Jason over at Number Warrior, an excellent blog for math teachers, has a short but fascinating post on trying to analyze why students make careless errors when it comes to negative fractional exponents.
I hope he doesn't mind if I repeat my comment over here - I think it raises some important issues for all of us who are trying to help students overcome these apparently 'careless' errors. I also recommend you visit his blog - fascinating stuff...
Jason's post:
So why would a student incorrectly evaluate 
I believe this is a case that the knowledge of negative exponents was stored somewhere back there, but because the first problem looked “easy” my students just went for the impulse answer. (Nearly everyone — even students who scored very high overall — got it wrong.) I wonder how I can get students to reach back there more often, because neither gentle admonishments nor fierce reminders seem to work.
My response:
Jason,
We can speculate about why students make errors, but I’ve learned there are usually several reasons. I found it helpful to simply ask them to explain how they got that result (if they can!).
Some thoughts:
Your 2nd example procedurally involved fractional exponents, but ended up raising the base to a negative integer, not a negative fraction. This is a minor distinction, one extra step, but you never know. Also, I found it helpful to encourage them to write the extra step or two rather than do it mentally. Thus, 16^(-1/2) = 1/(16^(1/2)) might help. in other words, when they have to cope with both the negative and the fraction, make them always do the negative first. Some individuals are simply not detail-oriented and have trouble with precise procedures. I believe left-brained people have fewer of these issues because they are wired to do step-by-step procedures!
Finally, although none will admit to this, some youngsters know how to study for a math test and some simply don’t practice sufficiently. The “I think I know the material” students who didn’t review enough usually get burned on these procedural problems that have that one extra step. Ok, I’m probably over-analyzing all of this - it’s just a darn common error! Happy Holidays!
Dave Marain
My gut feeling is that these kinds of issues which math teachers have to confront daily, beg for considerable dialog. I know I benefited from asking more experienced teachers for advice when so many of my students struggled with certain types of questions. Asking students themselves to analyze their own errors is rarely a waste of time in my opinion. We always want to encourage self-reflection and it's usually good practice to have students correct their errors after receiving their tests back. And, of course, this kind of dialog also serves as a window into their 'mysterious' minds!
I hope this generates some further discussion about 'careless errors' and what we can do to help students cope!
Happy Holidays Everyone!
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