In the midst of a very good post, Julie writes:
They also distributed the exponent to both numbers inside the parenthesis. I have no idea why they would do this, because I don’t even teach them the power to a power exponent rule. I have them expand any set of parenthesis with an exponent. They do love the distributive property, but we-e have never, ever, ever, distributed an exponent. Sigh. I failed.
There are a lot of wonderful reflective moments in that post, but I find the above fascinating. Kids are distributing exponents inside the parentheses. You might be tempted to explain this mistake as confusion between the power rule for exponents and the distributive property, but seeing as the kids haven’t yet learned the power rule that explanation seems false, at least for Julie’s kids.
Distributing exponents is tempting for kids. Why?
Julie has a strategy for helping her kids move past these sorts of mistakes:
So, starting Monday, I am going to have one problem of the day for both classes posted on the board. It will look like a variation of this. Evaluate -3x^2 – 2x + 5 when x = -2. I’ll throw in fractions, decimals, and any other basic, easily forgettable concept. This should help them quickly practice evaluating algebraic expressions, exponents, and the order of operations, EVERYDAY. And we will go over it, together, everyday.
Will repetition work? Will she need something else? Here’s hoping that there’s a follow up post from Julie letting us know how her experiment goes. The question for me is, what are the returns on any given genre of problem? Do they diminish with repetition? How different does a problem need to be in order to promote real learning?
Productive Struggle 
In my experience, we automatically recognize important differences in where thing sare placed and we expect our students to… here’s a theory as to “why in the world”: That exponent is a number that’s just outside parentheses, that are around an addition thing. Oh, and when you’re doing exponents, you multiply. The importance of the exponent being little and up in the air might be missing? (I also remind often that it’s not like a regular number — you don’t do *anyting* with it as meaning “the number two.”)
I find that repetition *does* work if I actually do it *enough* — as in, in little shots, every day, to automaticity… and I don’t know whether or not it has to include the “why” part, but I always include it. If somebody misses it, I’ll prove it doesn’t work with numbers. Lots of folks don’t really believe that this letter stuff has anything to do with numbers; when I show them that five squared (25) isn’t equal to four + 9 (two and three squared and then added), it helps. I’ll even make up a drama about the exponent not having the power to tear those numbers apart; that the big ol’ multiplying 10 *does* have that power, since he’s down at their level. (They’ll have already seen my explanation of the distributive property per http://www.resourceroom.net/math/distribthree.swf (one of these days I’ll learn to make the visuals more clear…)
Just some ideas —
xiousgeonz, I love your ideas. I’ve been thinking about this ALOT since Friday night when I graded these tests. Today I gave them a problem and many students got different answers. They realized WHY we needed to work on this. All of us getting a different answer to the same problem can’t be a good thing. lol!
The more I think about this, the more I am coming to think that the problem isn’t the individual pieces, it’s that in this type of evaluation problem, we are putting all of those pieces together. Students have much to process, and have a tough time pulling it all together when many of these concepts are relatively new to them.
That’s what I’m going with anyway! I’m going to keep at it for the rest of the year with the problem of the day. I’m going to mix it up a bit too, adding other things that confound them, such as fractions and decimals. I’ll report back to you all! Thanks for the post Tina! 🙂
Michael (@mpershan) actually wrote this one. I’m playing with the theme so that you would actually have a way to know that. Thanks for being willing to share!
Anytime! I fail (read learn) a lot. Lol!