 General Principles:
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What does Maria need to learn?
To a casual observer, Maria appears to be expert at decimals, because
she deals adequately with decimals in everyday life. The interviews show
how competent she seems. In reality she has very little understanding
of decimal place value, and works only on analogy with money. Other money
thinkers might interpret decimals as (whole number) percentages or from
measurement etc. Maria understands place value and the base ten relationships
but only across a limited range.
Maria has learned facts such as "ten hundreds make a thousand" as isolated facts, without generalising to the relationships between the place value columns that are expressed in the endless base ten chain. She needs to learn about the density of the decimal number system, to change her idea that the real numbers are discrete and separated (like the whole numbers are).
Lesson Ideas:|
This game reinforces ideas related to the endless base ten chain, such as when a decimal (or whole number) is multiplied by 10, the digits move into the next largest column. Number slides are also useful. |
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Use Maria's homework in your class for stimulating discussion which may demonstrate to some money thinkers in your class what is wrong with their thinking. (More info on this activity) |
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Work beyond two decimal places |
Don't always round to 2 decimal places. This reinforces her wrong ideas. Teach rounding well. |
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Forces discussion of numbers between other decimals. Play the game well beyond two decimal places. Hidden Numbers using numbers such as 5.38 and 5.3824 is useful for stimulating discussion. |
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For Maria, it is essential to introduce the largest piece first (the one) and then discuss how to cut into smaller and smaller pieces. Don't start with a small piece and work upwards. Be sure to have Maria imagine pieces which are too small to make and link to place value. |