i was in a recent conversation with a college friend and we were talking about our high school math experiences. and we could not settle something. i was in pre calculus and he was in allgebra 3. what are the differences? it has been so long ago that it is hard to decipher what we each learned and how it differs. how is a more advanced algebra compared to beginners calculus?
- - James Patunas |
Precalculus has many names but usually not Algebra 3, that's pretty
much nonexistent. So-called "integrated math" programs speak of 1, 2,3, 4 and it is impossible to know what is contained in which (if at all) without looking closely at the program itself. One of the most successful in terms of student performance is the Saxon series and there the third book is marketed under different names depending on what the school wants to call it. One of the problems of our country is the lack of an official and clear set of mathematics content standards on a course-by-course basis even though we all know it in our heart of hearts. A decade ago California did that and they remain the best in the nation: http://www.cde.ca.gov/ci/ma/cf/documents/math-ch2-8-12.pdf Unfortunately, two decades ago the National Council of Teachers of Mathematics muddied the waters by putting out a document which it called "Standards" but it doesn't contain any and the education wing of the NSF has misspent millions of dollars in pursuit of that ed school vision of mathematics education that does not coincide with mathematics education itself. The bottom line to your question is that it is impossible to answer it accurately without knowing more about the programs in question. In the unlikely case that you can name the publishers and series title or principal authors of the books in question, we might be able to give you some insight. Wayne At 09:31 AM 3/29/2008, James Patunas wrote: >i was in a recent conversation with a college friend and we were >talking about our high school math experiences. and we could not >settle something. i was in pre calculus and he was in allgebra >3. what are the differences? it has been so long ago that it is >hard to decipher what we each learned and how it differs. how is a >more advanced algebra compared to beginners calculus? > >- - James Patunas > > >-- >No virus found in this incoming message. >Checked by AVG. >Version: 7.5.519 / Virus Database: 269.22.1/1350 - Release Date: >3/30/2008 12:32 PM |
Of course, it's also just vaguely possible that Dr. Bishop's post
offers nothing more than highly prejudiced comments about Saxon (which he favors), NCTM and the NSF (which he does not), and caps it off with some very peculiar statement about what "we" might be able to do if "you" provided certain information, names book titles and authors. But I wouldn't rush to find that information: if the authors are like-minded individuals, they will be praised. If not, they will be attacked. What you won't get is anything useful. Note well that the knee-jerk dismissal of "integrated math" makes it sound like some crazy idea, rather than the standard used in most countries other than the US. And last I checked, the US wasn't leading the world in high-quality mathematics teaching. Not even in California. Part of the on-going puzzle of the Math Wars is trying to make ANY sense of the irrational opposition to integrated curricula in the face of overwhelming evidence that we are off our collective rockers in doing so. Our insistence upon separating mathematics in high school into arbitrary blocks such that after the sophomore year we are in the unenviable position of having to invent things like "Algebra II," "Algebra III," and "Precalculus" (or, as Ann Arbor's high schools call it, "Analysis") and then decide which topics go where (don't even bother looking for any sort of rational explanation about why), is akin to our insane inability/refusal to convert to the metric system. I guess Americans really are too stupid to handle something so simple and elegant. Or maybe we have a few too many conservative and reactionary thinkers worrying about trivia and calling it "back to basics." On Mar 30, 2008, at 8:17 PM, Wayne Bishop wrote: > Precalculus has many names but usually not Algebra 3, that's pretty > much nonexistent. So-called "integrated math" programs speak of 1, > 2,3, 4 and it is impossible to know what is contained in which (if > at all) without looking closely at the program itself. One of the > most successful in terms of student performance is the Saxon series > and there the third book is marketed under different names > depending on what the school wants to call it. One of the problems > of our country is the lack of an official and clear set of > mathematics content standards on a course-by-course basis even > though we all know it in our heart of hearts. A decade ago > California did that and they remain the best in the nation: > http://www.cde.ca.gov/ci/ma/cf/documents/math-ch2-8-12.pdf > > Unfortunately, two decades ago the National Council of Teachers of > Mathematics muddied the waters by putting out a document which it > called "Standards" but it doesn't contain any and the education > wing of the NSF has misspent millions of dollars in pursuit of that > ed school vision of mathematics education that does not coincide > with mathematics education itself. > > The bottom line to your question is that it is impossible to answer > it accurately without knowing more about the programs in question. > In the unlikely case that you can name the publishers and series > title or principal authors of the books in question, we might be > able to give you some insight. > > Wayne > > At 09:31 AM 3/29/2008, James Patunas wrote: >> i was in a recent conversation with a college friend and we were >> talking about our high school math experiences. and we could not >> settle something. i was in pre calculus and he was in allgebra >> 3. what are the differences? it has been so long ago that it is >> hard to decipher what we each learned and how it differs. how is >> a more advanced algebra compared to beginners calculus? >> >> - - James Patunas >> >> >> -- >> No virus found in this incoming message. >> Checked by AVG. >> Version: 7.5.519 / Virus Database: 269.22.1/1350 - Release Date: >> 3/30/2008 12:32 PM |
In reply to this post by James Patunas
Do you ask because you're planning on introducing Bishop 2.0, with new, improved conservo-compassion? If so, let me know
But clearly you missed the point about the silliness of breaking high school math into arbitrary chunks. And attacking integrated math as a concept just because you don't like reform books. Nctm And progressives don't own a patent on integrating secondary math. Does MC hold one on insisting upon rigid boundaries by grade and topic? Meanwhile, the porcine quality of air traffic in greater LA is not in my control. - ------Original Message------ From: Wayne Bishop To: Michael Paul Goldenberg Cc: Math-teach Teach Sent: Mar 31, 2008 7:23 PM Subject: Re: Algebra 3 - James Patunas At 07:11 PM 3/30/2008, Michael Paul Goldenberg wrote: >Of course, it's also just vaguely possible that Dr. Bishop's post >offers nothing more than highly prejudiced comments about Saxon >(which he favors), NCTM and the NSF (which he does not), and caps it And it's transparently possible that my response was in regard to the content of Algebra 3 and that MPG had (has?) nothing to offer Mr. Patunas. But maybe I'm wrong. Maybe MPG can give us a coherent description of his idea of the content of Algebra 3. Maybe without throwing in superfluous (and listserve-prohibited) personal invective. Maybe pigs fly. When Michael Paul Goldenberg 6655 Jackson Rd #136 Ann Arbor, MI 48103 734 644-0975 (c) 734 786-8425 (h) "Oh, bother," said Pooh, as he chambered another round. |
In reply to this post by James Patunas
On the subject of `integrated math'. Most countries have a division into algebra, geometry, trigonometry etcetera. These subjects are however divided into small pieces (a few weeks worth of teaching) and are alternated. So the math curriculum would look something like
Year 1: Algebra 1 for a month Geometry 1 for a month Trigonometry 1 for a month Geometry 2 for 2 months Algebra 2 for 2 months Trigonometry 2 for a month Algebra 3 for 2 months The `old-fashioned' American curriculum has much larger blocks of usually a year. The `new-fashioned' American curricula have smaller blocks of it sometimes seems 10 minutes.... |
James,
Does your summary mean that most countries have 10 months of school each year? Do you have any idea how much time is devoted to mathematics instruction during the day? The number of student content days varies from state to state in the US. I recall hearing the number of days ranging from 145 to 190 (9 months or less). How many days of high school are required in CA? Other states? Readers might be interested in looking at the high school integrated mathematics standards in Georgia. I do not know how many student contact days are required in GA. http://www.georgiastandards.org/math.aspx The College Board and Achieve (American Diploma Project) also have produced documents that summarize their high school mathematics expectations in sample integrated course formats. The three of these integrated course descriptions are quite different. Unlike the description below, they all included statistics and probability expectations. The Achieve document includes some discrete mathematics. Beth On 4/2/08 4:43 AM, "Mark Roberts" <[hidden email]> wrote: > On the subject of `integrated math'. Most countries have a division into > algebra, geometry, trigonometry etcetera. These subjects are however divided > into small pieces (a few weeks worth of teaching) and are alternated. So the > math curriculum would look something like > > Year 1: > Algebra 1 for a month > Geometry 1 for a month > Trigonometry 1 for a month > Geometry 2 for 2 months > Algebra 2 for 2 months > Trigonometry 2 for a month > Algebra 3 for 2 months > > The `old-fashioned' American curriculum has much larger blocks of usually a > year. The `new-fashioned' American curricula have smaller blocks of it > sometimes seems 10 minutes.... |
In reply to this post by James Patunas
Perhaps you could cite documentation to support your description below. Which countries did you have in mind?
- ------Original Message------ From: Mark Roberts Sender: [hidden email] To: Math-teach Teach ReplyTo: Math-teach Teach Sent: Apr 2, 2008 4:43 AM Subject: Re: Algebra 3 - James Patunas On the subject of `integrated math'. Most countries have a division into algebra, geometry, trigonometry etcetera. These subjects are however divided into small pieces (a few weeks worth of teaching) and are alternated. So the math curriculum would look something like Year 1: Algebra 1 for a month Geometry 1 for a month Trigonometry 1 for a month Geometry 2 for 2 months Algebra 2 for 2 months Trigonometry 2 for a month Algebra 3 for 2 months The `old-fashioned' American curriculum has much larger blocks of usually a year. The `new-fashioned' American curricula have smaller blocks of it sometimes seems 10 minutes.... Michael Paul Goldenberg 6655 Jackson Rd #136 Ann Arbor, MI 48103 734 644-0975 (c) 734 786-8425 (h) "Oh, bother," said Pooh, as he chambered another round. |
In reply to this post by James Patunas
Dear Beth,
It seems that you were responding to my message and not the one of James. Your comment is besides the point that I wanted to make, but I'll try to give you hopefully useful info on your point anyway. I know that in the Netherlands due to mandatory holidays the numbers of days that school are open is a maximum of 200 a year. Most schools however only reach about 180. Summer holiday is 6 weeks, Christmas holiday is 2 weeks, fall holiday is 1 week and spring holiday is 1 week. On top of that there are several isolated mandatory days off. In the lesson plan you would take the `big' holidays into account (leaving 42 weeks or about 10 months), but not the isolated days off. But the list a gave is fictitious, it was just meant to indicate that typically one coherent piece of mathematics tends to be taught for a period of 1,2 or 3 months and then the class moves on to another coherent piece of mathematics. This is in contrast to the American situation where this period is either much longer (the entire school year) or much shorter (sometimes not even the length of one exercise), depending on whether the curriculum is `old-fashioned' or 'new-fashioned'. The situation of blocks of 1,2 or 3 months is the curriculum as it was taught in The Netherlands some 20 years ago. The current curriculum in The Netherlands however looks like the 'new-fashioned' American curriculum. The situation of 1,2 or 3 month blocks does still exist in e.g. England. |
In reply to this post by James Patunas
Wayne:
> One of the problems of our country is the lack of an > official and clear set of mathematics content > standards on a course-by-course basis even though we > all know it in our heart of hearts. On the contrary, locking ourselves in to some national set of standards on a course-by-course basis would be an assault on local autonomy, plus be another big step in the direction of mono-culture, away from diversity, a biological no-no in swiftly changing times. What we need are competing curricula, developed cooperatively but ultimately adapted to conditions on the ground by those closest to the action. Watching over the shoulder of a UK guy, fighting for reforms, gives me new appreciation for the freedom we have to try new things. This attitude of "don't experiment with real children" is hogwash because every curriculum is already an experiment, a trial. Some do relatively better than others by certain measures, but that doesn't support the argument that we all should jump on the same bandwagon. Those who call for school choice, real choice in public schools, should by logic accept diversity as a consequence i.e. the math curricula at schools A, B and C in the very same town might differ quite drastically (as is the case in Portland, Oregon, especially in K-8). Parents and students have real choices at last. Top-down national standards, and a call for more choice, seem contradictory policy goals. Kirby |
Kirby is dead-on.
There's no consistency from curriculum to curriculum, from book to book, from state to state, and probably not from teacher to teacher as to how to break down topics and shove them into courses, or exactly what content belongs, what is optional, what is antiquated, etc. And while button-down minds no doubt thing that's a horrid state of affairs and fear that the trains won't run on time as a result, the problem is too much standardization even so. And the trains failing to run on time isn't really about whether synthetic division is an algebra 2 or precalculus topic or whether we should be teaching it at all. On Apr 4, 2008, at 11:25 AM, Kirby Urner wrote: > Wayne: > >> One of the problems of our country is the lack of an >> official and clear set of mathematics content >> standards on a course-by-course basis even though we >> all know it in our heart of hearts. > > On the contrary, locking ourselves in to some national > set of standards on a course-by-course basis would be > an assault on local autonomy, plus be another big step > in the direction of mono-culture, away from diversity, > a biological no-no in swiftly changing times. What we > need are competing curricula, developed cooperatively > but ultimately adapted to conditions on the ground by > those closest to the action. > > Watching over the shoulder of a UK guy, fighting for > reforms, gives me new appreciation for the freedom we > have to try new things. This attitude of "don't > experiment with real children" is hogwash because every > curriculum is already an experiment, a trial. Some do > relatively better than others by certain measures, but > that doesn't support the argument that we all should jump > on the same bandwagon. > > Those who call for school choice, real choice in public > schools, should by logic accept diversity as a consequence > i.e. the math curricula at schools A, B and C in the very > same town might differ quite drastically (as is the case > in Portland, Oregon, especially in K-8). Parents and > students have real choices at last. Top-down national > standards, and a call for more choice, seem contradictory > policy goals. > > Kirby > > |
In reply to this post by Mark Roberts-6
On Wed, Apr 2, 2008 at 1:43 AM, Mark Roberts
<[hidden email]> wrote: > On the subject of `integrated math'. Most countries have a division into algebra, geometry, trigonometry etcetera. These subjects are however divided into small pieces (a few weeks worth of teaching) and are alternated. So the math curriculum would look something like > > Year 1: > Algebra 1 for a month > Geometry 1 for a month > Trigonometry 1 for a month > Geometry 2 for 2 months > Algebra 2 for 2 months > Trigonometry 2 for a month > Algebra 3 for 2 months > > The `old-fashioned' American curriculum has much larger blocks of usually a year. The `new-fashioned' American curricula have smaller blocks of it sometimes seems 10 minutes.... I'd love to learn more about what countries do this. And I strongly disagree about a lot of the new-fashioned American curricula. In fact one of my big objections to the "integrated" math books of roughly a decade ago (which is the last time I looked closely enough at them to know) was that those books did just about what you describe above: Chapter 1 algebra, chapter 2 geometry, and chapter 2 had essentially nothing to do with what was in chapter 1. It just reinforced the misconception that different fields of math had nothing to do with each other. What I like about "problem-based" approaches to math is that students can see algebraic or geometric approaches to problems and discuss them in the same class without feeling "out of place". For example, consider squares ABPQ, BCRS, CDTU, with A-B-C-D collinear and Q-S-U collinear and all on the same side of the line ABCD. What can you say about the side length of BCRS if AB = x and CD = y? It's very easy to use lots of different ideas here -- similar triangles, slope (and so we can see that they're really the same idea), or even area. So 10 minutes spent discussing this problem from all those points of view would be what I call an "integrated" lesson even if the next 10 minutes we're on to a different problem using different tools. Anyway, where do you see the curriculum split into blocks of a month like this? In my visit to Russia, and what I hear about the curriculum in lots of other European countries, I see something more like (maybe in 8th or 9th grade) Algebra for 2 hours a week, geometry for 2 hours a week, trig for 1 hour a week. Or if you go to the specialized math schools, algebra for 3 hours a week, geometry for 3 hours a week, calculus for 3 hours a week, and special topics for an hour a week (again probably in 8th grade). They may even have different teachers for these subjects, so to call this "integrated" the teachers have to work together (which, at least in the better schools, they do). - --Joshua Zucker |
In reply to this post by Michael Paul Goldenberg
And President Clinton wondered aloud (and in exasperation) why we
couldn't have a national algebra test, "Algebra is Algebra!" MPG's post is another that belongs in the "Anything but Teach" thread. Wayne At 07:59 PM 4/4/2008, Michael Paul Goldenberg wrote: >Kirby is dead-on. > >There's no consistency from curriculum to curriculum, from book to >book, from state to state, and probably not from teacher to teacher >as to how to break down topics and shove them into courses, or >exactly what content belongs, what is optional, what is antiquated, etc. > >And while button-down minds no doubt thing that's a horrid state of >affairs and fear that the trains won't run on time as a result, the >problem is too much standardization even so. And the trains failing >to run on time isn't really about whether synthetic division is an >algebra 2 or precalculus topic or whether we should be teaching it at >all. > >On Apr 4, 2008, at 11:25 AM, Kirby Urner wrote: >>Wayne: >> >>>One of the problems of our country is the lack of an >>>official and clear set of mathematics content >>>standards on a course-by-course basis even though we >>>all know it in our heart of hearts. >> >>On the contrary, locking ourselves in to some national >>set of standards on a course-by-course basis would be >>an assault on local autonomy, plus be another big step >>in the direction of mono-culture, away from diversity, >>a biological no-no in swiftly changing times. What we >>need are competing curricula, developed cooperatively >>but ultimately adapted to conditions on the ground by >>those closest to the action. >> >>Watching over the shoulder of a UK guy, fighting for >>reforms, gives me new appreciation for the freedom we >>have to try new things. This attitude of "don't >>experiment with real children" is hogwash because every >>curriculum is already an experiment, a trial. Some do >>relatively better than others by certain measures, but >>that doesn't support the argument that we all should jump >>on the same bandwagon. >> >>Those who call for school choice, real choice in public >>schools, should by logic accept diversity as a consequence >>i.e. the math curricula at schools A, B and C in the very >>same town might differ quite drastically (as is the case >>in Portland, Oregon, especially in K-8). Parents and >>students have real choices at last. Top-down national >>standards, and a call for more choice, seem contradictory >>policy goals. >> >>Kirby >> > > >-- >No virus found in this incoming message. >Checked by AVG. Version: 7.5.519 / Virus Database: 269.22.7/1361 - >Release Date: 4/5/2008 7:53 AM |
Save yourself and the rest of the list a lot of time, Wayne: just
state that all posts not in agreement with your micron-wide viewpoint belong on the "Anything but Teach" thread. And Wayne Bishop quoting Bill Clinton (who knows nothing about mathematics teaching) for support? That would have to be like Haim citing Joe Stalin for backup of one of his typical social views. I suspect that an appropriate thread to contain the vast majority of Wayne's posts would be "Anything but let kids actually do mathematics." On Apr 5, 2008, at 8:17 PM, Wayne Bishop wrote: > And President Clinton wondered aloud (and in exasperation) why we > couldn't have a national algebra test, "Algebra is Algebra!" MPG's > post is another that belongs in the "Anything but Teach" thread. > > Wayne > > At 07:59 PM 4/4/2008, Michael Paul Goldenberg wrote: >> Kirby is dead-on. >> >> There's no consistency from curriculum to curriculum, from book to >> book, from state to state, and probably not from teacher to teacher >> as to how to break down topics and shove them into courses, or >> exactly what content belongs, what is optional, what is >> antiquated, etc. >> >> And while button-down minds no doubt thing that's a horrid state of >> affairs and fear that the trains won't run on time as a result, the >> problem is too much standardization even so. And the trains failing >> to run on time isn't really about whether synthetic division is an >> algebra 2 or precalculus topic or whether we should be teaching it at >> all. >> >> On Apr 4, 2008, at 11:25 AM, Kirby Urner wrote: >>> Wayne: >>> >>>> One of the problems of our country is the lack of an >>>> official and clear set of mathematics content >>>> standards on a course-by-course basis even though we >>>> all know it in our heart of hearts. >>> >>> On the contrary, locking ourselves in to some national >>> set of standards on a course-by-course basis would be >>> an assault on local autonomy, plus be another big step >>> in the direction of mono-culture, away from diversity, >>> a biological no-no in swiftly changing times. What we >>> need are competing curricula, developed cooperatively >>> but ultimately adapted to conditions on the ground by >>> those closest to the action. >>> >>> Watching over the shoulder of a UK guy, fighting for >>> reforms, gives me new appreciation for the freedom we >>> have to try new things. This attitude of "don't >>> experiment with real children" is hogwash because every >>> curriculum is already an experiment, a trial. Some do >>> relatively better than others by certain measures, but >>> that doesn't support the argument that we all should jump >>> on the same bandwagon. >>> >>> Those who call for school choice, real choice in public >>> schools, should by logic accept diversity as a consequence >>> i.e. the math curricula at schools A, B and C in the very >>> same town might differ quite drastically (as is the case >>> in Portland, Oregon, especially in K-8). Parents and >>> students have real choices at last. Top-down national >>> standards, and a call for more choice, seem contradictory >>> policy goals. >>> >>> Kirby >>> >> >> >> -- >> No virus found in this incoming message. >> Checked by AVG. Version: 7.5.519 / Virus Database: 269.22.7/1361 - >> Release Date: 4/5/2008 7:53 AM |
In reply to this post by James Patunas
> Kirby is dead-on.
> On the other hand, there's no guarantee that variety, in and of itself, will produce winning combinations. The entire petri dish may go bad. Giving local faculties more control is another way to filter *out* a lot of the bogus garbage that passes for math teaching in the USA today. More elucidation of my views: http://worldgame.blogspot.com/2008/04/national-standards.html Kirby |
I agree with what you wrote on the blog, but I do wish for ways to
get more ideas into the brains of the teachers we've got. Yes, ways exist to spread good ideas, but that doesn't mean that many teachers have any idea they're out there, have time or motivation to look for them, or take advantage of what they find if they do look. Is there a way to push teachers to keep more abreast of what's available? Is that desirable? I think about how the Japanese structure does allow for ideas to spread from the local to the national and back again. I recognize that there is also a definite potential for that process to not be so good, and in this country there's enormous resistance to any sort of uniformity as well as top- down control of education. Yes, that's probably a good thing in many ways, but I see way too many places where there's not much going on with content or teaching or technology that's worth doing and students are being robbed of the opportunity to get on the STEM train. It's pretty mind-numbing and depressing. I'm not willing to trade autonomy for standardization, but I think there are useful components of what the Japanese do that we could adapt to our needs here and still remain free to innovate. Indeed, some structure is likely necessary to "riff off of." On Apr 5, 2008, at 9:57 PM, Kirby Urner wrote: >> Kirby is dead-on. >> > > On the other hand, there's no guarantee that > variety, in and of itself, will produce winning > combinations. The entire petri dish may go > bad. > > Giving local faculties more control is another > way to filter *out* a lot of the bogus garbage > that passes for math teaching in the USA today. > > More elucidation of my views: > > http://worldgame.blogspot.com/2008/04/national-standards.html > > Kirby > |
In reply to this post by James Patunas
Look at the following syllabus for England:
http://www.aqa.org.uk/qual/pdf/AQA-5361-6371-W-SP-08.PDF This is a syllabus for the last two years of high school. Students typically take Pure Core 1-4 (each lasting a semester) and two modules from statistics, mechanics and decision. The better students (who not only take `maths' bit also `further maths') typically do Further Pure 1-4 (each lasting a semester) and two more modules from statistics, mechanics and decision. So a typical students who does maths and further maths does: year 1 semester 1: Pure Core 1, Further Pure 1 year 1 semester 2: Pure Core 2, Further Pure 2 year 2 semester 1: Pure Core 3, Further Pure 3 year 2 semester 2: Pure Core 4, Further Pure 4 and during each semester an elective from statistics, mechanics and decision. This is `typical', but variations are possible. If you look at the content of these courses (more detailed content is given later in the pdf-file) you see that it is certainly an `integated course'. The modules are assessed at the end of the semester, so teachers have to stick to the division of the material across the modules. Teachers are free to teach the content of each module in whatever way they like. The subdivision of the material of a module into 7 or so subtopics like Algebra and Functions Trigonometry Exponentials and Logarithms Differentiation Integration Numerical Methods suggests to teach each of these during a couple of weeks and in the given order. And as I understand it this is indeed what almost all teachers do. The system is set up so that connections between subtopics have to made, although often only in a later module. |
In reply to this post by James Patunas
James,
it is hard to say, one way or the other. The US lacks an official Mathematics curriculum, so district to district can vary a good amount. chances are the algebra three class was some sort of integrated course. - - Jason Stalnecker |
In reply to this post by James Patunas
I believe Algebra 3 is a regional name for College Algebra, which used to be a half year algebra course beyond Algebra 2. It used to be offered with a trig course for the other half year. Today most high schools have trig/pre-calc courses. The difference is that college algebra could be taken without taking trig. It was basically a functions course and with linear programming, it completed the pre-requisites for business calc. Most colleges still offer a College Algebra course and a Pre-Calc course because an Intermediate level Algebra is not as rigorous as a good high school Algebra 2. In a supplement to the Math Panel's final report on The Major Topics of School Algebra found at
http://math.berkeley.edu/~wu/NMPalgebra5.pdf, you will note that Dr. Schmid and Dr. Wu consider a 2 1/2 year time frame for covering all the topics. That extra 1/2 year is still called Algebra 3 in some regions of the U.S. |
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