Positive numbers and absolute value

They say that when you’re a hammer, everything looks like a nail. Since I’m currently thinking about conceptual vs procedural teaching, I’m noticing examples.

Here’s a good definition of absolute value: “the magnitude of a real number without regard to its sign; the actual magnitude of a numerical value or measurement, irrespective of its relation to other values.” These are two definitions, and an obvious difference is how the absolute value of a complex number is treated, but let’s stick with real numbers for this discussion.

Conceptually, we can correlate any position on the number line to either a signed value (positive or negative) or an unsigned one (absolute value). Except for 0, there are two signed values for every unsigned one. So |-5|=|5|=5. We could, for the sake of clarity, always use + to indicate a positive signed value and ± to indicate an unsigned value: |-5|=|+5|=±5, in which case it would be clear that +5 does not equal ±5. (When considering complex numbers, we would need to modify this.)

Procedurally, though, it’s not unusual to see an instruction like Math is Fun’s: “So in practice ‘absolute value’ means to remove any negative sign in front of a number, and to think of all numbers as positive (or zero).”

Khan Academy likewise says, “If you don’t care too much about the concept… whether it’s negative or positive, the absolute value of it’s always going to be positive.”

Cool Math puts it in ALL CAPS!: “What do absolute values do to negative numbers? MAKE THEM POSITIVE!”

And while Purple Math, like these other sites, talks about absolute value in terms of distance, the site’s authors still imply that positive and absolute value are the same thing: “Given –| –3 |, I first need to handle the absolute-value part, taking the positive of the insides (the ‘argument of’ the absolute value) and then converting the absolute value bars to parentheses.”

Eduplace comes closer to avoiding this issue, but still has a subtle implication: “This is why we don’t say that the absolute value of a number is positive: Zero is neither negative nor positive.”

No, we don’t say that the absolute value of a number is positive because the absolute value of a number is unsigned. The distinction between magnitude and signed quantity is wholly a conceptual one. I certainly agree that, procedurally, there’s no difference between |-5| = 5 and |-5| = ±5. And we can see the procedural language in the examples above: MAKE THEM POSITIVE! Remove any negative sign.

The problem with the procedural approach can be seen with how students respond to -|5| vs. |-5|. The direction to “remove any negative sign” leads to both of these being interpreted as +5. Purple Math’s additional clarity (“taking the positive of the insides”) is a step in the right direction, but it still implies a technically incorrect concept.

The rebuttal to this may well involve choosing our battles. In practice, there’s no important difference between +5 and ±5, so what does it hurt to teach a procedure that results in the right answer, even if it glosses over a trivial, even nit-picking, distinction? Fair enough for most students, and these sites (the first five, other than Wikipedia, to show up on a Google search*) do all mention or even highlight that absolute value is about magnitude or distance. Even so, this strikes me as another example where we gloss over an important concept and fall back on teaching a procedure.

* Wikipedia’s hive mind also says: “As can be seen from the above definition, the absolute value of x is always either positive or zero, but never negative.”