What does it mean to be “good at math” today?

Mathematics and computing are not the same thing, but they are closely related, and have been throughout history. Indeed, the way most people find themselves using their school mathematics (if at all) in the workplace after they have graduated involves computation. Indeed, computation may be the only thing they do at work that makes direct use of their school math skills.

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How computing was done before 1965.

Prior to 1965, that meant that if someone who worked for a large company or organization told you their job involved math, you could reasonably assume they spent much of their working day in an environment like the two shown above. (For a smaller company, it would be a smaller room, maybe even a small, one-person office.)

They might have some form of mechanical, arithmetical calculating device on the desk in front of them, or perhaps easy access to a larger mechanical calculating machine, as in the forefront of the picture on the right. But for much of the time, they would be working with pen (or pencil) and paper.

The one exception were scientists and engineers. Their workplace might look more like the one shown here.

True, this 1961 photo of NASA space scientists working on a giant blackboard was staged for a LIFE magazine cover, but the blackboard was real. While its size was unusual, more modest blackboards were regularly covered in chalked symbolic equations by scientists and engineers all around the world.

While the complexity of the math differed enormously between the two use cases, what both scenarios had in common was that it was people who were performing the calculations.

That changed dramatically around 1965, with the growing availability of modern digital computers that handled the calculations. All of them! Numerical, algebraic, calculus, it made no difference. If a calculation could be specified by a step-by-step procedure, it could be turned into an algorithm that a digital computer could execute. What had hitherto (throughout all of history) required humans to do, could suddenly be handed off to a machine.

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The IBM 360, which first went into use in 1965, was the first commercial, general purpose digital computer. Expensive drives for the magnetic tape data storage occupied a lot of the space the computer required.

With large corporations and university research labs using digital computers to do much of the procedural mathematics that had previously been handled by the human brain, it was soon only in smaller companies and schools that humans still toiled away with paper and pencil.

That state of affairs came to an end in the late 1980s, when software systems started to appear that could handle all procedural mathematics on a personal computer (and in due course on a smartphone).

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Starting with the appearance of the computer spreadsheet in 1979, the graphing calculator in 1985, and all-powerful, versatile Wolfram Mathematica in 1987, by 1990 it was no longer necessary for a human to ever again execute a mathematical procedure. Machines could do it much faster, with far greater accuracy, and for much larger datasets, than any human could. Note that the paper and pencil has not gone away. Mathematicians still use that old tool! (But in general not for computation.)

With the arrival of Wolfram Alpha in 2009, everyone with at least a smartphone had access to a free cloud tool that can handle any procedural math and is no more difficult to use than Google. (Check it out. It does way more than math, but it’s the math offering that is relevant here. Pick a problem that interests you, edit the example already there, and then hit RETURN. In an instant, not only is your problem solved, the system presents you with a range of information about the solution.)

Given the widespread availability of tools like Wolfram Alpha and the others in the above figure, “doing math” in today’s world means being able to make effective use of those tools to solve problems — either problems within math or, more generally, problems in the everyday world. That answers the question posed in the title of this post.

Next question: How do you (or your students, or your children) learn how to do that? What is involved?

Throughout history, the goal of teaching math has been to provide the next generation with the skills they need to use math effectively in the world they will live and work in.

But as I showed above, that world has changed a lot over the past sixty years.

Unfortunately, a lot of school math is still focused on preparation for the workplace shown in the very first images. It needs to catch up with the rest of the world, as represented by the final image.

This is where BrainQuake comes in. BrainQuake was founded with the overarching goal of helping prepare the next generation to be successful in today’s mathematical world. Independent university studies of our launch product Wuzzit Trouble have shown that we are succeeding in meeting that goal. (See the previous post.)

I’ve written about Wuzzit Trouble in previous blogs, so you can find out about it there. For now, let me leave you with one thought.

Maybe what you have just read in the post has left you thinking that, if computers do all the calculations, all person has to do is push a few keys and sit back while the machine does the math. That’s not an uncommon reaction. If so, let me suggest you download Wuzzit Trouble and play through all 75 puzzles. (Or, on iOS, download the new BrainQuake app and start to play through it as far as you can get.) Both games start gently, so at first you should have little difficulty. But don’t let your guard down. As I explained in the April 11 post to this blog, that seemingly simple puzzle (which is one of three puzzles in the BrainQuake app) has some challenging math under the hood.

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BrainQuake’s Gears puzzle presents daunting-looking algebra problems as a simple-looking puzzle.

As with all the digital math tools on that smorgasbord diagram above, our app will do exactly what you instruct it to do — it executes the procedures you tell it to do. But as you get further into the game, you will discover you are having to work harder and harder to get it to do what is required to solve the puzzle. You’re no longer “sitting back.” You’re probably starting to sweat. “This is hard!” you think. (But by now you’re hooked!)

At that point, what you are experiencing is an example of what it means to do math in a world where tools execute the procedures. Welcome to 21st Century mathematical thinking.

To be continued …

— Keith

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Developing children’s true math proficiency

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