"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.
with comments on mathematics education for breadth rather than for speed through the standard curriculum, by a Fields medalist.
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.
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.
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).
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?
* 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.
* 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.
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.
Do you really believe that Morgan Stanley or Citigroup suffer from a lack of understanding of statistics?
Of course, it is unwise to limit your involvement in government to merely voting, but that is a separate topic.
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.
And yes, I do vote.
> 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.
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.
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 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.
But that's a ridiculous thing to assume.
You overextrapolate from an overstated initial cause, and offer no evidence whatsoever for any of your bald assertions.
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.
But even more, you say:
> ... taught axiomatically, Algebra and Geometry
> ARE interesting in and of themselves
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.
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.
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.
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.
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.
"Advice to Mathematics Teachers on Evaluating Introductory Statistics Textbooks" by Robert W. Hayden
"The Introductory Statistics Course: A Ptolemaic Curriculum?" by George W. Cobb
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:
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.
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.
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.
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.
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.
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.