## The Mysterious Mind of the Algebra Student

So why would a student incorrectly evaluate $16^{-\frac{1}{2}}$ to be -4 but manage to correctly get on the very next problem that $5^{\frac{1}{4}}\cdot5^{-\frac{9}{4}}$ is $\frac{1}{25}$?

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.

### 5 Responses

1. perhaps — a culminating class (groups) exercise after fractional exponents have been introduced, with emphasis on what may seem like subtle differences with placement of signs

-16^(1/2), -16^(-1/2), (-16)^(1/2), …etc with clear student presentation of the differences between these expressions as the goal of the exercise…

Ah… the art of teaching… always a challenge to not only get students to understand, but to get them to apply the understanding in a rigorous and careful manner come test time [when the sweet students are riddled with “test anxiety” 🙂 — at least that is what they tell ME when they perform poorly]

2. I may be making an elementary mistake here, but isn’t the result of 5^(1/4) * 5^(-2/4) = (1/5)^(1/4), or in other words, the 4th root of 1/5?

3. Tristam,
I think the 2nd exponent was -9/4 not -2/4.

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 issue 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

4. Jason,
I hope you won’t mind (I should have asked you first) that I repeated this excellent post and my response over at my blog. I encouraged my readers to visit your outstanding blog as well. Happy Holidays!
Dave Marain

5. Tristam, due to font blurring issues I can see how the 9 might look like a 2. I don’t think there’s anything I can do about the font readability of LaTeX in WordPress unfortunately, although if there’s a way of tweaking the size up I’d love to.

Dave, thanks for the post!