The purpose of this reflection is to clarify what I mean when I use the terms model, drawing, symbols, and pictures. I use these words frequently in my lessons, as well as with my students.
Model- a model can be a physical representation using manipulatives or a drawing of what this representation would look like
Drawing- shows a mathematical concept (not in the way an artist would) in a simple picture that usually represents models using manipulatives.
Symbols- numbers, variables, and signs (operations) usually in the form of an equation or expression.
Pictures- students are encouraged to use mathematical drawings rather than pictures (the way an artist might draw something) because they are more efficient. However, it is important to allow students the option of drawing a picture when they need it to help make sense of a problem.
When working with fractions, I will steer my students away from modeling fractions with circles. Fraction bars (tiles) and number lines are much more helpful for the students because fractions have to show equal parts. Often students have a hard time breaking circles up appropriately if it is a more challenging denominator (anything over 12, odd numbers etc).