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!
Monday, September 26, 2011
The state of mathematics education in this country is not as hopeless as we are led to believe. I’m here to say that good math teachers are alive and well in the United States. Critical thinking, intrigue, enthusiasm, real applications, and in-depth problems abound in high school math classrooms across the country. I’ve seen it, and I will testify to its existence. So, let’s make some noise. Contribute, share, grow. That’s what I have in mind.
My first thought is to start with big ideas and see where that goes. The Understanding by Design (UbD) framework calls these big ideas “Essential Questions.” A google search of ‘essential questions in math’ led me to a list of 150+essential questions in math. Seems like a good place to start, but alas, as I read through these, my suspicions are confirmed. Many of these 'essential' questions seem to fall into one (or both) of these two categories:
- I can rattle the answer off in a couple of sentences.
- I don’t care.
How about a list of questions that is intriguing to me AND to a typical high school kid? How about some questions that rise above a particular piece of content and apply to a whole variety of mathematical topics? I think the list will be shorter than 150, but who knows?
So, I start this blog with a healthy dose of humility and a shout out to all those who have mentored and inspired me in this particular endeavor:
To Ron, whose excitement and enthusiasm for math and teaching led thousands of his students and colleagues to a love of mathematics. To Rich and John, who showed me that the supply of deep and intriguing problems is endless and not as elusive as I thought. To Isaac, Mary, Steve, Andrew, Margie, Rich, and so many others who never let me forget that collaboration is so much more fun (and effective) than independent work. To Mark, who shows me that there is no shortcut through hard work. To Alice, who teaches me that real genius is not an act; it’s an action. To Colleen, who constantly reminds me that I can be better. To Jen, who showed me how to make anything look fun. To Diana, Bob, Abdellah, Brian, Ivan, Yuri, Ann, Nick, and Joan who taught me how to make math sound reasonable. And to my parents, who taught me that if higher knowledge is the goal, then critical and creative thinking is the path.
And thanks to you too. I look forward to working with you.