“For some definition of …”

I think the best thing I’ve learned in prosem this semester (our “introduction to the math ed world” course) is the importance of definitions.

Definitions in Mathematics

In math, definitions are important. Definitions can build on axioms or other definitions, but basic defined terms rely on “primitives” that are assumptions about the “basic facts.”

Note: In order to be perfectly precise, we would need to define everything. But definitions rely on words.  One word cannot be defined without relying on other words.  For those other words to be defined, more words need to be used. This is why we use primitive terms even though they are not at all precise. This means we have to be very careful about assumptions we make regarding primitive terms – sometimes we will assign properties to the primitives but unless that happens we can’t assume properties.

It’s how mathematicians set the stage to do math. In mathematics, being precise is everything. Some choices when creating definitions are based on conventions and others are based on strategy. A mathematician could arbitrarily assign any meaning to a symbol or term, but they need to communicate to others so paying attention to conventional use is important. (Although some of the most interesting mathematics results from mathematicians choosing to ignore certain assumptions or to change elements of definitions.) Mathematicians also make strategic choices to make later theorems weaker or stronger and (often) easier or harder, respectively, to prove. So, definitions are important in mathematics to communicate and to set up later arguments.

Definitions in Mathematics Education

Choices about definitions of terms in Mathematics Education are similar to those in Mathematics. Still, a math ed researcher needs to make decisions based on convention and strategy, to communicate and support arguments. A difference is that it is more difficult to be precise in language, when that language has to describe the complexities of life, teaching, and human beings. It is still important to reflect on definitions and attempt to be precise, however. It is difficult to look at prior research that has “taken-as-shared” meanings of terms or processes.

For example, in my reading, I see many studies that attempt to measure the effects of technology in teaching. As one example, many studies measure the effects of “use of graphing calculators to support student learning of functions.” But those studies do not always describe how the graphing calculators were used: Did students use them only to calculate values or graph functions? Did students use graphing calculators to explore the parameters of functions, making and testing conjectures? Did students gather data with calculator-based rangers (CBRs) and the explain the effect of changes in the movement on the function? These are important differences.

Arguing without Defining

I have often found myself in heated “discussions” (and by “discussion,” I mean “argument”) that ultimately end when we realize we’re both fighting for the same idea, but using words in different ways. Taking the time to define terms is important in research and in real life!

Communication is so difficult in everyday life because we all come from different communities (even communities of one) where different norms are accepted as the only norms and where meanings remain unexamined. But, by examining our meanings, we can be more aware of others’ different meanings. This awareness can make communication easier and more satisfying, and can open up potential communities. (Well, I think so.)

Notice that I did not define terms in this blog post. Oops.