One of the most important things students need to understand to reason proportionally is the difference between comparing two quantities in relative (multiplicative) versus absolute (additive) terms. Students often struggle with making the move to thinking multiplicatively.

How can we begin to help them make this transition?

The research suggests one way:

**Make the two ways to compare quantities explicit.**

What does this look like?

Here’s an example of how ratios are introduced in Chinese classrooms (Cai, 2004; Cai & Sun 2002):

*Miller Middle school has 16 sixth-grade students and 12 of them said that they are basketball fans. The remaining students are not basketball fans. How could we describe the relationship between the students who are basketball fans and those who are not?*

Once students determined that there are 12 fans and 4 non-fans, they described the relationship in the following ways:

- There are 8 more fans than non-fans.
- There are three times as many fans as non-fans.
- For every three fans, there is one non-fan.

The first description is an additive comparison, while the next two are multiplicative comparisons. Through continued exposure to the different ways to compare quantities, students begin to move flexibly between additive and multiplicative reasoning.

Of course that’s only the beginning of developing proportional reasoning. But being aware of the different ways to compare the situations is an important first step. Knowing why one comparison might be more useful than another is the next step.

Cai, J. (2004). Developing algebraic thinking in the earlier grades: A case study of the Chinese elementary school curriculum. *The Mathematics Educator 8(1) *107-130*.*

Cai, J., & Sun, W. (2002). Developing students’ proportional reasoning: A Chinese perspective. In B. H. Litwiller & G. Bright (Eds.), *Making sense of fractions, ratios and proportions: National Council of Teachers of Mathematics 2002 Yearbook *(pp. 195-205). Reston, VA: NCTM