Friday, March 9, 2012

Exponent Negativity

An 8th grade student approached me yesterday after third period. I had just taught composition class - the composition class that conflicts with the Algebra I class I have been attending in order to get over my math anxiety.

"Mrs. Lahey, I we did some hard stuff in math today, and we were really worried that you'd get behind, so I agreed to take notes for you and teach you the lesson. I can come by during lunch today and explain what we learned, if you want."

This reversal just slays me. My students are worried for me. Worried about my ability to keep up in math class. How sweet is that? I was really touched. One of the things I love most about my school is the sense of community, but until yesterday, this community had always been a "them" and "us" community. We adore our students, mind you, but as much as we'd like for our community to be one, big, fuzzy "us," it's not. Students are students and teachers are teachers, and never the twain shall meet.

Until yesterday.

Before I start getting attacks from readers who think I am trying to create inappropriate relationships with my students - relationships I am supposed to view as rigidly hierarchical and hopelessly lopsided as a power structure - know that I am not trying to be my students' friend. I just think it's good for them to see adults not know things, and not be afraid to not know, and not run from not knowing.

I talk and talk about the importance of viewing education as a lifelong process rather than a means to some calligraphy-on-parchment end, and my attempt to work through my math anxiety is proof of statement. I really mean it. I love to learn - and not just the stuff that comes more easily. Even the stuff that makes me want to give up and run screaming in the other direction.

Stuff like this:

As I am a newly minted Algebra I student, let me break this explanation down for you.

[Silence, eventually the sound of fingers tapping idly on the keyboard, as I attempt to think of words appropriate to the material contained in the scanned image, above.]

Okay. Here's what I know. I must accept the fact that any number to the power of zero = 1. Ellie, my English-student-cum-Algebra-mentor explains that concept in her red notes above. I don't understand it, but I accept that it's a rule, and follow it.

The other thing I learned is this: if an integer is negative, you can stick it at the bottom of a fraction under a one, and it magically becomes positive. I don't know why, but if I simply accept this and apply it, my homework answers are right. I got through a difficult problem set today through the blind application of these two rules.

I'm not proud of this reality; I'm just owning it. I just don't get it. Alison Gorman, my colleague, friend, and math teacher is patient and kind and generous with her time, and I feel as if I get it for a second or two after she explains it to me, but then, poof, it's gone.

This has always been my problem with math. I try to understand the whys and wherefores of the rules, I really do, but the why just goes over my head. I don't know if it's because I don't care, or because I know that in the end, if I just decide to accept the rule and use it, I can get by through sheer grit and application of the rules I don't really understand. That's how I got through high school math.

Alison is one of the most effective, organized, creative, and dedicated teacher I have ever met, and yet, I think she may have met her match in me. She's extremely patient, and more than a little entertained by my efforts, but I'm afraid she's going to realize that I have severe limitations where numbers are concerned.

I will continue to try to understand, because I hate that I don't. I hate not knowing. I hate butting up against my limitations.

It's time to bust through the negative and transform those integers into their positive form once and for all.

Monday: Powers of 10 and Scientific Notation.

Part V of my math odyssey can be found here.


  1. The why is actually fairly manageable.

    When you multiply two exponents with the same base you add the powers. x^2 * x^3 = (x*x)*(x*x*x) = x*x*x*x*x = x^5 = x^(2+3)

    When you divide, you subtract the exponents. 3^5/3^3 = (3*3*3*3*3)/(3*3*3) = 3*3 = 3^2 = 3^(5-3).

    x^0 can be written as x^(2-2) = x^2/x^2 = 1, because any value divided by the exact same value is always 1. Negative exponents work the same way. x^-3 = x^(0-3) = x^0/x^3 = 1/x^3.

  2. Nothing triggers my compulsion to try to explain things more strongly than a phrase like "I accept that it's a rule", so here's a go:

    I hope that doesn't end up just increasing your confusion and/or frustration.

  3. x^0 = 1 is a requirement in order that the exponent rules you've developed make sense. See

    The same is true of negative exponents. The "why" is because it extends the rules from the other cases in ways which make sense. This happens often in mathematics. We develop rules, and to make the rules work, we extend them in the "edge" cases.

  4. Wow! I have math peeps! And I get it! Thanks, everyone!!!

  5. Jessica, I posted this on Twitter, because you are like any of my students who don't feel like "math is their thing," but you are able to articulate your struggles and frustrations so eloquently, honestly, and understandably. Please keep learning math for the sake of us math teachers!

  6. I recently tweeted about how (many, not all) adults have no shame in bragging about how terrible at math they are or were when they were in school. However, we never hear an adult brag they can't read. This devalues numeracy.

    I think you are spot on. It's human to not know everything. Too many teachers are fearful they will be 'exposed' and wind up passing along bad information or posturing in a manner that turns them off to their students. Kids see through that stuff.

  7. I feel that this comment summarizes the struggles of many of our Algebra students:

    "The other thing I learned is this: if an integer is negative, you can stick it at the bottom of a fraction under a one, and it magically becomes positive. I don't know why, but if I simply accept this and apply it, my homework answers are right."

    Many students are taught a series of maneuvers in algebra, with no understanding of why they occur. Because of this, Algebra often comes across as a series of magic tricks which must be mastered.

    Much of algebra is just substitution; replacing a scary-looking expression with something somewhat simpler and more recognizable. As we help students understand this, the less Algebra seems like dis-connected rules.

    In the case of the exponents, we don't want students to think that the exponent simply moves down below and becomes positive automatically. Rather, we want students to understand that 2^(-3) is the same as 1 / 2^3 and see that they can be interchanged.