putting place value in its place

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© ATM 2007 • No reproduction except for academic purposes • [email protected] for permissions
PUTTING PLACE
VALUE IN ITS
PLACE Ian Thompson
An important conclusion of this study was
that children are able
to add two-digit
numbers successfully
using partitioning
without needing to
have an understanding
of place value.
14
A: What happens when you multiply a number by
ten?
B: You add a zero.
A: No, we say that ‘the digits move one place to
the left.
The context for this exchange could be any one of
several scenarios: a primary or secondary classroom,
a teacher training establishment, an LEA CPD course
venue or a national numeracy strategy consultants’
training session. In each context the ‘teacher’ would
be keen to avoid the acquisition of any misconceptions on the part of the ‘learner’ because, as we all
know, the ‘adding a zero’ rule breaks down in the
case of decimal fractions. I would like to argue in
this article that the ‘teacher’ may well be acting in
too pedantic a manner in this situation, particularly
if the ‘learners’ are young children. I have found
that the ‘digits moving one place to the left or right’
mantra has become more commonplace since the
advent of the national numeracy strategy.
Let us try to unpick what is involved in an
understanding of the concept of ‘moving one place
to the left’. First of all, children need to realise that
in mathematics we impose an invisible column
structure on multi-digit numbers (unlike in literacy
where a word like ‘cat’, even though it comprises
three individual letters, does not have such a
structure imposed on it). They also need to appreciate the following:
● that the quantities represented by the individual
digits are determined by their positions in the
number;
● that as the digits are moved to the left their
value increases by an appropriate power of ten
(depending on the number of places moved);
● that the value of an individual digit is found by
multiplying its face value by the value assigned
to its position;
● that the quantity represented by the whole
number is the sum of the values of these individual digits.
These properties of place value understanding
are described by Ross [1] respectively as the positional, base-ten, multiplicative and additive properties
of place value. They obviously demand a fairly
sophisticated level of understanding.
So, what do we currently know about young
children’s ability to meet this demand? A recent
investigation by Thompson and Bramald [2]
explored the relationship between young children’s
understanding of place value and their ability to add
two-digit numbers. The study took the form of a
series of one-to-one interviews with a sample of
144 children aged 6 to 9 from eight primary
schools. On the oral question ‘What is 25 plus 23?’
(accompanied by a card with 25+23 written on it),
63% of the sample answered the question correctly
using a strategy that partitioned either (or both) 25
and 23 into 20 and 5 and 20 and 3 respectively.
However, on two other practical questions that
involved important aspects of place value the
children were less successful. On the ‘milometer’
question (Brown, [3]) [The reading is 6299, what
will it be after the car has travelled one more mile?],
24% gave the correct answer. On the ‘bricks’
question (APU, [4]) [How does the value of a brick
change when it is moved one column to the left?] only
10% were correct. The total number of children
who were successful on both questions was just 4
(all from Y4), compared to the 91 who performed
the addition correctly using partitioning. An
important conclusion of this study was that children
are able to add two-digit numbers successfully using
partitioning without needing to have an understanding of place value.
In the context of this article it seems fair to
conclude that children in Y2 and Y3 have little or
no understanding of what we conventionally call
MATHEMATICS TEACHING 184 / SEPTEMBER 2003
© ATM 2007 • No reproduction except for academic purposes • [email protected] for permissions
place value. In Thompson’s [5] terms they have
some appreciation of the ‘quantity value’ aspect of
place value (73 is 70 plus 3), but not the ‘column
value’ aspect (73 is 7 in the tens column and 3 in
the units column). Consequently, it would appear to
be somewhat over-optimistic to expect young
children to understand the concept of ‘moving the
digits one place to the left’, which, as argued above,
appears to demand a fairly sophisticated understanding of place value1.
There are other instances of mixed or confusing
messages concerning aspects of place value in the
Framework [6]. For example, whenever ‘know the
value of each digit in a number’ is mentioned it is
exemplified in terms of ‘623 is 600 ⫹ 20 ⫹ 3’
(quantity value) rather than in terms of 6 hundreds,
2 tens and 3 ones (column value). However, the
latter interpretation is emphasized when the
Framework suggests that children as young as 6 or 7
should be able to ‘represent 14 on an abacus’: surely
one of the most formal representations of the
column aspect of place value. And, why on earth is
the word ‘exchange’ listed in the Y1 section of
Mathematical vocabulary [8] when written
methods for addition introduced as late as Y4 are
described as being only about ‘preparing for
carrying’ (my italics)?
The reader may well want to argue that if children
are taught to ‘add a zero’ they will be learning a ‘rule’
that they will have to unlearn later. My response to
this would be that:
1 We already do this in the area of calculation and
the number line: initially a young child ‘cannot
take five from three’ nor ‘divide 3 by five’, and
much later a secondary pupil ‘cannot find the
square root of negative four’.
2 Much current emphasis on errors and misconceptions focuses on ‘cognitive conflict’ (see
Swan [9]). The ‘add a zero’ issue provides a
perfect context for such discussion work.
3 I would also like to see some research (rather
than anecdotal) evidence that might provide
information about the percentage of children in
Y6 (the first time multiplying a decimal by 10
appears in the Framework) who are likely to
write 0.8 ⫻ 10 ⫽ 0.80. Personally, I would put
money on this percentage being very high!
In conclusion, the 144 children in the
Thompson and Bramald [2] research were asked
what 8 ⫻ 10 was, and were then asked ‘What
happens when you multiply a number by 10?’ The first
part was answered correctly by 85% of the children.
The second part generated a range of answers: 18%
of the sample were unable to answer the question;
50% of Y3 and 63% of Y4 said ‘Add a nought (or
zero)’; others said: A tens number; It has to be a
MATHEMATICS TEACHING 184 / SEPTEMBER 2003
number in the tens; It highers it; Count up in tens; You
come up with the -ty of that number; It changes the
front number; and You share it (!). Not a single child
mentioned ‘moving places’, ‘shifting into columns’
or ‘place holders’ despite the fact that many of the
teachers interviewed said they had addressed the
topic using such language.
Ian Thompson is Visiting Professor at the University of
Northumbria.
If you would like a free copy of the Nuffield Report referred to
in this article contact [email protected]
stating whether you require a Word or pdf version. A downloadable pdf version is available on the MT website www.atm.org.uk
Footnote
1 Exploring the ‘moving digits’ slogan further we find that the
situation is more complex than it at first seemed. It is not
enough just to say that the digits move, given that, visually,
they appear to be no different after this movement. What we
obviously have to do is signify that they have indeed been
moved. So, perhaps we should change A’s response to: No, we
say the digits move one place to the left, and then we insert zero
as a place holder. This action would seem to be supported by
the NNS Framework [6], which includes the following key
objective in its yearly teaching programme for Y2 children:
Know what each digit in a two-digit number represents, including
zero as a place holder (p.10), and the national curriculum [7]
which states that Key Stage 1 children should be able to:
recognize that the position of a digit gives its value and know
what each digit represents, including zero as a place holder
(p.63). So, an appreciation of the concept of ‘place holder’ is
prescribed for children aged six or seven. However, having
explained the shifting of digits and the use of a place holder,
it would be difficult for a teacher not to smile at the ‘smart
Alec’ response from ‘B’; So, as I said, to multiply by 10 you
add a zero!
References
1 S. Ross: Place value: Problem solving and written assessment,
Teaching children mathematics, March 2002.
2 I. Thompson, I. and R. Bramald: An investigation of the
relationship between young children’s understanding of the
concept of place value and their competence at mental
addition. (Report for the Nuffield Foundation). University of
Newcastle upon Tyne, 2002.
3 M. Brown: Place value and decimals. In K. Hart (ed.),
Children’s understanding of mathematics: 11-16. London:
John Murray, 1981.
4 APU: Mathematical development: primary survey report no.
3. London: HMSO, 1982.
5 I. Thompson: Place value: The English disease? In I. Thompson
(ed.), Enhancing primary mathematics teaching.
Buckingham: Open University Press, (in press).
6 DfEE: The National Numeracy Strategy: framework for
teaching mathematics from reception to year 6. London:
DfEE, 1999.
7 DfEE/QCA: The National Curriculum: handbook for
primary teachers in England – Key Stages 1 and 2. London:
DfEE, 1999.
8 DfEE: The National Numeracy Strategy: mathematical
vocabulary. London: DfEE, 1999
9 M. Swan: Making sense of mathematics. In I. Thompson
(ed.), Enhancing primary mathematics teaching.
Buckingham: Open University Press, (in press).
15
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