That’s not correct. You need Non-Euclidean Geometry to summon Cthulhu!
I can’t find exactly what has been described, but here are illustrative examples of some algebraic principles rendered in pictorial form:
Damn, I never got that one in school!
(We had a conference about it in college, but it wasn’t coursework. Trying to picture that kind of stuff was eye-crossingly difficult fun).
That’s the biggest difference right there between European education and U.S. Education. Most of Europe thinks its perfectly acceptable to track kids into the trades at a fairly young age. Most of the U.S. wants to make sure the door is never shut on college - even if you can’t write in the passive voice or “call it x.”
In the U.S. you pretty much need to finish all twelve years of school to get a high school diploma - which means 18. Some states have several types of diplomas - one that implies college readiness, one for trade readiness - but others only have a CCR diploma (College and Career Readiness). In those states, in order to get your diploma, you are supposed to have competency to start college without the need for remedial work. And here in my state they also have to passing Algebra II (which is a not required in other states with a CCR diploma). Even the states with a trade diploma, you are still generally required to finish out all twelve years (thirteen with kindergarten, which has become mandatory in most states) - there aren’t a lot of 16 and out with a degree states.
But I agree, one of the biggest problems in the U.S. is that we haven’t figured out how to deal with the huge gap between seventeen year old kids doing multivariable calculus (that was second year of college stuff for my boyfriend the chem engineering major - 30 years ago) and the ones that haven’t figured out “call it x.” AND simultaneously deal with the majority of kids who are in the middle.
Do you get the idea that if 3 × 2 = 6, then that must mean that 6 ÷ 2 = 3?
Because what Derleth posted is the exact same thing, but with letters in place of the numbers. m × v = p, so p ÷ m = v. The rest is just notation (mv means m × v, and p means momentum because Latin). That’s what makes algebra so powerful. It allows us to think about the relationships between numbers generally. We can work out arithmetically that 3 × 2 = 6 and 6 ÷ 2 = 3, but that only proves that this relationship is true for those three numbers. Algebra lets us say that if a × b = c, then c = b ÷ a no matter what numbers we use.
I do sympathize a lot with “notation anxiety.” In math a huge body of knowledge may sometimes be wrapped up in a single symbol. And since the subjects are cumulative (at least up to the undergrad level) if you see a dense block of notation and aren’t confident in what each symbol means, it can cause a lot of stress. A major stumbling block for me when learning proofs was getting in the habit of making sure I understood exactly what each individual symbol in each line meant before attempting to proceed further. One thing that I think students often struggle with is the idea that in math, each individual symbol is often an entire “word” rather than a letter.
And yet, you must be capable of learning symbols and syntax, because at some point you learned the Roman alphabet and the conventions of English syntax.
That helps a bit, but I still find myself looking at stuff like “(2x4)x3” and thinking "Two times four is eight. Why isn’t the equation 8x3? Why are you making me do extra maths when maths is already hard enough? Because you suck, maths. (Try and imagine that being said in Sterling Archer’s voice. It’ll help. :))
Yes, I do. Like I said, I’m fine with arithmetic. But as soon as we
Because what Derleth posted is the exact same thing, but with letters in place of the numbers. m × v = p, so p ÷ m = v. The rest is just notation (mv means m × v, and p means momentum because Latin). That’s what makes algebra so powerful. It allows us to think about the relationships between numbers generally.
[quote]
We can work out arithmetically that 3 × 2 = 6 and 6 ÷ 2 = 3, but that only proves that this relationship is true for those three numbers. Algebra lets us say that if a × b = c, then c = b ÷ a no matter what numbers we use.
[quote]
Part of the challenge I have is wondering why I (or anyone else who doesn’t do actual science for a living) would ever need to do that.
The thing is, I know English extremely well and to my mind you can’t do arithmetic with words. 1 + 1 does not = “A long, long time ago, in a galaxy far, far away” any more than “Cat + Owl = 3.1415926274”, if that makes sense. They’re separate things entirely.
I can’t find the precise thing either (although they’re easy to make). But algebra tiles are very close (though I have to laugh, because their picture gets it wrong).
And it’s not shut for us either… several of my brother Edu’s classmates in Mechanical Engineering had FPII degrees as Mechanics. Translating to the current model,
his track: ESO in Sciences, Bachillerato in Applied Sciences, University entrance exam, Grado in Mechanical Engineering.
their track: ESO in Mechanics, FP in Mechanics (we don’t have FPII any more, what used to be FPII is now FP), University entrance exam, Grado in Mechanical Engineering.
Their university entrance exams would be different due to coming from different tracks, but the door isn’t closed, if you decide that after learning how to fix motors you now want to be able to design them.
Because I guarantee you already do it, albeit without the symbols.
If I give you a distance, can you make a guess at how long it would take you to travel it in a car? Probably. In fact, if you’re anything like the people around here, you already express amounts of distance in units of time when it comes to something you might drive, saying things like “That’s five hours down the road.” or similar.
How do you know how to do that?
Well, you divide the distance by however fast you can travel down the road. You can apply that formula to any distance and any speed and get the right answer: If the distance is zero, it takes zero time because you’re already there. If the speed is zero, it takes infinite time because you’re not moving. If both distance and speed are zero, you can start a fistfight between math geeks. Anyway, you don’t have to discover that basic fact every single time. You can memorize it, and apply it with different numbers over and over again, and know that it always works.
That’s fully half of algebra. Moving from words to letters, changing distance to d and speed to s, for example, is just further laziness. It’s easier to write and type single letters, so that’s what we do. It doesn’t add any abstraction beyond using the word “speed” instead of always having to come up with some specific number. Other symbols are similarly convenient, once you’ve gotten accustomed to them. Mathematicians tend to get a bit overly-familiar, if anything, leading to some rather bizarre abuses of notation, but that’s a story for another time…
The other half is turning one piece of knowledge into another, just by mechanically shuffling symbols around. We know that time is distance divided by speed, so we also know that speed multiplied by time is distance. If you travel at 100 kph for five hours, you’re going to end up 500 km from where you began. Intuitive enough, but the point of algebra is that we don’t need intuition. We just need symbols and rules and we can get a machine to do all the scut-work for us.
My point is, you already do algebra, you just don’t do it in terms of little letters and symbols dancing around. The letters and symbols aren’t any more abstract than what you’re already doing, they’re just more convenient, and easier to get computers to work with. Mathematics can get totally abstract, but algebra is no further removed from reality than normal language.
The problem is that now it seems like the expectation is that all students be above average rather than each student reach his/her potential. No teacher is going to be judged on the number of kids who underachieve in the classroom but ace standardized tests (like my kids; and I’m not implying it’s the fault of the teachers they don’t do the work), but they may if a kid who comes to kindergarten two years behind doesn’t become “proficient”.
yeah, I have to resist the urge to punch the radio in my truck any time I hear ads for Kumon.
I actually think this is a great argument for why we shouldn’t be having this argument - it’s silly to just declare something not useful. When we’re talking about wiping something out of the curriculum, we ought to do a little more than just say “I don’t think this is useful.”
Yes, we should keep Algebra II, just like we should keep geometry, because you probably use both in your daily life without ever realizing it. Geometry teaches you spatial reasoning skills. I don’t recall taking Algebra II in high school, but from Shagnasty’s link to a sample curriculum, I’d say it’s pretty dang useful to know how to mathematically model something. And it’s just plain cool to know about Pi and e. (Read this if you don’t believe me, or even if you do: The Baffling and Beautiful Wormhole Between Branches of Math | WIRED)
The argument behind most “it’s not useful” statements seems to be that that person wasn’t good at it, and maybe that person didn’t get the full value out of it. That’s ok - it’s ok to be bad at something. I was bad at some things too - my poetry teacher forbid me from rhyming after I handed in my first poem, and I still don’t understand why - but that didn’t mean I wanted to clear those things out of the curriculum.
Are you saying that remembering how long it took to travel a certain distance and comparing it to the distance proposed is always mentally figuring time equals distance divided by speed?
I kinda hate this article a lot.
Not really, sorry. That’s how the GPS does it, and there are many situations in which it sucks at it. A GPS or maps program will give the same time from my mother’s house to my grandmother’s when everybody is out on vacation and you can actually drive in town at the posted speeds, when everybody is rushing out or in at the ends of vacation, when it’s sunny or when four drops of water have turned every driver into a bumbling imbecile.
A human will do it that way if he doesn’t have previous memories to draw on. If those memories are available, he can retrieve records of previous trips to that place, if possible under conditions similar to those which are being contemplated, and perform an average (if you have memories of similar conditions then it’s a weighed average).
Martini, I had a similar problem to you. I’m very good at geometry, but most people aren’t; my brain reckons that “why the fuck should I come up with three differential equations to figure out which family of curves this question is about, when I already know it from the description?” The answer is, one, most people can’t do the kind of reasoning I can, so there is a need for a way for them to be able to solve that kind of problems, and two, algebra is actually a language. It’s a translation of other things into a compact written form. And the compact part is superimportant, because it takes a lot less time and paper to write down an equation than to describe it, so long as everybody involved happens to know the language.
How much analysis can you do with just graphs?
You can do everything with geometry.
I take it you mean analytic geometry. But you do know that algebra and geometry are basically one and the same. And analytic geometry simply makes a geometric expression measurable within a cartesian system, allowing you to extract quantitative data USING algebra and calculus.
Without a cartesian system, analysis through synthetic geometry is very limited.
With the difference that one is completely intuitive to me and the other one a pain in the ass, yes (partly from having had a lot of bad math teachers, of the “just learn it” type - cake still goes to that one who claimed that “there is no logic in math”). It wasn’t until I realized that algebra was a different method to say the same things that I was finally able to swallow it.
The rest of your post I’m afraid I didn’t understand it.