Teaching Division Alternatives

The Partial Quotient Algorithm

Nov 10, 2007
Greg Cruey

Division: Step One - Greg Cruey

Traditional long division is a mechanical, abstract process that many elementary students oftem find difficult. The partial quotient method is one popular alternative.

The traditional approach (or algorithm) for large number division is the most abstract and difficult approach to division. Yet many adults think it is the only approach. Take a simple problem like 7,248 divided by 3 and try to explain the traditional method: three goes into seven twice, write down the two over the seven, multiply two times three and write the answer, six, under the seven, then subtract six from seven. That leaves 1,248. Since three won't go into one you have to move over a column and divide three into 12... You get the idea. Students often have no idea why they do what they do in this process. They do it because the teacher said to, if they can do it at all...

A Conceptual Approach with Easy Numbers

The partial quotient method of solving large division problems has two main advantages. First, it allows elementary school students to see the problem in a less abstract form. They actually ask concrete questions like "So, how many nines are there in 2,079?" Division becomes an idea instead of a long division template. Second, it allows the students to work the solution flexibly, using numbers they're comfortable with, instead of requiring a rigid mathematical process that the student may not find comfortable. (How that works will become more obvious as the method is presented.)

Step One: Setting Up the Problem

The first step in the partial quotient approach is to set up the problem. This looks much like a traditional long division problem except that a horizontal line gets drawn along the right side of the problem to create space for the student to track "partial quotients." (See "Step One" image below.)

Step Two: Picking Easy Multiples

The question in the mind of the student is simple: "How many groups of nine are there in 2,079?" The partial quotient method tries to get the student to the answer through basic logic. So the next question the student asks is, "Well, are there at least blank number of groups of nine in 2,079?" Fill in the blank with a number the student is comfortable with - let's say 100. So the question becomes "Well, are there at least 100 groups of nine in 2,079?" The student does the math and figures that 100 groups of nine (100 x 9) is 900. The student writes 100 in the partial quotient column and writes 900 under 2,079 in the problem template; then he does the subtraction to see how much is left. (See "Step Two" image below.)

Step Three: Can I Do That Again?

With 1,179 left in his dividend, the student should ask this basic question: "Can I take that many out again?" If the answer is "yes" (like in this case), the student should do that. If the answer is "no," the student has to find a smaller easy multiple to take out. In this case the student writes 100 in the partial quotient column again and writes 900 under 1,179 in the problem template; then he does the subtraction again to see how much is left. (See "Step Three" image below.)

Step Four: Can I Do One More Time?

With 279 left in his dividend, the student should ask that basic, logical question again: "Can I take that many out one more time?" If the answer is "yes," the student should do that. But in this case the answer now is "no," so the student has to find a smaller easy multiple to take out. Let's assume this student is not very good with "nines" yet and decides to just take 10 "nines" out. The student writes 10 in the partial quotient column and writes 90 under 279 in the problem template; then he does the subtraction to see how much is left. Since the dividend left is still more than ninety, the student can repeat this step two more times. (See "Step Four" image below.)

Step Five: Add up the Partial Quotients

When the student has taken this process as far as possible (the dividend left is less than the divisor) the final step is easy: add up the partial quotients to get the whole quotient. The student in this particular case will add 100+100+10+10+10+1 and get 231. There are 231 groups of nine in 2,079. Or, phrased more traditionally, 2,079 divided by 9 is 231. (See "Step Five" image below.)

For more on teaching division, see What To Do When Johnny or Suzie Can't Divide...

Greg Cruey - Greg Cruey is an educator and journalist. He works as an educational interventionist specializing in a small, rural school in Central ...

Comments

Aug 22, 2008 7:16 AM

Guest :

Lazy lazy lazy...no wonder we lag behind in math!

Aug 22, 2008 10:02 AM

Greg Cruey :

Actually, the problem is that for years we've taught one AND ONLY ONE way to "do it." That discourages any thinking. The goal is to make math more conceptual. And I see no particular reason why it should be hard. The goal is to get a right answer...

Aug 24, 2008 6:10 PM

Guest :

It might not make us the fastest, but it builds confidence, and many of our kids lack enough confidence to be successful in math.

Sep 18, 2008 3:11 PM

Guest :

Partial division is the easiest way for children to understand what division is all about. Why is this lazy? Don't you get the same quotient?

Sep 26, 2008 11:15 AM

Guest :

This is OK to show to someone having *extreme* difficulty, but I agree, very lazy method. :(

Sep 26, 2008 11:26 AM

Greg Cruey :

LOL. I really don't get the "lazy" stuff. Are you saying it's just not enough work and we should make getting the answer harder, for the sake of good morality? Most people who use this method don't teach it exclusively. It usualy helps concrete thinking elementary students better visualize division. Is "lazy" just a synonym for untraditional here? Why should division be hard? "Lazy" is a moral term; why is the traditional method more moral than the partial quotient method? You don't have to answer all my questions, but it would be nice if someone would explain WHY it's lazym instead of just saying "Hmmm, that's lazy." - Greg Cruey

Sep 30, 2008 3:06 PM

Guest :

Teaching my student's how to solve division using the partial quotient method has given them the confidence they need to know they can succeed doing long division. There is no laziness factor here, just a different method for getting the same outcome. It is a proven, successful to teach division to struggling and highly achieving students. It is a great method, it is not lazy. Remember, there is more than one way to do things.

Oct 1, 2008 5:15 PM

Guest :

Those of you who think that the partial quotient method is lazy fail to realize that the traditional algorithm, or the 'goes-in-tos' method little to bolster an understanding of what it means to divide the whole number. The traditional algorithm is simply a series of memorized steps; if you not understand that you are dividing 7423 into four equal groups, then how does the traditional method support comprehension and logic? You are not dividing 7 by 4 - you are dividing 7000 into four groups. There's a big difference. The partial algorithm method requires a true understanding of the concept of division, not just the memorization of steps. That is why so many of us were poor in more advanced math - lack of conceptual understanding. Math is not about the memorization of a series of steps, but about logic. In fact the traditional method could be described as lazy because there is no requirement to explain/demonstrate one's thinking - it's just the step-by-step dance of memorization. Rote memorization does not equate understanding.

Oct 16, 2008 11:57 PM

Guest :

I looked this up because my niece was doing it and all she had was a worksheet with no instructions. At first I thought it was odd but reading the comments it made sense how it gives a better grasp in seeing sets of numbers in larger numbers. I am not sure it would be easier to do in my head but it's nice seeing an alternative that is basically based on what I do in my head anyway with long division... figuring upwards from the divisor. With really large numbers that are harder to keep track of the quotient method seems easier because you can use rounded quotients to get basic figures and these rounded quotients are easier to remember and add up than to keep track of 9 different digits in various sets. And I agree that it leaves (at least me) with a better understanding of about how many parts are in a whole. Great for quick estimations for sure.

Oct 21, 2008 5:45 AM

Guest :

It may seem as if this method is lazy but for students that struggle with math, this helps to reinforce place value concepts and it also helps students that are struggling to be able to divide using numbers that make more sense to them. I teach this method in my class along with the traditional method and column division so that I can meet the needs of all of my learners. Not all students learn in the same way and as long as they have the answer and are able to show me how they got the answer, I am fine with it.

Oct 23, 2008 4:39 PM

Guest :

I'm sorry. This is just flat out stupid. You teach the kid how to multiply, barely in the new curriculum, then throw it all out and go for the "CONCEPT". If you teach them multiplication WELL, and then teach them that division is just the opposite, they have a foundation to work from.
My generation, with our Old Math, put men on the moon. The modern math generation has managed to crash several with few survivors.
1st you teach a good foundation, 1st through 4th grade, addition, subtraction, multiplication and then division.
They don't do that anymore and that's why the grades are plummeting. Too much got published and too few perished.

Nov 6, 2008 3:44 PM

Guest :

it is in my opinion a very confusing and i prefer not to use it. with my 5th grade class they find it hard and they do not like i.

Nov 12, 2008 1:17 PM

Guest :

I don't wouldn't use the term lazy, but my daughter is having no easier time learning this method than the traditional method, and I certainly can't help her. So I don't think it is easier nor is it building her self esteem and if the goal is simply to get the right answer why can't she just use the old method instead of being forced to learn a new one that she doesn't get. And by the way memorization is not a dirty word either-seems like test scores were a lot higher in "the good old days"-you don't have to be a genius to see that.

Nov 12, 2008 3:17 PM

Guest :

This is a very, very good explanation of the method, especially given that the first website I accessed (NYU's) made the method seem unapproachable - and I'm an 800 on my SAT sort of math guy. Thanks for your help.

Nov 15, 2008 8:28 PM

Guest :

I really hope that individual that commented on this strategy being a lazy way to divide is not an educator. What an IGNORANT comment!!!!

Nov 20, 2008 5:17 PM

Guest :

Its kinda confusing

Nov 22, 2008 6:49 AM

Guest :

I have been teaching 4th and 5th graders this method for the last 5 years--my kids are grasping this algorithm quicker than the "traditional" algorithm. It also reinforces place value and mental math skills! I have had several students who were able to solve these types of problems without pencil and paper in a matter of seconds! It is far from lazy--it becomes a more efficient manner of solving for the quotient. Try it you'll like it :)

Nov 30, 2008 10:13 AM

Guest :

In the UK we call this the chunking method. It's really popular with lots of kids.

Dec 1, 2008 5:31 PM

Guest :

I'm not sure how you can find this method confusing. It breaks the numbers down conceptually. I'd like to see some of the adults who commented on this site, how to explain the long division method conceptually. Isn't the idea of learning to understand, not to become machines and memorize rules?

Dec 9, 2008 6:24 PM

Guest :

As a left handed individual forced to write with my right hand in Kindergarten to 5th Grade, It is very difficult to express the logical sequence of my mind. As a child to ninth grade I calculated answerers in my head, seconds after seeing the question using; reciprocals converting the quantities to other numbers and then converting back. and other relationships, yet was unable to express them on paper. Back then there was no ADA or Dyslexia everything was right or wrong. Even further back people placed Galileo types in jail for thinking differently. At 50 I struggle with division and conventional classroom environment. So open your mind and motivate unconventional students.

Jan 15, 2009 6:51 PM

Guest :

I am a teacher of students with learning disabilities, and I for one will be teaching my students division using this method. Many students (even those without learning disabilities) have difficulties with multiplication and division. As long as my students get the right answer with having to use a calculator, I am all for trying alternative methods. This method will be especially useful when the divisor is greater than one digit. For those of you who think this method is lazy, maybe you are the ones that "don't get it" and therefor you "poo-poo" any method that you think is unconventional.

Jan 21, 2009 4:33 PM

Guest :

After about 20 minutes of using this method, my students said, "This is fun . . . can we keep doing problems?" It shows the students that there are many different ways to do the same thing. It allows students to work with "friendly" numbers they feel comfortable with and it makes "long" division more accessible. They're still doing division, so how is it lazy? The students are simply chipping away at the division problem at their own rate.

Jan 23, 2009 3:20 PM

Guest :

There are a few things I really like about the partial quotient method. The first is that it transparently integrates place value concepts into the process. The student thinks and documents the place value in each step.
Secondly, the partial quotient method promotes and makes apparent the experimental nature of long division. It helps students keep track of each of his/her experimental steps in the process. That is invaluable.
Third, as a retired teacher (4th, 5th, and 6th grades) I remember students struggling mightily with long division without some kind of device such as partial quotient division.
Fourth and 5th grades when many students are deciding that school is a think that they are not particularly interested in. I think much of this is because of the high degree of abstraction that students face. There is nothing wrong with abstraction. What is bad pedagogy is when abstraction is not introduced appropriately - developmentally appropriately. This kind of approach to division is one of those appropriate steps.
We (the United States) is not known for broad high achievement in mathematics. We have a few who achieve very high. But we have many who do not perform at a level at which they are able to make a contribution to society. This can help those people.
When I was teaching I do remember parents would sometimes object more informal algorithms and 'guess and check' strategies. I think largely because they had learned a strict algorithm and had a hard time thinking conceptually and experimentally about math.
I am very encouraged to read in this comment section from teachers reporting that students regard this method as fun and want to keep doing it.

Jan 26, 2009 3:55 PM

Guest :

This is why our school slag behind!!!! What advantage is it to teach a partial products when you can do division the old fashioned way and get the correct answer every time! There is much more room for error in partial-quotients. Kids don't get abstract thinking. Do elementary teachers and administrators keep up with the news? Do you want your anesthesiologist competent in math or do you want them to "guesstimate!" I know your answer!!! Let's face it people when we look for genius in math we don't look to elementary school teachers!
One final note, a comment below states the teaching math the "ONE and ONLY way discourages any thinking." I can assure him that the highest level of math that so few will achieve is where you have the real ability to think and see the beauty in math - quantum physics - By instituting bad math programs early on you shut so many capable people of attaining this - 99% of you never get to experience this beauty! Pity!

Jan 26, 2009 4:06 PM

Guest :

Jan 26, 2009 4:09 PM

Guest :

Greg, What do you really know about teaching math? Are you sending children on to the top colleges to study in math and sciences? Do you really understand what it takes to achieve higher level math? Stop dumbing down our children! What percentage of doctors around the country are foreign?

Jan 27, 2009 8:21 AM

Guest :

Unbelievable! I encountered this "method" while tutoring a 5th grade student who doesn't know her math facts. I retired after 31 yrs. mostly inner city middle school reading and language arts. Having learned my math facts early, however, I can calculate mentally faster than most students can type problems into a calculator.(Ironically, this is how I earned my "street creds" with my students.)
Hey, Greg C. Didn't anyone teach you short division? There is long division AND short division. To do short division, however, one must have know his/her math facts. Divide 4/798=? 4/7=1 4/39=9 4/38=9 R.2 answer 199R2 It is simply KNOWING division, multiplilcation, and subtraction facts.
As educators, let me re-phrase, as enablers, instead of insisting our students LEARN math facts, we try to dumbdown everything. We have become a nation of brain numb button pushing adults who teach the young that answers are vague generalities and that it takes a group of six to figure out what an individual used to be able to. Tests are still individual endeavors! No wonder cheating is a "no brainer" for the young.

Jan 27, 2009 7:00 PM

Greg Cruey :

To the guest from Jan 26... Wow, talk about melodrama! You said: "Do you want your anesthesiologist competent in math or do you want them to guesstimate!" Surely you're aware that there are, like, 20 years of school (half or more of them college) and several board exams between 4th grade math and life as a practicing anesthesiologist. I don't think teach new ways to do math is bad. I don't think knowing more than one way to do it is bad. I don't think the traditional method is necessarily the best method. But most of all, I don't think I'll be the one responsible if a kid learns partial quotient method in my room but manages, years later, to kill a patient while practicing bad anesthesiology... Give me a break!

Feb 3, 2009 2:41 PM

Guest :

This is a horrible way to teach students. Discipline and method should be taught the traditional way. The concept should be "don't reinvent a wheel that has worked for centuries"

Feb 4, 2009 9:47 AM

Guest :

I am truly amazed at the amount of ignorance floating around here. As a fourth grade teacher with a masters degree from one of the best education universities, I'd like to pointout that partial quotients is the basis for higher/level division thinking. In fact, it is THE BASIS for our 'traditional' methods. The same is true of partial products, partial sums and partial differences. Just because we can represent math concepts in a shorter more concise way (traditional method) does not mean it teaches the concepts behind math theory. My students prefer partial products and in fact, it leads to some students being able to accurately do long division mentally. Can't say the same of the 'traditional method'!

Feb 4, 2009 11:56 AM

Greg Cruey :

Re the "I am truly amazed..." comment: THANK you. - Greg

Feb 6, 2009 9:53 AM

Guest :

I have been teaching this method for almost 5 years and there is nothing lazy about it. Not everyone learns the same way and this method provides an alternative for different thinkers. The reason we lag behind in math in this country is because we continue to cookie cut our delivery of instruction and do not differentiate enough. This method allows for that. Who wouldn't want schools to create free thinking and independent learning? Please, open your minds to new things. It is how we are going to excel in this world and allow our childrent to do the same.

Feb 7, 2009 9:44 AM

Guest :

If the student only knows the partial quotient method, what does he do when he gets to Algebra 2 and needs to divide a polynomial by a polynomial? Will the partial quotient method work? I'll have to think about it, as it isn't obvious to me.

Feb 7, 2009 10:25 AM

Greg Cruey :

Kids learn division in the third, fourth and fifth grades. They're as young as 8 and as old as 11 during this time. Algebra II comes four or five years later. Considerable amounts of math instruction take place in between...

NO ONE has suggested teaching ONLY the partial quotient method. (I think several people have suggested teaching ONLY the traditional method.) The partial quotient method serves in part to teach kids to actually THINK about division in a manner that the traditional method does not. But...

x+1 / x2-9x-10
......x2+1x ......(taking x+1 out x times)
......----------
.........10x-10
........-10x-10.. (taking x+1 out -10 times)
......----------
..............0

(pardon the dots, butthe comment section won't just create spaces)

I took x+1 out x times on the first line and -10 times on the second line.

So the quotient is x-10...

Feb 8, 2009 2:36 PM

Guest :

partial division is no help at all Greg i hope all of you know that

Feb 12, 2009 9:53 PM

Guest :

Thanks,Greg, for showing me how one would use the partial quotients method to divide a polynomial by a polynomial. I'm a math tutor. I wonder how to figure out which method to teach a given child. Unfortunately, in my school district, long division (or partial quotients method or any kind of division) is not required in the elementary curriculum. It is up to the elementary teacher whether to teach it---some do, some don't. It is not part of the middle school curriculum either!

Yes, children vary a lot in learning styles. One of my sons picked up long division after my showing him 3 examples in one 5 minute sitting. With my other son, it took me several weeks of monopoly money, buttons, etc. before he got the hang of it. I think the latter son would have done very well with the partial quotients method. I wish I had known about it 10 years ago. By the way, he is no dummy in math--he got a 5 (highest score) on the AP Calc test.

To me, long division and partial quotients method are just different ways of keeping track of the same thing. I don't think one is lazier than the other. It's kind of like in sewing...there are many ways to sew a zipper in. Some people prefer one way, some another. The end result is the same.

Thanks, Greg, for an interesting way to do division. Do you have a book about ways to do other math procedures in alternative ways?

Feb 13, 2009 4:01 AM

Greg Cruey :

My school district uses Everyday Mathematics, a curriculum developed at the University of Chicago. Alternative methods to perform all operations are presented in their materials...

http://everydaymath.uchicago.edu/

Feb 14, 2009 9:40 PM

Guest :

I can see how it would be somewhat of an easier way to do long division in your head, but really on paper it takes just as many steps, so really what's the point?

Feb 14, 2009 9:53 PM

Greg Cruey :

The point is that it teaches students to THINK about division differently. Long division often doesn't teach younger student to think about division AT ALL - just to memorize the steps. It also provides students with more than one approach to solving a problem. They can use long division (which they do eventually also learn) or partial quotient. And if there's more to one way to solve a division problem, maybe there's more than one way to solve MOST PROBLEMS - in life or math...

Mar 21, 2009 11:37 AM

Guest :

A master's in education from the best university teaching program does not ensure one's teaching competency or skill. Nor does it ensure that a teacher is prepared to meet the needs of individual students. That only comes with comes with experience.I have taught elementary and middle school math using both the Everyday and Traditional math algorithms. The use of a variety of algorithms is now common practice, due in part, to recent emphasis on having students articulate and demonstrate the steps they take in finding solutions to math problems. If a child can arrive at the correct answer and can understand the reasoning behind the steps taken, then it doesn't matter which algorithm is employed.

Mar 28, 2009 5:13 AM

Guest :

My 3rd grader has just revealed this to me. I'm confused. He is confused. Just teach and be done with it. I'm just as irritated and frustrasted as my child is. Learning new things is fine and wonderful but don't pressure the kids into learning it for a test they are required to pass in order to go to the next grade level. Teach it for fun and those who get it great and those who don't don't punish them by sending to summer school because they can't grasp each concept in life. Aren't we all still learning? I consider myself successful and have managed just fine in life. I didn't do as well in school as I should have. Simply because I didn't get it and still until this day could careless what x+y=. We are making it too hard in school and not hard enough at home.

Apr 6, 2009 3:38 PM

Guest :

ah i hate this

Apr 28, 2009 3:27 PM

Guest :

Ridiculous! This is not easier than long division.

Apr 29, 2009 8:37 PM

Guest :

The argument in this discussion in fact demonstrates the value of offering more than one algorithm. Some say the partial method is easier, and others claim that it is more difficult. You disagree. This disagreement is evidence that we think differently just as our students do. If you teach both methods, without a biased motive behind the lesson, then you will find that partial quotients is easier for some, while the traditional method is easier for others. Students will have the opportunity to employ the method that works best for them. The way I see it, you have a win-win situation. Furthermore, since we use partial algorithms in advanced mathematics, and this method works for many in foundational mathematics then why not use it? It is not a matter of lazy or not lazy. Do you discourage students from cross canceling because they won't have to multiply the larger values? (And yes, it may be called 'canceling') Would you discourage students from graphing linear functions by recognizing the slope and y-intercept and insist that they use a table of values even after they have learned to use y=mx+b? It is a matter of what makes sense to students. I teach pre-algebra through Precalculus in high school. Students come to me as 14 year olds, still unable to perform operations on fractions.(Criminal yes, but a fact.) I show them other methods and let them choose which one works best. For the first time, they get it, and they find a new sense of confidence. This attitude about it being lazy brings to mind the old adage 'when I was your age I had to walk to school barefoot in the snow.' Looking for alternate routes to the same destination is essential to innovation and invention.

Jun 3, 2009 12:21 PM

Guest :

Wow. That was a lot to read, but I'm glad I put in the time to get through it all. I want to thank you to everyone for their comments, even the ones I don't agree with, because the previous comment is right. This board is a perfect example of why we need to teach alternative algorithms to students: we think differently.

As for the Traditional Method being the best ... well let's just say I find it funny that it's called "traditional" in the first place. Long division was put in place as a short cut for other algorithms to be used by mathematicians and math students who already knew what they were doing. You didn't treat hundreds like ones or magically "bring down the 5." Instead there was math happening with real numbers that could be represented using base ten pieces, blocks, beads, etc. There was also a much more concrete understanding of place value.

Did I learn fine with long division, standard multiplication, and every other "traditional" algorithm taught in schools? Sure. I'm a math teacher myself. But I wish alternatives had been provided when I was learning how to do this stuff because I probably wouldn't have hated math at the time.

Jun 8, 2009 8:42 AM

Guest :

How would you use this method to find repeating decimals??? The reason we need students to use the standard method of long division is because it can be expanded upon in higher grades. I wish elementary educators could think past the 5th grade and realize that students NEED to learn the standard methods in order to continue in higher math.

Jun 17, 2009 5:20 AM

Greg Cruey :

Greg Cruey:
Dear June 8th Guest, Children in the elementary grades often work with remainders instead of decimals. If the problem in the example had been 2082 divided by 9, the remainder would be three. Nine would have gone into three 0.3 times (if we were working with students proficient in the concept of decimals), the remainder would have been 0.3, and nine would have gone into that 0.03 times, with a remainder of, well, you get the idea. Decimals are a relatively new skill for most third and fourth graders. But if they know what decimals are, the algorithm still works fine. THAT SAID, no one is arguing that we should teach this method EXCLUSIVELY. We begin in kindergarten or first grade having kids divide by giving kids nine counting bears and telling them to make sure all four of their group members get the same number of counting bears. We progress to something like this partial quotient method. And before they get to you they learn the traditional method - and maybe one or two other approaches to division. Our goal is NOT for the kids to be able to work a formula. Our goal is for them to have a cognitive grasp of the concept of division that allows them to work word problems and apply the idea in real life situations. If we do our job under this curriculum, they know quite well how to use the tradition (or standard) method you're concerned about. Plus they can think... - greg

Sep 21, 2009 11:44 AM

Guest :

I'm a SAHM, so I'm not a math "expert"; however, I found math fun and exciting when I was a kid. I was fortunate to understand math, so division was never a problem for me. I personally find the partial quotient algorithm a confusing way to learn division. Yet, I also realize that not everyone learns in the same way or at the same rate. My childrens' elementary school has moved to the "Everyday Math" curriculum this fall, and I'm finding a wide variety of ways the teachers are teaching addition, subtraction, multiplication and division. My son is in fifth grade, so he learned 'longer' division last year (single digit divisor, multiple digit dividend). I spent a fair amount of time with him on the 'longer' division unit because he and his teacher just didn't click on this unit. (This is not a slam on his teacher; just a fact that he was not understanding the methods she used to teach this specific unit). He understands division using the "traditional" method of division. I am concerned that the partial quotient method will only confuse him and make math the 'dreaded' subject, again. From what I've seen on this curriculum, the kids are tested on the various methods of division. I'm curious why there is a desire to have the kids 'know' the method (become proficient) in the method if the method doesn't work for them? I'm not against teaching a different method. I applaud the teachers for using a variety of methods to get kids to understand math problems. However, if this is about teaching them different alternatives, why the emphasis on tests of 'knowing' all the concepts?
As I was reading the example of the partial quotient algorithm above, I was amazed/dismayed/stunned/all of the above to read, "Let's assume this student is not very good with "nines" yet and decides to just take 10 "nines" out." (Step 4) Why would a student NOT be profecient with their 'nines'? Are students NOT proficient in multiplication before they are taught 'longer' division? I cannot imagine why a student would not be proficient with their nine multiplication facts before they are doing longer division?
Is this because not enough time is spent on multiplication at school? Should someone be spending more time with this child on the multiplication facts? I don't expect a response to all these questions. But from the outside looking in (into the educational world), it appears we have bigger problems that what methods to be utilizing when teaching division.

Sep 21, 2009 7:19 PM

Greg Cruey :

Hi Stay at Home Mom, Not everyone in a class learns at the same speed or in the same way. The trend in curriculum today is to spiral through concepts - teach something, watch some kids get it, move on to something else, teach that, and come back later (maybe weeks or months later) to the first topic. For some kids it's a developmental issue. They haven't developed abstract thinking skills yet and some of their classmates have.

If we waited until EVERYONE had mastered "nines" before we so much as introduced double digit division, we might never teach it...

No, not all students are completely proficient i multiplication before division gets introduced. Everyday Math tends to see the two skills as complimentary and doesn't put one much before the other. - greg

Sep 24, 2009 7:23 AM

Guest :

Dear Greg,

Ok, ok...maybe proficient wasn't the correct word to use. Students should be more than "exposed", so maybe they should have a working knowledge of their multiplication facts. I guess my point is that regardless of what curriculum is being taught, or what methods are being utilized, a fundamental knowledge of your basic math facts is necessary. If a child doesn't know that 3+4=7, then it doesn't matter if you teach the traditional method of addition or the partial sums method, neither method is going to make much sense if they don't understand the basics. Even if a student is weak with their math facts, (they don't recall facts instantly), that student with a little bit of time should be able to recall their math facts.

Without a solid foundation, a house falls down. Without a solid foundation in math (i.e. knowing your math facts), it doesn't matter what method of addition/subtraction/multiplication/division is taught. Math is much more difficult.

Sincerely, Your SAHM

P.S. By the way, if spiraling is the new way to teach concepts, why wouldn't the teacher show the student that 9*10=90, 9*20=180; 9*30=270--so take that way from 279. The multiples of 10 concept is an everyday math concept....why not utilize it here??

Oct 1, 2009 5:28 PM

Guest :

the problem I see is that you still have to teach traditional division once the student begins learning decimals. I'll agree that this is helpful to get you started, but ultimately, they still need to know long division.

Oct 18, 2009 8:17 PM

Guest :

As an educator I am appalled at the ridiculous/ignorant comments that some guests have left. I pray that few of them are still in the classroom, or better yet, I pray that my children never set foot in their classrooms!

When I first began teaching, I taught only the traditional method of long division. I dreaded it every year. My students didn't know their facts coming into 5th grade, and I had state standards that required me to teach students to divide 4 digits by 2 digits. How in the world was I to do that when they could barely recall multiplication facts (especially the 6, 7, 8, and 9 facts)?

About 5 years ago I was ready to give up. I hit the internet searching for a better way to help my students and came across Partial Quotients. What a relief! Since then, I have taught BOTH algorithms. My students who are ready to embrace the traditional method do just that. However, my students who aren't ready are still given a chance to be successful. Just last week, I had a student who said, "I can't divide." He was nearly in tears at the thought of starting division and shut down...until I started dividing using P.Q. By the end of a 45 minute class, he was dividing independently...and understanding what he was doing!! It frustrates me to read all of your comments that slam P.Q. when Greg has said over and over that it is not the only method you teach.

I am a firm believer that our entire approach to teaching math is wrong. Memorization isn't working. I have students who come to me and share some little riddle they learned to help them remember a fact, but have no earthly idea what it means.

Elementary teachers do lay the foundation - in concrete concepts. It is the teachers who move straight to the abstract and require students to memorize facts without the concrete representation that fail our students. It is those teachers who are probably the ones complaining about P.Q. and the other algorithms of EDM. P.Q. paints a much better picture of what is happening in division. And, to the SAHM, eventually the students do begin to use their extended facts - when they are ready.

Sounds like many of you need to spend a day or two in a classroom where the majority of the children come from undereducated, impoverished homes and see how far you get with your "dead men serve burnt donuts" or whatever trick you use to help the kids memorize one more thing!

Oct 18, 2009 8:25 PM

Guest :

As an educator I am appalled at the ridiculous/ignorant comments that some guests have left. I pray that few of them are still in the classroom, or better yet, I pray that my children never set foot in their classrooms! When I first began teaching, I taught only the traditional method of long division. I dreaded it every year. My students didn't know their facts coming into 5th grade, and I had state standards that required me to teach students to divide 4 digits by 2 digits. How in the world was I to do that when they could barely recall multiplication facts (especially the 6, 7, 8, and 9 facts)? About 5 years ago I was ready to give up. I hit the internet searching for a better way to help my students and came across Partial Quotients. What a relief! Since then, I have taught BOTH algorithms. My students who are ready to embrace the traditional method do just that. However, my students who aren't ready are still given a chance to be successful. Just last week, I had a student who said, "I can't divide." He was nearly in tears at the thought of starting division and shut down...until I started dividing using P.Q. By the end of a 45 minute class, he was dividing independently...and understanding what he was doing!! It frustrates me to read all of your comments that slam P.Q. when Greg has said over and over that it is not the only method you teach. I am a firm believer that our entire approach to teaching math is wrong. Memorization isn't working. I have students who come to me and share some little riddle they learned to help them remember a fact, but have no earthly idea what it means. Elementary teachers do lay the foundation - in concrete concepts. It is the teachers who move straight to the abstract and require students to memorize facts without the concrete representation that fail our students. It is those teachers who are probably the ones complaining about P.Q. and the other algorithms of EDM. P.Q. paints a much better picture of what is happening in division. And, to the SAHM, eventually the students do begin to use their extended facts - when they are ready. Sounds like many of you need to spend a day or two in a classroom where the majority of the children come from undereducated, impoverished homes and see how far you get with your "dead men serve burnt donuts" or whatever trick you use to help the kids memorize one more thing!

Oct 21, 2009 1:44 PM

Guest :

I am a high school math teacher and often see a student reach for a calculator for a problem like 18/1. It is so frustrating to see students that have no clue about what division means. I was trying to find on-line poster that explain this process and other algorithms that I do in my head faster than students using their calculators. What I would ask elementry math teachers is why would you just teach one way of doing things if the students in two years will just reach for a calculator because they cannot do the problem mentally. It would be equivalent to showing students how to do synthetic division because it is faster without teaching polynomial division. I think the big thing that critics of this method need to realize is: 1) It works 2) It is the same as the traditional method if your math skills are good enough to pick the biggest chunk possible.

Jan 3, 2010 7:29 PM

Guest :

Thanks. I'm and engineer with a son with learning disabilities. Specifically he has a problem with recall of math facts. I have been searching for a way to teach him division that is intuitive and builds on understanding the concept rather than memorizing steps to solving it. This also has the benefit of not relying on math facts. I don't get why people would call this lazy.

Jan 12, 2010 4:59 PM

Guest :

That way is the best

Jan 20, 2010 3:43 PM

Guest :

I agree, the goal should be to get the right answer. Some schools actually discourage this method too.

Jan 25, 2010 2:01 PM

Guest :

This method is new to me. As a parent, I had to look up the method on how to do it so that I could support my child. I wouldn't call this the "lazy method" as others have said. The way it's described it's like making a bunch of guesses towards the answer and then adding up the guesses. To me long division makes perfect sense, or even short division as another poster offered.

To me, the frustrating thing is to have to learn all these new methods of mathematics when I learned just one method. Partial sums, partial products, lattice method, etc. I think all these different methods over-complicate things.

Jan 25, 2010 2:29 PM

Greg Cruey :

Hi Jan 25, The thing to consider is this. We grew up in a time when learning the method was sufficient.. Today, the reality is that kids need to graduate with an ability to think through any problem (math or otherwise) from a variety of angles, because that's what today's workplace requires - thinking, and problem solving. It's not JUST about getting the right answer to a division problem. If it were just that, we could hand out calculators...

Jan 26, 2010 2:08 PM

Guest :

confusing , time consuming but challenging.

Jan 29, 2010 2:53 AM

Guest :

Big applause to Greg Cruey. I was searching the web for the same exact method. I remember a classmate from Jamaica showed me this method during our College Calculus class years ago and I was totally DUMBFOUNDED!! I am also from Jamaica but left too early during elementary school to be taught this method. This is the absolutely the best way to teach division because it reinforces what division really is SPEED SUBTRACTION on the number line back to zero. Clearly those who consider it lazy DON'T understand math. And if you do you SHOULD NOT be teaching. Math is a concept and not everyone can learn the concept the same way. For anyone to chastise a teacher who goes out of their way to bring alternative AND effective methods to a classroom and then turn around and call a 'right brain' orientated student lazy because they learn different has no place in a classroom.

Jan 30, 2010 11:15 AM

Guest :

I think it is great that someone is willing to help teach a variety of methods and feel that the comments here could stop people from helping find alternative ways to help our children. I appreciate Greg that info is out there and sorry you have to take abusive rude comments. People if you are not willing to say something on a megaphone in front of a group of your peers don't say it. If you don't like it don't use it.
As for me, struggling child who normally gets A's, I'm thankful for a fresh perspective to see if it may work for my child.
Of course most who wrote nasy things probably will never care to read what others say. Most only wish to get there own witty opinion out there and that is it.
So thanks Greg.

Feb 9, 2010 7:17 PM

Guest :

First I would like to thank you for this article, I have several students that never grasped division and this will give me another resource for helping them. My question is, what do you do when you end with a remainder?

Mar 3, 2010 6:58 AM

Guest :

I find this discussion about "lazy" fairly absurd. I am an intervention specialist. Sometimes a person will remark that something I'm doing with a student is "babyish". Good teachers find the way that works for each student. It is not a matter of laziness or maturity; teaching in the way that facilitates learning is the right way to teach. Students who have one success are more likely to work harder next time. Teach, don't moralize.

Mar 3, 2010 4:38 PM

Guest :

the article is informative; the method is ridiculous - just another excuse for students not to learn their multiplication tables or place value. traditional long division makes perfect sense when the teacher includes place value concepts.

Mar 3, 2010 4:41 PM

Guest :

Mar 3, 2010 4:43 PM

Guest :

The article is informative but the method is ridiculous; just another excuse for students not to learn their multiplication tables or place value concepts. Traditional long division is efficient when taught using place value concepts. Partial quotients could take students a very long time to complete and then they have to add all of the partial quotients together to get an answer.

Mar 10, 2010 3:48 AM

Guest :

Pretty sad when we resort to guesswork, which is the basis of partial quotients. The kids do not find it any easier than long division and actually are able to use the old method in about five minutes when I explain it to them in a manner that they can VISUALIZE. With partial quotients they seem to throw out any random numbers. Example "what is 27 /3?" "24" "no" "30" "no" "ohhhhh 14". I swear I have that sort of conversation in every elementary class I substitute in. When I help kids and give them the traditional method they say "OHHHH! Now I get it!"

Mar 10, 2010 4:17 AM

Greg Cruey :

Hi March 10th,

What you want to call guesswork here, most reasonable people woudl call logic and reason. My view is that from teaching the memorization of forumlas that give a kid an answer through a process he doesn't undertand we've progressed to using logic and reason to obtain an answer that is just as accurate and that the child better understands....

Apr 22, 2010 7:48 AM

Guest :

Well Greg I am in a Concepts Math course needed so I can become a teacher and between the method listed above and another method called Double- Division I think both ways are a good way to teach division and they are likely to get the answer correct more often than the traditional way that we were all taught to divide. Yes, we should still teach them the traditional way but if teachers can fit the other 2 methods into their schedules I think it will benefit all children in the long run.

Aug 30, 2010 3:41 PM

Guest :

Ok, how about this. My daughter is in 5th grade and is doing partial quotients. She has a problem 93 divided by 10. Her teacher didn't show the class how double digits were done...only single digets divided by double. We are banging our heads on how to show the child how to do this. I have an idea somewhat but I am not sure. How do we do this?

Aug 30, 2010 3:51 PM

Greg Cruey :

There is no single correct way. Part of the point of partial quotient is to let a child use numbers that can handle. You right down 93. If you're child knows that five 10's is fifty, subtract 50 from 93 and make a not to the side that you took 10 out five times. Do the subtraction and you've got 43 left. The hope is that the child will draw the conclusion themself that they can't take five 10's out again. Maybe they'll try four 10's; maybe they'll take 10's out one at a time. Eventually they get to a remainder fo 3 and decide that they can't take 10 out anymore. So they count of their tally in the margin to see how many times they took 10 out of 93 and they note that they had three left over.

The process is no different for single digit divisors and double digit divisors.

Sep 9, 2010 2:23 PM

Guest :

this is the dumbest thing ive ever seen or tried to explain to my child!!!!!!!!! Would never require my kid to dumb themselves down from only having to know division of the number 100 and 10 for double digit multipication. PLEASE DONT DUMB THE WORLD DOWN TO TRY TO MAKE EVERYONE HAVE FAULSE HOPE!!!!!

Sep 9, 2010 3:01 PM

Greg Cruey :

Hi Sept 9 Guest,

Sorry, but I don't see this as dumbing down anything. Here's why. The traditional method of long division is mostly about rote memorization. In contrast, the partial quotient approach does two things. First, it makes it easier for students who (for whatever reason) have a hard time memorizing math facts to still get a right answer. Second, it makes the process of division more rational - in the sense that students THINK about the process instead of simply applying a memorized method. Anytime I can get students to THINK about math, that's not dumbing it down.

Sep 28, 2010 6:17 PM

Guest :

I have no problem with alternative methods of learning but in this example long division takes three steps; the partial quotient method takes five. How is that simpler or less confusing?

Oct 4, 2010 8:16 AM

Guest :

I just taught my fourth grade class this method because the Florida standards have changed and now state that there will be no need for long division but the students may be assessed to divide 3 digits by 1 digit. I want them to be ready for anything and tried giving this method a shot to further enhance their understanding of division. We solved 820 / 3 using this method and it was extremely tedious for all of us. I'm now thinking that I will give them problems in this method that can be solved in a reasonable number of steps and will then go ahead and teach them long division so they are ready for anything the FCAT may throw at them. What do you think? Would anyone suggest any other methods? I miss long division.

Jan 4, 2011 7:40 PM

Guest :

Kids need to know their multiplication facts thoroughly or they will constantly get wrong answers with any division algorithm. Estimation is also an essential skill, probably more essential than long division. Let's face it, someone in the real world, at work, will use a calculator to do a hard l-o-n-g division problem. But for quick thinking on the fly or to verify the calculator's answer is in the ball park they'll have to estimate. So to say that the partial quotient method allows you to divide with limited knowledge of multiplication facts and estimation is like saying you can learn division without learning what its components are. Hey, wake up, math is sequential! What I find most frustrating is my daughter's school district where in grade 4 they insist kids use the partial quotient method and by jr. high they insist everyone use the standard algorithm and act like you're a dumb redneck if you tell them it's confusing the hell out of your kid.

Jan 8, 2011 7:45 AM

Guest :

Even though this method is different than what I learned when I grew up I don't understand the lazy comments throughout this string of messages. Children learn in different ways so a new way can be helpful to many.

From reading the comments below one can see many different types of people are reading this - some open minded, some incredibly small minded. The high level of ignorance in this country really scares me.

Thank you to the teachers out there who are open to new methods of teaching and who try to help all their students.

Mar 5, 2011 12:18 PM

Guest :

When a student picks up a pencil and paper to work out a problem, regardless of how they are doing it they are practicing their math skills. If a student thinks this is fun and wants to do more math problems they are learning to see the math patterns that they will need in the future. No one learned to drive by getting behind the wheel of a car, in everything we do to learn a skill we need to practice. When a student asks for more problems they are practicing a skill. If a student finds that this method takes to long to complete they will find ON THEIR OWN that if they combine and use higher numbers they will get the answers faster. They will not lose interest in division as some students do with the traditional methods

Apr 5, 2011 7:38 PM

Guest :

It reflects the general misunderstanding of what education is about especially in the Maths area. We try and help the children understand what they are doing so they don't rely on rote they can take the problems apart and put them together. This is what made the West so innovative in the science and maths areas we develop that kind of thinking. We have been doing it since the introduction of the Hindu Arabic numbers. It doesn't work for everyone I remember a survey done in 1975 that showed only 25% of the population could do long division and only 75% could do subtraction involving decimals. So much for the glories of the past we always hear about but shouldn't believe. There have been only marginal improvements but they come form developing understanding not from learning meaningless chants and processes. We switched to decomposition subtraction from equal addition after the survey (Victoria - Australia) because very few people (including teachers) understood how equal addition worked, it was never explained to the children properly, so they never understood. And by the way we don't lag behind in Maths and neither do the European Countries that share the approach, keep pushing the boundaries Greg make them think and understand.

Apr 27, 2011 12:02 PM

Guest :

If i can ever work out the kinks consistently (I always end up dropping a 10 somewhere) I will teach my kids this method. The traditional Divide, multiply, subtract, bring down, do it again, doesn't click with all children. Sometimes they just need to have 'another' way to do it. It's not lazy, just special, which is what the children in special ed need. Thanks. It's only lazy if you don't think there's more than one way to skin a cat.

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