The Death of Calculus
One of the standard moans is that no one takes physics anymore, and when they do, “calculation has been replaced by writing”.
Far be it from the bunny to be serious, but have you folk realized that algebra and calculus are in pretty much the same position that arithmetic was in 1970 when HP's calculator killed it. Why sweat integrating functions when Mathematica/MathLab/Maple can do it for you? To the same extent, you have to know the rules in order to spot problems (ill stated inputs, etc), but you don’t necessarily have to master the minutia.
The time is here when students take their computers into the test to run the symbolic math applications for their test, and in that case calculation will have been replaced by writing and the students will have to explain why and what they did. Not a bad thing IEHO.
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those of us who studied phyics with a math emphasis are aghast at you, Eli :)
WV: preambe - a hastily written preface, but in pentameter.
Don't be stuck in the last century Marion.
Eli
Add Wolfram Alpha to your list.
It's such a shame that students no longer have to wrack their brains manipulating complex integrals.
Some of Horatio's fondest memories (from the 70's/80's) involve late night integration marathons with Abramowitz and Stegun and the CRC handbook.
What fun.
And Marion: don't be stuck in the last millennium.
I've long forgotten the calculus I learned in high school, but if I had learned it at the same time that I had easy and continuing access to useful software, I might still be able to do something with it.
My two bits.
I was awful at calculus when it was memorizing rules, but as soon as it was applied and used to answer a question, or when I could play with MathLab, it made sense and I learned it.
I am all for being able to understand and use concepts, and being able to use a computer for calculation. It still requires you to figure out if the output was reasonable or the result of an error on your part.
I found my old slide-rule the other day... Last used in 1972. And I've forgotten all the fluxions I ever learned. :-(
I don't know if it's a good idea or not... but here the calculator is band for the first year of math... not the most advanced one but a good start IMHO.
If I could get out of this rocker, Horatio and Eli, I'd whack you aside your fool heads with my slide rule.
In the meantime, get off my damn physics lawn!
Hmm. My teaching experience is not favorable to the calculators. When they first came out, I figured that no big deal whether one arrived at 2+2 = 4 by way of device or pencil. One (echoing Anne) still had to know what to be adding, why you were adding, rather than some other operation, and to consider whether the output was reasonable.
Instead, what I saw was a transition to GIGO meaning 'garbage in, gospel out'. Once an answer came up on the screen, it must be correct. Significant digits? However many the calculator carries. Sanity checking? Nonsense, if the calculator says 2+2 = 1, that must be the answer. (Hitting the / instead of the +? Nahh, can't happen. Besides, the test has too many questions to check your work. After all, you can work so much faster with a calculator.)
The outcome seems to be that those who can get good answers by calculator are exactly and only those who could get them by hand as well. But the fraction who can't get them by hand is staggering.
Still have my slide rule, and my grandfather's :-)
I'm currently studying maths and physics (first year Uni). I'm enjoying it a lot. I don't see the calculator affecting the work I do - I use it all the time in physics, but it's bleedingly obvious when the answer is wrong due to input error. In maths, I don't use it much. Text book writers seem to have this habit of making answers come out in rational fractions :)
"The time is here when students take their computers into the test to run the symbolic math applications for their test, and in that case calculation will have been replaced by writing and the students will have to explain why and what they did. Not a bad thing IEHO."
No. I hate writing.
-- bi
Heaven forbid that anyone that ever takes physics should ever have to solve a real problem in the real world without a computer at hand.
Frankly, I find that when the shit hits the fan, computers are the first thing to stop working. Then some basic math skills can help you pump the water out and get the power back on. Nobody has good data at that point, so the calcs do not have to be too precise, and understanding the math lets you do "quick and dirty" calcs.
The guy with math skills says, "Send me an 8 inch pump." The guy without math skills says "send me the biggest pump you got." Is that the 4 inch pump that is already on the helo, or the 16 incher back at HQ?
So we will compensate for lack of math by just passing a law that says, "Mother Nature is not allowed to do anything that will impair computer functions or telecommunications in the vicinity of critical infrastructure or large populations." While we are at it lets go ahead and outlaw hurricanes and global warming.
Dear bi,
In that case don't blog:)
Eli
This is just another instantiation of linear (human) behavior running afoul of exponentials in computer performance and pricing.
At the Computer History Museum, I do occasional docent tours. Among things that we have are:
- one the world's 2 working Babbage Engines, which would have been a great leap forward in the production of mathematical tables in in the early 1800s ... had he been able to deliver it.
At home, for some odd reason, I still have a 1964 CRC Standard Mathematical Tables, 46th Ed. I have not used it for a very long time. I'm not sad about that.
- A good collection of slide rules. When I take college kids through past those, they look at them in wonder.
-A good collection of electromechanical desk calculators, like the Marchants or Fridens used (en masse) to help design the first A-bombs.
- A good collection of handheld calculators, which course were sometimes viewed as bad for pedagogy when they appeared.
Let's see: how much time *should* a student practice doing square roots manually?
The problem is that it takes *time* for people to adapt long-honed teaching methods when technology changes quickly. One needs continued examination of the inherent tension between:
- making sure someone understands, well enough
and
- not wasting masses of time doing something a computer can do quite well, and that in a real job, would almost never be done by hand.
I don't think there are easy answers, but I've seen people trying very different approaches to teaching, problems, and tests,and I suspect there will be a lot more.
For example: suppose every student has a laptop they can use *all* the time. How does that change the sorts of questions you ask on an exam?
People might look at Problem Based Learning, especially as practiced (in extreme form) at Republic Polytechnic in Singapore. Fascinating to see.
-A good collection of electromechanical desk calculators, like the Marchants or Fridens used (en masse) to help design the first A-bombs.
Ah, back in the good old days, when a "computer" was the person who operated the calculator ...
Still have my slide rule, and my grandfather's :-)Hmm. My teaching experience is not favorable to the slide rule. When they first came out, I figured that it was no big deal whether one arrived at 2+2 is about 4 by way of device or pencil. One (echoing Anne) still had to know what to be adding, why you were adding, which scale on the rule to use, and to consider whether the output was reasonable.
Instead, what I saw was a transition to GIGO meaning 'garbage in, gospel out'. Once an answer came up on the slide rule, it must be correct...
-----
(and you should see what the old fart mathematicians said about the invention of pencil and paper!)
Mechanical assistance for doing arithmetic or calculus is a Good Thing. Engineers or scientists who can't reliably do arithmetic or calculus without mechanical assistance are No Bloody Good. Schools which grant degrees to such people, purporting to indicate that they have a grounding in useful mathematical skills, are also No Bloody Good.
As an employer, I can and do insist on basic mental mathematical skills. If you can't reliably multiply two-digit numbers in your head, or differentiate cos cos on paper, or tell me instantly - within 20%, say - how many microseconds a computation takes if we do 17 thousand of them in 2.5 seconds, don't bother applying.
I'd like to add mental extraction of square roots, but it's a fairly obscure and nowadays little-used skill, so only true geeks bother to learn it.
When I was a student in the Late Cretaceous, engineers in the class were thrilled with the new-fangled calculators because they could get oh so many decimal places in their answers.
Recall Isaac Asimov's classic 1958 short story, The feeling of power, which of course appeared well before handhelds.
However, I still think that pedagogy hasn't yet generally caught up with the existence of widespread cheap computing. Optimal mixes of skills and testing change.
Put another way, Augmented Intelligence (human/machine combination) has almost always done better than Artificial Intelligence.
A few examples from other areas, not all exactly the same issue, but related:
1) History.
Merely knowing (historical, for example) facts might once have been good enough, as it implied you'd spent enough time chasing them down in libraries for much to have soaked in.
Now: OK, Google. How do you find things? How do you assess conflicting interpretations? What sources are likely to be more accurate?
Old skill:find the data and sort out what's important.
New skill: even more crucial to be able to sort out what's important.
2) Art.
A decade ago, I attended a talk for supporters of a local art museum. John Lasseter (Pixar; event was at SGI, so we could get him) spoke. During questions, someone asked him "We have a nephew dying to get into this area. Which computer tools should he know?"
Lasseter:
(Sigh): doesn't matter much, we have a lot of our own internal tools, but tell him to be able to see, draw, and tell a story. Computers can take care of the mechanics, but if he can't do those other things, he'll have a tough time working on the creative side.
Old animation skill: good drawing, day-in, day-out.
New skill: use the tools, but apply creativity other ways.
Needless to say, Lasseter's worldview has something to do with Pixar's track record.
3) Spreadsheets.
Everyone is now a programmer. I see no end of business plans where someone generates growth forecasts just by plugging in a growth percentage. VCs are unimpressed.
4) Programming.
A decade ago, they asked me to give a talk for a computer class at Penn State, then answer questions from students about "programming in the real world."
A student asked: "We've studied C, C++, Java, etc. In the real world, which language is most important in business?"
me: "you're not going to like the answer... English." *(student surprised) "Everyone programs at much higher levels than they used to, and that's great, but if you cannot talk to people, figure out requirements, express your ideas, present them, no matter how good a coder you are, you will not be very effective."
(student next to first nudges him, whispers "I told you so.")
"3) Spreadsheets.
Everyone is now a programmer. I see no end of business plans where someone generates growth forecasts just by plugging in a growth percentage. VCs are unimpressed."
Disinvestment bankers OTOH.....
So long as students can understand the how and the why of the basic rules of calculus, including limits, then how they actually compute the integrals is not terribly important - any more.
On the other hand, at university level I would expect serious mathematics or physics students to be comfortable at manipulating expressions involving integrals and derivatives. Otherwise they aren't going to have much hope at proving things or developing new ideas (that rely upon calculus). If they use a computer algebra package to assist with this and to reduce the human error-rate, then good on them. Still, some practice with pencil and paper helps burn in the rules; having a well-developed gut knowledge of these things helps motivate the more abstract topics that follow.
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