## Why is calculus considered so complex?

For the discussion of math. Duh.

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Yakk
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### Re: Why is calculus considered so complex?

The above presumes, however, that the material wealth and ability to technically innovate in a society are immaterial to the well being of a society.

Calculus, as mentioned, produces exponential dividends.

A merely linear boost is insane: when someone invents or designs a widget that makes 1 billion people's lives 0.1% better (heck -- not even better. Let's call it richer!), that person is boosting world wealth by 1 million times more than the typical person.

And when one person invents something that makes the entire world's wealth grow 0.001% faster, or makes innovations a fraction of a percent more likely, the results are even more ridiculous.
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voidPtr
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### Re: Why is calculus considered so complex?

Yakk wrote:The above presumes, however, that the material wealth and ability to technically innovate in a society are immaterial to the well being of a society.

Calculus, as mentioned, produces exponential dividends.

I don't think anyone would disagree with this, especially the engineers, scientists, and social scientists who highly depend on calculus in their working lives. We're not talking about those people though -- they will take calculus because they need it -- we're talking about the non-math orientated person and the basic necessary maths they need to be a well-rounded educated person and perhaps excel in other non-mathematical areas more suited for them.

Virtually everyone needs a background in basic statistics in their daily lives. We're constantly bombarded with studies and probabilities and asked to make informed decisions based on them. (A hypothetical situation: Imagine that a news report announces that randomly selected individuals at airports will be selected for a Swine Flu test that is 99% accurate. What does that really mean? If someone tests positive on the test, they have a 99% chance of having swine flu? That's very doubtful. ) For those people, although calculus may be a nice intellectual pursuit, if they don't use it in their working lives it probably will not carry the same amount of value to them as an individual or society as a whole.

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### Re: Why is calculus considered so complex?

Right. A working knowledge of standard deviations and Bayes' theorem would be worth so much more to the average person than calculus. (Again I'm going to go out on a limb here and isolate the most important sub-things.)

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### Re: Why is calculus considered so complex?

The problem with making higher math non-essential is that you then miss people who avoid it because they think they won't like it. You have to expose them to that sort of mathematics before you know whether or not they'll be good at it, and some won't take the opportunity.
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### Re: Why is calculus considered so complex?

I don't think I know a single person who became interested in higher math because of their high school calculus class. The usual routes I can think of are through extracurricular activities: the local math club, competitions, reading, etc. There are a lot of ways to expose kids to interesting mathematics, but what I'm saying is I think the educational system has other priorities. The needs of the 0.1% of the population who would really benefit from an intense math curriculum can be met through other channels (summer programs and the like).

Edit: I do know people who became interested in higher math because they had a really good teacher.

achan1058
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### Re: Why is calculus considered so complex?

Sir_Elderberry wrote:The problem with making higher math non-essential is that you then miss people who avoid it because they think they won't like it. You have to expose them to that sort of mathematics before you know whether or not they'll be good at it, and some won't take the opportunity.
Tell them this helps them win at poker, it probably will get many of them interested. Just don't let the parents or ministry know. In fact, I wish the combinatorics/probability section in our curriculum can push further into this. Instead of just count how many ways can you get a flush out of all 52 possible cards (which is in our curriculum), ask them how many ways given what you have and 4 cards down the river, and what is the probability of winning.

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### Re: Why is calculus considered so complex?

TheQntty wrote:If you don't want calculus to seem complex then do some real analysis

Ironically in complex analysis things wind up much simpler because every differentiable function is analytic.

DuncanMcG wrote:
headprogrammingczar wrote:You mean vector calculus? I consider myself very good at calculus, but god that class was horrible.

That was pretty much my experience. Multi-variable was tough, especially when you get to all the different fundamental theorems.

In my experience one of the major barriers in multi-variable calculus is that people don't keep straight the fact that there are 3 ways of specifying any given surface (as a function, as a level curve, or parametrized), and you can solve most problems with each approach. Those solutions will look entirely different, and so you have to keep the separation clear. If you don't, it can get very confusing.

It doesn't help that people who know the subject will have a pretty good idea which approach will work best for each problem. And so will use that one. Leaving students who try to read the solutions asking themselves how they are supposed to figure out which approach to use.

When I was a TA for that class I put out the effort to write a solution set that solved all of the problems using all 3 solutions. That way people could see that one approach was better for a given problem, sure, but they could also see how to finish it with whatever approach they tried. I got a lot of good feedback from that.
DuncanMcG wrote:As a math major at my university, I always get questions about why I bother to study such a horrible subject. Maybe people are so afraid of math, and especially calculus, because figuring it out just takes repetition. You get better by doing more of it. Not to say that this isn't true for other subjects, but other subjects get fun faster. At least in physics, music, English, and petty much everything else you get to do stuff (often with your hands) other than stare at a chalk board.

My answer to that is that I prefer to work my brain really hard for a short time than to have to work for a long time. There is a minimum amount of time needed to write a 20 page essay. A set of math problems can be much faster if you are on. (If your brain doesn't engage, then you can spend forever being stuck as well...)

voidPtr wrote:I think twenty-thirty years ago people probably spoke of "algebra" or "trigonometry" as the dreaded end-all be-all math course.

You'd be wrong about that. In my memory it was Calculus that was talked about with dread. In comments in books from the 50s, it was still calculus.
voidPtr wrote:I suspect calculus pedagogy has vastly improved over the last few decades and calculus is becoming more "mainstream" as time passes, and it will eventually be the norm that it is taught in high schools to regular level students and assumed knowledge by the time students start university-level maths. Maybe a couple decades from now most people will talk about the dreaded "DEs" class.

My impression is that pedagogy has not improved. In fact, if anything, I suspect it has gotten worse. The move these days is towards having calculus taught in high school by teachers who don't really understand it to students who didn't really learn algebra solidly enough.

achan1058 wrote:Truth be told, (some) professors are notoriously bad at arithmetic and algebra. Anyways, for most of my experience, (I did tutoring before and am currently a TA) it isn't that people are not capable of learning it, or that there are too many rules to remember, but instead they are trying to remember too many rules. With so many different rules, of course one is bound to get confused. If they can only remember less, (but remember the important bits) their results will improve. In fact, I personally never remembered the quotient rule, nor the derivative of arcsin/arccos/arctan, and have to re-derive them all the time. But seeing how it only takes 2 minutes to derive these, it was no great loss of time to me, but it eased my memory a lot. In fact, one can probably summarize the whole course in a dozen or so pages, which is probably impossible for a course like biology.

My brother wanted to take a course that had a calculus prerequisite and didn't have it. The professor was willing to waive the prerequisite, but he wanted to learn some calculus. I taught him what he needed to know in about an hour.

As for the original question, I think that calculus is the meeting of two problems for most people. The first is their belief that simple = easy. Well math can be simple and hard at the same time. The second is that many students try to just learn the formulas for the test rather than understanding the subject, and calculus has so many more formulas than previous subjects that this strategy falls apart for a lot of people. They then blame the subject rather than their learning strategy.
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### Re: Why is calculus considered so complex?

DuncanMcG wrote:
headprogrammingczar wrote:You mean vector calculus? I consider myself very good at calculus, but god that class was horrible.

That was pretty much my experience. Multi-variable was tough, especially when you get to all the different fundamental theorems.

I've never taken multivariable, but do your teachers really not even hint that all of the fundamental theorems are a special case of one totally awesome theorem? Differential geometry is not only a beautiful subject, it's also the most elegant language in which to talk about, say, Maxwell's equations (the only thing I've had to use multivariable for so far). Divergence, gradient, and curl are all also special cases of a single operation.

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### Re: Why is calculus considered so complex?

I'd say the main reason Calculus is considered so complex is the fact that it revolves around the concept of infinity. Throughout every level of math up to Calculus, there was almost always a definite answer or number. Suddenly calculus comes in with the slope between two infinitely close points as a derivative, and the integral of the sum of infinitely smaller sections underneath of a graph. It might make calculus easier if derivatives were at first described as simply "The slope of a graph at exactly one point" so as to understand the basic meaning, and then the mathematics of taking the limit of the slope as the two points converge infinitely is later explained once they understand what the answer is supposed to be. Same with integrals, if it was simply explained first as "the area underneath the graph, which is the reverse of the derivative", it'd be a lot simpler than "the sum of an infinite number of infinitely smaller sections of space underneath the graph.

TL;DR: Calculus' constant use of Infinity is what separates itself in complexity from other levels of mathematics.
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### Re: Why is calculus considered so complex?

Andioje wrote:I'd say the main reason Calculus is considered so complex is the fact that it revolves around the concept of limit.

Fixed. To me, the two concepts are only slightly related. But I will agree that the definition of limit is more complicated than definitions seen by most students before calculus - hell, just look at the number of quantifiers (3!) in the definition.
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Cleverbeans
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### Re: Why is calculus considered so complex?

I think Calculus gets it's reputation as difficult because it's the first set of college math and the public education system does an abysmal job of preparing students for it. I doubt I could if even one in ten thousand high school seniors could give a reasonable definition of 'number'. Building a curriculum devoid of definitions, proofs, rigor and motivation is mindlessly short sighted, and it shows.

As an alternative hypothesis, it could be that people as a whole are just incredibly stupid. The empirical evidence is definitely biased toward that conclusion....
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### Re: Why is calculus considered so complex?

Cleverbeans wrote:I doubt [...] if even one in ten thousand high school seniors could give a reasonable definition of 'number'. Building a curriculum devoid of definitions, proofs, rigor and motivation is mindlessly short sighted, and it shows.

As an alternative hypothesis, it could be that people as a whole are just incredibly stupid. The empirical evidence is definitely biased toward that conclusion....

a) I think we would get several hundred conflicting definitions on this board. I'm starting a thread to find this out.
b) We already knew this, as you stated, empirical evidence abound and all that. I am curious to see if anyone in the arts / media will be pro-math to finally overturn all of the propaganda that allows people to excuse themselves as people who just "hate math." All we really need to do is show parents how to teach their children everyday math stuff (cooking, measurements, etc.), and generation after generation, we might be able to turn the tide. It would certainly be easier and more compassionate than attempting to perform acts of Social Darwinism on the less math-prone groups. Of course, the lottery (a tax on those who do not know/understand math) already does that well enough.
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### Re: Why is calculus considered so complex?

Cleverbeans wrote:As an alternative hypothesis, it could be that people as a whole are just incredibly stupid. The empirical evidence is definitely biased toward that conclusion....

Stupid != Not liking/understanding calculus or higher math in general
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### Re: Why is calculus considered so complex?

Well I read the first 15ish posts.

My 0.02¢: some people are just better than others at math. Specifically higher levels of math. I've tried helping close family with math many, many times. I just get it, and person P does not. In my opinion, a large amount of people can only solve problems very similar to those they've been taught to solve. These people dread "word problems" and complain of tests over material "they've never been taught". From this I say that "good at math" just means "logical, intuitive, and resourceful". I can look at an equation I've never seen before, and solve it. I'm thinking small here. Say I see ax^y + bx = c. Maybe I can solve it. Person P has never seen a formula of that kind, and therefore doesn't know some predefined recipe to solve it. I'm not trying to be a dick, but I don't get why people don't get math.

On other news, I can't tell you which way is right and which way is left (you might be surprised). I can't figure out how to get from point A to point B without either GPS or a map. The only way I know how to get from place to place is by using an imaginary map in my head. If the location or crossroad is not on my mental map, I will not be arriving.

I think some people are just "good at" certain things.

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### Re: Why is calculus considered so complex?

andrewxc wrote:Of course, the lottery (a tax on those who do not know/understand math) already does that well enough.

A common misconception. Obviously you haven't heard of nonlinear utility functions before.

I don't understand this tendency to oversimplify...
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### Re: Why is calculus considered so complex?

crzftx wrote:I think some people are just "good at" certain things.

I think this is a very self-destructive philosophy. It's certainly true that talent and skill aren't distributed equally, but nobody would say that some people are just "good at" swimming and other people will just "never get it" (or at least, I don't think they would). I'm pretty sure most people are of the opinion that anyone can learn how to swim if they put time and effort into it. (Full disclosure: I can't really swim, but that hasn't stopped any of my friends from trying to teach me.)

The problem is that most Americans had such a poor mathematics education that they have become convinced that mathematics is something that they just cannot do and therefore that they shouldn't pay attention to it. I've been told that when mathematicians talk about their work to Americans, they say "oh, I was terrible at that at school," whereas in eastern Europe (for example) people will say "oh, mathematics! That was my favorite subject in school!" Even the taxi drivers!

The bigger problem is that some adults literally never learned how to think abstractly, and this isn't just true of mathematics: it's simply true of how they think about anything.

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### Re: Why is calculus considered so complex?

andrewxc wrote:b) We already knew this, as you stated, empirical evidence abound and all that. I am curious to see if anyone in the arts / media will be pro-math to finally overturn all of the propaganda that allows people to excuse themselves as people who just "hate math." All we really need to do is show parents how to teach their children everyday math stuff (cooking, measurements, etc.), and generation after generation, we might be able to turn the tide.

Unfortunately there is a lot more money to be made by just letting them be stupid and producing shows that whip them into a sexual fervor so we can hock them cheap, useless garbage on credit while charging 10% interest compounded biweekly. If I seem somewhat bitter, well, it's been a rough week. I found out they don't even teach prime numbers till high school in California if at all, and I spent the better part of an afternoon trying to teach a college student what a string was because they refused to acknowledge that they'd heard of an alphabet. I literally had to walk them through their ABC's because they were so afraid of the word 'math' they completely lost their mind.

As for turning the tide, I really don't think we need to teach them more 'practical' math at all, just show them the theory behind what they already know and teach them the axiomatic structure at an earlier age. Certainly you can start on Euclid's Elements by age 10 or introduce some basic abstract algebra and linear algebra in junior high. Consider the following groups - [imath]\unicode{x2124}/12 \unicode{x2124}\[/imath] and[imath]\ \unicode{x2124}/60\unicode{x2124}[/imath] Look familiar? Almost everyone learns to manipulate them in kindergarten if not before. I can teach a four year out about the symmetric groups with a couple colored blocks, and equivalence relations and partitions to anyone who know the word 'same', hell my two year olds can sort blocks by color, size and shape. So why is the theory put off for nearly two decades? It seems backwards to put off the foundations till after people have forgotten how to think on simple things. The expectation of complexity is blinding, so best to show it to them before they learn helplessness.

Oh and as for this....

Sir_Elderberry wrote:Stupid != Not liking/understanding calculus or higher math in general

Of course, let me rephrase... [imath]Stupid \rightarrow not\ liking\ or\ understanding\ higher\ math \rightarrow perception\ of\ complexity.[/imath] Better?
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### Re: Why is calculus considered so complex?

t0rajir0u wrote:The problem is that most Americans had such a poor mathematics education that they have become convinced that mathematics is something that they just cannot do and therefore that they shouldn't pay attention to it. I've been told that when mathematicians talk about their work to Americans, they say "oh, I was terrible at that at school," whereas in eastern Europe (for example) people will say "oh, mathematics! That was my favorite subject in school!" Even the taxi drivers!
This.

In those countries, and Asia, it is culturally unacceptable to hate math. Even if you aren't exactly good at it, you can't say you hate it or you are never good at it. I think we need to reinforce this kind of attitude in North America.

romulox
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### Re: Why is calculus considered so complex?

For me the term "good" as applied to mathematical skill is too ill defined. For one, there should be a distinction between proof based mathematics for college/university and above, and the calculating mathematics that is usually seen in American high school and equivalent.

Regarding the process:

There is something to be said for people responding well to positive reinforcement. If mathematics is taught in a way that is aligned with how a person effectively learns, in general, they will do well. This idea should not substitute teaching and/or the stimulation of thought toward a problem. In some sense mathematics is only learned by doing.

Personal interest in the material is also a strong motivating factor for success with the material. To some extent this can overshadow short run negative or null reinforcement. Though, in the long run, constant negative reinforcement should eventually overcome personal interest. Similarly, strong personal interest should eventually overcome short run negative reinforcement

Willpower is another aspect which can sufficiently motivate persons to success, though after some point, willpower will not solely be enough to be mathematically successful.

Regarding calculation, while important, I would say the ability to calculate and decipher contrived word problems is less logical prowess and abstract thinking and more finding which equation is being discussed and applying some formula to receive some answer. This is useful come tax season, or in calculating a tip, but is not useful for dealing with Hilbert space or 3-spheres.

Note: There are exceptions (i.e. the engineer, the accountant, the foreman, etc...) who need to understand how (or that) complex mathematical objects behave in certain ways, and be able to calculate various items correctly. These are not mathematicians in the proof based sense of the word.

Regarding proof based mathematics: Mathematics presents a collection of relationships (usually equivalence) of different objects using a particular language. To a large extent this language has been molded by similar (in a very broad sense of the word) minds. There could be an argument that people whose brains naturally interact using this language will appear adept at understanding these relationships more quickly than others who would require a translation. In this sense, the language implicitly accepts or discards persons based on the positive or negative response, personal interest, and how your brain interacts with the language. That is, there is the assumption that if persons who are interested in these relationships of objects could have them translated to the language their brain interacts best with, they could understand and be successful interacting with these relationships.

To some extent, one must be able to discard the "everyday" reality as being necessarily in correspondence with mathematical reality. Sometimes these two worlds collide sometimes they don't. For proof based mathematics collections of rules govern whether or not objects exist. The ability to deal with this separation psychologically will be useful to the successful mathematician.

For some people, the notion of using logic to discuss relationships between objects is absurd. To others it is the most natural way to interact.

Regarding the initial question I would say, calculus is a broad sieve. It begins to incorporate portions of proof based mathematics, but in general, is still heavily determined by calculation. I would note that it takes a while to truly understand the intricacies of calculus and why they are important. It introduces concepts which begin to stretch the intuition, like limits, and convergent infinite series. Calculus also emphasizes the use of abstraction. Each of these additions will slowly discard persons who are not sufficiently interested, comprehending of the material, motivated, or well enough taught. The next iteration of the sieve will be an introduction to proof style class.

If success is measured by a grade at the end of the term, then I would say there is no one key to success. Some people will be taught well (read, in the same manner in which they interact with the world) others not, some will be interested enough to overcome poor teachers while some won't be, some are just lazy and don't care or aren't interested. If success is measured differently then I would say some of the other comments regarding proof based mathematics begin to kick in as one progressively moves beyond calculation.

These are just some thoughts I've gathered over the years. It should be noted that in High School (American) I flunked algebra and geometry (got credit by taking a course equivalence test) and entered college (university) as an English major. After switching majors many times (English, economics, history, computer science) I finally settled on mathematics after taking a Number Systems course taught using the Moore method which axiomatically developed the reals via limits of Cauchy sequences. I'm currently a grad student in mathematics.

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### Re: Why is calculus considered so complex?

romulox wrote:After switching majors many times (English, economics, history, computer science) I finally settled on mathematics after taking a Number Systems course taught using the Moore method which axiomatically developed the reals via limits of Cauchy sequences.

What do you think about the Moore method specifically? From what I've heard, it seems unnecessarily restraining. Being presented axioms without motivation is just not how mathematics is actually done. I suppose it's healthy to force students to reconstruct things by themselves every once in awhile, but did your teacher go over the context and the history after all was said and done?

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### Re: Why is calculus considered so complex?

t0rajir0u wrote:What do you think about the Moore method specifically? From what I've heard, it seems unnecessarily restraining. Being presented axioms without motivation is just not how mathematics is actually done. I suppose it's healthy to force students to reconstruct things by themselves every once in awhile, but did your teacher go over the context and the history after all was said and done?

It may not be how math is done, but as a learning tool I can see the merit. Moore's method reminds me of the means Kató Lomb used to learn languages. Rather than using texts and dictionaries, she would pick a up novel or textbook and try to figure it out on her own. She found that the words she learned without help stuck much more forcefully.

My experience with math is much the same, those ideas and theorems I've come to discover on my own have a much more enduring place in my memory. The epiphany experience rewards the learner, and trains the mind effectively. Context and historical motivation are of course important to original work, and can encourage someone to pursue a certain line of learning. However I don't really need to know anything about closed form solutions to polynomials to enjoy and prove a sizable portion of the basic theorems on groups once I've decided I want to learn about them. Thinking earnestly on a problem builds intuition, that's the main advantage I see for employing Moore's method. The sort of reasoning skills built in this manner are far more practical then being handed the quadratic formula for a 'plug and chug' session.
Last edited by Cleverbeans on Mon May 25, 2009 8:22 pm UTC, edited 1 time in total.
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### Re: Why is calculus considered so complex?

In my experience, there's always something "harder" than what you've done before. After algebra, there's formal calculus, then functional analysis, after that there's measure theory, probability, etc... Calculus is as a good point as any other to be considered complex.

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### Re: Why is calculus considered so complex?

t0rajir0u wrote:I can't really swim, but that hasn't stopped any of my friends from trying to teach me.

Better get on that if you're planning to graduate (or did MIT get rid of that requirement? That would make me sad).

I've been told that when mathematicians talk about their work to Americans, they say "oh, I was terrible at that at school," whereas in eastern Europe (for example) people will say "oh, mathematics! That was my favorite subject in school!" Even the taxi drivers!

This has broadly been my experience when I have travelled to Europe and China. Not everyone, but a lot more often than I get it in the US.

The bigger problem is that some adults literally never learned how to think abstractly, and this isn't just true of mathematics: it's simply true of how they think about anything.

Yes.
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### Re: Why is calculus considered so complex?

What do you think about the Moore method specifically? From what I've heard, it seems unnecessarily restraining. Being presented axioms without motivation is just not how mathematics is actually done.

I've had four courses which used the Moore method so far: Number Systems, 2 Analysis courses, Metric Spaces.

Each used a modified form of the method. The key components in each of them from the students perspective was self discovery (i.e. don't look at other peoples' work, books, papers, don't ask other people, etc...), and presentation. From the professor's perspective the theorems, propositions and lemmas should be arranged in an achievable manner for the students while not hindering students mathematically.

In the first three courses, the professor would maybe minimally introduce an idea, and hand out a 1 page sheet of definitions theorems and propositions. Students are only allowed to use theorems which they (as a class) have proven. As a homework assignment each class we would be assigned the next two or three theorems from the page for the next class. During the class, a student is required to defend his proof of one of the particular theorems. Other students should be the main source of criticism with the professor there only to refocus or broadly guide the discussion and only when necessary. Ideally the classes should be no larger than 6-8, but can be larger depending on how the presentation/defense of proofs is undertaken. The atmosphere of the class should be a mathematical one, and the students should have similar mathematical backgrounds.

In the last course, metric spaces, we used a book. The professor provided a great amount of historical background for most of the topics and would introduce them in great detail often answering any questions. During the course of the discussion, theorems and propositions would arise concerning portions of the theory; these would become homework problems. The class was twice a week, and lecture would be one class period, while the presentation of proofs and their defense was another period. Much of the above concerning presentation and self discovery still applies to this course.

For my personal interpretation, I think the moore/modified moore method provides a much more intimate course of action with particular mathematical ideas at the cost of visiting less material. Much of what may seem arbitrary about mathematics is often removed when one is required to prove each of the steps up to a major theorem, instead of simply reading someone's proof. As students mature in mathematical ability I think it is possible to relax some of the restrictions as was done in my metric spaces course. Also success requires both the professor's belief in the system and the students strong engagement in the class.

As an example, the exercises for Ch. 9 of Principles of Mathematical Analysis by Walter Rudin, discuss counter examples and places where analysis of the real line fails to apply to R^n, as well as, some use of theorems taught. Chapter 9 as taught in a traditional setting would assign exercises as a homework component and have tests covering the meat of the theorems, while class time is just lecture. Chapter 9 as taught in a moore method setting would focus on the theorems as the exercises, possibly including the counterexamples as false theorems throughout the course. The focus for the professor would be for example decomposing the Inverse Function Theorem and building a trail of propositions which develop into it. One could probably make 50-70 small theorems out of the 20-30 Rudin uses to introduce the IFT. The focus for the students would be proving these theorems and working toward a componentized version of the IFT, which they may or may not know they are proving.

Also having to present work and defend it is a much stronger motivator for most people than simply turning in exercises to have them graded.

This method will not work well in every course and is not suited for all subjects in mathematics. It is not the end all for teaching mathematics but it can provide a concrete foundation on which to build research in some areas. As students mature mathematically some of the restrictions can be relaxed. The main focal points of self discovery and presentation should remain throughout a mathematical career be it in research or teaching. What is lost as the student develops mathematically is the professor providing the nice path of theorems which lead to the big theorem at the end.

These are just some thoughts from my personal experience.

-romulox
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t0rajir0u
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### Re: Why is calculus considered so complex?

Interesting. Self-discovery is certainly a healthy part of the mathematical process; I attended a summer program that consisted in large part of a guided tour towards the proof of quadratic reciprocity via us working through the basics of elementary number theory mostly by ourselves. I can see how that would be a valuable experience for other subjects if supplemented properly.