Computer Science and Math

http://deanzchen.com/computer-science-education-and-math

by nkurz • 12 years ago

65 54 12 years ago

14113• 12 years ago

One quote that struck me was this:

For too long, we have taught computer science as an academic discipline (as though all of our students will go on to get PhDs and then become CS faculty members) even though for most of us, our students are overwhelmingly seeking careers in which they apply computer science.

This attitude strikes me as fundamentally wrong - universities are not and should never be trade schools. The point of a university should be to put you in contact with the cutting edge, and allow you to broaden your horizons, not train you for some specific software development job!

hmpc• 12 years ago

Agreed. The problem is not CS, it's the popularisation of CS as a gateway into the wonderful and high-paying world of software ninjas/rockstars/gurus/[hype du jour]. Transforming CS into preparation for software development careers would be akin to redoing engineering schools to teach students machining skills: important, yes, but that's what vocational/trade schools are for. If you're going into college, suck it up and learn your math. For everything else there's the documentation.

Ntrails• 12 years ago

I think there is a balance. Usually the smartest guy in the room still needs to explain how his brilliant insight can be applied to the situation at hand.

scott_s• 12 years ago

It is well accepted that a student in, say, mechanical engineering is far more likely to go on to be a practicing engineer, and not an academic.

nathas2• 12 years ago

Then why do other engineering disciplines go onto practice in industry?

There should be computer science, and then there should be software engineering. One is in the sciences school, one is in an engineering school.

I bet the engineering enrollment would be significantly higher.

acbart• 12 years ago

There should be a College of Computing, offering several different majors: "Computer Science", "Software Engineering", "Human-Computer Interaction", and, damn it I want this even if there's only 3 or so people qualified in the world to teach it, "Computer Science Education". They should also offer a 3-credit general elective named "Computational Thinking" that's comparable with an English 101 course.

snowwrestler• 12 years ago

One can get a university degree in physics, which exposes one to the edge of knowledge and prepares one to proceed toward a Ph.D. and research.

Or, one can get a university degree in engineering, which is basically training in the practical application of physics to solving real problems. A physicist can be a good engineer, but they would probably need an on-the-job apprenticeship first.

I think there is a parallel with the study of computer science vs. software development.

ssdfsdf• 12 years ago

I had the opposite problem. I took my degree because I was hoping to become a CS academic or at least be involved in theoretical work. I ended up with a vocational course. I was/am extremely unhappy.

jroesch• 12 years ago

I've had similar feelings over the course of my education. Many students show up wanting to be taught how to do game dev, web dev, iOS or what ever the "hot" thing is. It is unfortunate because technologies (like Rails or iOS) aren't difficult to learn (in the grand scheme of things). Though a detailed understanding of CS both from the theoretical and engineering standpoint is something that is hard to impart.

tegeek• 12 years ago

I had the opposite experience. I went to study Software Engineering (in 2009) and during first semester I happened to attend a class Programming Paradigms (topic was Functional Programming that day).

The lecture was being delivered by John Hughes, one of the researcher & designer of Haskell. In 1 hour class he blew my mind and I was sold to Computer Science in general and Programming Languages in specific. Next day I registered myself in Introduction of Functional Programming class. The Haskell language broke me from inside out and I failed the course. But I got one of the biggest pleasures of my life. It was about thinking in terms of mathematics, lambda calculus and building a life long love for mathematics. I then studied courses in Logic, compilers, and frontiers of programming language technologies. I started job during the education and couldn't complete my masters degree but that 1 year of studies in CS changed my mind forever. And I'm very thankful for the 1 hour I spend with John Hughes.

stiff3• 12 years ago

Professors or not, this is a quite mediocre discussion. For one, "discrete" and "continuous" mathematics are not disjoint things, and the "discrete" math that you can do without touching "continuous mathematics" is for the most part too simplistic to serve as a model of any real world phenomena, probably including most practically useful algorithms. Almost any "discrete" problem at the real-world level of complexity requires tools from real analysis, and in fact more often than not from complex analysis.

For example, the "Concrete mathematics" book by Knuth that is praised so much is mainly a book about generating functions in the form of power series of a complex variable. Despite the problem being discrete, you only get to get answers by setting up those power series with the discrete coefficients in front of the complex powers, and then obtain answers by differentiation and whatnot.

You do not get any "mathematical thinking" without understanding calculus, this is non-sense. The theory of real variables is not only one of the most ubiquitously practical one, but it is also the model for many other theories. Modern probability theory is largely advanced calculus in a particular setting, for example.

That is not to say that calculus is all there is. You just need to know a whole lot of math to apply it fruitfully in modern complicated settings, and calculus is some of the math that you need to know if you want to work on machine learning or robotics. If you want to work on webapps on the other hand, you do not need almost any math at all, but then you also do not need a university to learn in in the first place.

sanxiyn7• 12 years ago

How does, say, encoding and decoding error-correcting codes require analysis in any way?

I protest that formal power series and formal derivative should not be counted under calculus or continuous math. You can define and prove properties of formal derivatives without any reference to continuum whatsoever.

Math has unity, real numbers are great, but I think people are overselling continuous-discrete connections. Dividing line between continuous math and discrete math is one of the most clear boundary in math, and there are huge amounts of useful discrete math which does not require continuous math.

If you work on machine learning, learn continuous math. On the other hand, if you work on programming language, learn logic and discrete math instead of continuous math. If you work on web applications, well, good luck. :)

ColinWright3• 12 years ago

My PhD is in discrete math and graph theory, and yet I find calculus and probability to be indispensable tools. I devise algorithms that work on images and streams of text, and yet I use power series and vector spaces over the complex numbers to analyze them.

Writing code to implement error-correcting codes doesn't use continuous math, but some error-correcting codes are best understood as working in polynomial rings over finite fields, and many of the ideas and tools are directly linked to similar structures in infinite continuous spaces.

Limiting yourself to learning only about discrete math that does not require any continuous math is like limiting yourself to using only one style of programming, or one language, or one operating system, or one hardware platform. If you don't mind being so limited, fine. If you want to understand things in greater depth and thus give yourself more opportunities and have more tools in your arsenal, learn both.

They support each other in ways you can't imagine without actually learning them.

yodsanklai• 12 years ago

There are tons of areas of CS where "continuous maths" are needed. Not only ML. And even if not needed per se, "continuous maths" can help to build intuition about discrete results.

lambdaphage3• 12 years ago

Here is a non-rhetorical answer to your rhetorical question:

http://www.inference.phy.cam.ac.uk/itila/

See how far you'll get without calculus.

stiff3• 12 years ago

Whether you are interested in coefficients or in the main variable, a complex power series is a complex power series, and you use the theory of complex functions. You don't just take derivatives of generating functions, you use the full package, including for example convergence criteria of infinite series - that's where the power of this approach comes from in the end, that large parts of the powerful apparatus of complex analysis can be used for solving discrete problems. I don't really see any significance in the fact that formal derivatives can be defined as discrete operations, especially this only works if the powers are natural numbers, which is not the case in general.

All this is tangential in the end, there are certainly some specialized areas of CS that require less calculus than others, in general though you encounter calculus a lot in almost all areas of science and engineering, and the university is supposed to give you a broad foundation for your future career, so it should teach calculus to prospective scientists and engineers.

sanxiyn7• 12 years ago

I agree that having calculus and continuous math in CS curriculum is deeply unfortunate. There are lots of more relevant math, such as first-order logic, combinatorics, and information theory. I would take logic class over calculus class any day.

stiff3• 12 years ago

I would love to see how far you can develop combinatorics or information theory without using any calculus. Even in combinatorics "continuous math" has been used for centuries already for difficult problems:

http://en.wikipedia.org/wiki/Analytic_combinatorics

sanxiyn7• 12 years ago

Coding theory is largely developed without any calculus.

Dewie• 12 years ago

That might well be for sufficiently advanced combinatorics or information theory.

But I have done just fine without calculus tools in my computer science degree (not purely computer science or math, though), which I'll be starting on the fifth year of this fall. This is isn't to say that you don't need calculus in computer science; it is to say that you don't seem to need calculus for basic and introductory computer science courses, and things related to that; discrete math has been much more relecant.

In spite of that, my department has calculus as its math course (you can also take stats in parallell), while discrete math is in the second semester. They also advice students that haven't taken the most advanced math in high school against taking the most theoretical/mathematical computer science degree. This was due to the calculus course building on that high school math experience. But then the discrete math course - which is much more directly relevant - does not build much or at all on calculus, and you can do just fine without the most advanced high school math in that course. Maybe the only thing you'll suffer from is having a bit less mathematical maturity at the offset.

But since they start with calculus, they discourage people like me who, while not having had the most advanced calculus (or pre-calculus) in high school from taking a more math-heavy degree, even though I've found to like a lot of the math involved in some subfields of computer science!

kenster07• 12 years ago

The fundamental problem is actually the arbitrary separation of the practical and theoretical. They should be informing one another.

Consider this: many great "theoretical" discoveries in history, including those in CS, were not theoretical at their inception. They were the result of people trying to solve practical problems -- not the product of a bunch of debt-ridden students trying to maximize GPA in an ivory tower. This cannot be overstated.

At the end of the day, the distinction between academic and practical is largely self-imposed, at least partially ego-driven, and probably highly inefficient for society as a whole. If you want "practical" people to engage "theoretical" problems, then make it practical for them.

dkural4• 12 years ago

On the other hand, the Turing machine (a machine capable of universal computation.. ie a computer) was invented as a method of solving a highly theoretical, seemingly useless mathematics problem called the Entscheidungsproblem, in Hilbert's list:

10. Determination of the solvability of a Diophantine equation. Given a Diophantine equation with any number of unknown quantities and with rational integral coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers. The Entscheidungsproblem is solved when we know a procedure that allows for any given logical expression to decide by finitely many operations its validity or satisfiability ... The Entscheidungsproblem must be considered the main problem of mathematical logic.

Computers may seem obvious in hindsight but it was formally defined to solve a theoretical mathematics problem by the mathematician Alan Turing.

_tb• 12 years ago

This topic always makes me bitter, if what you want is write some fancy apps or games, and don't like math, please don't go to CS.

You don't need a college degree for that, you just want to be a software bricklayer-monkey, learn it by yourself.

anyways, CS is full of guys like that. ...I should have gone full math.

thomasahle• 12 years ago

Yup, I think this is also good advice. If you are young, good at math and trying to choose between CS, CS+Math and Math; don't go with CS. You'll find much more challenge doing a Bachelor with other people who actually get maths. The few things you miss not doing full CS are much easier to read up over a holiday than the fundamental concepts of university math.

nathas2• 12 years ago

Except for all of those employers that want a CS degree that are writing web apps.

wisienkas• 12 years ago

For computer science, yes you need to know your math, since it will involve math and expect you to know math.

If you however were going into a 3½ IT programmer degree, it is a whole other story.

Paul_S• 12 years ago

The article keeps using webdevs as an example of programmers who don't need maths. I really don't get that and it reminds me of that humorous programmer hierarchy diagram. Maths will make you better at your job no matter the field and the idea that webdevs are some lesser breed that couldn't benefit from maths because all they do is bang rhythmically on their keyboards is rather patronising and not explained in the article.

ottocoder• 12 years ago

Hmmm, I would agree that some areas of mathematics are more "easily applicable" to programming.

However, mathematics is really all about reasoning and dealing with abstractions. A proof in linear algebra is not much different than proofs in real analysis or topology, in my opinion (the content might be very different, but the methods of reasoning are the same). So I think it really comes down to the answer to two questions: why study CS and why study math?

If we are a little more precise, and split "CS" into "computer science - the study of the theory of computation" and "programming" while also splitting "math" into "mathematics - the art of mathematical reasoning" and "a tool to solve problems" we can cross those two sets (to be a little math-nerdy :) and we get the following tuples:

(computer science, math for reasoning), (computer science, math as a tool), (programming, math for reasoning), (programming, math as a tool)

The problem is that people who have little or no exposure (and some people who have lots of exposure) may not think about/be aware of the difference between each of the tuples. Furthermore, by the time you understand the differences and what your personal preference is, you may have invested too much time into one particular school/program of study/however you're learning to switch.

cessor• 12 years ago

Let's ask this question in a more general fashion...

I believe I share this view on maths and cs. There appears to be a gap between computer science and software-making (writing, coding, engineering, "bricklaying", "applied cs"). The first thing is academic and scientific, the second is a craft. I know educated computer scientists who couldn't program to save their mothers and I know programmers in the industry who never went to CS class and taught themselves, and they keep reinventing the wheel over and over again (poorly). I believe these are different disciplines, although they are related. The academic part surely benefits from understanding maths, but for the "making" part, where people write rails apps, rest apis or websphere portlets, these skills aren't necessary (although I'd say one always benefits from deeper knowledge).

I am making the same experience in another field: Psychology. Most of my costudents wish to become Psychotherapists. Yet the bachelors degree is mostly limited to scientific examinations of the field. There is a gap between "psycho-science" and psychotherapy as well. I believe one can become a very good, practical, experienced and effective therapist without reading hundreds of papers from all different fields such as social, cognitive, work. Instead they could specialize in the things that require treatment such as depression etc. Again, knowing the science behind the practical stuff might help a therapist to gather deeper insights, but aren't always necessary.

The seats to study Psychology in Germany are scarse and limited to students with the very best grades. Most students appear to get very bored by the scientific aspects but have to go through the whole system in order to get a degree, which is required if you wish to work as a therapist. Therefore they keep blocking the spaces for people who are genuinely interested in the scientific aspects.

There - at what point in time may a practical discipline emancipate itself from its scientific origin, thus freeing up resources and gaining through specialization? Should programmers really have to study cs, or whould a new "software crafting" subject be sensible?

dkarapetyan• 12 years ago

You don't need a CS degree to be a programmer. If you want to be a programmer but still need to go to school then learn something that you consider fun just so you can say you have a degree and just program on the side. I don't see how schools are getting in the way of people becoming programmers when the craft of programming is accessible to anyone with an internet connection or even a local library.

dkural4• 12 years ago

What do you mean by programming?

Mathematics professors can't do accounting, because they don't know the relevant regulations etc.

Similarly, A CS professor may not know the popular javascript library of the day etc, but I find it to be a myth and stereotype that educated computer scientists can't program. When pressed, no one is able to give real world examples. Every actual CS professor is a counterexample though. To give a famous one, Donald Knuth is prolific for instance:

http://www-cs-faculty.stanford.edu/~uno/programs.html

henrik_w• 12 years ago

I found "Do software engineers need mathematics?" quite interesting: https://www.maa.org/external_archive/devlin/devlin_10_00.htm...

conformal• 12 years ago

if anyone thinks that a few semesters of introductory mathematics or science courses is "not relevant" for a CS program, they have completely missed the point. this is a fool's complaint, like saying that "because i'm a political science major, i don't need to learn anything about actual science".

in reality, the point of introductory level math and science courses in college is to show you a rather narrow skillset: you are given what amounts to a toolbox with a couple dozen tools and asked to use this small set of tools to solve problems that require application of, at most, 2 or 3 of these tools in succession. a fair analogy is that it is a multistep "put the round peg in the round hole" problem.

if assembling a small toolbox of skills and trying to apply the correct tools in short succession is too much for you, just quit university and code some stupid app. is it really that hard to understand why basic math and science is relevant?

DISCLAIMER: i was a teaching assistant for undergrads for 3 years during undergrad and grad school.

sanxiyn7• 12 years ago

If math is relevant for a CS program mainly as a problem solving training, I really think calculus should be dropped. Logic provides the same problem solving training, and content is actually relevant.

The article is not arguing "math is too much". It is arguing "if math is for problem solving, why not choose math with some relevant math content instead"?

jnbiche• 12 years ago

So, as I think most of us can reasonably agree, there's computer science, and then there's software engineering/software development. Two separate endeavors with a few shared skills.

I'm continuously frustrated by the number of people on the "software engineering" side of the fence who confuse the two, and who demand computer science skills in applicants to develop web apps.

By the same token, it appears here that we have someone on the other side of the fence (computer science in the academy) making the same error. He refers repeatedly to "programming" and its application as a job skill. He then rightly states that few programmers need calculus for their job. But from there he extrapolates that computer scientists don't need calculus for their job, which is not only a huge logical jump, but also makes this age-old error of confusing computer science and "programming" as a job skill.

klunger• 12 years ago

You said basically what I was thinking. I just want to add the importance of _experience_ in differentiating the two.

With more experience, I think you can become a better software engineer/developer. With computer science, it seems to be more of an "either you get it or you don't" situation, rather like higher level math. This is why you will get some really great developers that were just willing to put in the sweat equity, but maybe have a hard time with the only tangentially relevant math requirements.

my3681• 12 years ago

Imagine trying to teach philology, while also teaching them their first language! This is exactly the problem that CS program directors face.

I work as a researcher in the computer science department at the University of Alabama, and I can tell you this is an ongoing conversation/debate not only here, but at many schools across the southeast. Unlike the valley, we have a tremendous shortage of developers, and that pressure ends up being felt at the university level.

To combat this, classes have been opened that expose the students to software engineering principles while not losing the theory that differentiates computer science from programming. This is largely helping, but is still, in many respects, an experiment.

Also at play is the notion that there are people in computer science that fall more into the design disciplines (i.e. HCI, UI, UX people). Right now, we have nothing for these people except double majoring with art/phycology and the occasional HCI class, but this may change soon. There are certainly frontiers in this area yet to be explored!

Either way, it's a big problem worth discussing, so I am happy I see it here.

*Edited for clarity

webmaven• 12 years ago

"Imagine trying to teach philology, while also teaching them their first language!"

It's a bit worse than that. Imagine selecting that first language based on past popularity (Latin), universality (Esperanto), internal consistency (Lojban), precision and concision (Ithkuil), etc.

shabadoop• 12 years ago

Also at play is the notion that there are people in computer science that fall more into the design disciplines (i.e. HCI, UI, UX people). Right now, we have nothing for these people except double majoring with art/phycology and the occasional HCI class, but this may change soon. There are certainly frontiers in this area yet to be explored!

Whenever I read articles about social network-type sites that are doing UI changes it seems like there's pretty extensive use of data analysis based on A/B testing for different changes that are being made. Wouldn't a practitioner still need solid coding skills and a strong base in statistics?

reader5000• 12 years ago

Calculus as typically taught is just a series of symbolic-manipulation rules. It's not taught as math. Ironically CS is probably many students' first exposure to actual math, by which I mean reasoning about (and proving things about) abstract entities.

The requisite calculus could very well be covered in an Intro to CS class itself "hey by the way there are these numerical functions and they have rates of change. Moving on..."

Much more important mathematics for CS is Logic. Linear algebra. Combinatorics. It's true at some point "it all ties together" and there are some combinatorial problems that can be solved with complex analysis.

But to me, calculus as it is generally taught (heres a polynomial calculate its derivative using the symbol-manipulation rules you memorized), is the complete opposite of what CS and actual mathematics is. I think logic or maybe combinatorics are the easiest-to-grasp domains that capture the actual type of reasoning problems CS people encounter.

brudgers• 12 years ago

Learning math doesn't limit anyone's options. Not learning math does. Not providing people with access to the tools needed to actualize their potential is a real injustice, and a computer programmer/engineer/scientist with less understanding of mathematics will be more limited on average than one with more.

And the problem is that second semester college Freshmen aren't really in a position to know what will really matter in the long term. Professors of course aren't either, it's just that personal study and research and the 900 years of institutional experience inherent in the university meme are likely to produce slightly better intuitions.

Can math be taught better? Hell yes, and we will see a significant change as Common Core bubbles upward. We'll also see changes as university education continues to become less elite and the institutions more 'customer' driven to less rigorous requirements. I mean the gist of the article is: standards are too high.

There's some truth to that. Widening the funnel inherently means lowering the standards - business schools used to mostly train accountants, now they produce a lot of marketing degrees. What is really needed is colleges of computing.

Of course those 900 years come back to bite. Someone loses the turf war: Engineering, Mathematics, or Business. Maybe the best place to teach the computing crafts would be the studios of the College of Art, particularly if mathematics is pushed to the periphery.

Why would a computer scientist need calculus? Because what calculus provides is training in replacing equations from one domain with equations of another - it provides a background in high level symbol manipulation. Doing it well requires competence in algebra, and algebra is the cornerstone of analysis of algorithms, and analysis of algorithms is what distinguishes computer science as a subset of all programming related activities.

The question is where does the line get drawn between programmer and software user when we talk about the person at the keyboard - or which side dominates when we are talking about building something with Rails: knowledge of relational algebra or ~rake db:migrate~?

mh_yam• 12 years ago

Programming is not the same thing as computer science...you can get very far in programming without knowing much math, but not in computer science.

ivan_ah• 12 years ago

The hidden assumption behind the "you don't need math" camp in the discussion here is that math is somehow very difficult to learn. Nope. The ideas from high school math have been around for thousands of years. How hard could they be to learn?

And calculus, calculus is just calculations on functions. Not very profound, but not very difficult either. Interestingly, I find the procedures of calculus to be very similar to iterative programming procedures. A limit is kind of like a while loop, a sequence is an iterator, etc... For example

   lim_{n->∞} n    <==>   while(true){  n++; }

Perhaps the best "pitch" for acquiring math knowledge is the power you'll gain for modelling the real world. In that respect physics is useful too, since it also deals with models for the real world (kinematics model, momentum model, energy model, etc.)

dkural4• 12 years ago

Mathematics is the only way I know how to solve technical problems from first principles, and not get stuck when existing solution X doesn't fit problem Y.

If people want to start programming trade schools, that's fine, but that's not "Computer Science", or "Software Engineering" for that matter. Engineering means being able to solve novel problems in an applied manner. You can't do that without mathematics.

Calculus is ultimately symbolic manipulation, as another commenter said. Mathematica / Matlab can do calculus. So should you.

Computer Science and Math

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