[cdsmith]

Basically, the difference is that in a functional programming language, words like "variable" and "function" have the same meaning they do in mathematics.

In C or Python, a "variable" is a storage location in which you can place different values at different times. This is completely different from mathematics, where "x" doesn't equal something different just because you progress to the next step of computation. (But "x" might refer to something different in a different context, such as when evaluating a function with a different argument; that's the sense in which it varies.)

Similarly, in mathematics, there's no such thing as a function that returns something different each time you evaluate it, or that gives one result on Wednesday but a different result on Friday, or that even gives no result at all but causes letters to appear on a nearby computer screen. Yet in C or Python, all of these things are called functions.

At first, it might seem surprising that you can do much programming without the C or Python version of functions and variables. But:

1. You can do more than you think without reassigning variables or using functions that aren't "real" functions in the sense of mathematics, and

2. Of course, in the end, all languages - including functional languages - do have a way to create boxes to store values over time, or to perform actions that make things appear on computer screens. It's just that in a functional language, these things are not the basic bread and butter tools of programming to the extent that they are in C or Python. So you have variables and functions (in the mathematical sense), and you also have other things that act like storage locations and actions, but you only use the latter when you have a real need for them.

That's the general character of the difference. There is, of course, a lot of detail that I am leaving out in a response on a web site.

The reason people were speculating about whether this would work better with people not yet accustomed to programming is that it's often thought that experience in an imperative language (like C or Python) leads you to naturally try to do things with actions and storage locations, even when it's not necessary... and then it can get frustrating and confusing to work in a language where most things are actually not done that way. There's a theory that says if you maybe learned the functional way first, you wouldn't find it as frustrating.

I'm of two minds about that. I spent a year teaching Haskell with this system (I'm the Chris Smith mentioned in the article), and I found that it's partly true that students don't get as frustrated... sometimes. But there are other cases where students naturally fall into trying to do things in imperative ways, despite not having past experience in a programming language. I actually think, computer programming aside, that this is one of the things that trip up kids in math classes. So it's not just programming languages where this comes up: a lot of teachers present math as step-by-step processes, as well. Essentially, they introduce imperative programming without the computer. This is what happens when teachers tell students that parentheses are about what to do "first", instead of about which sub-expressions are meaningful on their own. Partly, I think this is just how the human mind works, too, and it's an inherent tendency that we have to overcome to really "get" algebra and other mathematics in a deeper sense than knowing how to do specific problems. It's still frustrating, then, for kids to adapt to having to describe processes using equations and expressions, which at their core do not list things to do, but instead describe and state facts about relationships between quantities.

I still think it's an important skill to learn. Indeed, I'm a lot more interested in teaching that kind of reasoning than I am about teaching computer programming. That's why I made changes in the new CodeWorld (the site discussed in this article) that are designed to deliberately thwart students who try to think of their programs as sequences of actions, and encourage those who think in terms of compositional expressions.

[p1esk]

Thanks for your response. I've read it several times, but unfortunately it didn't click for me.

>>>Similarly, in mathematics, there's no such thing as a function that returns something different each time you evaluate it, or that gives one result on Wednesday but a different result on Friday

I don't get this example. If the function is f = t, where t is time, then f will give different results on Wed and Fri.

In both math and Python, a function such as Square = x*x will return exactly the same thing as long as x does not change.

>>>This is completely different from mathematics, where "x" doesn't equal something different just because you progress to the next step of computation

When I do math (such as algebra, or calculus), I think of a variable x as a container that can hold any value. This is exactly how I think of a variable in C or Python.

For example, I can build a graph of a parabola in Python. The way my code does it is pretty much the same as if I did it by hand with a pencil: I assign multiple values to x, and find corresponding values for y.

I can't quite pinpoint the source of my confusion...

[cdsmith]

> I don't get this example. If the function is f = t, where t is time, then f will give different results on Wed and Fri.

I assume you meant something like f(t) = t?

If so, then right, if you pass in the current time as a parameter to the function, then the result can depend on it. But in Python, you don't pass in the current time. That's because "function" means something entirely different in Python.

In math, at the simplest possible level [1], a function is a set of ordered pairs, with the property that no two different ordered pairs have the same first element. That's all there is to it. So when I write f(x) = 3x - 5, that's just shorthand for saying that f is the infinite set { (0, -5), (1, -2), (2, 1), (1/2, -7/2), ...}. That's all there is to it. There's no sequences of operations, no process, etc. It's just a set of ordered pairs.

Python's functions, on the other hand, are at their core a sequence of instructions. Some of those instructions (like loops or conditionals) have other sequences of instructions as components. They share an environment, which consists of storage locations called local variables. A Python function is actually a pretty complex thing, when you get down to it.

You could model Python's functions using standard mathematics, if you play some tricks: add hidden parameters, give interpretations to the results in terms of physical effects, etc. But this just tells you that math is powerful enough to model Python functions (and, well, anything else) in terms of its much simpler ideas. It doesn't mean that those ideas are the same as the things in Python that go by the same name.

> For example, I can build a graph of a parabola in Python. The way my code does it is pretty much the same as if I did it by hand with a pencil: I assign multiple values to x, and find corresponding values for y.

Okay, but you're describing the process of drawing the parabola. In math, f(x) = 3x^2 + 2x + 1 isn't a process for drawing a parabola. It is a parabola. So f, here, is itself the set of ordered pairs, which make up a parabola if you treat them as points on the coordinate place. That's the basic difference.

[1] Okay, for the pedants out there, this depends on your choice of foundations. I'm assuming the standard interpretation using ZFC, but a similar point would apply if you prefer a different foundational theory.

[p1esk]

Ok, I think I can now clearly see the difference between a variable or a function in math, and in Python.

However, math and computing are not the same. How can you possibly represent a concept of an infinite set in a computer language? After all, any computer is a Turing machine, and the sequence of instructions is the very foundation of computing (at least that's how I think about it).

It's interesting: I see many replies to my question, however no one has been able to show me a clear example of some code in FP language, side by side with the same code in Python or C, and point out how the FP code is better (in any way).

I learned programming concepts by studying computer architecture, and after I wrote bunch of MIPS assembly, I could clearly see how C makes my life easier. Then I looked at Python, and I could see, for example, how a concept of a class/object makes my life easier compared to C. So now I'd like to see how FP can make my life easier compared to C or Python. I still don't get it.

[cdsmith]

There are various arguments for the benefits of functional programming. But that's really beside the point here.

In this particular case, the point was that I created an educational tool for teaching mathematics. It happens to involve a programming language; but the goal isn't to train computer programmers. It's to teach math. So the language used is one that lets you describe functions and quantities the same way as you would in math, and mostly avoids the need to think about a computer doing some sequence of operations.