Is Learning Detailed Math Still Necessary in the Age of Computers?

[djosey]

Well I've just installed Mathematica on my laptop, and have been playing with it for a while. And this made me realize one thing: lots and lots of things that we learn in math lectures can be done very easily with a computer! derivation, integration, and probably much much more...

This has left me wondering, why do we also spend a lot of time to learn all this stuff? I can off course see the usefulness in learning what an integral is for example, it helps you make sense of all problems and phenomenons which use them. But why do we learn it in such details? All the different methods for integrating, many many exercises...

Now for all i know that may be a provocative question, let me make it clear that I'm not trying to be lazy or anything, I'm just curious. I enjoy doing this kind of maths and calculus, I'm just surprised that I'm spending so much time on that. Is it because computers can't solve some things? Do people working in math fields still use those rules?

 

[VanOosten]

i used to think these programs to do math were genius but they cannot do all problems and often give answers that can be simplified a lot if done by hand.

 

[VanOosten]

i don't know if mathematica is any better then Maple (math program which i use) but the integral of 1/x*e^-x dx easily stumped it, but i can do that integral by hand

 

[Integral]

Calculators do arithmetic wonderfully, should we quit teaching that as well?

 

[djosey]

Calculators do arithmetic wonderfully, should we quit teaching that as well?

Humans can harvest grain by hand perfectly well, should we stop having machines do that for us?

More seriously, you misunderstand me: i never said one should not learn integration, i explicitly said i fully support learning what an integral is, I'm merely asking why one learns it in such detailed and time-consuming way. VanOosten said it's because programs aren't always very good at it, which is a very good reason. As for arithmetics, clearly it's usage is so widespread that it would be more than foolish to stop teaching it

 

[p1ayaone1]

If we stopped teaching calculus today, who would write the new versions of Maple tomorrow?

 

[VanOosten]

If we stopped teaching calculus today, who would write the new versions of Maple tomorrow?

its true i have come to expect at least one new version a year

 

[chiro]

Well I've just installed Mathematica on my laptop, and have been playing with it for a while. And this made me realize one thing: lots and lots of things that we learn in math lectures can be done very easily with a computer! derivation, integration, and probably much much more...

This has left me wondering, why do we also spend a lot of time to learn all this stuff? I can off course see the usefulness in learning what an integral is for example, it helps you make sense of all problems and phenomenons which use them. But why do we learn it in such details? All the different methods for integrating, many many exercises...

Now for all i know that may be a provocative question, let me make it clear that I'm not trying to be lazy or anything, I'm just curious. I enjoy doing this kind of maths and calculus, I'm just surprised that I'm spending so much time on that. Is it because computers can't solve some things? Do people working in math fields still use those rules?

I think the crux of the argument is that at least for the foreseeable future, humans need to make the final decision and are responsible for the decision that they make.

Here are a few examples:

A medical specialist comes in looks at a patient. He might have a computer that has an expert system installed. It gives its diagnosis given the various data that it has obtained. However in the end the doctor needs to make a decision and sometimes no matter how advanced that expert system is, he may find the diagnosis to be flawed based on some nugget of wisdom or knowledge that the system has missed.

An engineer designs a bridge, the computer software package which does most of the grunt work does most of the calculating. Despite all of the work that the computer does, the engineer still has to make sure that the computer has the right assumptions and a recheck might be necessary. The engineer like the doctor has the responsibility of giving the "a-ok" sign-off, not the computer.

If people do not understand what is going on, there will be chaos. We can program computational devices to figure all this stuff out, but in all jobs, its the human that gives the a-ok for the results to be deemed correct. If you don't understand what's going on, there is going to be the potential for something to screw up, and screw up drastically.

Its kind of like say a normal person getting sick and googling the remedies on the internet and calling themselves a doctor. There is a reason why doctors have to spend years in medical school instead of months learning how to use an expert system.

 

[CRGreathouse]

A medical specialist comes in looks at a patient. He might have a computer that has an expert system installed. It gives its diagnosis given the various data that it has obtained. However in the end the doctor needs to make a decision and sometimes no matter how advanced that expert system is, he may find the diagnosis to be flawed based on some nugget of wisdom or knowledge that the system has missed.

Or the doctor might think (s)he has something that the expert system missed, but the expert system was right and the doctor was wrong. I hear that studies done on the subject have actually shown *better* results from expert systems than from a panel of experienced doctors (though I admit I have not read the studies themselves).

 

[p1ayaone1]

Or the doctor might think (s)he has something that the expert system missed, but the expert system was right and the doctor was wrong. I hear that studies done on the subject have actually shown *better* results from expert systems than from a panel of experienced doctors (though I admit I have not read the studies themselves).

Maybe, but someone still needs to build the expert system

 

[chiro]

Or the doctor might think (s)he has something that the expert system missed, but the expert system was right and the doctor was wrong. I hear that studies done on the subject have actually shown *better* results from expert systems than from a panel of experienced doctors (though I admit I have not read the studies themselves).

The point I was trying to make is that ultimately humans have the final word and responsibility of their decisions. If a doctor used an expert system to diagnose the patient and they misdiagnosed the patient and they died, they can't blame a computer for the result: they have the final responsibility and say on the decision.

Even if expert systems make better decisions than doctors some of the time, a court of law (and common sense) won't allow a computer to give the diagnosis.

 

[CRGreathouse]

Even if expert systems make better decisions than doctors some of the time, a court of law (and common sense) won't allow a computer to give the diagnosis.

I understand the courts, but why would common sense not allow an expert system to give a diagnosis if it was shown to be better than doctors?

 

[chiro]

I understand the courts, but why would common sense not allow an expert system to give a diagnosis if it was shown to be better than doctors?

I actually agree that expert systems have their place, I'm not arguing that. My point is that a human has to be the authority figure who makes the final decision.

In the case that expert systems are doing better than experienced doctors, then based on that it would make sense to have wider adoption of such systems. This will help in the case where the doctor uses the system, gets a suggestion complete with the logic behind that decision, which influences the doctor to change their mind. In this case expert systems have helped save lives.

However situations present themselves where the expert system may overlook something. An expert system may present a solution that seems logical, but may actually be wrong. In this case the doctor "sticks to their guns". If the doctor is right, they may have saved the life of a patient where the system may have not done so. If the doctor is wrong, the medical community will take note and update their literature.

I think if you asked anyone, no matter what kind of educational background, they would probably agree that an expert system will at least for the time being never replace a humans judgement.

 

[CRGreathouse]

I actually agree that expert systems have their place, I'm not arguing that. My point is that a human has to be the authority figure who makes the final decision.

In the case that expert systems are doing better than experienced doctors, then based on that it would make sense to have wider adoption of such systems. This will help in the case where the doctor uses the system, gets a suggestion complete with the logic behind that decision, which influences the doctor to change their mind. In this case expert systems have helped save lives.

However situations present themselves where the expert system may overlook something. An expert system may present a solution that seems logical, but may actually be wrong. In this case the doctor "sticks to their guns". If the doctor is right, they may have saved the life of a patient where the system may have not done so. If the doctor is wrong, the medical community will take note and update their literature.

If I recall correctly the study tested three options: the expert system alone, the board of doctors advised by the expert system, and the board of doctors alone. The expert system outperformed the board advised by the expert system.

Now I can't at present speak to the applicability of the study (at the very least, I'll have to read it in full first) but supposing for a moment that it was as I represented it, I don't think it would be advisable to risk lives just to keep 'the humans' in charge.

OK, source time: my understanding of the general issue comes from Ian Ayres, "Super Crunchers", chapter 4. I'm not immediately sure which of the papers they cite is the appropriate one; if someone is interested I can try to find out.

I think if you asked anyone, no matter what kind of educational background, they would probably agree that an expert system will at least for the time being never replace a humans judgement.

I suppose that's a microcosm of the entire argument, isn't it.

 

[Office_Shredder]

It's also as much about training yourself to get ready for the next level. You would be screwed trying to study non-linear PDEs if you never even learned how to solve linear ODEs because computers can solve those

 

[G037H3]

is the OP an engineer?

 

[AlephZero]

The reason you think Mathematica is so useful is BECAUSE you have learned to do some what it does by hand. If you gave Mathematica to somebody who knew nothing about math beyond how to count using their fingers, they would find it useless.

To quote Hamming (if you don't know who he was, see Google), "The purpose of computing is insight, not numbers". The same is true for manipulating symbols, as well as number crunching.

 

[dav2008]

I think the crux of the argument is that at least for the foreseeable future, humans need to make the final decision and are responsible for the decision that they make.

Here are a few examples:

A medical specialist comes in looks at a patient. He might have a computer that has an expert system installed. It gives its diagnosis given the various data that it has obtained. However in the end the doctor needs to make a decision and sometimes no matter how advanced that expert system is, he may find the diagnosis to be flawed based on some nugget of wisdom or knowledge that the system has missed.

But what if the medical specialist has developed his own model to determine the total area under a curve?

http://care.diabetesjournals.org/content/17/2/152.abstract

In Tai's Model, the total area under a curve is computed by dividing the area under the curve between two designated values on the X-axis (abscissas) into small segments (rectangles and triangles) whose areas can be accurately calculated from their respective geometrical formulas. The total sum of these individual areas thus represents the total area under the curve.

 

[jasonRF]

Humans can harvest grain by hand perfectly well, should we stop having machines do that for us?

More seriously, you misunderstand me: i never said one should not learn integration, i explicitly said i fully support learning what an integral is, I'm merely asking why one learns it in such detailed and time-consuming way. VanOosten said it's because programs aren't always very good at it, which is a very good reason. As for arithmetics, clearly it's usage is so widespread that it would be more than foolish to stop teaching it

A practical answer:

I am an engineer so usually end up needing numerical evaluations of whatever analysis I am doing. My experience is that "nice" closed form solutions of complicated integrals usually run faster on a computer than a brute-force numerical approach (eg. Mathematica's numerical integration is great and pretty much always works but is way too slow to be useful for something I do millions of times ...). However, I have frequently found the answers Mathematica gives for something like an integral are useless, even though they are correct. There are many ways to represent any given function. Giving me a complicated hypergeometric function with wacky arguments does not help here. I usually end up doing manipulations by hand in order to represent as some collection of "simpler" functions, or to get to an integral that is more amenable to a "nice" answer from mathematica or to a direct numerical approach.

I guess this falls into the "programs aren't always good at it" category, but is also a "programs can't read your mind and know what you want to do with the result" kind of question. You may want a result represented in terms of functions you already have numerical routines to compute, for example. Mathematica cannot do that for you.

So even in the applied world of engineering, I find the ability to do integrals - way beyond just typing into mathematica - to be very useful.

Plus, figuring out an integral is like solving a puzzle, which can be fun if you want it to be!

jason

 

[slider142]

As far as I know, Mathematica cannot yet create or solve a loosely defined problem; it can only manipulate or calculate what you ask it to (and in the newest version, aggregate all information and calculational engines it has related to a search term).
The fact that the calculations you ask Mathematica to do are the ones which solve the problem you are working on (assuming you're doing something more interesting than rote busywork for mechanical calculation experience) is decided by you, not the computer.
In most mathematical work, you will usually quickly search through the relevant field's work to find specific theorems and build structures of your own that depend on your ability to note applicability and generality.
It is not usual that you will need to do rote calculations; for applied and the complicated problems in pure math, these are for computers. However you will quickly get a feel for the numerical side of the problem by your past experience in doing busywork and rote calculations in class. It's a bit like muscle memory in learning to play an instrument.
As the previous poster noted, the ability to quickly approximate integrals and other quantities without resorting blindly to a computer will allow you to quickly change tactics and survey a range of possible solutions and failed paths.

 

[jasonRF]

As the previous poster noted, the ability to quickly approximate integrals and other quantities without resorting blindly to a computer will allow you to quickly change tactics and survey a range of possible solutions and failed paths.

Yes, and I must say that on several occasions the leading term of the asymptotic expansion of the integral provided all the insight I needed. Much more than an uninterpretable closed-form solution or a bunch of numerics would. Of course, to learn how to do that you need to know how to do the stuff in basic calculus, followed by "applied" complex variables.

Yes, I think every engineer should spend many hours learning how to do integrals!

jason

 

[Dr Lots-o'watts]

...why do we also spend a lot of time to learn all this stuff?...Is it because computers can't solve some things? Do people working in math fields still use those rules?

It's because :

1. Those computer are built by people who know how to do this stuff. Do you want to be left behind and depend of those few programmers?

2. Not every school can or wants to spend their meager budget on software licenses.

3. Schools must be independent from specific (software) corporations. The mathematica algorithms to solve integrals is proprietary, the pen and paper techniques are not.

4. Kids must be occupied by something that is agreed upon by the majority of adults/parents. Child labor is not allowed in many countries, running around in the streets is dangerous. All that's left that's within the average taxpayer's budget is math and languages.

5. Learning this stuff is actually useful. When you're familiar with it, you know where you're going with the software. You'll have more fun with it.

 

[jsgruszynski]

Well I've just installed Mathematica on my laptop, and have been playing with it for a while. And this made me realize one thing: lots and lots of things that we learn in math lectures can be done very easily with a computer! derivation, integration, and probably much much more...

This has left me wondering, why do we also spend a lot of time to learn all this stuff? I can off course see the usefulness in learning what an integral is for example, it helps you make sense of all problems and phenomenons which use them. But why do we learn it in such details? All the different methods for integrating, many many exercises...

Now for all i know that may be a provocative question, let me make it clear that I'm not trying to be lazy or anything, I'm just curious. I enjoy doing this kind of maths and calculus, I'm just surprised that I'm spending so much time on that. Is it because computers can't solve some things? Do people working in math fields still use those rules?

Without learning the guts of math you can't really formulate problems well enough to use these tools. For the same reason that "Why learn this? I can always just Google it!" is Epic Fail.

 

[djosey]

Well I'm pretty much convinced! Some very good arguments here, thanks for the replies. I think the crux of the problem is that integration is a very complex thing to do, not like, i don't know, calculating the fraction of two real numbers. I was convinced the first time someone said that.

edit: there are some arguments i still disagree with, for example the idea that we shouldn't depend on the few programmes who can integrate or the "who we build the next generation of mathematica" argument. Who here doesn't depend on the relatively modest portion of humanity with the knowledge to recycle water, or make paper... Reasonnable specialisation is for the most part a good thing.

 

[CRGreathouse]

But what if the medical specialist has developed his own model to determine the total area under a curve?

http://care.diabetesjournals.org/content/17/2/152.abstract

Wow. It's terrifying to see how many articles cite that.

I like how (1) the author names the method after himself, (2) the method is compared to the other methods known by the author, which are terrible, and (3) the trapezoid method is so much better...

Of course reason #0 is that the method is many hundreds of years old... anyone know just how old?

 

[PhysDrew]

Of course reason #0 is that the method is many hundreds of years old... anyone know just how old?

I'll guess about 200 years

 

[ManishR]

Now for all i know that may be a provocative question, let me make it clear that I'm not trying to be lazy or anything, I'm just curious. I enjoy doing this kind of maths and calculus, I'm just surprised that I'm spending so much time on that. Is it because computers can't solve some things? Do people working in math fields still use those rules?

the real reason,
lot of people will lose their jobs & recession will hit us again. simple as that.

the other reason being, governments are lazy.

if there is only addition in math & it is proved that no more discoveries can be made from addition then there is absolutely no reason for teaching addition to a kid.

but since there is no proof that no more discovery can be made, then math (or any other subject) should be taught because one might find an interesting thing and end up making an amazing discovery.

i firmly believe that there is absolutely no difference between a computer and a human brain except
> computer compute faster then a human brain so every process weather it is addition or amazing discovery like relativity or calculus will be computed faster then human brain. (however modern computer can't figure out relativity itself (in human language), because we don't know how to ask them.)
> computer's "imagination" depends upon lesser variables (the user, power off, water on motherboard (in this case we (not computer scientists) tend to ignore what computer is imagining)) etc then a human's "imagination" (i.e. your childhood, education, punch of Ramboo, and another billion variables).

so until we find a "functional" super brain equivalent to infinite brains of einstein,
"we need you people"

 

[Amanheis]

Stephen Wolfram's brother Conrad makes some pretty valid points arguing against everyone in this thread.



Short excerpt: Nowadays is all about problem solving, which usually involves the following steps:

1) Identify the problem
2) Make it so that you can calculate the solution
3) Do the calculation
4) Interpret and apply the results.

Step 3) is the ONLY step that can be done by a computer, and a computer can do it much better, faster and more accurate than any human, yet we spend most of the students time on teaching exactly that.

Also, this Blog entry is pretty interesting, too:
http://blog.wolfram.com/2010/09/27/do-computers-dumb-down-math-education/

And no, I don't work for Wolfram ;)

 

[daviddoria]

Haha Manheis, as I was reading this thread I was thinking "how has no one posted the Worlfram TED talk??". I don't work for Wolfram either, but he is definitely right. Frankly, I am quite shocked that people here do not support this concept.

The people who will build the next-gen Maple style software should of course be trained in these "how to do the calculation" techniques, but there is no reason for every high school student (1/1000000000 of whom will write Maple style software) to learn it.

 

[Jamma]

What about the following very simple answer:

What if we don't know the values about what we are integrating, i.e. what if we are using an integral in a proof but we need the role of the integral to stay general. What if in fact we are trying to prove something very general about the integral itself.

A simple example would be Greene's Theorem, but I am sure that you need to really understand how to deal with integrals left in their general form a lot of the time in pure and applied mathematics to prove some nice theorems involving them.

Having a computer which can merely approximate numerically what the answer to a given integral is will be of no help here.

 

[daviddoria]

This is for mathematics students. Certainly if you study mathematics as your "field" then you should know these things. But engineers need not know them, and certainly people studying/practicing non-technical disciplines need not study them in high school.

 

[djosey]

Great video Amanheis, thanks! That's exactly what i was thinking about!

 

[jasonRF]

This is for mathematics students. Certainly if you study mathematics as your "field" then you should know these things. But engineers need not know them, and certainly people studying/practicing non-technical disciplines need not study them in high school.

I must say that as an engineer I disagree, as I noted in my previous posts. Could I be "dumb" and simply push everything through a black box such as Mathematica to do everything for me? Yes, I suppose I could. But as I noted above, in my experience as a practicing engineer the complete generality of Mathematica also makes is SLOW - certainly too slow to get numerical answers for things I need to do millions or more times. I need to be smarter than that, and sometimes that means doing analytical work. Perhaps I just don't know Mathematica well enough, but I don't know how to tell it "give me yet another representation of that function that I think I may be able to evaluate numerically much faster than you can with minimal numerics/programming/research work on my part ..."
or "give me yet another representation that gives me more insight without doing millions of numerical evaluations at all ..."

This doesn't mean that I don't agree with much of the blog. More realistic models of reality should be taught in college, and that requires numerical solutions of things. This could indeed make courses like freshman physics both more interesting and more instructive, and Mathematica or something like it could help. I wish my instructors had done such a thing. I also think calculus courses should introduce basic numerics, and even have a couple of simple "labs" using something like Mathematica to provide the students with an introduction to the resource. When I took it we did indeed spend a little time on numerics (as the prof said - most integrals cannot be solved analytically), but not the "labs", and I suspect that isn't uncommon. I guess the few weeks we apparently "wasted" on techniques of integration could have replaced with a half-hour intro to mathematica and the remainder spent learning something else I missed in school ... but for me the cost would have been high.

So while I agree my education, and presumably the education of others, could have been better by using these software tools more effectively (granted they were more primitive in my day - Macsyma was what I usually used!) I do not think skipping how to do serious analytical work on your own is justified. Perhaps one day the computers/software will be advanced enough, but in my experience they are not close to there yet, even if Wolfram (who of course wants everyone to buy Mathematica) says otherwise!

jason

 

[einsteinoid]

You're not going to get full use out of computational software without a full understanding of the computations.
Just as one won't get much use out of a Russian thesaurus unless they speak Russian.
Ti ponemaesh?

 

[jasonRF]

This is for mathematics students. Certainly if you study mathematics as your "field" then you should know these things. But engineers need not know them, and certainly people studying/practicing non-technical disciplines need not study them in high school.

One further comment. As an engineer, I have no interest in hiring engineers that cannot do analytical work, since that would mean I would have to do it all for them. I already have plenty of work without having to do part of theirs as well. No Thanks!

jason