Kids just can’t seem to figure out absolute value. Pat writes,
“I’ve posted a mistake I see ALL THE TIME from my students when working on Absolute Value Equations. Looking for any advice on how other teachers are handling this issue. Thanks!”
Incidentally, I think that there’s a very good argument for this sort of thing being struck from the curriculum in its entirety. It’s entirely isolated in the curriculum, unconnected to anything else.
Thanks to Pat John for the submission!
8 replies on “Value of Absolute Value”
The only time I’ve seen absolute value used to a real purpose is in the definition of the limit. Perhaps we could make sure students are willing to only use logical steps when they get to that? I do like Kate’s lesson on it (http://function-of-time.blogspot.com/2010/06/absolute-value-both-rigorous-and-in.html).
I understand the argument from striking this from the curriculum, and perhaps agree. But let me play Devil’s advocate for a moment. The expression |x-y| represents the distance between x and y. Perhaps in problems like that above, this point of view could be useful. For instance, if a student were actually convinced to draw a the number line with 3 marked and to write out “four more than the distance from 2x to 3 is 7”, it would be clearer to the student that we are looking for a point 2x whose distance from 2x to 3 is 3, and from the picture it would be recognized that there are two such points and what they are.
The issue, I think, is that it is an equation. Absolute value inequalities, in my experience, are much more important (i.e., definition of limit). And other arguments involving inequalities can be useful in many places. For instance, when learning about continuous variables in probability theory, one is given a variable X and a function of X, say X^2, and one wants to identify the density function of X^2 from the density function of X. The most robust method is computing the cumulative distribution function for X^2 from the cumulative distribution function of X, and this requires giving a logical argument that derives one inequality from another. While the inequalities in question often won’t involve absolute values, absolute value inequalities generally provide a fertile domain for students to test and develop their understanding of how to give arguments involving inequalities.
I just love the error. A wonderful case of “hive mind” math. Don’t think, just blindly follow a learned procedure. And don’t check the results, either, because that might cause a little cognitive dissonance.
This isn’t all that unrelated to the derivative of a constant example in which the student goes through some elaborate calculations to conclude something that could have been arrived at with no calculation at all: just a few seconds of actual thought.
The more I see this kind of thing, the more convinced I am that we need to completely revamp how we teach mathematics in this country. While some people would react to these examples as reflecting on students (and of course, they do, in specific and individual ways), the real significance to me is that we continue to train students NOT to think, and we do it REALLY well.
Present company excepted, of course ;^)
Agreed that this could be largely struck from the curriculum–we teach it in 8th grade, and Barry makes an excellent point that absolute value has elegant applications, but blindly practicing absolute value without context doesn’t help get at the subtleties of its importance and use in mathematics.
Also agree that this is a biggie for creating a mindset that math is about memorizing nonsensical rules, especially among our lowest-skilled students–and they aren’t going to remember what they learned to apply it later anyway.
There are some practical applications, too. Anything where you need to keep within x units of some center. Like not wanting a plane to bank too many degrees in either direction from straight ahead. Or wanting temperature to stay within certain bounds. Or detecting a rise or drop in air pressure in a building, etc., etc.
barryrsmith gets at the key here: we need to be able to think about distances properly, so at least at first, absolute values should be taught in terms of distance and shown geometrically rather than algebraically or numerically.
Yes, I prefer geometric approaches to come earlier than algebraic/numeric ones, but why not ground the whole idea in some authentic real-world problem? Otherwise, for many kids, it’s just more irrelevant symbol pushing.
Combining the geometric interpretation with something real (as I suggested last night), is my preferred way to go. And then you can look at transformations, absolute value inequalities, etc.
I don’t think the biggest mistake here is applying linear techniques in a nonlinear situation, actually. The biggest mistake is right at the beginning, when they split it into two cases:
|2x – 3| + 4 = 7 and |2x – 3| + 4 = -7
They seem to have learned that absolute value has something to do with plus-or-minus, but are applying this “nonlocally”.