Mathematics software/advanced calculators and the learning of mathematics.

[Mr.Watson]

in this day and age, when there are software like Maple and Mathematica and all the fancy graphical calculators, I often wonder how much time we waste even in higher learning when we use old paper and pen-methods to do our math. I mean that why do we learn and use step by step methods to take for example derivatives and integrals, when we could just solve them with calculators/software.

What even the point of learning to use quadratic equation when you can always just solve it with calculator? Does it really give you any more understanding about the math beyond?
Atleast it really is a waste of time. I mean think how much time for example physics student could use to really learning to understand physics, if they wouldn't waste their time by mechanically crunching differential equations step by step, when they could just get the values out of computer/calculator.

Or is there something important in this that I miss? Because sometimes I see even professional physicist solving calculus equations step by step when they could easily get the value of x out of the equations with calculator in no time?

 

[Diffy]

in this day and age, when there are software like Maple and Mathematica and all the fancy graphical calculators, I often wonder how much time we waste even in higher learning when we use old paper and pen-methods to do our math. I mean that why do we learn and use step by step methods to take for example derivatives and integrals, when we could just solve them with calculators/software.

What even the point of learning to use quadratic equation when you can always just solve it with calculator? Does it really give you any more understanding about the math beyond?
Atleast it really is a waste of time. I mean think how much time for example physics student could use to really learning to understand physics, if they wouldn't waste their time by mechanically crunching differential equations step by step, when they could just get the values out of computer/calculator.

Or is there something important in this that I miss? Because sometimes I see even professional physicist solving calculus equations step by step when they could easily get the value of x out of the equations with calculator in no time?

It helps to understand what you are actually doing.

Sure MATLAB can derive functions. But if you are taking a calculus class, you should learn calculus, no? Part of calculus is learning how to take the derivative of a function.

Do you honestly think that if people were just shown how to push a button on a calculator they would have mastery of a subject? That is a pretty simple and naive view.

Why do we have to learn history if it is all on wikipedia? we can just look stuff up.

Why do we have to learn Chemistry if machines can mix chemicals for us?

Why do we need to learn to write English if software will just produce text from my voice.

Why do I need to learn a foreign language if I can have my phone translate for me?

Do you honestly not see why someone should learn to do something before they just rely on technology?

 

[Mr.Watson]

Sure, I can see how it can help students how are fast time learning calculus, but what wonders me more is that I can see even seasoned professionals like physics professors doing this. Or is there something useful about solving the equations step by step instead of just using software? Or why is it that even physics professors solve derivatives with step by step instead of using calculators/matlab? Or am I overestimating MATLAB capacity to perform hardcore-calculation? Because after all, solving step by step is so time consuming, there sure has to be something good about it, because even professional do so . Or do they? :D

Somehow I just can't see value of wasting time by step by step approach, after you have learned basics. Offcourse it is sometimes faster to do it by hand, but not always.

 

[Diffy]

Sure, I can see how it can help students how are fast time learning calculus, but what wonders me more is that I can see even seasoned professionals like physics professors doing this. Or is there something useful about solving the equations step by step instead of just using software? Or why is it that even physics professors solve derivatives with step by step instead of using calculators/matlab? Or am I overestimating MATLAB capacity to perform hardcore-calculation? Because after all, solving step by step is so time consuming, there sure has to be something good about it, because even professional do so . Or do they? :D

Somehow I just can't see value of wasting time by step by step approach, after you have learned basics. Offcourse it is sometimes faster to do it by hand, but not always.

Well then that makes a little more sense.

But honestly I don't know. There is a lot of things Matlab and software can solve. But not everyone knows how to use Matlab. I know a lot of professors at my old university didn't know too much about computers let alone programing in a sofisticated environment such as Matlab.

Additionally there could be other reasons for calculating things by hand. Perhaps a sense of satisfaction. Perhaps as a means of proof or a sanity check. Perhaps a matter of accuracy where Computer's can lose accuracy in operations like division and are limited by hardware.

Maybe it is interesting or they can gain insight into the problem by seeing each step, who knows I guess it depends on the problem.

I would say, if you see someone working something out on pen and paper, when you know it can be done otherwise, maybe ask them why. I would bet they would answer something to the effect "Well, this is the way in which I know how to solve the problem".

 

[Mr.Watson]

http://www.ted.com/talks/conrad_wolfram_teaching_kids_real_math_with_computers.html

I think that this TED-talk by Conrad Wolfram sums up pretty nicely what I am trying to articulate here. Mathematics is so much more than calculating, so why are we wasting so much time learning to calculate things that computers can do, when we could use this time doing real mathematics?

 

[Number Nine]

http://www.ted.com/talks/conrad_wolfram_teaching_kids_real_math_with_computers.html

I think that this TED-talk by Conrad Wolfram sums up pretty nicely what I am trying to articulate here. Mathematics is so much more than calculating, so why are we wasting so much time learning to calculate things that computers can do, when we could use this time doing real mathematics?

Not being able to look at a function and see its derivative is a hindrance to doing real mathematics. How are you supposed to maintain a train of thought throughout a problem if you have to go back and forth to the computer for every simple calculation? For that matter, how is anyone supposed to program the computer to do those calculations? And how is anyone supposed to verify that the calculations are correct?

 

[coolul007]

It's the journey, not the destination.

 

[BloodyFrozen]

It's the journey, not the destination.

Exactly!

 

[Andy Resnick]

in this day and age, when there are software like Maple and Mathematica and all the fancy graphical calculators, I often wonder how much time we waste even in higher learning when we use old paper and pen-methods to do our math. <snip>

It's a tricky balance- for example, a lot of physics labs have been 'outsourced' to computer simulations, especially including data analysis, and I think that reduces the value of labs. OTOH, as you point out, there are lots of tools available and it's important to learn how to use the tools, for a variety of reasons.

Something to think about is the overall learning objective: for example, a business student should learn what compounded interest means and should perform a few detailed calculations in school, but that student will *always* use a calculator. A science/engineering undergrad should learn the algorithms needed to compute and simplify various expressions, and a math undergrad has specialized, additional, needs.

In practical terms, knowing what the math software is doing is important because the user can do troubleshooting and consistency checks.

 

[brimacki]

in this day and age, when there are software like Maple and Mathematica and all the fancy graphical calculators, I often wonder how much time we waste even in higher learning when we use old paper and pen-methods to do our math. I mean that why do we learn and use step by step methods to take for example derivatives and integrals, when we could just solve them with calculators/software.

What even the point of learning to use quadratic equation when you can always just solve it with calculator? Does it really give you any more understanding about the math beyond?
Atleast it really is a waste of time. I mean think how much time for example physics student could use to really learning to understand physics, if they wouldn't waste their time by mechanically crunching differential equations step by step, when they could just get the values out of computer/calculator.

Or is there something important in this that I miss? Because sometimes I see even professional physicist solving calculus equations step by step when they could easily get the value of x out of the equations with calculator in no time?

What happens when you run into a problem that a computer can't solve? What happens when you want to solve a problem no other human has ever solved?

For example, a computer cannot tell you what x^2 + y^2 when x and y are extremely large numbers is (a formula that is extremely useful in calculating astrophysical distances). Without a knowledge of the math behind that formula we couldn't tell a computer to do (1 + (y/x)^2)*x^2 instead.

 

[lurflurf]

What happens when you run into a problem that a computer can't solve? What happens when you want to solve a problem no other human has ever solved?

For example, a computer cannot tell you what x^2 + y^2 when x and y are extremely large numbers is (a formula that is extremely useful in calculating astrophysical distances). Without a knowledge of the math behind that formula we couldn't tell a computer to do (1 + (y/x)^2)*x^2 instead.

What if you run into a problem a computer can solve? Computers can solve problems no other human has ever solved like the four color theorem. The x^2 example is beyond silly. If your calculations take 2^24 hours by hand, supposed accuracy is not very helpful. We can learn from William Shanks that hand calculations are error prone.

 

[JonDrew]

It's the journey, not the destination.

This is a very arrogant view of why someone should learn to do mathematics by hand. So, so, many people are completely ignorant of mathematics because everyone is stuck on the meritocratic system which modern mathematics was build by.

Todays education-system is designed for the 1900's when education was something societies relied much less on. I think the world would be a much better place if we started realizing that to study scientific and engineering disciplines, actually understanding the calculations behind mathematics isn't necessary for 90% of the task these disciplines entail. I do believe some level of understanding is necessary but I don't know what the merger of digital and physical mathematics should be.

Schools should be a place to replace ignorance with valuable skills not a weed out process. I personally, would still enjoy studying the math behind the sciences. But I also think it forces to much of our society into a scientifically ignorant abyss when we demand everybody learn the calculations.

The journey is to costly, its like forcing people to walk to there destination when you have a bus ready to take them. You can walk if you want, but don't expect everyone else to do the same.

 

[Devils]

Todays education-system is designed for the 1900's when education was something societies relied much less on.

See

As somebody with an education degree, the talk by Conrad Wolfram is condescending. He appears to have no idea how math or other subjects are taught these days.

Some modern pedagogy paradigms
- 'authenticity' - students solve problems that are derived from real world examples
http://www.learner.org/workshops/socialstudies/pdf/session6/6.AuthenticInstruction.pdf
- 'mathematics literacy' using a multiliteracy paradigm. This includes explicitly teaching problem solving & explicitly teaching the language of math. It doesn't mean being able to blindly regurgitate textbooks in exams.
- 'ICT integration' - most schools in the western world have mandatory computers for high school students or are rapidly moving that way. Classroom teaching means teachers & students are immersed in a digital world & are digital natives.

Here in Australia Grade 10 students get industry-recognised certifications & grade 11 students can take university courses (while still at school). This seems well beyond Conrad Wolfram's understanding of education.

 

[Ackbach]

Computers make fast, very accurate mistakes. They are a tool, and not always a very good one, at that. They are very stupid, because they do what you tell them to do (most of the time), not what you want them to do.

I once had my HP-50g calculator give me a numerical approximation to a rather complicated expression that had cubes in it. It gave me the wrong answer! But when I first asked it to simplify algebraically, and then approximate, it gave me the right answer. But how did I know it gave me the wrong answer? There's no shortcut to having checks on the machine other than thoroughly knowing what you're doing.

 

[Sankaku]

The answer is both. You should be able to do it by hand and you should be able to use a machine. To be able to do only one makes you half a scientist. I should qualify that, though. There are some things computers can't do, and some things humans can't do (in one lifetime). The question is, do you know which is which?

I think the world would be a much better place if we started realizing that to study scientific and engineering disciplines, actually understanding the calculations behind mathematics isn't necessary for 90% of the task these disciplines entail.

Hmm. So the world needs a bunch of clueless drones who can punch numbers into a screen but have no idea what they are actually doing? I hope that anyone with a real passion to understand the world doesn't settle for this.

 

[lurflurf]

^No one understands everything, that does not make them a mindless drone. It devalues people who study something for their entire life to presume you can understand it. That is why we have specialists. Even a pocket calculator can out perform any human, understanding how one works is difficult and not particularly useful.

 

[JonDrew]

The answer is both So the world needs a bunch of clueless drones who can punch numbers into a screen but have no idea what they are actually doing? I hope that anyone with a real passion to understand the world doesn't settle for this.

Again, complete ignorance to the topic at hand. The world will always have "clueless drones", as you call them. But there are many, many people who are not clueless drones who have no understanding of mathematics and will never take the time to invest in an education of mathematics in its entirety. Teaching people how to do complex calculations with a calculating software whilst showing them what those calculations mean has the potential to allow many more people to even begin pondering an understanding about the universe which was never accessible to them before because it was hidden behind a gigantic block of difficult equations (for humans).

If you ask todays average high school student what a differential is used for, they'd have no idea. With calculating softwares these days there is no reason why a typical high school student can't extensively learn how calculus is used even if they don't have the skills to solve the equations by hand.

And a bunch of "clueless drones" punching numbers all day is far better than just a bunch of "clueless drones" not punching numbers all day. Because in the end, those are our options for the people who choose to be such "clueless drones".

 

[micromass]

Again, complete ignorance to the topic at hand. The world will always have "clueless drones", as you call them. But there are many, many people who are not clueless drones who have no understanding of mathematics and will never take the time to invest in an education of mathematics in its entirety. Teaching people how to do complex calculations with a calculating software whilst showing them what those calculations mean has the potential to allow many more people to even begin pondering an understanding about the universe which was never accessible to them before because it was hidden behind a gigantic block of difficult equations (for humans).

But understanding the universe IS understanding the math. You can't do physics without knowing the math. If you don't know what the equations mean or represent, then how can you possibly understand the universe?? If you can't derive equations, then you're not understanding the universe.

 

[WannabeNewton]

You seem to be under the naive impression that higher math and physics is all calculations. I find it silly that this position is even being argued. Crack open an proper math or physics text and you'll see that most of the problems can't even be done by software applications. Until you can write a program that can solve all the excercises in a graduate differential topology, analysis, GR, QFT, classical mechanics text. etc., your argument has no real ground. I can't even conceive of the idea of schools not teaching rigorous math and physics courses just because software applications can solve calculations.

 

[JonDrew]

If you can't derive equations, then you're not understanding the universe.

That is what I accuse of being ridiculous. Solving the equations can give you a more in depth understanding, but solving the equations is not some sort of prerequisite to understanding the universe, especially now.

You seem to be under the naive impression that higher math and physics is all calculations. I find it silly that this position is even being argued. Crack open an proper math or physics text and you'll see that most of the problems can't even be done by software applications. Until you can write a program that can solve all the excercises in a graduate differential topology, analysis, GR, QFT, classical mechanics text. etc., your argument has no real ground. I can't even conceive of the idea of schools not teaching rigorous math and physics courses just because software applications can solve calculations.

Nobody is offering the argument that schools shouldn't teach rigorous math. They are however saying that more students could learn high maths with the help of software applications.

"Until you can write a program that can solve all the excercises in a graduate differential topology, analysis, GR, QFT, classical mechanics text." this is far beyond the scope of this conversation.

 

[micromass]

That is what I accuse of being ridiculous. Solving the equations can give you a more in depth understanding, but solving the equations is not some sort of prerequisite to understanding the universe, especially now.

So what do you propose? Just accept the equations as god-given?? This is not how science works. Scientists actually need to derive equations. And computers are rarely helpful with very advanced equations.

Nobody is offering the argument that schools shouldn't teach rigorous math. They are however saying that more students could learn high maths with the help of software applications.

Please show me how one would learn analysis, abstract algebra or topology with the help of software.

 

[MarneMath]

I have a question for Jon. Do you believe that people who wish to be mathematician/physicst should go through your course? Or do you believe a course that you seem in favor of should exist for people who hold an interest in science but do not wish to pursue it on the professional level?

 

[JonDrew]

So what do you propose? Just accept the equations as god-given?? This is not how science works. Scientists actually need to derive equations. And computers are rarely helpful with very advanced equations.



Please show me how one would learn analysis, abstract algebra or topology with the help of software.

Most people are not scientist and never will be. And If you think that most of todays high schools aren't teaching equations as god-given then you are in for a reality shock.

All of the math courses you just listed are not even required of most engineering or physics bachelor degree programs. So I will fail to validate you in your request.

 

[WannabeNewton]

I think there is a bit of a misunderstanding here as to what demographic and what education level this concept of making software the method of teaching the core of the curriculum is aimed at. Maybe if that is cleared up then it would be easier to proceed without argumentation.

 

[JonDrew]

I have a question for Jon. Do you believe that people who wish to be mathematician/physicst should go through your course? Or do you believe a course that you seem in favor of should exist for people who hold an interest in science but do not wish to pursue it on the professional level?

I think that this sort of material should be taught to everyone in an effort to raise scientific literacy. While a young student learns how to derive pre-algebra equations they could also learn to manipulate calculus on Mathematica. That is the sort of vision I see.

This way a graduating high school student could have a lot of exposer to maths involving Differential Equations, Gradients, optimizations, etc. before even choosing an engineer/science discipline and therefore boosting the scientific literacy rate.

 

[MarneMath]

It appears the problem for most people is that they believe you want people who want to be professional scientist to go through those classes, where what a future scientist needs is a solid foundation in the 'hard stuff' where computers are not so helpful.

If you envision younger high school age kids being exposed to some more complex ideas and these ideas demostrated via computers, then that's a bit less extremely than what they seem to think you mean.

 

[symbolipoint]

The answer is both. You should be able to do it by hand and you should be able to use a machine. To be able to do only one makes you half a scientist. I should qualify that, though. There are some things computers can't do, and some things humans can't do (in one lifetime). The question is, do you know which is which?



Hmm. So the world needs a bunch of clueless drones who can punch numbers into a screen but have no idea what they are actually doing? I hope that anyone with a real passion to understand the world doesn't settle for this.

Sankaku seems to have the best thoughts on the topic. The discussion has otherwise become a mess.

 

[lurflurf]

The usefulness of computers in algebra, topology, analysis, GR, QFT, classical mechanics, and electromagnetism among others is beyond obvious. Besides being ridiculously error prone and slow hand calculations do not even encompass the same methods as computers use superior methods that are not practical to perform by hand. Sankaku thoughts are particularly contradictory, even though Sankaku acknowledged the existence of problems computers can solve that humans cannot, using them makes one a clueless half scientist drone without understanding. Apparently computing square roots by hand is more worthwhile than computational topology.

 

[635nm]

The problem with learning mathematics using software, is computers will always do what you ask, which may not always be what you want. If you don't understand the math, how can you be sure you are asking the computer the solve the right thing?

 

[lurflurf]

So your pen and paper does what you want? It is a common concern that a fool with a computer might make an error, as if there is no risk of mistake in hand calculation. Also that the computer user will not understand what they are doing and by taking it away she will know exactly what to do. Let's assume our computer user knows exactly what to do and has spent a thousand hour gaining a deep understanding of the relevant subject and ten minutes doing hundreds of pages of preliminary calculations correctly with the help of the computer. Our pen and paper user has spend the thousand hours doing the same preliminary calculations by inferior and less accurate methods while making 4378 mistakes and ten minutes attempting to learn the relevant subject. In this realistic scenario and with other things being equal (prior to the year in isolation performing the thousand hours and ten minutes of preparation the two were clones with the exact same genetics, knowledge, and experiences) who is more likely to correctly solve the problem first?

 

[MathWarrior]

From an educational standpoint it would make sense to do it both the old fashioned way and with computers. This way you are double exposed to everything and you will hopefully understand it better.

Why should one way be preferred over the other? If the goal is education, you should be learning as much as possible not the bare minimum.

 

[Sankaku]

Sankaku thoughts are particularly contradictory, even though Sankaku acknowledged the existence of problems computers can solve that humans cannot, using them makes one a clueless half scientist drone without understanding. Apparently computing square roots by hand is more worthwhile than computational topology.

Please, could you explain this statement? I cannot tell what you are criticizing here.

 

[lurflurf]

^You say

You should be able to do it by hand and you should be able to use a machine. To be able to do only one makes you half a scientist.
So a scientist who cannot do their work without a computer (most of them) is a half scientist which I take it is bad. However many in this thread would praise a technophobe scientist who never uses a computer which you would not you go on to say
There are some things computers can't do, and some things humans can't do (in one lifetime).
Which seems to be in conflict with the previous statement as we must know how to do by hand that which can only be done by computer. Finally
So the world needs a bunch of clueless drones (who do not actually understanding the calculations behind mathematics for 90% of the task in their disciplines) who can punch numbers into a screen but have no idea what they are actually doing?
Yes the world needs such people they can do their work perfectly well without understanding how their roots, linear algebra, Fourier transforms, multiplication, statistical analysis, group theory, or whatever else are actually calculated. The experts in those fields know. Like wise they do not understand most of their own field much less any related field they might use results from. This is also not a problem. This is why we have specialists, no one person can understand everything.

 

[lurflurf]

From an educational standpoint it would make sense to do it both the old fashioned way and with computers. This way you are double exposed to everything and you will hopefully understand it better.

Why should one way be preferred over the other? If the goal is education, you should be learning as much as possible not the bare minimum.

It is about learning the best tool for the job. The reason learning useless hand calculation is not better than not is because in the time one spends learning them cannot be used to learn something useful. Pilot do not spend a portion of their training trying to fly with there arms and carpenters do not practice pounding in nails by hand. I suspect most people here agree and do not themselves do square roots and trigonometry by hand, but they think doing integrals and algebra by hand is important.

Until you can write a program that can solve all the excercises in a graduate differential topology, analysis, GR, QFT, classical mechanics text. etc., your argument has no real ground.

This is wrong in so many ways.
-All? You would not be impressed by a program that solved 99% of the exercises?
-Are all these exercises solvable by humans? Many books intentionally or unintentionally include very difficult or impossible exercises.
-Does the program need work alone? What about a program that is very helpful to a human in doing the exercises?
-Probably the exercises you are thinking of are meant to be done by humans. What about a books whose exercises are meant to be done with a computer? You must realize that by limiting exercises to those that can be done by the average reader without a computer in a short time greatly limits the learning. All those books have simple contrived problems. Consider a spherical cow.

 

[Sankaku]

^You say

You should be able to do it by hand and you should be able to use a machine. To be able to do only one makes you half a scientist.

So a scientist who cannot do their work without a computer (most of them) is a half scientist which I take it is bad.

This is a straw-man argument. I was trying to make the point that it is not an either/or situation. This thread was veering off toward "pencil vs machine" arguments.

However many in this thread would praise a technophobe scientist who never uses a computer which you would not you go on to say

There are some things computers can't do, and some things humans can't do (in one lifetime).

Which seems to be in conflict with the previous statement as we must know how to do by hand that which can only be done by computer.

We must know how to do it by hand, yes. This doesn't mean that all the work needs to be done by hand. It means you understand the theory of what you are doing on the computer.

Finally

So the world needs a bunch of clueless drones who can punch numbers into a screen but have no idea what they are actually doing?

Yes the world needs such people they can do their work perfectly well without understanding how their roots, linear algebra, Fourier transforms, multiplication, statistical analysis, group theory, or whatever else are actually calculated. The experts in those fields know. Like wise they do not understand most of their own field much less any related field they might use results from. This is also not a problem. This is why we have specialists, no one person can understand everything.

That is fine, if your aspirations are to be a semi-competent technician. I was assuming that people here wish to become scientists who seek to understand the world. Of course, as you say, no one person can understand everything. However, if you are interested in discovery, you should be more curious about how the tools you use actually work.

Let me ask you for your recommendations for mathematical software and what parts of mathematics you find them particularly useful for. I am happy to tell you what I like, but I think you can go first as the advocate of machine solutions to "99% of the exercises" in my differential topology textbook.

 

[lurflurf]

The whole thread is a pencil vs machine argument. What do you mean by "understand the theory of what you are doing on the computer. " You object to those "who do not actually understanding the calculations behind mathematics". If a computer tells me
sin(1)~0.84147098480789650665250232163029899962256306079837106567275170999
I certainly do not actually understanding the calculations. I also do not know how to do it by hand. I cannot imagine that knowing how would be helpful. I think doing lots of tedious useless hand calculations will result in less discovery.
Many software packages are helpful for different things
Latex
For typsetting
mathematica
For integrals and such
Magma
for algebra
Matlab
for something sometime
Mathics
http://www.mathics.net/
Which is quite basic
Sage
which include many helpful packages including
R
for statistics
PARI/GP
for arithmetic
LinBox
for linear algebra
Maxima
for symbolics

 

[MarneMath]

You really don't know how to do sin(1) by hand? That's a bit odd. Interestingly enough, knowing how a calculator obtains answer is helpful knowledge if for no other reason than a roundoff error. It was a simple roundoff error that allowed the patriot system to fail and get 28 people killed. I'll hate to think some engineer thought to himself, "round off errors! Who cares abou that!"

I think much of Sankaku is a simple, you be aware of the limitation of the technology you us. Graphs can be misleading, computers are known to have bugs, and sometimes certain tools shouldn't be used for certain problems. To be able to figure out work around for these problems, sometimes it requires a good grasp on theory or the 'old fashion way' to solve a problem to see if the computer is giving you a reasonable solution.

 

[MathWarrior]

One of main reasons you need to know how to do it by hand, is someone has to write the stuff that performs the calculations. A computer can do the work, but whose making the computer work? And making sure its doing it right?

Also, floating point computer calculations come with some error due to floating-point arithmetic.

For example a computer cannot contain the exact value of PI in purely a floating point value. This goes for all irrational numbers.

A computer also does not understand what is meant by infinity. Since it is a finite machine. So all applications in physics that use the concept of infinity in their design will not be able to be represented in a machine. This results in further error.

Minor errors might not matter much to some people but if you scale your calculations to something larger, it now has a much larger impact.

There is also the problem that many of the theorems that are from mathematics and physics, are made through proofs. How would a computer prove something when it has no understanding of what is meant by basic language.

 

[lurflurf]

No one is arguing round off errors should be very large, that is ridiculous. I know how to compute five or ten digits of sin(1), but not hundreds. No it is not the same thing. Even if I was world champion at computing sine by hand a computer would easily beat me. Doing four by four problems in a high school algebra book is not the same as doing million by million problems. Again with the old someone did a calculation on a computer one time and made an error. The person still made the error not the computer. People make errors in hand calculations all the time. If a calculation is impossible for a human that is an accuracy of 0%. In fact the patriot system was the result of a huge blunder, not any inherent problems with computers. That is the lesson, don't make huge blunders. The patriot system did not kill anyone, a scud did. The patriot failed to save them, which is not the same thing. The patriot system should have been able to save them, but it would be impossible without computers. A pad of paper and pen would be entirely useless, that is technology with limitations worth being aware of.

 

[MarneMath]

I think you're missing the point entirely in some vague attempt to standfast by your position. I don't think anyone here would argue you should be able to do ALL problems without a computer or that a computer cannot help you solve a lot of problems. This isn't a computer all the way or by hand all the way argument. In fact, there shouldn't be argument. You should know how things work. You should be aware that computer programs are designed by people and you should aware that oversights can occur and be able to see if the answer is reasonable. You should be aware of inherent limits of programs.

You seem to believe that people here think you should just do everything by hand. That's not the case at all, the argument is more along the lines that you should know it can be done by hand to some extent and be aware of the technique if for no other reason than to check reasonability. Is there any harm in this? Does learning a simple technique really eat up so much of your time that you rather just be ignorant of it?

If so, then you can carry on your merry well and just believe whatever answer a computer gives if you believe that's all you need to be successful. I, as someone who does most if not all my works on computers (a lot of variable order makov models), would like to be aware of possible errors in the programming and limits of it. But, that might just be me and my need to give a better answer to my boss than simply "uh this is what the computer gave me."

*As a side note about the scud, you can get bogged down in terms and phrases if you wish. I don't care to play that game. Clearly the computer didnt make an 'error' but blind trust in the machine allowed it to fail. The fact of the matter is that it was a simple error. Something that could've and should've been stopped earlier. The computer did what it was programmed to do, and sadly it was programmed poorly. Thus, this only supports my points, you should be aware of how a program works and be aware of possible problems.

 

[Devils]

A computer also does not understand what is meant by infinity.

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

This thread needs to be closed.

 

[lurflurf]

π=1 (base π)
A pad of paper cannot contain the exact value of π in purely a floating point value (base 2,7,10...). Leaving aside the details of translation of hand calculations into computer algorithms the same methods are not used. Knowing how to calculate something by hand and how to calculate it with a computer are entirely different things. A computer is a useful tool.

 

[MarneMath]

Is anyone saying that a computer isn't useful?

 

[Sankaku]

The whole thread is a pencil vs machine argument.

No. All your posts are pencil vs machine arguments. The rest of us are trying to point to an appropriate balance between the two. I think I am done here.

 

[Opus_723]

I used to worry about whether I was was investing too much time in learning how to do math by hand.

Until I took a Mathematical Computing class.

There were plenty of kids in that class who had VASTLY more experience in programming than me. Most of them had already done projects in Mathematica and just wanted to solidify their knowledge of its ins and outs. I knew next to nothing about programming. But the teacher is notoriously difficult, and he gave us problems that no one managed to do on the first try. Everyone's programs bugged up and did nonsensical things until you dug into them good.

And that's when I discovered that, although I had little experience programming, all my time with pencil and paper math had prepared me better for debugging than anyone in the room. Since I wasn't as comfortable with code, I had Mathematica plot and animate all sorts of visuals representing intermediate steps. I got comfortable creating visuals early and used them to debug rather than searching through lines of code. I took derivatives by hand and then plotted them to check how many roots I SHOULD be getting, and which ones were missing. At one point we were dealing with a function that behaved so badly that FindRoot couldn't keep up with it as parameters changed, so I had to construct a fairly intricate "guess" function that approximately tracked the behavior of the function while being much simpler computationally. I expanded expressions into Taylor series by hand to see what the computer should be telling me in certain extreme cases.

This may sound unsophisticated to people who know all sorts of impressively accurate numerical techniques, but I produced working programs faster than anyone else who I saw working in the lab, and I spent half my time doing things by hand in my notebook. It was mostly due to it being the only thing I felt comfortable with, but it seemed to work really well.