Stop belaboring the easy stuff and redirect that time to the hard stuff

Recent claims about how students don’t understand the “equals” sign (and therefore require lessons on the underlying concept of mathematical equality) remind me of yet another old post.

(Perhaps the biggest reason why kids appear not to understand the equals sign is because they’re having trouble making sense of the stuff on either side of it).

Conceptual structures vs. learning conceptual structures = not homomorphic 

It’s become an obsession. Everywhere I look now, I see cases where people dwell on basic concepts that are relatively easy to understand in isolation and challenging only within complex situations. Time and again I wish that instructors would move on more quickly and save time for the more complicated applications that lie ahead.

Typically, this concept-dwelling happens at the beginning of a course or unit and takes up the first couple of classes. For example, in an MIT OpenCourseware course on basic chemistry that I’m having J sit through in preparation for Chem 101 in college, the instructor begins with an overly long discussion of what’s known about the basic structure of atoms. While she asks the students whether they’ve ever had plum pudding, I think about the challenging problems that lie ahead and the time that could have been reserved for working through some of these in class.

The problem, I think, is a combination of the difficulty that many instructors have remembering what it’s like to be the student, and the default assumption that the conceptual structure of a given subject maps neatly onto to the process of learning that subject. What’s foundational to a subject, in other words, may not be the same as what’s foundational to learning that subject. Now I don’t remember much at all about chemistry. But I’d venture to guess that, while learning the structure of atoms is part of what’s foundational to learning chemistry, but there are a number of other key concepts that are also foundational to learning chemistry, and more time-consuming to master (including, for example, some basic algebra).

Another fallacy that afflicts introductory classes is a sort of ontogeny recapitulates phylogeny assumption. This is the idea that the history of the field’s development over time maps onto the student’s personal trajectory of understanding it. But while the history of the various notions of elementary particles is interesting in and of itself (a great topic for a history of science class), it probably doesn’t replicate the student’s own developing understanding of elementary particles, and going over that history may or may not actually facilitate that understanding.

Of course, all of these are empirical questions that could be addressed by applying the foundational concepts of yet another field–in new and complex ways.

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