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A typical student mistake

You see this:

$$\frac{15}{4\sqrt{15}}=\frac{15}{4\sqrt{15}}\cdot\frac{\sqrt{15}}{\sqrt{15}}=\frac{15\sqrt{15}}{4\cdot15}=\frac{15\sqrt{15}}{120}.$$

You can see that the student tried to rationalize the denominator. They did it the hard way, and made a mistake. You want to teach them a better way, that would have helped them with this problem.

Trying to get them to see the mistake(s)

You say this is not right. They look at the thing and say, oh, you're right. $4\cdot15$ is $60.$ Yes, you say, but do you think perhaps there was a way of avoiding the offending calculation altogether? Perhaps there was a way to do it in such a way that you couldn't have made a mistake because you wouldn't have had to calculate anything.

They look again and you see a mixture of impatience and anxiety in their eyes. They know why they made the mistake. It's because $4\cdot15$ is $60.$ Why are you bothering them and implying they're stupid?

You say, look, it's always a good idea to avoid calculations. It saves you time and spares you mistakes. Look at the two fifteens, the one at the top and the one at the bottom. What's better, cancelling them out, or multiplying the $4$ by the $15$ first?

They agree it's the former, but you can see it's not sinking in. They've done it this way for so long and it's automatic. This is the way they know. They understand your way is better, but they're unwilling to follow it. It's difficult for them to take in a new idea if they have one that works.

Unfortunately for them, you have one more thing to add. Look at the previous part of your solution to this problem. You showed there that $\dfrac{15}{\sqrt{15}}=\sqrt{15}$ Can you see why that is? Yes, of course they can. And they show you how to multiply the fraction by $1$ written as the quotient of two square roots of $15$. You'd like to make them appreciate the fact that $\sqrt{15}\cdot\sqrt{15}=15$ by definition and make them able to see the result without having to write the complicated algebraic manipulation. But you've tried that too many times and know what their reply will be. They'll explain to you that you're a mathematician so you see these things and they're not so they don't. You decide against even mentioning the thing this time.

Is there a better way to teach this?

This simply happens all the time. How do you deal with this? How can you make a student appreciate a better way of doing something they already know how to do? How do you conquer their mental inertia?

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    $\begingroup$ Well, I don't know what 4 times 15 is either. It's now corrected from 90 to 60. $\endgroup$ – ymar Mar 4 '15 at 21:52
  • $\begingroup$ There's a lot of exposition here that makes this seem like a wall of text with a vague question. Can you make it a more explicit, concise question? $\endgroup$ – Chris C Mar 4 '15 at 21:59
  • $\begingroup$ @ChrisC I don't think I can make it less vague because the question I have in mind is rather general. I could delete the exposition altogether I guess. Or delete the question if it's a bad one. $\endgroup$ – ymar Mar 4 '15 at 22:06
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    $\begingroup$ @ChrisC Well, that's an example. There are lots of other things like this. When you tell a student to calculate 1+2+3+4+5+6, they will do the math and tell you the result. When you tell them how to do it differently, they'll feel embarrassed or just shrug their shoulders, and in either case they won't take it in. And God forbid you tell them the Gauss story. That'll just give them an excuse. $\endgroup$ – ymar Mar 4 '15 at 22:26
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    $\begingroup$ I just remembered a problem I encountered when I prepared for the high school finals myself. It was to calculate the circumference of two tangent circles (the figure "8") given the area they enclose and the sum of their radii. The solution in the book was to solve the system of equations for the two radii, calculate the circumference of each circle and add the two. So it's not just the students who do things automatically. But I guess the authors of the book would actually appreciate someone telling them the right way, so there's a difference. $\endgroup$ – ymar Mar 4 '15 at 22:54
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Use a positive approach, instead of a negative approach.

The approach in the original post is:

  • Not only did you make a mistake,
  • But you need to go figure out what you could have done instead (that you did not see)
  • Oh by the way, you need to figure out something else that you could have done instead (that you did not see).

That approach provides three pieces of negative feedback before the learner gets to solve the problem to the teacher's satisfaction.

A positive approach might be:

  • Yes, you know how to rationalize denominators. You multiply by a fraction that equals one, where both the numerator and the denominator equal the relevant square root.
  • But that involves a lot of math, and you might make a mistake.
  • Sometimes you can use this shortcut. If the numerator already includes a power of the square root, rewrite the numerator to show it in terms of the square root. For example: $$\frac{15}{4\sqrt{15}}=\frac{\sqrt{15}\sqrt{15}}{4\sqrt{15}}$$ Then you can cancel out the square roots right away.
  • You won't always be able to use this shortcut. But when you can, it makes things easier.
  • Here are some example problems. Circle the ones where you can use this shortcut.
  • Practice solving the problems.

And tell them that it is OK if they don't remember the shortcut. They can always do it the hard way -- if they are careful. If they do it right, they will get the same answer either way.

Study habits

Also, encourage your students to maintain a "cheat sheet" of the things they learn in your class. Tell them that if they think that this is a good trick, they should put a note about it (in their own words) in their "cheat sheet". Encourage them to read through their "cheat sheet" the night before an exam.

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The answer is: If the student isn't willing, you cannot. The problem you describe roots in the difference between breadth-first search and depth-first search. We're psychologically trained to use depth-first search if we only care for the result. And most students only care for the result of having done the task. So, if they see a task for which they know a way, the do it and have it done. We as math educators or mathematicians see the problem from a breadth-first search point of view. We try different paths in our head and find a very quick and easy was of solving. But the students don't and they know it.

So, don't try to make them take a breadth-first search point of view, if they are unwilling. It'll give them nothing. If they only want to have done the task, all you do to postpone this point of having it done is only creating frustration with them and you. Instead, try to make them willing. Showing off doesn't help. Make them have fun and get a positive feeling with math.

If they are willing, first show them all the paths and make them understand the relationship between them. Then, let them try out the different paths and find the one that suits them best. Repeat with similar exercises until they know, that the best path depends on the problem.

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    $\begingroup$ You've intrigued me. Know of any key papers on the subject? $\endgroup$ – Chris C Mar 5 '15 at 0:33
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    $\begingroup$ Thank you for this answer. That one cannot do that is something I'm afraid is true based on my experience. But still, I need to try because that's my job. These students will face exams which they have to pass with good grades to have good lives. And they will make mistakes like this and run out of time on the exam, all for a reason I know. It's frustrating not to be able to overcome this problem. Trying to show different paths is what I do but few students have patience for that. :( $\endgroup$ – ymar Mar 5 '15 at 0:48
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    $\begingroup$ I've just thought that maybe how you show the different paths is very important. I'm not sure that my ways are the best. I'll have to think about it. $\endgroup$ – ymar Mar 5 '15 at 1:08
  • $\begingroup$ If your aim is simply, that the students are fast enough in solving during the exams, than you can do mock exams under heavy time pressure. Take some students out of the group and instruct them intensively on how to do the taks in the mock exams fast. Then compare their results and the results of the rest of students. Yes, this is extreme teaching to the test, but you're not doing this for the real exam and it helps to visualize. $\endgroup$ – Toscho Mar 7 '15 at 13:31
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I'd try to create a need for simplifying first.

$\frac{15\sqrt{15}}{4\cdot15}$ doesn't motivate a student to simplify because $4\cdot15$ is easy to do anyway.

$\frac{259\sqrt{259}}{137\cdot259}$ on the other hand is a right pain if no calculators are allowed!

I'd have a starter (maybe a game or competition) that involved simplifying a mix of $\frac{15}{40}$ styled questions and $\frac{2\times5\times7\times11}{5\times5\times7\times11}$ styled. I'd ask how they did each and why?

Hopefully this will empower the students to make life easier for themselves later on.

Ultimately though I'd leave them to do it their own way as you only truly improve by making the connotations yourself.

Hope this helps.

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Students will adopt 'better' methods and rigorous screening for simplification opportunities when it benefits them to do so. Taking the given example, how do you think your students would feel if presented a worksheet with 10 such problems? 30? 100?

Somewhere there's an appropriate number and difficulty level of questions such that if it were approached naively it would be an extremely boring, arduous task, but if approached with a keener eye would become quick work, and moreover become more exciting and better educating at the same time.

Give them (something approximating) this work, and prime them with the tools to cut through it like mature mathematicians instead of slogging through it like grunts. Be sure to throw a few in there which aren't susceptible to any great shortcuts as well, or else you may accidentally teach that 'problems from ymar always have 1-step solutions'.

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