The Saxon texts strategically distribute the content typically found in chapters and units within more traditional texts. This allows more time and practice for skills and concepts to be mastered before the next increment of rigor is added. The daily, cumulative practice is the glue that holds the approach together. Doug Rohrer from the University of South Florida has done some interesting research regarding the advantages of mixed and spaced practice, the method used in the Saxon approach.
________________________________ From: "email@example.com" <firstname.lastname@example.org> To: email@example.com Sent: Thursday, May 5, 2011 4:40 AM Subject: Re: [math-learn] Other Comments On High School Algebra 2
I took my own high school Algebra 2 course back in 1966, and I can think of several differences between the course I had that year and the Algebra 2 courses I see today.
First and foremost in my mind, the subject of matrices was not covered in the Algebra 2 course I took in 1966. I'm curious. Does anybody here know roughly when matrices began to be taught in our high school Algebra 2 textbooks and curriculums?
Next, if my memory is correct, the subject of conics was not covered in as much depth in 1966 as they are today. To cite one example, I do not remember having to graph hyperbolas, whose centers are not at the origiin, and then having to calculate the equations of their asymptotes.
In regard to the subject of imaginary and complex numbers, I do not remember this topic being taught in my own Algebra 2 course in 1966. Perhaps, my memory is failing me. However, I have a copy of a 1990 Algebra 2 textbook, in which this topic is presented in considerable depth.
One thing I find to be rather curious is that the Algebra 1 textbooks I see today make no mention of imaginary and complex numbers. When a problem such as finding the square root of negative 4 is presented, the Algebra 1 textbooks quickly explain that there are no real solutions and leave it at that.
Then fast forward to Algebra 2, during which students learn that there really is an "answer" to finding the square root of negative 4, which is 2i.
Another difference is that the topic of Probability was not taught in any of the junior high or high school math courses of the 1960s.
Another thing I see are inconsistencies in the manner in which various topics are taught and presented in the Algebra 2 textbooks and their teachers.
For example, some teachers completely ignore the topic of solving systems of 3 equations in 3 variables. Others go into it at great depth.
The Glencoe Algebra 2 textbook teaches both the diagonal method and the expansion of minors method for finding the determinant of a 3 X 3 matrix. In contrast, the expansion of minors method is completely ignored by the Algebra 2 textbook from McDougal-Littel. In addition, McDougal-Littel does not present the proportion method for solving problems with direct, inverse, and joint variation.
One of my students at a different private school uses the brand new Saxon Algebra 2 textbook (the one with the blue cover). I find the manner in which it jumps from one lesson topic to another, with little or no continuity or connection between successive lessons, to be very confusing. I found the same type of problem with the new Saxon Geometry textbook with a student I tutored last year.
In addition, this private school teaches Algebra 2 only as a one-semester half-year course. Personally, I do not think justice can be served for the majority of high school students, in order to achieve an adequate and thorough instruction of Algebra 2 course during only one semester.
As a result, their Algebra 2 teacher "cherry picks" the Algebra 2 lessons of her choice and skips over many of the textbook lessons. So far, she has skipped each and every single lesson on the topic of matrices.
At least, in our local public schools, taking Algebra 2 as a one-semester course is an option for the more accelerated students, with its instruction during an entire school year being the "norm" for the vast majority of students.
Okay, I'll stop now. I just wanted to share these additional comments and observations.
In a message dated 5/4/2011 1:01:52 P.M. Eastern Daylight Time, firstname.lastname@example.org writes:
Given all the posts about Algebra 2, it seems that the situation is roughly the same as when I took the sequence of courses Algebra 1, Geometry, Algebra 2, more than a few decades ago.
Given that, and this is what I was really getting at, I think having complex numbers in Algebra 2 is inappropriate. That is too sophisticated a topic for a third mathematics course. Very few such students would have the maturity to understand complex numbers.
Lacking such depth, the subject would degenerate into mindless formalism and button pushing. That is the antithesis of education.
--- On Wed, 5/4/11, email@example.com_ (mailto:firstname.lastname@example.org) <email@example.com_ (mailto:firstname.lastname@example.org) > wrote:
> From: email@example.com_ (mailto:firstname.lastname@example.org) <_starca email@example.com_ (mailto:firstname.lastname@example.org) > > Subject: Re: [math-learn] RE: "Algebra 2" problem > To: email@example.com_ (mailto:firstname.lastname@example.org) > Date: Wednesday, May 4, 2011, 8:10 AM > Hello, > > This was no misprint. > > The first equation in this system was written exactly as > below, with the > negative 1 on the right hand side. > > The teacher did say that there would be one problem which > would involve an > imaginary number solution to be written in ordered pair > form. > Apparently, this was that problem. > > Dennis > > In a message dated 5/4/2011 7:43:23 A.M. Eastern Daylight > Time, > email@example.com_ (mailto:firstname.lastname@example.org) > writes: > > > > > math-learnHello all, > > I assumed there was a misprint in that x^2 + y^2 = -1 > equation, and > that the "-1" was "supposed" to be 1. > > In my eight years of teaching/tutelage etc I've never > come across an > equation of an imaginary circle / having negative > radius. > > Meanwhile, I am curious about how students would > react to the > equation sin^2 x + cos^2 x > 1. Hmmm..... > > -John Morse > > hang glider pilot near Albany NY > > [Non-text portions of this message have been > removed] > > > > > > > [Non-text portions of this message have been removed] > > > > ------------------------------------ > > Yahoo! Groups Links > > > email@example.com_ (mailto:firstname.lastname@example.org) > > >
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