How do I get them to appreciate learning a new way of doing that thing?

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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'.