Every year I promise myself that I'll be better at organization and post-lesson analysis, but yet, I always seem to be too wrapped up in production to have time to reflect. So this year, I find myself with a new focus: take a look at what I've already produced and do some serious self-reflection.
And so, in this spirit, I pulled out an old lesson I created for my beginning algebra students on using the lowest common denominator to solve rational equations. It's a nice lesson. The student's always chuckle at my creature graphics and at the end of the day, they can do the math.
But with fresh eyes, I wonder: this particular skill is useful, because the 'trick' can take a muddy equation and clean it up a bit. But it isn't essential, because the students already have methods that can solve these types of problems. And yes, I know they'll really need this method when they get to more difficult rational equations with variable expressions in the denominators. But they don't care about THAT! So what's the big idea? And then it comes to me...
It's about elegance. What makes one solution more elegant than another? And I love this question, because I don't have an easy answer, but I care. I WANT to be able to write things that others will look at and say, "Wow, that's really elegant." Plus, it opens up doors in all directions: multiple methods for solving systems of equations, solving by graphing, calculating area of unusual shapes, proofs, trig identities... my brain is already jumping all over the place.
How many times do we ask our students to make choices about using the best method or presenting the best argument? But this concept of 'best' is elusive. Ask any student, and they will surely tell you that the best solution is the correct solution. But I know that not all correct solutions are created equal. And this is where mathematics resembles art. While there are many paths that are correct, we can evaluate the merits of one solution over another by considering craft, execution... and elegance.
Disclaimer (and shameless plug): I am proud of my original materials, especially in their revised forms, and so as I review and analyze them, I am posting them to share on the online marketplace: Teachers Pay Teachers. Some are free and some have a nominal fee. You can see them all by clicking the link to my store on the right. If you wish to become a seller too, click here and you will award me referral points. Happy sharing!