| Summary of Misconceptions about Decimal Numbers
Misconceptions can be diagnosed by listening and watching carefully when a child answers strategically designed tasks. One of the simplest and best tasks for this topic is to select the larger from pairs of decimals. Because this is such a good task, the misconceptions have been organized in three groups according to how the child orders decimals. Other tasks useful for revealing students' thinking are presented in teaching.
These students generally pick longer decimals to be larger numbers. There are a variety of reasons why they do this. Some children have not adequately made the decimal-fraction link and others have place value difficulties. The most common reasons for longer-is-larger behavior are outlined below. Longer-is-larger misconceptions are most common in primary school, with about 40% of Year 5 students interpreting decimals this way, diminishing to about 5% by Year 10 (see research data).
Whole Number Thinking
At one extreme, some children see the decimal point as separating two quite separate whole numbers. For example, instead of thinking of a decimal number such as 4.8 or 4.63 as a number between 4 and 5, they may see the numbers as two separated whole numbers 4 and 8 or 4 and 63. If asked to circle the larger of the two numbers, such a child might circle the 63 only, instead of either 4.8 or 4.63. These children are rare and need individual remedial help.
More commonly, children who have not completely made the decimal-fraction link will think of two different types of whole numbers making up a decimal such as 4.63:
Read more about analogies to money, sport and remainders in division.
Whole number thinkers are likely to expect that the number after 4.9 (4 wholes and 9 parts) is 4.10 (4 wholes and 10 parts). See how such a child is likely to count. They are also likely to have difficulty coordinating the number of parts and the size of the parts in a fraction, because they do not understand the decimal-fraction link. If the predominant discussion in the classroom is with decimals of equal length, the misconception is not challenged, and may continue to secondary school.
There are some variations in the way whole number thinkers order decimals. Sometimes these students select just on length alone, e.g. they will pick 0.021 to be larger than 0.21 just because it is longer. Other students look more carefully at the decimal part as a whole number, so that they will think that 0.21 and 0.0021 are equal, because the two whole numbers 21 and 0021 are equal. See a case study of 'Caitlin', who is a whole number thinker like this.
Column Overflow Thinking
Column overflow thinkers interpret 0.35 as 35 tenths, 0.149 as 149 tenths and 0.678912 as 678912 tenths, 0.035 as 35 hundredths, 0.0149 as 149 hundredths and 0.0043 as 43 thousandths. This thinking generally leads to choosing the longer decimals as larger except when there are zeros in the first decimal places.
These difficulties are like the difficulties shown by small children learning to count who often say:" . . sixty six, sixty seven, sixty eight, sixty nine, sixty ten, sixty eleven, sixty twelve...".
Similarly when children first learn to add, they may put more than one digit in each place value column:
Understanding how to rename this number from "sixty twelve" (arrived at by the addition) to seventy two depends on understanding the relationships between the place values of the columns. Ten in the ones column gives one in the tens column. Column overflow thinkers may have mastered this idea for whole numbers, but need to learn it again for the decimal positions.
Column overflow thinking also arise simply by "forgetting" which column name to take when describing the decimal as a fraction. Instead of getting the name from the rightmost column (in this case the hundredths, as 0.35 is 35 hundredths) the student may just take the name from the leftmost column (the tenths).
See a case study of 'Brad', a column overflow thinker.
Zero Makes Small Thinking
Therefore 0.428 may mean 4 tens + 2 hundreds + 8 thousands, or possibly 4 ones + 2 tens + 8 hundreds if another ones column has been inserted after the decimal point "for symmetry". These students might select 0.35 as larger than 0.41 because 53 is larger than 14 or because 530 is larger than 140. See a case study of a reverse thinker, 'Tuyet'.
There seem to be two causes for this thinking pattern. A few children, especially younger children, may have very little idea of fractions and have not begun to appreciate numbers less than one. More likely, hearing difficulties or language background is the cause. Often a child with this misconception has not heard the "th" sound in the column names; so rather than tenths, hundredths and thousandths, they believe that there are more tens, hundreds and thousands to the right of the decimal point. Judgments about the size of the decimal number are affected by what are perceived to be the columns with the largest value, that is the most-right columns.
The final "ths" sound is often missed by children from language backgrounds where a final "s" or "ths" is not a normal sound. Tenths sounds very similar to tens, hundredths to hundreds etc. Teachers must be very clear in speech and writing. More information.
These students generally pick shorter decimals to be larger numbers. There are a variety of reasons why they do this. The first reason is inability to coordinate the numerator and denominator of a fraction; the others are related to students drawing false analogies with fractions and negative numbers. Our research data shows that at any one time, about 10% of students in all years from 5 to 10 have shorter-is-larger misconceptions.
Denominator Focussed Thinking
Students who use denominator focussed thinking are aware of the place value of decimals, but cannot readily move between the various forms of decimals that are evident using expanded notation. For instance, Nesher and Peled (1986, p. 73) report an interview with an Israeli denominator-focussed child who was explaining why 4.45 was chosen as greater than 4.4502:
"Up to here (points to the 4.45 part) it's the same number, this zero doesn't make a difference, but here the 2 (4.4502) is ten-thousandths and here (4.45) it's hundredths, so hundredths that's bigger."
So the child has a strong vision of the decimal parts of the numbers as 45 hundredths and 4502 ten thousandths but doesn't also see the latter in the partially expanded form 45 hundredths + 2 ten thousandths.
The percentage of denominator focussed students in our Australian sample is about 4% in Years 5 and 6 and then decreases to 1% of Year 10.
Students with this misconception can be distinguished from others when they are asked to choose the larger of two decimals of equal length such as 0.3 and 0.4. They choose 0.3 (because 1/3 is larger than 1/4). For this reason, we call them reciprocal thinkers. (See glossary for reciprocal)
Such a student may interpret 2.6 as being like two and one sixth or as 2/6. For a question which asked students to write a decimal to tell what part of a region was shaded, more than 25% of Year 7 students in a national survey of students in the USA wrote 1.5 for 1/5 and 1.4 for 1/4. (Hiebert, 1985). Many students exhibit confusion between fraction and decimal notation. Carpenter et al (1981) report the results of a large sample of 13 year-old children in the USA On a multiple-choice question which asked for the decimal equivalent of 1/5 only 38% answered correctly, whilst just as many students (38%) chose 0.5.
Because decimals do not explicitly show the denominator, (the value is instead indicated by the place), it is likely that some students will assume that the numbers written represent the denominator, rather than the numerator of the associated fraction.
Confusion in High Places
The population of the USA is in fact slightly less than 5% of the world's population, not one-fifth at all. Momentarily, President Clinton has probably confused one fifth and 5%.
(Source: Guardian Weekly, June 28 1997, p3)
"It is a long way, but in the other direction".
Voula was confusing the length 0.9m, with 9 metres in some sort of negative/opposite direction.
Anita, a tertiary student, explained why she had chosen 0.20 to be larger than 0.35
"I was thinking along a number line and considering decimal numbers to be equivalent to negative numbers. Therefore -20 was larger than -35".
With the pair 2.516 and 2.8325, Anita explained:
"I felt more comfortable selecting the number with the least digits as I though the longer the number, the further it was down the number line in the negative direction."
Irwin (1996), working in New Zealand, also described children who were confused about negatives and decimals. When they were asked to put numbers on a number line:
This confusion is obviously more likely to occur after students have worked with negative numbers at school (Year 7/8 on), but it also occurs in younger children. Elizabeth, a Year 6 girl whose understanding of decimals otherwise appeared very sound placed the numbers 0.149 and 0.65 on a number line in the positions of -0.149 and -0.65.
Like reciprocal thinkers, these negative thinkers will generally choose shorter decimals as larger.
Why might decimals and negatives be confused? We speculate that the reason for a confusion of decimals/fractions and negatives is psycholinguistic in origin. They both arise as opposites, as "inverses" of cognitively "positive" operations which make numbers bigger. Decimals (and fractions) arise from division, the inverse of multiplication. Negatives arise from subtraction, the inverse of addition. So, in a sense, negatives and decimals/fractions are both ways of being opposite of positive and big. Both 1/3 and -3 arise as "opposites" of 3, the primary quantity.
To stop this confusion, be sure that children's ideas of decimals become well consolidated, e.g. by using decimals in many areas of mathematics. When teaching about negative numbers, be especially sure not to use whole numbers only (i.e. -3, -4, -10) but be certain to include a wide range of numbers ( -3.6, - 2/3, -0.01, -118.6) so that the different concepts are juxtaposed. Paradoxically, keeping concepts isolated one from the other can be a cause of confusion, rather than helping students to keep them separate in their minds.
Negative thinkers may have forgotten about the decimal-fraction link; this having been overtaken by interference from new knowledge, rather than have never having known about it. Why should students confuse decimals and negatives? As noted elsewhere, the place value names are, to an extent, symmetric around the ones column. This seems to remind some older students of the way in which the positive and negative parts of the number line are symmetric about zero. This may dispose some of them to interpret decimals as negative numbers. (More information). Another way of reducing this confusion is to use vertical as well as horizontal number lines.
To separate students using negative thinking from those using reciprocal thinking requires the inclusion of comparisons with zero in a task. The Zero Test was devised to do just this, and complements the original Decimal Comparison Test. Our research data was collected using the original Decimal Comparison Test, however, so only the combined prevalence of these 2 groups can be reported, accounting for 5% to 8% of students from Years 5 to 10.
Students in this category can generally decide which of two decimals is larger. Of course, many of these students are true experts, with a good understanding of decimal notation. Other students follow one of the two correct rules, (by equalizing with zeros or comparing from left to right - see below) but discussion with them shows that the rules are not supported by understanding. Others (such as the money thinkers and students who have special difficulties with zero) may have good pragmatic skills, but in reality very little understanding. A variety of tasks are needed to decide how much understanding they possess. Our Australian research data shows that about a quarter of Year 5 students are experts but this rises to only about two thirds of Year 10 students.
Equalising Length with Zeros
In our research, we have seen many well-taught children who correctly follow this rule but talking to them reveals a wide range of misconceptions. They know the rule, but do not understand it. Some will forget the rule fairly quickly if it is not taught with understanding.
Left to Right Comparison
An example: to compare 23.873 with 23.86:
Like the other correct strategies, this strategy can be taught as a rule to follow without understanding. The Hidden Numbers computer game enables a teacher to see whether children are using this strategy.
Money thinkers apparently have a good understanding of the first two decimal places, but are not sure of the order of other numbers on the number line. One tertiary student, for example, when asked to place numbers between 3.14 and 3.15 on a number line drew this, not realizing that she had omitted 3.141, 3.142, 3.143 and 3.144:
She repeatedly omitted some numbers in several similar tasks, and admitted that she was unsure of her answers. Furthermore, she had little idea about the general relationships between the place value columns. Other students have told us that numbers such as 4.45 and 4.4502 are really equal. These students (in fact some are adults) may believe that the extra digits on the end are 'mis-hits' and shouldn't really be there; in effect their number system is discrete with integer numbers of cents. See a case study of 'Maria' who relies on the analogy with money.
Not all of these students think of money - some have other similar models such as percentages. Many of them will not realise that they have a problem with decimals. They do not appreciate that there are an infinite number of decimals between any two others (density).
Money is a useful but limited way of thinking about decimals. Using the money analogy can mask misunderstanding. Care needs to be taken in teaching decimals with money. More information on money as an analogy for decimals.
Special Difficulties with Zero
These students may observe that the number 0 belongs to the "ones" column in place value terms, and since this column is to the left of the decimal columns (tenths etc) it is larger than numbers such as 0.6, which start in the tenths. These students may think that 0.6 is less than 0, but may know that 0.6 is greater than 0.0 or 0.00 etc. For these students, the whole number 0 is different from the decimal 0.0 or 0.00. This demonstrates the importance of teaching which presents a variety of examples and numbers in many forms. The Zero Comparison Test was created to detect any such difficulties that students may have.
It is often very surprising how closely students' answers follow the predictions made above across a range of tasks. Many children are "hooked" onto their wrong ways of thinking because, as has been shown above, they produce right answers to a lot of questions. Students (and also teachers) can think that they have just "made a careless mistake" on the other questions, without realizing how seriously flawed their ideas are.
Although children may have a particular interpretation of a mathematical topic, they usually do not appreciate all of its consequences. So their thinking may appear to be inconsistent. For example, a column overflow thinker may think 0.03526 is 3526 hundredths and 0.35 is 35 tenths. However, they might decide that 3536 hundredths is smaller than 35 tenths because they cannot coordinate the size of parts and the number of parts of a fraction.
Often children hold a range of ideas - sometimes mutually contradictory - using them according to circumstances. This makes diagnosing a child's difficulties more tricky, but interesting. Partially formed ideas can change in the course of an interview with a researcher or a discussion with a teacher.
The misconception categories described above account for a very large proportion of the students. However, there are other ways of combining ideas and drawing analogies with other learning that are not fully described in this summary (several others are given by Stacey and Steinle (1998)) and others that may not yet be known.
Some students complete tasks such as the Decimal Comparison Test using rather vague guiding principles, which vary from item to item and from the beginning of the task to the end. Thinking about the task may make them adjust their ideas, so their thinking at the end of the task is different to that at the beginning. About 30% of students completing the Decimal comparison Test seem to waver between ideas, so that their thinking cannot be classified (See research data.)