A Natural Approach to Negative Exponents

For today’s class in our beginning algebra course, we took a different approach to negative exponents. The decision to do something different is partially rooted in my conviction that most of our textbooks are wrong about what negative exponents mean.

To set the stage, the first thing we did was a little activity on basic properties of exponents. The activity is based on this document Class 22 Group Activity Exponents

This activity uses the type of approach many of us use for a more active learning classroom. I suspect mine is not as polished as many; several students found the ‘long way’ a bit confusing. As usual, I did not present any of the ideas before students got the activity and worked in their small groups.

One of the problems on this activity is the problem shown here. In the ‘long way’ method, students easily wrote out the factors and found the answer. Quite a few of them used the subtraction method to create a negative exponent. In a natural way, we noticed that m^-4 is the same as having m^4 in the denominator.

Negative exponents indicate division!

We did not create negative exponents in order to write reciprocals. We started using negative exponents in order to report that we divided by some factors. I find it troubling that we have focused on a secondary use for the notation, when the primary use makes more sense to students.

If you want to see what is so important about this, give a problem like this to your students.

Direction: Write without negative exponents.

Almost every student focusing on the reciprocal meaning will invert the fraction — making the 4 a multiplier instead of the divisor it really is. Most students focusing on the division meaning will see that the m cubed needs to be in the denominator.

In part of this activity, students also dealt with a zero power. In doing the long way (write it all out), quite a few students wrote that variable in their work; it made sense, though, to omit that factor because it said “zero factors” … and then we can talk about what value that ‘zero factor’ has in a product (one).

As we shift towards more work with exponential functions, it becomes critical that students understand the meaning of all kinds of powers. A core understanding of negative exponents is part of this; fractional exponents are important too (though we tend not to cover these in either our Math Lit course or beginning algebra).

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Learning math, Student Success & Support | Jack Rotman | November 10, 2014 2:21 pm

7 Comments

By schremmer, November 10, 2014 @ 9:24 pm

I do not see how this is particularly natural.

Why not just say that ax^+n is code for “a multiplied by n copies of x”, ax^–n is code for “a divided by n copies of x” and ax^0 is code for “a left alone”.

Students find the idea of code entirely natural, particularly as it is a shorthand. The welcome question the students tend to ask is why not code multiplication with • and division with ÷? After doing (the long way since at this point that is all there is) instances of ax^+m • ax^+n and ax^+m • ax^+n they are happy to discover that “oplussing the exponents does the job for multiplication and “ominussing” does the job division. (The “o” in oplus and ominus is to remind them that they are dealing with signed numbers)

To see how this articulate with other developmental matters, see, once again,
How Content Matters

To see how this comes handy with power functions in Precalculus I, see
Exam 2 Discussion

Regards
–schremmer
By schremmer, November 10, 2014 @ 9:31 pm

Oops!

instances of ax^±m • ax^±n and ax^±m ÷ ax^±n

Embarrassed regards
–schremmer
By Eric Neumann, November 11, 2014 @ 4:22 pm

I’ve always proved the exponent formulas by having students work through the “long way.” Unfortunately, the exponent formulas are one of the many topics in Developmental Maths that the majority of our students vaguely remember, and therefore tune out all explanations in favor of trying to recall the formulas from rote. Many of them, therefore, wonder what in the world I’m doing (or worse, am expecting them to do), get afraid that they will be asked to do things the long way on the exam (which seems to them an insurmountably cumbersome amount of work), get upset when I reveal that they won’t be forced (on the exam) to do it the long way after all, proceed to try to merely memorize the formulas as if they had no logical basis, and finally proceed to mix them up and/or leave problems blank that could easily have been done the “long way” but were frustrating because they knew they couldn’t keep the formulas straight, so that even problems as straightforward as those on your activity sheet appear hopelessly complicated.

It is not enough for us to prove mathematical formulas, or even to help our students “discover” those formulas. We must somehow help them truly believe that these discoveries/proofs are logical, easy to understand and redo on the fly, and in fact aid memory of the formulas rather than hinder it. I have used worksheets very similar to your “Class-22” only to have the exact same problems left blank or done by using the formulas improperly on a test one week later.

Yet I do continue to have the students work through the derivations of the exponent formulas because it does help some students – and leaving it out wouldn’t benefit anyone. I really just wish it were handled better the first time around before so many of their minds were fixed in “If only I could remember the dang rule” mode.
By Jack Rotman, November 12, 2014 @ 7:26 am

Eric:
Very good points … our focus on ‘understanding’ can be perceived as a burden by students. Our classroom culture needs to place a high emphasis on reasoning and ideas from the first minute of the first day; our own attitudes need to support the type of learning we want students to engage in.
I don’t have any magic for doing this, and I know I fall short several times in the semester. Every day as I get ready to go to class, I deliberately think about the concepts of meaning, properties, and choices; those seem to capture the essence of learning mathematics. This attitude on my part seems to help some students break out of negative attitudes about learning math.
By Eric Neumann, November 12, 2014 @ 9:01 am

“meaning, properties, and choices.” I like that!
By schremmer, November 14, 2014 @ 9:19 am

I am clearly a terrible teacher else Neumann and Rotman are terrible students. Let me try again.

(1) I assume that we all agree that, somehow, both natural and negative exponents have to be introduced. My point was that we might as well introduce them together as signed exponents which, in fact, takes less time than introducing separately natural exponent and then negative exponents.

(2) A secondary point—that they seem to have missed entirely but which I think is rather important—was that the definition should involve a coefficient because it makes exponents easier to read and deal with. (See Exam 2 Discussion.)

(3) I am not proving anything and I certainly do not “have the students work through the derivations of the exponent formulas”. I assume that we all agree that, somehow, exponents have to be used and my point was that exponents might as well be used to make easier to multiply / divide these very expressions that exponents were introduced to “shorthand” in the first place.

(4)I do not see how any of this is a “burden” to the students nor how it “place[s] a high emphasis on reasoning”.

Regards to all.
–schremmer
By schremmer, November 14, 2014 @ 9:28 am

Oops. Not only am I a terrible teacher, I am also a terrible reader of my own stuff. Another typo!

Exam 1 Discussion

(Question 2 and following.)

Embarrassed regards
–schremmer

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