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What Math do You Use?
e_lehman asks: "I've been associated with MIT's introductory 'Mathematics for Computer Science' course for a number of years. The course has emphasized different topics in different years: logical foundations, proofs, probability, combinatorics, etc. But this is at the whim of instructors. What mathematical topics should we be teaching to budding computer scientists? What mathematics do you actually use or need, working in the computer industry? Here are some candidates: boolean logic, graphs, number theory, combinatorics, proofs, set theory, relations and functions, approximation methods, solving recurrences, generating functions, analysis of state machines, asymptotic analysis, and addition of small integers."
A solid grounding in maths is always good. When pushed far enough it caus you to develop different thinking patterns that address problems in new ways. Which are always useful for any problems.
If a CS student is to ever make something useful out of his maths, you should be teaching him analysis:
... and so on.
- Logic
- How to set up proofs
- Full induction
- Rows (pardon me if this is not the correct english word, something like Sum[v=0,eternity] of 1/v!).
- Functions (differential, integral)
- Differential Equations
- Function theory
Basically anything that will teach them what maths is really about and give them the ability to get along with maths in a scientific way on their own.
&& aemula C. ab stirpe interiit
I believe that Computer Science != Software Engineering. The job I actually ended up in is a programming job, which means that not all the CS I had is relevant. But that is good - they aren't meant to be the same thing.
For instance, in CS, to calculate the time efficiency of some algorithm, some really hairy analysis may be needed to decide it's actually O(n log log n). This is important as a part of CS. What is important for CS is not defined by what is important for getting a programming job. The science is important for its own sake, as a branch of math.
That said, logic, sets, graphs, relations, discrete algebra - the "discrete stuff" so to say, is both what I like best and what I think is central to CS.
I believe posters are recognized by their sig. So I made one.
I know it is supposed to be an academic study, but this is just going to be too demanding. Where could you even find people qualified to teach such exotic stuff?
I think it's less about how much math you will use and more about how math changes the way you approach problems. Exposure to math teaches people to approach problems systematically and logically. You may think that you're never going to use matrices and eigenvalues in The Real World, but math classes are worth more than mere knowledge.
My dingo ate your honor student.
Statistics. I strongly wish I had a deeper grasp of statistics in almost everything I do. Finite state automata and directed acyclic graphs may be all the rage in compsci, but if I compreheneded confidence intervals and probability distributions I'd do much better work than I currently do.
Their analysis, NFA to DFA transformation and the applicability of FSMs to most stateful problems
(protocols, lexical analysis, communications, etc.)
Use of graph representations for the analysis of many of the same problems mentioned above
Methods for modeling continuous phenomena discretly.
(Euler Integration, FEA, Meshing, etc.)
All these thing consistently make my job easier, more interesting and, continue to provide a level of insight to tricky problems (especially the first two) that exceeds simple 'programming'.
The two most under appreciated courses of all time in Computer Science education have to be the Theory of Computation (FSMs etc) and the Discrete Mathematics (Graphs, Numerical Analysis, etc). An alarming number of new graduates cannot phathom how to apply this stuff. It's powerful and once you start using it, you'll always see things a little bit more ''completely''.
I am a student at the University of Minnesota. I have been working with faculity from the physics and computer science departments on and off over the last 4 years. Most of the math we have been taught is not really used in computer science. You have some analysis of run times and memory requirements, but so far nothing that required higher mathmatics. Differental equations have been the most useful to me. They are used in all kinds of different science and analysis of situations (fiscal models are big on them). From what I have seen undergrad CS is an alright intro to the topic, but you do not get to any of the really interesting stuff(iw you need higher level math) until you hit grad school or start working with researchers. Then it is really dependent on who you are working with. I am currently taking a grad course titled "Intro to Parallel Computing" and the most mathmatics we have used so far is stats to look at an omega network... not really exciting or informative.Good class and learning lots of stuff, but not quite there yet. I really think that most of the mathmatics that the students should learn are dependent on the topics they are being taught. Discrete math is very important, but was not very challenging to most of the students that took it. Mostly the mathmatics that are needed for a paticular topic should be taught in the class the topics are presented in. Think along the lines of self contained courses. I know diversity of ones education is important, but sitting through discrete math was a wasted semester. Sets, unions, intersections, and the pigion hole principel are not that hard to understand... and I can not remember when "proving" something was correct was actually useful. (I read a story about how this researcher proved this very short peice of code correct. Three other researchers found MANY errors for how small the code was.) Proving code may have its place but it is not something 99.9% of us are going to be doing with any frequency. What I would have rather seen is software enginering concepts introduced eailer and the chance to actually USE and explore the ideas presented there. /me is still bitter about the poorly run software eng course he took. It was so bad they are totally changing the course.
Anyways:
-diff eqs are good
-software eng good
-discrete math good but we DONT need a whole semester
-get involved with your profs it is one of the best thing you can do!
-The UMN is a great cs school. They listen to their students, have great instructors (we have many that have worked in industry and know their stuff) and is generaly a good place to be (ignoring the code, you are cuddled up next to your sparc station right?)
Ben
Set theory is extremely important in "the real world", especially for developers who write in any kind of query language (and really, what developer hasn't had to do that at some point, think SQL, EJB-QL, etc).
If every developer had a formal background in set theory, I wouldn't see quite so many bad SELECT statements, misuse of joins, etc. Bad queries can be a huge bottleneck in a DB drive application.
If I had to pick something else, how about a brief (but mathematical) introduction to cryptography? Public key schemes are easy to learn, and very interesting to the average computer science student. We covered this in one of my first math courses in University, and it interested me enough that I went back to take the dedicated crypto course in later years. The knowledge I gained there has been very useful in settig up servers, evaluating products with crypto, etc.
After 40 years using math for EVERYTHING in my day to day life, including my professional life, I'm amazed at what is most useful.
The fact is that, after all this time, the most useful and frequently used math isn't algebra, or geometry or calculous, or statistics. The amazing thing is that most of the math is on my fingers.
My experiences from work...
There are all kinds of embedded applications that require a lot more knowledge about resonances, loop gain, bode diagrams, filters, etc. than I got in CS.
Any sort of feedback loop can oscillate or ring if it's designed improperly, and there's a whole science to designing them properly that I wish I'd studied.
Any sort of modem (includeing cable, DSL, and radio) requires FFTs and filters. Cepstral analysis seems to be incredibly cool given how often it comes up when breaking audio watermarking schemes, but I don't know much about it.
Group theory comes up a lot in error-correcting codes.
3-D graphics requires a solid grounding in linear algebra and trigonometry. (And games these days involves feedback loops of NPCs responding to players and each other - see above.)
Trigonometry, and its applications to linear equations, are important to anyone doing 3D graphics work. It's surprising how many people can't visualize sine, cosine and tangent functions nowadays.
In grade school I had one CS teacher want me to help a fellow student animate a sprite-based ferris wheel on the screen; I used a simple loop from 0 to pi/2 in small steps, with sin and cos functions to place 4 cars on the wheel. The teacher was amazed at how smoothly the wheel ran, and didn't seem to understand how it all worked...
A long time ago, in a galaxy far, far away ... I was admitted to the CS/SE honors program at the Univ of Houston. Had some of it paid for, but finances were not the issue - all Freshman honors students had a mandatory 6 hour 'kick your ass' English / History class designed to weed out the wannabes.
... but it was about -learning- those maths. Face it, Differential Equations, the higher layers of Statistics, the higher thought planes of Calculus - those are all some friggin'eh difficult concepts to grasp. DiffEQ is the Star Trek of math - all I remember about that was taking a very evil math equation and saying 'Ok, we morph this through the time-space continuum and in an alternate universe we use their math to solve an equivalent equation and then we bring it back to our universe by applying the Heisenburg principle ... voila - Solved!'
It worked. Neither English nor History were my strong suits and I wasn't about to subject myself to that kind of hammering when I felt that as a software engineer I should be focusing on other things. I didn't enroll at UofH and went to another University in Texas on a full ride scholarship. No 'freshman destroying' 6 hour English class required.
Perhaps all the math (more hours of math than the math major, as I recall) wasn't about -knowing- diff'eq, four semesters of Calculus, eight discrete maths, three semesters of statistics, etc
Think about it - in the event you actually do work for NASA or the NSA or whatever, do you want your group of peers made up of 6 week wonders with just enough MFC training to develop C code on Windows, or do you want a guy that can learn four chapters of advanced calculus in three weeks? Well I generally learned it overnight because I neglected the 'daily attendance' part of a few courses, but you catch my drift.
Right knowing that somebody got through college with their BS/CS under the Department of Engineering tells me that they do not know FEA (finite element analysis) but it also tells me that they can learn it in a few weeks. Ditto pretty much any language, or OS, or platform. It tells me that they can optimize a boolean expression in their head and if they don't know the answer to something they know where to find out.
Hard math in college is good - push em until they break. Then push em some more.
Glonoinha the MebiByte Slayer
The phycist's approach has changed over the last few decades with the improvements of the personal computer. While there used to be "the theorists" and "the experimentalists", there is now a new group: "the computationalists".
Phycisists are usually well trained in applied mathematics, calculus, applied PDE's, group theory, linear algebra... you name it, but that's not OUR main problem: we suffer from not being great coders. As a computationalist, I can tell you that many a phycisist will apply brute force and rely on the power of computers to get them the answer by the end of the week, rather than writing elegant, efficient code that might give you the answer in an hour! I suppose this is a little off-topic, but keep on focusing on execution time and algorithms, and spread the good word of computer science to other branches of science. Physics, Chemistry and many of the physical sciences rely on computation, and make huge Beowulf clusters and MPI to do their calculations for them. Perhaps they could save a buck or two, OR get results faster by learning a healthy dose of solid computer science.
Secondly, I just heard a quote from an unknown source: "There's the right answer and the wrong answer. In modelling, there's the third option: the irrelevant answer". Focus on teaching people what is relevant, and what should be solved by other means than sitting behind a computer.
The most interesting course I've ever done is Quantum Computing.
A very interesting but very difficult course.
I've been a generalist for the last 21 years. I've worked multiple indiustries, been everything from architect to team lead, grunt, QA and CM Nazi.
Most of what I use is boolean algabra. Actually a logic class in the philosophy dept was a killer and helped a lot.
Anything with word problems. The old junior HS if a train going east.... Or the first semester Calc differential equasions. Anything where a problem is stated and a solution has to be found. anything where you have to get from point A to Point B and show how you got there.
Set theory is needed.
The Math learned from "The Dragon" book on compilers was good.
Being able to do hex/octal/binary math is required with debugging sessions and TCP/IP programming.
Where a lack of math failed me: