Math Education-Is There More To It Than Just Numbers? 30
Rasha asks: "I am taking a class this semester which discusses different models of the human mind. One topic that has caused much debate is the nature of math education and its goals. I am writing a paper for this class which will attempt to discover what exactly we try to accomplish by teaching math to young students. Are we trying to give them skills or is there more? Is math our attempt to access the more abstract parts of students' brains and develop them? I have sent a survey to a bunch of teachers (mainly elementary and high school) but I am curious what Slashdotters think. I think that most Slashdot readers are probably more mathematically inclined than the average and might have a greater insight into the issues that I'm addressing. Also if anyone knows any previous research on the topic let me know (this is not the focus of the paper though, I'm just curious)." (There's more...)
"Here are some of the questions I sent to the teachers so you can get a feel for what I'm looking for:
- Of all the subjects which is the most important for the development of the student? That is, which subject gives the most skills to the student beyond the actual information taught? Why? What is the goal of teaching Math to children? Is it to give them skills to manipulate numbers or does it accomplish something else (or maybe both)? What are those skills?
- People often say that math teaches abstract reasoning. Is this so, how and why? Could there be a better way to accomplish this?
- With the development of small computers and calculators do you see the role of math education declining? Why or why not?
- Why are children often forced to memorize multiplication tables and do long division?
- Why is it that students who have some deficiency in math are stigmatized as "not so bright" more often than children who fail to do well in other subjects? Conversely, why are children who excel at math considered gifted (more so than other subjects)?"
Math is good (Score:1)
Aid to abstract thinking? (Score:1)
If it is true that people teach maths to aid the development of abstract thinking, I wish someone had told this to my high school maths teachers. Unless they were actually trying to stifle such thought... At my high school, maths was taught with only reference to the real world (like geometry was 'how many square metres of carpet do you need for this odd shaped room'). As a result, I did very well in high school, and failed completely when I approached real maths at a higher level.
If only teachers thought it through before they adopt fashionable new techniques. Or you could look at it as a left-wing way to teach - who cares if you lower the standard of the more able students if you can raise the standard of the less able - after all, all good socialist members of society should be the same... I don't know about the rest of the world, but here in England most teachers are socialists.
Re:importance of H.S. geometry (Score:1)
If the people I teach stopped me and said they wanted to figure it out themselves, I'd have the same reaction. Let's say the eight-year-old next to me on his computer, trying to learn how to edit XPilot maps.
I don't say things like that, just so that I can glean any extra hints and tips from my teacher. However, I'd greatly enjoy a learner who wanted to learn, not just watch someone else. (Ever seen a child yaaawn while you're solving the problem he just gave to help him with? Argh.)
-lf
Re:Interesting Statement I heard once (Score:1)
I would have been severely hampered in my calculus classes if I needed a calculator for basic arithmetic. Computers don't SOLVE problems, they can brute force solutions to problems. The real value of solving problems is the insights gained along the way. We didn't gain any real insight when computers proved the 4-color therom by brute force.
We don't need to do rote processes as much, that's why I like computers, but arithmetic is as essential as good spelling and grammar skills are.
(My spelling and grammar skills need work, but I still believe that they are essential in our society.)
Wrong place to ask this question (Score:1)
You are asking a crowd which will be biased in one of two ways, and probably not well versed in the field you're asking about. You should go do some research with real experts.
You also shouldn't put too much stock in what teachers themselves say, since they are no more experts in that particular field of education and brain research as the trolls here are. Those with lots (20 or 30 years) of experience might have something useful to contribute, few others will. These people are out in the trenches doing a job, not doing research. Besides, how many math teachers would you expect to say no, what I do isn't important?
Brain research is in its infancy, and mostly we don't know much more than we do. It's generally accepted that puberty and the advent of abstract reasoning tend to coincide, but it's not just a given that it happens that way with everyone. nor does anyone have any idea why that might be, or what gives rise to the development of abstract reasoning. Maybe it's all that rote memorization of math tables (I hope not, because with the rate at which people are doing that nowadays, we'd have an awful lot of non-abstract adults soon!), maybe it's some other combination of things. No one really knows.
The fact of the matter is that to live in the real world, you need math. You can't carry a calculator into 7-11 every time you buy a big gulp to see if you got the right change. Algebra (at least basic stuff -- solve for X kinds of things) is important, too. I'd argue that to really intuitively understand the world around you, basic calculus is important, too (rate of change, etc.)
No, people who are math-gifted aren't necessarily smarter than those who are language-gifted. The notion of placing one discipline above another in iportance is a failing of our (the US) educational system. These areas are all important. But the lack of standards in the US is another failing. I think that just because you aren't good at math shouldn't give you a pass to avoid it.
Many things taught in school (lond division, siagramming sentences, etc.) are done to drive home structure. That there is a process behind what the calculator says the answer is. rote memorization of division tables facilitates the performance of long division, and the performance of long division performed enough times allows a student to develop a more intuitive, quicker method. (Short division?) They compliment one another. Learn the basics, drive home the rules for the non-basic stuff. Then you can do anything.
There has been shown to be a strong correlation between music education and mathematical skills. Something like music helps with math. Duh. Music is math, at least organizationally. Helps order thinking, helps develop intuitive sense for order -- which orders make sense and which don't. Language, thought itself all thrive on order, or the ability to order and associate things. The most brilliant people can order and associate things in ways that are new, unique, but often obvious to everyone else, AFTER they've been shown the association.
Way more than numbers (Score:1)
Better yet, and even more poorly taught (at least in the USA) is being able to apply what is taught to a new but simple problem, a skill that even many of the top high school graduates don't have. (This is a large part of why many college freshman physics courses have high failure rates.)
These skills require a level of understanding the math and of abstract thinking which is hard to teach, but most teachers don't even bother to try. It's just a whole lot easier to say "this is how to do this sort of problem...". If this was taught consistently, then math would be one of the most useful classes in an education. Without it, it's largely replacable by a calculator.
The students with a natural aptitude for math and abstraction (and those whose parents work to ensure their children do understand and can use what they learn) will do well, find math useful and end up with superior problem solving skills.
Math is good (Score:1)
I'm a graduate math GSI at Berkeley, and have even participated in a fellow grad student's survey in an attempt to understand why teachers do various things (e.g. why did you ask for the answer from the class instead of just telling them?)
So here's my take on your questions:
Of all the subjects which is the most important for the development of the student? That is, which subject gives the most skills to the student beyond the actual information taught?
These are two different questions. As one of my professors noted, most people only use basic computation in their daily lives, so that is the most important topic. The most transcendant topic is, of course, the problem solving skills students derive from doing their homework. They (hopefully) learn to attack problems in all areas of life. Some of the things they learn are incredibly basic: What are the preconditions? What do I want? What tools do I have at my disposal? How can I break this into a sequence of simpler problems?
What is the goal of teaching Math to children? Is it to give them skills to manipulate numbers or does it accomplish something else (or maybe both)?
Yes. Everyone needs to be able to manipulate numbers in the grocery store. This is all that is taught until about ninth grade, when most people take geometry. Geometry introduces the conept of a proof, which is at the core of all abstract thought. Most students do not ever understand proofs, but only those who do can pursue any sort of advanced career. For example, programmers write proofs in their heads all the time, such as, "This function will not dereference a null pointer because such-and-such."
People often say that math teaches abstract reasoning. Is this so, how and why? Could there be a better way to accomplish this?
As mentioned above, by teaching proofs, math teaches abstract reasoning. A better way to teach it would be to emphasize proofs in other classes, such as chemistry, biology and physics.
With the development of small computers and calculators do you see the role of math education declining? Why or why not? Why are children often forced to memorize multiplication tables and do long division?
No. Suppose I ask you to prove Fermat's Little Theorem, and you are allowed to use all the theorems from a basic course in abstract algebra (I even hand you a book on the subject). The problem is, most likely, still intractable to you. However, if you had memorized the theorems from the course, you would quickly reply that FLT is a mere corollary of one of those facts. So it is with any tool. Caclulators are cumbersome, slow computational tools and if all students become dependant on them, then problems which should be easy will become burdensome. Even bright students would have to do stupid things like stop in the middle of a large problem and use a calculator to determine if 11/5 is bigger than 2, slowing them significantly. Furthermore, students would be unable to do rough approximations in their head to get an idea of what the answer should be.
Why is it that students who have some deficiency in math are stigmatized as "not so bright" more often than children who fail to do well in other subjects? Conversely, why are children who excel at math considered gifted (more so than other subjects)?"
I'm not sure this is even the case, since the gifted math and english classes at my h.s. consisted of mostly the same students. The students who were good at math were good at other stuff, and the good writers and artists were competent enough to be in the advanced math classes, too. There is, perhaps, a societal belief that math is the closest one can get to pure thought (I agree with this view), and hence good mathematicians are good thinkers.
Re:Interesting Statement I heard once (Score:1)
Re:Aid to abstract thinking? (Score:1)
Here's a free hint: George Orwell, author of Animal Farm and 1984 - two of the 20th Century's greatest greatest warnings about authoritarianism - was a dedicated socialist.
From arithmetic to ethics (Score:1)
This isn't a serious question, but more of a social quandary; should we let these people continue to screw things up in the world at large, or force them into remedial education as soon as humanly possible to try to arrest, or at least slow, the damage?
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Re:Get real! (Score:1)
Of course I did. As did everyone else in my calc classes. Hell, that's what the professor told us to do.
Re:Elementary school teaches Arithmetic, not Math (Score:1)
Thanks (Score:1)
Re:Math is good (Score:1)
The skills at effective communication are very similar to understanding mathematics. Writing a convincing essay is very much like proving something in math: you have a hypothesis, state some 'theorems' ('facts' from some other author, or from your experience), and using logic, draw conclusions.
Of course, this is not the same as being poetic, and being a great literary writer (requiring artistic talent, etc., which I don't have) but communicating your ideas clearly requires the same skills if you are talking about math or anything else.
math is the closest one can get to pure thought
I share this opinion. Unfortunately, I cannot get that close, so I'm studying CS
It has to be in your head (Score:1)
Of all the subjects which is the most important for the development of the student?
The subject that makes the student think is the most important one. I don't belive there is a particular subject that is the "most important" for every person. Everyone is better at some things than other people, whether it be playing the violin, running a business, mountain biking, or brain surgery. Each of these tasks takes different training, and for each person, a particular subject will be more or less important. The skills taken from a subject often depend on the teacher, since artists can relate things to artworks or techniques while chemists can relate things to equations or particles.
Math does still need to be taught to every student, just like reading will never go out of the schools. Yes, you can find a job in which you won't need a pencil and paper or a spreadsheet to figure things out, but most people don't want to work at a fast food chain or a retail store for the rest of their lives. Spreadsheets will calculate everything out for you, but you have to know how to use it, what formulas to put in it, and why you're using them.
People often say that math teaches abstract reasoning.
Math problems are often solved using a set of steps or a pattern; this type of reasoning (breaking the problem up into pieces, relating steps to other concepts, etc.), which I believe to be quite logical rather than abstract, can be very effective in solving many problems, not just mathematical ones.
With the development of small computers and calculators do you see the role of math education declining?
Math education is more important than ever. When most people worked in factories or in fields, calculus was probably not a priority. Today's society needs people who are more educated in technical fields -- people who can create and improve those small computers. I'm pretty sure that kind of work isn't done with elementary school division and multiplication.
I have tutored a few junior high kids in algebra, and during my sessions, I do not allow calculators except for nasty decimal stuff. I'm the "hard tutor," but I'm making my students more comfortable with thinking, rather than just flying on autopilot with a calculator. Calculating 3*25 is hard until you ask how many cents are in three quarters.
Why are children often forced to memorize multiplication tables and do long division?
These are simply the first steps (building blocks) toward understanding a subject. After all, you start reading by looking at individual letters, rather than words like "aneurism," "pseudonym," or "carboxylic acid." The fundamentals have to be understood before you can go on to more advanced topics.
Why is it that students who have some deficiency in math are stigmatized as "not so bright" more often than children who fail to do well in other subjects?
I don't believe this is true, especially in later years (high school/college). Students who don't excel in math are told that it's hard stuff, and not everyone understands it, so it's okay. This is why people don't continue taking math courses: it's hard, not everyone understands it, and it's okay to give up. This is a society that produced a talking Barbie(TM) doll that said, "Math class is hard!" (I know the doll was pulled from the shelves, but the fact that it was produced at all is a clear indication of the mindset in this country.)
Conversely, why are children who excel at math considered gifted (more so than other subjects)?"
Along the same lines, I think that because society deems math to be difficult, kids who do well in the subject are praised highly. There is nothing wrong with this notion, as long as the genius of these kids in other fields isn't overlooked and other kids in other fields aren't overlooked either.
The Role of Calculators in Math Education (Score:1)
Re:The Role of Calculators in Math Education (Score:1)
I wouldn't say the use of calculators has made math harder, just more realistic. Math is not about nice, round numbers; it is about problem-solving. Calculators have allowed for mathematical exploration and experimentation and, consequently, enhanced ability to problem-solve and understand. Students and non-students alike are no longer inhibited by the limitations of their ability to perform tedious paper-and-pencil calculations. Calculator-use allows students, teachers, and even laymen more time to develop mathematical understanding and reasoning skills. And while math tests may have gotten "harder," the mathematics itself has become more realistic, accessible, and problem-solving-oriented.
A few answers on math education (Score:2)
Math=logic (well, not really) (Score:2)
Eventually, maybe this stuff won't be very necessary. When there is ubiquitous and omnipresent computing ability (chips in our heads?) and if we feel we can always trust these, then complicated arithmetic will be unnecessary. But kids spend a lot of time learning 5+3, and when you become an adult you need to be able to figure out simple math like that without an interruption in thought. These things that must be learned are mostly "math facts" - the basic bits of math that we must learn to solve without reducing them, i.e., you can't be very functional if you figure 6+3 by using your hands, but you must simply Know that 6+3=9.
Many (most?) kids don't really get much else than that. They are introduced to other things, but the introduction is poor and their attention is difficult to maintain on something that requires hard thought.
For instance, I say 6+3=9, but really people think "6 and 3 make 9". The deeper notion that 6+3=9 implies that 9-3=6. If you really understand this, answering 4x+2=10 is pretty easy (though is still requires a certain gestalt to achieve the specific value of x from the infinitite possibilities). But equality, timeless and hopelessly nonimperitive, is a very difficult concept. C's x=x+1 is far easier, though from a mathematical perspective is looks terribly confused.
Really, all these hard bits of math are philosophy, not skills, and certainly not science. There's not a lot of philosophy until college (and even most of that is dumb, but I won't digress). That's probably not the way it should be... and that the most inaccesible bit of philosophy -- math -- is the most emphasized is perhaps a bit peculiar. A question like "can killing be justified" is something you can relate to life. But when you really start thinking about probability, say, and the fact that there's a 50% chance that a coin will flip heads before you flip it, but that chance becomes 100% one way or the other after you've finished... it's awfully abstract.
So people say math is about teaching abstraction, and maybe it is. Being able to resolve the infinitude of possibilities into one solution.
So I think good math education is about exploring the intuitive (gestalt/right-brained) solution of problems, abstract (numerical, symbolic) and concrete ("real world"). Ironically, the most successful math students are rigorous and left-brained, because teachers like this and give problems that can be solved this way -- even though real problems that people actualy want to solve are seldom so easily solved. (this bias is by no means isolated to math, though)
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My answers (Score:2)
Of all the subjects which is the most important for the development of the student? That is, which subject gives the most skills to the student beyond the actual information taught? Why?
Reading. Developing ability in reading and the habit of reading makes learning a lifelong activity instead of something you're forced to do in school, that ends the moment you graduate.
What is the goal of teaching Math to children? Is it to give them skills to manipulate numbers or does it accomplish something else (or maybe both)? What are those skills?
For the large majority of students, the goal is to get them to learn math skills and how to apply those skills in real life situations. For a minority of students it's more, and it's crucial to make sure that all the students who might benefit from more get the opportunity to show it.
With the development of small computers and calculators do you see the role of math education declining? Why or why not?
A few years ago, I made a $16 purchase in a supermarket. I gave a $20 to the assistant manager (!) working the register. He rang it up as $200, and started counting out the $184 the machine told him to give me. That's why you need to learn to do math yourself.
Why are children often forced to memorize multiplication tables and do long division?
In theory, you could learn math by principles rather than by memorization. In reality, that's a disaster because 1) many, if not most, students are not capable of learning math that way and 2) many, if not most, elementary school math teachers don't understand the principles well enough to teach that way.
I find it odd that people are so convinced current methods of math teaching are wrong. This isn't something new, like "diversity education". Formal math education has been systematically refined since Euclid. Why is it so hard to believe it works well?
Why is it that students who have some deficiency in math are stigmatized as "not so bright" more often than children who fail to do well in other subjects? Conversely, why are children who excel at math considered gifted (more so than other subjects)?
I'm not sure that's true. On the contrary, at least for educated American adults, saying "I'm terrible at math" is not considered embarassing. If anything, people take a sort of pride in it. But you'll never hear an engineer or physicist brag, "I have poor reading comprehension." or "I'm a terrible speller." Well, except for Rob Malda...
Real life use for math (algebra & geometry) (Score:2)
One can usually get fairly close via figuring out how many jelly beans will fit into the area of the jar's cross section (sometimes they make the jar round to make it more difficult, but hey, A=PI(r^2) for a circle - no sweat), then multiplying by the number of "thicknesses" of the jellybeans that the jar is in height (if you understood that, then you know what I am saying), to get a number that is close. Add a little fudge factor ('cause those damn jelly beans never manage line up properly in even layers, like they would in an ideal system), and you're set.
Sometimes a jar/vase is used with varying area cross sections - these can be figured out individually, then totalled at the end.
Heaven forbid they use a sphere (oh no - spherical volumes - don't make me work).
Of course, I may be just too much of a geek...
Then again - anybody got ideas/info on obtaining better values for this kind of close fit problem?
Re:The Role of Calculators in Math Education (Score:2)
//rdj
Get real! (Score:2)
Because public education in this society was designed around the time of the Industrial Revolution, when factories needed just a few simple things from their workers:
1) Do what you are told without question
2) Do it again and again and again (ie repetitive tasks) without getting fidgety
Bullshit! Don't try passing off your political views as knowledge about education. Arithmetic is learned by rote because is so simple and basic that it should be done quickly with little thought. Or do you really want to do 5*2 as (((((((((1++)++)++)++)++)++)++)++)++) ?
The basics have to be memorized so you can move on to more interesting things. What's ridiculous is how much time is spent memorizing - nay, trying to teach kids "how it really works". Think "hash table".
As a CS/Math major, I feel that arithmetic is basically useless, in the sense that nobody needs to know long division (or what 11 * 15 is, etc). That's what calculators are for. I feel the same about Calculus (yes, somewhat more deep and much more complex than addition and subtraction but basically just computation; no actual thought required if you know the right techniques). The interesting problems are the ones that computers can't solve.
Then you're a fool and a stooge. If you can't do arithmetic then you're a slave to however can do it and uses that knowledge to make the calculators and computers. The reason you think "no actual thought is required" is that you have learned it by rote. You've done exactly what you think shouldn't be done. Yeah, most anyone can compute a derivative algorithmically, but that's not the point and you miss completely any appreciation of either the mathematics or the real world. Go read Newton, philistine!
As for "higher level math" which you imply starts after arithmetic, everyone uses it every day. If we didn't then every dog would be completely unique; you could never form a concept of dog - i.e., you could never define a variable to be an element of the set "dog" and then recognize memebers of that set. The math professor you heard was dead wrong and probably pushing his own political agenda to get more funding. High schools should be starting with, at least, calculus; that they don't is indicative of what a failure the education in the USA is.
So now get religion. (Score:2)
If you're a math major, you should get religion; start seeing how all the parts tie together.
religion - res ligare - things tied
-or-
religion - re ligare - regarding ties
(I don't remember which; and my latin grammar is poor from lack of use.)
For example: Study the history of calculus. I think it starts with the method of exhaustion of Eudoxus. Learn what problems they were trying to solve and why and what they tried. See how the methods have evolved. Look into their connections with other sciences - not just today, but throughout history. Even delve into the lives of the people. See what they learned that led them to try what they did. Find what non-mathematicians did for the science and what mathematician did for the other sciences.
All of this is your education. (voice of yoda) "Not this base computation".
ex - out from
ducere - to lead
Education is a leading out: of your notions into ideas, and of yourself from slavery to the minds of others.
A branch of knowledge is more than just lumber; it is part of a living thing and a home to other things.
Don't forget your looking glass.
Social and political implications of innumeracy (Score:2)
There are so many important things in everyone's lives which really require algebra or better that it's sad most people can barely handle arithmetic. For instance, the optimum first-year depreciation deduction for a business vehicle may not be either the flat deduction or the straight-line or double-declining balance figure, but some proportion of one and the balance of the other to reach the depreciation limit. To determine what the proportions should be, you need algebra and differential calculus. It's a simple formula, but you need to understand what you are doing. Another example is home mortgages and retirement planning. If you don't know what your loan balance will be 5 years from now, you have no way to plan. If you can't calculate your IRA portfolio's value based on projected rates of return, and the income you can expect to take from it, you have no idea what you have available to live on and what kind of lifestyle you can expect... nor what to do to get to where you want to be. This requires knowledge of compound interest, which is a fairly simple derivative of the formula for the sum of an infinite series.
This last is very important in politics. A great debate is going on in the US presidential race, and it is almost entirely uninformed by numbers. Only a tiny fraction of the populace would understand, so the news media does not publish them, and ignorance is perpetuated. This certainly does not serve either the body politic or posterity.
Algebra and calculus would be useful to a huge number of people, far more than have a good command of them. Unfortunately, those who have the need for it often have no idea what their problem is or that some knowledge of these matters would improve their lives.
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Interesting Statement I heard once (Score:2)
Why are children often forced to memorize multiplication tables and do long division?
Because public education in this society was designed around the time of the Industrial Revolution, when factories needed just a few simple things from their workers:
1) Do what you are told without question
2) Do it again and again and again (ie repetitive tasks) without getting fidgety
Doing stuff like mutiplication tables is good practice for #2. And at the elementary school my younger brothers went to, there was a rule: "You will obey an adult without question". My Mom got mad at that one: an adult? Any adult? One who just walks of the fscking street?!?!? "Come with me, little girl?" "OK, I wouldn't want to break the rules". So you see #1 is well handled in most lower education (and in some cases taken to a dangerous extreme).
As a CS/Math major, I feel that arithmetic is basically useless, in the sense that nobody needs to know long division (or what 11 * 15 is, etc). That's what calculators are for. I feel the same about Calculus (yes, somewhat more deep and much more complex than addition and subtraction but basically just computation; no actual thought required if you know the right techniques). The interesting problems are the ones that computers can't solve.
importance of H.S. geometry (Score:2)
In a standard high school geometry course, theorems and proofs are introduced well for a HS level. Childeren are taught to look at complicated problems/theorems and then break them up into small steps in order to prove it. One of the main concepts is to teach the students to solve a problem by breaking it into a linear series of steps all leading towards one goal. This is a major part of the abstract reasoning that you mention in the second bullet.
I have four years of tutoring experience at a college level and in every college class from basic algebra to differential equations, all problems can be solved/understood by breaking them into a linear series of smaller problems. (This is also true of the classes I took that were past Diff Eq.) Something that should have been developed in HS geometry.
To answer your other point about Multiplication tables and long division...
I believe students should memorize them merely to teach them that some things they have to sit down and rigorously go over until they know it. What other area in education forces students to rigorously memorize anything? In schools around Okemos (MI), they have discontinued multiplication tables from the curriculum, and I fear that the students are never going to learn that sometimes they need to sit down and go over something until they know it. There's no easy way to learn multiplication tables and that is it's strength.
One last interesting point is that I have tutored a wide range of math classes at college, and the biggest difference between the math-challenged people in College Algebra 101 and the engineering students taking differential equations is self confidence. My observations show that Mathematical self confidence almost always correlates to personal self confidence. In the Diff Eq room, you help a student by guiding them, and they would trudge through the details, making mistakes, but ultimately solving the problem. In the Algebra 101 help room, many of the students need you to solve the problem while they watch, they seem afraid to perform basic steps by themselves.
One of my most rewarding experiences in the math help room was with a girl that would come in 2-3 times a week. In the beginning of the semester, she was constantly frustated because she didn't understand things, and I wouldn't just sit down and solve it for her. I would tell her how to do it but make her do the thinking. One day in the middle of the semester, she stopped me in the middle of a problem that I was helping her with. She said that she wanted to figure the rest of it out herself. I was happy for about 3 days because of that, and she probably never even realized it.
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Math and English (Score:2)
Math is very important because it develops abstract thinking. It is where you learn to break a problem down and think logically. Yes, the average person probably does not use much more than arithmetic during their day, but the skills of critical thinking are used all the time.
Recently there has been reforms in the math curriculum, at all grade levels, including the misnamed 'Harvard Calculus'. These reforms aim to encourage people to devote more time to 'understanding' and 'exploring' the math, without being rigorous: relying on calculators and estimation.
Although, on the surface, these reforms sound like a good idea, they do not work as expected. Students become reliant on these techniques, can pass all their exams by graphing a curve and looking for the maxima, and never gain any real understanding of what is going on.
It is only through working through laborious problems (be they long division or finding integrals) that a student will finally, really understand what is going on. Students do a few problems, step by step, until it just 'clicks' and they then understand the purpose of each of those steps, and what the problem really 'means'. Just typing something into a calculator, and looking for the minima, does not really teach anything: it does not explain why there is a minima, and why it is at that point.
English is the other very important skill. It teaches communication, which is the most important skill one can have. Being able to read with understanding, and communicate ones thoughts clearly, is a skill surprisingly very few people have.
It is only through spending a lot of time reading until you gain comprehension, or repeatedly re-writing something until it is clear and concise, that you can learn to communicate clearly.
So often, I am surprised at office memos (and even newspaper articles) that re logic-less and make absolutely no sense.
Clearly, these are the two skills most important for success, and the teaching of them should not be softened.
Sentience == Abstract thought === Math (Score:2)
Granted, practical subjects, which teach you actual physical skills in how to manipulate objects, are useful, but it is essentially true that you could also teach a monkey to do those things. The difficult part is the thought behind those actions; knowing whether to build a table, rather than how to build a table.
Students who are good at math (and other abstract subjects, like music for example) are treated as more "intelligent" because they are in fact more sentient, assuming it is possible to be "more sentient".
I would also say that students would be better prepared, and better educated, if more attention were paid to developing their capacity for abstract thought with :-)
(a) greater attention paid to math
(b) some or more attention paid to topics like music and, yes, computing where abstract, logical thought is often key (unless you're using a Microsoft product
Elementary school teaches Arithmetic, not Math (Score:3)
1)
Reading. Period. After that, she can teach herself. But "A, B, C" isn't enough, which is why English classes are so key, they give practice in reading. English class doesn't teach anything about reading; for that see "How to Read A Book" [amazon.com] by Mortimer J. Adler.2)
So that "the future of America" will be able to live in the "America of the Future(TM)".3)
Again, until high school, it's just coping skills. Then, higher thinking is slowly introduced. Slowly.4)
Math teaches abstract reasoning, arithmetic does not.From "Mathematics Dictionary" 5th Ed. [loc.gov]
Thought that I would clear the definitions up a bit. Basically, you don't hit mathematics until high-school. So elementary 'math' doesn't teach abstract reasoning, though it may teach reasoning on some level.5)
You always need a gut-level check of whatever you are doing. If you don't know that 1882*1000 should be bigger than 1.9, you won't realize that you divided instead of multiplying.In engineering we occasionaly finish a complex analysis which has many possibilities for making mistakes by doing a "sanity check" where we use a less precise but simpler method to check our answer. Stress analysis of a spring using elasticity methods is a good example. I had a 3/4" stack of paper for my analysis, with the pages covered in calculus and static analysis. When I was all done, I checked my spring constant equations against a handbook equation, and I was close. So I assume that I was 'right'. Without the sanity check, I wouldn't really know.
6)
Because it is actually useful. Not just for engineering students like me, but for checking the high-school dropout who is ringing up your groceries: if he puts the decimal in the wrong place, your loaf of bread is $10, not $1. That is much easier to check if you know that $10 is 10 times $1, and that multiplication by 10 can be done by moving the decimal point. An ability to do basic arithmetic cannot be thought unnecessary when our society is ruled more and more by numbers. (Politicians use polls, we all use prices, homeowners use mortgages, nearly everyone uses credit cards. To understand all of this, we must understand arithmetic so well that we don't have to check to see if we did it right; arithmetic must be nearly second nature.)7)
Because it seems that our society thinks that math is hard (to quote Barbie), so if you can do math, you must be smart. That one is mostly societal.BTW,
Are the people in your class primarily from the sciences or the humanities? I ask because I have noticed a trend at my university [calpoly.edu] that the students who use math in class regularly (physics, engineering, chemistry, etc.) think that math is an essential life skill for everyone to know, and the students in the humanities (psychology, english, history, etc.) see math as useful in balancing a checkbook, but beyond that, it seems to have little point. "Why did I have to take algebra? I've never used it?" When this comes up, the science types insist that math is an essential skill, but are hard pressed to find "real life" examples of how algebra or geometry could be useful. And we aren't even up to basic Calculus in the discussion! I would like to find a way to convince people that math is useful, not just arithmetic.
Louis Wu
Thinking is one of hardest types of work.