# My Calculus Rant **URL:** \ **Category:** Great Debates\ **Created:** [March 12, 2019, 7:34pm UTC](https://boards.straightdope.com/t/my-calculus-rant/831020 "2019-03-12T19:34:29Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![Derleth](https://avatars.discourse-cdn.com/v4/letter/d/b9e5f3/32.png) [@Derleth](https://boards.straightdope.com/u/Derleth)\ **Post date:** [March 12, 2019, 7:34pm UTC](https://boards.straightdope.com/t/my-calculus-rant/831020/1 "2019-03-12T19:34:29Z")
> [@Why isn't implicit differentiation using infinitesimals taught earlier?](https://boards.straightdope.com/t/why-isnt-implicit-differentiation-using-infinitesimals-taught-earlier/830933/56): > > To play devil’s advocate, why should this be taught at all? Here’s the post which sparked it. In the interest of not hijacking that thread, I’m making this one. Historically? Because calculus is the royal road to physics and engineering. It’s impossible to do physics and engineering without calculus, so if your Applied STEM field is pretty much engineering, routing all of your little collegelings to calculus is a pretty safe bet. Sure, the few people who go into pure math and focus exclusively on algebra will be peeved, but you’ll have plenty of bridge-designers. From there, it’s a straight shot to making calculus the place where you ensure your freshmen can communicate mathematically, the equivalent of undergraduate English courses making sure everyone in the school knows what a paragraph is. But, of course, the world has changed since then, and now Applied STEM leans more programming-centric, which is a fundamentally algebraic field. Being able to do the old-fashioned Advanced Counting is more important than being able to integrate a continuous function in terms of figuring out algorithmic complexity. However, calculus is still the intake course, so everyone’s forced to take it because nobody else wants a huge number of The Great Unwashed in their lecture sections. Personally, I think grade school math should build towards a good understanding of statistics, simply because statistics is useful for everyone and morally necessary to prevent people from being mislead, and that college courses should diversify their course progressions so CS students don’t have to waste time with the chain rule. ---
**Author:** ![Bryan\_Ekers](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bryan_ekers/32/183_2.png) [@Bryan\_Ekers](https://boards.straightdope.com/u/Bryan_Ekers)\ **Post date:** [March 12, 2019, 8:01pm UTC](https://boards.straightdope.com/t/my-calculus-rant/831020/2 "2019-03-12T20:01:05Z")
Reached your limit, have you? ---
**Author:** ![Stranger\_On\_A\_Train](https://avatars.discourse-cdn.com/v4/letter/s/13edae/32.png) [@Stranger\_On\_A\_Train](https://boards.straightdope.com/u/Stranger_On_A_Train)\ **Post date:** [March 12, 2019, 8:03pm UTC](https://boards.straightdope.com/t/my-calculus-rant/831020/3 "2019-03-12T20:03:25Z")
You understand that integral calculus is inherent in any statistics involving continuous distributions, right? Even if you assume that a working knowledge of calculus is not necessary to perform the turn-the-crank operations of statistical analysis, it is necessary to understand why particular distributions are or are not appropriate for a specific type of problem. Being able to interpret statstical analysis is certainly a useful ability that anyone who makes decisions based on quantitative data should be able to do, but it is also important to understand the limitations and assumptions of those applications. From a more fundamental standpoint, integral and differntial calculus are inherent in all physical phenomena, and are also the first (and often only) non-analytic geometry applications of trigonometry that most people experience in stem fields. Understanding applications of calculus and differential equations to different areas of science prepares students to understand the theory behind ideas ranging from chemical reaction rates and thermodynamics to digital signal processing and climate circulation modelling. From an intellectual standpoint, calculus is the first introduction into the idea that mathematics isn’t just a mechanical tool for performing calculations but is a different way of learning and thinking. It is to STEM fields /what the Classics and Shakespeare are to literature; you don’t need to have them to know how to read and right, but you are missing almost all of the context about where our language and semantics of Western culture come from. Programmers and software engineers (which is what most computer science departments seem dedicated to churning out) may not need to know any calculus to do their jobs, but then, they don’t actually need to know much actual computer science theory like discrete mathematics, either, unless they are programming at a bare metal level (and even then very few programmers actually have to write out or interpret instructions at the assembler level). The notion of higher education is to prepare students to do more than just grind through rote rules to produce a product, but instead to be able to understand the fundamentals behind what they do so that even if they aren’t advancing the state of the art they can at least understand and evaluate whether a concept is fundamentally workable. > [@Bryan\_Ekers](#): > > Reached your limit, have you? _golf clap_ I should have anticipated such a derivative comment from you. Stranger ---
**Author:** ![Airbeck](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/airbeck/32/6574_2.png) [@Airbeck](https://boards.straightdope.com/u/Airbeck)\ **Post date:** [March 12, 2019, 8:04pm UTC](https://boards.straightdope.com/t/my-calculus-rant/831020/4 "2019-03-12T20:04:28Z")
Meh, struck me as derivative ---
**Author:** ![Airbeck](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/airbeck/32/6574_2.png) [@Airbeck](https://boards.straightdope.com/u/Airbeck)\ **Post date:** [March 12, 2019, 8:06pm UTC](https://boards.straightdope.com/t/my-calculus-rant/831020/5 "2019-03-12T20:06:22Z")
Hey I just got reverse ninja’d by an edit. That’s not fair… but maybe it’s not rational to be upset by that… ---
**Author:** ![Pleonast](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pleonast/32/1183_2.png) [@Pleonast](https://boards.straightdope.com/u/Pleonast)\ **Post date:** [March 12, 2019, 8:22pm UTC](https://boards.straightdope.com/t/my-calculus-rant/831020/6 "2019-03-12T20:22:08Z")
Here is what I think the priority on mathematics education should be.1. Arithmetic. More measurement and estimation. Less fractions. 2. Financial. More price-per-units, interest and rates, and double-entry book-keeping. 3. Algebra. More set and group theory. Less proofs. 4. Statistics. More counting, probability, priors, confidence, etc. 5. Geometry. More construction. Less proofs. 6. Computation. More numerical analysis and error propagation. 7. Calculus. More multi-dimensional. Less proofs.Every grade level should get all of these, because they’re interdependent. Emphasis should always be on problem-solving, not proofs for the sake of proof. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [March 12, 2019, 8:36pm UTC](https://boards.straightdope.com/t/my-calculus-rant/831020/7 "2019-03-12T20:36:37Z")
> [@Derleth](#): > > Here’s the post which sparked it. “Why should this be taught at all?” implies that maybe it shouldn’t be taught at all—that no one, anywhere, should learn Calculus. This position is silly and indefensible. Are some people being required to study Calculus that shouldn’t be? Yeah, I guess, but I’d like to see some specifics. It would make for a more fruitful debate if we could specify _of whom_ or _for what_ Calculus shouldn’t be required. I think there’s a growing consensus nowadays that _too many_ kids are taking Calculus in high school. See, for example, [here](https://www.edweek.org/ew/articles/2018/05/23/calculus-is-the-peak-of-high-school.html) or [here](https://www.maa.org/the-changing-face-of-calculus-first-semester-calculus-as-a-high-school-course). > [@](#): > > The pressure to take calculus in high school is understandable. Competition for admission to the best colleges and universities is fierce. It has helped to create strong growth in AP programs across the board. Many mathematicians deplore this movement of calculus from the college to the high school curriculum, but the pressures are too strong to stop or even substantially slow it. What we can hope is to shape it. > With this in mind, the presidents of the MAA and NCTM issued a joint statement in 1986 [3] with two strong recommendations which I paraphrase here: > > 1. In spite of the pressures to take calculus while still in high school, students should never short-change their mathematical preparation in subjects such as algebra, geometry, or trigonometry. Solid mathematical preparation is far more important than exposure to calculus. > 2. When calculus is taught in high school it should be a college-level course. This means that the goal of the course should be to give students the same breadth of topics and mastery of calculus obtained by students taking such a course in college. It means that the course should be taught with the expectation that students who perform satisfactorily will be able to place into the succeeding college calculus course. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [March 12, 2019, 8:42pm UTC](https://boards.straightdope.com/t/my-calculus-rant/831020/8 "2019-03-12T20:42:26Z")
> [@Pleonast](#): > > 1. Arithmetic. More measurement and estimation. Less fractions. How are you supposed to do measurement and estimation without fractions? Are all measurements supposed to be whole numbers? > [@](#): > > 1. Financial. More price-per-units, interest and rates, and double-entry book-keeping. How much of this is actually math, and how much is economics, accounting, or record-keeping? > [@](#): > > 1. Algebra. More set and group theory. Less proofs. ? > [@](#): > > 1. Statistics. More counting, probability, priors, confidence, etc. How are you supposed to understand probability without fractions? > [@](#): > > 1. Geometry. More construction. Less proofs. Why? > [@](#): > > Emphasis should always be on problem-solving, not proofs for the sake of proof. What do you have against clear thinking, logic, and being able to show why something is true? ---
**Author:** ![Broomstick](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/broomstick/32/246_2.png) [@Broomstick](https://boards.straightdope.com/u/Broomstick)\ **Post date:** [March 12, 2019, 8:58pm UTC](https://boards.straightdope.com/t/my-calculus-rant/831020/9 "2019-03-12T20:58:19Z")
> [@Pleonast](#): > > Here is what I think the priority on mathematics education should be.[indent]1. Arithmetic. More measurement and estimation. Less fractions. Er… what’s wrong with fractions? I mean, sure a fundamental competence in measurement, estimation, adding, subtracting, multiplication, and division is essential to going any further but fractions (and a few other things) are also important. Not only that, but fractions have real-world applications in measuring (something you want more of) and tasks like cooking. Which people do every day. > [@](#): > > 1. Financial. More price-per-units, interest and rates, and double-entry book-keeping. Double-entry bookkeeping? Seriously? Are you going to require competence in using an abacus and a slide-rule, too? Who the hell does double-entry bookkeeping anymore? Absolutely more instruction in financial topics are a must - interest, compounding interest, saving strategies, budgeting, etc. The stuff people actually need to know. > [@](#): > > 1. Algebra. More set and group theory. Less proofs. Also more story problems and practical applications. And what’s wrong with proofs? They teach logical thinking, which is an important life skill. > [@](#): > > 1. Geometry. More construction. Less proofs. Again, proofs teach logical thinking. And you’re not going to get far with teaching “construction” if you neglected fractions earlier. If you don’t understand the utility of proofs in mathematical instruction I dare say you don’t understand teaching math. I agree that teaching practical applications of all this stuff is important, but students should at least be introduced to abstract though and theory even if they don’t wind up on a first-name basis with those concepts. ---
**Author:** ![Ruken](https://avatars.discourse-cdn.com/v4/letter/r/f475e1/32.png) [@Ruken](https://boards.straightdope.com/u/Ruken)\ **Post date:** [March 12, 2019, 9:04pm UTC](https://boards.straightdope.com/t/my-calculus-rant/831020/10 "2019-03-12T21:04:38Z")
I often wonder about what should or should not be included in a curriculum without reaching any strong conclusions. Some schools will offer math classes tailored to a specific major. I’ve seen it with physics. Not chemistry. Calc 3 was a requirement for us but we really didn’t use it much. Looks like chemical engineers typical have a lot more math required. I do agree that more stats would be good for a lot of people. Although I’m told taking AP statistics in high school looks bad, so the kids taking calculus in 10th grade need to be careful. ---
**Author:** ![Hellestal](https://avatars.discourse-cdn.com/v4/letter/h/3ab097/32.png) [@Hellestal](https://boards.straightdope.com/u/Hellestal)\ **Post date:** [March 12, 2019, 9:24pm UTC](https://boards.straightdope.com/t/my-calculus-rant/831020/11 "2019-03-12T21:24:00Z")
I’d be interested in the OP’s perspective of statistics without continuous probability distributions. I mean, sure, probability starts out discrete with most real-world applications. But the whole point of stats is that a discrete distribution can and will converge to a continuous distribution given the right conditions. Stats without convergence to a normal (“central”) distribution in the limit doesn’t seem much like stats to me. > [@Broomstick](#): > > Double-entry bookkeeping? Seriously? Are you going to require competence in using an abacus and a slide-rule, too? Who the hell does double-entry bookkeeping anymore? Everyone. Every real accountant, everywhere in the world, uses double-entry bookkeeping. The ledgers are electronic today, not fine paper bound in leather and scribbled into with a copperplate hand, but even in our digital age, bookkeeping remains firmly double-entry. The fundamental principle has not changed at all. I’m somewhat surprised that anyone would think it archaic. It hasn’t been discarded. Double-entry remains the absolute backbone of accounting. It’s hard to imagine that ever changing. ---
**Author:** ![Velocity](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/velocity/32/18006_2.png) [@Velocity](https://boards.straightdope.com/u/Velocity)\ **Post date:** [March 12, 2019, 9:26pm UTC](https://boards.straightdope.com/t/my-calculus-rant/831020/12 "2019-03-12T21:26:43Z")
If every high school student studied nothing but basic algebra and math that is related to personal finance, I think that would really suffice for 99% of the population. There is little point in studying something in high school that you won’t need; the attention of high schoolers is taxed enough already as is. ---
**Author:** ![CelticKnot](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/celticknot/32/3392_2.png) [@CelticKnot](https://boards.straightdope.com/u/CelticKnot)\ **Post date:** [March 12, 2019, 9:29pm UTC](https://boards.straightdope.com/t/my-calculus-rant/831020/13 "2019-03-12T21:29:07Z")
I didn’t even read the entire OP of the first thread before I was on my knees thanking God I never took Calculus. I was an elementary teacher. It was completely unnecessary in my life. ---
**Author:** ![Andy\_L](https://avatars.discourse-cdn.com/v4/letter/a/c67d28/32.png) [@Andy\_L](https://boards.straightdope.com/u/Andy_L)\ **Post date:** [March 12, 2019, 9:30pm UTC](https://boards.straightdope.com/t/my-calculus-rant/831020/14 "2019-03-12T21:30:03Z")
> [@Velocity](#): > > If every high school student studied nothing but basic algebra and math that is related to personal finance, I think that would really suffice for 99% of the population. There is little point in studying something in high school that you won’t need; the attention of high schoolers is taxed enough already as is. High schoolers don’t necessarily know what they won’t need, so a certain amount of exposure to a variety of subjects is good, to give them the chance to learn enough so that they can make an informed decision about whether to go further. ---
**Author:** ![Pleonast](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pleonast/32/1183_2.png) [@Pleonast](https://boards.straightdope.com/u/Pleonast)\ **Post date:** [March 12, 2019, 9:33pm UTC](https://boards.straightdope.com/t/my-calculus-rant/831020/15 "2019-03-12T21:33:31Z")
_How are you supposed to do measurement and estimation without fractions? Are all measurements supposed to be whole numbers?_ There’s this wonder invention called decimals! And, not “no fractions”, but “less fractions”. _How much of this is actually math, and how much is economics, accounting, or record-keeping?_ It’s applied math that can be used by anyone who has money. _Why? What do you have against clear thinking, logic, and being able to show why something is true?_ Nothing, which is why focusing on problem solving is better than focusing on mathematical proofs. Proofs are a tool to decide formal propositions and are rarely used by anyone outside academia. Using previously proven theorems to solve problems is an extremely useful skill, however. And is exactly the sort of thinking that students should be learning. _Er… what’s wrong with fractions? I mean, sure a fundamental competence in measurement, estimation, adding, subtracting, multiplication, and division is essential to going any further but fractions (and a few other things) are also important. Not only that, but fractions have real-world applications in measuring (something you want more of) and tasks like cooking. Which people do every day._ Fractions are important and necessary, but I feel like they’re over emphasized. Estimation can’t be done without them. But most arithmetic can be done more easily with decimals and an understanding of estimation. _Double-entry bookkeeping? Seriously? Are you going to require competence in using an abacus and a slide-rule, too? Who the hell does double-entry bookkeeping anymore? Absolutely more instruction in financial topics are a must - interest, compounding interest, saving strategies, budgeting, etc. The stuff people actually need to know._ Double-entry bookkeeping is a fundamental skill that helps anyone who uses money. Trying to teach any financial topic like saving, borrowing, and budgeting without double-entry bookkeeping is like teaching biology without evolution. It’s possible, but everything makes a lot more sense if you understand it. \*And what’s wrong with proofs? They teach logical thinking, which is an important life skill. Again, proofs teach logical thinking. \* Logical thinking is good, but there’s other ways of teaching it. Like with problem solving and other applications of mathematical knowledge. _If you don’t understand the utility of proofs in mathematical instruction I dare say you don’t understand teaching math. I agree that teaching practical applications of all this stuff is important, but students should at least be introduced to abstract though and theory even if they don’t wind up on a first-name basis with those concepts._ I’m not a math teacher, but I took 10 semesters of math classes in four years of high school and double-majored in mathematics and physics as an undergrad. Abstract thought and theory is very important, but proving mathematical theorems is not a useful skill for anyone not working in academia. I’d like to see abstract thought and theory taught in other ways. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [March 12, 2019, 9:43pm UTC](https://boards.straightdope.com/t/my-calculus-rant/831020/16 "2019-03-12T21:43:09Z")
> [@Pleonast](#): > > Abstract thought and theory is very important, but proving mathematical theorems is not a useful skill for anyone not working in academia. [Locke, Jefferson, and Lincoln would disagree.](http://nautil.us/issue/41/selection/euclid-as-founding-father) > [@](#): > > Gulliver asked Lincoln about his remarkable oratory skill: “I want very much to know how you got this unusual power of ‘putting things.’ ” According to Gulliver, Lincoln said it wasn’t a matter of formal education. “I never went to school more than six months in my life.” But he did find training elsewhere. “In the course of my law-reading I constantly came upon the word demonstrate,” Lincoln said. “I thought, at first, that I understood its meaning, but soon became satisfied that I did not.” Resolving to understand it better, he went to his father’s house and “staid there till I could give any propositions in the six books of Euclid at sight.” ---
**Author:** ![Broomstick](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/broomstick/32/246_2.png) [@Broomstick](https://boards.straightdope.com/u/Broomstick)\ **Post date:** [March 12, 2019, 9:46pm UTC](https://boards.straightdope.com/t/my-calculus-rant/831020/17 "2019-03-12T21:46:35Z")
> [@Hellestal](#): > > Everyone. > > Every real accountant, everywhere in the world, uses double-entry bookkeeping. The set of “everyone” and the set of “every real account” may overlap, but not as much as you think. The vast majority of people are NOT accountants, not even fake ones. They have no need to understand the theory and inner workings of double-entry bookkeeping. ---
**Author:** ![monstro](https://avatars.discourse-cdn.com/v4/letter/m/ba9def/32.png) [@monstro](https://boards.straightdope.com/u/monstro)\ **Post date:** [March 12, 2019, 10:43pm UTC](https://boards.straightdope.com/t/my-calculus-rant/831020/18 "2019-03-12T22:43:32Z")
> [@Velocity](#): > > If every high school student studied nothing but basic algebra and math that is related to personal finance, I think that would really suffice for 99% of the population. There is little point in studying something in high school that you won’t need; the attention of high schoolers is taxed enough already as is. 99% of the population don’t need to know a lot of the stuff taught in high school, if by “need” you mean “can’t function well in modern society”. But the point of college prep curriculum has always been loftier than that. High school students learn how to write a basic five paragraph essay even though most adults don’t have to write five paragraph essays to make a living. But it’s kind of hard to make it through college if you don’t know how to write an essay. At any rate, college prep curriculum doesn’t require calculus. At my HS, you were only required to take trigonometry to be considered sufficiently “college prep”. I do think that calculus, physics, chemistry, etc. are useful courses that are also used to gatekeep STEM fields. But even though I’m generally against “gate-keeping”, I guess I don’t have a major problem with it in this context since I kind of think problem-solving is a basic requirement of STEM. You know you can’t hack engineering calculus? OK, maybe you need to take the calc at the community college and transfer the credits. If that’s not an option, maybe you work with a tutor. Maybe you study a STEM-adjacent major (environmental studies instead of environmental science). Maybe you prepare yourself for a “D” in calc, but you compensate by acing statistics. There are ways to do STEM without studying calculus. Humanities majors are generally required to take “seemingly useless” courses, so why shouldn’t STEM majors be subjected to the same thing? STEM curricula shouldn’t necessarilly be more “vocational” than other kinds of university/college curricula. Instead of freeing students from the burden of calc, I think we should be focused on making calc instruction more engaging and accessible. I think half of the struggle I had with calculus was staying awake during the part when the instructor would prove whatever formula we were learning that day. I don’t know what would make it better, but I suspect current technology offers some alternatives to the old-school approach of “Let’s scrawl a bunch of stuff on the chalkboard with our backs to the class for thirty minutes and then feign surprise when students say they hate calculus.” ---
**Author:** ![Translucent\_Daydream](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/translucent_daydream/32/16170_2.png) [@Translucent\_Daydream](https://boards.straightdope.com/u/Translucent_Daydream)\ **Post date:** [March 12, 2019, 11:06pm UTC](https://boards.straightdope.com/t/my-calculus-rant/831020/19 "2019-03-12T23:06:34Z")
Calculus was very hard for me, but I didn’t get good at mathematics until I actually started using them on a daily basis. Statistics in college (junior level 300 series) and calculus (sophomore level 200 series) were the hardest classes for me to pass. I have to admit looking back at college 15 years later, the classes I struggled with were the ones where I was learning something I could not apply to my life. Now that I am an operations director for a company that basically couldn’t function without an understanding of Algebra, Statistics, and Geometry, I wish that all the humanities classes could have been taken then and the STEM classes were taken at my mid-thirties. I would have aced every one of them now. ---
**Author:** ![Sam\_Stone](https://avatars.discourse-cdn.com/v4/letter/s/ecccb3/32.png) [@Sam\_Stone](https://boards.straightdope.com/u/Sam_Stone)\ **Post date:** [March 12, 2019, 11:29pm UTC](https://boards.straightdope.com/t/my-calculus-rant/831020/20 "2019-03-12T23:29:17Z")
> [@Velocity](#): > > If every high school student studied nothing but basic algebra and math that is related to personal finance, I think that would really suffice for 99% of the population. There is little point in studying something in high school that you won’t need; the attention of high schoolers is taxed enough already as is. So do you apply that to history? Chemistry? Does an understanding of covalent bonds really apply to most people’s lives? How about biology? Do I really need to understand mitosis or RNA to be a functioning person? We don’t just teach people the minimum they need to function. We teach them enough to understand a complex world, to make good judgments by drawing on a wide range of knowledge, to have shared understanding between people to make society more cohesive, etc. You may never have to do an integration by hand for the rest of your life. You may also never have to do a quadratic equation, or diagram a sentence, or understand what the Iliad is. But knowing those things makes you a better citizen and helps you lead a more rewarding life. We also teach a lot of varied things in grade school so doors aren’t closed to you later. If you take nothing but ‘business math’ in high school, you are going to be out of luck if, in your 20’s you decide you want to go to college and learn to be an engineer. You learn the basics of biology so if you decide to be a doctor later on, you have the foundation needed to tackle pre-med. If you want a vocation, take a trade. Nothing wrong with that. If you want to go to college, be prepared to learn a lot more than is absolutely necessary to function as a drone in your field of choice. [Next page](https://boards.straightdope.com/t/my-calculus-rant/831020.md?page=2)