### Decimals are easy to teach and to learn, aren't they?

To some extent this is true. It is very easy, for example, to learn to add and subtract with decimals. The rules for whole number addition and subtraction can be applied with very little change. In contrast, learning to add and subtract fractions is a new and challenging task. For this reason, decimals may seem an easy topic to teach.

However, evidence from mathematics education studies across the world indicates that both children and adults experience fundamental difficulties with decimals. This is serious because decimals are so widely used in practical situations. The main difficulties are

 fully understanding the meaning of the decimal numbers, being able to choose what operation with decimals to apply in a given situation.

The best indication is that only about half of the population is fully mastering decimals by the end of Year 10 (See research results).Difficulties with decimals cause trouble solving problems, making sense of answers to problems, rounding numbers off sensibly and reading scales. Even though the rules for calculating with decimals seem very simple, few children seem to be able to carry them out correctly just by rote - they need to understand what they are doing.

### Why do you put the decimal point in the "ones" column. This doesn't look neat.

Many teachers explain the decimal point as the separator between the whole number part and the fractional part of the number. We think it is better to stress that a decimal is ONE number, not two numbers put together. The role of the decimal point is to mark where the ones column is.  This is why we put it in the ones column. It also helps students to understand where the symmetry is. See here, for example.

 OUR MOTTO: The decimal point marks the ones column.

### I have found teaching decimals using money is very successful. Is this the best way?

Money can indeed be very useful, but there are important traps. With money, although 26 cents is 26 hundredths of a dollar, people often do not think about \$5.26 in this way. They think instead of 5 dollars and 26 cents with the dollars and cents representing two separate systems of units, albeit with a conversion between them. Although people know that there are 100 cents in a dollar, the fractional relationship is not uppermost in their consciousness and so often, 5.26 is thought of as two numbers, not as one. As a teacher, if you use money to teach about decimals you need to be careful not to teach two parallel systems of whole numbers, without strong links between the dollar and cent components.

There is also some evidence that teaching which relies too much on money encourages children to be "money thinkers". These children do not appreciate that there are decimals beyond a fixed number of places (usually 2). This is reinforced by too many questions in secondary school requiring rounding to two decimal places.

Another point for caution with using money to teach decimals is that is \$5.26, the 2 as a decimal represents 2 tenths. As money it represents 2 tens (of a different unit).

### Comparing decimals by adding zeros

I teach students to compare decimals by adding zeros until there are the same numbers of decimal places. Then they can compare them as whole numbers. For example, to compare 0.186 and 0.3 change it to a comparison of 0.186 and 0.300. It is easy then to see that 300 is bigger than 186 so 0.3 is bigger than 0.186. Is this the best way?

This is a mathematically correct strategy for comparing decimals. However, Resnick et al (1989) suggest that using this approach can mask other misconceptions that students may have about decimals. We have interviewed some Grade 5 and 6 children who completed the Decimal Comparison Test completely correctly with the adding zeros strategy. However, difficulty with some interview questions reveals they had an incomplete understanding of decimal notation.

The strategy also encourages some children to think in a whole number fashion. It is important to reinforce the size of the "fractional part" of the number.

### Why should children learn to compare decimals of different length? In real life this isn't needed.

Calculator multiplication and division can result in answers with more than two decimal places (for example, 2/7=0.2857142857...). The conversion of metric units can also lead to decimal measurements with different numbers of decimal places. A person intending to borrow money for a home loan could be presented with interest rates given to different numbers of decimal places, eg 4.6% and 4.35%.

Have you ever noticed signs in large libraries which say "do not reshelve books"? Well, now we know why... Many people are unable to reshelve a book in the correct location, as it requires comparing decimal numbers of different lengths. A university librarian told us of the difficulties experienced by some new staff (employed as "reshelvers") during their training.

### I wonder if primary school children need to be exposed to decimals longer than one or two decimal places.?

For many children it takes several years for full understanding of the decimal number system to develop. Children need to work with decimal numbers that contain more than two decimal places to foster this understanding. They need to be able to see the general patterns of the endless base ten chain, not just a few isolated facts they can learn off by heart. There are many contexts (in measurement topics, for example) where such numbers are encountered and this should be encouraged in the primary school classroom.

### Can the rule for multiplying decimals be explained to children?

To multiply decimal numbers, you get the number of decimal places in the answer by adding the number of decimal places in the two numbers being multiplied. For example to multiply 3.1 (one decimal place) by 4.02 (two decimal places), first multiply 31 by 402 and put the decimal point in so that there are three (1 & 2) decimal places in the answer. Can this be explained to children?

The algorithm described here is often used in schools to carry out pen and paper calculations of decimal long multiplication. Following the rule does not "justify" or "prove" why it works. A full justification of the rule is given in Foundations for Teaching Arithmetic (Marston and Stacey, 2001).

It is important to develop the meaning of decimals by numbers less than one. Area models are a useful way of showing how decimals can be combined through multiplication, while estimation helps children to decide the correct order of magnitude of a result. For example, 0.5 x 0.8 cannot equal 4 or 40 because both the multipliers are less than one. Also 0.17 x 65.983871 must be less 65 but more than 6.5 (one tenth of 65), so answers of 1.12172 or 112.1725 cannot be correct).

### How do you divide by a decimal number? Can it be explained to children?

One difficulty that children have with dividing by decimal numbers is that when dividing by a decimal between 0 and 1, the result is larger than the original amount. This is counter-intuitive to many children who associate division with "getting smaller". Being able to use various interpretations of division may assist in making sense of division by a decimal. For example, 0.6  0.2 may be read as "0.6 divided by 0.2" or "how many point 2's in point 6?" (the latter is more likely to help a student find the correct answer 3). A pen and paper algorithm that can be used to divide two decimals involves converting the decimals into whole numbers and then doing a usual short or long division. So long as the decimal point is moved an equal number of places in each number than the ratio of the number stays the same and the division result will therefore be correct. For more information, see Foundations for Teaching Arithmetic by Marston and Stacey..

### Can I omit decimals if students aren't good at whole numbers?

The first priority of teaching mathematics is to make sure children can work properly with whole numbers. Is it OK therefore to leave complicated work with decimals out of the primary curriculum?

Many of the difficulties that we have encountered with children in the field of decimals result from over-generalisations about whole numbers. This would suggest that students could benefit from being introduced to decimals and the meaning of decimal notation earlier rather than later in their primary school lives. It is clear that this area is complicated for children, but the importance of the fundamental numeracy represented by understanding decimal notation means that they must have as many opportunities to practise as possible.

### Why don't builders and other tradesmen use decimals more?

Why do builders measure in millimetres rather than metres? For example they will give the dimensions of a room as 3150 mm by 4280 mm instead of 3.15m by 4.28m.

One reason for using millimetres is probably to avoid decimal numbers! However the most important reason is to make the precision in measurements clear. If building measurements were quoted in centimetres then it may not always be clear that a measurement to the nearest mm is required. For example, reporting the shelf width as 35 cm does not tell you whether the shelf is exactly 35 cm, or if it is actually 35.2 cm rounded down. Writing 352 mm makes it clear that the measurement is to the nearest millimetre, a suitable precision for most building tasks. (See more information about indicating the accuracy of measurements)

### What is a good teaching aid to show equivalences?

What is a good teaching aid to show equivalences such as 0.387 = 3 tenths + 8 hundredths + 7 thousandths = 387 thousandths = 3 tenths + 87 thousandths = 38 hundredths + 7 thousandths, etc.?

Number expanders are great! (Full details and instructions on how to make/buy and use are provided.)

### Is decimal order like alphabetical order?

I use the analogy of alphabetical ordering of words to assist my students to order decimals. Is this a good strategy?

The process of ordering words alphabetically (eg bear, ear, eat, eats, edible) is similar to that of ordering numbers, however, care needs to be taken with this analogy. While letters in words are identified with a position relative to the start of the word (left-justified), digits within whole numbers are identified with columns determined by the position of the ones column at the end of the number (right-justified). The following table demonstrates how the correct alphabetical ordering of words corresponds with a correct ordering of decimals but an incorrect ordering of whole numbers. (Here the nine letters used are replaced by the digits 1 to 9.)

 Alphabetical order:  Correct Whole number order:  Incorrect Decimal order:  Correct bear 2416 0.2416 ear 416 0.416 eat 418 0.418 eats 4187 0.4187 edible 435294 0.435294

Other analogies such as money, sports scores, remainders in division and the Dewey Decimal System (used to arrange books in libraries) are discussed in the Background section.

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### Why do you call the units column the ones column?

A very hard choice. If we use numbers (hundred, ten, tenth) etc for the other column name it seems odd not to use "one". The disadvantage with "ones" is that the abbreviation O as in Hundreds, Tens, Ones (H, T, O) must NOT be used. Too many students confuse one and zero (bizarre, but common, see Stacey, Helme and Steinle (2001))

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### Why do you never just write .5 (point five) etc, but always 0.5 etc?

This is good practice because sometimes the decimal point is hard to see. As my wonderful Grade 4 teacher said, "You can never tell if it is a decimal point or just a spot made when a fly crawled out of the inkwell and onto your page". Not such a problem since the invention of ball point pens, but still a good idea!

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### I have a colleague who makes a big deal about the words "digit" and "number". Is this just being pedantic?

We expect children to know the difference between letters and words! Any child who did not understand this difference would have trouble when their teacher (or parent) was using these terms in later teaching.
Yet we find children in interviews who do not know the difference between digits and numbers! These children do not know whether to focus on the individual digits or the entire number when their teacher is explaining some mathematics.

 The 26 letters (a to z) are the building blocks used to form a word (the complete quantity) The 10 digits (0 to 9) are the building blocks used to form a number (the complete quantity)

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