The most basic idea about decimals is that, like fractions, they enable us to describe parts of a unit quantity. They do this not by creating new "sub-units" but by creating new numbers. Decimals and fractions are two solutions to one problem. Many young children and some throughout secondary school do not make the decimal-fraction link correctly. Others exhibit the same cognitive difficulties that are encountered with fractions in their thinking about decimals. Fundamental understanding of fractions, such as dealing with equivalent fractions, is critical to decimals as well.

 Not seeing decimals as representing part of a unit quantity Associating decimals with the wrong fraction Cognitive difficulties common to understanding both decimals and fractions Coordinating number of parts and size of parts of a fraction. Partitioning, unitising and reunitising

### Not seeing decimals as representing part of a unit quantity

Perhaps the most basic knowledge about decimals is to understand that both decimals and fractions are designed for the same purpose; to express parts of the unit quantity. Even into the early years of secondary school some students do not fully appreciate this.

Some children see the decimal point as separating two whole numbers. At one extreme, they might see two quite separate numbers in one decimal number. 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 as 4 whole numbers and 63 of another unit, rather like 4 goals and 63 behinds in Australian rules football). These children will tend to select longer decimal numbers as larger. For example, they would pick 4.63 as larger than 4.8. Many of them will be categorised as whole number thinkers. Over-reliance on money as a model can also lead to this "two parallel whole numbers" interpretation, for those students who think about dollars and cents, without fully appreciating the role of a cent as a hundredth of a dollar.

A group of older children who show no evidence of understanding the decimal-fraction link are those who interpret decimals as negative numbers. These students may have forgotten about the decimal-fraction link; having been overtaken by interference from new knowledge, rather than have never having known about it. More information about this thinking.

### Associating decimals with the wrong fractions

Common fractions provide more information explicitly than do decimals. The fraction 2/5, for example, indicates that the reference unit (the "whole") has been divided into 5 equal parts and 2 of these parts make up this fraction. In decimal notation, the denominator (glossary) is hidden, just as the place value (glossary) of the columns in whole number numeration is hidden. Just as it is simpler to see the Roman numeral XXXII as 3 tens (XXX) and 2 ones (II) than it is to see this structure in 32, so it is easier to interpret the fraction 4/10 than the decimal 0.4 where the size of the parts (tenths) is indicated only by the place value.

Some students who link decimals and fractions may nevertheless not associate a decimal with the correct fraction. For example, a student may interpret 2.6 as two and one sixth or write 1.4 as the decimal for one quarter (Hiebert, 1985). Interpreting the decimal part of a number as the denominator of a fraction is referred to as reciprocal thinking.

### Cognitive difficulties common to understanding both decimals and fractions

Coordinating number of parts and size of parts of a fraction
Because decimals and fractions are both used to describe parts of a unit quantity, some of the difficulties that students show in understanding fractions are evident in understanding decimals. To understand the size of a fraction, the numerator and the denominator must be considered simultaneously. The denominator indicates the size of the parts into which the referent whole has been divided and the numerator indicates how many parts there are. Not being able to coordinate these two factors is a major developmental difficulty in understanding both fractions and decimals.

Some students simply consider how many parts there are in the decimal and do not consider the size of the parts. They conclude that 0.621 with 621 parts must be larger than 0.7, which has only 7 parts. They do not think about what these parts, or extra bits or remainders etc might be. More information about whole number thinking.

On the other hand, other children concentrate on the size of the parts and do not simultaneously consider how many of them there might be. They conclude that 0.621, which is made up of thousandths (very small) will be smaller than 0.7 (which is made up of the relatively large tenths) but also smaller than 0.5. These children generally think shorter decimals are larger numbers. Peled and Shahbari (2003) found that 78% of the denominator focussed students in their study were unable to correctly compare common fractions. More information about denominator focussed thinking.

Partitioning, unitising and reunitising
Partitioning, unitising and reunitising are three cognitive processes that are required for dealing with common fractions and they also affect students' understanding of decimals. There is general agreement (Behr et al, 1992) that many students' difficulties relate to changes in the nature of the unit that they have to deal with. For example, to find three quarters of 24 counters, the counters are first thought of as individual units, then the 24 counters need to be perceived as a whole so that one quarter can be taken. Then three of these new composite units need to be taken to make three quarters.

Decimals present problems especially with re-unitising between tenths and hundredths etc. For example to see 2 strips of a 100 square as representing

• 20 hundredths of one unit OR
• 2 tenths of one unit OR
• 0.2 or 0.20 of one unit

requires several cognitive steps. The initial counting units are the small squares (see diagram below). Two different composite units are created from these - a tenth is a new composite unit made from a strip of ten small squares and the whole square is a new composite unit, made of the 100 small squares. Seeing the square as being composed of 10 strips (each a tenth) requires the idea of a unit-of-units. Finally to talk about 0.2 of the square means that the square itself is a measure unit. (Baturo and Cooper, 1997)

 Hundredths: 100 of the 1-unit (little squares) is unitised as one 100-unit (big square) One 100-unit becomes the measure unit to which the shaded part is related. Hundredths to Tenths: Ten 1-units is unitised as one 10-unit (i.e. one strip) One hundred 1-units is perceived as ten 10-units Ten 10-units is unitised as one group of ten groups of 10 units One group of ten 10-units becomes the measure unit to which the shaded parts are related.

When the whole is partitioned into tenths only, students need to only unitise once. There is only one measure unit invoked. Similarly if hundredths only are to be considered. However, when hundredths need to be perceived as both tenths and hundredths, as they are for recording decimals and for renaming places (e.g. 2 tenths = 20 hundredths), the cognition required is more complex.

This depth of cognitive processing involved in dealing with multiple units underlies the difficulties that students have with the basic place value ideas and can be a cause of students' resorting to the simpler but erroneous misconceptions.