This headline is a lie. It’s not that I think algorithms are bad. They’re not. Honestly, I think that’s how many of us move through our days without killing ourselves or someone else. We habitually take the medications prescribed by our doctors; we cook our eggs (and avoid salmonella); we follow the steps for safely backing our cars out of the driveway; we put on our socks before our shoes.

Even certain mathematical algorithms are very useful, like the order of operations (or PEMDAS).

But in the end, I think that dictated algorithms are not so great for people, especially people who are learning a new skill, and especially when the algorithm has little to no meaning or context.

Don’t know what an algorithm is? Check out my earlier post defining algorithms. 

People Aren’t Machines

There are many different educational philosophies that drive how we teach math. For generations, teachers worked under the assumption that young minds were tabula rasas or blank slates. Some educators took this to mean that we were empty pitchers, waiting to be filled with information.

This is how teaching algorithms got such a strong-hold on our educational system. Teachers were expected to introduce material to students, who were seen as completely ignorant of any part of the process. Through instruction, students learned step-by-step processes, with very little context.

In recent years, however, our understanding of neurology and psychology has deepened. We understand, for example, that children’s personalities are somewhat set at birth. And that their brains develop in predictable ways. We are also beginning to realize that certain types of learning and teaching promote deeper understanding.

The result is a better sense of students as individuals. Instead of a class filled with homogeneous little minds, we know now that kids (and grownups) are wildly different–in the way they digest information and approach problems. (To be fair, this is closer to John Locke’s original theory of tabula rasa, in which he states that the purpose of education is to create intellect, not memorize facts.)

In terms of a moral, there’s not much I recommend in this Pink Floyd video, but I can certainly identify with the kids’ anger at being treated like cogs in the educational system. Besides, it’s cool.

A Case for Critical Thinking

Certainly critical thinking is not completely absent in the teaching of algorithms. It’s marvelous when kids (and adults) make connections within the steps of a mathematical process. But critical thinking is much more likely, when the process is more open-ended. Give kids square tiles to help them understand quadratic equations, and they’ll likely start factoring without help. Let students play around with addition of multi-digit numbers, and they’ll start figuring out place value on their own.

You can’t beat that kind of learning.

See, when someone tells us something, our brains may or may not really engage. But when we’re already engaged in the discovery process, we’re much more likely to make big connections and remember them longer.

That’s not to say that learning algorithms is bad. But think of the way you might add two multi-digit numbers without a calculator. Instead of stacking them up and adding from right to left (remembering to carry), you might do something completely different, like add up all of the hundreds and tens and ones — and add again. In many ways, you’re still following the algorithm, but in a deconstructed way.

And in the end, who cares what process you follow–as long as you get to the correct answer and feel confident.

Teaching Algorithms is Easier, Sort Of

So if discovering processes is so much better, why does much of our educational system still teach algorithms? Well, because it’s more efficient in a lot of ways. It’s easier to stand in front of a group of kids and teach a step-by-step process. It’s harder–and noisier–to let kids work in groups, using manipulatives to answer open-ended questions. It might even take longer.

But I say that based on what we now know about kids’ personalities and brains, we’re not doing them much good with lecture-style classes. So in the long run, it’s easier to teach with discovery-based methods. Kids remember the information longer and get great neurological exercise. This allows for many more connections. At that point, the teacher is more of a coach than anything else.

In the end, we all use algorithms. But isn’t it better when we decide what steps to follow, through trial and error, a gut instinct or discovering the basic concepts underlying the process? That’s where we have a big edge over machines. After all, humans are inputting the algorithms that machines use.

Photo Credit: teclasorg via Compfight cc

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.

Photo Credit: andywalton7 via Compfight cc

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.