Mathematics Education Research Journal
1997, Vol. 9, No.1, 25-38
g
Changes with Age in Students'
Conceptions of Decimal Notation
e
Kevin Moloney and Kaye Stacey
The University of Melbourne
'1
t
This study examines Australian students' conceptions about ordering decimals. It
builds upon previous work which established three common misconceptions. A
longitudinal study of 50 secondary students over twelve months showed little
change in their misconceptions. A second study traced the incidence of each
misconception from Years 4 to 10 in a sample of 379 students. It was found that the
whole number misconception was important in earlier years but disappeared with
time. The fraction misconception persisted, being displayed by approximately
twenty per cent of Year 10 students. The zero-rule misconception was uncommon.
The diagnostic test, which substantially improved on one used in previous research,
may be very useful for teachers.
A thorough understanding of decimals is critical in our society, with its
widespread use of computers and calculators using decimal digital displays, and of
metric measurement for scientific and everyday purposes. Understanding decimal
notation is a multi-dimensional task. Students need to coordinate place-value
concepts with aspects of whole number and fraction knowledge. Making the
transition to understanding decimals relies on having a thorough understanding of
previous concepts fully integrated with new information. This study looks at the
nature and prevalence of the conceptions that students construct for themselves
when they are making their way through this transition. The results show that
many students have misconceptions which remain with them even to Year 10. In
this study, the term "decimals" will refer to "decimal fractions" and "fractions"
will refer to "common fractions./I
Although students learn about decimals in primary school, it is well known
that secondary and some tertiary students in many countries use decimals without
an adequate knowledge of the concepts involved (Brown, 1981; Carpenter, Corbitt,
Kepner, Lindquist & Reys, 1981; Grossman, 1983; Nesher & Peled, 1986). In
particular, some students acquire skills in the use of and operations on decimals
without understanding the comparative size of the numbers involved. Given a pair
of decimals they may be unable to identify the larger and they may not have a good
sense of the size of an answer if this is a decimal. Most commonly, problems arise
when the decimals are of unequal length. Comparing decimals of unequal length is
a common task. It arises, for example, with calculator use. A student with difficulty
ordering decimals would not, for example, be able to use a calculator (or division)
to decide whether a car travelling 3 kilometres in 4 minutes (0.75) is going faster
than a car travelling 4 kilometres in 5 minutes (0.8).
Sackur-Grisvard and Leonard (1985) investigated French students' ordering of
decimals and found three common systematic errors, which provided evidence of
the different incompletely-formed conceptions of decimal notation that students
have. Resnick, Nesher, Leonard, Magone, Omanson and Peled (1989) investigated
26
Moloney & Stacey .
the occurrence of these broad classifications of errors for small samples of students
from Israel, France and the United States; they named them the "whole-number
rule," "fraction rule," and "zero rule." In all of the samples, significant proportions
of students were seen to be consistently following these erroneous rules, which
they had constructed for themselves when trying to assimilate new knowledge
about decimals into what they already know about whole numbers, place value
and fractions. Resnick et al. used this as an example to demonstrate how
overgeneralisation of concepts (in this case, whole number, place. value, and the
size of fractions) can cause systematic errors.
Sackur-Grisvard and Leonard (1985) observed that there was no widespread
problem in comparing decimals when the integer parts were different and so the
previous studies and the present one deal only with the implicit rules which
students use to compare positive decimals with the same integer part.
The present study reports some Australian data on students' conceptions of
decimal notation. Although it has been well established in previous studies that
many students are not able to order decimals, and the ways in which they think
about the ordering task have been classified (in terms of the three mal-rules above
or expert behaviour) the prevalence of these rules and the ways in which a
student's thinking changes from one to the other have not been documented in the
,past. The present study reports on a longitudinal study which showed surprising
stability in secondary students' conceptions and illustrates what movement
between rules might occur. A second cross-sectional component of the present
study shows the distribution of the classifications across a population of Year 4 to
Year 10 students and indicates that many students probably move from one malrule to another.
Students' Conceptions
Whole-number rule
Sackur-Grisvard and Leonard. (1985) found that some students systematically
chose the number with more digits after the decimal point as the larger. This is
called the whole-number rule. They would say that 4.125 is greater than 4.7
because the whole number 125 is larger than 7. The decimal point is recognised but
only as a separator and the decimal portion is often read as a whole number, for
example "four point one hundred and twenty five," rather than the conventional
"four point one two five." Resnick et al. (1989) suggest three reasons why this rule
may occur. These students have implicitly constructed a number line such as the
one shown on the left in Figure 1.
Nesher and Peled (1986) conducted two studies involving Israeli students from
sixth to ninth grades. They observed that whole number students were basing their
judgement of decimals on their knowledge of whole number and suggested that
this previous knowledge was, in fact, deficient and that, for these students, place
value was not a consideration even with whole numbers where they also simply
reasoned that longer means bigger.
Changes in Students' Conceptions of Decimal Notation
27
EractionRule
$~C~l.1r-Grisvard
.and Leonard (1985) found that some students consistently
<1lOse tltenUJl\berwith fewer decimal places as the larger. This is called the fraction
J:'t.11e''l11e~esU1del1ts would say that 2.3 is greater than 2.67. These students reason
th(lfth~3jst~l1ths/the 67 is hundredths and tenths are larger than hundredths.
·*"lte J:'l1c:JtJ,vely,$ome confuse .3 with 1/3 and .67 with 1/67, giving the same result
~gl'\~diff~rentreason. It may also be the case that some students adopt a "longest~~sIl1.(l!Iest" rule simply as a response to feedback that the whole-number rule is
iJ.'l,¢orrect. In effect, these students have constructed a number line like one of the
. liii~sor\ the right of Figure 1. In offering these number lines we do not wish to
iJ:t).ply that students could offer such a number line for themselves and we know
!hat students will often have other special knowledge (e.g. that 0.5 is a half) which
<:l1tsacross the application of their more general rules. They do not see all of the
logical consequences of their ideas, and they do not recognise when some of their
igeasare logically inconsistent. The number lines are offered principally to assist
tl1.ereader to clarify the ideas behind student thinking.
Nesher and Peled (1986) observed that fraction-rule students recognised the
digits after the decimal point as relating to fractions but did not connect the size of
the parts and the number of parts. For example, when asked to write 3/4 in
decimal form some fraction-rule students wrote 3.4, 0.3 or 0.34. Nesher and Peled
suggested that this might be because the size of the parts is normally written as the
qenominator with fractions but is not explicitly given in the decimal, instead being
hnplied by location.
Zero Rule
The whole-number rule does not predict where decimals with one or more
zeros immediately after the decimal point will be ordered. Sackur-Grisvard and
Leonard (1985) found that students who otherwise applied the whole-number rule
(the longer decimal is larger) often obtained a correct answer when there was a
zero placed immediately after the decimal point. These students know decimals
beginning with zero are small. Thus, a zero-rule student, ordering the three
decimals 3.214, 3.09 and 3.8 would correctly choose 3.09 as the smallest but then
would use the whole-number rule to order the other numbers getting 3.09, 3.8,
3.214 in increasing size l .
Nesher and Peled (1986) see the zero-rule classification as only an improved
variation of the whole-number rule. In contrast, Sackur-Grisvard and Leonard
(1985) proposed that it was even more advanced than the fraction rule. With the
whole number and fraction rules, the length of the string of digits determines
magnitude without consideration of the actual value of each of the digits after the
decimal point but zero-rule students see the zero as a place holder and begin to
show recognition that the number of parts as well as the size of parts is important
when comparing decimal portions of numbers.
One of the reviewers of this paper suggested that the zero rule may have roots in
children's familiarity with the use of decimal-like notation for measures and money.
1.
Moloney & Stacey
28
2
2
2
1.9
1.8
tenths
1.1002
1.1001
1.1000
1.999
1.101
1.100
1.99
1.21
1.20
1.19
1.12
1.11
1.10
1.9
1.8
1.7
1
1.3
1.2
1.1
1
Order of decimals
fora wholenumber student
2
1.1
1.2
1.3
1.4
1.5
1.1
1.99
1.98
1.91
1.90
1.89
hundredths
1.02
1.01
1.999
1.9
1.10
1.11
1.12
1.20
1.21
thousandths
1.002
1.001
1.9999
1.9998
1.99
1.100
1.101
1.0001
1.99999
1.999
1.1000
1.10001
1
1
Order of decimals for a fraction-rule
student (two possibilities)
Figure 1. Students' ideas of the order of decimals between 1 and 2.
,
29
Changes in Students' Conceptions of Decimal Notation
The Diagnostic Test
.!~dl~~.~ify the conceptions of students by examining their use of these
ert~l'\efw~s.r1.des, Sackur-Grisvard and Leonard (1985) asked students to order sets
·~f;~~~i:~f;?giJpals. Resnick et al. (1989) simplified this task, asking students to
c~9~~~tJ).~·larger number from pairs of decimals. The comparison pairs and the
~~~J.I).~9~~d.by Resnick et al. to classify students (with a correction) is given in
'I'(l~l.~J/;'I1}~questionasked was "For each pair, circle the number that is bigger."
'lJl"i~i.f>~irs~f;?l'egiven in a random order and whether the larger number was given
fi~~()J."i~~~9I1d. was also randomised. A consistent whole-number rule student
~~91.d.¢h.()()Se as the larger number that with the longer string of decimal places
a1"l.~~piW()tlld be incorrect (as shown in Table. 1) for the first six items but give a
c9rJ."fact al}Swer for the last two items. A zero-rule student would do likewise except
wllereazero is involved as in the fourth, fifth and sixth items and would then give
a9q~rect answer. A fraction-rule student, always choosing as the smaller number
tb.~()ne with the shorter string of decimal places, would give the correct answer for
tliefirst six items but give an incorrect response for the last two. An expert is
always correct. Resnick et al. .were able to classify 88% of subjects. They used
s1..l.pplementary questions to confirm whole-number rule and fraction-rule usage
bu.f·l1ot for confirming zero-rule usage.
Table 1
Chart of Comparison Items and Patterns of Correct (VJ and Incorrect (X) Student
Responses Used by Resnick et al. (1989) for Classification
Number pair
Wholenumber rule
Zero rule
Fraction rule
Expert
Whole-number-rule items
X
.J
.J
.J
.J
.J
.J
X
.J
.J
.J
.J
.J
.J
.J
.J
.J
.J
.J
.J*
.J*
. X*
.J
.J
4.8
4.63
X
X
0.5
0.25
0.36
X
X
0.100
X
4.7
4.08
X
2.621
2.0687986
X
4/100
0.038
Zero-rule items
Fraction-rule items
4.4502
4.45
0.457
4/10
* Corrected from Resnick et al. (1989).
X*
30
Moloney & Stacey
The Longitudinal Study
Method
A written test was administered twice to 26 Year 7 students and 24 Year 9
students in a middle-class, non-government school with classes of mixed ability.
The test was first administered in mid-1992 and then one year later to the same
students. The test used the Resnick et al. (1989) comparison items (Table 1) and a
written form of the supplementary items which had been used by Resnick et al. for
follow-up interviews. Immediately before the testing, students in Year Thad done a
unit on decimals, revising decimal" place value but mainly concentrating" on
operations with decimals. In Year 8 they had studied the conversion of fractions to
percentages and decimals. The course in Years 9 "and 10 assumed knowledge of
decimal notation, so there was no particular preparation for the test.
Students were classified by their answers to the comparison items in Table 1.
Where their answers to the other items did not confirril this classification, they
were listed as unclassified.
Results
The number of students found to be in each classification is shown iIi. Table 2.
Eighty percent of students were able to be classified. The fraction-rule and expert
classifications were the largest. Only one student consistently used the zero rule.
There was only a small change in understanding of decimal notation during the
course of the year. Table 3 provides an overview of movement of individuals
between classifications over the year. It shows that only six of the fifty students
changed classifications. Five moved to the expert classification and the other was a
previously unclassified student who came to use the fraction rule consistently.
Most students who had misconceptions about decimals retained them throughout
a full school year.
Table 2
Numbers of Students Using Each Rule in 1992 and 1993
Classification
Whole
Number
Fraction
Zero
Expert
Unclassified
Class 1 (N=26)
1992 (Year 7)
1993 (Year 8)
4
4
9
8
9
0
0
9
5
4
Class 2 (N=24)
1992 (Year 9)
1993 (Year 10)
2
2
6
5
1
1
8
12
7
4
Subjects
Changes in Students' Conceptions of Decimal Notation
31
TaBl~3
b.Jittf,be,.s of Students in 1992 Classifications by 1993 Classification
·1992
. Ciassification
... Whole number
.. Fraction
Zero
···Expert
Unclassified
Whole
Number
6
1993 Classification
Zero
Fraction
13
Expert
Unclassified
2
1
1
16
3
8
Conclusion From the Longitudinal Study
This study demonstrates that misconceptions about decimals are a significant
p~oblem, at least at this particular secondary school. It is of great concern that half
Qfthe students at Year 10, and a greater percentage at other year levels, did not
¢ompare decimals correctly. Their teachers had previously been unaware of the
.¢xt€mt of this problem. It is also very surprising that only 6 out of 50 students
~l}()wed any movement between classifications over a full year of schooling. This
provides evidence that students were responding to thp items in a way which
reflected a stable thought pattern and strengthens the claim that the items used in
the study do test for an important set of misconceptions. Use of the zero rule was
not detected in significant proportions. In this study, only unclassified and fractionrule students moved to being expert. The whole-number students, in particular,
qid not change their thinking.
The relatively large percentage of unclassified students suggests a need t9 try
to improve the reliability of the supplementary items used to confirm the
classifications made with the initial decimal comparisons. The adaptation of the
supplementary items from Resnick et al. (1989) from interview to written form
needed to be improved for the cross-sectional study and further items were needed
to try to detect the zero rule more reliably.
The Cross-Sectional Study
The second study reported here is a cross-sectional study of students from
Years 4 to 10. The longitudinal study had indicated little change in students'
conceptions about decimals. Only a few unclassified and fraction-rule students had
moved into the expert classification. Therefore a cross-sectional study was used to
gather some data related to the way in which students' thinking about decimal
numbers develops across the years of school when decimals are taught.
Method
A second test (see Table 4) was constructed in order to classify a greater
percentage of students and with more certainty. The Resnick et al. (1989) test
(Table 1) classified students using their responses to only eight comparison items:
three pairs related to the whole-number rule, three to the zero rule and only two to
Moloney & StaceJj
32
the fraction rule. A response to a single item in each section could alter the
classification. Also, two items required students to convert fractions to decimals.
Resnick et al. acknowledged this as a potential source of error. In the new test, these
two items were changed so that only decimal notation was involved. The number
of comparison items was increased to fifteen with five distinguishing each of the
rules. A criterion of four items out of five was set as appropriate to demonstrate a
given rule 2 . Table 1 displays the comparison items and the scheme used for
classification in the cross-sectional study. The question asked remained the same,
that is, to circle the larger of the two decimals. Again, the pairs were presented
randomly with the correct response randomly first or second.
Table 4
Chart of Comparison Items and Patte;ns of Correct
for Classification in Cross-sectional Study
Number pair
(0 and Incorrect (X) Responses Used
Wholenumber rule
Zero rule
Fraction rule
Expert
X
X
X
X
X
X
X
X
X
X
...j
...j
...j
...j
...j
...j
...j
...j
...j
...j
...j
Whole-number rule items
4.8
4.63
0.5
0.36
0.25
0.100
0.08
0.75
0.37
0.216
...j
...j
...j
...j
Zero-rule items
4.7
4.08
X
2.621
2.0687986
X
3.72
3.073
0.04
0.038
8.514
8.0525738
X
X
X
...j
...j
...j
...j
...j
...j
...j
...j
...j
...j
...j
X
X
X
X
X
...j
...j
...j
...j
...j
Fraction-rule items
4.4502
4.45
0.457
0.4
17.353
17.35
8.24563
8.245
...j
...j
...j
...j
5.736
5.62
...j
...j
...j
...j
...j
The classification of students was carried out on the basis of the 15 comparison
items given in Table 4. Further questions were devised to link the misconceptions
to other aspects of understanding decimals. The hidden number item and zero
insertion task from Resnick et al. (1989) were trialed but found to be unsatisfactory
and they were omitted. Other test items were either taken from, or based on, those
We now believe we should have omitted the comparison of 0.04 and 0.038 altogether
as it is not clear whether zero-rule students will be incorrect or correct on this item.
2.
Changes in Students' Conceptions of Decimal Notation
33
found in a range of literature (Brown, 1981; Carpenter et al., 1981; Resnick et al.,
1989) to investigate possible links between particular misconceptions and other
errors that students make about relationships between place-value positions,
writing the decimal for a given fraction, giving equivalent fractions, identifying a
decimal on a number line, giving a decimal between two given decimals and
stating the value of a certain digit in a given decimal. The analysis of these links is
not fully reported here but is discussed by Moloney (1994).
The test was administered to 379 students in Years 4 to 10. All students were in
classes of mixed ability. The secondary sample of 208 students was from a single
school (a middle-class, Catholic girls' school) and the primary sample of 171
students was from one of its feeder schools (a middle-class, co-educational Catholic
school).
All the students in the primary sample had been exposed to decimals and
fractions but were yet to study them as a particular topic in the year of the testing
except for those students in Year 4. In Year 4 students were expected to be familiar
with recognising decimals to hundredths place value, perform simple addition and
subtraction and perhaps multiplication by a single digit. There had been some
ordering of decimals and relating them to fractions in common usage and money.
Year 5 students were expected to use decimals to measure, approximate by
rounding, and to use decimals to count, order and measure in practical
applications. Year 6 students were expected to recognise place value to
thousandths, add, subtract and multiply decimals and to relate decimals to
fractions and percentages. In Years 7 and 8 students' knowledge of decimal
notation was assumed and the emphasis was on operations with decimals and
converting decimals to fractions and percentages. For Years 9 and 10 decimals were
not covered as a specific topic and knowledge again was assumed.
Students undertook the test during normal class time under the supervision of
their own teachers and were asked to write their answers according to their
understanding of the question and to leave out anything not understood.
Results
The students were classified based on responses to the fifteen items in Table 4.
Most students gave responses completely consistent with the schema shown in
Table 4. The criterion of four out of five consistent responses in each of the three
subsections enabled other students to be classified. In all, 85% of primary school
students and over 95% of secondary school students could be confidently
classified. The number of unclassifiable responses decreased steadily through the
year levels to insignificant proportions. Not all of the students became expert but
Jhey did become more consistent.
Table 5 shows the distribution of rule use in each of the year levels. Line graphs
illustrating changes in expert, whole-number and fraction-rule usage from Years 4
to 10 are given in Figures 2, 3 and 4.
In looking at overall trends, the percentage of students classified as expert rises
reasonably steadily (considering the small number of classes sampled), but is still
only at 73% by Year 10 (see Figure 2). Although most secondary students in this
classification answered the other items on the test paper correctly, there were
Moloney & Stacey
34
Table 5
Percentage of Students in Each Classification by Year Level
Year level
Number
in year
level
Wholenumber
rule
%
4
5
6
7
60
52
59
58
56
49
45
3
42
19
7
4
2
0
8
9
10
Classification
Fraction
Zero rule
%
rule
%
60
33
25
43
28
27
20
0
0
3
14
0
2
4
Expert
%
Unclassified
%
Total
classified
%
22
8
41
31
15
17
12
5
4
0
2
85
83
88
95
96
100
98
64
69
73
significant proportions of incorrect responses on some items. Belonging to the
expert classifieation does not necessarily mean a student is an expert in all decimal
concepts.
100 -I--_ _.L-_-.L_ _...a..._ _..&..._......_ _......_ _..a.-_-+
80
1::
Q)
Q.
x
Q)
60
I
Q)
0>
.$
cQ)
~
40
Q)
Q.
20
O-+---~-_,.--_r_--~-......,~-_r--_r_-_+
3
4
5
6
7
Year level
8
9
10
11
Figure 2. Percentage of expert students by Year level.
Use of the whole-number rule was significant in earlier years but reduced
through the secondary years (see also Figure 3). Sackur-Grisvard and Leonard
(1985) claimed that this was developmentally the simplest rule. In accordance with
the findings of Nesher and Peled (1986), the responses of whole-number rule users
on the other items showed they had very underdeveloped concepts of place value
Changes in Students' Conceptions of Decimal Notation
35
and fractions, and the decimal point emerged as nothing more than a separator of
whole numbers.
The fraction rule is prevalent at all levels, constituting 20% of the sample in
Year 10 (see Figure 4). Fraction-rule students were more successful on the other
items than whole-number rule users and were beginning to integrate fractional
concepts with decimals.
The zero-rule classification was very small in size except at Year 7 (14%). We
attribute this to sampling error. Students in this classification tended to answer the
other items like the users of the whole-number rule, but the small number ofzerorule users prohibits confident generalisation.
As would be expected from a sample of only 14 intact classes, there are some
apparent anomalies in the data. In the secondary sample, there are steady trends
and a low number of unclassified students. There seems to be gradual movement
toWards the expert classification, although significant numbers of students persist
with the fraction rule. However, trends in the primary school data are less evident,
as the Year 4 students seem to have a better understanding than the Year 5
students. Between Years 4 and 5 there was a large increase in the use of the wholenumber rule and a decrease in both the fraction rule and the expert classifications,
contrary to the trends evident elsewhere. One of the Year 4 teachers attributed this
to the fact that the Year 4 students had commenced their study of decimals just
prior to the testing whereas the Year 5 students had not had any decimal work for
some months prior to the testing. It may be that for the Year 5 classes the simplistic
whole-number rule had asserted itself in the absence of instruction and practice.
100
<I>
......
+-_ _
_ _"""'_ _..I.-_--'_ _- I -_ _-L-_
I--_~
_+
80
'S
<I>
.0
E
::J
C
60
<I>
(5
.c
;:
I
<I>
40
C>
<tl
C
<I>
~
<I>
Q.
20
o+---.lIII'----r--'"'T""--,...---..,--~:::::=-~-_+
3
4
5
67
Year level
8
9
10
11
Figure 3. Percentage of whole-number-rule students by Year level.
36
Moloney & Stacey
100
+-__"---_-.1.._ _...1-_ _
..&.-_-+
.&.-._--1._ _......_ _
80
<J)
"2
c:
0
U
60
.rg
I
<J)
OJ
~
40
c:
<J)
~
<J)
0.
20
O-t---r---,.---r---T"""'---.---r--.,...---t"
4
10
11
9
3
5
6
7
8
Year level
Figure 4. Percentage of fraction-rule students by Year level.
Discussion
This study confirms the utility of the classification developed in earlier studies
(Nesher & Peled, 1986; Resnick et al., 1989; Sackur-Grisvard & Leonard, 1985)
about students' misconceptions of decimal notation. It has also tracked the
prevalence of rule usage over seven year levels using a considerably larger sample.
Links between conceptions and other decimal concepts and procedures have been
summarised here and are available in a fuller report (Moloney, 1994). The
procedure for classifying students' conceptions has been improved and tested in a
format which is suitable for teachers to use with their classes. In addition, the
longitudinal study provided preliminary evidence that there is very little
movement between misconceptions in the secondary years and more extensive
investigation is warranted.
As in the earlier studies, a large proportion of the younger students
demonstrated the whole-number rule misconception. There was a slow decrease in
the fraction-rule misconception but it remained prominent in higher year levels
(20% in Year 10 in this study). This misconception, in particular, has the potential to
remain with students into adulthood unless it is challenged.
It is clear that there are two main rules, the whole-number rule and the fraction
rule. They are generated by the two ideas that lie behind decimal notation: parts of
a whole expressed in fractions (e.g. tenths, hundredths) and place value. We see the
zero rule only as a variation of, or slight improvement on, the whole-number rule.
There was little difference in performance between zero-rule and whole-number-
Changes in Students' Conceptions of Decimal Notation
37
rule students on the additional tasks. In our future work with teachers and in
designing instruction to assist students to move towards expertise, we plan not to
feature the zero rule (e.g., Moloney & Stacey, 1996). However its existence
highlights the fact that qualifying for expert status with decimals requires
coordinating complex procedures. It is not just a case of avoiding two erroneous
rules. The transitional phases, such as within the zero rule, may be rich sites for
future exploration because that is where adjustment to understanding is
happening. In this context, it is interesting to note that 3 of the 5 students who
moved to expert status in 1993 in the longitudinal study had been unclassified in
1992. What is the nature and extent of these transition phases and what is the
catalyst for rethinking? For these reasons, a more substantial longitudinal study
should be conducted,· at least over Years 4 to 7' where most movement between
rules seems to occur.
The overall finding of this study, that a very significant proportion of Year 10
students cannot reliably decide which of a pair of decimals is the larger, is
disturbing. It is a clear limitation of this study that it was conducted in only two
schools, but we know of no reason why these schools would be unusual. We
believe that one of the implications for this work may be that teachers need to
understand the common misconceptions so that they can be alert to clues when
students interpret numbers wrongly. We also believe that teachers should offer
teaching that targets, exposes and hopefully corrects misconceptions. At present,
many teaching practices are designed to avoid confronting difficulties, for example
by dealing firstly with decimals with one decimal place exclusively, then with two
places some years later. Unfortunatel)', research does not yet conclusively support
this implication. We do not know, for example, whether it is valuable to confront
the misconceptions of individuals, or whether general teaching to all is just as
good. Research which gives guidance for teachers about the usefulness of
classifications of student thinking is clearly required.
The curriculum implications of a study like this are also only partly clear. The
Victorian Mathematics Curriculum and Standards Framework (Board of Studies, 1995),
for example, assumes competence in ordering decimals to thousandths after the
end of level 4, which th~ "average student" (not precisely defined) is expected to
reach at the end of primary school. At higher levels, the curriculum focus turns to
operations and use of decimals, assuming ordering. This study found that the
percentage of experts did not exceed fifty percent until Year 8 and reached only
73% by Year 10. It therefore seems to us to be clear that teachers throughout
secondary school must be aware that their students may not understand decimal
notation and that they must be prepared to address this routinely in their teaching,
even though they will not find it in the curriculum goals for higher levels. So is
understanding decimal notation a reasonable goal for level 4? We believe that it is.
Many students already achieve expert status in primary school and we believe
(again with little evidence) that many more could do so. In our metric, calculatororiented world an understanding of decimal notation should be a high priority for
.
all students.
Acknowledgement
The authors would like to thank the teachers and students who generously gave their
time to participate in this study.
Moloney & Stacey
38
References
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and implications from national assessment. Arithmetic Teacher, April, 34-37.
Grossman, A. (1983). Decimal notation: An important research finding. Arithmetic Teacher,
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Moloney, K (1994). The evolution of concepts of decimals in primary and secondary students.
Unpublished Master of Education Thesis, 'University of Melbourne.
Moloney, K, & Stacey, K (1996). Understanding decimals. Australian Mathematics Teacher
52(1),4-8.
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Sackur-Grisvard, c., & Leonard, E (1985). Intermediate cognitive organizations in the
process of learning a mathematical concept: The order of positive decimal numbers,
Cognition and Instruction, 2(2),157-174.
Authors
Kevin Moloney, St Columbia's College, PO Box 89, Essendon, Victoria 3040, Australia.
Kaye Stacey, Department of Science and Mathematics Education, The University of
Melbourne, Parkville, Victoria 3052, Australia.
E-mail: [email protected]