Mathematics As the Most Misunderstood Subject 680
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."
Re:he's right (Score:3, Interesting)
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.
Re:he's right (Score:5, Interesting)
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.
Being a mathematics undergraduate... (Score:5, Interesting)
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.
Re:he's right (Score:5, Interesting)
>>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.
Not just maths (Score:5, Interesting)
Re:he's right (Score:5, Interesting)
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.
Re:he's right (Score:4, Interesting)
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.
Re:he's right (Score:5, Interesting)
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.
Re:HERE IS WHAT YOU NEED, KIDS !! (Score:4, Interesting)
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!
Math is just...math. (Score:4, Interesting)
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.
Poor Math Education Hits Close To Home (Score:5, Interesting)
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.
Re:he's right (Score:5, Interesting)
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.
Re:Poor Math Education Hits Close To Home (Score:5, Interesting)
I logged on for the sole purpose of replying to your post as our situations are so similar I couldn't let it pass without comment.
I realized in 1st grade that my son was the same as yours. His IQ doesn't test quite as high, somewhere around 130, but he has an intuitive grasp of certain things that's almost breathtaking. I remember when he, at 5, described to me the mechanics behind a lunar eclipse! It wasn't even a topic of conversation, just out the blue. Apparently he had been mulling it over and had worked it out. Anyway, back to the subject.
Let me say you rock as a dad, not only for noticing the problem but working with your son. My son has also been subjected to the "spiral curriculum" and it's alternately made me want to rage or laugh. Far too much time is spent teaching different ways to accomplish the same tasks and there is no way to speed it up for those who are bored. I solved this problem by advancing the curriculum at home. When my son got bored with addition and subtraction I made the numbers bigger, when that became trivial I made them harder by including decimals, then harder again by using fractions. When he became bored with multiplication and division I started teaching him Algebra. When his class moved on to kiddie Geometry and he grew bored with it I started him on Geometry I. You get the idea. It was in Geometry this year where the teacher caught on to me.
His teacher and I had a major blowout when one of his Geometry papers was returned with a score of zero. My son was freaked and so was I. What did I do wrong? I went back and forth through that paper for two hours looking for what had happened and couldn't find it. I called in the wife who has a Degree in Math and she couldn't find anything. I called in the Grandpa with dual Masters (Chemistry and Physics) and 45 years experience as a High School teacher and he didn't find anything. I went to the school the next day and had his teacher explain why and you know what the answer was? He forgot the damn degree symbols. Yes, that's right a 10 year old doing math work years ahead of his level received a zero with a page full of correct answers and a companion page showing all of the work because he FORGOT THE DAMN DEGREE SYMBOLS.
Further she told me that she didn't like me teaching him this stuff because my way was different than hers which made it difficult for her to grade his papers and he confused other children when he tried to help them! I didn't know whether to cry or murder her. The depth of willful stupidity on display at that moment still staggers me. In the end I politely told her that I wasn't going to stop doing it because his education was more important than her classroom. I left shaking my head and wondering how our education system got this screwed up.
So you keep rocking on Dad, you keep pushing his curriculum and teaching him. What the idiots at the school won't do for him is your privilege, and responsibility, to provide. When he grows bored you up the ante and make it more challenging by showing him the "Big Boy" way and giving him something new to explore. Someday when he outgrows your ability you can sit back and proudly tell him "Son, I don't have anything left to teach you." and then watch him start learning it for himself.
Re:HERE IS WHAT YOU NEED, KIDS !! (Score:4, Interesting)
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.