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Dismantling the calculus pyramid (wildaboutmath.com)
37 points by timwiseman 3007 days ago | hide | past | web | favorite | 45 comments



Professor John Stillwell writes, in the preface to his book Numbers and Geometry (New York: Springer-Verlag, 1998):

"What should every aspiring mathematician know? The answer for most of the 20th century has been: calculus. . . . Mathematics today is . . . much more than calculus; and the calculus now taught is, sadly, much less than it used to be. Little by little, calculus has been deprived of the algebra, geometry, and logic it needs to sustain it, until many institutions have had to put it on high-tech life-support systems. A subject struggling to survive is hardly a good introduction to the vigor of real mathematics.

". . . . In the current situation, we need to revive not only calculus, but also algebra, geometry, and the whole idea that mathematics is a rigorous, cumulative discipline in which each mathematician stands on the shoulders of giants.

"The best way to teach real mathematics, I believe, is to start deeper down, with the elementary ideas of number and space. Everyone concedes that these are fundamental, but they have been scandalously neglected, perhaps in the naive belief that anyone learning calculus has outgrown them. In fact, arithmetic, algebra, and geometry can never be outgrown, and the most rewarding path to higher mathematics sustains their development alongside the 'advanced' branches such as calculus. Also, by maintaining ties between these disciplines, it is possible to present a more unified view of mathematics, yet at the same time to include more spice and variety."

Stillwell demonstrates what he means about the interconnectedness and depth of "elementary" topics in the rest of his book, which is a delight to read and full of thought-provoking problems.

http://www.amazon.com/gp/product/0387982892/

See also

http://www.math.sunysb.edu/~mustopa/thurston_edu.pdf

with comments on mathematics education for breadth rather than for speed through the standard curriculum, by a Fields medalist.


I would second that, and add a recommendation for "Chapter Zero" which is a book targetted at the high school/early college level which provides an introduction to those fundamentals that are often skipped in the rush towards calculus.


This Chapter Zero,

http://www.amazon.com/Chapter-Zero-Fundamental-Abstract-Math...

right? I have not read this particular book, but in general most students who desire to study math should take a look at a "transitions course" textbook or two, to fill in the gaps left by studying only in the fast lane to calculus.

http://www.amazon.com/Mathematical-Proof/lm/R3LUNI80ZOUYK4/

http://www.amazon.com/Mathematical-logic-and-foundations/lm/...

http://www.amazon.com/How-solve-prove-books-useful/lm/31B4QV...


I agree agree that probability and statistics are much more important in life than just calculus. That is why in college I am studying Artificial Intelligence and Machine Learning - these are all math and probabilities.

Yet, I think Arthur Benjamin is right but for the wrong reasons:

1. People should know statistics instead of Calculus.

To know probability well you need to have mastered calculus. All the distributions, all the formulas build on things like sequences, and integrals.

2. If everybody knew statistics we wouldn't be in the economic mess.

Plain speculation. The majority people who have important part in the management of banks know statistics.

3. from continuous mathematics to discrete mathematics...

Probability and statistics involves tons of continuous mathematics and Analysis. Is a Gaussian distribution discrete? Is the Central Limit Theorem discrete?

The reason probabilities and statistics is important is because it lets people better understand why stuff around them happens and lets them reason about uncertain processes. Life is behavior under uncertainty.


I believe it is important to remember that Arthur Benjamin is referring to High School Mathematics.

To know probability well you need to have mastered calculus.

I won't argue that point itself. However, given that the audience for this mathematics education is High School Students, we won't have to go that far. Regression lines, standard deviations, and Simpson's paradox can be understood without Calculus. Personally, more time should be focused on the analysis and application (again, at a High School Level).


I would go further and not only teaching probability and statistics, but also teach basic calculus with simple polynomials, programming, Monti Carlo simulations, sines and cosines, computer simulations, all starting around age 13.

Advance algebra, geometry, and full trigonometry are not required for any of that.

Why make students wait until senior year or college to experience the fun stuff?


Bad idea. Some consequences:

* People would stop voting.

* Crime would probably increase.

* People would be less friendly/helpful to strangers in large cities.

* People would be less willing to drive (causing chaos in cities without sufficient public transportation).

* People would no longer care about environmental issues.

* Entrepreneurship would decrease.

* Stock trading would decrease.


Following the same "logic" you have:

* People would follow healthier lifestyles.

* People would avoid gambling.

* People would avoid bad marriages.

* The government would be more efficient.

An understanding of statistics wouldn't realistically change all of these things. In practice, people can always be swayed by a powerful rhetoric, emotions, and cultural habit. Statistics lend more weight to arguments, but only a tiny fraction of people will go back and verify a statistic for accuracy.


Losing democracy is a big deal though.

While it is true that many people vote due to social pressure, I suspect that this social pressure would decrease with a better understanding of probability and statistics.

You could try mandatory voting, but people could just vote randomly to avoid wasting any time thinking about the matter.


I disagree, I think once people understood voting they would be much less tolerant of the government acting in anyway and would require massive majorities to get anything done.


The vast majority of stock trading is institutional.

Do you really believe that Morgan Stanley or Citigroup suffer from a lack of understanding of statistics?


The rational reason to vote is not because your vote actually has an effect on the election's outcome. Rather, it is because if nearly no one voted, then politicians would have no incentive to take care of the needs of the people. The reason why representative democracy works (when it does work) is because the people have the power to remove politicians from office (or prevent their reelection).

Of course, it is unwise to limit your involvement in government to merely voting, but that is a separate topic.


This is a question about how much influence an individual has on others in terms of networking effects.

Moreover, it may be sufficient to have a positive attitude toward voting without actually ever voting.


> Moreover, it may be sufficient to have a positive attitude toward voting without actually ever voting.

Nice. Sort of like an election typhoid mary.

> This is a question about how much influence an individual has on others in terms of networking effects.

I agree that the influence an individual has on his social contacts is relevant here. However, I still insist that the individual's decision to vote is itself important.


Actually there is no rational reason for you, personally, to vote. However voting is a level of commitment that is sufficiently small that many are willing to put the effort out to be heard.

And yes, I do vote.


> Actually there is no rational reason for you, personally, to vote.

[snip]

> And yes, I do vote.

You believe there is no rational reason for you to vote, yet you vote anyway? So you believe that acting irrationally is a good thing to do?

I can see how we might differ in opinion about what is the rational action in a given situation, but I have a hard time understanding how we can agree on the correct action to take, but disagree as to its rationality.


Rationally the effort of you, personally, going to vote only matters if it changes the outcome. However the odds of a single vote doing that are very, very low. The odds of it changing the reported statistics even by 0.1% are also low. At a national level the odds of doing either are effectively non-existent.

Therefore rationally there is no point in an individual voting.

I know this. But emotionally I feel better knowing that I tried to use my voice, even though logically I know that the act was meaningless. I long ago came to accept that my emotions do not respond to logic, and therefore I have come to let emotion drive my goals, and then use logic to achieve those goals.

Therefore I am willing to engage in a logically pointless activity in return for the emotional satisfaction I derive. Even though I am aware that the emotional satisfaction is illogical.


> Rationally the effort of you, personally, going to vote only matters if it changes the outcome.

As above, I disagree with you about what the real purpose of an individual's vote is.

> Therefore rationally there is no point in an individual voting.

If you find that rationality and logic indicate that an individual should not vote, then you haven't used enough of them.

Specifically, if everyone was rational according to the definition you're implying, then bad things would happen. That sounds to me like it's the wrong definition.

> Therefore I am willing to engage in a logically pointless activity in return for the emotional satisfaction I derive.

Ok, that makes sense.


You're doing a lot of assuming you're right, and then concluding that anyone who disagrees with you is wrong. That's poor reasoning. In particular you say

If you find that rationality and logic indicate that an individual should not vote, then you haven't used enough of them.

Specifically, if everyone was rational according to the definition you're implying, then bad things would happen. That sounds to me like it's the wrong definition.

You have reasoned from the premise to an apparently absurd conclusion, and therefore conclude that the premise was wrong. But in fact what you should do is look farther to see whether the premise is, in fact, RIGHT! To that end I strongly recommend that you read The Logic of Collective Action by Mancur Olson. This sets out the classic theory of public goods which, among other things, concluded that very often bad things happen if everyone acts in their own self interest.

Now rather than seek a reason to reject the premise, analyze it. My claim is that if your reason for voting is X (for any particular X you want), then unless your act of voting has a chance of affecting X, you logically shouldn't spend energy voting. Why would that be? Well when you vote you have a cost (your effort) and a reward (something you care about becomes more likely). If your expended effort has essentially chance of resulting in a reward, then logically that effort was wasted.

This applies whether X is "get the election outcome you want" or "makes politicians care about the electorate" (which is the reason you gave for why people should vote) or anything else. If your vote isn't the deciding factor one way or another, then it is a waste of energy for you to vote. And that applies to both of the criteria I just listed.

In fact if you want to make politicians care it is far, far more effective to send an email, make a phone call, or send a piece of snail mail. Then you get to not just make them aware that someone is out there, but you actually get to tell them your specific concerns. Now odds are that you won't sway them. But the effort involved is on part with voting, and it is much more effective. So if you're willing to vote to make them care, you should send email about something every weekend!

When you look at things this way, my voting actually is logically defensible. After all the warm fuzzies I get from my emotional reaction to voting is a guaranteed return on effort that justifies the effort I expend. There is no question that the act of my voting makes the difference in my getting that feeling. So it makes sense for me to vote.


Are you sure you replied to the correct story. This story is about teaching more statistics and less calculus to students as it is vastly more practical for them.


These are consequences of a greater understanding of probability and statistics.


No, not really. You seem to be assuming that a greater understanding of probability would lead to decision-making that takes only probabilities into account and not, say, one's morality.

But that's a ridiculous thing to assume.


Your argument follows the same form as the one that says a world without a god is full of evil.

You overextrapolate from an overstated initial cause, and offer no evidence whatsoever for any of your bald assertions.


Hmm,

Oh where to begin?

The guy's rhetoric is pretty divorced from the sad state of actual math education today. The way that the standard math curriculum today heads up to a point in a pyramid is a problem in itself since a lot of the topics are made dull in themselves ("This is just something you need to learn for calculus" is a terrible answer for "why should I learn this?" but substituting "Statistics" wouldn't change things much). It doesn't really matter what the point of the pyramid is when most people never get there.

Just as much, the latest economic meltdown was engineered and believed in by statistics experts. For every Mandlebrot debunking the events, there were ten Myron Scholes basking in the glory of validating Wall Street's delusions with some mathematical magic. Statistics doesn't protect from wishful thinking outside of controlled, experimental situations.

And statistics in daily life? One might use some basic probability but the only other use it would have would sorting the pseudo-statistical rhetoric used by the media. A simple course in mathematical literacy with an emphasis on fallacies would be best for sorting this stuff. BUT again, no course can protect from wishful thinking, can protect people from the fallacies that let them ignore possible later dangers for immediate apparent gain. Further, a non-calculus-based statistics or mathematically literacy course isn't a basis for further scientific study the way calculus is, and believe-it-or-not some students still become physicists, chemists and engineers where calculus is indeed the foundation.

I could narcisistically say that my favorite, evolutionary game theory, would make a much better "point" for the curriculum pyramid but really, what is needed is to make every math class interesting in and of itself. With TV-dazed kids and math-apathetic teachers, I don't know if any curriculum could change things BUT I would want to have the curriculum of every class interesting and mentally challenging - taught axiomatically, Algebra and Geometry ARE interesting in and of themselves and a student needs no background at all for them. Math should be rigorous, conceptually challenge and optional past the basics. It seems like we'd need a different world for this but small steps are being made.


Firstly, Art is actually a math professor, and dealing with the low standards of incoming students is a major concern. I don't think he is "divorced from the sad state of actual math education today."

But even more, you say:

    > ... taught axiomatically, Algebra and Geometry
    > ARE interesting in and of themselves
Perhaps to you, but I deal with students for whom this is absolutely not interesting. They're not interested in puzzles, they're not interested in challenges, they're not interested in anything except texting their friends and talking about films, TV, clothes, football, etc. They don't want to be challenged.

I think we all agree that classes in general should be stimulating and interesting, andshould be better tailored to suit the needs of the individuals, but that's the problem. For a given child we don't know what they'll need, and we don't know what they'll like, and every answer will be different.

The problem is "one class fits all" and that's not going to change simply by re-writing the curriculum.


Perhaps to you, but I deal with students for whom this is absolutely not interesting. They're not interested in puzzles, they're not interested in challenges, they're not interested in anything except texting their friends and talking about films, TV, clothes, football, etc. They don't want to be challenged.

Uh, yeah I've dealt with those students. Notice the last part I add - optional. The thing that kills math interest utterly is those "required" classes which teach nothing to the uninterested.

Modern schooling drags the uninterested through a process of making motions towards understanding - we all know its a waste of time. It really would be better to give up until the students are interested. A motivated student can learn more in a day than the bored learn in the semesters of basic math. That might put you out of a job but those jobs just shouldn't exist. Sorry.


Won't put me out of a job - you've jumped to an incorrect conclusion. I'm not a teacher. I run two companies, and I go out to schools to give talks on why math is fun, interesting, useful, and occasionally exciting. Most of the students I deal with are motivated and interested, but even then, some don't like puzzles, and don't like starting from the ground up axiomatically.

This Saturday I'm talking about the Banach-Tarski theorem, and I'm starting from the result, then wroking backwards, deciding what we need to know as we peel it back. I've found that working backwards from a surprising result can create motivation to understand, but sometimes it causes the students to dismiss the whole thing as useless, pointless and irrelevant.

Sorry, I'm rambling. Reply if you're interested, ignore me if not.


Sounds like fun, I'm happy to hear about it...

I shouldn't be dogmatic about asking for an axiomatic development.

At the same time, it seems like the social attitude towards mathematics has reached the point where it would be useful for schools to ask students to put aside some of their initial attitude towards math.

The best teachers I've had often demanded more than I was initially capable of accomplishing. It's true that such teachers risked losing some of their audience. But if we don't have such teachers we risk even more.


As an aside, your profile says you're interested in contract work, but gives no way to contact you. Is that deliberate?


Best advice of this century in it's class.

I don't know why we are still wasting our time with calculus while I see a lot of people suffering in their daily life because of their lack of understanding in probability theories and statistics.

Statistics is easier and even more fun to learn than calculus.


Statistics is a very interesting subject, and it is a distinct subject from mathematics proper. Here (in what is becoming a FAQ post for HN) are two favorite recommendations for free Web-based resources on what statistics is as a discipline, both of which recommend good textbooks for follow-up study:

"Advice to Mathematics Teachers on Evaluating Introductory Statistics Textbooks" by Robert W. Hayden

http://statland.org/MyPapers/MAAFIXED.PDF

"The Introductory Statistics Course: A Ptolemaic Curriculum?" by George W. Cobb

http://repositories.cdlib.org/cgi/viewcontent.cgi?article=10...

Both are excellent introductions to what statistics is as a discipline and how it is related to, but distinct from, mathematics.

A very good list of statistics textbooks appears here:

http://web.mac.com/mrmathman/MrMathMan/New_Teacher_Resources...


Sadly, calculus is a pre-requisite for anything more than very basic probability and statistics.


I think this is directed mostly at the people who will never take another math class after they leave high school. Speaking from a canadian perspective, I had Grade 11 Math, Grade 12 Math, and Calculus classes. about half of the work in grade 12 was directly preparatory for the calc class. Students persuing a trade/vocational school or no post secondary education would only have taken 11 and 12 math. if the half of the math class that was 'wasted' calc prep was replaced with 'probability in the real world' type material, you'd still be in the basic precalc stats area, but possibly giving something more useful to the kind of people that are not well represented at a site like this.


True, but you don't need a lot of calculus to do a large amount of interesting statistics and probability. And the little you do need is probably better being taught as needed in response to actual problems showing up. Most people learn better when they see a clear reason for what they are being taught.

Start with discrete probability, no calculus needed. Then when you move to continuous case show how integrals show up naturally in place of sums, and hey presto a clear and obvious intro calculus which far more people will understand the use for.


Because calculus is beautiful, and if you are even remotely interested in the physical sciences, absolutely necessary.

I can't even imagine a high school curriculum without calculus. It's what made some of the decisions I took about grad school possible.


Calculus is beautiful and it is absolutely necessary for all physical sciences and has serious applications for financial sectors, computer science, and many other fields as well.

However, for the average person who is not going into any technical field, probability will have a far greater impact on their day to day lives than calculus. Probability is useful for everyone in this world.

Anecdotally, I have a bachelor's in mathematics. It occurred to me several years later when I went back to start on my master's that I had virtually never used calculus after graduating and had to review a lot of my basic calculus notes.


At the high school level you don't really know what you'll be doing 15 years later? You should have a complete grounding, so that you can make those decisions and not have to play catch up later on. I am not saying probability and stats are less important (probability is as important and increasingly stats), but you need calculus as well, at least a good basic grounding.


Agreed, but that is not the question. I would love to see everyone holding a diploma walking out with some knowledge of both calculus and statistics.

The question is given limited time in school, which is more important and useful for the average person? I think the answer is statistics and probability. People confront probability and statistics in the news all the time and make practical decisions based on what they think is more likely. Most people can apply probabilities in their daily life, few people in nontechnical fields will use calculus.

I brought up my own experiences to point out that as a former military analyst and current DBA and programmer I have used basic statistical knowledge on a regular basis. I have used knowledge of Calculus so little in my daily life and job that I had to give myself a very thorough refresher when I returned to an academic study of mathematics.


As a counter point, I have a masters in math, but highschool calculus almost drove me away from math completely. It was very obtusely taught and by far the dullest course I took in highschool.

The amount of calculus you actually need for high school science is both a lot less than what is taught at high school and different calculus than is taught at high school. I agree that most of calculus should be left to university since they're going to have to re-teach most of it anyway to undo the damage done by a bad high school curriculum.



Why? There are no discussions there.


Hmm. Maybe what I really meant was: this has been submitted at least twice before, three and four months ago or so, and in neither case did it generate any discussion.

I hope it does this time, because it's been shown to be interesting enough to be "discovered" by more than one participant.

And I do. I've never understood the US schools emphasis on calculus, and building it up to be such a huge deal. I first did calculus aged 16 in Year 10, and it was just another part of the syllabus. One more step to understanding the way stuff works.

Replacing it by another "BIG IDEA" seems not necessarily to be a Good Thing(tm). Perhaps there should be more of a plain, and less of a pyramid.

I agree that more elementary stats done earlier would be a really Good Thing(tm), but replacing one pyramid with another does not seem to me to be so.


First there is the recognition that there is no pyramid. Since math is so highly interrelated, learning about one area almost always helps understanding in a different area. Granted some areas require a certain prerequisites, hence you will encounter peaks, but I agree aiming for only one peak does not make all that much sense since calculus is only useful in certain situations.


Maybe it's built in "pyramid" like this because: 1. Knowledge is hierarchical 2. Calculus has a longer list of skills that needs to be mastered before learning the subject. 3. Historically, calculus came around first (am I right?). This might not be a coincidence.


Understanding (the path to knowledge) might be hierarchical but knowledge in general isn't. Organizing knowledge in hierarchies is a conventional teaching technique, and that has some self-fulfilling qualities: if you teach people in hierarchies, they'll see the world that way.




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