# Teaching Factoring – Should we?

There is a point of view that says that factoring is over-emphasized in algebra. I disagree strongly. Proponents of marginalizing or eliminating factoring make two major arguments.

1. Students don’t need factoring in order to do anything else. You know? Real-world wise, I buy it. But real-world wise, I can’t make a good argument for studying past per cents or making change. And I can’t make much of a real-world argument for most of what we teach, in most subject areas. But we expect knowledgeable adults, we expect young adults who can pursue studies in multiple areas, including those that are math-dependent, and we expect young adults who are conversant with a body of knowledge most of us share.

By deemphasizing and marginalizing factoring, we cheat our students.

2. The other argument is slick. Factoring, they put forward, is not necessary. The people who use this line agree that math is necessary, but usually focus on problems requiring numerical solutions: Set up the quadratic, use the formula or read approximate values from a graph. Let’s agree with them, but then ask, is factoring a useless skill? Of course not. Can it be avoided? Up to a point, perhaps, but you need to make a conscious effort to avoid it. Why not teach the skill?

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Here’s what’s happening. 1. Factoring is hard. 2. We have a societal attitude that says innumeracy is okay. 3. We have an undercurrent that says cut out as much math as possible. 4. We have another trend that says challenge everyone (a reaction to racist tracking) and so 5. instead of saying, forget about math, we have anti-math people saying: forget about hard math. And 6. We have teachers who either find some topics difficult themselves, or who have limited flexibility in relation to teaching or reteaching more difficult topics. And the result: factoring and geometry proof are marginalized.

It ends up looking like this: more students do more math at the level of algebra and up than ever before, but watered down. The critics of difficult, procedure-heavy math have given us easier, procedure-heavy math. But more of the procedures are calculator keystrokes. And they are winning. Factoring in particular has been reduced to a secondary place in the curriculum. Most text books are written with factoring coming after parabolas and the quadratic formula, ie, intentionally shunting factoring aside. And the problems are too few and too basic.

This is, of course, the wrong answer. We should be teaching real factoring. Our curricula should develop the mathematics in a natural way, (the ordering in most books supports, instead, early introduction of topics without appropriate background, in order to do graphing earlier. Bravo TI! See what money buys?) with factoring preceding quadratics, and with both of them preceding parabolas. We should do our best to teach real math to all of our students, and we should recognize that some of our students will not have great success with some of the more difficult topics.

We should approach each student as if s/he has potential to study much more mathematics, and offer that student appropriately challenging work. When a student stumbles, we should provide support. And if a student eventually reaches harder math than they can handle, well, we have brought them as far as they will go (at that time). Far better to push a student to their limit, then to assume lack of ability in advance.

So I do teach factoring, fairly substantial factoring. In subsequent posts I will describe how I organize the topic (for bright high school students who don’t necessarily like math, and for weaker college students who are looking for the minimum amount of mathematics to graduate).

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Excellent points! I agree with everything you wrote and could not say it better if I tried.

this looks like a pretty good summary

of the situation to me; of course if

i’d have said it it would have come out

a great deal rantier.

i’d like to see more “factoring” at the level

of the natural numbers — i.e., number theory.

i mean, not only the “tricks” (digit sums

for multiples of three, e.g.), but the reasons

they work (proofs! — some informality

might be called for here, but proofs just the same).

LCM and GCD could stand to be a lot better understood

*before* one goes about extending ’em to polynomials …

oops. time for class. i’ll check in again later.

Vlorbik, I’ve learned to rant calmly. It’s really a skill that I had to practice at. I also found it necessary to read very carefully things I fully disagreed with so that I could poke at them without caricaturing them.

Btw, if you haven’t looked before, peek at my pedagogy philosophy thing. I don’t have much time for the back to basics crowd, either.

And thanks for the kind words, p.o. I can think of no higher praise than that of successful veteran math teachers.

And I love factoring numbers, and playing with the topic with kids. I squeeze bits into my algebra course: 3599 is a classic, but we also figure out which unit fractions terminate, and why, bits of LCM and GCM ab = LCM(a,b)*GCM(a,b) – they don’t know this but can well understand it, and other stuff, where I can squeeze it in. Little bits of modular arithmetic are nice, and while not directly factoring, learning to represent multiples and more than mulitples (ex 3n or 5n + 1) helps develop those lines of thinking.

This year when I challenged kids to figure out this number trick we picked up on that same line of reasoning.

I absolutely agree that learning factoring is important. It provides a fundamental underlying many other processes in math, and even the process of learning it is beneficial, in my opinion.

3599 is a classic?

news to me.

you mean because it’s 60^2 – 1^2?

(in case for some reason some

*non* math-head is reading,

i’ll point out that this fact “easily”

implies that 3599 = (60-1)(60+1),

i.e. 3599 = 59*61 [and that this is the

*prime* factorization …].)

i seem to remember having to work out

(a,b)*[a,b] = ab for an abstract algebra class.

how it waited that long, i’ll never know.

maybe somebody had shown me and

i’d simply failed to pick up on it.

anyhow, the proof of this fact

(exponent juggling, of course)

was very satisfying. still is, actually.

it’s just a very cool fact.

LCD*GCD I learned somewhere in a math ed course. Of course it made all the sense in the world, I’d just not actually seen it before.

> “But real-world wise, I can’t make a good argument for studying past per cents or making change. And I can’t make much of a real-world argument for most of what we teach, in most subject areas.”

The problem with this line of thought is that it leads to a ‘math for math’s sake’ mentality – the very mentality imposed on long-suffering students world-wide in a (dare I say it) clever attempt to retain math courses in schools where there may be no need. In Australia, for example, most students study calculus in senior high school (17 & 18 yrs old). But most won’t ever use this calculus.

I used to teach math in the belief it was “good for them” until I started to concentrate on its utility. This was motivated by years and years of student retorts: “Why do we gotta do this?”. And I never really had a good answer.

I started to examine the difference between “understanding” and “meaning” in math education. Most math classes concentrate on the “understanding” part, but very few bring any “meaning” to its study.

I’m all for teaching factoring – but not just because…

> “Can it be avoided? Up to a point, perhaps, but you need to make a conscious effort to avoid it. Why not teach the skill?”

Math is abstract. That’s what it is. Numbers are not physical objects. The moment you substitute physical objects for numbers, it’s not really math, and to teach whatever you wanted to teach, you need to abstract the quality “number” from the physical objects, eg pull the concept “four” out of the collection “paperclip, paperclip, paperclip, paperclip” It only gets worse from there. Objects can illustrate math, can help teach, but the moment they stand in the stead of the abstraction, the abstraction is lost.

Math is for math’s sake if you want to be able to reapply it. Otherwise we get “how many lemons in seven groups of three?” “Sorry, we only learned pears.”

Waa???

I’m going to agree with Zac.

You have a straw-man argument, jd2718. Just because you use a utilitarian teaching style does not mean that students don’t learn to generalize. You seem to assume that challenging students equals teaching (only) abstract/esoteric math concepts, which is fortunately NOT the case. (Not that factoring polynomials is all that hard.) Check out any research on the art of teaching.

As for the specific topic of factoring polynomials, you’ll be happy to know that the upcoming Common Core (scheduled to be implemented in NY) will include factoring.

I came to the sight looking for a good reason for so much factor but still haven’t found one.

As an engineer with a degree in math I find that I never factor anything. In the real world it is highly unlikely that I will ever run into such nicely contrived values that I would attempt to factor. I would do it numerically (or for a quadratic use the quadratic formula). There are so many other interesting approaches to take and mathematical ideas to explore, ones that could really help kids learn more general problem solving techniques. I have to disagree.

I think too much factoring, turns people off from math. It really is a waste of time.

P.S. in the “real world” we do a lot more that percents! but we don’t factor!

No one factors, except mathematicians. But we use factoring as we learn more advanced mathematics.

You do everything numerically? And are convinced factoring is useless? Good for you. But don’t pretend you don’t have an opinion.

And at least pretend to read the post and the comments next time.

Hi jd2718 I enjoyed reading this but have a question:

I am curious why you favor factoring before graphing? My sequencing is as follows: real life parabolic behavior 2)basic Quadratics graphed, variations on quadratic eqns and their effects on parabolas 3) visually finding roots 4) factoring 5) quadratic formula 6) completing the square. The appreciation of factoring to find the root is a bit abstract without the visual of the x-intercept on the graph. At the freshman level where algebra 1 is generally taught I still find it important to begin with the visual side. Otherwise why factor? Introducing it as that which give the funciton no value I feel would be difficult in the beginning.

For algebra I we manipulate polynomials early (after learning to handle exponents). It takes time to get good at this.

We graph lines later (parabolas, too). Mid-year.

Late in the year, after learning to handle radicals, we tie all the strands up with solving quadratics by graphing, by factoring, by taking square roots, by formula.

I note that you have work with two variables early. I also note that you have the quadratic formula before completing the square. You may have chosen (or your book may choose) to introduce topics that are easier and easier to motivate earlier.

My focus, otoh, is on 1) justifying each topic mathematically and 2) introducing topics after students have had time to master requisite skills. The sequence you propose, imo, would not allow for either of these.

Am I missing something? How do you simplify rational expressions without being able to factor? And this leads to finding holes in graphs of rational functions. Factoring is not just about solving quadratic equations.

I’m in my 5th year of teaching and I’m beginning to struggle more and more with the usefulness of what I’m teaching, but I am determined that it has value.

I don’t see many people who lie down and lift something heavy above their heads for a living, but I know millions of people who do bench presses religiously to build all sorts of muscles. Maybe math is this kind of exercise for the brain. An exercise in methodical, logical, sequential, rational thinking which we definitely could use more of (especially in politics). So a kid doesn’t ever factor again in his life. So what? She’ll use that logical, trial and error, scientific method a lot….I hope.

I taught Algebra last year, and factoring has been cut from our Algebra 1 curriculum. And Proofs are not taught at our school at all. It’s in the curriculum, but since it’s not on the tests, the teacher doesn’t teach them.

This year, I am now teaching Algebra 2 and Advanced Math and my students can’t factor at all.

I think it’s terrrible! For me, factoring is a way of understanding number relations and understanding the basic concepts of math.

I am definitely teaching factoring until my students get it! And am looking forward to starting it soon.

Have you seen CME? It’s a new series of text books. Get at lots of real math. Doesn’t have enough practice problems, and at times can be a bit obtuse. But overall amazing. I just taught their Quadratics unit (from Algebra I) to an Algebra II class. Started with a numberic look at difference of perfect squares, moved on to factoring quadratics, then completing the square in a method that used difference of perfect squares along the way. Finally, they move on to graphing, and build on all the groundwork, making the vertex form seem obvious, since the kids are so used to it. And if you push it, you can even get them to see how the constants in the factors relate to the x-coordinate of the vertex. Check it out.

-Debbie, math teacher, Waltham, MA

We make comments on here as if all Algebra math courses are the same. We all serve different populations. For my group of kids, spending a month explaining, understanding, and working on factoring would be a waste of a month that could be better spent on other aspects of the math curriculum. Why? Because I have far too many algebra students who still are confused by negative numbers, dividing fractions, and exponents.

Who’s fault? Many factors, but I’m not going to lose 80% of the class’ attention by forcing the guess-and-check factoring simply because it’s the month of the year where we normally work on that.

Part of teaching to depth, to mastery, may include shortening or running out of time to teach certain math aspects. It’s just life in some school’s situations.

If I had a bunch of self-motivated students with computer access at home and a solid history in their previous math courses, then yes, factoring – if nothing else- would be a great brain exercise (reasoning, organizing, etc) that could later be applied in other realms of life.

What population do you serve?

There are real issues with forcing students who are not algebra-ready into algebra. Part of the reaction in California, if I understand correctly, did exactly that.

I work at a school where we help students that for whatever reason weren’t able to stay at the traditional high schools. Some have kids, others are homeless, some were in trouble with the law.

However, it seems that even in “normal” algebra courses, too many kids were of the “skimmed by and got a C” pre-algebra kids.

It does seem help is on the way with the upcoming streamlined teaching standards.

This is the best I ever saw!!!!!!

Factoring is communist garbage and should be removed from the curriculum. I can do any other kind of mathematics, but factoring is bloody impossible, ridiculously complicated; it requires a lot of guessing, switching numbers around all willy nilly, and all sorts of other things that are taboo in every other kind of math. Above all it is utterly pointless. I shouldn’t have to have my chance at a Bachelor’s degree ruined by such a roadblock.

You still failed to come up with a valid application for factoring, and a justification for EVERYONE knowing how to do this. In order for something to be important, there has to be a reason for it, otherwise the importance is invalid.

Yours is the weak argument, and what gives you the right to accuse people of being “innumerate” just because they don’t want to force otherwise academically sound people to waste their time with needlessly complicated busy work that has absolutely no application, none that would ever be used in any realistic context. Do you accuse a decorated, brilliant soldier of incompetence for not wishing to bring down an enemy tank armed only with a herring? Being unable to factor does not equal an inability to problem solve.

And i mean factoring polynomials, not the factoring of real numbers, just to clarify.

Really? Factoring is important because it is useful in upper-level mathematics, particularly Calculus and Trigonometry. It seems nobody stops to ask “when am I ever going to use this?” in the English class that reads Hamlet or writes cute poetry for Valentines day. In spite of this, most of them never do use it. But when it comes to math, people FREAK OUT. A degree from a university (or high school for that matter) is designed to be well-rounded, offering curriculum from all aspects of academia. The fact that you may not factor in a physics class doesn’t mean factoring isn’t important in a mathematics class. Indeed, it is the inverse of multplying polynomials, which is a useful mathematical process.

Incedentally, nothing about factoring is “willy-nilly”. What’s nice about factoring is it’s strictly process oriented. Follow the steps, and you find the answer.

A couple quotes caught my eye:

“Follow the steps, and you find the answer.”…but only if the quadratic is factorable.

“No one factors, except mathematicians”….actually, that should read AMERICAN mathematicians. Does anyone outside the US teach factoring of quadratics? Most teach the quadratic formula and move on.

Where I teach we use the CPM books and spend a large amount of time explaining algebra tiles, generic rectangles, diamond problems, etc. just to lay the groundwork for factoring quadratics. Couldn’t that time be better used?

As for it being mental “bench pressing”, wouldn’t sudoku or kenken puzzles produce similar results?

How did you get stuck with CPM?

As an adult attending college, I feel that I have to speak up about this. I always suspected half of this math we are forced to learn had little if any real world application; and now I have seen teachers admitting as much. I love how whenever a student asks a teacher what the use of learning this is, when we won’t even be using this kind of math, is always the same. The argument never changes: “We want our students to be knowledgeable,” “students have to have a well rounded education,” “It teaches problem solving skills,” etc. etc. The excuses never change.

As an adult, I should have the right NOT to take these classes that have no use whatsoever in my chosen field. I am trying to attain a degree in psychology, and I will never use half of the math I have been forced to learn. Even when I thought I wanted to be an RN, the most advanced math we had to use in clinicals was solving for an unknown variable. Yet, to get a Bachelor’s degree, I have to get into calculus? W H Y?

None of the excuses hold water. As an adult, who are you teachers to tell me that I should be “knowledgeable?” Are you saying that if I don’t know how to find the equation of two sets of variable expressions and graph them, that I’m not knowledgeable? Are you saying that if someone doesn’t know how to divide a polynomial by a binomial, or that if someone doesn’t know how to factor the sum or difference of two squares that they aren’t knowledgeable? Who are teachers to demand that we need to know these things that really have no bearing on MOST professions. As a psychologist, I won’t need to know any of that. As a DOCTOR, I wouldn’t need to know this advanced math that somebody somewhere decided every adult who goes to college needs to learn.

I think it’s time that we students stopped having this stuff that we will never use shoved down our throats. We have to spend OUR valuable time and MONEY learning how to do things that we will NEVER use; and for what? Just because somebody decided that we have to have “well rounded educations?” Excuse me, but after going through 12 years of school PLUS another 4-6 of college, I think my education is well rounded enough. I don’t need to learn this math that everyone claims is essential for us to know to graduate.

You math teachers need to stop trying to justify these advanced math classes being required for every degree, no matter what the profession. If students WANT to take these classes, fine; but it should be an option not a requirement.

~SilverWolf~

“Yet, to get a Bachelor’s degree, I have to get into calculus? W H Y?”

For the same reason calculus was a pre-requisite for early childhood education, and algebra is for becoming a business major at some schools (Indiana-Purdue, I believe is one example): Other programs use math courses to weed applicants out of their programs before they even have a chance to get in.

I kid you not, I was heading into a Bachelor of Mathematics program, in my last year of high school in calculus class. The class as a whole was doing poorly. After one particularly poor test, the teacher asked the class why they were doing so poorly. One of the students explained that she was only in the course to get into early childhood education. The teacher was notably less enthusiastic and pushy after that, and I was noticably hobbled in university calculus (admitedly my fault for not being more self-motivated in the first place).

It sucks, but it’s an easy way to cut the applications to sort through by 50-90% for a program, and that’s why it’s done.

I agree, and that is hella wrong. We pay to be educated. Not be weeded out. And the sad truth is that Math is 90% a weed out course (most times) and about 10% essential.

For example. I am teaching myself Algebra. After going through many teachers, i have found that I am the best one. Because Algebra is SUPPOSED to teach 3 things.

1. It is like the “engine” that drives advanced math. If you want to do higher math, you need Algebra because it is imbedded into all other forms.

2. Algebra makes math simple. That is mostly what it is. You CAN do math without Algebra, but it becomes more convoluted and difficult. Like 3X3X3X3 vs 3^4

3. It should teach logic and understanding. But how we actually teach it it only accomplishes memorization reinforced by repetition. “Why is X= 5? Because the formula says so.”

Hence why the US is falling behind in math and many hate it. The way we use it, OF COURSE everyone will hate it. Just look at these comments. Why can’t a math teacher give the answers as to why you need algebra? I gave 3 VERY good reasons and I had to find it myself. And I am only in pre-Algebra in College. And why do I have to sit by myself and figure out WHY the math adds up and understand the logic behind it? Hence why I stopped attending class. I went from failing to scoring 95 of my quizzes and tests.

Funny thing is that my math teacher is pressuring me to return to class. Why would I do that!? What I am doing works. I am learning and scoring high. Why go back to what has not worked for me for near 18 years? And that is the situation the country is in. Instead of finding solutions (what math is SUPPOSED TO TEACH) math experts simply double down on what is clearly not working, And Math is supposed to teach problem solving? I think not. The proof is all around us.

I shall finish with this. Philosophy teaches you to question life and seek truth. History teaches you about the mistakes and triumphs of the past. Sociology teaches you tolerance and opens your eyes to different cultures. English beyond English comp teaches an appreciation for literature. However, that is OPTIONAL. Many courses math people try to point out we do not need are OPTIONAL. If Algebra was OPTIONAL we wouldn’t have this problem. Imagine making it a requirement for all students to learn to read classical Shakespeare even if all they wanted was to be an Engineer? That would be a good comparison. Since all Algebra is mandatory whether you will use it or not.

@SilverWolf I completely agree with you silverWolf. I share the frustration. I just interviewed for two jobs. They use a computer test to weed out applicants. Unfortunately, the tests were so arbitrary, so poorly worded, so *irrelevant* to the position that all they really tested was the ability to do well on exams.

Unfortunately, science (namely neurology) really hasn’t gotten to the point were we can create a test that accurately measures “intelligence”—in part because scientists are still figuring out what it means to be intelligent. So until we get to the that point, we’ll continue to use mastery of esoteric content as one measure of “intelligence.”

I know it’s not fair, but that’s how it is.

Playing the devil’s advocate, our society is becoming increasingly dependent on complex technology—just take a look at your pocket. You probably have a smart phone that’s more powerful than your PC from 8 years ago. What this is telling me is that the bar to succeed is getting higher every day, i.e., the good jobs are increasingly complex. Mastery of “esoteric” content (like factoring) is a good indicator that you have what it takes to succeed in a demanding career—where we define “demanding” as requiring intelligence.

However, in your particular case, it seems that you are required to know “too much”—the mile wide, inch deep problem that’s been plaguing most schools. You are better served by focusing on less topics, *in greater depth*.

Jd, you don’t seem to make an argument as to why you think factoring is important. Can you give an example of an important algebra problem where factoring is necessary? Why should students learn to factor? Looking forward to your response.

Certainly you are not talking about taking out greatest common factors. That skill is used throughout mathematics. You meant factoring trinomials and uglier, right?

1. Factoring trinomials and uglier is the inverse of multiplying binomials and uglier. It would be odd to tell my students that we can do ~, but even though the process is reversible, it is too hard for them.

2. Factoring trinomials and uglier (I’m not responding to you here, but thought I’d throw it in) is not as hard as many make it out to be.

3. Factoring trinomials and uglier (Again, I know this is unresponsive), practices many manipulating skills that are useful in algebra.

4. Creating perfect squares is necessary to solve a range of problems in algebra, starting with deriving the quadratic formula. While it may be sufficient to teach completing the square without the students knowing factoring, it becomes exactly the kind of “trick” independent of other knowledge that we should not be teaching. And for those who would present the quadratic formula without deriving it – I don’t know what to say.

5. Factoring trinomials and uglier is used trigonometry, used in more advanced algebra, used in precalculus, used in basic proofs by induction, used in calculus, and used here and there in higher mathematics. I am currently taking grad courses in Number Theory, Logic, and Combinatorial Theory, and in the course of the term I have encountered factoring in two of the three courses.