# Mathematics software/advanced calculators and the learning of mathematics.

## Main Question or Discussion Point

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?

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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?

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.

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".

I think that this TED-talk by Conrad Wolfram sums up pretty nicely what Im 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?

I think that this TED-talk by Conrad Wolfram sums up pretty nicely what Im 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
Gold Member
It's the journey, not the destination.

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.

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
Homework Helper
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.

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.

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.

- '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.

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Ackbach
Gold Member
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.

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
Homework Helper
^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.

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".

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.

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.

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/physcist 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?

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