Heuristic

I’m only four weeks into my internship at the Conte Center for Computational Neuropsychiatric Genomics at the University of Chicago, and I’ve already heard about three different people tell me the mathematical aphorism, “All models are wrong, but some are useful.” Of course, it would be ridiculous to suggest that a model, graph, or diagram of any sort, by its own nature, should or could completely represent everything about reality. But students and researchers in areas of mathematics and statistics understand very well that data and information can often be misleading. We often look for theory in data without knowing what data actually represents, and the way we communicate scientific information can have on how another person perceives it.

When I explored the origin of this quote, I discovered it was by the famous mathematician George E. P. Box:

Since all models are wrong the scientist cannot obtain a “correct” one by excessive elaboration. On the contrary following William of Occam he should seek an economical description of natural phenomena. Just as the ability to devise simple but evocative models is the signature of the great scientist so overelaboration and overparameterization is often the mark of mediocrity.

(Box uses William of Occam to reference Occam’s razor by stating that scientists must make the fewest assumptions, or an “economical description”, when making scientific hypotheses.) I would think that Box’s statement to encourage scientists to put focus on elegance along with the actual information that one is attempting to convey when creating a model. Especially in the biological sciences, it’s easy to be inundated by the amount of information and complexity of the explored phenomena. A move towards some type of a specific design of a model, even if for the utilitarian sake of conveying information effectively, would be welcomed.

Though the way we structure the models we write in scientific journals and publish in newspapers is very important, the idea that our models are always wrong may have similarities to a deeper meaning in communication: the rhetoric we use is not always perfectly clear. During the first day of my Organic Chemistry I Lectures, our professor explained to us the way we were going to learn the material for the course that would help us be as successful as possible (both in the course and in our futures). He drew a graph on the chalkboard that looked something like this:

As students in the STEM fields, were all too familiar with this moral lesson: focus on application rather than memorization. I think it was safe to assume that most of the students in the class understood how it is much more useful to learn how to apply problem-solving techniques, and other concepts rather than memorizing definitions and facts in a science course. You should try to understand the reasons why a chemical reaction occurs in an organic chemistry course rather than trying to memorize the placement of electrons and arrows. You should be able to identify how to use theories how to solve a mathematics problem rather than being able to memorize equations. Without starting a discussions of whether or not this is actually a good model (after all, this drawing is not a serious scientific model), there a few things that are important to discuss when talking about this distinction between “memorization” and “application.”

The first issue that comes to my mind is that memorization and application are not two opposite, separate entities. In order to be able to “apply” concepts, it is implied that one must already have those concepts or something about those concepts “memorized.” When I understand how to “apply” Schrodinger’s Equation to predict the potential of particles in a physics equation, I must have “memorized” what each variable means. If I were to categorize facts and pieces of information necessary for an exam into those that must be “memorized” and those that must be “applied, then it would be clear that there is a huge overlap between the two and we would be left in vaguely-defined terms. This ambiguity can be observed in the tests of various classes, as well, as we see professors defend their tests and curricula as emphasizing “application” rather than “memorization” without actually exploring the skills necessary to do well in those courses.

But maybe there are better (or less wrong) models for explaining how students learn? Some of my professors have tried using Bloom’s taxonomy to alleviate some of these issues.

Named after educational psychologist Benjamin Bloom, this model attempts to view learning as a hierarchy in which one starts with the most basic skills at the bottom and moves up towards the “higher-level” skills. One must first begin to memorize information, then understand it, then such-and-such until he/she reaches whatever level satisfies him/her. By viewing “Apply” as simply a level above “Remember,” it is clear that one must be able to remember information and details before applying them. This seems to get rid of the first issue that I described. But it’s still a shame why so many students and professors spread around Bloom’s taxonomy without taking a look at what Bloom himself had to say about it:

The phenomenal growth of the use of the Taxonomy can only be explained by the fact that it filled a void; it met a previously unmet need for basic, fundamental planning in education.

Indeed, when Bloom created a model of the different ways students learn, it wasn’t built upon any logic or rationale, but, rather, was only meant to serve as a way for teachers and professors to classify different ways that students learn. It’s also not exactly clear how a certain test in a university course may require distinct processes that can be easily placed into this hierarchy, although, it would be very difficult to adequately explain these processes through any type of model.And, as researcher Richard Morshead points out:

I call attention to it because it functions as an important part of the rationale currently used by the authors to support their entire triparted taxonomic project. Here, they point out that when distinguishing between affective and cognitive objectives, they are not to be interpreted as suggesting that there exists a parallel distinction built into the basic fabric of behavior. (p. 45) They assert that the Taxonomy is purely an analytic abstraction. Its division into three domains, cognitive, affective, and psychomotor, is an arbitrary arrangement that seems to best reflect the way in which educators have traditionally classified teaching objectives. (p. 47) It does not reflect intrinsic separations within behavior….

As the authors themselves point out, all too frequently our descriptions of the behavior we want our students to achieve are stated as nothing but meaningless platitudes and empty cliches. (p. 4) If our educational objectives, they continue, are actually to give direction to the activities of both students and teachers, we must “tighten” our language by making the terminology with which we express our aims more clear and meaningful.

In David Flinters’ review of “Bloom’s Taxonomy: A Forty-Year Retrospective” by Lorin W. Anderson, Lauren A. Sosniak, he asserts:

Moreover, a dilemma common to both communication theory and educational practice is that the more similar a “message” (in this case the Taxonomy) is to the beliefs of its target audience, the more likely it will be embraced. 

So, should we throw away Bloom’s taxonomy completely? Probably not. But professors and students need to discuss it within the context that Bloom himself laid out for it. It’s important for us to not limit our abilities to think and learn simply based off how we want or believe the way human beings think through course material.