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What Math Courses Should We Teach CS Students?
Coryoth asks: "If you're taking computer science then getting as much mathematics as you can is probably a good idea. Ultimately, however, there are only so many math courses you an squeeze in. Given that, what areas of mathematics should we be teaching CS students for maximum benefit? Traditionally university math courses are structured around the needs of the physical sciences and engineering, which means calculus is what gets offered. While a decent calculus course can teach a certain amount of formality in reasoning, wouldn't CS students be better served with a course in mathematical logic and foundations with its greater degree of formal reasoning and obvious connections to fundamental concepts in computer science? Are courses in abstract algebra and graph theory going to be useful to CS students? Should courses in category theory (yes, it applies to computer science) be required of students going on in theoretical computer science? In short — what areas of mathematics are going to be the most useful and most applicable to computer science students? What courses were of the most value to you?"
Whichever branch of maths you follow it encourages logical thought.
init 11 - for when you need that edge.
Well, perhaps because it has something with what I do, but I was surprised computational statistics isn't on that list. Perhaps it's the other way around, statisticians need to learn to program. Regardless there is quite a bit of overlap. This is also one of the few areas remaining where the speed of your program actually matters.
It has been statistically shown that helmets increase the risk of head injury.
Personally, I found a great deal of education in my oenology courses. Well, they weren't really courses. Or at school. Mostly at home with friends.
But to get back to your question, I don't find any of the math that I took applicable to any of the work I do. I don't write 3D software, music synthesizer software, or calculate missile trajectories, so all that math is lost on me. The most applicable mathy CS thing I learned was covered in an hour regarding big O notation. Along with a good course on data structures, that has been the most beneficial thing I can say I learned.
But that is besides the point. What you do in the real world doesn't necessarily have to be relevant when choosing curriculums (curriculi? curricula?)
Give them the works! Everything from Calc through Topo and everything in between. Not all of them are going to grow up to be rocket scientists, but so what? The more knowledge you give these kids the better. I'm not saying you need to wipe them out or flunk them out or even use the math as a weeding tool, just that what you want to do is give them the tools to look at other sciences and not be befuddled.
There's nothing worse than a CS graduate that doesn't have a full grasp of math.
being able to use asymtotic notation for bounds on algorithm running times, and a good basis in proofs in order to prove them once you've come up with something. Also solving recurences, and proving them is invaluable. There are some other things that are very usesfull on a day to day basis, like linear algebra (spesifically coding theroy), geometry, graph theroy, counting and probability (but thats more of an ai thing)
I was a math minor at Penn State, and I decided to concentrate on Stat because that is what interested me, but everyone is different. I would say that if you are interested mostly in "pure"(not pure!=better) CS, then courses like graph theory and combinatorics are probably best. If you are interested in applying your CS degree to problems in engineering and science, then differential equations and numerical analysis is your best bet. If you want to go into the business/actuarial side of things, statistics is obviously best. The most important thing is that you take a lot of math, and hopefully math that actually interests you.
Monstar L
As a current student majoring in both (CS-Math), I've found useful all of my Math courses to CS. I'd say that Analysis, and Measure Theory have helped too. :-) )
However, we should not forget that it's suposed to be a CS major, so I'd recommend as "priority" ones: Calculus (one-many variables), Linear Algebra, Probability, Statistics, Linear Optimization, Non-linear Optimization, Combinatorial Optimization, Numerical Analysis, Computational Complexity, Graph Theory and Information Theory. As subjects of further study I'd choose some Logic, Abstract Algebra, Functional Analysis, Set Theory and Category Theory.
It is important that a CS student could get as much Math as he/she can, because it helps to provide a useful mental framework for thinking, and because helps to get chances of right use of some mathematical tools.
As I see it, many engineering majors are as teaching a student to battle with some weapon, teach some mathematical tecniques related to the subject, and some practical things about the tecnique ("the weapon"). But learning as much as Math as you can, is like to learn how to use a swiss knife: if you know how to use it properly, sky is the limit (you can be a "science McGyver"
I think the first question you need to answer is why you're doing a CS degree in the first place. Personally, I don't see any point in them. It's not the best way to learn to program (how can you really learn Java in a lecture theatre? It just doesn't work. Just get a good reference book, find some good code to copy the syntax from, and work it out as you go along.), if you want to know the maths side of it, do a maths degree (picking courses that are useful for the job you have in mind - you might have to pick a uni accordingly), you'll understand it far better (doing a few courses in maths is much harder than doing lots, because so many parts of maths interlink). If you want to be a Systems Administrator, or something, then I can't see why you would need a degree at all, it's experience that counts in those kinds of jobs.
Can anyone name a job for which a CS degree is the best qualification?
Wow. I do have an undergraduate degree in mathematics, and I'm not sure it covered everything you described there. You'd certainly be lucky to get things like Galois theory taught routinely at undergrad level these days, at least here in the UK.
If you disagree, post your argument. (-1, Overrated) isn't your personal censorship tool for views you don't like.
"Statistics" (2 semesters at least) and "experimental design". "Modeling and simulation" is closely related, but is somewhat covered if you take the stats and experimental design courses.
Here's why...
When starting on my PhD research, I pretty naively thought I'd just write a network simulator to try out my idea and to compare its performance to other network protocols. That would be fairly acceptable in today's CS climate, but STUPID.
People using simulators face a number of questions that they often don't ask, and therefore make their conclusions nearly meaningless. Are there specific hypotheses they're trying to test? How do they know they've performed enough simulation runs to draw conclusions at an acceptable confidence level? Exactly what is the distribution over which the inputs are randomized, and why was distribution chosen? To what extent is the model even validated (ok, this is more of a Modeling and Simulation issue than a stats issue)?
Psych and biology majors have been forced to rigorously answer these questions for a long time. We, the supposedly "mathematically superior" CS majors, have often ignored these details as though they're irrelevant. But if left un-tackled, we can produce crap research whose conclusions have little clear connection to reality. These is even true for when we can afford to do real-world tests and thus are less at the mercy of simulation model inaccuracies. How many real-world tests do we perform before we draw our conclusions? How do we randomize the inputs?
Much of today's network-related research sucks. Not because the ideas being generated are bad, but because the analysis of the new ideas and their comparison to the performance of pre-existing ideas is crap. Without taking stats and experimental design courses, even the reviewers of these papers don't realize that those weaknesses exist.
If you want your network research to be meaningful, test your ideas with meaningful experiments and analysis.
For CS students, the fundamentals of discrete math and set theory are probably some of the most important. More focus on these in a CS context would be helpful to people such as myself who find math more interesting when it has a purpose. I did not mind the physics and statistical math courses. I could see the point behind them clearly. But doing math just for the sake of doing math never seem to have a purpose.
I only look human.
My mother is a halfling and my dad is an ogre, so that makes me an Ogreling
Proofs. Proofs. Proofs.
Algebra, geometry, calc, who cares. It's the Proofs that make math apply to Comp Sci. Having obscure formulas memorized means squat. But being able to look at a problem and break it down into the most simple of building blocks, that is a critical skill.
-Rick
"Most people in the U.S. wouldn't know they live in a tyrannical state if it walked up and grabbed their junk." - MyFirs
For students who just want to get a job as a programmer. . .
I'd recommend a different major.
As with all the hard and applied sciences the undergraduate course is really just foundation building and the course of study should reflect that. CS is a life study, programming is a trade.
Even with that it's going to take you about 10 years to become a decent programmer; and 15 to become a decent CS; there's really no way around that. Why is everyone so fixated on becoming mediocre in a hurry? Learn the shit you need to learn, in a well ordered manner, instead of trying to cram half of everything into the arbitrary four years.
KFG
I'm a Computer Science major at RIT - our math sequence is Calculus -> Discrete -> Probability and Statistics. Discrete and Probability of course being the most relevant for computing.