Butt-head:
I’m angry at numbers.
Beavis:
Yeah, there’s like too many of them and stuff.
I like stupid number tricks, and in fact when I help people with algebra, the first thing I say when learning a new rule is “let’s try it with numbers.” And in the case of this rule, we can demonstrate that if you have 9 - 4, which should be 5, then you can rewrite that as 3[sup]2[/sup] - 2[sup]2[/sup], which according to our rule is the same as (3 - 2)(3 + 2). And lo and behold, if you multiply that out, you get 9 - 6 + 6 - 4 = 5.
And this kind of exercise helps clarify that the seemingly arbitrary and mysterious rule just says something about numbers which we can demonstrate is true.
I’m a person who doesn’t hate math, but was never particularly good at it. I’ve always been somewhat dubious of people who criticize math curricula for being too abstract or arbitrary. The whole point of math is that it is an abstraction.
You can hold three potatoes in your hand, and you can give two potatoes to someone else and have one left. This is a useful concrete exercise. But you need to think abstractly to know that 3 - 2 = 1 for all objects that you will ever see, not just potatoes. Even simple integers are an abstraction.
I think improvements can be made in pedagogy by clarifying the relationships between concrete, “real-life” examples and their abstractions (try it with numbers.) Demonstrating derivations is also good, as explaining where rules come from and why they work is always useful. But no matter what you do, you can’t teach algebra without eventually saying a[sup]2[/sup] - b[sup]2[/sup] = (a - b)(a + b). And more importantly, you can’t learn algebra without doing that kind of stuff, no matter how many word problems or concrete examples are involved.
There’s another aspect which I think is harmful, which is what I perceive as a reactionary attitude among some educators about the value of tedious, repetitive exercises. These people seem to over-emphasize the importance of conceptual learning, as if they can hammer Mathematical Insight into the brains of all students through enough demonstration and explanation.
But the truth is, for most people, gaining insight requires doing the work. Even if you don’t quite understand what’s going on when you begin, doing the tedious, repetitive exercise is often what makes a particular skill click. The student then says, “holy shit, I get it now! Why couldn’t the teacher explain something so simple to begin with?” That person then grows up to be a math teacher who wonders why his students don’t understand his brilliant explanations of such a simple concept. Why? Because they need to do the work.
Some people will need a lot more practice than others. A major problem is that curricula are designed to move at an average pace and if you fall behind in the beginning, you’re basically fucked forever, unless you can figure out a way to make up lost time, maybe by repeating a class or going to summer school.
The thing is math builds on itself. And most schools fail to realize this.
For example, I can take American History and let’s say I’m really interested in the Civil War but I’m bored with Reconstruction. I can get all A’s on the Civil War section and D’s on the Reconstruction section. Then by the time we get to WWI, I get all A’s again.
Why? Because history doesn’t (always) build on itself.
But math does. If you get a “B” in basic math you’re probably going to be deficient in some areas otherwise you would’ve gotten an A. But you still move on to algebra. So know you’re starting at a handicap, because algebra is assuming a thorough knowledge of basic math. So you are struggling from the start with algebra. Then you manage to get a B, in that. Now it’s on to geometry. But you still lack the basic math understanding and you are missing parts of algebra. Now you are even more behind starting with geometry, but the class assumes you have full mastery.
And you can see how the whole thing can easily fall apart. If you don’t get solid building blocks at the start, you can’t (or shouldn’t) move to the next level.
Your last sentence is true, but the first quoted sentence is false. Sudoku doesn’t involve calculation, but it certainly does involve deductive reasoning, patterns, logical consideration of possibilities, and things like that, that are a lot closer to the heart of mathematics than calculation is.
This agrees with my experience, as both a learner and a teacher.
Perhaps one of the main problems with math is that so many people identify it with arithmetic. It seems only natural to me for kids to find arithmetic rather boring…
You’re in luck. Words are the backbone of mathematics. Most mathematics isn’t about numbers, although numbers do admittedly come up often because they are useful concepts. Mathematics is having conversations and arguments where one thinks and talks about things abstractly.
For example, we all, at some point, stop playing tic-tac-toe as adults because it begins to be boring. No one ever wins. Well, you might wonder why that happens. And so you start thinking… “Well, for one thing, there’s no way for the second player in tic-tac-toe to guarantee that they’ll win [because if there were, the first player could just place down a piece at random (knowing that extra pieces on the board never hurt), and then pretend to be the second player, co-opting the guaranteed winning strategy for themselves]. But since there’s no way for the second player to guarantee a win, it follows that the first player can always keep open the possibility of not-losing, and thus guarantee that they never lose. So, at the very least, this explains why the second player never wins if the first player knows what they’re doing. Come to think of it, most of this argument doesn’t really depend on the game being tic-tac-toe… what else does it apply to? Hm…”. That’s a fine slice of mathematics right there, and it doesn’t use any numbers (apart from mundane things like talking about the “first” and “second” player).
The use of numbers isn’t what makes Sudoku math; the fiddling around reasoning about abstract symbols and rules is what makes Sudoku math. Sudoku isn’t arithmetic, but it’s math, although of a very particular sort.
It’s actually the kind of math exercise that you might expect many people would find particularly boring (semi-mechanically generated “Figure out how to place symbols to satisfy these rules” fiddling with small bits of semi-mechanical reasoning here and there, which any computer can readily do for you if you actually needed the results), but it is some sort of math and clearly many people do enjoy it. I wouldn’t try motivating the potential joy of mathematics to most people through Sudoku type puzzles in particular, because I imagine as many would be turned off as turned on, but clearly Sudoku does appeal to some who might nonetheless claim in other contexts to hate the grind of solving random math problems.
That’s a pretty broad definition of mathematics. A cryptogram requires deductive reasoning, patterns, and logical consideration of possibilities but very few people would regard that as a math puzzle.
/me raises hand
Me. I moved from one school to another in the summer between third grade and fourth grade. I learned to add and subtract, and was pretty good with math, then the first day at the new school rolled around. Pop quiz from Mrs Blank to see how much we remembered after summer. I did all the addition, and all the subtraction, and was sitting there. I got yelled at for not doing multiplication or division, decimals and fractions. I got yelled at and punished every math class for the next 2 weeks for not doing the class work [other than addition and subtraction] By the end of it, I absolutely refused to go into her classroom. I got sent to the principals office and my Mom got called down to school [ah for the days of the mom stays at home family] where they finally sorted me out. Fucking teacher never got in trouble for abusing me, mom ended up teaching me what I needed to get caught up to fractions.
Next jackass teacher I ended up with was 9th grade, back in the same school system. Regular teacher, Mr Penapento broke a leg and was out for the year. Head of the math department ended up subbing for him. He was a physics teacher. The guys found out he could be derailed by asking questions. I learned in great detail all about gear ratios in car transmissions. Something totally fucking useless to me because I as all girls got pink ghettoed into Home Eccch class. As if I needed to learn how to sew a fucking apron and make a cake from a box of mix. It also fucked my entire summer because all except for 3 people failed math and had to do summer school. It was absolutely not like Mark Harmon’s summer school. Oddly enough I transferred into a private school again [YAY] but managed to dodge math until my senior year … when the dean of students noticed I didn’t take any math … so I got saddled with geometry, algebra and trig all in the same year I did AP chem. Oddly enough I did just fine at regular chem, despite having no math past 9th grade. So my senior year, when you are supposed to slack off with nothing more than gym and english included 3 math classes and university level chemistry.
Did I mention I absolutely detest math? Not only do I detest math, I had been doing art and working as a machinist. Give me the tools and I can draw any fucking shape you want, give me a lathe, or a mill and I can make any form you want if I have a blue print of it … ask me to explain why dots and swooshies and lines make this object? Fuck if I know, I just make the shit. I detest math, and I worked as a forensic accountant for 2 years … I just need to know how to find the hidden assets, and plug numbers into journals.
I got my ass roasted here when I dumbly asked a question about why a particular facet of math was so important, and apparently I peeved the math nuts on the board for being totally underwhelmed by the answers.:rolleyes:
And I say yet again, for the average Joe, all they need to do is be able to memorize a handfull of math, enough to do their job. 90% of math as taught in school is totally useless. There is no reason to memorize shit, I can pick up a book and look up anything I need to or go online and google it if I am not near a book. If I am stranded on a desert island, I have no need to be able to derive a log table as without my meds i will not survive long enough to need one. [and the Skipper and Gilligan should have had the brains to patch the hole in the side of the minnow without the lame ass scientist who could do anything with a coconut and piece of bamboo except patch a freaking boat]
I don’t think most schools fail to realize this. Quite the opposite; I think most schools not only feel math builds on itself, but are locked into particularly rigid ideas of how the math curriculum must be structured, which aren’t actually necessary.
For example, the idea that one cannot understand the concept of a variable until first understanding decimal notation for integers or long division algorithms, or that one must learn calculus before linear algebra, or that thinking about permutation groups or directed graphs is intrinsically college-level material, none of which is true.
Math does build upon itself, the same way any field builds upon itself, but not at all linearly, not even in the supposed early stages; there are many strands which one can explore in many orders, not just those of historical discovery or curricular tradition.
Are you sure it is people or just Americans? This book has been selling well in the UK, even though it is full of equations and you can down load it for free from the author’s web site.
http://www.amazon.co.uk/Sustainable-Energy-Without-Hot-Air/dp/0954452933
The author’s basic premise is that you can’t discuss energy issue intelligently without using math.
No, people got annoyed with you because they worked hard to answer your question and then you promptly dismissed their answers as pompous and irrelevant.
I agree that all the average person has to do is memorize a handful of simple math, and that’s what they’ll need for their whole life. But that’s not the point. By that standard, we might as well not teach history, literature, science, cake-baking or gym. None of these classes are especially “useful” to an average 21st-century human.
The reason we teach history is because it helps us think about the present day and gives us useful context by which to judge the events happening in our life. You don’t have to be an expert in the Second World War to know that someone yelling about how “OMG GEORGE BUSH IS JUST LIKE HITLER” lacks a sense of perspective. We teach literature and philosophy and all the mushy shit because it helps us reason intelligently about the human condition. We teach science not because we expect every child to grow up to be a scientist, but because the ideas of the scientific method, skepticism, and investigation are useful tools for everything in life.
Math is the same. It’s only the bullshit teachers who actually try to come up with examples to satisfy the one jackass who always asks, “but when are we going to need this?” The correct answer is: you probably won’t. That’s not why math is taught. Math is taught because learning it helps us learn how to think. Learning abstractions gives us an intuition for creating our own abstractions. Learning how quantities relate to one another and how equations work allows us to reason instinctively about all sorts of things that don’t involve numbers or computation.
Your average Doper is probably a pretty good reader, by necessity. Reading (and reading English in particular) is actually an extremely complicated skill. But we do it so much that we don’t think about it. My opinion on math education is that even if you hate math and didn’t go very far, you probably use a lot of those math-type skills, in the form of abstract thought and reasoning, than you realize. You’re just not “doing math” so you don’t think about it at the time.
I am not sure I have anything to contribute to the question, but I’ll try. First a confession. I love to think about math (if some Limey wants to say “maths”, that’s OK with me, but I’ll stick to my abbreviation) and produce new math and publish it and so on, but let me confess that I absolutely hate to read other people’s math. There are mathematicians who appear to love reading math, but I am not one of them. So I feel considerable sympathy for people who hate math in general.
But there are things in math that absolutely enthrall me. I once said to a class in History of Math that I had always been intrigued by the fact that every positive integer is the sum of four squares (0 is allowed). Several students sniggered. I was no amused.
I have a taste for abstraction (I disliked math, even while doing well at it until I discovered abstract algebra). My wife is extremely intelligent and is able to follow quite complicated arguments, until they get abstract. I don’t understand that, but she wants concrete. There is psychology experiment that demonstrates quite convincingly that people fan on abstract exercises when they readily solve isomorphic concrete ones. (There are four cards on a table displaying 4, F, 1, E, which two do you have to turn over to verify the claim that a card with an even number on one side has a vowel on the other? The concrete problem: you are a cop and walk into a bar where there are four students, two drinking coke, two drinking beer. There are four IDs on the table, two underage and two over 21; which combination of ID and student do you have to look at to see if it is all legal. Everyone understands that in the second you need only look at what the underage students are drinking or at the IDs of the students drinking beer. But in the first case, it is apparently hard to see that you need turn over only the F and the 4.)
You can’t memorize math, and thinking that you can is probably a large part of the reason why so many folks struggle with it. You need to learn the process, not the results.
Quoth Little Nemo:
Maybe few people do regard cryptograms as math, but they should.
Take a square of any size (call the length of one side ‘a’), and cut a square out of it from one of the corners (call the length of the side of the square you cut out, ‘b’), leaving an L-shaped area. Now cut the diagonal of that L-shaped area, to create two identical shapes. Put those shapes together in a rectangle, and you’ll find that the rectangle measures (a-b) feet by (a+b) feet. The area of the original square was a squared, the area removed was b squared, and we’ve learned that a squared less b squared eqauls (a-b) times (a+b).
P.S. Upon rereading I think I sound condescending here - sorry about that; it was not intended.
For the same reason I hate dancing: Cause it’s so stupid.
For the record, this kind of stuff confused me and turned me off as a lad.
I liked the algorithms. Those I got. This stuff with the manipulating of the blocks and whatnot left me completely cold.
A little later, I came to see how the algorithm related to the manipulation of squares on a grid–and I thought it was pretty cool too–but this was a later aha moment. If you had hit me with the manipulations first, I would have been like “this is pointless and boring. Let me do some algorithms! That’s the fun stuff!”
Which is all just to say what we all already know, which is, different kids are different.
(A fetish for abstraction has continued to burden me–and lead to criticisms from my academic betters–all the way through my graduate education.)
That’s not a “concrete, real-world example.” that’s exchanging algebra for geometry. It’s a perfectly good and useful demonstration, but I don’t think it’s what people are looking for when they say they want concreteness.
It would be if you replaced a and b with concrete numbers.
See, once my physics teacher showed me what trigonometry was for I actually become pretty darn good at. Still wasn’t speedy at solving problems (and perhaps I should point out this was before calculators were standard - yay log tables). Stats… not so much.
It’s not just a matter of good teachers, or good learning environment, or practical uses, it’s also a matter of inherent aptitudes and interests.