Tuesday, July 31, 2012

The Necessity of Algebra

On Sunday, the New York Times published a piece by Andrew Hacker with the provocative title “Is Algebra Necessary?”. Several people have emailed me the link, so it seems like it has garnered some attention. It is worth reading.

While I disagree with most of Hacker’s anti-advanced-math argument, the statistics about the barrier students face when they are not proficient in high school math is important to consider. Failure in math does profoundly affect a young person’s options when it comes to college and career choices. How many people do you know whose math ability was a significant factor in their choice of college major and career? Just the other day, a friend admitted that she’d always dreamed of being a doctor, but she wasn’t good at math so she went into journalism instead.

Rather than lowering the bar for students, we should look at how we can improve mathematics education. Algebra is about abstraction, which is one of the most fundamental ideas across disciplines. When I was a software engineer, much of my day-to-day work was about designing abstract functions with variables to hold unknowns. It was about finding clean, elegant solutions to complex problems through thought, collaboration, trial and error. We conjectured, tested, proved. I couldn’t have learned concrete “job skills” ahead of time. I had to keep learning, inventing and working with big ideas. 
Great algebra and other math classes are excellent preparation for students, as are great science, literature, writing, and history classes.

Hacker’s article prompted many thoughtful responses. I’ve collected some here along with excerpts for your reading pleasure. I’ll end with a nice video of Paul Lockhart talking about the wonderfulness of math.

The New York Times Letters to the Editor: A National Conversation on Math
Andrew Hacker is right: most students will never need to use algebra. Many will struggle to learn it nonetheless. But the answer is not to let students quit as soon as they begin to struggle.

I myself hated mathematics for many years. Through algebra, geometry and trigonometry, I cursed a system that compelled me to take such “useless” courses. Eventually, I was required to take calculus, the most dreaded of all math courses. I prepared for the worst.

It came as a surprise, then, that I quickly found myself enjoying the class. The reason was that I had finally encountered a talented math teacher with a passion for the subject. His passion proved infectious, and now, a year later, I’m looking to study mathematical biology at an Ivy League university.

It’s an outcome I would have never predicted just a few years ago. It could have never happened if I had been allowed to quit when I first struggled with math.

Kitty Hawk, N.C., July 29, 2012

Motherlode: Adventures in Parenting by Jessica Lahey
...I know precisely where I lost my battle with math, the moment I was informed clearly and unequivocally that I simply wasn’t “a math person.” My seventh-grade math teacher, an otherwise lovely man, called each of his students up to his desk one by one in order to write a “1” (for the honors track) or “2” (for the standard track) on the school’s official math placement forms. As I watched from over his hunched and courduroyed shoulder, he wrote a beautiful, decisive and neat “1” on my form.

There it was, in permanent ink. I was good at math.

“Jess, could you come back up here for a minute?” he asked as I floated back to my seat.
He reclaimed my form, and carefully overlaid that beautiful “1” with a dark, clumsy “2,” pressing hard with his black pen in order to make sure the ink obliterated any evidence of his indecision.

And from then on, I wasn’t good at math anymore....

Scientific American: Abandoning Algebra is Not the Answer by Evelyn Lamb
Mathematicians are recruited by hedge funds, consulting firms, and technology companies not because they already know how to balance portfolios, what the best corporate strategies are, or how to optimize user interfaces, but because their mathematics degrees indicate experience and acuity at problem solving. It’s easier for companies to teach someone with a strong mathematics background how to do their specific work than to teach someone who knows the company business how to solve problems. And, like it or not, algebra is one of the first places students start to learn these problem solving skills....

Math education needs to improve, but if illiteracy were on the rise, I don’t think we’d be talking about eliminating reading from the curriculum.

Rob Knop
Algebra is fundamental to nearly all of "higher math". Even if you want to do more than the most basic of things with statistics, you need to know some algebra. To give up on that would be right on par with the giving up on the teaching of history as anything other than memorizing the occasional date, and to give up on the teaching of English literature as anything other than being able to read a short document for simple surface content and to put together a simple declarative sentence. If you want people to be educated beyond elementary school and beyond "job training", then algebra is one of the intellectual foundations of our civilization that simply cannot be neglected.

Harvard University Press Blog: Is Mathematics Not Beautiful?
For all his focus on the pain and fear of mathematics, Hacker has little to say about its beauty. He does suggest that we should treat mathematics as a liberal art, “making it as accessible and welcoming as sculpture or ballet,” but in the service of rationalizing its marginalization rather than encouraging its embrace.

Thursday, July 12, 2012

What must be and what can't be

Have you heard of KenKen puzzles? They are logic puzzles like Sudoku but with an extra element of math calculation. They are great challengers for people of all ages and abilities, since there is such a great range of levels of difficulty. I’ve given them to students, teachers, and friends. 

My dad first introduced them to me. He must have discovered them when they showed up in the New York Times on the same page as the crossword puzzle. He does them in pen, with many cross-outs. The answers to the previous day's puzzles are always there, but they are unnecessary. You know when you've got it right.

I have figured out how to do KenKen puzzles and can do an OK job, especially with the easier puzzles. I want to be better than I am now, so I know I need practice. My dad and mom are visiting me in Vermont this week, so I decided to see if I could get some help with my KenKen chops.

Here is my dad’s advice to me: “First you look at the arithmetic, put in some candidate numbers. Then you start looking for what must to be and what can’t be. Remember to see the whole puzzle even as you work on the pieces. Don’t quit. Keep looking for what must be and what can’t be. If you look at it a little bit you will see it. If you get really stuck, go pour yourself a glass of Pinot Noir or something and the answers will become evident.”

Dad is all about creative problem-solving perseverance, which means he is down with the Math Common Core without knowing it. I was stuck on the harder of the two New York Times puzzles today but he freed me with one little tip. I filled in the rest of the numbers almost as fast as I could write. The tip went something like, “This has to be a 5 and that can’t be a 5 so it must be a...”.

KenKen puzzles are great for young mathematicians. My youngest KenKen student was in first grade. I think most first graders are too young, but she nailed it. I like giving out packets that start with the easier 3x3 grids and are limited to addition and subtraction, and then progress to larger grids with multiplication and division. Most students can quickly find a just-right puzzle and dive in.

The best part of this story is that the marvelous Phia S., a student entering fifth grade, loves KenKens so much she decided to write her own. Here is a photo of the treasured book of KenKen’s given to me by Phia. 

Phia's amazing KenKen book
So, enjoy your summer and take your KenKens (and Pinot if you are of age) to the lake or wherever you might be relaxing. The New York Times publishes two a day on the crossword page (they always include the directions) and more online. Cheers!