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The American Diploma Project is and will be impacting on what is being taught in both Algebra I and II in the 15 states who have joined the ADP Consortium. The classic flow from Standards to Assessments to Course Content is leading to the type of content standardization in our schools which I envisioned decades ago. A natural part of this process is deciding what topics in our traditional courses need to be deemphasized or eliminated to allow more time for the study of linear and non-linear function models, one of the central themes of the new Algebra standards.This leads to curriculum questions like...
How much time should be spent on factoring quadratic trinomials in Algebra I?
My assumption is that factoring ax2+bx+c where a ≠ 1 is still taught in Algebra I. Please challenge that assumption if wrong! If we also assume there is sufficient justification for teaching this, then we move on to the issue of how much time should be devoted to instruction. Two days? More? Time for assessment?
Here are some arguments pro and con...
PRO
(1) It is required by the ADP Standards (see below).
(2) Learning only simple trinomial factoring of the form x2+bx+c is not sufficient for solving more complex application problems.
(3) The various algorithms, such as the "ac-method", which have been developed for factoring quadratic trinomials, are of value in their own right; further, the "ac-method" introduces or reinforces the important idea of factoring by grouping.
(4) Students gain technical proficiency by tackling more complicated trinomials.
(5) Students should be given the option of more than one method, not just the quadratic formula.
CON
(1) The AP Calculus exam generally avoids messy quadratics in their problems. If such occur, students normally go directly to the Quadratic Formula.
(2) The SATs generally avoid asking students to factor such quadratics directly, particularly since it is easy to "beat the question" by working backwards from the choices. Instead, they ask the student to demonstrate an understanding of the process.
Here's a typical question they might ask:
If 6x2 + bx + 6 = (3x + m)(nx + 3) for all values of x, what is the value of b?
(3)The ADP standards for Algebra I do include this topic but it does not appear to be stressed. The following are taken from the ADP Algebra I standards and practice test:
(3) Do other nations teach our traditional methods of factoring or are students told to go directly to the quadratic formula?
(4) Current Alg I texts seem to have deemphasized factoring in general and some have moved this topic to later in the book.
So I am opening the floor for your input here!
(a) How much time is spent on factoring quadratic trinomials in Algebra I in your school?
(b) Do you teach the "ac-method"? If yes, do you motivate it or teach it mechanically?
(c) Do you believe factoring quadratic trinomials is essential or should it be deemphasized?
By the way, here is an example of the ac-method:
Factor completely over the integers: 6x2 + 13x + 6
Step 1: Find a pair of factors of ac = (6)(6) = 36 which sum to b = 13.
Hopefully, students think of 9 and 4 without a calculator!
Step 2: Rewrite the middle term 13x as 9x + 4x (works in either order)
Then 6x2 + 13x + 6 = 6x2 + 9x + 4x + 6
Step 3: Group in pairs and factor out greatest common monomial factor from each pair:
3x(2x + 3) + 2(2x + 3)
Step 4: Factor out the common binomial factor 2x + 3:
(2x + 3) (3x + 2)
Step 5: Check carefully by distributing.
Here is a "proof" of this method (some details omitted like the meaning of h and k):
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5 comments:
Hey Dave!
I find that the "proof" at the last part of this blog entry pretty interesting and useful for me and my students (I'm a tuition teacher in Malaysia anyways).
Your blog has been useful for my students especially those who are taking the Additional Mathematics. I frequent your blog to get some inspiration to teach the subject. (:
By the way, since you're a blogger, maybe we can have a deal here. I saw that you have advertisements on your blog.
I'm helping a friend (on a part time basis) to find bloggers who are interested in putting paid text links on published articles. Bloggers get paid an amount of money for putting these links.
If you're interested in this idea, do feel free to contact me via e-mail. My e-mail address is kevlinefm1910@hotmail.com
Keep up the good work, Dave!
I think factoring is a useful skill that should remain in the Algebra 1 curriculum. However, it is usually taught independently of other concepts, such as connections to graphs of polynomials and solutions of polynomial equations. Many textbooks omit the fact that after you obtain the solutions of a quadratic equation with the quadratic formula, you can then relate those solutions to the factoring of the corresponding quadratic expression. Opportunity is lost for emphasizing the connection between zeros and factors.
Last semester, my junior math majors thought that all factoring happened over the integers since this is all that they have practiced. Since most of these students were future hs math teachers, driving home the Fundamental Theorem of Algebra and connecting it to the factoring techniques they already know was a big emphasis in my Intro to Proofs course.
(See my Geogebra applet:
http://www.mymathspace.net/geogebra/poly5.html)
Factoring should be emphasized as a skill, but should also be accompanied by its deep connections to solutions of equations and zeros and x-intercepts. If these connections appear later in the year, that's okay. But its should be tied up all together at some point in Algebra 1 or 2.
Reva,
Thank you for your profound and very important reply.
"Factoring should be emphasized as a skill, but should also be accompanied by its deep connections to solutions of equations and zeros and x-intercepts."
This quote from your comment should appear in every teacher's guide for Alg I or II!
Several thoughts come to mind here:
(1)If I were hiring a new secondary math teacher, I would check to see if you were the professor! How lucky these students are.
(2) Let me know if there's an opening in your school! I'd love to co-teach with you. If not, then let's collaborate on a guide for prospective high school math teachers. Email me!!
(3) Your theme of helping preservice teachers to recognize the interrelationships among math concepts is the most crtical piece and one I've been attemipting to share on this blog from its inception. We can't expect our teachers to do this unless they are shown how!
Many textbooks omit the fact that after you obtain the solutions of a quadratic equation with the quadratic formula, you can then relate those solutions to the factoring of the corresponding quadratic expression. Opportunity is lost for emphasizing the connection between zeros and factors.
Wait, really? Why else would students be factoring if not to find zeros? Students are just taught to factor for the sake of factoring, without knowing why one would want to do such a thing? That's abusrd. :( The understanding of what it's for is much more important than the acutal skill, IMO (especially since calculators/computers can do this for us now).
Mathmom,
There are many outstanding math teachers I've had the pleasure to work with who understand the importance of connecting procedures to understanding and who try to make sense of the WHY in math.
But, textbooks have forever compartmentalized mathematics which leads to teaching skills in isolation. It takes knowledge, experience, and the utmost dedication to develop lesson plans which help students understand the motivation for and the reasoning behind procedures.
This is why preservice education for prospective math teachers is so critical and one of the reasons I started this blog. I'm hoping in some small way to influence young teachers.
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