For years I’ve been telling people that allowing people to discover mathematical concepts is way better than teaching an algorithm. And a few months ago, a smart friend of mine asked, “What’s an algorithm?”
Duh. I should probably explain that part, right?
It wasn’t that she didn’t have a vague sense of what algorithm means. But in some ways, I was using the term as educational jargon. That’s not cool, so I’m here to correct my bad habit.
Is it better to show kids a step-by-step process for solving problems? Or should we give kids the space to discover mathematical concepts and how to apply them?
In its most basic sense, an algorithm is a set of steps. These steps might be followed by a computer or by a person, depending on the situation. In some cases, you can think of an algorithm as a formula.
Algorithms in Everyday Life
You encounter algorithms all the time. On Facebook, an algorithm determines which posts and advertisements you’ll see in your feed. In Weight Watchers, an algorithm outputs the points value for the food you eat and another spits out your weight loss trajectory. Google uses algorithms to determine search rank. (The more popular the site, the higher its rank.)
Algorithms can make your life easier (or harder, depending on how you look at it).
In these cases, you might consider the algorithms to be formulas. And they are proprietary. There’s no way Facebook, Google or Weight Watchers is going to share these processes.
At the same time, these algorithms can make your life easier (or harder, depending on how you look at it). Certainly, before computers, crunching these kinds of numbers was way more tedious.
Take the enigma decryption project during World War II. (This is the story told in The Imitation Game, a new movie starring Benedict Cumberbatch as the mathematical genius, Alan Turing.) Enigma was a rather brilliant German code that was considered impossible to break. That’s because the code changed every day. Before it could be cracked, the code was altered slightly, always leaving the allies a little bit behind.
Once Turing built his code-breaking machine, the process sped up considerably. With a few standard clues, his invention could spit out the decoded messages in a matter of minutes. Suddenly, the allies had an advantage, which ultimately saved millions of lives.
But Turing likely had a greater effect on our modern lives. He published a paper considering the reliability of certain algorithms–an underpinning of Google’s search algorithms. Turing was one of the first to see the benefits of building machines to follow algorithms that were too complex or tedious for humans.
Algorithms in Math Education
But as a math educator, I’m not so keen on algorithms. That is, I don’t think that teaching certain algorithms is very productive in the classroom. And this right here is one of the cornerstones of the Math Wars: Is it better to show kids a step-by-step process for solving problems? Or should we give kids the space to discover mathematical concepts and how to apply them?
I would say that we need both, but we should rely more heavily on discovery.
So what is an algorithm in the math classroom? The classic example is long division. Most grownups have this process down cold. But it’s incredibly difficult to explain to young students. In fact, it takes most students several years to really internalize the steps.
So what’s the problem? Well, the algorithm isn’t intuitive, and it doesn’t have meaning. That’s no big deal when a machine is doing the calculation–or when the algorithm is so ingrained that the human brain goes on auto-pilot to find the solution. But that doesn’t happen quickly during the learning process. It’s like learning a new language through rote memorization.
In addition, division is a tool that allows us to solve more meaningful problems. When the tool is difficult to learn how to use or must be learned completely out of context, we risk losing kids’ attention in the process.
I’m not completely against teaching mathematical algorithms. I’ve certainly employed long division from time to time as a grownup. But I’m more likely to give that task to the little computer in my smart phone. And at some point, kids should too.
What do you think? Can you describe any mathematical algorithms that you use in your everyday life? When do you let the machine do the work? And when do you do the calculations by hand? Share your ideas in the comment section.