Blog

by Joshua A. Taton, Ph.D. | August 14, 2023 | 3 min read

I have something embarrassing to admit.

I began learning about the meaning of the term "average," probably around 10 years old. In the Common Core State Standards (CCSSs), today, instruction on "measures of central tendency" (i.e., mean, median, and mode, which are three different types of average) begins in Grade 6.

But I didn't understand it well—at least not conceptually—until I was, um let's see, age 34? At that time, believe it or not, I was teaching a graduate-level probability and statistics course for educators.

And I already had earned one degree in mathematics (while in graduate school, well on my way to two more).

At the time, as I was preparing for the "mean-median-mode" lesson, it dawned on me: I knew how to solve (and passably teach) all sorts of problems that involved calculating each of these averages. Including the conditions that lead to their various biases.

But I had no real, concrete idea why we used them.

As the lesson approached, I remember being flummoxed and grasping for meaning in all sorts of texts and resources. But nothing readily emerged—nothing that would allow me to provide a straightforward context or motivation to my students. Before we dug into performing the calculations and discussing the related standards and pedagogies.

Looking at samples of elementary school word problems turned out to be the key that unlocked the mystery for me. It seems rather obvious and trivial now, but it certainly didn't at the time.

I don't recall the specifics, exactly, but I remember reading a word problem that involved calculating the "average" (in this case, mean) height of a group of students in a given classroom. I do not believe that the problem made the context obvious, but it somehow dawned on me that the only reason someone would calculate the average height of a group of students in one classroom was, well, to compare classrooms to one another.

Let me say that again: The only, true reason to calculate an average—whether a mean, median, or mode—is to compare one group to another group. And, of course, by "group," I am referring to some collection of individual items that exhibit variation (don't think too hard on this last point).

Comparing individual, discrete items is easy: a jar containing 4 oz. of blackberry jam has less jam than a jar containing 7 oz. of blackberry jam. But comparing the amount of jam on a shelf of partly used jars with the amount on another shelf of partly used jars—why? I have no idea—requires some other approach.

Before you squawk at me, yes, you could find the total of the amounts of jars on both shelves or find the total height of the students in a given classroom. But that would fail to allow for the identification of a representative amount of jam in a jar on each shelf (or the representative height of a student).

Which brings me to the Common Core State Standards and curriculum programs.

I think that the CCSSs obscure the fundamental notion of the comparison, its main purpose, when averaging. In Grade 6, under the domain of Statistics and Probability, they ask students to be able to "develop understanding of statistical variability" and "summarize and describe distributions."

To do so, students should be able to:

• "Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number." (6.SP.A.3); and

• "Summarize numerical data sets in relation to their context, such as by...Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered." (6.SP.B.5.C).

But—I ask you—where do we see anything indicating for the purposes of comparing sets of data?

In Part 2 of this blog post, I will tackle two additional topics: 1) The notion of variability, referenced in the standards; and 2) I share my research on how this failure—to identify a comparative purpose for—is replicated in standards-aligned curriculum materials.

I welcome your thoughts.