The decimal-fraction link
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 quantityPerhaps 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 fractionsCommon 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 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
Some students simply consider On the other hand, other children concentrate on
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
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. |