I'll also admit that my opinion might have something to do with trying to come up with an answer to "When will I ever use this in real life?" from one of my Trigonometry students that would be honest, while not sounding like I was saying that education was a waste.
A moot point, since I quit later in the year to join the Air Force after 9/11.
Sorry, but if you couldn't come up with answers for that for trig or calculous, I wouldn't want you as a teacher. A simple trip down to the shop class you can get a buch of examples with carpentry, and DIY home remodeling was a booming industry in 2001, growing in popularity on television. Construction is a major industry, and whether they know it or not they are using trig every day.
I can easily think of tons of uses for calculus, depending on the field they want to go into. You have your common uses in pretty much all fields of engineering, medicine, and sciences, but then also more interesting ones in accoustics or photography, or more mundane areas like ecconomics. Heck, I can even think of decent cooking applications for calculous - what is the optimal temperature to cook a thanksgiving turkey. You can even ask qualifiers like how this changes based off the size of the bird or how thawed it is.
There is a serious problem with how math is taught in this country that causes people to not understand how it can be used to improve their everyday lives. A lot of it has to do with how we teach it in a silo and then expect people to know when to apply it. The reality is much of what people take for granted has a start in math, and people don't bother looking at how to understand it.
Okay, seriously, is there any way you can think of that makes sense to introduce multiplication or division with negative temperatures?Because a lot of people don't understand that -$10 is a real amount of money. Even accountants don't use that formulation, which is why a ten dollar debt is written as ($10), often in a different colored ink.
Everyone understands that -5 is just as valid a temperature as 0 or 25. Making that connection, that debts are negative money and losses are negative gains, is one of the big things that needs to be taught. Can you think of another situation other than temperature where you see actual negative numbers in common practice?
Averages. Change of temperature over time. If my freezer is broken and gains one degree every four hours, how long will it be before the frozen pizzas thaw?
Heck, just converting from F to C requires multiplication and division.
Sure, accountants use notation ($10) <-- red, or whatever, but it's very simple, conceptually, to understand that that equals -$10. The math is indistinguishable because it's the same thing. I can't think of any adult that deals with money on a real basis who can't understand that owing $10 can be thought of as negative $10. If I run into someone who can't understand that concept pretty much upon its introduction to them, I wonder if they have no concept of money at all, or have serious other developmental handicaps.
Er, no, no, it isn't. There's a reason that accountants use a different notation, because that's the only way to make clients understand how debt works. (One issue, for example, is the confusion between magnitude and direction. Everyone understands that ($10,000) is "bigger" than ($5,000) [it's a bigger debt], everyone also understands that -10,000 is smaller than -5000 [it's further to the left on the number line]. The reason for this is that debt and credit go, mentally, into two separate boxes for most people; they're different things.
On the other hand, most people intuitively understand that there's nothing magical about the temperature zero, especially in Fahrenheit. It's just a point on a scale.
In fact, maybe it's the weird disconnect I can barely conceive of that subtraction is somehow different from or simpler than adding a negative that is the issue. Maybe a better introduction that it's just a difference in notation would make things click better for people.
That would probably help. I got my first introduction to negative numbers when I asked my father at age three or so about counting down ... what is the next number after zero? As a result, I could add negative numbers before I could subtract, because "negative one" just made sense as another number. A dollar of credit and a dollar of debt aren't on the same mental scale any more than a cup of flour and a pound of apples are.
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My father explained negative numbers in terms of sandhills on the beach. "Okay, this one's as high as the pail, this one's as high as your brother; there's no sandhill here, so it's not high at all. Now what happens if I dig a hole in the sand?"
The important part is that the absolute zero is coldest temperature. Those negative temperatures are in reality hotter.
Got it.
Curiously and on topic with calculus the negative temperature exist cause 1/T is related to the derivate of the etropy, 1/T = dS/dE.
For most systems dS/dE >0 (that is, the system get more "disordered" when you give it energy)
But for some systems dS/dE < 0 (this is cause the system have an energy associated with an state of maximun "disorder", so when you give it more energy beyond that its "disorder" decrease).
You lost me. In my limited scientific understanding, adding energy = hotter. :)
My father explained negative numbers in terms of sandhills on the beach. "Okay, this one's as high as the pail, this one's as high as your brother; there's no sandhill here, so it's not high at all. Now what happens if I dig a hole in the sand?"
an interesting way to go about it.
There you have it, everything a child really ever needs to get along in life. No need for actually learning HOW to make these calculations, or when to apply them, right?...
They need to
1) Understand what words/ideas correspond with which button on the calculator.
2) be able to see if the answer is reasonable in case they hit the wrong button
3) Do math when there's no calculator around. Kirth is never going to sit down to calculate a drop in well salinity without having at least a laptop around.
Calculus is not useless at all. Just the fact that you are not going to apply the knowledge you garnered to solve any specific situation in life dosen't make that knowledge "useless"
Yes actually, it does.
If its a mind expanding, spiritual journey for you then by all means take it. I don't think i need a study to show that you're in a small minority however and that most people are NOT getting that benefit, so trying to trot a large segment of the population into the dull tedium of random number and letter changing doesn't seem worth it.
Someone asked when i took calculus.
Admittedly its been a bit over 10 years. However I'm skeptical of the claims of some new insightful and revolutionary way of looking at math actually existing because from 7th grade on we were told it was coming... and it never did. I've also been helping my niece through her home work over the years, and its been the exact same do this process to those numbers without any sight that it was for me.
There you have it, everything a child really ever needs to get along in life. No need for actually learning HOW to make these calculations, or when to apply them, right?...
They need to
1) Understand what words/ideas correspond with which button on the calculator.
2) be able to see if the answer is reasonable in case they hit the wrong button
3) Do math when there's no calculator around. Kirth is never going to sit down to calculate a drop in well salinity without having at least a laptop around.
Calculus is not useless at all. Just the fact that you are not going to apply the knowledge you garnered to solve any specific situation in life dosen't make that knowledge "useless"Yes actually, it does.
If its a mind expanding, spiritual journey for you then by all means take it. I don't think i need a study to show that you're in a small minority however and that most people are NOT getting that benefit, so trying to trot a large segment of the population into the dull tedium of random number and letter changing doesn't seem worth it.
Anyone who wants to do any financial planning can use calc to make their lives easier.Want to figure out how long it will take you to pay off a loan, or save up enoguh money to make a down payment on a house? How about retirement planning? How long will it take you to develop the necessary nest egg? - Calculous is the basis for this math
Negative temperaureIf you always use the Kevlin scale, you'll never have a negative temperature.
You learn something new every day.
Thank God I never have to worry about it.
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Well, all I can say is that the adult woman who couldn't understand arithmatic with negative numbers when presented with the example of temperature immediately understood it when I presented her with the example of debits and credits.
I couldn't even follow the examples in that text book with temperature. It made no real world sense to me at all. While it's valuable to have multiple ways to explain a mathematical concept using real world examples, I think temperature is much less useful than debits and credits to explain arithmatic using negative numbers. I also think it's more likely to help people useful life skills.
Well, all I can say is that the adult woman who couldn't understand arithmatic with negative numbers when presented with the example of temperature immediately understood it when I presented her with the example of debits and credits.
I couldn't even follow the examples in that text book with temperature. It made no real world sense to me at all. While it's valuable to have multiple ways to explain a mathematical concept using real world examples, I think temperature is much less useful than debits and credits to explain arithmatic using negative numbers. I also think it's more likely to help people useful life skills.
unrealistic scenarios and poor pacing and worse explanations are serious problems endemic to math books in my experience.
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I'm sure someone needs it, but I for the life of me can't see why they insist that most people need it.
Most people in the world will get by without calculus just fine, but... the society with the highest ratio of advanced math education will probably be amongst the wealthiest nations and most easily maintain that position.
Advanced mathematics is the key to understanding and advancing science and technology. We don't need science and technology to advance, but I'm sure thankful it has so far.
Laptops and calculators are useful tools, but understanding math is understanding how to use those tools. I don't know calculus. If Kirth handed me the raw data and his laptop, I would not be able to build the excel spreadsheet to arrive at the correct solution. If someone told me what functions to use I could do it, but that would require them to know how those functions worked to know they were picking the right ones.
The closest program I know that can do independent computation is Wolfram Alpha. It's pretty neat and can do a lot of stuff, but it's still limited by both human input and it's ability to collect data. You need to tell it to look for the right things to get the correct answer.
Also, @BNW, SOMEONE has to know calculus in order to program the calculators and computers as to how to figure it out. So it can't be useless, it's just that, for you, it has no use because someone else is doing the heavy lifting for you.
Also, @BNW, SOMEONE has to know calculus in order to program the calculators and computers as to how to figure it out. So it can't be useless, it's just that, for you, it has no use because someone else is doing the heavy lifting for you.
I don't mind someone knowing calculus. Its not philosophy where I'm saying convert the calculus wing into an extra dining room or something. I just don't see why its included in general education.
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Because it means more scientists and engineers later on.
I don't mind someone knowing calculus. Its not philosophy where I'm saying convert the calculus wing into an extra dining room or something. I just don't see why its included in general education.
The main argument for calculus in general education is procedural thinking, i.e. how to figure out an effective step by step process for doing things, in an environment where there's a clear right or wrong answer at the end so you know if you did it right.
Calculus can provide that, and in most universities today so can statistics and in many cases a computer programming course. In some universities, even algebra will work. The important thing, though, is to demonstrate the ability to deal rationally and procedurally with hard data. There's a lot of room in the humanities for intuition and creativity, but the flip side of that is almost any idea can be justified if you work hard enough, even if it can eventually be proven wrong. Also a valuable skill for lawyers to have -- what's important is not that my client be innocent, but that I can muster a persuasive argument that he is.
But when that lawyer is trying to work out a financial plan with her accountant, she also needs to understand things like compound interest and she needs to understand enough to get the sums right, because the IRS isn't going to be happy with a spurious argument that she only gets taxed at three-quarters the normal rate.
+1. The U.S. has already pretty much ceded the field of stem cell research to Singapore, renewable energy to Norway, precision steel and metallurgy to Germany, and last I checked was behind Sweden and others in terms of aerospace technology. Our technical edge, hard-won during WWII and immediately after, is all but gone. And I have a feeling that, in the 21st century, the countries with the best technologies will be the most powerful countries in terms of wealth and politics. Why flush that down the toilet? Anything we can do to encourage or help our kids along to become scientists and engineers is probably to our benefit.
The main argument for calculus in general education is procedural thinking, i.e. how to figure out an effective step by step process for doing things
Its main argument bites because it does not in fact do this. It becomes a mindless exercise in translating one set of numbers by multiplying exponents by coefficients and dropping an exponent , or mindlessly plugging equations into that annoying to read f of the g of the f of the x and then trying to multiply them out without an error. There's no thinking going on there, its just a conscientiousness test.
Calculus can provide that, and in most universities today so can statistics and in many cases a computer programming course. In some universities, even algebra will work. The important thing, though, is to demonstrate the ability to deal rationally and procedurally with hard data.
I have never seen data in a calculus class or a calculus textbook.
Because it means more scientists and engineers later on.
I don't know how much it needs to be in the sciences in the age of excel. But either way its college math, the time for "latter on" is over.
Our technical edge, hard-won during WWII and immediately after
And this is why we need history.
We got that technical edge from being a better place to live for German scientists looking to defect, by scooping them up as we went to meet russia at berlin, and by being the only industrialized nation that wasn't bombed back about 20 years. It was hard work the same way turning the chess geek upside down for his lunch money is hard work.
Its main argument bites because it does not in fact do this. It becomes a mindless exercise in translating one set of numbers by multiplying exponents by coefficients and dropping an exponent , or mindlessly plugging equations into that annoying to read f of the g of the f of the x and then trying to multiply them out without an error. There's no thinking going on there, its just a conscientiousness test.
That's just the start - you teach kids fingerpainting before they learn the good stuff and this is exactly the same. Those drills you describe as mindless are not the goal. Performing lots and lots of similar caluclations is not 'doing calculus' but it's a necessary step in getting there.
.Attempts to teach maths without the tedius, rote learning are generally unsuccessful (or too labor intensive to be practical for mass eduction).
Because it means more scientists and engineers later on.I don't know how much it needs to be in the sciences in the age of excel. But either way its college math, the time for "latter on" is over.
Do you not do calculus in high school?
That's just the start - you teach kids fingerpainting before they learn the good stuff and this is exactly the same. Those drills you describe as mindless are not the goal.
But that's as far as it goes. If its not going to go any further why bother getting it that far, since its not going to get to the goal? If you need to do from 7th grade to freshman/junior college just to START a mind expanding process then I'd have to question if there isn't something else that might be a bit more eye opening that can be taught in less than 7 years.
Its main argument bites because it does not in fact do this. It becomes a mindless exercise in translating one set of numbers by multiplying exponents by coefficients and dropping an exponent , or mindlessly plugging equations into that annoying to read f of the g of the f of the x and then trying to multiply them out without an error. There's no thinking going on there, its just a conscientiousness test.
Perhaps you don't know what's being tested? If you like, you can treat it as a cryptographic assignment; there are a set of patterns and a set of rules to follow in for the various patterns. Recognize the pattern, apply the rules, and the correct answer will pop out.
And, yes, there's quite a bit of thinking involved
I have never seen data in a calculus class or a calculus textbook.
Have you ever seen a calculus textbook? "Data" doesn't mean "tables of numbers," but every calculus class includes problems of "here's a formula, apply it to this specific situation," and most have problems of "figure out a formula describing this situation and figure out what patterns to apply."
Here's an example I cribbed from the web:
The management of a restaurant assigns each waitress between 7 and 15 tables. One waitress finds that when she is assigned seven tables, each table brings in $30 per week in tips. If she is assigned more tables, the amount of attention she gives to each table decreases. Thus, each table brings in less per week in tips. Suppose each additional table causes every table to bring in $2 less per week in tips. Use calculus to determine how many tables she should be assigned to maximize her weekly earnings from tips. Show or explain how you find each formula you use.
Yes, that's hard data; you're given a specific well-defined setup and a set of operations to solve it. Without going into too much detail, you need to:
1) be able to describe her earnings as a function of the number of tables she's assigned.
2) determine at what value this function is maximized
2a) the maximum of a function is obtained at where the derivative = 0 (or at an edge of the interval), therefore
3) find the derivative of the formula you got in step 1 and
4) determine the zeros of the derivative
That's just the start - you teach kids fingerpainting before they learn the good stuff and this is exactly the same. Those drills you describe as mindless are not the goal.But that's as far as it goes. If its not going to go any further why bother getting it that far, since its not going to get to the goal?
It doesnt get to the goal of 'doing calculus' it does get to the goal of mind expanding. (Kids who learn their multiplication tables are better off than those who dont, no matter whether they actually use multiplication in their life - doing chin ups makes you fitter even if you dont actually ever replicate that exact activity in life).
it that far, since its not going to get to the goal?It doesnt get to the goal of 'doing calculus' it does get to the goal of mind expanding.
Again, it does not. You're using equivocation.
If doing calculus means "doing that thing you do in calculus I" then its not mind expanding (for reasons you agreed to above)
If doing calculus means the good stuff, people don't get there. When exactly does that kick in? Calc II? Calc III ?
(Kids who learn their multiplication tables are better off than those who dont, no matter whether they actually use multiplication in their life - doing chin ups makes you fitter even if you dont actually ever replicate that exact activity in life).
And how much of that is causation rather than correlation? The smart kid who applies themselves is more likely to be able to memorize their multiplication tables. I think its an indicator more than a cause.
Again, it does not. You're using equivocation.
It doesnt get to the goal of 'doing calculus' it does get to the goal of mind expanding.
Again, it does. That's one reason that many colleges will accept algebra instead of calculus for gen-ed credits; you don't get as much mind expansion, but you get some. With algebra you may not be able to tell the waitress how many tables makes her the most money, but you will at least be able to structure the formula to tell her how much money she makes for various numbers of tables.
From that, you can do the "maximize this" calculation by hand or spreadsheet, but if your complaint is about tedious and mind-numbing exercises, you should favor the direct solution rather than doing what amounts to solving the same problem a dozen times and looking for the biggest answer.
If doing calculus means "doing that thing you do in calculus I" then its not mind expanding (for reasons you agreed to above)If doing calculus means the good stuff, people don't get there.
Well, for me, it kicked in in calculus I, when I learned how to solve maximization problems. Among other things, it showed me that if the calculations are tedious and repetitive it means you're probably solving the wrong problem -- the solution is almost always better design instead of better execution of a bad design. That's a lesson that serves well in most of life.
Also, @BNW, SOMEONE has to know calculus in order to program the calculators and computers as to how to figure it out. So it can't be useless, it's just that, for you, it has no use because someone else is doing the heavy lifting for you.I don't mind someone knowing calculus. Its not philosophy where I'm saying convert the calculus wing into an extra dining room or something. I just don't see why its included in general education.
Maybe I'm just not getting your meaning.
What do you mean included in general education? As opposed to...there being special schools which are the only place mathematics higher than algebra is taught?I graduated high school without calculus, I got my first degree without needing calculus, I got my second degree without needing to take calculus, and now that I'm working for my bachelor's degree about 90% of the departments don't even require it. I happen to be studying economics, so a basic understanding, enough to find the area under a curve, is necessary.
What I'm saying is, even for people pursuing post-secondary education, the need for calculus is pretty rarefied.
Meatrace:
What i mean specifically is that its expected for a four year degree.
I needed it for a forestry degree. So if you needed it for that I'm pretty sure it was/is wide spread as a requirement. It was also on the list of required courses for just about every major I saw. Your school may be odd, things may have changed, or my schools could have been odd.
Again, it does not. You're using equivocation.
It doesnt get to the goal of 'doing calculus' it does get to the goal of mind expanding.
I don't think so.
If doing calculus means "doing that thing you do in calculus I" then its not mind expanding (for reasons you agreed to above)
Err, no. I thought I explicitly said "that thing you do in calculus I" was mind expanding? (Despite not being calculus).
If doing calculus means the good stuff, people don't get there. When exactly does that kick in? Calc II? Calc III ?
Post graduate maths (in my opinion). But I don't really see the point - most people don't do calculus (ever) and don't get to the good stuff. They nonetheless benefit from the drills one has to do first.
I really don't understand your objection here. My view is that those rote calculations are labelled "calculus" but are not the actual thing. They're necessary steps to getting there and they have benefits, even for the majority who don't actually go on to do actual calculus.
What's the alternative? If you don't do it in school, it's too late. There hasn't been a better option found for teaching maths that doesn't involve a large dollop of rote learning (at least not a cost effective one). How can you tell which ten year olds are going to go on to need higher maths?
(Kids who learn their multiplication tables are better off than those who dont, no matter whether they actually use multiplication in their life - doing chin ups makes you fitter even if you dont actually ever replicate that exact activity in life).And how much of that is causation rather than correlation? The smart kid who applies themselves is more likely to be able to memorize their multiplication tables. I think its an indicator more than a cause.
Sure, but the benefit shows up even for those at the lower end of the mathematics-aptitude bell curve. They won't go on to be professors maybe, but they'll think better. The benefits in IQ performance* from studying elementary maths, for example, show up for both the "smart" and the "dumb" kids.
Do you not do calculus in high school?Because it means more scientists and engineers later on.I don't know how much it needs to be in the sciences in the age of excel. But either way its college math, the time for "latter on" is over.
Only if you're advanced placement, so its possible but if things haven't changed, not standard. I got college credit in highschool for biology and history, math... not so much.
We may well be speaking at cross purposes. I did elementary calculus (involving lots of not-really-calculus) as a thirteen year old (first year high school). I think nearly everyone in the class benefitted, no matter what they went on to study.
I guess in a society where that hasn't happened, forcing every eighteen year old to do it might be akin to locking the stable door after the horse has bolted. I still think they'll benefit, but probably not by as much as if they studied courses more directly related to their major.
Because it means more scientists and engineers later on.I don't know how much it needs to be in the sciences in the age of excel. But either way its college math, the time for "latter on" is over.
I really want to see someone who doesn't know calculus use Excel to do calculus. You seem really convinced that this is super easy and common, I really want to see an example of it.
Just because you have a tool in front of you doesn't mean you know how to use it properly.
I've used Excel and programs like Access. It's been a long time, but I've set up complicated spreadsheets and databases. I would NOT be able to do the kind of work that Kirth does without explicit instructions. I wouldn't actually understand a lot of what I was doing unless I spent time looking up what it meant and trying to understand it. I can't do it because I don't know the math behind it.
Meatrace:
What i mean specifically is that its expected for a four year degree.
I needed it for a forestry degree. So if you needed it for that I'm pretty sure it was/is wide spread as a requirement. It was also on the list of required courses for just about every major I saw. Your school may be odd, things may have changed, or my schools could have been odd.
I'm a student at the University of Wisconsin, which is one of the most well-respected land grant schools in the country and a research one college. Some pioneering work on stem cells, for example, is being done on campus and because of it Madison has a burgeoning biotech industry. It's not Bob Jones University or something.
A fairly significant portion of my family and friends have at least an undergraduate degree. Off the top of my head, about 15, counting only immediate family and people I see on a weekly basis. Almost all of them are in biological sciences, although some also have social science degrees (psychology, sociology). My mother, father, and girlfriend all have postgraduate degrees. Of all of them, the ONLY one who had to take calculus is my friend who TEACHES it in high school. And even then he only had to take through Calc 2, which he passed with a D.
Of the 10 most popular majors at my school, only two require calculus: Economics and Engineering. The former only requiring a single semester.
Just saying, I have some serious quibbles with BS gen-ed requirements, but Calc is not only rarely required, it's actually useful and interesting (to me).
Unlike this Indigenous World Literatures class I'm taking right now...don't get me started...
If we eliminate calculus it could start a chain reaction that could cause the elimination of important things that people could not live without like art or, god forbid, philosophy...
* Duck and cover*
Just as chemistry should not be vilified for the excesses of alchemy, and psychology for the wrongs done its name, or the field of medicine for centuries of saw bones and leaches, philosophy shouldn't be dismissed because some guy 300+ years ago calling himself a philosopher was wrong.
The presence of the Twilight books doesn't diminish the works of Shakespeare or Hemingway, even though they're all printed on the same dead trees and available a shelf apart at the local bookstore.
I really want to see someone who doesn't know calculus use Excel to do calculus.
In my first calculus class i did do this a fair bit. Got a lot of points off for "right answer no work" .
You seem really convinced that this is super easy and common, I really want to see an example of it.
Thats kind of the problem for calc I isn't it? Actually getting a word problem.
Just saying, I have some serious quibbles with BS gen-ed requirements, but Calc is not only rarely required, it's actually useful and interesting (to me).
I wish I'd gone to your school. My forestry degree took Organic chem, physics I, Calc I, and one of the above taken to II. (i really thought i was going to like organic chem. Absolutely hated it. Apparently random interactions with an even more random IUPAC naming system)
Unlike this Indigenous World Literatures class I'm taking right now...don't get me started...
Bahahha.. yeah got a lot of that. My magic and religion teacher refused to see any practical reasons for a culture adapting to their environment- thought culture was its own independent force even when I could tell her when Indian trees spirits didn't let you cut down their trees and where one islands agricultural calender matched up exactly with an integrated pest management system.
Just as chemistry should not be vilified for the excesses of alchemy
I don't see why everyone rags on alchemy so much. Just about every device they had in an alchemist lab actually did SOMETHING, and most of the processes they described could easily be chemical processes by a different name. They were doing a little bit right at least...
and psychology for the wrongs done its name, or the field of medicine for centuries of saw bones and leaches, philosophy shouldn't be dismissed because some guy 300+ years ago calling himself a philosopher was wrong.
So when is philosophy right? Or even close enough to work? (which is my answer to that inevitable question about science)
I've used Excel and programs like Access. It's been a long time, but I've set up complicated spreadsheets and databases. I would NOT be able to do the kind of work that Kirth does without explicit instructions. I wouldn't actually understand a lot of what I was doing unless I spent time looking up what it meant and trying to understand it. I can't do it because I don't know the math behind it.
And, frankly, that's the issue and why formal math is needed, and the more the better.
Let's return to our little waitress word problem:
The management of a restaurant assigns each waitress between 7 and 15 tables. One waitress finds that when she is assigned seven tables, each table brings in $30 per week in tips. If she is assigned more tables, the amount of attention she gives to each table decreases. Thus, each table brings in less per week in tips. Suppose each additional table causes every table to bring in $2 less per week in tips. Use calculus to determine how many tables she should be assigned to maximize her weekly earnings from tips. Show or explain how you find each formula you use.
I assume that she wants to be able to make as much money as possible, and do as little work as possible herself in figuring it out.
With basic arithmetic, she can figure out that if she has 7 tables, she will make $30/table in tips, for a total of $210. She can also figure out, as a separate problem, that if she has 8 tables, she will make $28/table, for a total of 8*28 which is a little more, and so on. But that's a lot of raw number crunching.
If she has algebra skills, she can figure out that she makes (7+x)(30-2x) dollars and can even set this up as a cell calculation in a good spreadsheet, so she need only type the numbers in and see which one gives her the most money. She's solving as many problems but they're easier to solve if you have the skills and more amenable to computational support.
With calculus, she can solve directly for the maximum of the formula and determine how many tables she wants, solving one problem instead of a dozen.
Formal math means you use spreadsheets better, more effectively, and with fewer errors. It also means you're less dependent on them, which is always nice....
So when is philosophy right? Or even close enough to work?
Science helps you make decisions. Philosophy informs you about how those decisions should be made.
As a simple example, the joint idea that government comes from the consent of governed, but natural rights are inherent to all men is deeply philosophical (and quite controversial). If you like the United States but dislike Saudi Arabia, you're expressing a philosophical opinion.
If you think that you can justify your dislike of Saudi Arabia -- for example, looking the economic conditions in both countries as a measure of quality of life -- you're also expressing a philosophical opinion in the measure you choose.
Science can do the ground work and actually measure the economic conditions. But science can't tell you whether you picked the right measurement.
So when is philosophy right? Or even close enough to work? (which is my answer to that inevitable question about science)
Easy answer: empiricism--the philosophical school of thought that gave form to the scientific method and modern scientific thought :)
Logic: the study of formal and categorical logic has pretty much defined how computers work.
But a lot of philosophy isn't right or wrong, much like art isn't right or wrong or literature isn't right or wrong, and yet I don't hear you decrying their practice.
But really, this is a conversation for another thread :D
What i mean specifically is that its expected for a four year degree.
This is somewhat outdated.
In the 1950s and 1960s, calculus (along with Shakespeare and Western Civ, and at many schools ROTC) were general education requirements.
Most of the requirements have been broadened over the years; now what is typically required is literature-but-not-specifically-Shakespeare, cultural-studies-but-not-specifically-western-civ, and quantitative-studies-but-not-specifically-calculus.
At the Johns Hopkins University, for example, students need to take at least twelve credits in either natural sciences, quantitative studies, or engineering. You can complete a history degree without ever setting foot in a math class, but in that case you're learning a lot of chemistry or something.
And the "quantitative studies" area includes courses like "statistical analysis," "computing for engineers and scientists," "logic" (from the philosophy department), or "econometrics."
And ROTC is now optional.
Right or wrong? Well, that depends on the field. For example: Are our traits dependent on our genes or our environment? Due to the fact that every human being has both genes and an environment, and that these vary wildly, this question is a philosophical one. We can certainly not rule out that our traits are solely dependent on our environment, but we CAN know what happens if this is the only explanation you tolerate. Trophim Lysenko managed to be the cause of major starvation in Soviet because of his firm belief that environmental traits were inheritable. He believed that if a crop was exposed to cold enough climates, it would grow cold-resistant. Normally, a small-scale test would be made, and could be quite justified if you had some kind of support for the theory, but Lysenko thought on a far bigger scale than that. IIRC, it was wheat he wanted people to grow, everywhere, so that the place where the crop adapted could be found. Because of this: Millions starved. It may not be easy to say when a philosophical theory is right, but sometimes it's not difficult at all to see when it is WRONG.
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For your edification a link to my school's forestry school. UW Madison's school of Wildlife Management was the first in the world and it was created (and headed) by none other than Aldo Leopold. It requires no math higher than basic statistics, algebra, and trigonometry. You've been had, sir.
As for the other thing, I guess I lucked out. I had a professor for Anthropology of Myth, Magic and Religion who was singularly awesome. Not only did he acknowledge temporal influence on religious thought, but his line of reasoning was that was ALL that it was: all religious thought is either a reaction to external stimuli or navel-gazing.
My Indigenous Lit teacher on the other hand:
It came up in class that Koyukon Athabascans have a flood myth that is eerily similar to Noah's flood (raven built a raft and put two of every animal on it) and I suggested it was probably religious syncretism. In other words, they probably had a flood myth, since the rivers they live and subsist on flood every spring, but that early missionaries (which are historically verified!) probably brought with them the bible and they incorporated the idea of a raft and animals into their existing mythology.
Well, aside from getting uppity about my calling their beliefs "mythology", she said that so and so Koyukon elder says the story is older than the white man and she has no reason to doubt an elder. I was gobsmacked! She later insisted that she doesn't believe in objective reality. Well, there you have it, I think we've found our problem!
Also, a good deal of our homework assignments/projects have been art projects. This is a lit class and requires comm B to be fulfilled before you take it so I assumed there would be writing. Which is fine, because I'm a decent writer. But it's not a freaking art class, why do I have to draw a "deep, meaningful, COSMIC representation of the relationships between X, Y and Z"?!