Discuss Math Education: Wolfram's Viewpoint

[Spiralshell]

I tried searching to see if this had been discussed before, but instead I found a lot of people using Wolfram Alpha to help with homework etc. If this has been discussed before can someone link me to the thread? I am unable to find it.

http://www.ted.com/talks/conrad_wolfram_teaching_kids_real_math_with_computers.html *edit I can't post the link so I left spaces because I feel it is an important subject to discuss*

So, in the link above is a video of Conrad Wolfram talking about math education. His main point is basically that students should be taught to use the best tools available and learn how to apply math with those tools rather than spending their time learning it with pen and paper, however, he is not saying we should completely get rid of hand based calculation because it is helpful in some regards, but it should not be the prime focus of math education like it is. He gives some very good examples and asks the question why are we teaching math in this way. He also goes into the fact that with computer based math we could give harder and more realistic problems and it allows for different types of learning because the students will be able to interact and see math. He goes into a lot of detail. It is a very powerful speech and I could never do it justice, so watch it if you haven't.I am really confused by this video because I find myself agreeing with him. I am however not a math graduate and have not had experience with really high level math courses. I sort of wanted to test his idea on myself so after watching the video I decided to do all of my math homework in java and created programs to solve all of my homework problems. I felt good about it and I was able to do it fully and it worked. I had a lot less errors too... I felt he was more right after this because we do make errors. It is really easy for us to make errors, but a computer is going to do the same process again and it is only up to us to make sure that process works the first time.

I remember one of my precalculus teachers telling me that there are problems computers can't do... But I feel that he is wrong because anything with a process can be put into steps, but perhaps at the end of that process is where the human chooses the proper output is what he was talking about. That would still be a vital part of computer based education though...

One other thing he said that I felt really hammered in his point was that with hand based calculation it is really easy to make a mistake and find that you've made a mistake and then on a test forget which way was the right. I have done this before and it is extremely frustrating because it comes down to many things like burnout, boredom, etc.

What does everyone think? It would be nice to here from those of you who have completed high level math, physics, engineering, etc... courses to respond to to this. Is/Was learning by hand the correct way to teach math?

 

[QuarkCharmer]

You can't write the code to solve the problem until you learn the steps involved by hand.

...But I feel that he is wrong because anything with a process can be put into steps...

What if there are problems that cannot be put into steps, or there is no systematic way of deriving a solution? Computers just do calculations, it's up to the user to actually do the math. In that respect, I can agree with what he said for the most part, that computers can be a vital tool in more realistic problems, but I still feel that "doing it on paper" is the way to learn. I can't help but think that Mr. Wolfram might be biased on that argument also.

 

[Spiralshell]

What if there are problems that cannot be put into steps, or there is no systematic way of deriving a solution? Computers just do calculations, it's up to the user to actually do the math.

If there is no systematic way and no way of putting it into steps then you can't solve the problem by hand to begin with. How do you derive a solution to something that is underivable? simple, you don't derive anything because you can't. In that respect it would also prove Wolfram's point that students need better conceptualization and understanding of the problem then they would know what to do in that situation.

Also, when we do a problem by hand, we are doing calculations too, aren't we? Except that a computer can do them better.

Then again, I suppose there are proofs. I think it would be interesting to see programming based math proofs rather than by hand. Also, just as doing calculation by hand has grown so would doing computer based math, e.g. programming languages will be much more powerful in the future especially if our focus is on them.

In that respect, I can agree with what he said for the most part, that computers can be a vital tool in more realistic problems, but I still feel that "doing it on paper" is the way to learn. I can't help but think that Mr. Wolfram might be biased on that argument also.

I agree that he may be biased, but my problem with doing it on paper is utilizing only 1 type of learning. What do we do for visual or auditory learners for example? Math is entirely philosophical and Wolfram sort of found a way to express that in ways we can see.

Also, is it fair to people who want to study something like psychology? They have to learn statistics by hand at first, but then for the most part they have to master some type of software e.g. SPSS and it seems to me unfair that they are forced to learn hand based calculation.

 

[jgens]

Also, is it fair to people who want to study something like psychology? They have to learn statistics by hand at first, but then for the most part they have to master some type of software e.g. SPSS and it seems to me unfair that they are forced to learn hand based calculation.

In my opinion, the job of a math class is to teach students how to do the pure mathematics. It is the job of the psychology class to teach how to use that math in psychology.

 

[Spiralshell]

In my opinion, the job of a math class is to teach students how to do the pure mathematics. It is the job of the psychology class to teach how to use that math in psychology.

Everyone has to learn the basics at some point so how should they be taught? Should it be hand based calculation or is that a waste of time considering technology that grows and changes daily. And if we started with hand based calculation when should someone be introduced to computer based learning? Also, many colleges do not even allow graphing calculators for some math courses like trigonometry and precalculus, etc... Why not make harder questions that require their use? It always seemed strange to me. Shouldn't we adapt and use the best tools available? Perhaps we are cutting ourselves short in the long run if we don't. Wolfram mentions that real life problems have "hair all over them"... well learning a formula like D=RT (distance = rate * time) is really pointless because it is unrealistic, so why do we teach it?

Math seems to change a lot from say a high school algebra class where you say y=mx+b, to say a more advanced course where you might say f(x)=mx+b, to another class where you say ý=ax̄+b. Isn't this inefficient for someone who is not into the technical details about mathematics and cares more about other things like neurons firing? Wolfram says we should be focused on on the understanding of something like y=mx+b and what we would use that for rather than learning a bunch of different hand calculations. I still remember asking my professor in high school, 'why is being able to calculate the slope and such things useful?' He couldn't really answer and that is probably because he was not a very good math teacher, but I think we have to accept that may be largely the case for most high schools. Maybe if the class was all about understanding and why such things are vital and how to use them in realistic ways than a lot more people would be interested in mathematics because doing something over and over again not only becomes repetitive and boring but makes one prone to mistakes. We could be potentially losing people who would be great at math simply because of the way it is taught. It seems to me so one sided with something like mathematics. What about the other learners?

 

[jgens]

Everyone has to learn the basics at some point so how should they be taught?

I think they should be taught the plain mathematics. As I said earlier, if they need to apply that math in a particular setting, then that course can explain how to do that.

Should it be hand based calculation or is that a waste of time considering technology that grows and changes daily. And if we started with hand based calculation when should someone be introduced to computer based learning? Also, many colleges do not even allow graphing calculators for some math courses like trigonometry and precalculus, etc... Why not make harder questions that require their use? It always seemed strange to me. Shouldn't we adapt and use the best tools available? Perhaps we are cutting ourselves short in the long run if we don't. Wolfram mentions that real life problems have "hair all over them"... well learning a formula like D=RT (distance = rate * time) is really pointless because it is unrealistic, so why do we teach it?

Wolfram's suggestion is based around teaching math to future engineers and biologists and the like. But, in my opinion, his suggestion is nowhere near helpful for teaching math to future mathematicians or physicists. So you have to ask yourself: Do we teach math for the future engineers and biologists and the like, or do we teach it for the future mathematicians and physicists? I am more inclined to teach it for the latter.

 

[micromass]

I disagree with wolfram. Of course, I admit that using computers can provide a very good bonus to the math class. Letting students experiment with computers can be very valuable. I absolutely love web applets that illustrate some theory. But wolfram really goes too far.

First of all, he goes against the very philosophy of mathematics. In my opinion, mathematics starts with very few basic axioms and derives all the rest. This is what makes mathematics so beautiful, that it is possible to derive everything.
If you just get an equations and solve it using a computer, then you basically skip the knowledge of how the equation was solved in the first place. I find such a thing very unsatisfactory and it goes against the very spirit of mathematics.
It leads to sad situations where high school students calculate 3*5=15 and have no idea what that even means. Yes, this happens.
Not only goes it against the spirit of mathematics, but it's also quite dangerous. If you don't know how the solution was arrived, then you don't know the mistakes and assumptions the program made.

Second, there are some things a computer just cannot solve. And if you want to be able to solve these things, then you better be acquainted with the easy examples and methods. For example, given the function $f(x)=e^{-1/x}$ if x>0 and 0 if x<=0. Is this function infinitely many times differentiable at 0?? Good luck solving this with the computer. Good luck if you're trying to solve this without having done basic differentiation problems.
And yes, this fact is very important: it gives us so called bump functions and partitions of unity. Things which are extremely important in geometry, analysis and physics.

There is something wrong with math education though. High school math is extremely boring. Part of the solution is indeed computers. But also read this: www.maa.org/devlin/LockhartsLament.pdf

 

[nucl34rgg]

The point of mathematics education is not to train you how to use available tools to solve a problem in a quick and dirty way. Anyone can put in boundary conditions and press "run." Most of the problems we encounter as undergraduates are trivial and have been solved thousands of times in thousands of different ways. The point is to learn how to "do mathematics" -- how to think mathematically! For many, mathematics is a way to rid oneself of sloppy thinking. You can easily learn to use the available tools on your own with minimal effort, but to learn mathematical reasoning and to learn the framework upon which the aforementioned tools are based requires careful study.

Computers SUPPLEMENT mathematical techniques by making computations easier or allowing us to numerically approximate things that aren't solvable analytically. They do not and should not be used as an attempt to replace these techniques.

Many rely on the software to do everything for them without having any clue how the software works. This is a ridiculous strategy in the long run. It's basically equivalent to cheating your way through a topic.

 

[AlephZero]

This is an old question. It certainly goes back to the early uses of computers in engineering (round about 1960), when the debate was whether it was better for engineers to get wide but shallow experience of a range of situations from computer modelling, compared with the deep understanding of a few situations that was necessary to do any useful hand calculations. And of course the time engineers spend creating and "playing" with computer models is often time NOT spent doing any engineering.

There were probably similar debates when slide rules and the mechanical calculators were invented!

My own view is that it's not an "either/or" choice. Now that good computer tools are available, you have more stuff to learn, not less. You need to learn how to do that math (or engineering or physics or whatever) just as you always did, and you also need to learn how to use the tools effectively.

 

[genericusrnme]

Using mathematica can be pretty handy as a tool when you start maths, it's handy for being able to plot graphs out and helping people understand polynomials when they first stand out.

When you get into the more abstract maths though, mathematica becomes pretty redundant, it's not really that handy for abstract algebra at all.

For numerical work though, knock yourself out!

I agree with micromass here though

There is something wrong with math education though. High school math is extremely boring.

The problem I think is that high school maths isn't really maths, it's just fancy arithmetic. Most high schoolers where I'm from will leave high school for university without ever encountering a set or knowing what injective and surjective mean.
Highschool should at least start to give students some 'mathematical maturity'.

 

[vkash]

Wolfram Alpha is a corporation. When they say "buy their product and it will be beneficial for you" it means they are advertising. Using mathematics software is good but there is a maturation state after which you have to quit doing lengthy calculation and start calculation and switch to math software For a kid who is just learning multiplication division etc and you give him calculator to solve homework then it is definitely dangerous. So all it depends on your age(educational age).

If you know perfectly the thing you are doing in a calculator or with math software then it's good for example. If you are checking your work like you have drawn a curve and not sure that you are correct or not then this software is your free all time teacher. However, if you are doing your homework in it then it is troublesome condition and will make your mathematical concepts too weak.

 

[DeadOriginal]

If there is no systematic way and no way of putting it into steps then you can't solve the problem by hand to begin with. How do you derive a solution to something that is underivable? simple, you don't derive anything because you can't. In that respect it would also prove Wolfram's point that students need better conceptualization and understanding of the problem then they would know what to do in that situation.

Also, when we do a problem by hand, we are doing calculations too, aren't we? Except that a computer can do them better.

Then again, I suppose there are proofs. I think it would be interesting to see programming based math proofs rather than by hand. Also, just as doing calculation by hand has grown so would doing computer based math, e.g. programming languages will be much more powerful in the future especially if our focus is on them.



I agree that he may be biased, but my problem with doing it on paper is utilizing only 1 type of learning. What do we do for visual or auditory learners for example? Math is entirely philosophical and Wolfram sort of found a way to express that in ways we can see.

Also, is it fair to people who want to study something like psychology? They have to learn statistics by hand at first, but then for the most part they have to master some type of software e.g. SPSS and it seems to me unfair that they are forced to learn hand based calculation.

If you can't visualize it yourself, how can you expect the computer to understand what you want it to do?

Fermat's last theorem was seemingly impossible and had no systemic way of solving but it was done.

 

[Sankaku]

This poll presents a false dichotomy. It is not an exclusive-or question. You use a computer for the things it is good for. However, there are many things that it is not good for. The trick is know which is which.

Wolfram's suggestion is based around teaching math to future engineers and biologists and the like. But, in my opinion, his suggestion is nowhere near helpful for teaching math to future mathematicians or physicists.

I agree.

If you just get an equations and solve it using a computer, then you basically skip the knowledge of how the equation was solved in the first place. I find such a thing very unsatisfactory and it goes against the very spirit of mathematics.

Unfortunately, this is connected to jgens point above. The basic approach of much physics and engineering mathematics is to hand people tools without understanding them. Of course, intelligent students want more, but plug and chug classes make people dream of computers.

It is beneath the dignity of excellent men to waste their time in calculation when any peasant could do the work just as accurately with the aid of a machine.

The subtle point here is that mathematics is not really about calculation. That is just one of its useful side-effects. Wolfram actually uses something like this phrase, but in a very different way. My view is that mathematics is not about "solving problems", it is about understanding pattern and structure at a fundamental level. Certainly, over-emphasis on manual calculation doesn't help with this insight. However, if you hide most of the complexity behind a screen, your fundamental understanding of mathematics will have a gaping hole.

 

[Sankaku]

If you can't visualize it yourself, how can you expect the computer to understand what you want it to do?

I should say that I am a very big supporter of using computers as visualization tools. Translating your equations into code, and then seeing how changes affect what you see, can be a powerful way of gaining visual insight into your calculations.

However, much of pure mathematics is not easily "computable" and the parts that are will always be a small subset of the interesting stuff. There is a reason that pure mathematicians don't need supercomputers. As much as I like playing with good CAS programs, the most computer-intensive mathematics application I use now is Latex.