The Importance of Being Wrong

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The Importance of Being Wrong

Written by John Bragelman, Super Tutor for Enhanced Prep

This may seem obvious, but math students always focus on getting the problem right. 

Think about it – who comes to the board to show their solution to the class? A student with the right answer! When a teacher works through a problem for the class, they’ll work towards the right answer. When a student asks another student how to do a problem, they look for someone who knows how to get the right answer. 

Solutions are important, but incorrect solutions are actually just as important.

Whether you are in math class or preparing for the SAT or ACT, your wrong answers are a huge resource. Wrong answers can provide profound insight into your learning trajectory. They can even guide you towards a higher grade or score! 

The trick is to figure out what you did wrong, why you got the problem wrong, and how you can improve. 

I’ll use the following problem to demonstrate this: 

Here is the solution:

This is a somewhat straightforward distribution or FOIL problem, but there are so many ways to get it wrong! I’ve seen students arrive at the incorrect answer on this question by skipping steps, making an error on one crucial step, or simply choosing the wrong process.

 

Yet as I’ve reiterated, such errors are still valuable for introspection and growth.

 

Skipping Steps

 

On timed tests like the SAT or ACT, you need to move quickly. Because of this, students have a tendency to skip steps, especially when they know how to solve the problem. It’s not uncommon for high-performing students to make many careless errors on low-difficulty questions.

 

Here’s what skipping steps looks like with this problem:

What was the error? 

I’m missing the middle term.

 

Why did I make it? 

I only squared the first and last term! I didn’t square the entire binomial (4x-5).

How do I improve? 

SLOW DOWN! Especially on easy problems! Take your time, write out each line, and don’t be a step skipper!

An Error on One Step

Sometimes an error occurs when students follow the correct procedure but make a mistake on one step or part of one step. For example:

What was the error? 

The sign of the last term is a negative instead of a positive. Specifically, on the third line, when I multiplied (-5)*(-5), I missed one of the negative signs. 

Why did I make it? 

This part is trickier. Did I make the error because I was working too fast, or did I make the error because I struggled with positive and negative numbers?

How do I improve? 

If I made the error because I was careless, I need to slow down again. If I made the error because, well, I’m not the best at positive and negative numbers, then that’s exactly what I need to review. After reviewing those, then I may come back to try a few problems similar to this one.

A Process Error

The last type of error is related to the problem. I simply don’t know what to do. For example:

What was the error? 

I had no idea how to start the problem.

Why did I make it? 

I couldn’t remember how to FOIL. I may not have been able to work through the problem because I haven’t learned FOIL yet, or I may not have been able to because my mind blanked. 

(Again, it’s important to ask yourself why.)

How do I improve? 

Out of all the errors I talked about today, this is the only one that suggests I need to spend time reviewing how to distribute binomials (i.e. FOIL). 

Our impulse is to immediately perceive wrong answers in a negative light. But when you approach them in this fashion, asking what, why, and how, you’re well on your way to improving your understanding and performance!

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