Is Haskell good at teaching fundamentals of mathematics?

Question

I'm involved in teaching mathematics to classes with requirements such as AS and A level Mathematics specification.

Class books normally provide examples of computer software applied to mathematics tasks, but they are normally based on software applications like mathlab, mathematica or derive.

Even when a real programming language is used, it is normally old fashoned language like Pascal. And this is really sad in my opinion.

Because, Haskell is famous for his feeling with mathematics, and because I'm interested in it as well, I'd like to ask whether Haskell is a good choice as extension exercise method for students of a first year university class. Typical subjects are functions, caclulus, limits, but also linear algebra for example.

I've intermediate knowledge experience in programming (c#, ruby, powershell, javascript, tex) and also have already approached functional programming with XSLT.

Answers will be upvoted responsibly. Verbose answers including also some basic examples of Haskell code applied to mathematics, such as linear algebra (e.g. vectorial space), polynomials, solving linear equations are much appreciated and targeted as the wanted one.

Answer 1

Haskell is growing in popularity among mathematicians. As one blogger put it:

"after my involving myself in the subject, one thing that stands out is the relatively low distance between thought expressed in my ordinary day-to-day mathematical discourse, and thought expressed in Haskell code."

So here's some collected views, that I feel says you can't go terribly wrong taking this approach.

• Discrete math and logic taught via Haskell, in The Haskell Road to Logic, Maths and Programming, Kees Doets and Jan van Eijck

• Also, Discrete Mathematics Using a Computer, Cordelia Hall and John O'Donnell.

• Eleven Reasons to use Haskell as a Mathematician, Dan Piponi argues that there's really no other choice for mathematical thinkers when it comes to programming languages.

• Mikael Johansson is a mathematician at Stanford, who teaches higher level courses in math via Haskell. You can read his blog or more about his courses.

• Math packages on Hackage, including several large libraries.

Finally,

• Collected knowledge on Haskell for mathematicians, on the Haskell wiki.

Answer 2

While I'm a great fan of Haskell for teaching mathematics, it might not be appropriate for all topics at AS/A level.

You'll have no problem with teaching the idea of a function, and linear algebra can be nice in Haskell. Linear algebra is often taught in a very imperative way with in-place updates of arrays. It can all be translated to Haskell, but it might require a little bit of thinking differently.

But it gets tricky when you want to talk about limits and calculus. I've had great success doing calculus in Haskell with a non-conventional approach. But that might not be a great idea when teaching to a fixed syllabus. Alternatively you can play with symbolic differentiation but then you'll start having to build datatypes for expressions which may be tricky for A/AS level. It's not a lot of code to get started, barely a couple of lines, but it might be scary for students. Unlike Derive and Mathematica, you don't get symbolic expressions for free.

Basic combinatorics is easy to explore with the List monad. This might be useful for teaching probability theory.

Haskell is fine for numerical applications of the sort that might appear at A/AS level, eg. applying Simpson's rule or investigating sample means and variances.

Do you do elementary group theory? That's easy to code up and play with in Haskell.

Overall, if you take care around limits I suspect that it's straightforward to do much of A/AS level mathematics with Haskell.

Answer 3

I'm not sure if Haskell is a good choice. Sure, it has a very "mathematical feeling", deals with infinite structures etc. But you can appreciate these features only if you already know the language. Learning Haskell in order to learn mathematics stacks one hard task on top of another. And learning Haskell is hard, even if you already know programming. So it really depends if your audience is bright and curious enough, else you lose half of them because of mathematics, and the other half due to Haskell.

Answer 4

No computer language will teach you mathematics. Only mathematics will teach you mathematics. That is: pen and paper approach (or chalk and blackboard), axioms, theorems and proofs. This is mathematics. No matter how sexy the language is, a program written in it is not mathematics, it's just an application of mathematics. To apply something, you need to learn it first.