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I see many teachers use slow methods to solve a given problem where there's another faster methods that doesn't demand much more effort. I'm not looking for mistakes like saying that $a$ is the slope of $y=ax-x$ or that $x<\sqrt{2}$ is the solution of $x^2<2$ (yes these are real examples), I'm looking for methods that are correct but slow and inefficient. Here are some examples.

Simplifying radicals

Simplify $\sqrt{2^6}$. I see most middle school teachers do it like this $\sqrt{2^6}=\sqrt{2^2\times 2^2\times 2^2}=2\times 2\times 2=8$ instead of teaching students to take half of the exponent.

Table of signs

To solve the inequation $\displaystyle \dfrac{(x^2-9)(2x+1)}{(x-2)(x+1)(-x^2-2)}>0$, we study the sign of $\displaystyle f(x)=\dfrac{(x^2-9)(2x+1)}{(x-2)(x+1)(-x^2-2)}$. We can do it the long way studying the sign of each factor

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or the short way

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We take the highest degree term of each factor: $x^2$, $2x$, $x$, $x$ and $-x^2$. The product of their coefficients is negative so we start with a minus on the right and change the sign each time we encounter a simple root (or root of odd multiplicity) or pole. This way is more efficient not only because the table is much smaller but also because students don't have to memorize the table of signs of $ax+b$ and of $ax^2+bx+c$ when $\Delta$ is $>0$, $<0$ or $=0$.

Integration by parts

Writing $u$, $v$, $u'$ and $v'$ three times to find $\int x^3 \mathrm{e}^x \mathrm{d}x$ is tiresome and takes a whole page whereas with tabular integration it takes 2 lines.

Differentiation

To find the derivative of $\displaystyle f(x)=\dfrac{(x^2+x-1)^3 \sqrt{x^2+1}}{\left(\ln\left(\cos x\right)\right)^2}$, I prefer to do it this way :

$$f'(x)=\dfrac{\left((x^2+x-1)^3 \sqrt{x^2+1}\right)'\left(\ln\left(\cos x\right)\right)^2-\left((x^2+x-1)^3 \sqrt{x^2+1}\right)\left(\left(\ln\left(\cos x\right)\right)^2\right)'}{\left(\ln\left(\cos x\right)\right)^4}\\=\dfrac{\left(3(2x+1)(x^2+x-1)^2 \sqrt{x^2+1}+\frac{2x(x^2+x-1)^3}{\sqrt{x^2+1}}\right)\left(\ln\left(\cos x\right)\right)^2+2\tan x \ln(\cos x)\left((x^2+x-1)^3 \sqrt{x^2+1}\right)}{\left(\ln\left(\cos x\right)\right)^4}$$ Although it is long (~ half page), it's still better than naming $u$ and $v$ each time. The best method to do this is the logarithmic derivative.

These examples are from school but feel free to add examples from undergraduate mathematics. $\color{red}{\text{I'm not asking why do teachers still use them}}$; I'm asking for other examples of such methods.

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    $\begingroup$ For all these examples, there are reasons to do it this way, at least at first when learning a new topic. If you are looking for examples of problems that are hard to solve at first but easy after some time studying the topic, almost any textbook should contain some. $\endgroup$ – Dirk Jul 4 at 14:41
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    $\begingroup$ I am not impressed with tabular integration. It doesn't save much writing and it completely obscures the thought process. I'd rather have students use technology than use tabular integration. $\endgroup$ – James S. Cook Jul 5 at 0:36
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    $\begingroup$ Sometimes the teacher knows what the students need to hear. If you are one of the students, do you know the levels of all the other students as well as the teacher does? As a teacher, I look for ways to reach each student, so they can understand. In your first example, the 'quick' way may not reach many of the students. I sure don't want them thinking they have to memorize, when this makes so much sense. $\endgroup$ – Sue VanHattum Jul 5 at 3:04
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    $\begingroup$ I find this question silly. In a way, all of math is about coming up with more efficient methods which use more involved machinery, but then "efficient" and inefficient" are relative with respect to that machinery. If the question is $19 \cdot 21 =?$, an elementary school kid will add $21$ to itself 19 times, a middle school kid will perform long multiplication, and a math student will do $20^2-1^2$. So is the long multiplication an "efficient" or an "inefficient" method? $\endgroup$ – Torsten Schoeneberg Jul 5 at 17:35
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    $\begingroup$ @Jasper: Do any real-world applications have a functional form like in the derivative example? I think this is analogous to weight-training for tennis players rather than serving practice for tennis players. $\endgroup$ – Dave L Renfro Jul 7 at 8:40
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The three methods for solving for the zeros of a quadratic equation -

  • Factoring (and perhaps realizing that when the discriminant isn't a perfect square, no integer solution)
  • Completing the Square - and the process for this, which many students struggle with.
  • Quadratic Equation - Their 'goto' method, often despite a potential simple factored result.
  • Use Calculator Solver - Ironically not 100% as students can still have a typo entering the original numbers.
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  • $\begingroup$ Oops, that's four methods! :-) $\endgroup$ – Namaste Jul 11 at 18:19
  • $\begingroup$ Also missed the specific case of "difference of squares": $x^2 - y^2 = (x+y)(x-y)$, $\endgroup$ – Namaste Jul 11 at 18:42
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To solve the system $$\left\lbrace\begin{array}{l}\dfrac{1}{x+5}\geq\dfrac{1}{x-2}\\ \dfrac{(x-3)^2}{3}\leq (x-1)^2-8\\ -x(x+5)<0\end{array}\right.$$ many teachers draw a table of signs for each inequation and then find the intersection of the three partial solutions on an axis. It's faster to use one table to solve it.

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Simplifying equations in one unknown. I seem to remember some formulaic system of seven steps or so, including showing explicitly operations on each side (e.g. subtract 4 from each side, rather than toss over the equals and change sign). Note, the justification is more understanding, as well as less errors. Probably benefits to doing it long way at first, before learning faster method (similar to integration by parts).

Hope this is not a segue, but one similar thing is teaching of Excel design, where the advice is to show each operation with a new column or row. "Rows are cheap." Instead of some super long computer program like formula in the cell formula, use a new column or row for each operation. Makes future debugging or modifying easier.

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    $\begingroup$ Could you edit your answer to give an explicit example of this? The original post has some pretty detailed ones, so probably to be useful to them, that might be helpful. You can put basic math in dollar signs to make it look nice. $\endgroup$ – kcrisman Jul 5 at 1:51
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One explanation is that the teacher doesn't know any better. I think this is going to be more common at the earlier the education levels, i.e. elementary school teachers are going to be less fluent than middle school teachers, etc.

Another reason is that the student may not yet know the concepts that you think are better. Your radical simplification example stood out as an example of this. I've never taught middle school math but I've had children at that age and I'm pretty sure rational exponents are still years down the road. In this situation, the teacher is working with the tools available.

A third reason, that comes up more often in my college classes, derives from the nature of undergraduate math education. The vast majority of my students, even into the calculus series, aren't going to become mathematicians. Their interest in the material is almost certainly never going to extend beyond using it to solve non-mathematics problems. In that sense, the real goal of the class is to give them the tools that they need to succeed in their future classes in physics, engineering, chemistry, computer science, etc.

With that goal in mind, my goal is to give them as wide a selection of tools as possible. Some of them will be quick and easy to use, e.g. l'Hopital's Rule, where others may be less straightforward, e.g. evaluating a limit with radicals by rationalizing the numerator, but by putting both methods in their toolbox I'm maximizing the options that they have going forward.

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    $\begingroup$ To be fair, you really did not answer the question, to provide more examples of this issue. Instead, you gave an excellent dancer to the underlying origin of the issue. And I voted this up. $\endgroup$ – JoeTaxpayer Jul 5 at 15:24

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