# Mathematics and Education

A slow blog

## Unfinished Business from 300 BC

The double meaning of “ratio” in U.S. school mathematics is a phenomenon that goes back as far as Euclid. On the one hand, a ratio of two numbers is written as a pair, e.g., 3 to 1. On the other, a trigonometric ratio is a single number, e.g., the sine of pi/6 is one half—not 1 to 2. (Ratios of more than two numbers do not have this double interpretation, so are not part of this discussion.)

These two meanings parallel usage in Euclid’s time. According to the historian of mathematics Hans Jahnke, there were two kinds of ratios: ratios of what we now call the natural numbers (1, 2, 3, etc.) and ratios of magnitudes (e.g., sides of triangles).

You might think that the relationship between these two meanings would now be clear in school mathematics. However, the status of 1 seems to be have been—from a modern perspective—a major stumbling block. For Euclid, 1 was a unit and a number was a collection of several units, so 1 wasn’t a number. It wasn’t until at least the 15th century (probably longer) that 1 got full status as a number in Europe. According to the historian of mathematics education Kristín Bjarnadóttir, a major contribution to making this happen—and to erasing the distinction between number and magnitude—was Simon Stevin’s book L’Arithmétique, published in 1585.

But, Bjarnadóttir notes, it took another four centuries to complete the work of embedding “discrete arithmetic” (for example, 2 + 2) into “continuous magnitude” (e.g., interpretation of 2 + 2 on the number line).

That’s for the field of mathematics. For mathematics in school, in particular, for ratio, that work seems to be incomplete. The change in meanings from middle to high school often seems not to receive explicit attention. Middle school teachers need to be aware that their students will eventually understand trigonometric ratios as rational numbers. High school teachers need to be aware that entering students may understand the ratio of two numbers to be a pair of numbers. Surely many are, but is this result of hard-won experience or carefully designed teacher education?

Questions like this come to the fore because of the Common Core Standards for Mathematics. Consistent with U.S. tradition, in the Common Core a ratio of two numbers is a pair of numbers in grades 6 and 7, but a single number in high school trigonometry. What is not consistent with U.S. tradition seems to be the attention given to this change in associated professional development efforts. (See, e.g., the discussions of ratio in the Ratio and Proportional Relationships Progression, Functions Progression, and Tools for the Common Core blog.) Ultimately, this attention may pay off in improved student learning.