Slashdot Mirror
Promoting Arithmetic and Algebra By Example
Capt.Albatross writes "A couple of months ago, the New York Times published political scientist Andrew Hacker's opinion that teaching algebra is harmful. Today, it has followed up with an article that is clearly intended to indicate the usefulness of basic mathematics by suggesting useful exercises in a variety of 'real-world' topics. While the starter questions in each topic involve formula evaluation rather than symbolic manipulation, the follow-up questions invite readers to delve more deeply.
The value of mathematics education has been a (recurring issue on Slashdot)."
Aside from the obvious ones in engineering, where few kids will participate...
There is the issue of "how much paint will I need to paint my house?"
Doing the math will save you money.
Unfortunately for the counter-argument, every one of those examples can be replaced by simply visiting a website online that does the math for you.
I must bravely agree with the author completely. We must prepare our children for the future. Clearly we need a few intellectuals. But these folks all wear grey and work far too hard. The semi-intellectuals are about the same. Let's have our school system produce more of us, the ones who have the truly best balance in our work. We are smarter than the manual labor groups but don't have to work or study as hard as the elites.
See my journal for slashdot ID's by year. Mine created in 2005. http://slashdot.org/journal/289875/slashdot-ids-by-year
Every story like this reminds me of the movie Idiocracy. Maybe we should just encourage people to have lots of kids. We could even use the useless ones as food!
Try doing something as mundane as payroll or cash flow estimates without algebra, and you will know why. Even better, try to explain it to someone who DOESN'T understand basic algebra, and you will lament why they don't know algebra. Well, unless you have no plans on ever employing or managing people, because you will be a political scientist all your life.
The reason is not algebra's application to daily life. The reason you teach algebra is because algebra teaches symbolic manipulation. Learning math teaches you not just how to add two numbers. Addition is almost unnecessary in daily life (we do have calculators). Learning algebra is critical because it teaches us to think in terms of abstractions, of models. We do not teach mathematics to teach you how to add, we teach mathematics to teach you how to solve, to teach you how to think.
... that most k-12 math teachers never left academia.
Most math teachers go directly from high school to university or a teaching college and go right back to k-12.
The ones that have seen the inside of a machine shop, the inside of a land-evidence vault, worked for a logistics firm, done bookeeping, been an actuary, or even looked through a theodolite, are few and far between. So even coming up with "real world scenarios" is next to impossible.
--
BMO
But is it recursive? And more interestingly, does it converge?
But teaching kids how to think is not desirable in an economy that can't provide any jobs where they need to think. Thinking leads to people not doing everything their "superiors" tell them to do, and that leads to unhappiness. You kids to grow up to be happy, obedient adults, don't you.
Andrew Hacker and nearly everyone else is missing the point.
Taking an algebra class for many students is not about the algebra, it's about learning to think. Even if you never use algebra again, the process of learning algebra is mental exercise that improves the mind. Taking a foreign language, studying biology, learning economics, studying history - it doesn't matter what the subject is, merely the more you learn the better a learner you are, and the better thinker you are.
In sports we see athletes perform all kinds of exercises that help develop skills used in their sport, but are never used directly. Ever see footage of a football player stepping through tires? Ever see one do that during the game? Ever see footage of a quarterback or pitcher throwing the ball through a hanging tire? Ever see them do that in a game? Athletics is filled with examples of training exercises done to hone one's skills for a game, yet we have difficulty accepting that mental exercises hone skills we need for life.
*jokingtroll*
I'm so sick of all these nerdy math / science posts... we need more patent litigation and mobile device war posts
*/jokingtroll*
Seriously though, I think that straying away from the mathematical fundamentals will lead to straying from linguistic fundamentals and historical fundamentals. Eventually the bulk of the education system will be 'Can you read well enough to use a computer? Congratulations you are a high school graduate".
The ability to follow through an entire equation and achieve the outcome is very useful in life. Perhaps not in the form you learn in Algebra, but in one form or another.
But lets just let our kids get less smart and more dependent on the technology that will eventually stop evolving because nobody is smart enough continue the evolution.
http://xkcd.com/1050/
This post cannot be rebroadcast without the express written constent of Major League Baseball.
The reason you teach algebra is because algebra teaches symbolic manipulation
The tool you should be using is symbolic and philosophical logic
How do you propose to teach someone symbolic logic without teaching them what symbols are, and how to manipulate them?
Learning mathematics is learning to solve problems. Moving math into symbology (algebra) teaches problem solving as a concept vs. a rote skill. Anyone who thinks math is a useless skill needs to seriously re-examine their thinking. Learning how to approach, analyze, and resolve a problem is a skill that many do not have and the country is worse for that lack.
While algebra is useful, how it is taught is counterproductive to certain learning styles. Many kids don't pass may because they don't see the point. I had that problem in school.
If they would make it practical, instead of teaching it through theory, it would be more enjoyable. Say for kids who like computers, give them a math class where you build a circuit board and use physics or something.
You can't teach philosophy to children. Given the philosophy classes I've taken, you shouldn't teach it to adults. It's a class in mental masturbation. The old philosophers were the Original Internet Trolls. The point of philosophy is to argue, not to learn, discover, or use. Sure, after a few millenia of arguing a truth may accidentally be stumbled upon, but that's a by product, like plastics from the space program.
Learn to love Alaska
I've been a developer for about 16 years, and have had a pretty spotty math education. I've generally taught myself what I need to know as I needed to know it - 3D programming? What's a matrix? How do I rotate things with it? Developing animate graphical charts? How do I scale from business coords to pixel coords, and animate? Draw box an whiskers charts etc...
Recently, I've decided to stop doing the corporate developer gig and to go to school. As part of that, I've needed to take math a lot more seriously, so I've bought some books and been going through a more rigorous program.
One thing I've discovered through this process is that I *really enjoy it*. I'm not being pressured to learn something for a test, I'm not worried about a grade. Instead, I take my books to a coffee shop and relax and think about fascinating things, like trying to visualize the complex plane, and what the value for i really is, and what dividing by zero really means.
Instead of memorizing the quadratic equation, I spent some time learning how to derive it from basic principals. Instead of memorizing that the vertex of a parabola can be found by -b/2a, I noodled around and tried to visualize the determinant (sqrt(b^2 -4ac)), it's effect on an equation, and what happens if you zero it out.
I spend a leisurely afternoon coming up with a visual proof of the Pythagorean theorem, and was pretty excited when I finally had it, and was even more excited when I googled it and saw the same basic proof has been derived by students for a really long time - I loved the notion that I was connected back through time with a whole bunch of other people who were going through the same mental steps.
This stuff is great! And I'm only scratching the surface. I'm in baby algebra - and I'm excited to keep going.
My point is - we go about this stuff all wrong. Forcing kids to memorize equations so they can pass an exam is absolutely pointless, if not masochistic. Exploring really interesting concepts about numbers, and what they mean - this stuff should be recreation. It's great!
I see my older son struggling through his algebra course, and he hates it. He doesn't care, and hates doing the homework. But when I get excited about some math problem I'm studying, he'll come over to look over my shoulder to see what I'm doing, and we'll puzzle it out together. He forgets that we're doing math, instead we're talking about concepts and challenging each other. We'll spend an hour or two going over something that's really cool, and we both have a great time.
Ask him about math, however, and he immediately relates it to school, and he'll tell you how much he hates it.
Drinking habits can be dangerous. You can choke on the cloth and the nuns will wonder where their clothes are.
The understanding of the simple physics of simple machines would benefit greatly from basic algebra.
Eg, calculating torque to RPM change over an arbitrary gear ratio, or how much energy is needed to push a 1kg weight with a 1 meter lever.
Granted, these would only ever be interesting or useful to people who like to build things, but I can't begin to state how grateful I am to have been exposed to algebra.
I agree though, the typical scenarios created to sell algebra textbooks (rather, the problems shown in said books) are horrible, unnantural contrivances.
For some things, like "i", (sqrt of -1), I still haven't found a useful application outside of complex physics and formal mathematical proofs. It's a very unintuitive concept. Sure, if you want to calculate the mass of a tachyon its a fun thing, but seriously....
But simple algebra? If nothing else, it helps you disassociate your values, and see the raw algorithm. The numbers can change, but what gets done to them does not. That is a very useful piece of learning.
The reall issue here is not "math is hard yo, and nones o' dis shit gonna get used nohow teach." But rather "all students need to be treated the same."
Not all students are the same. How you teach students shouldn't be cookie cutter. Some would take to algebra like ducks to water. Denying them that is a crime against mankind. Others would rather perform a self-clitorectomy with broken glass. Forcing them to endure is also unreasonable.
As a cultue, we are so afraid of segregating our children's education, because some of us feel threatened by the percieved successes of others. (Not enough that I succeed, others must fail.) Rather than own up to our owm mediocrities, we denounce them, and enforce a fiction that causes real pathologies.
This is one of them.
Look kids, if x out of y of you learn algebra and go on to earn s dollars a year, y-x of you will earn s*m dollars a year, where m<1.
systemd is Roko's Basilisk.
>How do you propose to teach someone symbolic logic without teaching them what symbols are, and how to manipulate them?
You are implying that algebra is the be-all and end-all of symbolic logic?
Let me introduce you to New Math. I was a "victim" of it.
It revolved around number theory, set theory, and logical operations - and, or, not. I knew venn diagrams and set notation before I knew how to find least common denominators.
New Math was widely derided by people who thought that arithmetic proficiency done through rote learning and timed tests (which was my third grade math) should be the goal of the early grades. The thing is though, when home computers showed up, I was able to teach myself programming, and a course in digital circuits wasn't as hard as it could have been, and all sorts of stuff including a summer course in Logic (with Irving M. Copi's book) that made one of my smart friends drop in a week.
My point being is that teaching philosophical logic as such is a better tool for teaching one how to think, because it's not just applicable to math. It's applicable to law and debate and other such things. It gives one an appreciation as to why, in a law, there is an "or" instead of an "and" linking clauses.
But hey, what do I know.
--
BMO
Addition is almost unnecessary in daily life (we do have calculators).
No it's not unnecessary. I have $5 - can I afford a $4 sandwich and a $1.50 drink? I'm not going to pull out a calculator for that.
Here's my argument for teaching algebra: The more advanced algebra courses teach exponential growth. That's exactly the kind of equation you would do well to understand if you were, say, taking out a loan to buy a home. Not that there have been any problems with people taking out mortgages they don't understand or can't pay back.
I am officially gone from
>Given the philosopy classes I've taken
And yet you skipped right over that Logic course.
--
BMO
The idea that people must know more math is ridiculous. Just because you know more math does not make you smarter, nor does it generally improve that quality of life, including ability to get jobs or perform daily tasks.
In the "google" era of instant information, knowing that you need to use some math principal to perform a task and being able to retrieve information on how and where to use it is more important than expecting to memorize hundreds of examples or theories about math.
However I will say that while in school learning math is important to developing problem solving skills. I may have forgotten 90% of the math concepts I learned in High School and definitely most of the crap forced on my in University, but I know that my problem solving skills were honed learning to solve meaningless mathematical problems and theories. Math courses should focus more on the how of math then the why of it.
Learning math is not useless, but knowing specific math knowledge that you generally do not apply to day to day situations is useless, period. Its like regurgitating random facts at a party, nobody cares. If you feel the problem you are trying to solve can be achieve through math, Google it.
I haven't thought of anything clever to put here, but then again most of you haven't either.
Why would you teach a symbolic manipulation system that nobody is going to use?
It doesn't make sense to have people go through algebra courses if the end goal is just to learn the concept of a variable, a formula, the properties of real numbers, basic logic, and a convoluted system in which we build giant equations to explain how to write down basic mental math most people find intuitive.
The way public schools in the USofA teach algebra is a crime against common sense. Algebra should be introduced with the simple concept of equality.
=
No matter what This Side or That Side look like, they are the same. From there follows the idea of equivalent operations and it is all gravy from there. I taught my wife's little brother more algebra in an afternoon then his teacher did in a semester. They tried to disqualify his final exam because he didn't show all his work (unnecessary steps, imho). We fought it and they eventually caved.
His teacher was a physical educator and didn't understand the concept that there was only 1 right answer in math but multiple ways to get the right answer.
I have to dumb down explaining math to my 2nd grader cause the teachers want them to use boxes and visual crap. Sad.
The amount of math used expands to fill the user's ability. The more math I learn the more uses for that math I find.
Not a sentence!
Why would you teach a symbolic manipulation system that nobody is going to use?
Ugh. Its to teach that symbolic manipulation is possible and how to think symbolically, not the training task of how to factor equations.
A literary comparison is better.
As a training item were the original Tom Swift books excellent training for my job? No they're Fing useless as training manuals unless you're building an actual repellatron or a tri-phibian atomicar (real titles, BTW).
As an education tool were the original Tom Swift books great engineering tools? F no, they were pretty soft sci fi, use science in place of magic and you're there.
Were they a good way to teach me to read about 4 to 6 times as fast as a yankee can speak, hell yes.
Same analogy with algebra. You're going to learn how to manipulate symbols. Not just equations but symbols in general. Statements in a computer program? People? Who knows. But if you're doin' it right, your mind itself is actually different after you learn algebra.
"Science flies us to the moon. Religion flies us into buildings." - Victor Stenger
Being skilled in math makes your brain a far more capable information processing system, even if you're not aware that you're using math in your day to day life.
Maths is the only logic that many people really come across in life. I'd expect most of the readers here to understand the if -> then sort of logic, but that is actually a mathematical concept. Day to day I just quietly shake my head at the lack of logical rigor people apply in their ordinary lives, but then I remember that at best they have management degrees and have never really had to think. Then, and more importantly, I am frequently disappointed in the logical processes of qualified engineers and scientists, but then many of those people never have gone past second year tertiary mathematics education, if they did any at all!
My point is that maths is as pure a form of applied logic as you are going to find. You can't BS or fudge maths, the mistake will always come back to haunt you. If you then apply that level of reasoning to other areas of life, then at the least you understand the limitations of your reasoning and stop saying such stupid things all the time.
I completely conceded to the argument that mathematics is so often taught terribly.
I suspect that algebra has value beyond its immediate application and utility. I think this is often overlooked in debates like this. When learning algebra, you are in effect modifying your brain in a particular way. You are training yourself to think a certain way. You can gain a feeling of mastery if you learn it well. And implicitly, you are taught the value of reason and logic. The very fact that you are asked to learn algebra carries the message that logic and rational thinking are valuable skills, and that people who are good at such things are particularly valuable to society. The pursuit of a topic like algebra encourages discipline and structure in the way you think about many other things.
To focus only on the immediate applications of algebra is small minded and unwise, in my opinion.
This and no other is the root from which a tyrant springs; when first he appears as a protector - Plato (423 to 327 BC)
If you made an effort to learn some math, IMHO you are more likely to be a life-long learner.
If the only effort you made toward math was avoiding it, IMHO you are more like to be a life-long idiot.
Circle the wagons and fire inward. Entropy increases without bounds.
There is no such thing as "philosophical logic". There is only logic. Philosophy uses logic to reason about the world, but you can't reason in the first place without logic. And yes algebra is the, mathematically speaking, the foundation of logic. Algebra introduces the entire concept of a variable. Tell me exactly how one can make a philosophical argument of any merit without understanding the fundamental of abstraction. It is not the be all and end all of abstraction, and of symbolic manipulation, but it pretty close. Most children grasp the concept of metaphor and simile precisely the same time they grasp the basic concepts of algebra for a reason.
Most people find their first symbolic logic course to be quite hard. I've seen college students actually break down in tears over their first logic course - it is quite frustrating at first, and would be a reall turn-off to the 95% of people who don't enjoy such challanges.
Algebra is just an easier way to teach abstraction and abstract reasoning. It's much easier to relate the symbols to what they symbolize, and the manipulation is more intuitive.
Socialism: a lie told by totalitarians and believed by fools.
Bear in mind the TFA is from a Political Scientist. That's just below an economist in my book. I'm sure we'd all welcome comments from maths teachers about Political Science (btw, why the hell is Political Science a science?)
"The greatest lesson in life is to know that even fools are right sometimes" - Winston Churchill
This has been tried in many europeean countries in the 80's and it has failed quite miserably. Pretty much all of them returned to the true an tried method, with a salt of the new method in the 90's.
I live in a world where a large portion of the population thinks the world is about 6k years old because an old book and most of the "leaders" they look up to told them so. Many of them also believe that "creation science" is a better explanation for the origin of humans than is evolution, or is at least an equivalent "theory".
If you want to live a happy life in a world like that it is best not to think too much, and by "too much" I mean "at all".
The human race is doomed. Our technology for damaging the earth and killing each other has exceeded our political ability to control it. It will destroy us, soon.
Does anyone know where I can get a Hummer with an engine modified to burn coal? I want to see thick, black trails of smoke behind me everywhere I go. I want to eat the last of the endangered species, drink the last drop of clean water, and wipe my ass with paper made from the last tree.
always uses the Math-Makes-you-Logical fallacy.
It's not a fallacy, if it's true. And you should be saying "More-Logical" instead since no one is claiming that knowledge of math somehow magically makes your quirks, psychoses, and personal relationships smooth out.
The human race is doomed. Our technology for damaging the earth and killing each other has exceeded our political ability to control it. It will destroy us, soon.
There's this thing called "follow through", that is, completing a motion. Getting as far as we have, and then declaring inevitable doom, is not good follow through.
For some things, like "i", (sqrt of -1), I still haven't found a useful application outside of complex physics and formal mathematical proofs. It's a very unintuitive concept.
I think the problem with complex numbers is the name, imaginary. It makes them sound like fantasy numbers which doesn't really exist, they are after all the "opposite of real numbers". At some point in history however, negative numbers must have felt equally strange. You can't have "minus five" apples in your hand, such a thing just doesn't exist. But today most people have no problem to understand that negative numbers are a useful tool to describe things in our life and universe, such as the difference between how many apples I have now compared to before or to describe the distance I have traveled when I'm moving backwards with respect to some reference frame.
Yet, when it comes to complex numbers, people (including me) seem to have a hard time to do the same, to realize that they are simply a useful mathematical tools to describe different things that happen in the real world. That they are just as real or imaginary as any other analytical tool we use.
"Ugh. Its to teach that symbolic manipulation is possible and how to think symbolically, not the training task of how to factor equations."
You said that the first time. Apparently you thought I didn't get it. So let me rephrase with an example.
There are many systems that use symbolic manipulation. Computer programming does this in a far more direct, freethinking, and obvious way than Algebra. Algebra does a poor job of teaching people logic. Instead they walk away with a bunch of recipes for solving problems and not a mind that is enhanced with the ability to understand and apply logic and abstract thought. Programming (as opposed to computer science) on the other hand teaches that logic, every program task is a logic problem. I would also contend that the importance of direct mathematical logic isn't fully appreciated without something like a critical thinking course that teaches the use of logic in day to day discourse and rhetoric. Without that many people will complete problem sets with recipes instead of true logic no matter what the system.
I think people are afraid of putting coherent, organized, logical thought combined with critical analysis skills in the hands of young people they are trying to manipulate and control so they save this stuff for college. I think they should teach simplified forms of critical thinking in grade school cultivating entire generations taught to question information and pleas to authority from the start. Kids who will challenge educators and parents and thereby force them to come up with a sound logical structure for how they govern children and god forbid actually share with them the "why" of the way things are.