Is "College Algebra" really just high school "Algebra II"?

[swampwiz]

I had learned everything in College Algebra in my Algebra II course in high school, and indeed (at least at my alma mater) in engineering, physics or math, no credit is even given for College Algebra.

Perhaps what is going on here is that colleges can't trust that someone who has passed (even done well in) Algebra II has really had that covered, and so anyone who doesn't do well enough on a tracking test (or by extension have a high enough score in Math on the ACT/SAT) can be told, "you need to take this as a prereq to everything else"?

 

[FactChecker]

Regardless of a student's history, having poor scores in a tracking test or the ACAT/SAT Math test should be enough to require some review. It means that they do not have a good working knowledge of the basics.

 

[mpresic3]

This probably depends on the university. My university did not offer "college algebra". At the same time, it did offer a course in algebra. This course treated group theory, ring theory, coset, ideals, principle ideal domains, geometric construction, fields and the beginnings of galois theory. Clearly, this is not "college algebra" at some other schools. You need to be careful what course you enroll in.
At the same time, I hope prospective employers know the difference. It may be hard to explain to an employer how you came out with a C in abstract algebra as a senior, and another applicant got an A in college algebra as a freshman. I had a similar experience when my employer saw courses in analysis, linear algebra, differential equations, functional analysis, mathematical physics, complex analysis, and probability theory, and then was asked, "But did you take calculus?" He did not know "honors" calculus (tests with proofs) was called analysis at my school.

 

[PhDeezNutz]

@mpresic that is frightening. What kind of job were you applying for if you don’t mind me asking?

 

[apostolosdt]

Louis Leithold's "College Algebra" has in many American universities been a benchmark textbook. Difficult to find a copy today, but there must be one in your university's Library.

 

[apostolosdt]

I checked Amazon for Leithold. The one advertised there doesn't look to be the original. The latter is a hardcover, published by Macmillan; its cover is the following:

Please, note that the book is about freshman's algebra, not abstract algebra.

 

[vela]

This probably depends on the university. My university did not offer "college algebra".

Same here. I recall that when I was there, some students were complaining that the university didn't offer such a course. The math department responded that a student admitted to the university supposedly had learned this material in high school and that they should go take college algebra at a local community college if they needed a refresher. I know one reason the department didn't want to offer a college algebra course is that none of the math faculty wanted to teach it. (I can't blame them.)

I think the university eventually gave in and offered a remedial math course, but it didn't count toward satisfying any requirements for earning a degree.

 

[FactChecker]

Often, college algebra is included in "precalculus" along with trigonometry, complex numbers, vectors and matrices, probability and combinatorics, etc.
The precalculus college course zooms through those subjects as a review.

 

[symbolipoint]

The original posted question: Is College Algebra really just high school Algebra 2?

NO. It is just not that. Just one common example how differ: Polynomial Functions, their graphs, roots, studied in College Algebra but not in high school's Intermediate Algebra. Another common example is that Rational Functions are studied much more deeply in College Algebra, and not sure if or to what extent their instruction is placed in high school Intermediate Algebra.

The two courses do for sure overlap. You could say that College Algebra is just a more high powered form of Intermediate Algebra; but really they are two different courses.

 

[mpresic3]

@mpresic that is frightening. What kind of job were you applying for if you don’t mind me asking?

Technical Writer, I grant you the employer wanted me to write spec sheets, so he may not have been famiar with the names of technical subjects and how they may differ from university to university.

 

[mathwonk]

here are sample books for these two courses, with viewable contents. hard for me to see much difference here: as usual it probably depends on the school, the course and the professor.
https://openstax.org/details/books/college-algebra-2e
https://www.wiley.com/en-us/Young+Intermediate+Algebra:+Advanced+High+School+Edition,+2nd+Edition-p-9780470504833#content-section

 

[vela]

The original posted question: Is College Algebra really just high school Algebra 2?

NO. It is just not that. Just one common example how differ: Polynomial Functions, their graphs, roots, studied in College Algebra but not in high school's Intermediate Algebra. Another common example is that Rational Functions are studied much more deeply in College Algebra, and not sure if or to what extent their instruction is placed in high school Intermediate Algebra.

That actually sounds like exactly what we learned about when I took Algebra II in high school (many decades ago). We also covered matrices, combinations and permutations, conic sections, complex numbers, and probably a number of other topics.

The two courses do for sure overlap. You could say that College Algebra is just a more high powered form of Intermediate Algebra; but really they are two different courses.

One big difference (to students) is the College Algebra is one semester whereas Algebra II is typically two semesters.

 

[mathwonk]

I think you will see all those topics in the high school text I linked in post 11. I also had all of them in my junior year high school course. But the same course name means something different in everyones experience.

 

[symbolipoint]

That actually sounds like exactly what we learned about when I took Algebra II in high school (many decades ago). We also covered matrices, combinations and permutations, conic sections, complex numbers, and probably a number of other topics.One big difference (to students) is the College Algebra is one semester whereas Algebra II is typically two semesters.

Maybe some regional changes or changes through time. A kind of course which had been available for high school students was something with an imprecise title of "Mathematical Analysis" which was something in-between actual Intermediate Algebra and "PreCalculus"; the true colleg precalulus being a combination of College Algebra And Trigonometry taught in community colleges and universities. This "Mathematical Analysis" course at the high school was a year long, and contained some stuff more advanced than Intermed. Algeb but not as full as "College Algebr"; and also contained some significant Trigonometry.

 

[symbolipoint]

One big difference (to students) is the College Algebra is one semester whereas Algebra II is typically two semesters.

My direct awareness was that at least for high school, Algebra 1 was full year. Algebra 2 (known as Intermediate) was a full year. Algebra 2 could easily be understood as a continuation of Algebra 1; in that some things were explored more thoroughly and a few new things were put in. Later, in colleges, Elementary Algebra was one semester, and Intermediate Algebra was one semester. For the more advanced students, "College Algebra And Trigonometry" alternatively called "Elementary Functions" was one semester.

 

[symbolipoint]

I thought about this again, and I might have said this before, but:
Intermediate Algebra is a subset of College Algebra.

 

[Math100]

Yes, because some students haven't taken Algebra 2 yet but they have already entered college, and this is why colleges label the name of Algebra 2 as 'College Algebra' to sound more difficult, but in reality, it's the same as Algebra 2 in high school. I don't know too much about other countries, but in the USA, high school students only need to take up to Geometry in order to graduate from high schools, this is the minimum requirement for math completion at US high schools, especially if high school students don't intend to major in STEM fields at college.

 

[symbolipoint]

Course descriptions? Syllabi? No other way to know for sure. What these might have been decades ago may have later changed. When I have insisted about what I said, that was for long ago; but currently, maybe have been changed.

 

[ohwilleke]

Usually, "college algebra" is high school Algebra I and Algebra II if it is a one year course, and high school Algebra II if it is a one semester course. It is remedial and often doesn't count towards the credits required for graduation. Sometime, some pre-calculus is sprinkled in.

 

[symbolipoint]

Usually, "college algebra" is high school Algebra I and Algebra II if it is a one year course, and high school Algebra II if it is a one semester course. It is remedial and often doesn't count towards the credits required for graduation. Sometime, some pre-calculus is sprinkled in.

Way way different from my understanding. OR have times changed?

Too much in "College Algebra" is missing from "Algebra 2"; and my understanding is that College Algebra is not a remedial course.

edit: I really have no memory that I actually made post #18.
edit: And no memory that I made other posts in this topic, too.

 

[Haborix]

Usually, "college algebra" is high school Algebra I and Algebra II if it is a one year course, and high school Algebra II if it is a one semester course. It is remedial and often doesn't count towards the credits required for graduation. Sometime, some pre-calculus is sprinkled in.

That's my understanding of the course as well. UT Austin's description for the course "College Algebra" seems to lineup with that understanding as well. UT even says it does not count towards a math degree and is usually only offered in the summer, which to my mind is indicative of a remedial course.

 

[Astronuc]

NO. It is just not that. Just one common example how differ: Polynomial Functions, their graphs, roots, studied in College Algebra but not in high school's Intermediate Algebra. Another common example is that Rational Functions are studied much more deeply in College Algebra, and not sure if or to what extent their instruction is placed in high school Intermediate Algebra.

The two courses do for sure overlap. You could say that College Algebra is just a more high powered form of Intermediate Algebra; but really they are two different courses.

Then I took College Algebra in high school 50 years ago! I would venture to say that my Algebra course of 50 years ago was more rigorous than many college algebra courses taught in most state universities and certainly community colleges.

Our Algebra II class included some analytical geometry, and toward the end, some differential equations, which we were also doing in Chemistry (chemical reactions and reaction rates) and Physics. The head of our high school math department taught Calculus with Analytical Geometry, and she coordinated with the Algebra II teacher, so that students taking Calculus were well prepared. The Algebra II class expected students to have had Geometry and Trigonometry, which I did at a different high school.

On the other hand, when I was graduate student teaching undergraduate introductory engineering courses, I encountered freshman students who struggled with basic word problems that seemed to me to be at an 8th grade level (that was 40 years ago).

Certainly, there are differences among schools, even among those in the same municipal/metropolitan school district, as well as across the state and from state-to-state.

As a result of my high school program, I started university at the 2nd year (sophomore) level.

 

[symbolipoint]

The Algebra II class expected students to have had Geometry and Trigonometry, which I did at a different high school.

Parts of that seem very strange. The ordering of Algebra II and Geometry can be switched. Geometry course may contain some beginner bits of Trigonometry but that which be contained as part of the Geometry course is not the same as a full-term Trigonometry course. An Algebra-II student already taken a full course of Trigonometry? Maybe this was not what you were saying.

 

[Astronuc]

Parts of that seem very strange. The ordering of Algebra II and Geometry can be switched. Geometry course may contain some beginner bits of Trigonometry but that which be contained as part of the Geometry course is not the same as a full-term Trigonometry course. An Algebra-II student already taken a full course of Trigonometry? Maybe this was not what you were saying.

I was referring to my experience Geometry+Trigonometry, which I did at one high school. I was in an honors program (which was called Major Works in that school district). My class did the Geometry course (normally 2 semesters) in one semester, then we did a year's worth of Trigonometry in one semester. A typical student in that HS would have done Geometry (10th grade), Algebra II (11th grade) and Trigonometry (and maybe Analytical Geometry, possibly pre-Calculus). I changed high schools between 10th and 11th grade, so I took Algebra II (including some trigonometry and analytical geometry) in 11th grade, and Calculus (with more analytical geometry) in 12th grade. The second HS was on the trimester system, so both years were essentially 1.5 years (condensed into 9 months) when it came to STEM. Classmates went to universities like Harvard, Yale, Princeton, MIT, Caltech, . . . . . , and some when to the state universities. I went to a private university (majoring in physics) for a few years then transferred to a state university with a nuclear engineering program. I probably could have gone to MIT, but I would have had to go into debt to do that; my parents couldn't afford it, and I basically paid my way through university, and helped my parents to send my siblings to university.

 

[Frabjous]

I probably could have gone to MIT, but I would have had to go into debt to do that; my parents couldn't afford it,

I remember my father’s reaction after I got in. “My God, they want a third of gross.” I went elsewhere.

 

[Vanadium 50]

MIT financial aid meets full need. However, what they think you need and what a sane person thinks they need can be quite different.

What? You want Macaroni AND cheese?

 

[sbrothy]

I'm again on thin ice here but from what I've heard from college mathematics introductions some start courses simply assume that pupils didn't understand or pay sufficient attention during high school and thus start from scratch quickly going through the basics just in case.

I was surprised when I saw a college math 101 course starting from stuff I already knew from the early years from what is called "gymnasium" here in Denmark.

Stuff like polynomial division and interval notation like "[2, ∞[" (Did I remember that correctly?).

 

[symbolipoint]

I'm again on thin ice here but from what I've heard from college mathematics introductions some start courses simply assume that pupils didn't understand or pay sufficient attention during high school and thus start from scratch quickly going through the basics just in case.

I was surprised when I saw a college math 101 course starting from stuff I already knew from the early years from what is called "gymnasium" here in Denmark.

Stuff like polynomial division and interval notation like "[2, ∞[" (Did I remember that correctly?).

"College Mathematics" might include many different instructional levels. Especially a community college could go as low as Basic Arithmetic, consumer Mathematics, which would be well below the high school level; and may offer Pre-Algebra, Geometry, Elementary and Intermediate Algebra, maybe some others, being about high school level but still are below the college level. Courses as Statistics, Trigonometry, College Algebra, "Elementary Functions", would be some of the college level courses; and then some others.

 

[Muu9]

I'm again on thin ice here but from what I've heard from college mathematics introductions some start courses simply assume that pupils didn't understand or pay sufficient attention during high school and thus start from scratch quickly going through the basics just in case.

I was surprised when I saw a college math 101 course starting from stuff I already knew from the early years from what is called "gymnasium" here in Denmark.

Stuff like polynomial division and interval notation like "[2, ∞[" (Did I remember that correctly?).

In the US, that would be written as "[2, ∞)"

 

[apostolosdt]

In the US, that would be written as "[2, ∞)"

Yes, and the US notation is still found in most texts; yet, I find the "[a, b[" more intuitive: Only one symbol is used and the impression that the interval is now "open" to the right is more apparent. And it's not new at all, for it has been taught in Europe well before the 80's.

 

[sbrothy]

I also find the US notation unintuitive.

 

[symbolipoint]

I also find the US notation unintuitive.

When it is taught, it very quickly makes sense and becomes quickly understood when seen written and easily producible when writing.

 

[gmax137]

[2, ∞[

Sorry for the tangent, but ...
What does the ,∞[ part mean? I understand if you write [2,b[ you mean everything from 2 up to but not including b. But when you put the upper limit as ∞, it is unbounded, right?

So I am asking, what's the difference between [2,∞[ and [2,∞] ?

Or [2,∞) and [2,∞]

 

[Muu9]

Sorry for the tangent, but ...
What does the ,∞[ part mean? I understand if you write [2,b[ you mean everything from 2 up to but not including b. But when you put the upper limit as ∞, it is unbounded, right?

So I am asking, what's the difference between [2,∞[ and [2,∞] ?

Or [2,∞) and [2,∞]

[2,3] is a closed set - near 3, there's a point (namely 3) which is in the set but all of it's neighboring points to the right are not.

[2, 3[ (or [2,3)) is an open set - near 3, even though the points to the left of three are in the set, three itself is not, therefore there is no right-most boundary point where you can say "this is in the set but any numbers to the right of this are not".

Personally, I don't like the bracket-only notation, as it seems to suggest that ]2, 4[ represents the complement to [2,4] or the complement to (2,4) (i.e all the number outside that range) rather than just the open set (2,4).

 

[gmax137]

Yes but my question was, does it make any sense for a closed set to have "infinity" as an upper bound?

 

[Muu9]

Yes but my question was, does it make any sense for a closed set to have "infinity" as an upper bound?

No of course not, it's just notation, like x "approaching infinity" in the case of some limits.

 

[symbolipoint]

No of course not, it's just notation, like x "approaching infinity" in the case of some limits.

If infinity is one of the limits for an interval, this is not a clearly understood precise number or value, so that end of the interval remains "open"; so this is why the parenth is used to show that side of the interval (the inner curve of the parenth facing the infinity symbol).

 

[bhobba]

I'm again on thin ice here but from what I've heard from college mathematics introductions some start courses simply assume that pupils didn't understand or pay sufficient attention during high school and thus start from scratch quickly going through the basics just in case.

That happened in my degree in Australia. To get admitted you must have done Math B and C roughly equivalent to UK A Level Math which is Calculus BC plus a bit more. Yet we had Calculus A, which was just a boring rehash of HS calculus. The real shock came with Calculus B which was real analysis. I loved it, but most detested it. The latest I heard is they don't do that any more, starting with Probability and Stochastic Modelling, Abstract Mathematical Reasoning, Linear Algebra, and a Second Major Elective (all math majors where I went do a second Major, which can be an area of math like Stats, Operations Research or Data Science) first semester. Real Analysis was a 3 credit course when I did it. They replaced it with Abstract Reasoning (4 credits) which includes Real Analysis plus a bit more. They got rid of Analysis entirely for a while, which I found a bit depressing.

As far as HS goes likely math or associated majors such as Actuarial Science, Mathematical Physics etc is accelerated a bit and do (at many schools) the equivalent of Pre Algebra, Algebra 1 and 2, and Geometry starting in year 7 in 3 years instead of 4. Then Math B and C years 10 and 11 and 4 year one university math subjects senior year. Taking two Subjects over the summer means you can complete the degree in 2 years instead of 3.

Then I found something interesting. In the US 7000 students take the Calculus BC exam in year 8 or less, with over 50% passing. I thought what? I suspect we can accelerate calculus even further for better students.

To answer the original question in the Australian context if you have not done Math B and C at HS you do Math B as one 4 credit subject then Math C as another 4 credit subject except they call it by different names like foundational math etc. Many degrees just require Math B, some none at all. The thing I find depressing is that 45 years ago (god I am getting old) everyone, and I mean 100%, did the same subjects in years 11 and 12. English, Math B and C, Physics, Chemistry and either Technical Drawing or Biology. Tech Drawing was for those interested in engineering. Biology for those interested in being a doctor, nurse etc. Mathematics guys like me were advised to take whatever appealed - I did Technical Drawing.

Now, and it is a big worry, less than 8% take even Math B and even fewer take Math C. Sad, very sad.

Thanks
Bill

 

[Muu9]

The Queensland (i.e. not all of Australia) A/B/C curriculum was replaced with General Mathematics, Mathematical Methods, and Specialist Mathematics, respectively, in 2019. Students in New South Wales can take Mathematics Extension 2, which is more advanced than Queensland's Specialist Mathematics (formerly C).

 

[apostolosdt]

Yes but my question was, does it make any sense for a closed set to have "infinity" as an upper bound?

Your remark is correct. I guess, no matter the context or the notation, including infinity limit requires special attention. Thanks for bringing it up.

 

[sbrothy]

Yes but my question was, does it make any sense for a closed set to have "infinity" as an upper bound?

I learned infinity is always written as an open-ended set. Thus [2, ∞] makes no sense. Is the same true with the American notation?

 

[Muu9]

I learned infinity is always written as an open-ended set. Thus [2, ∞] makes no sense. Is the same true with the American notation?

Yes, in American notation it would be written [2, infinity)

 

[WWGD]

Sorry for the tangent, but ...
What does the ,∞[ part mean? I understand if you write [2,b[ you mean everything from 2 up to but not including b. But when you put the upper limit as ∞, it is unbounded, right?

So I am asking, what's the difference between [2,∞[ and [2,∞] ?

Or [2,∞) and [2,∞]

In theory, there's this Compactification of the Reals; one-point or 2-point compactification that includes one of $\pm \infty$.
https://en.m.wikipedia.org/wiki/Compactification_(mathematics)

 

[sbrothy]

Correct me if I'm wrong but, as I understand it, it's pretty much convention. Like they decided that infinity can never be inclusive. Much like you can never divide by zero. It's just the choice that makes the most sense.