The Importance of Mathematics Courses in Computer Science Education

My issue with this article is the author's conflation of his own like and dislikes with what works for students. Many people go into computer science because they want to know how computers work or because they want to know how to program that demo they just saw, or more applied things along these lines. Very few (but definitely some!) go into computer science because they want to know about Turing Machines, or the Curry-Howard isomorphism, etc. Currently CS education serves the latter group very well, while neglecting the former group. This is a shame, because plenty of people who find math frustrating can become great programmers and find great satisfaction in understanding programming and computer science better. Many of the most interesting fields of computer science aren't very mathematical.

When I came to school I hated math, despite being relatively good at it. By now, my interests have changed and I'll graduate with a minor in math and a major in computer science, but largely in spite of my experience with collegiate mathematics classes rather than because of them. The way to get people to appreciate math isn't to force them to take classes on something they don't find interesting. It's to give them opportunities to see how it can be interesting or relevant to them and let them come to it when they're ready.

For myself I became more interested in math when I wanted to learn about graphics programming and machine learning. The first requires linear algebra and the second requires a lot of probability and statistics, which themselves require a solid understanding of calculus. By coming to mathematics this way I was able to see it's relevance to me and my interest in it grew the more I learned. Previously, when I was forced to take the standard multivariable calculus -> linear algebra sequence I did just enough to get by, since I really didn't care about the material. Once I had this small positive exposure to math that was useful to me it was much easier for me to appreciate more abstract math as well. But the important part is that no one made me learn this stuff! Had this been part of the curriculum I would have hated it, but since I was allowed to learn it when I was ready for it I found it very beneficial.

I once heard someone opine that computer science education plays a cruel joke on its students, and this blog post is perpetuating that joke. Computer science is often treated as an offshoot of mathematics when, in reality, you need to do far, far more writing than advanced math. Most students are going to be leaving school and working on professional programming, where they'll largely work with a group of people jointly maintaining systems. And these systems require good writing skills to effectively write documentation for the code, as well as writing the design documentation.

In contrast, you'll rarely need more than algebra, basic statistics, and maybe graph theory most of the time. All the advanced math stuff, such as SVM or Taylor expansions, are generally going to be found in the libraries that you'll use rather than code you'll write yourself. You may need math to understand what they're doing in detail, but with better writers on the documentation, you wouldn't need the advanced math to understand how to use them properly.

First of all, I want to say that this industry could do with a much higher level of mathematical rigor, and that a lot of the systemic problems are owing to the lack of it. But to me, this:

> It was a good discussion, that highlighted the distinction between the currently accepted view of mathematics as primarily about properties and relations, and the pre-nineteenth century view that it was at heart procedural.

...does not follow. Programmers don't view mathematics as procedural because they studied mathematical history, but because the Von Neumann architecture, from which the semantics of almost all programming languages derive, is highly stateful and procedural.

As I see it, this is the core problem that must be solved before many of the tools of mathematics which would really help us today can be meaningfully exploited.

Mathematics is used in education (schools, ..., universities) to weed out people. For those who perform this selection the ability to grok mathematical problems often is the only way to categorize human capacities. A self-fulfilling prophecy.

The University of Waterloo's unique by putting their Computer Science program in our Faculty of Mathematics (the first in North America, the largest in the world).

By studying a significant amount of Math, it definitely improved my logical and reasoning skills at least ten fold.

Computer science is a branch of mathematics. If you mean engineering and programming, then you can get by without any math, but I don't expect you to be writing compilers, operating systems, database engines either without solid CS education.

A practical math topic I think all computer science (and engineering, really) students should take is convex optimization.

Often a problem in engineering or computer science can be phrased as minimizing a cost function over some data and/or variable constraints. Take SVM for example, or sparse signal denoising, or template matching. And far too often, people will apply "advanced state of the art techniques" to solve these problems, e.g. neural networks of some exotic kind, graphical models, etc. And while these are great techniques, they need not be applied to convex problems of a reasonable size.

In the examples I listed, I can almost guarantee that a knowledge of how to phrase the model as a convex optimization problem and throw it into an interior point solver is going to be much more effective than running stochastic gradient descent on some user-defined model with way too many (or too little) parameters.

Because when you say a problem is convex, what you are saying is that it can be solved globally in polynomial time (usually on the order of ~20 least-squares problems of size corresponding to number of variables and constraints). There is a sophisticated convergence theory as well. And if you understand the mathematical theory, you get to use this mature technology to solve exactly a whole class of problems you wouldn't think you could.

It's also worth mentioning that a lot of very hard non-convex problems can be approximated and solved well using these methods, and this is a rapidly growing field of research. Here's an example: http://papers.nips.cc/paper/2979-efficient-sparse-coding-alg...

Resources:

http://stanford.edu/class/ee364a/lectures/examples.pdf http://stanford.edu/class/ee364a/lectures.html https://www.youtube.com/watch?v=McLq1hEq3UY&list=PL7A3953FD9...

The author conflates abstract and immaterial. Computer programs don't have to be abstract, and often are not. And that's precisly what makes them more approachable than mathematics: you manipulate actual structures, see the results in real time and get to see how the actual process is carried out without language ambiguity. In short you get a concrete feeling of things that you can build a mental representation of that's closer to the senses.

Now some software engineer should explain to the author how programming is central to his understanding of mathematics and see how things develop.

"Once you realize that computing is all about constructing, manipulating, and reasoning about abstractions, it becomes clear that an important prerequisite for writing (good) computer programs is an ability to handle abstractions in a precise manner. Now that, as it happens, is something that we humans have been doing successfully for over three thousand years. We call it mathematics. This suggests that learning and doing mathematics might play an important role in educating future computer professionals. "

The conclusion is not good. The logical conclusion would be that if we define mathematics as 'handling abstractions in a precise manner' then programming is part of mathematics. In this case his conclusion becomes:

'learning and doing some classical parts of mathematics might play an important role in educating people who will do this another part of mathematics'

Which might be true to an extent, but this statement is very broad.

It is not a question that some amount of math is helpful. But after a while there are diminishing returns, because you simply learn different skills than what you need.

After having some very basic math knowledge the best strategy is to try to learn a lot of practical 'engineering' from very good books/courses, and only go back to math on-demand.

For example if you want to have some knowledge on machine learning or computer graphics, don't start to learn math for years first and then start learning about machine learning/3d graphics. Start to learn abut the engineering field first (take Andrew Ng's coursera class, or some equivalent in 3D graphics) and only go back to learn some linear algebra or very basic calculus after you see why it will be useful and to what degree you need it (you mostly don't need most of the theorems if you are learning the basics of these engineering fields).

Being an engineer is very different than being a mathematician. If you want to be a good engineer then mostly learn a lot from good engineers (or researchers in the engineering field) and learn some math along the way, but that will be secondary in my opinion. It's good to learn tons of math, but only if you learn even more engineering besides the math (to be a good engineer).

Here's a crazy idea... let's stop treating curriculums like silver bullets and realize that no matter what we do we're not going to succeed with the current system of "hope your innate talents match up with the current education model du jour".

Education needs to be radically tailored to individual students or we're going to continually be trapped in this vicious cycle of "Current education paradigm X is broken, paradigm Y will solve all of our problems!" (where Y ∈ {"more math!", "more programming!", "more fundamentals!", "more application!", ...}).

Yes, highly customized education doesn't scale. Yes, it's very expensive. But sacrificing future generations on the altar of "efficiency" seems highly suspect to me, and the honest fact is if the incentives within the American (sorry non-US people!) education system were properly aligned we would be much farther along. Granted, that's a difficult task, but I didn't get into computers so I could spend my time working on easy problems.

> Some of the computer scientists, however, instinctively interpreted the statement “Let x be such that P(x)” as a variable declaration. This led them to give an “incorrect” answer to a question

"Let x be such that P(x)" is something like a variable declaration. It's the introduction of an argument to a function (think "function(x){ ... }"). I think the real conceptual problem lies in the fact that many mainstream programming languages allow subversion of the type system by allowing "uninitialized" variables, so that every type is automatically inhabited.

There is also the fact that a mathematician may just want some "x such that P(x)" rather than introducing a generic one, which I think the author mentions.

This is a quibble, but there are cases when the physics of the computer cannot be disregarded and must be considered. Two of my favourite examples are cosmic rays flipping bits [1] and printers jamming. [2]

Regardless, I agree with his overall thesis: the machine is just a hunk of atoms with no meaning of its own. It is us who place a meaning on the machinations of the computer. Computer people have come to accept the abstractions of computer science as tangible and consider the abstractions of mathematics intangible. I think both abstractions are comparable. Both end up being interpreted and executed by humans and by computers, and it is merely the overwhelming prevalence of CS abstractions being executed by computers and mathematical abstractions being executed by humans that is causing this divide between the two tribes.

There is much that computer scientists can learn from remembering that the machine is just a represantion of abstractions, just like mathematicians can benefit from remembering that occasionally the physics of the machine are relevant.

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[1] An entirealy reasonable explanation at times,

http://lwn.net/Articles/219983/

[2] http://nedbatchelder.com/blog/200811/print_this_file_your_pr...

I really enjoyed this article and the comments so far. I struggled with math heavily in high school and in college and believed I wasn't fit for programming despite being a 'computer head' since I was a kid.

I never struggled with statistics, I felt like it was the 'easy' math and now work in analytics at a large investment firm and do scripting and analysis with the big databases.

I don't think I'd excel at creating the databases I work with but I understand them and my ability with java and R and SAS and python have helped me be more efficient and a leader in the group.

My understanding of other math has come a long way since university with real tangible examples, I just don't learn from abstract examples I think.

I don't think there is a right answer here, I think there are enough opportunities in the field for there to be two branches of computer science much like there are two sides of business schools. The hardcore finance path and the businesss communication people tease so much.

i started a degree in cs last year and i noticed: most cs people i met @ my university have BIG problems with getting their hands dirty by writing some code. excellent in math, but not able to write some simple for-loops. finishing university without being able to code is kind of poor... but i agree - abstraction is very important to handle complexity.

Guy who started out in math and ended up with a blended math and CS degree here. Their definitely were some parallels, but even in a program where they tried to combine the two, it didn't always mesh. I felt more like upper level math was a springboard for algorithms and proofs, so that by the time I got into algorithms I could say "hey I already have been doing combinatorics proofs very similar to this."

This should not be conflated with hacking though and software development. You can do both of those without any strong background in mathematics. Alot of software development and hacking is just using things people with CS skills built. I do not have to understand quicksort to use the native library sort.

My comment is:

No-comp sci is not a branch of maths. Yes-you do need to know some maths to study it.

Programming != Math.

Logic is important, but that is mostly orthogonal to any course of math instruction I've ever been exposed to. Now, if your goal is to be an academic computer scientist, then yeah, you do have to know some advanced math, just because the frontiers of research are so far out there. If you're aspiring to be a software engineer/developer, then chances are, outside of a few specialties, you may go weeks without using any math that an elementary-schooler couldn't perform.

A developer who is overly mathematically-inclined is actually a small red flag for me, because I have seen too many people who are very good at math and seem to understand theory, but absolutely cannot write code.

Anecdotally, I have seen much better results from developers who have experience in hands-on trades like mechanics, plumbing, carpentry, etc. There's an element of visualizing the entire system and how the components interact that seems to map over well to programming.