How to help Caitlin
caitlinGeneral Principles:
Target examples so that Caitlin realizes that she has something to learn! (e.g. She incorrectly thinks 0.9 < 0.12, but can correctly order decimals of the same length).
Teach underlying place value principles with concrete models (e.g. LAB).
Incorporate decimals of various lengths in the one situation wherever possible.
Conduct class or group discussions on this and other misconceptions.
Provide opportunities for Caitlin to use her new understandings.

What does Caitlin need to learn?
Caitlin has quite good whole number place value knowledge. For example, in "Talking about Place Value" interview, she recognizes that 025 and 25 are the same whole number. (Other whole number thinkers may not have such a good grasp of whole number place value ideas, such as only one digit (0-9) can fit into each place value column, and the value of the columns increases as we move further to the left, by a factor of ten. They will need this first.)

Caitlin knows that both decimals and fractions involve dividing a unit into parts. However, she only sees the number of parts; she does not see the size of the parts. Our number system is very sophisticated and indicates the size of parts only in a hidden way through place value. This is what Caitlin needs to learn to see.

Lesson Ideas:

Marking Homework

Use Caitlin's homework in your class for stimulating discussion which may demonstrate to some whole-number thinkers in your class what is wrong with their thinking. (More info on this activity)
LAB or other concrete model

Make numbers with concrete materials, discuss how the length of the pieces represents the size of the numbers, and relate the concrete model to written symbols continuously.

Number Expander

A visual aid to remind Caitlin that
0.83 = 8 tenths + 3 hundredths = 83 hundredths.

Mix decimal lengths

Frequently use decimals of different lengths in the one situation. Caitlin cannot find out that she does not understand if she only deals with one decimal place or only with two etc.

Number Trails

Decimal Skip Counting

With any of the skip counting activities, Caitlin will soon find that there is something wrong, e.g. she would predict 0.10 as the next number in this sequence: 0.7, 0.8, 0.9, ___. This provides a good opportunity for class discussion and demonstration with LAB or other concrete model.

Number Between

In the Number Between interview, Caitlin suggested "zero point four half" as a number between 0.4 and 0.5. Playing in a group of friends can show her that 0.45 is exactly "zero point four half". Later we need to be certain that she understands why this is.

Canteen prices

When Caitlin's understanding improves, she will see what is wrong with the canteen price list.