This section presents about 30 items to use with your students to pinpoint topics to emphasise in teaching. Evidence is presented from studies around the world that demonstrates that decimals are not an easy topic for students to learn. The items chosen are typical of the questions that have been asked and the results that are generally obtained.
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Items about decimal notation |
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Items about decimal operations |
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Items on reading scales and rounding |
Items about decimal notation
Test Item: Ring the bigger of the two numbers: 0.75 or 0.8.
In a large UK study (Brown, 1981), only 57% of
children aged 12 years got this item correct, improving to 75% for children
aged 15 years. Some children were simply comparing the whole numbers 75
and 8, thinking that the digits after the decimal point represent a different
whole number which also has ones, tens, hundreds, etc. Children who make
this error are longer-is-larger
thinkers.
Australian data (Stacey and Steinle 1998; this site) shows that about half of children in Year 5 make errors like this.
Ordering single decimal place numbers is a relatively easy task. For example, a very large study of British 11 year olds carried out by the Assessment of Performance Unit and reported by Bell (1983: p. 105) found that 80% of students could do this correctly. (The others are shorter-is-larger thinkers). However, when the decimals have varying numbers of decimal places, the facility of the question drop sharply: only 21% of 11 year olds were able to correctly order 0.07, 0.23 and 0.1. Only about half of 11 year olds understand whole number place value well and many less understand decimals.
Test Item: Say the number 0.29
In the UK study (Brown, 1981), 19% of children
aged 12 years and 10% of 15 year olds read the number simply as "twenty-nine"
. These children either ignored the decimal point (quite a common reaction
to an unfamiliar symbol) or interpreted the digits after the decimal point
as a new number. When students say "zero point twenty nine",
it is worthwhile checking on their understanding of decimal notation.
Test Item: Ring the BIGGEST of the three numbers: 0.62 or 0.236 or
0.4
In the UK study of 98 children aged 12 years (Swan,
1983) 50% of 12 year olds chose 0.236 (longer-is-larger
thinking) giving reasons like these:
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"it is the biggest because it has got the most numbers" |
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"you can tell it is the biggest because there's no number at the beginning of all and its the highest number at the end" |
On the other hand, 28% chose 0.4 (shorter-is-larger thinking) giving reasons like these:
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"it has only one number which is a tenth and the rest are hundredths and thousandths" |
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"it is the biggest because it has the least number of diggets" |
Test Item: Which list shows smallest to largest?
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(a) |
0.345 |
0.19 |
0.8 |
1/5 |
|
(b) |
0.19 |
1/5 |
0.345 |
0.8 |
|
(c) |
0.8 |
0.19 |
1/5 |
0.345 |
|
(d) |
1/5 |
0.8 |
0.345 |
0.19 |
This item was used in TIMSS, the Third International Mathematics and Science Study (Lokan, Ford and Greenwood, 1996 and 1997) and data was collected in 44 countries in 1994. Australia was a little above the international average (44% correct) on this item, with 47% of 13-14 year old students selecting (b) correctly. In Singapore, 84% were correct. Answer (a), which would be correct if the decimal point was replaced by a fraction line (shorter-is-larger misconception), was selected by 25% of Australian students. Answer (c), in which the length of the decimal numbers increases (longer-is-larger misconception), was chosen by 18%. (TIMSS data held at Australian Council of Educational Research)
Grossman (1983) reports on the results of a mathematics assessment test used by the City University of New York . While over 50% of the 7100 students entering tertiary education could perform operations on decimals, only 30% could order them by size.
Test Item: Ring the BIGGEST of the four numbers: 0.19 or 0.036 or
0.195 or 0.2
The US National Assessment of Education Progress (NAEP) found that 46%
of 13 year olds were correct (0.2) but 47% chose the longest number (0.195).
(Carpenter, Corbitt, Kepner, Lindquist and Reys, 1983).
Test Item: Write a number in the space to complete the statement.
73.45 = 70 + 3 + 0.4 + ____
Children often have difficulty with zero as a place holder. Only
51% of 12 year olds answered this correctly with 24% giving 5, .5 or 0.5
as an incorrect answer.
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0.1 |
10 |
0.2 |
20 |
0.01 |
2 |
Only 44% of 12 year olds and 59% of 15 year olds answered this item correctly (Brown, 1981)
Test Item: Write 4/10 as a decimal.
Hiebert (1985) reports that 4.10 is a common error
amongst Year 5,6 and 7 students. For a question which asked students
to write a decimal to tell which part of a region was shaded, more than
25% of Year 7 students wrote 1.5 for 1/5 and 1.4 for 1/4.
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Australia was close to the international average on this item, with about 40% of 9-10 year old students selecting 0.2 correctly. More than 40% chose 2.8 (there are 2 columns shaded and 8 columns unshaded). Four high performing Asian countries (Singapore, Japan, Hong Kong and Korea) and Portugal did very well on this item with over 70% correct. This item is regarded as appropriate for 9-10 year olds in all Australian states and for 8-9 year olds in three states. (TIMSS study reported by Lokan, Ford and Greenwood, 1997)
Test Item: 0.4 is the same as .....
four, four tenths, four hundredths, one fourth
Australia with 48% of 9-10 year olds correct was well above the international
average of 39% correct. but well below Singapore with 90% correct. In
Australia, 29% of students ignored the decimal point and chose 4 and 16%
were confused with fractions (reciprocal
thinking) and selected one fourth. (TIMSS data held at ACER)
Test Item: Is 786 ÷ 987 negative, zero or positive?
This item demonstrates a deep confusion between fractions, decimals and
negative numbers. Hayes (1998) studied the teaching
of negative numbers in several Australian secondary schools. In a large
sample of 13/14 year olds, 54% of the students in a control group said
786 ÷ 987 is negative. He tested an experimental method of teaching
negative numbers, and still 41% of the experimental group said 786 ÷
987 is negative. This issue needed more attention. Some reasons for these
confusions are explored by Stacey, Helme and Steinle
(2001).
Test Item: How many different numbers could you write down which
lie between 0.41 and 0.42?
Most children do not have a sense of the "density" of decimals, that
is, that there exists an infinite number of decimals between any two given
numbers. Only 12% of 12 year olds and only 20% of 15 year olds answered
this item correctly (indicating there are an infinite number of numbers,
or at least "lots and lots") (Brown, 1981)
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Write down any number which is: |
Typical incorrect response |
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BIGGER than 3.9 but SMALLER than 4 |
3.10 |
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BIGGER than 6 but SMALLER than 6.1 |
6. |
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BIGGER than 0.52 but SMALLER than 0.53 |
0.52 |
In the first part, a variety of misconceptions lead to the answer. The answers to the second and third parts are very reasonable, but these children need to learn about the amazing way in which the place value system solves this problem.
Test Item: Four tenths is the same as .......... hundredths.
Only 28% of 12 year olds and 40% of 15 year olds were able to answer
this item correctly (forty hundredths, or 0.4 = 0.40) (Brown,
1981)
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Questions |
Typical incorrect response |
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0.2, 0.4, 0.6,____,____ |
0.2, 0.4, 0.6, 0.8, 0.10 |
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0.3, 0.6, 0.9,____,____ |
0.3, 0.6, 0.9, 0.12, 0.15 |
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0.92, 0.94, 0.96, 0.98,____,____ |
0.92, 0.94, 0.96, 0.98, 0.100, 0.102 |
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1.13, 1.12, 1.11,____,____ |
1.13, 1.12, 1.11, 1.10, 1.9 |
When students make these responses they are treating the decimal part of the number as a second separate whole number, and apply the same ideas as they would to a similar sequence composed only of whole numbers. A variety of misconceptions lead to this.
Items about decimal operations
Test Item: Write your own story to go with this sum:|
4.6 + 5.3 = 9.9 |
When asked to provide a realistic context in which decimals are used, some children may associate inappropriate contexts for the decimals along with misconceptions already noted above. Some typical inappropriate responses were:
| "James had 4.6 sweets. His best friend gave him 5.3 sweets and he has 9.9 sweets altogether." |
| "Chris has 4 boxes of crayons. 4 full and the other with 6 in it. Then his father gave him 5 full boxes and another with 3 in it. Altogether he had 9.9 boxes of crayons." |
| "John had 4 apples and 6 quarters. His mum gave him 5 apples and 3 quarters so he had 9 apples and 9 quarters" |
| "Terry had 4 portions of cake plus a 6th of a piece. His mother gave him 5 portions and a 3rd of a portion. Terry had eaten 9 portions and a 9th of a portion". |
Students' difficulties to write a story that matches the sum reveal their misunderstandings of decimal numbers. The first response is simply unrealistic. Does this child appreciate what 0.6 of a sweet might be? The second and third responses each show that the student has not appreciated the special role of ten in the decimal system. These students do not appreciate that the decimal point is not just a separator between different units (boxes of crayons and individual crayons or apples and quarters) as it is sometimes used in sport and other contexts. The fourth response associates decimals with the denominator of fractions, which is quite common.
Test Item: Ring the one that gives the BIGGER answer (in each line):|
a) |
8 x 4 |
OR |
8 ÷ 4 |
|
b) |
8 x 0.4 |
OR |
8 ÷ 0.4 |
|
c) |
0.8 x 0.4 |
OR |
0.8 ÷ 0.4 |
Many children incorrectly apply the "multiplication makes it bigger, division makes it smaller" misconception when multiplying and dividing by a number less than one. In a large British study (Brown, 1981), 50% of 12 year olds and 30% of 15 year olds believed that multiplication was the correct answer in each case. The"multiplication makes it bigger, division makes it smaller" misconception arises from over-generalising correct ideas that apply to whole numbers. Students in Years 5 to 8 need special attention to the differences when multiplying and dividing by numbers less than one, so that they can develop good intuition about how these numbers behave.
Test Item: Ring the calculation you would need to do to find the answer to this question: The price of minced beef is shown at 88.2 pence for each kilogram. What is the cost of a packet containing 0.58kg of minced beef? (These costs were realistic at the time of testing and in the units were used in shops.)
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0.58 ÷ 88.2 |
88.2 - 0.58 |
0.58 - 88.2 |
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0.58 x 88.2 |
88.2 + 0.58 |
88.2 ÷ 0.58 |
In the UK study (Brown, 1981), 18% of 12 year olds and 29% of 15 year olds correctly answered with multiplication. Many children (37% of 12 year olds and 42% of 15 year olds) chose division, most likely because they knew they needed an answer less than 88.2 pence and so chose division to make a smaller number. See comments for item above.
Test Item: Multiply 5.13 by ten.
This is an easy item to do mentally if a student has an understanding
of place value and the size of decimal numbers. Without this understanding,
it is hard to remember the rules about moving the decimal point or the
numbers. Only 37% of 12 year olds and 65% of 15 year olds answered this
item correctly. Many children just added a nought getting 5.130, some
moved the decimal point or the digits in the wrong direction. Most of
the children who tried to use long multiplication algorithm also made
errors, commonly when they multiplied by 0. (Brown,
1981)
Test Item: Add one tenth to 2.9.
This is an easy item to do mentally if a student has an understanding
of place value and the size of decimal numbers. Without this understanding,
it is hard to remember the rules. Only 38% of 12 year olds and 59% of
15 year olds answered this item correctly. Some children just added 10
to 9 getting 2.19, some add one tenth to nine tenths, getting 10 tenths
correctly but they write it as 2.10. (Brown, 1981)
Test Items: Divide 24 by 20. Divide 16 by 20.
In the UK study, the success rates for these two items were about 8%
for 12 year olds and 35% for 15 year olds. Some children thought there
was no answer, either because the division did not give a whole number
answer (24 ÷ 20) or because a number cannot be divided by
a larger number (16 ÷ 20). Many 15 year olds incorrectly
wrote the remainder of 4 as the decimal part (i.e. 24 ÷ 20
= 1.4 ). (Brown, 1981)
|
900 g |
9 000 g |
90 000 g |
900 000 g |
In the TIMSS study, Australia was close to the international average on this item, with 53% of 9-10 year old students selecting 9000g correctly. (Mullis et al, 1997)
Test Item: In a discus-throwing competition, the winning throw was 61.60 m. The second place throw was 59.72 m. How much longer was the winning throw than the second place throw?|
1.18 m |
1.88 m |
1.98 m |
2.18 m |
In Australia in the TIMSS study, 72% of 13 year old students were correct. The most common wrong answer was 2.18m, indicating problems with "trading" or maybe not looking beyond the whole number parts for an estimate. (Lokan et al, 1996)
Reading scales and rounding
Test Item: Read the following scales and write your answers in the boxes provided. Give your answers as decimals.
In the study by Swan (1983a and 1983b), answers
of 8.
and 2.3 were typical errors made by children. Another scale had endpoints
of 3 and 4 (with 5 divisions) and 28% of 15 year olds wrote 3.1 for the
first tickmark. Another scale had endpoints of 5 and 6 (10 divisions)
and 68% of 12 year olds correctly identified the 8th tickmark as 5.8.
This shows that students tend to associate one tick mark with one counting
number. Scale reading is an important skill, that requires a good understanding
of the meaning of notation.
Test Item: Give two identical numberlines, one with tickmarks on tenths
and the other with tickmarks on fifths. Put an arrow to the same place
on each, e.g. at 1.6 and ask the students to fill in each box.
Children can incorrectly link two different decimals to a common position
on a number line and believe the decimals name the same number (may be
due to their previous work with fraction equivalences using number line
representations). For example, a child might mark the first point as 1.6
and the same point on the next numberline as 1.3 and then link these two
decimals. This is similar to marking 1/2 and 2/4 on two different numberlines
when showing the equivalence of fractions.
|
100 |
90 |
89.1 |
89.06 |
89.064 |
In this TIMSS item, Australia was close to the international average (45%), with 44% of 13-14 year old students selecting 89.06 correctly. 28% of Australian students selected 89.064, probably indicating confusion about which place value column is the hundredths. (TIMSS data held at ACER)