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by Joshua A. Taton, Ph.D. | August 21, 2023 | 4 min read

In Part 1 of this two-part series of posts, I argued that that the Common Core State Standards (CCSSs), at best, obscure the importance of comparison when averaging. At worst, via one possible interpretation, they overlook providing the motivation or purpose behind averaging entirely.

Now, looking at curriculum materials, my concern is significantly but not completely alleviated. For example, let's consider the Grade 6 program of Eureka Math by Great Minds, lessons on statistics occur in Module 6. (Note that—unfortunately but with understandable reason—this is the last module of the school year.)

Before moving further, I recall that Eureka Math is the most commonly-used program in the U.S., and it is one of the most highly regarded by educators. And, for transparency' sake, I should state that—while I have attended several Eureka Math professional development sessions, which were quite good—I have no formal or informal affiliation with the publishers or writers of this program. In fact, to me, Eureka Math is not especially unique, as there are a number of standards-aligned programs in the U.S. with which I am familiar and that I generally appreciate.

Module 6 begins with an overview for teachers, describing the topics covered and that, in particular, students will be summarizing and describing features of distributions of data. The overview explains that—in addition to measures of central tendency (or averages), like finding the mean and median—students will be exploring measures of variability (or spread).

Understanding the spread of a set of data is crucial to making comparisons, as well. Just as Lesson 8 of Module 6 demonstrates, by looking at the distribution of monthly temperatures in San Francisco and New York City, two sets of data can have the same mean (or nearly the same mean) but different dispersions of results. The data can be clustered close to the mean or spread far away from the mean (or somewhere in-between).

Having the same mean temperature might suggest that the climates of these two cities are perhaps indistinguishable. On the other hand, unlike San Francisco, the wide variation in temperature in New York City (ranging from a mean low in January of 39 degrees Fahrenheit to a mean high of 85 in July) might suggest that it has a more extreme climate.

This focus on variability and on explorations, like the one above on temperature, is undoubtedly positive. It helps teachers (and their students) understand why the calculations we perform in statistics are meaningful.

Even further, the word "compare" (or a grammatical variation thereof) appears 131 times within Module 6. Students are asked to compare an individual data point (such as the height of one sample student) to a larger population, and students are asked to describe and compare populations by looking at graphs of their distributions and measures of central tendency and variation.

This is all very good and surprising, since the CCSSs are notably silent on some of these issues.

On the other hand, though, the techniques of finding a mean or calculating variability are introduced conceptually—e.g., finding a mean is like doing a "fair share" or "balancing" problem—their purpose is not explained at the outset. This leaves students to guess at why one might want to perform such calculations.

Further, nowhere in the module overview (nor in the topic or lesson notes) is the word "purpose" connected with an explanation of the underlying motivation for finding measures of central tendency or variability.

The overall theme is clearly present, as noted on p. 32 of Module 6: "Comparison of distributions is a focus of lessons later in this unit." At the same time, this theme is buried behind 32 pages of prefatory content, and it is omitted from the introduction to the concept and technique of finding a mean that begins on p. 66 of Module 6.

One might argue that comparison is offered as a key idea in the overview to Topic B. Here, the text states (p. 65 of Module 6):

"Lessons 10 and 11 give students the opportunity to use both graphical and numerical summaries to describe data distributions, to compare distributions, and to answer questions in context using information provided by a data distribution." (emphasis added)

I would argue, though, that—presented as a single sentence in a paragraph that simply provides the outline of topics covered—this note is easy to overlook. And, further, noting that graphical and numerical summaries can allow students to compare distributions, is different than offering an explicit declaration.

I would like to see the text explicitly say, at the outset, "We calculate and use the mean and mean absolute deviation [the measure of variability used in Grade 6 in the CCSSs], so that we can more easily compare sets of data to one another and for the purpose of making decisions about the data."

Why am I belaboring this point? For two reasons: First, the CCSSs also need to be explicit, because research has shown that teachers are disinclined or discouraged to use curriculum materials in lieu of designing their own curriculum from the standards. Second, because, I also believe, curricular murkiness has significant ramifications that extend into adulthood.

In particular, one of the most challenging ideas for school leaders to grasp, I would argue, are mean achievement and growth scores of students in their buildings or district. I believe that a stronger understanding of not just the techniques of calculating, nor even a conceptual basis for these techniques, but also the underlying motivations and purposes could help school leaders make sense of the voluminous data reports they receive. And, further, what are valid and invalid conclusions to draw.

But that's a larger, more complex topic that I'll leave for another day.

One final note of disclosure: I am only able to see the freely available, open-educational resource (OER) version of Eureka Math that was published in 2015. I am aware that a new version Eureka Math^2 is now available, but it has not been publicly released.

I welcome your thoughts.