MY PREFERRED MODELS FOR DECIMALS include areas that can be folded. That includes strips and rectangles. Number lines that can be partitioned into smaller and smaller equal parts.
Decimals are fractions. ****Read them as fractions. ****Build them by dividing wholes into equal parts. *****Hundredths are in every tenth. Thousandths are in hundredths are in tenths.
NUMBER IS IN NUMBER
This can be quite evident if students are expected to divide tenths into hundredths in area models and on number lines. This can be quicly lost if students are allowed to create a new set of models for the next unit of decimals. For example in the pictures below. The pink balls that represent tenths are separate from the pink balls that represent hundredths. Whereas in an area model or linear model, the decimal parts can be seen to be inside each other. DENSITY
The models we use to represent concepts in mathematics exist in space. They prompt visual spatial reasoning. For ease of discussion, I loosely sort models into the following categories:
Linear Area Volume
The key word here is loosely. I am simply trying to begin a conversation with others that will help us build some common understandings.
One of the things I study is the impact that a specific category of models may have on the development of the idea at hand. For example, if I am talking 2 dimensional shapes: quadrilaterals, triangles, multi-sided, regular and irregular polygons but I am holding a prism, what affect will my model have on the thinking and reasoning my students are able to generate from the object?
How will/do their perceptions and understandings change if they hold the object instead of me? How will/do their perceptions and understandings change if they place the object on a surface and trace the face I am describing since I am discussing 2 dimensional shapes?
The answer, significantly!
If we want our students to experience higher success, higher achievement in mathematics, then we must engage them physically in the learning.
Stop showing and telling and expect them to show and tell.
NO one model on its own is ever enough. All models break down somewhere. That is why we must COMPARE, COMPARE, COMPARE. Teach students to analyze: How are these the same? How are they different?