"I do not understand why we teach solving quadratic equations by factoring at all? I believe the lead up to this, namely the weeks of wasted time on factoring expressions, loses too much valuable learning time in the Algebra curriculum. I think we ought to reduce to near zero the bothersome practice of factoring, teach the quadratic formula right up front since what we hope to accomplish is a solution of the quadratic, and move on." -- @cbrownlmath
I would disagree for the same reason that we want students to understand the process of long division (for dividing polynomials to get oblique asymptotes)...because in Pre Calc we are going to want the students to be able to solve trig equations that require factoring. Something like (sin x)^2 + 2 sinx + 1 = 0. It is so easy when I can show the kids that this is simply x^2 + 2x + 1 = 0 if you take the trig "word" out of it...and they instantly know that they would/could factor it. Then they are able to put the trig "word" back in and factor (easily) the trig equation. I DEFINITELY do not want them to only have quadratic formula to rely on at that point!!
First, a disclaimer: I teach factoring. Not particularly well, but I teach it.
Second, one benefit I see in the way factoring plays out in most classrooms is that students (often? sometimes? occasionally?) have an opportunity to strengthen their understanding of variables and variable expressions. Is this enough to warrant the *huge* role factoring plays in most secondary algebra classrooms? Probably not.
With those two things in mind, I'm wondering...
@Chris, do you see any value in factoring for the general Algebra 1 or Algebra 2 student? If so, what is it? Where do you draw the line on reasonable/useful factoring tasks?
@Paula, to play devil's advocate for a moment... Why not have students rely on the quadratic formula for equations that are quadratic in form? Instead of x equals etc., could they use sin(x) equals etc.? And to push a bit further (not because I disagree, or because I know the answer myself, but because I'm curious to know what others think)... What is the value in solving something like (sin x)^2 + 2 sin(x) + 1 = 0?
Looking forward to your replies... :)
Reply
Chris Brownell
3/17/2014 08:42:16 pm
Paula, we have had this discussion before it seems :)
I understand your desire to have some amount of factoring in a trig students' background. What I am objecting to here is what seems to me to be an overwhelming chunk of the curriculum in Alg. 1 and 2 given to this practice.
If we expect 100% of the students in mathematics to spend so much time on a subject that 15% of them will make use of later, we do the bulk of the students a disservice, and sell our chosen subject (math) short.
Reply
Chris Brownell
3/21/2014 04:24:35 am
Michael,
Yes I do see some benefit to teaching it, but at a greatly reduced emphasis. Beyond an introduction to some of the basic forms (perfect squares being the only ones that come to mind to me) so few other forms have much benefit prior to their use in higher mathematics (if then). The perfect squares, I think are interesting because of their ability to demonstrate structure of the Real numbers (which are polynomials themselves).
I bring this up largely in response to my experience over the last several years observing new teachers going in to the profession. Many of these young teachers spend inordinate amounts of time and convolutions of speech developing new catchy ways for students to memorize tricks (Super X, Magic X, Bottoms-up, UnFOIL, etc.) all of these little tricks make use of the structure of the polynomials of course, but that never is the focus. The focus is on memorizing a trick. No emphasis on real understanding takes place.
In these situations it seems to me that the children are unintentionally viewed as machines that need to be programmed to follow, blindly, a set of coded instructions, instead of Fortran, Java, php, C++, the programming language is some odd form of school English. The pseudo code looks like:
Input: Quadratic Function ax^2+bx+c
If b=0 then ...
else If a = 1 (test this by the theory of the invisible 1) then search databank for similarity to some already predetermined form ->when found execute subroutine Y
and so on.
Math class should be a place of discovering reality, not a place to program human machines. I submit as evidence that this is what is happening that the bulk of the teachers I have worked with who claim lots of success (or even moderate success) in teaching factoring, that they do not see any relevance to the subject outside of, it is needed in the next school mathematics class. Most don't know the rich history of number theory that leads us to these forms, fewer still can name an authentic (non-contrived for textbook) uses of the practice.
Chris, perfect length rant. I'm left with two thoughts...
1. Even after all of my studying (MA in Education at FPU, MA in Mathematics at CSUF) I'm still so weak in my understanding of how elementary and secondary mathematics leads to and/or is extended in higher level mathematics. If only there was an 8th day of the week...
2. I need to find ways to honor the potential for mathematical discovery in JH and HS classes by limiting the amount of time spent "programming" my students with procedures and skills of questionable value. Yet I cringe every time I hear of a former student of mine performing poorly on a skills-heavy placement test (HS placement test, college entrance test, etc.).
James Key @iheartgeo gave a great presentation on factoring quadratics and having it become a little more concrete by substituting actual values in for the terms. He used 10^2 for the x^2, 10^1 for the linear term and 10^0 for the constant.
This presentation he gave made factoring and multiplying with quadratic expressions more natural. It was aided visually by a standard 2x2 grid used in [fill in the blank] method(s). His approach also had some collateral learning in revealing how product of the diagonals in the grid are always equal, which can lead into similarity and proportions.
We need to bug him to get more written up on this. Visit his website: http://iheartgeo.wordpress.com/ , and get him to blog about it. The presentation took place on bigmarker.com/globalmathdept
There's also potential for easy graphing of y=x^2+bx+c when thinking of it as y=(x)(x+b)+c with roots at zero and -b, and a vertical shift of c. Equations with a leading coefficient not equal to 1 would have a slightly more complex transformation.