Answer Standards Fight Mathematics

“Negative exponents are not allowed in answers.”

My intermediate algebra class has not had negative exponents in the material so far.  However, here is what most students remember:

I can not have negatives, so write the reciprocal.  No exceptions.

Given the ‘answer desperation’ of most students, any ‘rule’ about the answer gets added emphasis in learning.  As teachers, many of us try to make it easy for students … so we add our emphasis.  The result is that our standards (often sensible but arbitrary) fight mathematical knowledge.  Students focus on the form of the answer and our rules about that, and have less understanding of the mathematics involved in the situation.

In the case of negative exponents, the rationale for ‘no negative exponents’ is marginal at best.  True, in some cases, positive exponents are simpler; however, for the majority of situations, negative exponents are simpler — they often avoid the need to write a fraction.  The evolution of exponential notation and meaning is based partially on the idea that negative exponents are a simpler way to show division … and fractional exponents are a simpler way to show roots.

For my class, the previous emphasis on ‘no negative exponents’ distracts them from understanding simple division problems.  We do more polynomial arithmetic than is really needed, but these division problems are just dividing a binomial or trinomial by a monomial.  The student answer desperation and the negative exponent prejudice combine to distract them from the basic ideas of division.

A related issue came up in my beginning algebra classes … students wondered if they should finish a fraction problem by changing it to a mixed number.  The context was solving a linear equation with one variable; the form of the answer is a trivial matter compared to the mathematics.  Overall, one of the most common questions I ever hear is “how do we need to write that answer”.  Sometimes, this deals with algebraic concepts and the question is valuable; many times the question is a distraction.  I often tell students that I don’t care what form they give an answer as long as it clearly communicates a correct result.

Perhaps we should take a step back from all of our simplistic statements about ‘form of the answer’.  Many of them are conveniences for grading work, nothing more.

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2 Comments

• 

  By Sue Jones, September 25, 2012 @ 5:29 pm

  I often refer to that part of any given question as “okay, now we get the grammar right, but this means the same thing.”
  Negative exponents are one of those times when the students are inspired to say “and, of course, I will use this so often in my life!”

• 

  By Audrey, September 25, 2012 @ 6:51 pm

  I wholly agree with this. When students go on into science classes they have to use the answer to one part of a question in another and often the most ‘useable’ form may not be the so called ‘correct’ form. I tell my lower level students there is ‘back of the textbook’ correct and then there is just correct because it conveys the answer.

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