Slashdot Mirror
Mathematics As the Most Misunderstood Subject
Lilith's Heart-shape writes "Dr. Robert H. Lewis, professor of mathematics at Fordham University of New York, offers in this essay a defense of mathematics as a liberal arts discipline, and not merely part of a STEM (science, technology, engineering, mathematics) curriculum. In the process, he discusses what's wrong with the manner in which mathematics is currently taught in K-12 schooling."
Yes, the problem teaching Math(s) and programming (applied Math(s)) is that it's just about intelligence - which you can't teach. The smarter you are, the better you'll be able to figure it out. The problem with teaching is all the generalists who think because they have a "Degree in Education" they are able to teach any topic. Traditionally dry disciplines need to be taught by specialists with passion and enthusiasm for their topic, not by generalists who happen to have a gap in their timetable.
let A=1, B=1
A^2=B^2 because A=B, so
A^2=AB and
A^2-B^2=A^2-AB , next we factor
(A+B)(A-B)=A(A-B) , divide like terms
(A+B)=A
substituting our variables for their values we learn that
2=1.
The brain can be trained and the processes of problem-solving can be generalised - see Polya's How to Solve It. But it doesn't help much to just read the book: you've got to practice, and practice, and practice some more. You must make mistakes and learn from them. You must be prepared to accept multiple inputs rather than merely those which reinforce your strengths and/or prejudices. You must sometimes, as the old 9/11 troll used to say, get some perspective - don't count the angels on a pinhead while Rome burns, even while the most secure of academic positions involves the former and there's such an alluring spirit of mental masturbation in many disciplines and departments.
Meanwhile a good teacher has spent enough decades on some area that he knows both where to provide you hints on specific complex problems and which direction to guide you in when you're contemplating your whole professional life. But, again, don't just choose the teacher who happens to share your academic and ethical prejudices.
I can attest that "true" math is very removed from computation. The computational classes are all regarded as the "easy" classes. This is in contrast to the "hard" classes, real analysis and abstract algebra. Being thrown into real analysis after just one quarter of study in proofs is extremely rough going. If proofs were introduced as puzzles or just introduced earlier in education the whole of America would be better off for it.
My own motivations for being in math are for the challenge and because of the lack of concrete answers in calculus. Trigonometric functions especially are always treated as little boxes that magically calculate what you need.
In any case, at least math attracts the curious.
Eat sleep die
>>Mathematics is the foundation for philosophy
Eh, kinda. Advanced logic is the foundation for a lot of modern philosophy, but Wittgenstein and the rest of the 20th century analytics were just responding to the tremendous success of physics at figuring shit out, and wanted to smear some of that patina on themselves. Well, logic has always been a part of philosophy (think Socrates and his syllogisms) but reading the Tractatus is like reading a modern computer science proof.
Which isn't surprising, either, given that computer science is essentially applied philosophy in a lot of ways. (cf Bertrand Russell, etc.) If you've ever sat through a class where philosophers have sat there talking themselves in circles about how an object can't both be is-a and has-a at the same time, you (if you're like me) feel like leaping up and just telling them to fucking encode whatever paradox they're trying to create in a object hierarchy, and be done with it. I've long longed to write a book called "Computer Science has figured a lot of your shit out in practice, Philosophers".
It does kind of bug me though, that a person who graduates with a degree in mathematics (which is a fairly difficult, hard-nosed subject) gets a wishy-washy BA degree, whereas a hippie with a degree in "environmental engineering" gets a BS, but ultimately I think there's a lot of problems with our current conception with categorizing things into "science" and "not-science". Economics and Climatology are very analogous in terms of what they do - gathering tons of data, running analyses on it, and projecting things out into the future, and both are essentially "empirical studies of the world about us" (i.e. a sort of base level of science, though with the testing, replication and confirmation bits left out), but we consider one to be a social science and another to be hard science. There's also a huge debate now over Anthropology, after the American Anthropology Association dropped "science" from its official bits.
Basic math is easy enough for nobody to have an excuse for not knowing it.
No sig today...
(A+B)(A-B)=A(A-B) , divide like terms
Divide by zero error! After this point, every conclusion is invalid since the results are undefined.
Depressingly, some people (adults as well as kids) would not spot that.
Those who can make you believe absurdities can make you commit atrocities. - Voltaire
Okay.
Philosophy is the process of speaking greek and stroking beards. Therefore by stroking a grecians beard, I shall become a philosopher.
I have defined philosophy (badly) and applied a complete absense of logic. Is that not what you meant?
I've seen the following link in many a Slashdot thread before, but it certainly bears repeating here: "A Mathematician's Lament" by Paul Lockhart It's mostly known as an insightful critique of what's wrong with K-12 math education, but I've always liked it as an explanation of why people who enjoy math do it in the first place: it's satisfying in an artistic way. I think it would be great if more students saw math as something worth doing for its own sake, like art or athletics, and hey, it lets you do science and engineering too.
In fact, this summary sounds similar enough to "Lament" that I wouldn't be surprised if this Dr. Lewis was inspired by and/or cited it. But this is Slashdot, so I'll let someone else check that out.
"This algorithm runs in constant time. Come on, 2,147,483,648 is a constant..."
This is exactly the kind of thinking that has got us into the mess we're into now.
Learning math is just as difficult as learning any other subject or content material. Deciphering poetry, learning programming, studying psychological theory, and learning calculus all involve concentration, study, and struggle from the learner. No one is born knowing any of those things, therefore they all must be learned. The entire point of the OP is to say that the way we go about teaching math is wrong and that people need to reconceptualize how they teach the information because it doesn't make sense to the learner. In the end, its all difficult to some degree. It's when you have that "A-Ha!" moment, it clicks, and you get it. But if you have some terrible algebra teacher who doesn't understand advanced math or someone who doesn't care that you learn, only that you can complete problems 1-50 in a mechanic fashion, then of course it's going to seem difficult (or more difficult than it should be).
Carl Sagan quotes get you an automatic +5 on all posts.
It's time to stop posting.
I have a cousin who is great at mathematics, and really can see mathematics as an art. Whereas I am happy if I can solve a problem, he will look for an "elegant solution". I had a number of equations that I solved, trying to optimise the buffer size for various input queues. I shown him, and he quickly said that I had the right answer. A day later he came and shown me how he derived an equation that could simply solve all problems of this type. He also generalised it to allow buffer sizes that were complex numbers. The first part was very useful to me, the second absolutely useless - but to him it was all just interesting.
This is one way that mathematics as an art is unlike any other art. It gives useful results. I have heard time and time again about engineers going to the mathematics department of a University asking how they can solve a "new" problem - to be told that the solution had been discovered a century before. I am sure most of these solutions came from someone just wanting to find an elegant way of expressing something without thought of any use. So if its an art and is useful why do so few people follow it?
The answer is obvious, because its hard! In many forms of art you can slap anything down and convince someone that it has value and its art. This may not always have been true, before photography accurate representational art was highly valued - but today someone producing a lifelike portrait will not be values as much as someone slapping their name on an unmade bed! Mathematics has to be right, you can't just slap down a few numbers and call it an equation. This is the basic problem that anyone will have in persuading someone to follow maths for its art, there are a lot easier ways to become an artist.
Well, we would likely all be malnourished, due to lack of fertilizers, at least those of us who hadn't died at childbirth or soon after. There wouldn't be an Internet to talk on, but that would be okay, since we wouldn't have time to use one due to the lack of engines and the resulting need to do backbreaking labour 16 hours a day. In short, our lives would be miserable, but due to lack of medicine, they would at least be short.
Missing these kinds of little details is why I have very little respect of philosophers. As far as I can tell, most of them chose their field because it doesn't punish sloppy work. And then there's idiocy like the Chinese Room, which assumes that a system cannot have properties its components don't have, yet hasn't been laughed out like it should had been.
Philosophy means you accept the human condition. Technorcacy means you try to do something about it. Hope for a better world in the future lies on the latter, not the former.
Forget magic. Any technology distinguishable from divine power is insufficiently advanced.
I wish science in general was considered part of what a learned person has to know. I mean, if you want to pass for an intellectual you have to read your Dante, your Beckett and you at least need to know who Lautreamont was. But, apparently, you can very well get away with thinking that you can suck gravity out of a room the way you suck air, or with not having even heard about string theory. That divorce makes no sense, and it was impossible in the history of ideas till very recently. And Euler's formula is more beautiful than most poems.
The way math is taught in schools is atrocious. Most math texts that I've used with 5th and 6th graders emphasize learning processes and methods for solving a set of problems. The texts do not hold all of the blame, however. The texts are written to follow state and national standards. The standards are written in such a way to emphasize process and not necessarily apprehension of greater concepts. For example:
5th Grade Level Expectation 1. Differentiate between the term factor and multiple, and prime and composite (N-1-M)
While these vocabulary items are important and these skills are definitely useful, learning this skill in isolation (which most texts teach) is pretty useless as students do not connect these skills to a greater picture.
A revision of mathematics standards and teaching methods will go a long way to improving the quality of mathematics education. A holistic approach that includes some wrote learning of basic skills and lots of real application problems. Real application problems are not word problems. How many "real" word problems have you had to solve in the last ten years?
Some texts such as Every Day Math from the University of Chicago does a much better job at integrating all sorts of skills and teaching in a much more holistic method. It includes some excellent modeling exercises, games that rely on a real understanding of mathematical principals for mastery and interesting lessons. But even the best text can't help a kid if they don't have a good teacher that really understands mathematics. Watching an uniformed teacher try to explain what a prime number is, or a different method for division (such as repeated subtraction) is painful. They simply can't do it. Unfortunately, in my experience most of the teaching candidates that were in my classes thought that math was "hard" and "didn't really matter." They scraped by with the lowest possible scores in the required math classes and one even told me she "wasn't going to bother teaching math." While this is pure anecdotal evidence, the declining math scores in the US show that we really do suck and producing math teachers.
The problem stems from bad math teachers badly teaching math which of course leads to more poorly instructed math teachers. Placing a real emphasis on reading and mathematics, with highly qualified and well-supported specialists is the only way we're going to solve this problem. Unless we have some real political will akin to that found during the space race, we're not going to solve this problem any time soon. We'll just keep cranking out kids that think that math is done by computers and a few nerds that wave their magic math wand over problems to find solutions.
This one's tricky. You have to use imaginary numbers, like eleventeen... --Hobbes
This is by far the best defense of mathematics I've ever read. It's a shame that the poor quality of grade school math education has made it necessary, though. Can one imagine a similar essay on any other subject? Only math is so poorly taught.
So over the past two millennia we have cut the working day by 1/3rd and doubled the average lifespan at birth (if you ignore infant mortality, our lifespan hasn't increased that impressively).
Meanwhile we have turned the majority of Western humans from independent men into chair-warming consumers singing in lockstep for trinkets. We've made up for the opportunity to live a life of leisure surrounded by virtually infinite resources by blasting our population beyond 6 billion.
Technocracy is for the lazy man who wishes to be controlled and for the fascist who wishes to control others. The technocrat only has to think about one thing. But philosophy regards technology as one of many tools, not as a master. The philosopher-ruler (for philosophy is a basis for living, not an alternative) must not let prejudice cause him to dismiss the possibility that he can do better and for more.
I remember been taught differentiation at school – One lesson, lecturer puts a parabolic curve, x=y*y, on the board, and asks the problem, determine the angle of the line
Then, he didn’t say anything else.. Just, for the rest of the lesson, responded with ‘Yes’, ‘No’, or ‘Maybe’. So, after a frustrating 20 minute discussion, trying to work out how the hell to do this problem, someone came up with the idea of adding a ‘little bit’ of x, to x..
We worked out, as a group, the concept differentiation, with only the smallest bit of guidance from the lecturer. This is how things should be taught – allowing people to discover concepts themselves, rather than preaching the correct ways to do things.
...the processes of problem-solving can be generalised - see Polya's How to Solve It. ..
"How to Solve It" also talks about more general problem-solving than just mathematical problems - crossword puzzles, for example. Prof. Lewis's article talks about the universal question "Why did they teach me the quadratic formula when I will never use it?" and this is really the answer; doing mathematics (should) teach people how to solve any problem logically. Well, any problem that can be solved logically, of course.
Meanwhile a good teacher ...knows where to provide you hints
Heh. Although a bit dry, one fun part of the book is where Pólya talks about giving hints to students : "Yet the teacher should be prepared for the case that even this fairly explicit hint is insufficient to shake the torpor of the students; and so he should be prepared to use a whole gamut of more and more explicit hints".
While agriculture requires backbreaking labour, hunter-gatherer societies only worked a couple of days a week. Not that I advocate a return to it, but backbreaking labour all the livelong day was not universal in ancient society.
Philosophical journals have the same rigorous standards for papers as journals for the various sciences. Your view of philosophy is about as valid as a grizzled mountain man who mutters about hard science being all book-learnin' and mumbo-jumbo.
Even that is a statement of philosophy. Furthermore, you seem unaware that many calls for improving human lives came from works of philosophy: More's Utopia, Kirkegaard's questions of metaethics, even what is often called the beginning of the Western tradition, when Socrates hung out in the agora and asked passersby "What if what you comfortably believe is wrong?"
Economics and Climatology are very analogous in terms of what they do - gathering tons of data, running analyses on it, and projecting things out into the future, and both are essentially "empirical studies of the world about us" (i.e. a sort of base level of science, though with the testing, replication and confirmation bits left out), but we consider one to be a social science and another to be hard science.
Well, economics is, especially in its present state, largely influenced by individuals, who can be a lot harder to predict than wind currents. You may identify trends, constants and correlations, but mostly in hindsight. Accurate predictions are as scarce as in cartomancy and useful controlled experiments are hard to imagine. While Climatology shares some of those characteristics, I think we have a much higher chance of predicting a storm than the stock market. Unless tons of people start walking around with nuclear powered, oversized fans. If you catch my drift.
Philosophy means you accept the human condition.
No.. Philosophy means questioning the human condition. it's confronting the status quo and asking "why?"
So exactly the opposite in every way of what you think it is.
You're also wrong in your assumption that philosophy and technocracy are mutually exclusive, in fact if they aren't mutually inclusive, then as a technocrat you're trying to find solutions when you don't even know what the problem is.
Philosophy is a very powerful way of thinking, and in no way whatsoever does it represent conformity or acceptance, it represents freedom of thought to think critically.
In fact philosophy really should be tought in schools, it's the basis of how we view the world today, and if the future is bright, it will be philosophy to thank and the people who dared to question the status quo.
Why are people even debating philosophy vs technocracy? Why should someone have to choose one over the other? How do people get dragged into such nonsense? Here a new subject for you: tomatoes vs rainbows. Go.
If you've ever sat through a class where philosophers have sat there talking themselves in circles about how an object can't both be is-a and has-a at the same time, you (if you're like me) feel like leaping up and just telling them to fucking encode whatever paradox they're trying to create in a object hierarchy, and be done with it. I've long longed to write a book called "Computer Science has figured a lot of your shit out in practice, Philosophers".
I understand where you're coming from, but for many philosophers, what they're doing is not just trying create a practical solution to a problem, but describe reality. Your object model might solve the problems from your point of view, but it includes many built in assumptions about the thing modeled.
In a related way Wittgenstein later came to criticize the Tractatus. Part of the criticism is that if you assume the universe can be fully described with formal logic (logical atomism), then you are already subscribed to a certain type of metaphysics.
Missing these kinds of little details is why I have very little respect of philosophers. As far as I can tell, most of them chose their field because it doesn't punish sloppy work. And then there's idiocy like the Chinese Room, which assumes that a system cannot have properties its components don't have, yet hasn't been laughed out like it should had been.
There's plenty of philosophy-types who think that Searle is an idiot, too, for the Chinese Room and other things. Guy loves to position himself as a defender of rationality and realism because it lets him belittle poststructuralists with oversimplifications and straw men while acting like a hero of a scientific worldview that he clearly doesn't know that much about.
In some ways his antagonistic materialsm is quite similar to your dismissal of philosophy in general, actually.
Philosophy is the path by which every man continually asks questions of his condition and can thereby strive to improve it. It is something practiced while living, not instead of living (as "pursuit of happiness" is the ongoing enjoyment of happiness, not the singular and final goal of happiness). You may as well argue that man should not breathe because people who breathe are wasting their time only breathing when they should be doing other things.
Philosophy does not give a single solution to the world's ills and it does not force you to do anything or to make others subordinate to your will. I'm not sure what you're afraid of, but it's not philosophy.
You should have had Mr Burton, my maths O level teacher. He was brilliant. He was totally passionate about his subject and he was also a fantastic teacher. he encouraged us to think about maths rather than to just blindly follow formulae. I still vividly remember the lesson where he taught us differential calculus from first principles.
He encouraged us to study outside of lesson time and his door was always open during lunch, or after school. almost every one in his class passed their maths O level with at least a B, over half had A's
It's no exageration to say I owe my career as a developer to him and his enthusiastic teaching.
And the people shall be oppressed, every one by another, and every one by his neighbour Isaiah 3:5
I had teachers with a passion and enthusiasm for maths all the way through school. Condolences for your experience.
I had various teachers with passion and enthusiasm for chemistry, geography, German, English literature, physics, Latin, history, PE and maths, and none of them were PhDs or anything, just good teachers.
To have a right to do a thing is not at all the same as to be right in doing it
"Meanwhile we have turned the majority of Western humans from independent men into chair-warming consumers singing in lockstep for trinkets."
I suggest you take off your rose coloured glasses and go read some history, in particular just how "free" your average serf was in feudal times and even later. Don't like what your overload or king does? Tough. Complain and you'll probably at best end up homeless or at worst end up swinging from a tree.
People in the west have NEVER been as free as they are now.
So get yourself a fucking clue!
My highschool math teacher was a retired NASA programmer. According to her, teaching Mathamatics was about leaching logic and problem solving. If you forgot all the formulas taught in her class, she said, it wouldn't matter. The real skill learned was how to deal with an entirely new mathematical problem. WHY is area "height x width"? How to build your own sort of equations. Sure enough, decades later I have forgotten every single equation I had been taught there, but when faced with a logic problem I'm still able to work it out.
Missing these kinds of little details is why I have very little respect of philosophers.
They don't "miss" those details, they're not in scope.
As far as I can tell, most of them chose their field because it doesn't punish sloppy work.
Philosophy does punish sloppy work. relentlessly. Philosophical work is subject to more scrutiny and criticism than any discipline I know of, and that includes pure maths.
And then there's idiocy like the Chinese Room, which assumes that a system cannot have properties its components don't have, yet hasn't been laughed out like it should had been.
Laughing something out doesn't work in philosophy. Unlike whatever discipline you work in, it seems, in philosophy you have to show the reasons why something is wrong. And if you think the issue of emergent properties hasn't been considered in excruciating detail in connection with Searle's Chinese Room thought experiment then you clearly have no idea what philosophy is doing.
Philosophy means you accept the human condition.
Say what? Some philosophy is abstract, but so is some maths. Lots of philosophy (philosophy of science, political philosophy, ethics) is concerned with changing the human condition. Maybe you criticise philosophy because it didn't discover antibiotics (although it did lay a lot of the foundations), but do you criticise biology because it didn't invent democracy? Both changed the human condition, in ways appropriate to their respective disciplines.
Quidnam Latine loqui modo coepi?
I've long longed to write a book called "Computer Science has figured a lot of your shit out in practice, Philosophers"
Well, go on then, if it's that fucking simple and obvious. Put those silly old philosophers in their place, what do they know?
I'm thinking of writing a book called "Why do so many students of Computer Science think they have solved all the riddles of the universe because they know how to write a sorting algorithm?"
To have a right to do a thing is not at all the same as to be right in doing it
A Ph.D. tells you nothing except that the holder did some original research at an early point in their career.
America, Home of the Brave.
I can very well relate to this post.. the foremost reason for Mathematics being misunderstood is the problem with the way it is taught in schools. Right from childhood you are told to mug up the multiplication tables, formulas and everything is told be "it is like that.. just remember". The flaw is with the education system where stress is not to "understand" and see things logically but on how much can you mug up and pass those tests and get a so-called "good score". The teachers need to be trained to generate interest, talk concepts and not just ask to be ready with the multiplication table the next day for the test.
I very much disagree. Teaching is a specialty in its own right and a good teacher can teach almost any subject given appropriate support and resources. Of course some competency in the subject is necessary to provide insight when the lesson isn't hitting the mark but you don't need to be an expert.
I'll give an example: During high school I had two physics teachers. One was pretty talented at physics and had been teaching it for years. The other hadn't taught physics much before and wasn't that strong (if he did the exam some students would have got better marks than him).
The first taught a pretty neat syllabus. He did lots of well set out problem solving examples but the course was pretty dry. I think most students just got recipes out of the course and no real insight.
The second teacher followed the same syllabus but his examples weren't as well set out and he didn't necessarily get the right answer. Because he wasn't so confident in his answers he provided a lot of demonstration of checking techniques such as estimation and general sensibility checks (The ball doesn't roll up the inclined slope). His efforts to verify his answers, clean up his working and fix his answers taught more students about problem solving, physics and maths techniques than the rest of the course.
It's a shame I didn't realise this at the time. It must have taken real guts and a lot of home work to get up and teach that course. Brilliant teacher but not a brilliant physicist.
A Ph.D. tells you nothing except that the holder did some original research at an early point in their career.
There is also little if any correlation in being able to research, and being able to teach. Culturally, "everyone knows" the purpose of a phd is to become a professor and teach university students while collecting a $100K+ salary. The upper 50% to 10% cream of the crop actually get hired to do that. So, pretty much by definition, as a general cross section of the population, they are in the bottom of the barrel of teaching ability. So I'd be expecting, unless they're education phds, they're almost by definition probably not going to be good teachers.
"Science flies us to the moon. Religion flies us into buildings." - Victor Stenger
I honestly can not understand where there can be "beauty" in a mathematical expression that covers the entire blackboard.
No one else can, either.
The beauty is in the simple relations between apparently unrelated things that, while provably true, still seem magical and mysterious. One example:
You're probably aware that the ratio of the circumference to the diameter of a circle has been given a special name, pi. This is a practical, useful thing that seems purely geometric; you can measure the diameter of a circular hole, multiply by pi, and get the circumference of the hole. Fine.
Well, it was shown in the 17th Century (!) that pi is also equal to four times (one, minus a third, plus a fifth, minus a seventh, plus a ninth, ... , on out forever). In fact there are many series of integers that are related to pi.*
Now, why would this be? What does the ratio between the circumference and the diameter of a circle have to do with the "counting numbers"? Why should there be any relationship at all? After centuries of puzzling, no one knows. ... , on out forever. What does pi have to do with the inverse squares of the integers?!?
________
* My personal favorite was proven by Euler in the 18th Century:
pi squared, divided by six, is equal to one over one squared, plus one over two squared, plus one over three squared,
Philosophy is only a pretty word for wild speculation/daydreaming/brainstorming
You are exactly right, if your definition of "philosophy" is "wild speculation/daydreaming/brainstorming".
Unfortunately for you, words have generally agreed upon meanings, not just whatever brainshite you happen to vomit forth.
To have a right to do a thing is not at all the same as to be right in doing it
Well, reasoning is formalised by logic which is today usually regarded as a branch of mathematics. And reasoning is a requirement to practice philosophy. (N.B. even if you can somehow argue that you can come up with some philosophy without reasoning, you cannot practice philosophy in the general sense without reasoning.)
Moreover, mathematics in the most general sense is about formalising pattern-matching skills: recognising when and how to generalise. This crosses into philosphical (not mathematical) induction, well-understood as far back as Newton in justifying his theory of gravitation but little understood by dilettantes centuries later.
I maintain, then, that mathematics forms a basis for philosophy, regardless of those smartass xkcd comics. If you want you can turn it into a semantics game and argue that logic and philosophical induction are "not mathematics, but philosophy", and it is fruitful to argue whether either logic or philosophical induction can be justified without philosophy. But the same applies to anything, and the tools exist in their own right.
In practice, there are two forms of teaching. The first is applied subject matter in school. In this specific case, it is applied mathematics. They give you the calculation tools for describing a relationship and then they expect you to find similar relationships and apply that formula. The goal is to teach the use of a tool. It is no different than teaching one to write a coherent paragraph, communicate in a foreign language, or to be a good citizen in a democracy. Teaching applied mathematics is a necessary element of any school curriculum.
The second is one of discovery. My journey began as a teen, when I read about fractals in an article from Scientific American. Since then I've gone on and explored prime number theories, methods of calculation, the history of these discoveries, and I've gone looking for the blind alleys that may not have been explored as thoroughly as we might think.
We need to recognize that education is not about discovery. It is about teaching a person the tools of modern society. However, in our zeal to teach the applied aspects of these subjects, we need to realize that we are failing to nourish the creative spirit of discovery. Mathematics is no different than reading, writing, civics, history, geography, or language. Learning to write a coherent text does not make one appreciate literature.
Our schools are obsessed with application, not discovery. We spend ridiculous time teaching application, application, and more application. Then we sit and wonder why our children lack the will to explore...
Nearly fifty percent of all graduates come from the bottom half of the class!
This article frustrates me. He talks a lot about some particular thing, claims that it relates to maths, but doesn't really say what particular part of maths it relates to, nor does he get into specifics, nor does he spend much (if any) time on how to improve matters.
Okay, I'll try to explain my confusion with a parable. When I was fifteen, I did a school certificate maths exam. It had a whole bunch of questions, none of which we had ever answered earlier in the year, but somehow the examiner thought I could answer them, and unfortunately I was unable to answer all questions "correctly" according to the examiner.
What does that have to do with mathematics education over the past 25 years? Unfortunately a great deal. We were required to have exams for mathematics, because every subject had exams. The end result was that some people didn't do well in exams, even failing enough to be unable to continue on in their maths education in the next year. The truth is that exams cannot alone be used to evaluate a person's effectiveness as a mathematician. The only way to get around this is to teach mathematics properly, and make sure each person understands maths at all levels.
Ask me about repetitive DNA
Why are people even debating philosophy vs technocracy? Why should someone have to choose one over the other? How do people get dragged into such nonsense? Here a new subject for you: tomatoes vs rainbows. Go.
The slashdot hivemind divides the world into a series of either/or choices, e.g. emacs/vi, or pro/anti-Windows
To have a right to do a thing is not at all the same as to be right in doing it
In mathematics it is the truthiness of the statement creates "credit" and then we search back in history to find who said it first and then we give the credit to him/her and that is how reputation/respect is created. It flows back in time. Credibility accrues from the statement to the speaker.
In philosophy a bunch of people agree that some one was/is a great philosopher and so they give more value to a statement from such person. The credibility flows from the speaker to the statement.
sed -e 's/Chuck Norris/Rajnikant/g' joke > fact
Here, let me show you an even more beautiful mathematical paradox:
We try to solve this equation: x^2 - x + 1 = 0
We do that by adding x - 1 on both sides: x^2 = x - 1
We multiply both sides by x: x^3 = x^2 - x
Add 1 on both sides: x^3 + 1 = x^2 - x + 1
Recognize the first equation in the right side: x^3 + 1 = 0
Subtract 1 on both sides: x^3 = -1
Take the cube root on both sides: x = -1
Check the answer: (-1)^2 - -1 + 1 = 0
Have fun!
"How to Solve It" also talks about more general problem-solving than just mathematical problems - crossword puzzles, for example. Prof. Lewis's article talks about the universal question "Why did they teach me the quadratic formula when I will never use it?" and this is really the answer; doing mathematics (should) teach people how to solve any problem logically. Well, any problem that can be solved logically, of course.
Then why not teach logic and problem solving, possibly using mathematics as the language (but not necessarily)? When we tell ourselves that we're teaching maths, that's all people tend to teach (and learn, for the most part). I agree that teaching logic and deduction is valuable, more valuable than a lot of mathematics to many people (since with skills you can get the maths, but not necessarily vice versa)... but its rarely seen called out on a school curriculum. And that's a shame.
You're special forces then? That's great! I just love your olympics!
While agriculture requires backbreaking labour, hunter-gatherer societies only worked a couple of days a week.
Only thought to be the case by Europeans who didn't think that hunting was "real work".
Well, part of it does and another part doesn't.
Confucius say, "Find worm in apple - bad. Find half a worm - worse."
Really? At the University I attended it was crystal clear that Professors were hired primarily to do research and that teaching was a secondary consideration.
Oddly enough their teaching skills were distributed about the same as my high school teachers who were hired to teach and only to teach. That is to say a few were excellent teachers, some were good, the bulk were acceptable and a few were flat out terrible. What we learned from bad teachers was that a bad teacher or professor can't stop you from learning something you need or want to know.
That happens a lot in mathematics too - it has to, or mathematicians would have to spend all their time refuting amateur "proofs" of famous open problems.
The way to mastery typically involves teaching. =)
While teaching could be a speciality, I hold that it is an essential skill. If one cannot teach others, it is hard to imagine that this person could correctly teach himself correctly in the first place. In addition, teaching others helps remove personal biases and provides new opportunities to reconsider the original assumptions/axioms, without which we reach lower plateaus.
And so it is said that the good idea will stand the tests of time. I used to think that this required sheer technical correctness. Perhaps, at most, I was half correct. Now I believe that in addition to technical correctness, the rhetoric (aesthetic/attractiveness) of an idea determines reception. No idea matters if none listen. Form and function, rhetoric and logic... =)
Cheers
All philosophical theories suck - philosophy is the practice of coming up with explicit theories in order to poke holes in them. Any time you attempt to be really precise about anything in the world, you are going to be wrong. That's why "define X" is always a winning move in a discussion, if you are willing to be sufficiently pedantic. There is no metaphysics that "works" once you get precise about things, because nothing that claims to touch reality works if it has to be precise. It's not a shallow pool to draw from - there is no pool.
Really. Must we contextualize mathematics, or try to talk about what it is or is not? Do we really need to point to a particular cognitive framework as "the reason" why math is not taught "properly?"
To use a slightly loathsome phrase, math "is what it is." Instead of talking about how people should relate to it, I suggest a radical approach: just LEARN it. Teach it for what it is.
I struggled with arithmetic when I was in grade school, not because I didn't understand the rules, but because I kept making mistakes. And my teachers had the wisdom to know that those errors had to be drilled out of me before I could proceed any further. I suffered. I *hated* the tedium. We were asked to multiply two twelve-digit numbers with no assistance from any computing devices or tables; divide four-digit numbers into twenty-digit numbers, until we could do it with 100% accuracy every time. It didn't have to be lightning fast. It just had to be CORRECT.
And when I mastered that skill, it felt fantastic. We moved on to more advanced topics, and each time the teacher made sure we had firmly laid down the next conceptual brick of this vast mathematical edifice we were building for ourselves. It was hard but rewarding. To those critics who might say such an approach would discourage some students, and that some kids just need to be excited by what they learn, clearly you have never really understood what it means to build that foundation. It's got to be ROCK SOLID. No crap about trying to make math "fun" or "interesting" or "relevant." That sort of stuff comes when it comes; they are merely ornaments on the pillars. There's no point in making the structure pretty before you make it sturdy.
So then, how do you get students motivated? It's really quite simple. You challenge them and you force them to bust their asses, and when all their hard work pays off, that sense of accomplishment is better than any drug. To know that you did it on your own, and you have complete confidence in your mastery of the concept, is precisely what must drive them forward. You can't entice them with anything else. You can't try to swaddle the math in some cutesy real-world application, because that is going to be fake, and they know it.
That's the story of how I graduated with my BS in mathematics from one of the most prestigious scientific universities in the world. It was purely the early appreciation for persistence toward understanding mathematics for its own sake. I'm not saying everyone has to keep math "pure." If your goal is to apply it in some other discipline, go for it. But the learning process has to build upon that foundation of math for math's sake.
> The majority of hunter-gatherers only work about 4 hours a day.
They also shit where they live and move on from their "village" once they've spoiled the ground bad enough.
A Pirate and a Puritan look the same on a balance sheet.
My older son is in the 2nd grade and is gifted (IQ somewhere around 140). Right now, they're learning simple addition. There's only one problem. He already learned this last year. He was doing complex subtraction with my wife (a teacher) over the summer break. But the class is doing simple addition so that's what he's stuck on.
It gets worse. They're using a so-called "spiral curriculum" this essentially means they learn one way of figuring out that 8+3=11, then learn another way, then a 3rd, 4th and 5th way. My son gets it the first time, yet he has to sit through all of the other ways. He yearns for more advanced math. He asked me about multiplication and division and, when I showed him an example using Legos, he got the concept right away.
He already knows his times tables up to 5 and wants more. But school is boring to him because they don't push him. He isn't being challenged at all. He tends to act out when he's bored too which makes everything more complicated. If you have a child who is falling behind in school, there are resources to help them catch up. If you have a child who is gifted and wants to pull ahead, your kid needs to sit down, be quiet and learn for the fifth time what 8+3 equals.
My sci-fi novel, Ghost Thief, is now available from Amazon.com.
http://en.wikipedia.org/wiki/Divide_by_zero#Fallacies_based_on_division_by_zero
~ In Trust, We Trust ~
Well leaving aside the dubious notion that studying applied subjects is really pursuing "technocracy", I think we're engaging in a bit of false dichotomy here. You don't have to choose as an individual or as a society to pursue liberal arts or applied arts; to study philosophy or to study engineering.
The medieval liberal arts curriculum had two levels. The Trivium consisted of grammar, logic and rhetoric. These are the basic tools of expression, thinking and persuasion. A student versed in the Trivium can write a persuasive argument that is understandable and well thought out. He can likewise recognize errors in argument and techniques of persuasion that give poor arguments more weight than they deserve.
These are critical skills for an engineer, who must persuade clients to take his proposals, or argue for one approach to a problem over another based, not just on feasibility, but the goals of the organization. It's a very common complaint among engineers that they know what has to be done but they aren't listened to. It seldom occurs to them to study the tools of persuasion so they can understand how those tools are being used against them.
The higher level of study was called the Quadrivium, consisting of arithmetic, geometry, music and astronomy. This is a bit misleading, because the study of music was not the study of performance, nor was astronomy the study of observational astronomy. They were both effectively branches of applied math. Having mastered the basics of making sound and persuasive arguments in the Trivium, the student then studied advanced and logically exacting arguments in arithmetic and geometry, and practiced the application of these advanced skills in music and astronomy.
The choice of subjects in this program is probably not what we'd choose today. It isn't hard to come up with any number of reasonable ways to update this curriculum. For example, with arabic numerals we could roll the calculation aspects of arithmetic into logic in the Trivium, and replace the advanced aspects of arithmetic in the Quadrivium with mathematical analysis through calculus. But what we should not lose sight of is the *aim* of the Trivium/Quadrivium program: to produce a student who at age twenty one or so is well prepared to take on the study of *any* specialized field such as law, medicine or engineering.
That aim is gone from the modern conception of liberal arts, because in practice the aims of a modern liberal arts education are *vocational*. It is expected that when you get a bachelor's degree, you are prepared to take up a specialized career. In some cases such as engineering or labor relations, that expectation is *explicit*. In others, such as art history or English literature, I'd argue that there is an *implicit* assumption that this is job training. You are fitted to pursue advanced studies in one field, or perhaps to teach that field at the high school level, even though it is extremely unlikely that you will pursue those careers.
So we treat what we call a "liberal arts" education today as if it were a robust fundamental education, but we organize the actual instruction to pursue narrow vocational goals. We de-emphasize the rigorous mental discipline of mathematics and at replace that, if at all, with training in its handiest applied methods. The most serious drawback of this pseudo-vocational training is that its value fades with time. What you learned about psychology or electrical engineering in the 1980s may largely be obsolete.
This confounding of fundamental education and vocational training leads to a serious structural fault in how our society organizes education. We don't expect most people to continue educating themselves after the age 21 or so. That might be tolerable if we gave them strong training in rigorous thought along with using and recognizing the techniques of persuasion. But the education we give students at the bachelor's level does not really set them apart from somebody who gets a certificate from a trade school
Post may contain irony: discontinue use if experiencing mood swings, nausea or elevated blood pressure.
No, they debate fundamental questions (phrased in CS-speak): "Is a pointer to an object the same thing as the object?"
From a CS perspective, the answer is obvious, as is the relationship between a pointer and an object. But philosophers fill up books on this subject.
Just because the lack of a final answer to a problem is dissatisfying does not mean that there must be one. Some problems simply cannot be resolved absolutely.
The example Sartre used is a good one. How would you make a decision in that instance which ends the debate?
The world does not conform to your desire for resolution.
const int one = 65536; (Silvermoon, Texture.cs)
SJW, n: "Someone I don't like, and by the way I'm a fuckwit" - AC
plus the formula is flawed in my field of expertise and needs a bit of fine tuning to be accurate for variable photon flux on the same angles.
In the best case, what you're saying is something along the lines of, "multiplication is flawed, because in relativistic physics, F=mA needs a bit of fine tuning". In the vastly more likely case, you're full of shit, and every mathematician (and physicist) reading this is laughing at you.
Yo dawg, I heard you like the Ackermann function, so OH GOD OH GOD OH GOD
"People in the west have NEVER been as free as they are now."
Eh, that's pretty iffy.
It would be more accurate to say that people in the West have never been better off in terms of material wealth, true. We've never had as high a level of technology or cheap access to gadgets or advanced medicine.
But free? I guess it depends on your definition of freedom. We're certainly more free than the Russian serf of the 1700's or the Spaniard under the Caliphate of the middle ages or the Greek and Serbian living under Turkish rule before the 20th century. But the homesteader in 1800's Oklahoma or Nebraska had far more freedom than you'll ever have, simply because the laws that governed him could be read, from beginning to end, in a matter of minutes. He didn't live as long, have cars or the Internet, or run up a huge Mastercard bill. But also he didn't have anyone telling him how fast he could ride his horse, he didn't have a "homeowners association" suing him for the color of paint his chose for his humble home, and the government wasn't trying to "help" him by taking half of what he earned and spending it on services he didn't ask for. He had to face the big bad world all on his own, but they were his choices.
I don't think many people want to go back to a horse and buggy, but at the same time it's patently silly to talk about how free we are when our government has re-defined freedom from "freedom TO" do things, and now regards it's role as "freedom FROM" things, "protecting" us like a nanny looks after a child.
Life is hard, and the world is cruel
It sucks when you get 1 upped by someone with a sense of humor doesn't it? Don't worry, have a cyber hug, you'll feel better in the morning.
/hug
Motorcycles, Robots, Space Gossip and More!
>>I doubt philosophers give a rats ass about pointers, let alone fill up books on the subject.
From the Stanford Encyclopedia of Philosophy:
* Almog, J., J. Perry, and H. Wettstein (eds.) (1989), Themes from Kaplan, New York: Oxford University Press.
* Bach, K. (1987), Thought and Reference, Oxford: Oxford University Press.
* Bach, K. (2004), 'Points of Reference,' in Bezuidenhout & Reimer (eds.) 2004. [Preprint available online]
* Barcan Marcus, R. (1947), "The Identity of Individuals in a Strict Functional Calculus of Second Order," Journal of Symbolic Logic, 12(1): 12-15.
* Barcan Marcus, R. (1961), 'Modalities and Intentional Languages,' Synthese, 13(4): 303-322.
* Barcan Marcus, R. (1993), Modalities, Oxford: Oxford University Press.
* Bezuidenhout, A., and Reimer, M. (eds.) (2004), Descriptions and Beyond, Oxford: Oxford University Press.
* Brandom, R. (1994), Making it Explicit. Cambridge MA: Harvard University Press.
* Brueckner, A. (1986), 'Brains in a Vat,' Journal of Philosophy, 83: 148-167.
* Davidson, D. (1984), Inquiries into Truth and Interpretation, Oxford: Clarendon Press.
* DeRose, K. (2000), 'How can we know that we are not Brains in Vat?,' Southern Journal of Philosophy, 39: 121-148.
* Devitt, M. (1981), Designation, New York: Columbia University Press.
* Devitt, M. (1990), 'Meanings just ain't in the head,' in Meaning and Method: Essays in Honor of Hilary Putnam, Cambridge: Cambridge University Press, pp. 79-104.
* Devitt, M. (1996), Coming to our Senses, Cambridge: Cambridge University Press.
* Devitt, M. and Sterelny, K. (1999), Language and Reality (2nd edition), Cambridge MA: MIT Press.
* Devitt, M. (2004), 'The Case for Referential Descriptions,' in Bezuidenhout and Reimer (eds.) 2004.
* Donnellan , K. (1966), 'Reference and Definite Descriptions,' Philosophical Review, 75: 281-304. [Post-print online version]
* Donnellan, K. (1972), 'Proper Names and Identifying Descriptions,' in D. Davidson and G. Harman (eds) The Semantics of Natural Language, Dordrecht: Reidel.
* Evans, G. (1973), 'The Causal Theory of Names,' Proceedings of the Aristotelian Society, Supplementary Volume 47: 187-208.
* Evans, G. (1982), The Varieties of Reference, Oxford: Oxford University Press.
* Field, H. (2001), Truth and the Absence of Fact, Oxford: Oxford University Press.
* Fodor, J. (1990), A Theory of Content and other Essays, Cambridge MA: MIT Press.
* Frege. G. (1893), 'On Sense and Reference,' in P. Geach and M. Black (eds.) Translations from the Philosophical Writings of Gottlob Frege, Oxford: Blackwell (1952).
* Kaplan, D. (1989), 'Demonstratives: An Essay on the Semantics, Logic, Metaphysics, and Epistemology of Demonstratives and Other Indexicals.' In J. Almog, J. Perry, and H. Wettstein (eds.), Themes from Kaplan, Oxford: Oxford University Press.
* Kripke, S. (1977), 'Speaker's Reference and Semantic Reference,' Midwest Studies in Philosophy 2: 255-76.
* Kripke, S. (1980), Naming and Necessity, Cambridge: Harvard University Press.
* Meinong, A. (1904), 'The Theory of Objects,' in Meinong (ed.) Untersuchungen zur Gegenstandtheorie und Psychologie, Barth: Leipzig.
* Mill, J. S. (1867), A System of Logic, London:
But when the "amateur" proof turns out to be correct, the work is duly recognized in math, like that of the Russkie recluse. But no such luck in philosophy where "correct" is a malleable concept.
Fuck systemd. Fuck Redhat. Fuck Soylent, too. Wait, scratch the last one.
They do, it's called a university and the school of hard knocks. Now we could refine that and go with the European system of dual (or triple) tracks for secondary education but that would be admitting that some snowflakes are less special than others.
There are 4 boxes to use in the defense of liberty: soap, ballot, jury, ammo. Use in that order. Starting now.
5 out of 4 Americans have trouble with fractions.
There are no karma whores, only moderation johns
Let me try to explain why this appears to work but doesn't. The problem is with this line:
We multiply both sides by x: x^3 = x^2 - x
When solving an equation, there is an assumed logical progression. Suppose you want to solve:
Then, you want to find the set S1={x: x is a solution of (1)}. You do this by transforming the equation repeatedly until you get to a form from which it is easy to derive the solutions. But when you make a transformation of the equation, you need to think about what the set of solutions is after the transformation. Let proposition P1 = x is an element of S1. (Similarly Pn for Sn). If, as the next step, you write:
you are implicitly stating that:
(<=> means "if and only if") If you then write:
x^3 = x^2 - x, (3)
the set of solutions has changed: -1 is introduced as a new solution. In this case, this is because (2) was multiplied by x, which is not a non-zero constant, and thus the meaning of the equation has changed. Logically, you are now stating that:
In other words, if you find an x for which P2 is true, then P3 will also be true for that x, but not the other way round.
Normally when you solve an equation, you implicitly create a progression P1<=>P2<=>...<=>Pn. From this, if you can see that Sn is the set of solutions for (n), then going back by implication from Pn to P1 you can conclude that Sn=S1. However, if the chain is broken and you write P1<=>P2<=>...<=>Pj=>P(j+1)<=>...<=>Pn, you can only conclude that S1 is a subset of Sn. However, because you are missing an implication from P(j+1) to Pj, you cannot say that Sn=S1.
There are many operations that potentially change the set of solutions, such as multiplication of both sides by zero, squaring both sides, and others. At every transformation, you must make sure that the solutions stay the same. In solving other problems, the logical progression can become more complex and then cannot be implicitly assumed like this. Generally, it is always a good idea to know precisely what you are stating in terms of logic.
With all the problems identified in TFA, you think this is the worst part of today's mathematics teaching?
In high school, I spent one year homeschooling, taking correspondence courses from a community college. They had some absurdly simple problems -- probably algebra, but I'm not sure -- which I had to finish and practice to get a grade. So I did, and the entire time I spent on that math class was less than an hour a week.
By the time I got to college, experiences like this had given me an attitude similar to yours, only worse -- I thought I was such a smart person. And indeed, I could just read and understand what Calculus was about, especially after having some small amount of Calculus in high school. This class assigned homework, but it wasn't graded. The teacher warned on the first day of class that the homework, although not graded, is required.
I didn't listen, and I failed that class. I also ended up dropping out because of a similar attitude in Computer Science, English, and Philosophy.
Recently, I've gone back to college, and I actually put in the time to learn this stuff. In particular, Calculus 2, where we're first really taught to integrate, requires practice. Never mind that they didn't allow integral tables on the quizzes and exams -- even when you have them, you need to know which strategy you're going to apply to this problem, which formula might be even remotely relevant. The only way you can know this is if you've got enough practice that you can start to see the patterns -- so you see what's happening when your chosen u and v in an integration-by-parts aren't going to work, so you know not only that this problem will be a trig identity, but which one and how to apply it...
So it's one thing to have a general understanding of what integrals and derivatives actually are. I could probably even pull off a limit-of-the-difference-quotient integral if I had to, just from pure understanding, without needing endless repetition. But there is no way I could do integrals at all without hours and hours of practice, even though I understand perfectly well what they are, what they represent, and why they're so bloody difficult.
Now, it's possible that there are a few geniuses who actually don't need the practice. If so, they wouldn't have to do the homework in my college Calculus course, and they'd be even faster at breezing through any sort of practice without needing help. But if they exist, they're also fantastically rare, and I'd also guess they'd be the least likely to be discouraged from an education (self-taught or otherwise) in mathematics.
The people the system is most dramatically failing to reach are the people TFA is talking about, people who have no idea what math is really about, and who thus miss the relevance to their own field or to anything they want to do. They're not necessarily bad at math, but they do need practice, and they need just a bit more motivation for that practice than getting a passing grade in high-school algebra.
Don't thank God, thank a doctor!
Actually, there was this weird thing going on in Math as a field for much of the 20th century: reinventing Euler. Euler was so very far ahead of the field that odds were that anything you discovered for the next couple of centuries had likely already been discovered by him - thus the saying that theorems are named for the first person after Euler that discovered them.
But math didn't devolve into a "study of Euler", instead the field plowed ahead happy to rediscover ideas from first principles instead of just leaning on the "teachings of a recognized great mind". There was some practicality to this, as translating his notes was very difficult (and only really finished recently) given he didn't use modern notation (having pre-dated most of it), and he wasn't always right. It says a lot about post-Enlightenment Western thought that, even though the field might have advanced quicker by becoming "the study of the expert", mathematicians just weren't particularly interested in that.
Socialism: a lie told by totalitarians and believed by fools.
The mindset of the teacher reminds me of the Harry Chapin song "Flowers Are Red."
Teachers that are that narrow minded should be transferred to places where they can't do any damage to students. Perhaps a prison environment would be best for them. They could at least try to help some of the people they screwed over.
Some in their 50s or so may remember "New Math", which was an attempt to teach elementary math with more emphasis on the underlying theory. It's now widely considered to have been a disaster. The author of the original article seems to date from that era.
One of the approaches to fundamental mathematics is to start with axiomatic set theory and build up from there. (That's not the only approach; one can also start with the Peano axioms and build up to set theory via lists, as is done in constructive Boyer-Moore theory.) This is minimalist and elegant (which is why mathematicians like it) but it requires considerable theoretical development before you get to addition. Teaching kids arithmetic that way was a disaster.
Euclid's approach to axiomatic geometry is like that, too. There's a lot of abstract logical structure that has to be built up before you can do anything. That's how math was taught up to 1900 or so, and 7th grade geometry is still often taught that way.
That's the "liberal arts" approach to mathematics. It's an intellectual exercise forced onto little kids. Even if you use advanced mathematics in your work, it's very rare to need either axiomatic set theory or axiomatic plane geometry.
A completely different approach can be found in some math courses given during WWII courses to soldiers who needed to do technical work. These were utterly practical. Trigonometry was taught with direct applications to surveying and static structural analysis. After that trig course, you could calculate the size of the beams required for a truss bridge. The calculus course covered subjects like the ballistics of big guns. (I especially liked the "tables method" of integration, which taught you how to use those tables of integrals in the back of the book.)
There's a mindset in math teaching that math is about "puzzles". It's not. (Mathematics in England at the university level went off into that dead end for a century, with rated "wranglers" and "senior wranglers", until Hardy kicked them out of it.) But the school version of mathematics overstresses puzzles, because they're easy to assign and grade. That's a bigger problem than the "liberal" aspect.
For a non-puzzle curriculum, see PSSC Physics, which was taught in the 1960s. Lots of little experiments which required some calculation and data analysis.
You should have had Mr Burton, my maths O level teacher. He was brilliant. He was totally passionate about his subject and he was also a fantastic teacher. he encouraged us to think about maths rather than to just blindly follow formulae. ...
You were lucky to have such a teacher. But there are other ways that can work, too.
Back when I was a high-school sophomore, I decided that math was interesting, so I read that year's math text in the first month, then grabbed copies of the more advanced texts over the following months. By late winter, I'd run out of math texts that the high school had, and asked the teacher for more. The reply was the conventional "You're not ready for those yet", which was clearly BS, but was supported by the other teachers, too.
But I had a couple of friends at a nearby college who were willing to loan books to me, and I got several years worth of math texts through them. One funny aspect was that they were female, contrary to the stereotypes. I had some good math discussions with them. One especially funny case was when they gave me a copy of the text called "Calculus for the Practical Man". I asked if they were permitted to read it, and they basically said "Of course not; we girls aren't smart enough to understand complex stuff like that" with grins on their faces. They were taking more advanced math classes at the time, though.
I'd have thought that that title would be too non-PC to still exist, but I just fed it to google, and it pointed me to the amazon.com page for the current edition. I guess some things never change. I did generally prefer the more theoretical texts, but it was good motivation to read explanations of why it could be valuable in real-world situations.
Anyway, my high-school math teachers turned out to function primarily as gatekeepers who blocked my access to more advanced math than they understood. But it didn't matter, because there were ways of doing an end run around them and getting the information elsewhere. Nowadays, the Internet exists, so kids in similar situations can often just download the PDF for a text and learn that way. Assuming that the kids actually have Internet access, of course, which isn't true in a lot of the world (or even parts of the US and Europe).
Those who do study history are doomed to stand helplessly by while everyone else repeats it.
Okay, you need to look at Searle's arguments a little more carefully.
The Chinese Room is a direct response to the Turing Test, which says that an entity that talks like a human and thinks like a human is, in some sense, equivalent to a human. It is an attempt to prove that such an entity need not understand anything. Therefore, you need to look at it as a proof rather than a plausibility argument. It is necessary for Searle to prove that the Chinese Room cannot understand anything. To refute him, it is not necessary to show that it must understand something, but merely that his proof is fallacious.
Searle claims that there is no understanding of Chinese in the room. The initial proof is that nothing in the room understands Chinese, and therefore there can be no understanding of Chinese there. Searle doesn't consider the possibility of emergent phenomenon here. Interestingly, he argues later that consciousness is biological, despite the fact that anything true in biology and not true of individual atoms must be an emergent phenomenon.
GP's use of "begging the question" is the older, more precise and meaningful, definition. Searle claims he has proved that the Chinese Room doesn't understand Chinese. He then entertains several possible objections to his proof, and "refutes" them by having already proved that the Chinese Room doesn't understand Chinese. (This is different from the more common usage nowadays, using it to mean evoking a question.) By attempting to prove something by assuming the thing itself, Searle is begging the question.
As far as incredulity goes, "I can't believe that!" is not a valid philosophical argument. (It's not a valid refutation in most fields of argument either.) It is not necessary to believe that every algorithm leads to consciousness to believe that some algorithm might.
Searle is claiming that there is something biological about the human brain that makes it special, in a rather incoherent but dogmatic way. Whether you consider him arguing that there is depends on your definition of "arguing". Mine, in any scientific or philosophical field, doesn't include proof by blatant assertion, which pretty much sums up Searle's claims of biological specialness. Apparently, biologists find his claims unfathomable.
Therefore, while we haven't shown that a computer can understand things in the human sense, or deserves human rights, nobody's shown that that can't happen. Not with a respectable or valid argument, anyway.
"When you have eliminated the unacceptable, whatever is left, however improbable, must be the truthiness" - Holmes