This page shows the prevalence of the misconceptions about decimals and how this changes as students progress through school. The data was collected by Kaye Stacey and Vicki Steinle as part of the Learning Decimals Project conducted from 1995 to 1999. The 3204 students came from 12 schools in Victoria, Australia and ranged from Year 4 to Year 10.
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Prevalence of misconceptions - Major classifications |
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Prevalence of misconceptions - Fine classifications |
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Persistence of misconceptions |
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Teaching makes a difference! |
Prevalence of misconceptions: Major classifications
The graph above shows the percentage of students at each year level who perform expertly (red) and show the major misconceptions. The red regions show that the percentage of experts is very small in Year 4, but grows rapidly in Years 5, 6 and 7. Throughout the early years of secondary school the growth is slow and only about two thirds of students are expert by Year 10. The blue regions represent the longer-is-larger thinkers. Most of the younger students think like this, but this misconception quickly disappears. The yellow regions stay quite constant, indicating that at all ages about 10-15% of the population have a shorter-is-larger misconception. Similarly the percentage of students who do not answer according to a consistent known rule is about 20% and fairly constant.
The data was collected using a 30 item decimal comparison test and testing was conducted approximately every 6 months for four years. The data displayed in the graph is also given in the table below. This indicates the large number of students tested. The 12 schools were in areas with very low, medium and high socioeconomic status.
| Year level | Yr 4 | Yr 5 | Yr 6 | Yr 7 | Yr 8 | Yr 9 | Yr 10 |
| No. of students | 336 | 481 | 262 | 1147 | 385 | 310 | 283 |
| Task Expert | 4% | 24% | 31% | 48% | 60% | 65% | 70% |
| Other | 13% | 16% | 18% | 18% | 17% | 20% | 18% |
| Shorter-is-larger | 9% | 14% | 19% | 17% | 13% | 10% | 8% |
| Longer-is-larger | 74% | 46% | 33% | 17% | 11% | 5% | 5% |
Prevalence of misconceptions: Fine classifications
The major misconception groups used above clump together some very different ways of interpreting decimal notation. Understanding these differences is a key to helping students. Therefore it is useful to split the groups up into more precise ways of interpreting decimals. The following table lists the percentage of students using longer-is-larger and shorter-is-larger misconceptions for Year 5 and Year 10, as well as a breakdown into ways of thinking. A description of each of these ways of thinking.
Some of the ways of thinking have been detected in only a small percentage of the students by using the Decimal Comparison Test, but their inclusion is warranted as they have consistently appeared in interviews of primary, secondary or tertiary students. This is because different tasks prompt different knowledge, when it is not well consolidated.
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Misconception
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Year 5 |
Year 10 |
| Longer-is-larger (26% Overall) | 46% | 5% |
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whole number
thinking |
35%
1%
5%
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2%
0%
1%
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| Shorter-is-larger (15% Overall) |
14%
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8%
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focussed thinking reciprocal/negative thinking |
4%
6%
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2% |
| Apparent-expert behavior (47% Overall) |
28%
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77%
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expert thinking |
24%
2%
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70% |
| Unclassified (12% Overall) | 12% | 11% |
These figures are averages from the 12 schools involved in the Learning Decimals Project. Prevalence in individual schools and classes varies widely, reflecting teaching. For example
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65% of Year 5 at one school were whole number thinkers |
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15% of Year 6 at one school showed column overflow or zero makes small thinking |
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37% of Year 6 at one school had shorter-is-larger misconceptions |
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27% of Year 6 at one school showed reciprocal or negative thinking |
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8% of Year 8 at one school demonstrated money thinking. |
For technical reasons, this data underestimates the percentage of money thinkers:
"John" was a Year 7 student when he was first tested and in Year 9 for his 5th test. John tested as a task expert on the first test, but then a money thinker on the next test. On John's third test he again tested as a task expert and then as a money thinker on the fourth. His last test was answered again as a task expert. We think that John was probably a money thinker for all of this time. This is because the test we used was not able to identify money thinkers who make lucky guesses. For this reason, the prevalence of task expert is overestimated in the table above and the prevalence of money thinking is underestimated by the test: it might be double the percentages indicated. Even using this underestimating test, in one of the secondary schools, 14% of the 612 students were classified as money thinkers at some time in the project and this thinking appears to be very prevalent amongst adults.
Persistence of misconceptions
The data revealed that students with misconceptions tend to hold on to them. This section shows that over six or more months, there is a chance of between a third and a half that a student's misconception category is unchanged.
There were 63 students in the project who tested as Longer-is-larger (L) on 4 or more consecutive tests. Sally was one of these; her 6 test results are given below. Over three years, Sally's ideas have only oscillated around in the longer-is-larger group of misconceptions.
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Sally |
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Yr 4 |
Whole Number thinking |
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Yr 5 |
Column Overflow or Zero Makes Small thinking |
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Yr 5 |
Whole Number thinking |
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Yr 6 |
Whole Number thinking |
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Yr 6 |
Column Overflow or Zero Makes Small thinking |
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Yr 7 |
Whole Number thinking |
There were 27 students in the project who tested as Shorter-is-larger on 4 or more consecutive tests. Bob was one of these; his 6 test results are given below. From Year 6 to Year 9 his ideas were basically unchanged.
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Bob |
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Yr 6 |
Shorter-is-larger but unclear which type |
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Yr 7 |
Reciprocal or Negative thinking |
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Yr 7 |
Reciprocal or Negative thinking |
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Yr 8 |
Reciprocal or Negative thinking |
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Yr 8 |
Reciprocal or Negative thinking |
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Yr 9 |
Reciprocal or Negative thinking |
Overall, the likelihood of a student retesting as Longer-is-larger (L) on the next test was found to be 0.44 (based on 1257 pairs of tests). A breakdown by year is shown below.
Similarly, the overall likelihood of a student retesting as Shorter-is-larger (S) on the next test is 0.38 (based on 847 pairs of tests). A breakdown by year level is shown below. Note the rise to 0.49 at Year 8 - we think this may be due to the interference of new learning about negative numbers (negative thinking is one of the S-type misconceptions).
Teaching makes a difference!
The sections above show that many students continue to hold misconceptions for very long periods of time or they waver between different misconceptions as they struggle to sort out ideas. However, the data above also shows big variation between classes, which demonstrates that teachers can make a big difference. Misconceptions arise naturally, from students' being unable to assemble all the relevant ideas together or from limited teaching, but students can easily be helped to expertise.
Even a small amount of targeted teaching makes a difference. In one experiment (Helme & Stacey, 2000a & 2000b) all four teachers of the Year 5 and Year 6 classes at one school attended a one hour session on how to use LAB to teach the meaning of decimal notation. A total of 87 children from these four different classes were tested twice and the data below is only for these students. Data collected earlier by the Learning Decimals Project over a period of three years (1996-1998) indicated that, for this school, 41% of Year 6 students tested as expert in decimal notation. In the cohort of 1999, 46 children (53%) tested as experts on the pretest, a slightly higher proportion than previously.
As it happened, the teachers of Class M and S each gave a couple of lessons using LAB, whilst the teachers of Class X/Y and Class Z were unable to spare the time. About three months later the classes were tested again. As the table below shows, a lot of children in Classes M and S had become experts, but no-one in Classes X/Y or Z did. A little bit of teaching made a large difference.
Numbers and percentages of experts by class on pre- and post- tests.
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