Not All Careless Errors are the Same

Students often hear the warning, “Watch out for careless errors!” It’s a common refrain from teachers, parents, and test prep tutors alike. But really, what is a careless error? Most educators agree that careless errors are those mistakes that have nothing to do with lacking the knowledge or proper understanding of a concept. They are the self-inflicted wounds, the unforced errors that could, and should, have been avoided.

Careless errors include miscalculations that result from a failure to write out steps. They also include transcription mistakes that occur when one is rushing or not paying enough attention. One type of careless error, though, is different from the others in that, though it can be avoided, it is not fully attributable to pure carelessness. It is rather the result of a strange quirk in the way our brains work.

I have an embarrassing confession: I sometimes mix up left and right. It doesn’t happen when I’m driving or when I pick up a pen to write, but if someone says, “raise your right hand” or “point to your left,” I almost always screw up if I don’t stop and think for a second. I’m a righty. It’s probably more accurate to say I’m an extreme righty. If I try to dribble a basketball with my left hand, I look like a velociraptor trying to work a yo-yo. So my continued stumbling over left vs. right has nothing to do with being ambidextrous, nor do I think that I’m unintelligent because I have this problem. My directional confusion is based on  something called a dichotomy error.

One of the strongest features of the human brain is pattern recognition. To aid this, we are constantly categorizing. When a category set has exactly two subsets that are mutually exclusive, they are said to be in a dichotomy: good vs. evil, real vs. imaginary, Washington Redskins vs. Dallas Cowboys (okay, maybe not the last one, even though for many it is synonymous with good vs. evil).  These dichotomies serve us well, but they can also be problematic. For one thing, well-intentioned people create false dichotomies when trying to simplify an argument. Then there are classes of things that exist in true dichotomies, but are not as instinctively separable. This causes dichotomy errors.

Download dichotomy error example questions below!

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We see this often with graphing points. Every student I’ve worked with, at one time or another, has plotted a point like (0, 3) on the x–axis even though it belongs on the y–axis. When they do this, they have not forgotten which is the x–coordinate and which is the y–coordinate. The carelessness is a product of the fact that they consider a task like this easy and thus unworthy of extra thought. Math is a minefield of potential dichotomy errors. The radius and diameter of a circle, the formulas for Area and Circumference of a circle, parallel lines and perpendicular lines, and factoring and FOILing are all pairings of concepts that produce dichotomy errors time after time.

The keys to combating these errors are patience and humility. Take an extra second on questions where you are aware that dichotomy errors lurk, and rely on those tricks and pneumonic devices, no matter how basic and silly they seem, that are so helpful for memorization. You’d be surprised how many students still think about those two l’s in the middle of the word parallel when trying to remember what that relationship means. And is there anything simpler than matching the lengths of the words radius and diameter with the relative lengths of the segments they represent? Radius is a smaller word, so it is the smaller segment.

Tricks need to be both simple and consistent. Sayings like “i before e except after c” don’t always work. I’m constantly spelling the word weird wrong because the trick doesn’t cover that word (the saying is technically a false dichotomy because there isn’t mutual exclusivity). Use alphabetical order whenever appropriate, like how xcoordinate and horizontal both come before their relative partners ycoordinate and vertical in the alphabet, but, above all else, avoid making your memorization trick overcomplicated. After all, the act of raising your right hand shouldn’t require that you first solve the riddle of the Sphinx.


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