Longer-is-Larger Misconceptions These students generally think a longer decimal is a larger number than a shorter decimal	Whole Number Thinking Column Overflow Thinking Zero Makes Small Thinking Reverse Thinking
	Shorter-is-Larger Misconceptions These students generally think a shorter decimal is a larger number than a longer decimal.	Denominator Focussed Thinking Reciprocal Thinking Negative Thinking
	Apparent-Expert Behavior These students can generally decide which of two decimals is larger but sometimes not for the right reasons.	Equalizing Length with Zeros Left to Right Comparison Money Thinking Special Difficulties with Zero
	Consistency of Students' Thinking Do these misconceptions persist? Do they matter?

Longer-is-Larger Misconceptions

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
Learners with this way of thinking assume that digits after the decimal point make another whole number. They have not effectively made the decimal-fraction link. Our data indicates that 30% of Year 5 students are thinking this way, although figures as high as 60% of Year 5 at some schools have been recorded.

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:

	perhaps 4 "whole numbers" and 63 more bits of unspecified size,
	perhaps as 4 "whole numbers" with a remainder of 63
	perhaps as 4 "whole numbers" and 63 of another unit, rather like 4 goals and 63 behinds in Australian Rules football or even as 4 dollars and 63 cents.

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
Some students will usually choose longer decimals as larger, but will make correct choices when the initial decimal digits are zero. For example, these children will say 0.43 is greater than 0.5 but will know that 0.043 is smaller than 0.5. One group of these students, called column overflow thinkers, have made the decimal-fraction link but have trouble with fundamentals of place value. Column overflow thinkers have learnt the correct column names for decimal numbers, but attempt to write too many digits into a column. So 0.12 is 12 tenths (as there is no zero after the point) while 0.012 is 12 hundredths (as there is one zero after the point). In effect, they squeeze the number 12 into one column. This is why we call it column overflow.

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
Some children who order decimals in the same way as column overflow thinkers (above) actually seem to know little at all about place value. These zero-makes-small thinkers may have very little idea of the decimal as representing a fractional part. They respond to many of the questions as do whole number thinkers. They know, however, just one thing more than do whole number thinkers - that a decimal starting with zero in the tenths column is smaller than one which does not. For example, they will know that 0.21 is larger than 0.0021 or 0.012345. Unlike whole number thinkers, they therefore can choose that 0.0762 is smaller than 0.53 correctly. A child with this misconception will order decimals in the same way as a column overflow thinker, but talking to them will reveal the differences.

Reverse Thinking
Some children do not know that the place value of the columns decreases when we move to the right. They know there is a similarity between the patterns of column names on the right and the left, but may assume they are the same. Occasionally this results in a child thinking that the names for the place value columns are the same on both sides of the decimal point:

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.

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Shorter-is-Larger Misconceptions

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
These students often have a good knowledge of place value names, but they have difficulty coordinating the size of the numerator and denominator of a fraction. They understand, for example, that 0.4 =4 tenths and that 0.83= 83 hundredths. They also know that tenths are larger than hundredths. They wrongly conclude that 0.4 is greater than 0.83 because they think only about the size of the parts (the tenths or hundredths) and cannot simultaneously consider how many parts there are. This is why we call them "denominator-focussed" thinkers. These children need more help with coordinating the influence of the numerator and denominator for fractions (see decimal-fraction link). See a case study of 'Ricardo', a denominator focussed thinker. See how he is likely to count with decimals.

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.

Reciprocal Thinking
Another reason for shorter-is-larger thinking is that children are trying to interpret decimal notation in terms of the more familiar fraction notation. They have made the decimal-fraction link but, unlike the denominator-focussed thinkers above, they do not consider place value. They see the decimal part as the denominator of a fraction, with larger denominators creating smaller fractions. For example, they think that 0.12 is something like 1/12 (they may not think it is really the same) and 0.3456 is something like 1/3456. The consequence of this is that they act as if longer decimals give smaller numbers.

They know that	12 < 3456,
so they know that	1/12 > 1/3456
so they conclude that	0.12 > 0.3456

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.

See a case study of 'Courtney' who thinks like this. See how she is likely to count.

Confusion in High Places
Confusion between fractions and decimals/percents even happens in high places. When President Clinton was opening the G8 summit in 1997 he was reported to make the following statement about the USA, intending to show that it had more than a fair share of wealth in the world, but saying, instead, that it has approximately a fair share.

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)

Negative Thinking
When Voula, a Year 9 student, was asked to indicate how long 0.9 of a metre was, she paused for a long time before stretching out her arm and pointing to the left saying:

"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.

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Apparent-Expert Behaviour

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
Equalizing length with zeros is probably the most common strategy taught in Australian schools, although it is used infrequently in other countries, such as Japan. To find out which of two decimals is the larger, append zeros to the shorter until they have the same lengths and then compare as whole numbers. For example, to compare 0.4 and 0.457, append zeros to 0.4 to get three decimal places (0.400) and then compare 400 with 457. This strategy always works for comparing decimals.

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
This correct strategy is to compare columns from left to right, until a digit in one decimal is larger than the corresponding digit in the other (and the first will then be larger than the second), OR until one decimal stops (which will then be the shorter one, except in the case of zeros).

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 Thinking
Some students may appear to be experts, but in reality have very little understanding of decimal place value and its fractional aspects. These students are usually able to deal with decimals in everyday life because they understand one and two decimal place numbers well. Many of these students relate them to money. For example, they think of 4.63 as 4 dollars and 63 cents. They think of 4.8 as 4 dollars and 80 cents. With this as a model, they are able to carry out many tasks.

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
Amongst the group of students who seem to be very good with decimals, there are some who have particular trouble with zero. They may be able to correctly describe the relative sizes of all decimals except when one is equal to zero, when they reveal that they think that all "zero point something" decimals are less than zero. This may be negative thinking (described above) but it can also be due to over-generalisation of place value ideas and confusion of the place value columns with a number line.

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.

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Consistency of Students' Thinking

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.)

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