequences
Zero and the counting numbers
Numeral identification
References
Acknowledgements
Backward number word sequences
L0 Emergent
L1 Initial (10)
L2 Intermediate (10)
L3 Facile (10)
L4 Facile (30)
L5 Facile (100)
Cannot count
backwards from
10 to 1.
Counts backwards
from 10 to 1 but
cannot give the
number immediately
before a given
number.
Counts backwards
from 10 to 1 and gives
the number before, but
counts from one to do
so.
Counts
backwards from
10 to 1 and gives
the number
before without
dropping back.
Counts backwards
from 30 and gives
the number before
without dropping
back.
Counts backwards
in the range to 100
and gives the
number before.
Level 0: Emergent
The emergent level characterises children who have not yet developed knowledge of the backward
sequence of number words from ten to one.
Level 1: Initial (10)
At the level of the initial backward number word sequence up to ten the backward sequence of
numbers from ten to one can be produced but not the word for a number immediately before one
in this range.
Level 2: Intermediate (10)
At the intermediate backward number word sequence to ten, the number word just before a given
number word can be produced within this range but only by dropping back to count from one
when doing so.
Level 3: Facile (10)
At the facile backward number word sequence to ten the number word just before a given number
word can be produced within this range without dropping back to count from one.
Level 4: Facile (30)
When students are facile with backward number word sequences up to thirty the backward
sequence of number words from thirty to one can be produced and the number word just before a
given number word can be identified within this range without dropping back.
Level 5: Facile (100)
When students are facile with backward number word sequences to one hundred the backward
number word sequences can be produced in the range from one hundred to one and the number
word just before a given number word can be identified within this range without dropping back.
>
5
The Numeracy
Continuum
Aspect 1:
Counting
sequences
and numeral
identification
This student counts
from one to find the
number word before
thirteen.
Introductiona
Challenges in the ‘teens’
Forward number word sequences
Backward number word sequences
Number word before
Zero and the counting numbers
Numeral identification
References
Acknowledgements
>
6
The Numeracy
Continuum
Aspect 1:
Counting
sequences
and numeral
identification
Zero and the counting numbers
When children count backwards from 10, some learn to finish the backward count with “zero” and
others with “blast off”. Although there is value in counting down from 10 to 0, a problem arises
when some children begin a forward count from zero. In developing a one-to-one match between
objects and counting words, children can allocate the word “zero” to the first object counted. For
this reason, the descriptions of the levels of the counting words do not include zero.
Introduction
Challenges in the ‘teens’
Forward number word sequences
Backward number word sequences
Zero and the counting numbers
Numeral identification
Numeral identification
Our system of numerals has a long history of development, with the symbols derived from the
Indian Brahmi numerals, then adopted and popularised by the Arab empire. Initially the place value
system used nine numerals and a blank space called sunya in India and sifr in the Arab world, both
words meaning empty. The tenth symbol for zero appeared later. There are two uses of zero that
are both important but are somewhat different. One use is as an empty place indicator in our
place-value system and the second use of zero is as a number in its own right. There are also
different aspects of zero (and other numbers), namely the concept or quantity, the notation and the
name. In assessing numeral identification we seek to establish the link between the verbal and the
symbolic.
References
Acknowledgements
In using number, children must integrate many layers of verbal, procedural, symbolic and
conceptual meaning. To illustrate how these layers can coexist, consider the following video
showing some of the different meanings attributed to “five”.
>
7
The Numeracy
Continuum
Aspect 1:
Counting
sequences
and numeral
identification
Introduction
Challenges in the ‘teens’
Forward number word sequences
Backward number word sequences
Get me 5 counters…
Zero and the counting numbers
Numeral identification
References
Acknowledgements
Teacher:
Could you get me 5 counters from this pile, please?
Student:
(Grabs a handful of blue counters and then a handful of yellow counters)
Teacher:
OK. We will leave the yellow ones there – put the yellow ones back.
Now, would you like to tell me, how many blue ones have you got here?
Student:
(Pointing at different counters, leaving some out) One, two, three, four, five.
Teacher:
How many?
Student:
(Repeats the count again leaving out some counters) One, two, three, four, five.
Teacher:
How many was that?
Student:
(Holding up 5 fingers) Five.
Teacher:
Five. Um hmm.
>
8
The Numeracy
Continuum
Aspect 1:
Counting
sequences
and numeral
identification
Introduction
Challenges in the ‘teens’
Forward number word sequences
This child has learnt what five fingers are, and in that limited context, could be said to know five. He has
knowledge of the forward sequence of number words to five and recognition that the last word in a
count has special meaning. His counting procedure starts with the first item in a row and ends on the
final item in the row, but he does not match his counting words one-to-one with each and every item.
As well as the spoken word (e.g. “five”), number can be represented symbolically as a written word
(e.g. five) or as a written numeral (e.g. 5). Although numerals are the written and read symbols for
numbers, they can also play a similar role to letters in forming part of a name, as in licence plates and
telephone numbers. At its most basic level, numeral identification is a form of shape recognition, which
can result in a simple association of the word “two” with the symbol ‘2’ without a cardinal meaning
(Mix, Sandhofer, & Baroody, 2005). This means that numeral identification can develop at a different
rate to number knowledge.
Learning to identify, recognise and write numerals is an important part of early arithmetical
development. When a young child learns the name of a numeral it sows the idea that a symbol can
stand for a whole word (Mix, Huttenlocher, & Levine, 2002).
Backward number word sequences
Zero and the counting numbers
Numeral identification
References
Acknowledgements
Although this student
knows that the
symbols may identify
numbers, he is
uncertain as to which
numerals match with
which words.
Numeral identification
>
9
The Numeracy
Continuum
Aspect 1:
Counting
sequences
and numeral
identification
Introduction
Challenges in the ‘teens’
Forward number word sequences
Backward number word sequences
Zero and the counting numbers
Numeral identification
References
Acknowledgements
Numeral identification refers to being able to state the name of a displayed numeral.
L0 Emergent
L1 1-10
L2 1-20
L3 1-100
L4 1-1000
L5 1-10 000
May identify
some but not all
numerals 1–10.
Identifies all the
numerals 1–10.
Identifies all the
numerals 1–20.
Identifies
numerals in the
range 1–100.
Identifies numerals
in the range
1–1000.
Identifies numerals
in the range
1–10 000.
Level 0: Emergent
At the emergent numeral identification level the student may identify some, but not all numerals in
the range 1–10.
Level 1: 1-10
At the 1–10 numeral identification level the student can identify all numerals in the range 1–10.
Level 2: 1-20
At the 1–20 numeral identification level the student can identify all numerals in the range 1–20.
Level 3: Facile (10)
At the 1–100 numeral identification level the student can identify all numerals in the range 1–100.
Level 4: Facile (30)
At the 1–1000 numeral identification level the student can identify one–, two– and three–digit
numbers.
Level 5: Facile (100)
At the 1–10 000 numeral identification level the student can identify one–, two–, three– and
four–digit numbers.
>
10
The Numeracy
Continuum
Aspect 1:
Counting
sequences
and numeral
identification
References
Mix, K. S., Huttenlocher, J., & Levine, S. C. (2002). Quantitative development in infancy and early
childhood. Oxford, UK: Oxford University Press.
Mix, K. S., Sandhofer, C. M., & Baroody, A. J. (2005). Number words and number concepts: The
interplay of verbal and nonverbal quantification in early childhood. In R. V. Kail (Ed.), Advances in
Child Development and Behavior (Vol. 33). New York: Academic Press.
Wright, R. J. (1989). Numerical development in the kindergarten year: A teaching experiment.
Unpublished doctoral dissertation, University of Georgia.
Introduction
Challenges in the ‘teens’
Forward number word sequences
Backward number word sequences
Zero and the counting numbers
Numeral identification
References
Acknowledgements
>
11
The Numeracy
Continuum
Aspect 1:
Counting
sequences
and numeral
identification
Acknowledgements
The descriptions of the levels of the number word sequences, forwards and backwards, and the
levels of numeral identification are derived from the work of Professor Bob Wright. His contribution
and the contributions of his colleagues to our growing understanding of the role played by counting
in developing a rich sense of number are gratefully acknowledged.
Introduction
Challenges in the ‘teens’
Forward number word sequences
Backward number word sequences
Zero and the counting numbers
Numeral identification
References
Acknowledgements
>
12