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My understanding is that students find absolute value to be challenging to learn or understand. Off the top of my head, I can come up with two possible reasons for this.

  1. Absolute value is a piecewise defined function. Piecewise defined functions are more difficult due to increased abstractness: they are not a simple formula, but include a conditional. (I do not know if this is true, but it sounds plausible enough.)

  2. Absolute value is difficult, because it combines algebra (changing sign) with a geometric interpretation as a distance of number. At according to the thesis of Hähkiöniemi on derivative [1], it is challenging for students to change between perspectives in a fruitful way.

Is one of these the reason for the difficulties, or maybe it lies elsewhere? As always, answers using scientific literature are the most valuable and ones relying on explicit personal experiences are also fine.

[1] Hähkiöniemi, Markus. The role of representations in learning the derivative. No. 104 in Reoprts of university of Jyväskylä, department of mathematics and statistics. University of Jyväskylä, 2006. http://urn.fi/URN:ISBN:951-39-2639-7

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    $\begingroup$ I think your first answer is mostly likely the main reason. Students' understand functions as being things like x^2, because those are usually the only sorts of functions they've really had to deal with before. $\endgroup$ – Jessica B Aug 16 at 12:48
  • $\begingroup$ I agree with @Jessica B, and I think another reason (which is possibly included in her reason if interpreted broadly enough) is that absolute value manipulations are different from the standard algebraic manipulations they're used to (distributive property, combining like terms, etc.). Indeed, the concepts in your second reason are often a way of getting students to better "visualize" absolute value manipulations and to help keep students from making incorrect manipulations or incorrect deductions involving absolute value manipulations. $\endgroup$ – Dave L Renfro Aug 16 at 13:01
  • $\begingroup$ The main difficulties I see are when students are asked to deal with absolute value inequalities. Inequalities are very hard for most students to deal with. $\endgroup$ – Sue VanHattum Aug 16 at 16:10
  • $\begingroup$ Because it's intricate. Yes, even just one or two "if then"s still makes something intricate! We are meat, not silicon. Just explaining a rule or set of rules is not adequate for us, if we have to remember some intricacies. Instead of tacitly searching for some lock-key explanation idea (what is the hurdle and how do we adroitly remove it), I recommend to drill, drill, drill. And then drill some more. This is the way to familiarity and to making struggling concepts routine. It's no different in music or sports. That's the way us meat people are. $\endgroup$ – guest Aug 18 at 18:00
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    $\begingroup$ I suspect most students have no trouble with the notion of the absolute value of a number, and that those who have trouble with the absolute value function do so because more generally they have trouble with the function concept (the same reason they struggle with the notion of the square root function). $\endgroup$ – Dan Fox Aug 21 at 19:11
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(My answer is just a guess and not based on any formal research.)

I suspect the absolute value function may be difficult to understand because it involves "negative numbers that aren't negative." One way to define the absolute value of $x$ is:

$$|x|=\left\{\begin{array}{rl}-x, & x<0\\x, & 0\le x\end{array}\right.$$

I think the $-x$ confuses students, causing them to think that "sometimes the output of the function is negative."

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    $\begingroup$ Yes. In my experience, many students struggle to accept that for some values of x, it is true that |x| = -x. $\endgroup$ – idmercer Aug 22 at 20:05
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I am writing this based on pure observation (e.g., entering year four of teaching this topic to secondary school students, and having co-taught a minicourse for teachers on absolute value functions$^\star$).

There are a lot of definitions/interpretations of absolute values:

  • the (abstract) axiomatic one;

  • the piecewise or "case-based" one (provided by Joel Reyes Noche);

  • the colloquial function one (erase any negative sign and return the result);

  • the equivalent function one of $x \mapsto \sqrt{x^2}$;

  • the geometric one (where $|a-b|$ is the distance on the number line between real numbers $a$ and $b$);

  • the positive difference interpretation (subtract the lesser number from the greater one, i.e., $|a-b|=\max(a,b)-\min(a,b)$ for real numbers $a$ and $b$);

  • the graphical one communicated by a $\mathsf{V}$-shaped curve;

etcetera.

A complication that has already arisen in the above collection of definitions/interpretations is its ambiguity around whether we are defining a function whose input is one number or two numbers; the one number version can, of course, be viewed as the two number version where (at least) one of the entries is zero. Nevertheless, I think that sometimes the topic is introduced with multiple approaches and without drawing this distinction; e.g., if you try to help a student get a grip on absolute value functions by asking the difference in age between them and a friend, then you are effectively asking about the two input interpretation, which could be confusing if you had just introduced it as, e.g., a piecewise function.

There may also be an issue of timing: When I delve into absolute value functions, equations, and inequalities, it is in an Algebra 2 course; in our present course sequence, this means that students have done Algebra 1 already, and then spent a year on Geometry. In other cases, you may have this topic broached in an Algebra 1 course; so, students are in the throes of matching graphical representations, geometric representations, symbolic representations, and so forth.

As to covering absolute values pedagogically, I think there is a great value in viewing more general absolute value functions of the form $x \mapsto a|x-h|+k$ for real parameters $a, h, k \in \mathbb{R}$ in terms of transformations, and, in addition, connecting this to quadratic functions written in vertex form, $x \mapsto a(x-h)^2 + k$, to emphasize similarities and differences (and to reinforce terminology: intercepts, roots, vertex, concavity, end behavior, domain, range).

$\star$: I would be remiss if I failed to link our (my colleague/department chair, Liz Brennan, and my) freely available materials from a minicourse taught through Math for America:

There are a lot of materials in there, which range from a question I asked on MathOverflow (MO 301514) to a misformulated problem (p. 9 #6 is impossible!) to problems that participants formulated during our final meeting (Desmos link).

The three-meeting course linked above is by no means exhaustive: In fact, I have been thinking more about absolute values and their role in defining equations as one endeavors to impose domain restrictions; for example, see the sequence of tweets that begins here.

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At my school (a community college in California) the curriculum is set up so that students first take an algebra course in which essentially every function is a linear function, and only later do they do anything at all with nonlinear functions. Nonlinear functions are just more complicated, so they require more thought about the logic. Thinking about logic is harder than manipulating formulas according to recipes.

As an example, if I tell a student that $x^2=4$ and ask them to solve for $x$, the formula-manipulation approach would be "do the same thing to both sides," so they may get $x=2$. It requires an extra logical step to realize that there are two roots. One way to write this is $|x|=2$, but I'll see students do things like $x=|2|$, or $|x|=|2|$ (which is correct but kind of silly). Understanding why $|x|=2$ is the right choice requires some more logical thinking.

An example that comes up in physics is that students will memorize the fact that $a=-g$ for free fall, despite instruction to the effect that this depends on the coordinate system, and that they need to pick a coordinate system first. Some textbooks even encourage this. Picking a coordinate system requires an extra logical step, which is hard if you aren't used to thinking about math logically. I can tell them that $g$ is defined as $|a|$ so that we can put a value of $g$ in the book without reference to any coordinate system, but again, this requires some logical thinking about topics that they aren't used to thinking about, and have gotten the impression that they never have to think about.

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    $\begingroup$ "Picking a coordinate system requires an extra logical step, which is hard if you aren't used to thinking about math logically." — I think an extra complication is that in many pre-college environments the axes are drawn with two arrowheads, and it is assumed that values increase in the right or upwards direction; in this case arrowheads indicate that this is an infinite line. I believe that an alternative approach when an arrowhead indicates the direction in which value increases, and the infinite length of the axis is assumed just because it is not terminated with a point, is less ambiguous. $\endgroup$ – Rusty Core Aug 23 at 23:14
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Absolute value is difficult for students because they have difficulty parsing and simplifying logical statements.

Some of the results of working with absolute value statements seem to actually hide the inner mechanics of how the logic of an absolute value statement play out. Because of their piecewise nature, you must understand logical statements to work deeply with absolute values. Students often develop a superficial, rote approach to these expressions because they are simpler than actually describing what is going on (rote is also sufficient for most exercises that I have encountered).

The best anecdote that I can offer was the following question, which was asked if me by a student in 11th grade: When you are solving the inequality abs(x-2)≤3, the solution is -1≤x≤5, which is an AND statement (-1≤x AND x≤5); but the solution process involves an OR statement (x<2 OR x≥2) to address both parts of the piecewise definition. Where does this change happen from OR to AND?

It is a question that really requires one to logically express the solution process clearly to answer (the answer is actually that the statement is always an OR statement that just happens to be equivalent to the AND one... (x<2 AND x≥-1) OR (x≥2 AND x<5), where the outermost OR is the continuation of the two distinct possibilities from the solution process).

All of the notions of the different interpretations of absolute values are tools to aid students' abilities to intuit answers to problems involving absolute values, but do little to actually let students rigorously understand them. To do so, you would have to actually pick one as the definition and relate the others rigorously to that. This is an exercise that almost never happens in classes.

In truth, I think that students find absolute values difficult because the subject is legitimately difficult! Otherwise, why would we bother having so many ways to think about how to interpret such statements? These interpretations are shortcuts around the rather tedious legwork of working directly with the logic (my personal chosen definition). I think that most teachers probably don't know absolute values as clearly as we would like them to.

I offer the following questions as food for thought on the difficulty of absolute values:

  1. When solving equations, as opposed to inequalities, it is more common in secondary school to find a countable number of solutions as opposed to an uncountable number, but the solution set for the equation abs(2x-2)-abs(x)=2-3x is [0,1]. Why can you get entire intervals of solutions to such an equation?
  2. How do you graph y=abs(x-abs(x-1))?
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