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...
So why would a student incorrectly evaluate to be -4 but manage to correctly get on the very next problem that is ?
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.
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!).
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!
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!