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

[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

 

[zgozvrm]

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

My wife was a grade school teacher for many years (she's now an administrator), but when she was teaching math to 3rd graders in the early 90's, the curriculum was pretty much designed to teach kids how to use calculators to do math, basically neglecting to teach them how the operations really worked.

About 6 years later, when one of her students (the daughter of a friend) was in high school, she needed help with Algebra. My wife asked if I would tutor her. I recall going through a word problem with her and making sure that she could set up the problem, based on the information given. At one point, I asked something like, "what's 3X times 4?" and she immediately went to her calculator. I said, "You should be able to do that in your head." She just gave me a blank stare and I thought, "Wow, she's in Algebra, but can't multiply 3 by 4 in her head!?" My wife told me they don't make students memorize multiplication tables any more. Unbelievable!

So, while calculators are handy and can solve complicated calculations faster than we can by hand, they are ultimately slowing us down because students are being taught to depend on them.


Another example was while we were dating before we were married: we went to a store and the power went out. We had a single item that was priced at an even dollar amount, say $2.00 (if I recall the tax then was 6.0%). The "kid" at the register wouldn't sell us the item because the register was down. I told him that he could charge us and write it down, then enter it into the register when the power came back on. He said something like, "Yeah, but how will I charge you the tax?" I told him to use the calculator next to the register if he had to (which he shouldn't have needed). Ultimately, we walked out without our item.

 

[gmax137]

I don't know, I can kind of see the OP's point in some ways. Back in 7th grade I spent weeks 'factoring' quadratics to find the roots; then they showed me the quadratic formula and I don't remember 'factoring' anything since - I just plug the coefficients into the formula and get the answers. So, why all the emphasis on 'factoring'? Maybe, because back there in my 7th grade all we had was our fingers and slide rules, no calculators?

Maybe the factoring is more beautiful - in the sense that it allows you to dissect the equation into parts? OTOH, in the formula you still see how it all depends on 'b^2-4ac'

EDIT - sorry, once again I replied at the bottom of 'page 1' before realizing the thread has moved on to three pages. I guess I better read pages 2 and 3 now...

 

[zgozvrm]

I don't know, I can kind of see the OP's point in some ways. Back in 7th grade I spent weeks 'factoring' quadratics to find the roots; then they showed me the quadratic formula and I don't remember 'factoring' anything since - I just plug the coefficients into the formula and get the answers. So, why all the emphasis on 'factoring'? Maybe, because back there in my 7th grade all we had was our fingers and slide rules, no calculators?

Maybe the factoring is more beautiful - in the sense that it allows you to dissect the equation into parts? OTOH, in the formula you still see how it all depends on 'b^2-4ac'

EDIT - sorry, once again I replied at the bottom of 'page 1' before realizing the thread has moved on to three pages. I guess I better read pages 2 and 3 now...

Yeah, but why go through the trouble of using the quadratic, when some quadratics can be done in your head.

For example, I can factor X² + 14X + 49 "by hand" much faster than I can plug the values into the quadratic formula (and I don't even need a calculator, or do any "long" multiplication).

Plus, it's nice to have a good understanding of how the quadratic formula works.
And, if you haven't used the quadratic formula for a while and forgot it, you can easily derive it on your own.

 

[Office_Shredder]

I don't know, I can kind of see the OP's point in some ways. Back in 7th grade I spent weeks 'factoring' quadratics to find the roots; then they showed me the quadratic formula and I don't remember 'factoring' anything since - I just plug the coefficients into the formula and get the answers. So, why all the emphasis on 'factoring'? Maybe, because back there in my 7th grade all we had was our fingers and slide rules, no calculators?

Maybe the factoring is more beautiful - in the sense that it allows you to dissect the equation into parts? OTOH, in the formula you still see how it all depends on 'b^2-4ac'

EDIT - sorry, once again I replied at the bottom of 'page 1' before realizing the thread has moved on to three pages. I guess I better read pages 2 and 3 now...

Factoring quadratics maybe isn't necessary, but if you want to solve high degree polynomials a common strategy is to find some obvious roots (e.g. rational root theorem) and then factor it, allowing you to work with a smaller polynomial. You can't do that if you don't know anything about factoring, and learning how to factor quadratics is the first step of that

Furthermore, you need to know about factoring to understand things like why a polynomial has a multiple root if and only if its graph is tangent to the x-axis

 

[holomorphic]

Part of the problem here is that you're suggesting everything you learn in school as an end in itself, i.e. you learn integration by hand so you can do integration by hand, and you learn to solve a linear ODE by hand so you can also do that by hand. Naturally, we all know this is not the case. Of COURSE simple problems like that are easier to do by plugging them into Mathematica/Maple/MATLAB.

But REALLY, we take those classes to drill PROCESSES into our heads that can later be expanded upon to do things that machines cannot, at present, do. We're first trained to manipulate things on a very low level to solve homework problems, the answers to which are (usually) known. Then, in the real world, we're asked to manipulate things which are much messier to try to get a good result. From experience, we know that unless you learned to use your head in school, you won't be able to use it later.

As a last observation, let me just say that honestly, I could not have taken an advanced calculus/real analysis course without having taken the elementary calculus sequence.

 

[holomorphic]

To try to condense a little what I said...

It's not necessarily that we need the specific manipulations that we are taught (though these are totally helpful to remember for each successive level of whatever class) but the whole ability to manipulate things, which is a skill that has to be built and maintained through practice.

Just like musicians with their scales and etudes.

 

[djosey]

I think that we are thinking too generally: some things in maths should be learned by hand, some things would immensely benefit from mathematica-like software. The good question is what topic falls in what category, and for whom. People studying maths in uni, for example, need to really learn how to do stuff.

 

[Mark44]

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?

Да, я понимаю.