Oct. 24, 2026
In a previous post, I gave examples of how a person could understand the mean (average) across all six levels of Bloom’s Taxonomy.
But I later argued that Bloom only gives us categories of activities relating to knowledge, not necessarily levels of increasing complexity or depth.
To more clearly illustrate what I mean, below are examples of both shallow and deeper understanding at each level of Bloom…
Level 1: Remembering (Knowledge)
Activities relating to identifying and recalling concepts.
Sticking with the example of the mean (average)…
Shallow understanding can be demonstrated by writing out a basic version of the formula, Σx/n, or saying something like, “Add up the parts, then divide by the number of parts.”
While this is accurate, a person could write or utter these words without necessarily being able to clarify what any of it actually means.
Deeper understanding in this level might constitute memorizing a network of concepts within which the mean sits.
For example, you might know that this is only one type of average among many, including…
- Mean
- Median
- Mode
In fact, it’s not even the only mean out there. This category includes…
- Arithmetic mean (what people usually mean by “average”)
- Geometric mean
- Harmonic mean
At this level of Bloom, deeper understanding could include the ability to identify and recall definitions and formulas for each of these and others.
It could also include knowing different or more complex variations of the formula, which type of average is most appropriate for different types of analytic situations, and so on…
Ultimately, however, all these terms, as well as their definitions and formulas, are just words, which are merely labels for concepts.
As such, while it would be difficult to imagine someone having deep understanding of these concepts without knowing the relevant terms, that knowledge alone is still not sufficient to guarantee comprehension.
Imagine you memorized a brief passage in a foreign language by writing and speaking it daily…
At some point, you’d have it down pat. But unless you ever bothered to translate it, you’d never know what any of it meant!
In the same way, a person could memorize every relevant term and definition in a given subject domain until they could recite it all at will, and that still wouldn’t guarantee that any of it meant anything to them.
Level 2: Understanding (Comprehension)
Activities relating to explaining concepts.
Shallow understanding here might be demonstrated by an explanation like, “The mean is a typical or average score in a distribution.”
This is true, but it’s incredibly vague. It seems to overlook the existence of other types of averages (see previous section). There’s also no apparent realization that certain factors (e.g., outliers, distribution shape…) could affect how well it typifies the distribution.
Two common candidate explanations for deeper understanding would be…
- The mean as the balancing point of the distribution; half the “weight” lies below it, and half lies above it. This is why the sum of deviations from the mean is always zero.
- The mean as the equal share per unit; if we totaled the scores, then cut the total into equal parts (one for each observation in the distribution), the value of each part would be the mean.
This is usually how I introduce the concept to students.
But we could go even deeper.
Perhaps the deepest (?) level of any mathematical or statistical concept is the algorithmic level.
Here’s the idea: If you truly understand a concept, you should be able to create your own algorithm from scratch that reproduces the idea. (Doesn’t really matter whether it’s unique or not, so long as you generate it yourself.)
One way to take the arithmetic mean to this level of understanding is to realize it’s only one case of a more general process. There are more technical versions of this, but here’s the basic generalization…
Suppose you perform some operation (e.g., addition, multiplication) on a set of values, resulting in a total effect. For example, summing all the values results in a total; multiplying all the values results in a product; et cetera.
A mean is a single value that could replace each score in the distribution without changing the result of that operation.
Consider summation (adding things up)…
If our set of scores is [1, 2, 3, 4], the result of summation would be 1 + 2 + 3 + 4 = 10. The arithmetic mean, Σx/n = 2.5, is the only value that could replace each score in that operation, while still producing the same result: 2.5 + 2. 5 + 2. 5 + 2.5 = 10.
Or consider multiplication…
The product of multiplying all the values in that same distribution is 1 ⋅ 2 ⋅ 3 ⋅ 4 = 24. Again, the geometric mean, ⁿ√(Πx) ≈ 2.2134, is the only value that could replace each score in that operation, while producing the same result: 2.2134 ⋅ 2.2134 ⋅ 2.2134 ⋅ 2.2134 ≈ 24.
What makes this explanation “deeper”?
To me, it’s deeper in that it represents a more general principle about how things work. The mean is not (just) an arbitrary concept that people use by convention; it fits into the bigger picture of how math works.
Based on this, it becomes apparent that there are infinite number of possible means that we could define.
This allows for understanding an entire class of calculations with a single concept. For example, someone who understands the above explanation is in a far better position to next grasp the harmonic mean and various other means.
Viewed through this lens, the arithmetic mean is only one case of a single value that can represent a distribution of scores in a particular context. It’s not some magical statistic that we should always default to when summarizing a set of scores. (This has critical implications for other Bloom levels, like application, creation, etc.).
Another takeaway from this discussion is that deeper understanding can often be acquired by exploring different ways of explaining the same concept.
The more multidimensional your understanding of a concept, the more your mental representation of it begins to transcend any particular set of symbols on a page.
And it doesn’t have to stop here.
Yet another alternative explanation could be through a regression lens:
Using an ordinary least squares approach, the arithmetic mean is the predicted value of the distribution when no additional variables are taken into consideration (i.e., no predictors; intercept-only model).
By “predicted value,” I mean one possible “best guess” for what a randomly selected score from the distribution would be.
And on and on it goes…
Okay, this post is already longer than I expected it to be. More next time…
