I’ve been troubled ever since David Cox asked why students needed to learn the names of the algebraic properties (associative, identity, and so forth). Certainly I see misunderstandings: students confronted with
not aware they can combine the 3 and 2 with associative and commutative properties, or the classic
leading the student to cancel the x terms rather than consider the distributive property. (*)
Clearly these things are being taught, but what’s going awry when they are used in practice? And why do students learn the names of these things?
It struck me that students only get taught the definition in a positive sense, memorizing (for example) that
and identifying 2 + 4 = 4 + 2 or (2 + x) + 3 = 3 + (2 + x) as the commutative property, and leaving things at that.
From my experience students can eventually identify a particular sequence of symbols with particular name but with no transference to actual practice. I’ve seen books address this by forcing students to laboriously label the property with each step of an algebra problem, but even after such an activity the knowledge is compartmentalized and then forgotten.
What I believe is missing is for students to know when not to use a property, that is, the “negative space” of the definitions. The use of the distributive property, for example, is often done with pattern matching other than understanding, giving results like:
Rather than labeling each algebraic manipulation in a problem, I’d suggest students correct erroneous algebra while stating which rules are being used wrongly.
I feel the negative space gives sufficient justification for learning the properties of names, so it’s possible to say that “here you can’t use the associative law like you’re trying to” rather than trying to redefine the positive space just to talk about the negative space, or worse, establish as a specific rule “you can’t do that” individually for each type of mistake. It’s easier to use the positive definition as a tool to anchor the negative examples as a single category.
(*) I don’t understand why it’s rarely made explicit that the distributive law includes division. I have yet to find a textbook that does this. While a can be a fraction in , most students do not recognize this on their own.