Is Calculating Math? Part 2

The history of math bifurcated a few thousand years ago with the rise of cities. Humans have been using records and math to determine seasons long before the historical record shows otherwise farming would not have been possible. There is also evidence of games like Mancala and Nim going back very far into human history. There was no need for humans to play games or develop calendars for tracking the sun and seasons but we did because it was fun and useful.

As cities grew larger and empires formed, math served other purposes like machine/weapon making, business/taxation and other administrative tasks. Over time the prominence of calculation as math took hold and math as recreation became more and more esoteric. It was understandable that this would happen as calculation was necessary to get the necessary precision for engineering.

Flash forward to the early 20th century. Computers are capable of doing large calculations that would have been too time consuming for humans to attempt previously. Yet, there were very few people on Earth who knew how to program these machines. As time went on, new programming languages (e.g. Pascal, Basic) and the greater demand for software applications allowed more people to learn programming. Now we are a highly connected society, heavily reliant upon the Internet and technology to maintain "modern" society.

Wait, isn't this a post about math?

Over the last few thousand years, math has become more and more defined as the ability to calculate accurately and quickly. We know this because our classrooms reflect this in our curriculum. Almost all of our time is devoted to helping students become great at calculating. But wait...we have computers now. Do we even need to know how to calculate?

Gottfried Leibniz (co-discoverer of Calculus in the 17th century) said, "It is unworthy of excellent men to lose hours like slaves in the labor of calculation which could safely be relegated to machines."

In other words, let the machines do all of the hard work so we can use our incredible minds to be creative and inventive.

How do we do that, and how do we help our students to do that as well? Now more than ever, we are using software that we have no idea how it works. Whether it is guarding our bank information or the photos of our vacation, software is no longer on the periphery of our society. I have to agree with many who are calling Programming the new literacy. Clive Thomson says, We're All Coders Now and Douglas Rushkoff goes one step further with the title of his newest book Program or Be Programmed (Youtube interview with author).

Conrad Wolfram ,director of Wolfram Research, has a TED Talk that has gone viral. It is 17 minutes, but if you are a math teacher or interested in curriculum reform it is one of the best ways to spend 17 minutes. Here is the accompanying blog post with transcript.

Math has always been about four steps.
1) Posing the right questions
2) Taking a real world example and transforming it into something that can be described with math.
3) Calculating
4) Confirming that the results apply back to the real world situation and answer our original question.

Yet our math class spends 80% of it's time doing Step 3. As I previously stated, this was because computers and the software simply was not friendly enough for students to use so students had to learn how to calculate by hand.

I also appreciate his statement that

math!= calculating    math is greater than calculating 

Or translated from the computer terminology. Math is not calculating, it is greater than calculating.

Wolfram says that Calculus has typically been taught later because the calculations were too difficult, yet now we can teach it to students earlier because the computer can do the computation for us. The concepts and ideas of Calculus can be understood far earlier than when most students learn it.

I can hear the grumbling already. I have heard math teachers, department heads, and even students say that programming is cheating. If a student does not actually calculate the answer then they do not really understand the math.

Hmmm, well do you understand how Netflix suggests new movies to you? How about your car, if something big went wrong, could you fix it yourself? Yet you can benefit from these technologies. Let me clarify something, just because you don't understand how something works, doesn't mean  you cannot use the tool itself.

Calculus is not the chain rule or L'Hopital's rule. It is seeing patterns of slope and area/differentiation and integration in nature. If students cannot understand math until they understand the calculation, then the only people who understand math might be those with PhDs. Since we know that is ludicrous, lets stop implying to the students that math == calculating.

Let them use their calculators (better yet show them how to program their calculators) and spend that brain energy and class time analyzing graphs and patterns because that is what computers are not very good at doing. I used a programming calculator to make my own applications in High School which saved me a lot of time on homework or tests to derive and solve more complex problems.

Now, you might say, should a student never learn to add and subtract? Should we rely on calculators for everything and teach our students nothing about computation? Of course not, the pendulum shouldn't swing too far to the other side, but we are doing our students a disservice by not allowing them to spend more time developing their problem solving skills, creativity, and lateral thinking (and in my opinion, solving for the y-intercept in the point slope form is useful but not creative).

What are my suggestions for how we should begin to move forward?

1) Begin integrating computer science into the math/science/humanities curriculum. Note I am not saying separate computer science classes. I think we should continue to do those as well, but we need to begin integrating it into our STEM classes and allowing it to free up our time to do more interesting things as well as discover the joy of creating your own software.
  • I encourage you to start with Python as as programming language. It is free and open source so it can be implemented immediately no matter what your situation, be easy enough for students to learn, and still remaining powerful enough to do anything you want.
  • There is now one place on BrokenAirplane where you can find all of the programming resources, like tutorials, and software to get you started (all free). If you have any suggestions let me know but this section will always be updated.
  • You should certainly do something even if you yourself have never programmed until now. Even just a little introduction will whet their appetite for more. Don't feel like you have to jump in and do an elaborate project, just start with what you enjoy and feel comfortable with. Most students in America do not see programming until college and even then only if they are a CS, math, or science major. We must start as young as possible and you are the one to do it.

2) Less is more. We must shorten the curriculum of calculation so we can spend more time learning actual math. Calculating is unnecessary in large part so spend time analyzing data, finding patterns, discussing and writing about math, and so on. If your students begin creating programs and algorithms that do the calculating for you or confirm mathematical proofs, they will learn so much more deeply than any set of problems will ever accomplish.

3) Connect with others. I get my support from friends who teach computer science, fellow science/math  teachers, the Python Education mailing list, websites and people like Maria Droujkova's Natural Math, Allison's Infinigons blog, and many other relationships that you will develop once you start.

4) Realize that you are doing something important. We are the only developed nation that does not require computer science as part of the curriculum. I am not saying that I would want it to be, because then it might become rote and boring. This kind of literacy needs to be shared via motivated educators like you. I thank you and your students will thank you.

If you need any support, I am always available via comment, social networking, or email (phil@Brokenairplane). It is my goal to be a resource for all teachers and students seeking to implement technology and programming in their classroom and I thank you for reading and sharing. Your help makes what we do possible.

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