## In Defense of Factoring

I must say that I was rather surprised when an instructor at AMATYC suggested that we no longer teach factoring of polynomials in elementary algebra. I mean, to me factoring is about as essential a skill as breathing. But I thought about it, and although I’m not in agreement, I believe that it is important that we critically examine the topics we teach and make sure that they are still necessary.

The number one reason instructors give me for not teaching factoring is that we can use the quadratic formula to solve ANY quadratic equation. While that is true, that’s not enough for me. Solving quadratic equations by using the quadratic formula is often not the most efficient technique. Many students make arithmetic or calculator errors when finding the solutions. If a student can factor the expression, the solutions flow without trouble. Would you want to solve the equation $(x+7)(x-5)=0$ by multiplying the two binomials and then using the quadratic formula? How about the equation $(x-8)^2=-20$? Square $x-8$, add 20, then use the quadratic formula? Hardly the most efficient way to do it.

It’s true that students taking an introductory statistics course will not have to factor a polynomial, but that is not a reason to remove factoring from the curriculum. Students who struggle in introductory statistics do not struggle because of their mathematical deficiencies, they struggle because they have not developed their critical thinking abilities. They struggle to determine which hypothesis test is appropriate because they do not know how to take a look at the facts presented and determine which tool can efficiently be used in this situation. Learning to factor polynomials, and knowing when to use factoring, help to develop critical thinking skills.

Could there be some compromise? Absolutely. Factoring trinomials whose leading coefficient is not equal to 1 is often tedious and time-consuming. When we reach intermediate algebra, I tell my students to use a 10-second rule when it comes to factoring (and jumping to the quadratic formula). I still think that this type of factoring helps to develop intuition, but I could understand leaving it out. Perhaps we could leave out sum/difference of cubes? My students seem to like them, but they wouldn’t shed a tear. I teach them because being able to identify polynomial “types” is beneficial to students.

One argument that does not carry as much weight with me is “We need to factor in order to work with rational expressions and equations.” I’m not so sure that rational expressions could be pushed off until college algebra. It is a great way to practice factoring (as many as 4 polynomials per problem). It is a great topic for developing critical thinking – do I need a common denominator, am I trying to get rid of the denominators, …

One comment that really hit me was one made by “footmassage”: “It (factoring) should be in the air throughout whether your solving, writing lines in point-slope form, rewriting fractions, simplifying rational functions, manipulating transcendental functions, etc.  It is an essential basic tool that should be developed over a course(s) for fluency.” (Check it out here.)

Another comment, from the AMATYC session, that summed it up for me was “If we take out all of the topics that we have mentioned here today, I’m afraid we will be left with students who will be unable to think at all.” (Paraphrased, speaker unknown) That’s what we need to keep in mind as we transition into mathematics in the 21st century. We are not just teaching mathematics, we are teaching students to reason and think. I fear the day that mathematics is taught to students in the same way that students are taught to use word processing software.

I do promise to keep an open mind, and vow to always examine whether topics should still be taught.

-George

I am a math instructor at College of the Sequoias in Visalia, CA. If there’s a particular topic you’d like me to address, or if you have a question or a comment, please let me know. You can reach me through the contact page on my website – http://georgewoodbury.com.

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• 1. Whit Ford  |  January 2, 2012 at 6:54 pm

I vote for keeping factoring in the curriculum in a number of places, including some factoring of 3 and 4 term polynomials. The most important concept that factoring helps convey is that: a sum or difference can be rewritten as a product or a quotient, or vice versa. While such a change in appearance may be crucial to being able to simplify / solve / graph / differentiate / integrate, etc. – it is more significant at the high school level in making a higher level conceptual point: a quantitative relationship can take on many appearances, all of which are equivalent mathematically, and each of which may have its own advantages or disadvantages from an analysis or usefulness perspective.

If everyone is using automation come up with numerical solutions to things, then factoring may not need to be part of the curriculum – but at some point the potentially approximate answers provided by numerical techniques may not be enough, and a “closed form” solution may be desired – and that is when algebraic solution techniques become important.

Algebraic solution techniques can also be automated, but someone has to write the program… That may justify teaching algebra in college or graduate school, but not necessarily in high school. So I keep returning to the notion I mentioned at the end of the first paragraph above as being the most important algebra concept to convey and master in high school: you need to know how to transform an expression’s appearance in order to understand why it behaves as it does, and that understanding is critical to your being able to perceive quantitative relationships in the world around you, let alone do mathematical modeling or statistical analysis later on.

Whit Ford
http://mathmaine.wordpress.com

• 2. Lizzy  |  January 29, 2012 at 6:00 pm

I also vote for keeping factoring in the curriculum for all the reasons you and whit gave, but also for the simple fact that factoring shows students how polynomials and conics are structured. Understanding factoring is the key to understanding roots and completing the square. I feel strongly that these two processes- factoring to find roots and completing the square- unlock the mystery behind why polynomials and conics behave the way they do and isn’t that what we’re trying to do in high school math? Show students why and how functions behave, how to manipulate them to help us analyze problems? If one understands the connection between factoring and roots, and completing the square and shifting and stretching, then we can construct any polynomial or conic we want.