I am trying to decipher the "pre-algebra / algebra I / algebra II" system, and am having trouble figuring out what the differences between each of the steps are (what new information is taught in each step). Could anyone point me to a decent source or explain this for me?
Here's the first link google turned up:
http://www.mathematicallycorrect.com/kprea.htm
Seemed pretty good, but I didn't scrutinize very closely.
hadalittlelamb
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I don't know what math background you're coming from so if any terms don't make sense--just let me know.
A lot depends on curriculum/textbooks/teachers---but in pre-alg you get an intro to the systems of real numbers (whole, integers, rational) and the operations of each. There should be a lot of problem solving, and mastery of ratios, proportions and percents. You begin the study of variables, equations (solving for the unknown quantity) and then inequalities (regions of solutions). Some very basic geometry should be introduced/hopefully reviewed and then connected to the coordinate plane (think Battleship game). There should also be intro to probability. Mathematical sequences should also be intro there, as well.
Alg 1 reviews this stuff quickly and then should jump into properties governing real numbers and then should jump into polynomials and factoring, functions, linear equations and their graphs (lines) and then systems of linear equations. There once again should be a lot of problem solving, physics and business applications. If accelerated, the course should get to quadratic equations and probability and statistics.
Alg 2--if geometry is taught in between alg1 and alg2 there usually is a great need for a thorough review of basic alg concepts. Then jump to those systs of linear eqns, polys and functions in depth. Lots of factoring!!! You then move from the Real Number syst to the Irrational Number system, exponential functions and logs, trig and possibly vectors and sequences and series, and matrices. There should be a lot of graphing.
Hope this helps---remember so much depends on the school system one is in.
A lot depends on curriculum/textbooks/teachers---but in pre-alg you get an intro to the systems of real numbers (whole, integers, rational) and the operations of each. There should be a lot of problem solving, and mastery of ratios, proportions and percents. You begin the study of variables, equations (solving for the unknown quantity) and then inequalities (regions of solutions). Some very basic geometry should be introduced/hopefully reviewed and then connected to the coordinate plane (think Battleship game). There should also be intro to probability. Mathematical sequences should also be intro there, as well.
Alg 1 reviews this stuff quickly and then should jump into properties governing real numbers and then should jump into polynomials and factoring, functions, linear equations and their graphs (lines) and then systems of linear equations. There once again should be a lot of problem solving, physics and business applications. If accelerated, the course should get to quadratic equations and probability and statistics.
Alg 2--if geometry is taught in between alg1 and alg2 there usually is a great need for a thorough review of basic alg concepts. Then jump to those systs of linear eqns, polys and functions in depth. Lots of factoring!!! You then move from the Real Number syst to the Irrational Number system, exponential functions and logs, trig and possibly vectors and sequences and series, and matrices. There should be a lot of graphing.
Hope this helps---remember so much depends on the school system one is in.
go to www.dogpile.com enter pre-algebra hit fetch go to pre-algebra on about.com when you've read it, enter, on about, algebra and after that on
about enter algebra 2.
about enter algebra 2.
Jorghnassen
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Man, I'll never understand the American education system. Pre-algebra? Algebra 1? Algebra 2? I had math classes from primary until the end of secondary school. It's only in cégep with calculus and linear algebra that the classes were separated into specific sub-categories. Besides, until you study groups, rings and fields and things like that, "algebra" classes should really be called "symbolic manipulation" 
.
.What's to understand? Classes need names, don't they?
In the US education system, students are more typically grouped by ability level than by age after about age 11 or so. After all, introductory German is still introductory German regardless of whether the person learning it is 12 or 17, right? And similarly, the ability to solve 2x + 5 = 19 is still the ability to solve 2x +5 = 19; so there should be a single name for the class that covers that particular knowledge (as opposed to, say, the more advanced 2x^2 + 7x + 13 = -2).
If you don't want to call them "Pre-algebra" and "Algebra 1", I invite suggestions....
Jorghnassen
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Well that's the thing. We called those classes "mathematics" because algebra was only part of them. Though pre-algebra sounds like a non-course (as in "will only be useful if you take algebra afterwards"). I think this difference arises because the American system appears more modular (are those year long courses? I couldn't see someone spending an entire school year on trigonometry, that'd be either slow or overkill).
Those are usually year-long courses. But you can't call them all "mathematics". The different courses need different names, if only for the computer to keep them straight.
Pre-algebra is unfortunately named, but it's also a course that doesn't fit into any of the standard mathematical subdisciplines, so the name actually does make sense. Most students in pre-algebra have already mastered arithmetic, but have not yet understood the abstractions that underlay arithmetic -- they can do symbol juggling, but don't yet understand why. In many school districts, pre-algebra is the course most often skipped when students start being tracked by ability instead of age, precisely because the course requires more cognitive development than skill development.
Algebra -- usually taught as a two one-year courses, either Algebra I and II or as Algebra and Advanced Algebra -- covers algebraic manipulation of equations over the rational numbers and later over the reals, and lays the foundation for an approach to algebraic geometry.
Geometry tends to ignore the algebraic approach and simply be proof, proof, proof.
The next course in the sequence is sometimes called Triginometry but is more accurately pre-calculus; it covers non-polynomial functions (such as trig), but also lays a foundation of analysis in things like sequences, series, convergence, limits, &c.
And, of course, calculus is calculus.
Now, I suppose if you wanted to, you could call them math 1, math 2, ... math 6. But that's not a particularly helpful scheme of nomenclature.
Jorghnassen
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See, that's the biggest difference between the systems. In my case, though in rare instances some courses were offered in different grades (particularly sec 5 (grade 11) bio being available in sec 4 (grade 10)), all courses are fixed within a year. So it's always math xyz, where x is the (secondary) year (so all sec 5 math courses start with 5), z is the number of credits, and y is some other number defining the "stream" (back in my days, in sec 4 and 5, there were "strong" and "weak" math, more recently they've split into more streams, but governemnt reforms have changed the curricullum to cover more subjects in less depth, with disastrous results in interprovincial and international rankings).
joobz
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You find this a more useful description of course material?
That's a perennial debate in US education; whether or not to "mainstream" students of different ability levels. It tends to be very unpopular with parents at the secondary school level, though (results are mixed at primary).
College admissions in the United States are extremely competitive -- almost to Japanese standards -- and largely merit-based (which I consider to be a good thing). To get into Harvard today, for exampe, typically requires extremely good grades and very strong secondary school preparation. If you have not had calculus in high school, for example, you can more or less write off your chances of getting into Harvard, even for a non-scientific course of study such as history. On the other hand, most students simply can't handle calculus in high school as they lack the mathematical sophistication. Parents therefore get extremely annoyed if you hold back their prodigy by putting her in a class that can't handle her level of material.
One of the key differences between the US and most European systems of secondary education is that most European countries have an explicit separation between different types of students. A good example of that is the three-tiered Dutch system, where students are sorted by examination into the VMBO, HAVO, or VWO. In the States, all three groups would be taught in the same school building, and sometimes even in the same classes, but possibly at different years -- so the top (VWO) students would be learning algebra at age 14, while the VMBO students would learn in at 16 or 17.
How does your school system address the problem of students who are more advanced or slower than their grade-level indicates?
It works for many US colleges. I think everyone knows what English 101/102 means....
But on the whole, I agree. I think that the difference between English 101 and English 131 is (ahem) non-transparent.....
joobz
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True.. but they typically have a secondary title to provide clarification
Eng131:Intro to Spell check
Jorghnassen
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Well, "math" fits better than "some algebra, some geometry, some trig, some logic, etc".
Jorghnassen
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First, for the really slow ones, there's always the possibility of being held back a year, possibly repeatedly (this might not be as common as it used to, they don't like holding back kids anymore, it's bad for their self-esteem or some other BS). It is possible to skip grades (usually never more than once) in primary school (except for grade 6 I guess, as it is the last one before secondary school, and requires passing "ministry" exams, that is exams made and mandated by the government, which are uniform across the province). In secondary I don't recall seeing any skipping though it might not be impossible.
There may be one optional class in the first 3 years of secondary school, sometimes only offered to smarter kids, e.g. I took 3 years of latin, but the first two were "extra credits" for the top 40%, while the last one was just one option among many, though it was the only option with a prerequisite.
It's only in the last two years of high school (sec 4 and 5) that some real variety appears in the curriculum. There are "core" classes that are necessary to obtain a diploma, such as sec 4 history and English, so those who don't pass these courses the first time around (including supplemental exams in the summer) have to take them again the next year (without otherwise being held back a year).
It's in Sec 4 when "streaming" begins in math and sciences, "strong" for top 40% vs "weak" for the rest, at least back in my day, though there has been more possibilities since (usually something in between in contents, or the "strong" stream but with more time alloted to it). Sec 5 math is required for all but again, there a different types, the "strong" kind is a co-requisite for chemistry and physics classes and together all three classes are required for the science pre-university program in cégep. Those who are not going into science take other electives instead (such as history, journalism, arts, computer science, languages, etc).
Still, many schools offer sport/arts-studies programs where half the day is spent on academics (condensed, electives often reduced to what is required for science in cégep as they will be sufficient for any pre-university program), the other half on sport (or arts), often in seperate institutions (such as, say, a private music school). As expected, students in those programs are required to keep their grades up above a certain average (GPA). Other similar "advanced" programs include English immersion (in the French school boards, I think there are now third language programs as well) and IB. None of those approaches provide faster advancement toward cégep (and/or university), but their "enhanced" curricula are quite attractive to recruiters in competitive programs.
joobz
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Of course, because math is abroader term. You could call it "stuff" and that would cover all bases.
If I was to review a transcript and see 4 years of math listed. I wouldn't know what they learned. I would just know they took math.
If I saw on the transcript "algebra, geometry, calculus....", then I would know at least what they are familiar with. Maybe not the depth of their knowledge, but enough to have certain expectations. I find this more useful.
By pretending they don't and stuffing them into the classes anyway (well, for the slower anyway - which is why a lot of the kids put in Chemistry are reading at elementary school level, doing math up to e.s level and unable to understand most of the concepts they have to follow to do chemistry). A few years ago, the school I was in (numbers are not specific to disguise it in case anyone here lives where I do - but equivalent) over two hundred students took AP courses - and passed them. Of those, 1% passed the AP tests. In Virginia (I know because I looked up AP info) parents carefully study the ratio of takers to passers and it had best be close to 1:1 or they move to get to schools where it is). Here, it was assumed that low pass rates would be the case - but parents "get" AP (abject puerility) just by demanding it (with, fortunately, a couple of exceptions). Same for honors.
We like to say, our regular classes are for low achievers, our honors classes are for regular and higher end low achievers and our AP for honors and higher end regular kids - plus the actual AP level kids (and then watered down to honors level to have a decent passing rate).
Jorghnassen
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The thing is that here, everybody knows what Math xyz means, just like people in the US knowing the specific meaning of Algebra 1 and Algebra 2 and Calculus AB or BC. As one has to take math courses throughout high school, the only distinction needed is whether or not one has "strong" sec 4 and/or sec 5 math.
In fact, my girlfriend applied for a teaching job at a private school which follows the American system closely, and it's a part time job to teach AP calculus, so I finally learned how AP calculus (at least the version offered at that particular school) compares to the Calculus 1 class I took in the first semester of cégep. It turns out that this particular AP class does not involve trigonometric functions at all (limiting itself to polynomial, "rational", exponential and logarithmic functions), which, to me, was quite a surprise. I was, however, not surprised by the "applied" approach taken, though it's not the approach I prefer. This is just to say that not all calculus courses are equivalent, so having done "a semester of calculus" may not give sufficient information for a person who is unfamiliar with the program. From my perspective, not touching trig functions in a calculus class doesn't make sense.
This also reminds me of an "internet tough guy" anecdote from fark.com. Someone preparing for PhD quals in mathematics said the algebra section would be really hard. Then some internet tough guy called that person an idiot because algebra was really easy, and tough guy knew this for a fact because he had taken courses all the way to calculus in high school...
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Well, that's another difference, then. The United States does not have a strong centralized curriculum, so course names can (and do) vary from district to district, and course numberings have no rational basis whatsoever.
Saying someone has studied "Geometry," then, at least says that their coursework includes geometry.
Saying that someone has studied "Math 3" -- or math xyz -- says nothing to people who use a different numbering scheme.