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## Value depends on place: column names and sizes
One of the first tasks of understanding The structure of decimal numbers is the same. A number such 456.789 is
might then be a reverse thinker and select 0.35 as larger than 0.41 because 53 is larger than 14.
Error: Analogy with money confuses tens and tenths.Some children are confused about whether the first column after the decimal point is tens or tenths. This can come from analogies with money. Veronica in Year 5, for example, seemed to routinely think about this. She read the number 0.3 as "oh point three, that's thirty" and 3.5 as "three point five, that's fifty".
This is sometimes a language problem, sometimes arises by false analogy and sometimes is a reunitising problem.
## Endless base ten chainIn addition to knowing the names and size of the place value columns,
students need to know the relative value of the columns. They need to
know that the value of each column is ten times the value of the column
to the right (
dividing by ten or the powers of ten (glossary) is achieved by shifting digits into adjacent place value columns. Although this is very easy to carry out, it is not well understood and consequently many children cannot do it reliably. They often get confused about which way to "move the decimal point" and when to "add or take away zeros" - the result of trying to follow rules learned without understanding. This problem is widespread. For example Bell (1983)
reports that only 47% of a very large sample of British 11 year olds correctly
answered "How many times is 0.1 greater than 0.01?" and only 34% correctly
answered "What number is 10 times 0.5?" Performance on the item "ten times
100" was 71%, much better but not as good as might be expected. Equivalencies
like those shown below for 4 tenths and 376 thousandths are crucial to
understanding. Tables of Equivalencies:
- Only one digit (from 0 to 9) ever goes in one column
- Because all the columns have place value ten times as great as the column to the right, ten in any column gives one in the next column to the left.
Students who have not mastered this cluster of ideas will sometimes exhibit column overflow thinking. They interpret decimals as if more than one digit can go in each column. For example, Brad in Year 6, would interpret - 0.35 as 35 tenths,
- 0.678912 as 678912 tenths,
- 0.035 as 35 hundredths,
- 0.0149 as 149 hundredths and
- 0.0043 as 43 thousandths.
Brad's interpretation of 0.35 as 35 tenths instead of 35 hundredths may also have arisen simply because he has "forgotten" 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) he may just take the name from the leftmost column (the tenths). This is an important idea that needs definite consolidation, so that students are very secure with it. It is related to understanding equivalent fractions.
## Expanded form: reunitising tenths, hundredths etcAn idea central to dealing with the relative size of decimal numbers is to be able to interpret them in expanded form as decimals and as fractions. This section demonstrates the challenge of the cognitive processing involved. A decimal such as 0.639 can be interpreted in all the ways shown below. All these forms, except the last, can be obtained with a number expander.
To think of a number 0.639 is all the ways shown above requires a student to be able to deal with units made out of other units (unitising and reunitising). ## Which zeros matter?Many children have difficulty deciding how or whether zeros change the value of a number. With decimals, children need to know that the zeros which affect the place value of the figures are on the left and the insignificant ones are on the right: 3.250 and 3.2500000 are the same as 3.25 and 003.25 but 3.025 or 3.205 are different. The essential understanding of which zeros affect the place value of the digits and which ones do not is best demonstrated using concrete materials such as Multi-Base Arithmetic Blocks (MAB) or Linear Arithmetic Blocks (LAB). Making 3.25 and then 3.025, 3.250, 3.205 and 03.25 from blocks clearly demonstrates which zeros affect the size of 3.25, rather than just providing a list of rules for students to learn. Using the ideas of expanded notation are crucial.
It is very hard for children who do not understand the place value basis of decimal numbers to memorize the rules. Children generally decide which zeros change the value of a number according to their own interpretations of decimal notation. The erroneous misconceptions show many examples of this. However, the one principle above applies to both whole numbers and decimals. Answers to Exercises(a) 2.3 = 2 ones + 3 tenths (b) 5.82 = 5 ones + 8 tenths + 2 hundredths (c) 0.7411 = 7 tenths + 4 hundredths + 1 thousandth + 1 ten-thousandth |

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Last modified: Sun 16 April 2006

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