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I recently gave an exam problem that roughly 2/3 of the class made much more difficult by using the identity $1/x+1/y=(x+y)/xy$. Their work would have been much simpler if they hadn't done this. It seems like a symptom of the type of intellectual laziness that causes people to do things by pattern recognition rather than because they have a logical reason for doing them.

When students are taught this identity, are they told a reason why they might want to use it? What would the reason be?

I will ask them why they did it when I return the exam, but I suspect I'll get answers to the effect of "that's how you do that kind of problem in algebra." Maybe this is similar to stuff like multiplying out an expression such as $(x+y+z)(a+b+c+d)$, which I think they do because they think of the notation as a set of orders that they need to carry out.

Is this reflex similar to the one that compels them to rationalize the denominator in an expression such as $1/\sqrt{2}$? That one at least I think I understand the original reason for. If you're doing decimal arithmetic and performing division using long division (rather than log tables, a slide rule, or an electronic calculator), then rationalizing the denominator is a big win, so that would have sort of made sense before slide rules became common. Would the justification for habitually doing $1/x+1/y=(x+y)/xy$ be that division is a more expensive operation than multiplication if you don't have a calculator, slide rule, or log tables?

Or maybe math teachers want answers like $1/\sqrt2$ in a canonical form so that it's easier to check students' work? (But this seems like an anachronism now that we have WeBWorK et al.) But I don't see how $(x+y)/xy$ is more canonical than $1/x+1/y$, and in general it's preferable not to have the same variable appearing more than once, e.g., if we want to reason about the dependence on that variable.

Questions:

(1) Why is this identity taught?

(2) Why do students use it in inappropriate situations, and what can we do to cure this problem?

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    $\begingroup$ Students tend to think that in algebra there is a right way and a wrong way to write an expression, and their job is to do it the right way. Algebra is a set of rules for writing things the right way. The idea that some things are a matter of taste (whether to prefer this expression to that expression) or that some ways of writing an expression are helpful in some contexts but unhelpful in others is foreign to their way of thinking. Mathematicians think of algebraic rules for manipulating expressions as flexible rules. Students tend to think of them as rigid rules. $\endgroup$
    – John Coleman
    Mar 26, 2017 at 3:38
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    $\begingroup$ I haven't taught the identity here; in fact, I have given a problem in which knowing this identity would be very helpful, and had students not use it! "Compute the sum of the reciprocals of two numbers, if you know that the numbers have sum 5 and product 10." This problem is immediate using the identity above, but I find students set up two equations $x+y=5$ and $xy = 10$ and solve a quadratic to figure it out. Meanwhile, I've asked students to compute $1/84 + 1/126$ and they ess'ly use the identity you mention. My guess is you simply observe a lack of sense-making, cf. e.g. Schoenfeld... $\endgroup$
    – Benjamin Dickman
    Mar 26, 2017 at 4:06
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    $\begingroup$ The question is just slightly unclear. Critical detail: What was your written direction for the exercise? Is your criticism that the students simplified $1/x + 1/y$ at all, or that they simplified it via an unnecessarily complicated method? $\endgroup$
    – Daniel R. Collins
    Mar 26, 2017 at 13:37
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    $\begingroup$ @DanielR.Collins: It's a physics problem involving optics. They derive equations of the form $1/z=1/x+1/y$ and $1/p=1/q+1/(z+r)$, and they're supposed to express $q$ in terms of the quantities other than $z$. My criticism is that they needlessly complicated (antisimplified) these expressions by applying this identity. There is no simplification possible or required. All they had to do was substitute for $z$ in the second expression, and they were done in 3 lines of algebra. Instead they insisted on applying the identity, which made it into a huge, nasty mess. $\endgroup$
    – user507
    Mar 26, 2017 at 22:48
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    $\begingroup$ +1 for calling out rationalizing the denominator. I've always thought it was ridiculous and I was ecstatic when I had a calc 2+3 professor who hated it and insisted we don't do it. He would always say, "You wouldn't write $x/x^2$ instead of $1/x$, would you?" Granted I do recognize that, e.g., $\sqrt2-1$ is more simplified than $\dfrac1{\sqrt2+1}$, but they don't always work out like this. I just can't stand the "rationalize the denominator" mentality that constantly gets drilled into pre-college students in the U.S. Not sure how it is elsewhere. Oh, and +1 for mentioning WeBWorK. Yay. $\endgroup$
    – user6648
    Apr 12, 2017 at 16:53

4 Answers 4

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First, I would expect this identity not to be taught, but to be obtained by students as a particular case of the method of reducing fractions to the same denominator, which is sufficiently useful not to need an explanation why it is taught.

Second, the only way I see to take care of this problem (and many others) would be to teach less standardized exercises, and induce more sense-making. In particular, we should teach tactics and strategy of manipulating an algebraic expression in order to reach a certain goal (if we want to find roots of an expression, let it factorized; if we want to sum several expressions, a distributed form might be better; etc.) The problem is that teaching this is very difficult and very time-consuming (we cannot stick to a few exercises, we need several similar-looking exercises that needs different methods, to be able to point out which subtle differences could be used to construct different strategies), and needs more efforts from the students. It may feel like much effort for little gain, compared to focusing on a few standard exercises and have student turn out decent tests. Seeing student fail again and again when we try to test that kind of understanding is pushing us all toward standardized exercises and tests, and make it yet more difficult to change our ways. We should still try hard, because ultimately it is understanding and sense-making have value incommensurable with having mastered a handful of meaningless methods, but we should be warned that we will feel like Sisyphus all along the way.

A last remark (edit: which actually is a remark, and do not claim to be an answer about why these students would use some form over another): concerning $\sqrt{2}/2$ versus $1/\sqrt{2}$, one important purpose of normalized forms is to help recognize when two numbers are equal. There is even a lot of theory of normal forms in various mathematical settings, where one expect to be able to prove that two objects are equal if and only if their normal forms are equal, and to devise algorithms to produce a normal form from an arbitrary expression. Term rewriting is very much about formalizing this (but the same ideas appear in various subfields).

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    $\begingroup$ +1 for I would expect this identity not to be taught, but to be obtained by students as a particular case of the method of reducing fractions to the same denominator This is exactly what I was thinking, and I was considering how to say this in a comment before I got to your answer. For what it's worth, sums whose terms are the reciprocals of an arithmetic progression are called harmonic series, and in older algebra texts (at least 50 years old) harmonic series were studied side-by-side with arithmetic and geometric series, but the topic was mostly dropped by the 1950s. $\endgroup$
    – Dave L Renfro
    Mar 27, 2017 at 19:39
  • $\begingroup$ First, I would expect this identity not to be taught, but to be obtained by students There might be three levels of mathematical ability and skill to discuss, which would be, from lowest to highest: (1) students who are taught this identity and use it mindlessly, (2) students who are taught this identity and use it only when it is an appropriate tactic for the problem at hand, and (3) students who would be able to come up with this identity on their own. I would say that 80% of my students are at level 1, 20% are at level 2, and none are at level 3. I'm at a community college. $\endgroup$
    – user507
    Mar 27, 2017 at 20:11
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    $\begingroup$ @BenCrowell in my mind the point though is not "so that it's easier to check students' work." But rather to recognize that the numbers that can be obtained from the rational and a root using addition (subtraction), multiplication and division are the same that can be obtained using addition (subtraction) and multiplication only. This is not completely obvious and the direct generalization of this is a standard result in most intros to field theory. $\endgroup$
    – quid ♦
    Mar 27, 2017 at 21:36
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    $\begingroup$ @BenCrowell I can't speak for the US school system but in the Czech Republic, leaving results in normalized form is required for full points. Comparison between one values is one reason I believe, ease of approximation by hand is another. Lots of people can make a decent guess at how much $\sqrt{2}/2$ since they tend to have an idea how much $\sqrt{2}$ is and can divide by 2. Few people can make the same calculation when they try to invert $\sqrt{2}$. $\endgroup$
    – DRF
    Mar 28, 2017 at 9:23
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    $\begingroup$ @BenCrowell: a minor comment about you three levels of ability and skill. These levels assume that the identity is to be taught, which I find quite debatable. I would teach only a few identities, and alongside a few methods (including reducing to the same denominator). Considering that very many such identities should be taught is a stance that often comes in opposition to sense-making, while the motivation to make students able to deal with more situations without needing understanding feels to me a low-value goal. $\endgroup$
    – Benoît Kloeckner
    Mar 28, 2017 at 11:29
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For students at that level, routine algebraic calculations are an easy step, while thinking about what steps to take is quite difficult. So they'll often prefer to do familiar calculations - even long and tedious ones - and only turn to more extreme measures like stepping back to think about the big picture as a last resort. They also have an extreme aversion to backtracking (most of them think of trying an unsuccessful solution as a pure failure, rather than as progress towards finding the right one), so it's not surprising that, having backed themselves into a corner with an unhelpful algebraic manipulation, they push forward rather than trying something else.

One factor to consider is that students don't usually know how hard a problem should be: they can't tell the difference between "this problem get too messy, I should try something else" and "this problem just needs some more effort". (This works in both directions: students persevering through awful calculations, and also students who redo an easy problem repeatedly because they can't believe the answer isn't an integer.)

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    $\begingroup$ How to appropriately use algebraic tools is subtle. For one, a primary motivation for algebra is to replace otherwise complicated thinking with automatic procedures. Of course, the end result of that should be to free up mental space and energy to become able to solve more complex problems, not as a means to avoid thinking at all. But conveying that to students in a meaningful way (rather than just lecturing it to them) is generally quite challenging. $\endgroup$
    – Michael Joyce
    Apr 12, 2017 at 13:47
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I think that "rationalizing numbers" (clearing the denominators of radicals) is an anachronism: it was useful at one time, because if you want to evaluate $\frac{1}{\sqrt{2}}$, it is much easier to perform the division $\frac{\sqrt{2}}{2}$ if you are obtaining $\sqrt{2}$ by a table lookup.

So it was a useful skill at one time, but now it is just being passed along by rote.

However, I feel this way about a lot of math. For instance, I think most of what we teach students in a "Calc 2" course is similarly anachronistic.

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    $\begingroup$ @tilper The vast majority of functions you can write down (like, say, $\sin(x^2)$) have no elementary antiderivative. I think we emphasize the skill of finding antiderivatives ``by hand'' for basically the same reason we force them to rationalize denominators: at one time we needed to compute integrals by table lookups, so we had to transform many integrals into standard forms. With computers, we can numerically integrate with ease. I think most of the emphasis should be placed on getting rigorous estimates for numerical approx, rather than computing antiderivatives symbolically. $\endgroup$
    – Steven Gubkin
    Apr 12, 2017 at 17:01
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    $\begingroup$ @StevenGubkin I would strongly disagree with this approach. IMO the reason we teach 90% of calc is to force the students to at least try to think. Pretty much everything they do could be done by a computer, but that's not the point. The point is that unless they do a couple a hundred derivatives they don't understand what they are or how they behave. Being able to evaluate a derivative or an integral is the trivial last step which comes after understanding that you need to evaluate it and why. $\endgroup$
    – DRF
    Apr 13, 2017 at 9:32
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    $\begingroup$ @DRF I agree that students should have to think. I disagree that doing hundreds of derivatives and integrals involves much thinking, or gets students to understand what these operators are or how they behave. Maybe it makes them better at algebra. I think they should be modeling physical phenomenon, explaining why the differential equations they are setting up make sense, exploring connections, bounding errors, exploring pathologies, challenging definitions, etc. I think very little of the course should be mindless mathematical symbol manipulation at this point. $\endgroup$
    – Steven Gubkin
    Apr 13, 2017 at 12:24
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    $\begingroup$ Is teaching vocabulary anachronistic because the definitions of most words we encounter can easily be looked up on the web? $\endgroup$
    – user52817
    Apr 13, 2017 at 12:34
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    $\begingroup$ Similarly, if you find yourself thinking about things where computing hundreds of symbolic derivatives and integrals comes up a lot, then it might be worth putting the effort into learning to compute them with ease. If not, I see no harm looking them up. What are students supposed to get out of a calculus class? I think that they can perform the computations for that test, but a year later their "skill" has faded. I would much rather that they emerge with an understanding of the relationship between rates and total amounts, how to approximate, how to argue, some pictures, etc. $\endgroup$
    – Steven Gubkin
    Apr 13, 2017 at 13:28
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I certainly agree that there is a lot of "mindless" emphasis on various aspects of teaching algebra. However, to the extent that the mathematics community wants to encourage students to have a more "playful" way of looking at mathematics and to see the way that mathematics is the science of patterns, perhaps this "identity" could be used opportunistically to look at the phenomenon of Egyptian Fractions - the issue of writing p/q (p, q positive integers) as the sum of fractions of the form 1/x with distinct denominators.

So perhaps a student who added (1/3) + (1/10) in the spirit of (1/x) + (1/y) and found the answer to be 13/30 might find looking at (1/3) + (1/10) = 13/30 a source of "new" mathematical thoughts:

a. Can 13/30 be written in other ways as the sum of two unit fractions? as the sum of three unit fractions? What is the smallest denominator of unit fractions with distinct denominators that sum to 13/30?

b. Can any fraction p/q be written as the sum of unit fractions with distinct denominators? (Tradeoff between number of fractions involved and the largest denominator that appears?)

c. Study the solutions to: (1/x) + (1/y) = 1/z (e.g. (1/3) +(1/6) = 1/2)

So lots of current curriculum is not optimal, but if teachers have the freedom to do a bit "extra" and have a sufficient knowledge base perhaps some additional "positive" outcomes are possible.

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