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September 16, 2009
New York, N.Y.
I don't often identify with characters in novels, movies, or television programs, but when I do, it's probably not who you'd expect. Recently I've found myself feeling some affinity with Sally Draper on Mad Men. It might seem odd for me to "identify" with an 8year old girl, but when you consider that the year is 1963, it makes a little more sense: Sally Draper is approximately my age. In recent episodes I've become more interested in how Sally views the world of 1963, and more importantly, how the 1963 world views her.
Some younger fans of Mad Men may have been surprised when Sally Draper wasn't allowed to attend her grandfather's funeral, particularly considering how close they had become in the weeks before he suddenly died. It's even possible Sally didn't know that the funeral was taking place, or even that there was such a thing as a funeral. I suspect this was a common practice with children in that era. When I was six weeks shy of my 8th birthday, I didn't attend my father's funeral, and I'm pretty sure I wasn't given any choice in the matter, or even that I knew that it was happening.
As far as I've been able to determine, Mad Men is impeccably researched. Whenever something has appeared in the program that I've questioned, I've later discovered that they got it right. (For example, Jackie Kennedy's famous TV tour of the White House really was broadcast on Valentine's Day.)
But in last Sunday's episode, Don and Betty Draper are having a little talk in a classroom with Sally's teacher, and I was startled to see on the blackboard several rather gory exercises in long division — but not the way I learned long division!
The long division shown on the blackboard in Mad Men looked like this (except adorned with several downward pointing arrows):
This is called — I didn't know these terms until I did a little Internet research this week — the Column Division method. For each digit of the quotient you need to mentally find the largest number that when multiplied by the divisor won't exceed the remaining dividend.
In a public school in New Jersey I learned a technique of long division called the Partial Quotients method:
Each of the numbers to the right of the vertical line is multiplied by the divisor and then subtracted from the remaining dividend. The advantage of Partial Quotients is that you don't have to nail each digit exactly. You can build up to it because you're accumulating a list of numbers that sum to the total quotient.
I remember my mother attending a parentteacher evening where my teacher demonstrated this method to the parents as the latest and greatest way for kids to do long division. I know this because she later asked me about it, and then showed me the way she learned long division, and since that time I guess I assumed that the method I learned was ubiquitous in the early 60s — at least in schools in the northeast.
But I'm probably wrong about that. I suspect that different schools and different teachers taught long division differently, and it appears as if the Partial Quotients method is still sometimes taught, although I hope that Column Division is then introduced as an alternative, and Short Division as an option when the divisor is small. (Obviously, the more of the division process that is moved to the brain, the smarter the student becomes.)
Still, I think Matthew Weiner missed an opportunity here: A blackboard full of exercises in Partial Quotient long division might have seemed more unusual and "exotic" — and emphasized once again just how long ago 1963 really was.
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(c) Copyright Charles Petzold
www.charlespetzold.com
Comments:
I had never seen Partial Quotients until this post (product of early 80's grade school), and now I admit it makes me uncomfortable to look at. However, the thought process is so different from Column Division I think there is something to be gained in learning the process.
— Michael C. Neel, Wed, 16 Sep 2009 11:48:37 0400 (EDT)
Charles while we are about the same age I did learn long division by the Column Division way. At the time I can still remember everyone talking about "New Math" at the time I did not know what that meant. Now I wonder if this was part of the new system.
— Mike Sinnott, Wed, 16 Sep 2009 12:06:19 0400 (EDT)
I learned using Column Division, and I'm a bit younger than Charles. (I'm 32.) The Partial Quotients method took me a second to understand, but I like it  I could do long division that way.
— Brian Schkerke, Wed, 16 Sep 2009 13:51:10 0400 (EDT)
I'm a few years younger, but also went to school in NJ. I learned the Column Division method.
However, I don't see Partial Quotients being all that different from Column Division. If you guess right, the only real difference in the process is where you write the numbers.
— James Curran, Wed, 16 Sep 2009 13:53:54 0400 (EDT)
You're right it's not all that different, but I think the multiplication of the divisor by whole numbers makes the whole process somewhat clearer to the beginner. — Charles
My girlfriend was determining how much dinner would cost our group of friends a few months ago using this method. At first I thought it was incredible how bad she was at long division, but then she got the right answer using this method. I was glad I kept to myself, haha.
Thanks for explaining the method  I hadn't found the time to delve into it since I first saw it.
— Steve Adams, Wed, 16 Sep 2009 14:28:51 0400 (EDT)
It does not seem as though Partial Quotients can be extended beyond the decimal point. The thought process behind dividing whole numbers has to change when you get to fractional numbers. Also, there's just not room for it unless you know ahead of time how many decimal places you want in your final result.
— Chris, Wed, 16 Sep 2009 16:12:04 0400 (EDT)
I was born in 1955 and was taught the partial quotients method. (west coast Canada) Like Charles, my parents (father in my case) took one look and taught me the column method which I have used to this day. They tried to teach my daughter something even more convoluted (circa 1997) with even more emphasis on guessing so I maintained tradition and passed on the column method which she uses to this day.
— Brian Howden, Wed, 16 Sep 2009 17:05:15 0400 (EDT)
I was taught the Column Division method, without the minus signs, about 75 years ago  it hasn't changed much.
I had not seen the Partial Quotients method but if I had to teach long division I would start with the Partial Quotients method.
— Dave, Wed, 16 Sep 2009 19:26:06 0400 (EDT)
The first computer that I ever programmed used a variation of the Partial Quotients method. Each partial quotient was a power of 10, i.e. the leading digit was always a 1. That model might have been discontinued by 1963 though.
— Aunt Eake, Wed, 16 Sep 2009 19:40:15 0400 (EDT)
I was born in 1983, was a math major at UC Santa Barbara, and I've now been teaching high school math for 3 years.
Column division is the only method my students are familiar with, and the only one I used when I was in school.
I had not seen partial quotients until this post! It shares some methodology with the way I mentally divide numbers now, though. Now I'm frequently factoring both the dividend and the divisor and cancel out common factors. If this proves to be difficult with the numbers involved then I'll go for a calculator ;)
— Scott Farrar, Wed, 16 Sep 2009 21:29:39 0400 (EDT)
I'm 47 and was taught the Column Division. My children 13 and 16 were taught Partial Quotients along with a other methods as part of the "Everyday Math" program. At first I resisted but soon found the Partial Quotients easier to explain and help them with.
My kids were taught to build the Partials in any order they felt comfortable with, so in your example they could have chosen 2 first and then 20 next.
They "Everyday Math" program is very interesting since it gives the students alternative ways to both division and multiplication.
— bill baker, Wed, 16 Sep 2009 22:05:42 0400 (EDT)
I'm currently a Junior in a High School in Illinois. I learned longdivision with the Column method. Both are strikingly similar. With very little modification, you could even create a hybrid of the two methods that you are most comfortable with... Or you could just use one of the methods provided. It was interesting, though, that I haven't ever seen or heard of the Partial Quotients method. Thanks!
— Joe Burke, Wed, 16 Sep 2009 23:25:29 0400 (EDT)
I vaguely recall learning a similar method for dividing polynomials, but a quick skim through google came up empty.
This also reminds me of the way I was taught to convert from decimal to hex or binary and back again.
— Troy, Thu, 17 Sep 2009 01:15:25 0400 (EDT)
I'm 44 and learned the column method in Janesville, WI. Never saw the partial quotients method until today. My sister, younger by 5 years, learned a bunch of stuff that made no sense to me at the time, so maybe it was that.
— D, Thu, 17 Sep 2009 09:27:52 0400 (EDT)
I've been a Maths Teacher in the UK for the past 16 years. As a pupil I learnt what you refer to as the Column Division method. The other method, known over here as Repeated Subtraction Method (as opposed to the Partial Quotient Method), is meant to be taught initially in Primary school to reinforce the idea of subtraction linking into division, then as the student develops mathematically they can learn the Column Division as a more efficient method of dividing. There are similar methods of repeated addition which link into long multiplication (although we have a number of long multiplication methods between the repeated addition method and the traditional column method).
— Dave Ames, Thu, 17 Sep 2009 15:57:18 0400 (EDT)
I'm not sure it really matters how much of the division process the person /understands/. Both the systems described here are intended to deskill the job and remove the need for understanding, which was a laudable goal in the time before electronic calculators became ubiquitous.
As a child I was very late in learning long division, partly because different people had attempted to teach me their own favourite variants of what I now know is called column division. None of these people understood what they were doing, so all they could transmit to me was a set of rules to follow.
I'm OK these days. Some years ago I had to write "assembler code" to persuade a fairly primitive processor to do binary long division. Once I'd got to the point of realising that "shift all the ones and zeros one place to the right" means the same thing as "divide by two" I started to get the point of the whole thing.
I hope today's children, freed from the drudgery of needing to do long division "by hand" will have the chance to focus on why these systems work.
— Dominic Cronin, Thu, 17 Sep 2009 16:52:05 0400 (EDT)
Hi guy!
I'm 51, and I was taught the CDM just like Sally. I do however remember teachers briefly demonstrating alternate methods later on, including your Partial Quotients method.
— Eric Maffei, Thu, 17 Sep 2009 19:04:23 0400 (EDT)
Another 51 year old here. I was taught CDM as the only way to do long division in the Northern Indiana school system.
— Dave Quick, Thu, 17 Sep 2009 22:07:24 0400 (EDT)
I learned the Column Division method. I was born in 1956, and grew up in Syracuse, NY. I never heard of the Partial Quotients method until I read your post.
— Sheryl Canter, Sun, 20 Sep 2009 23:34:56 0400 (EDT)
Thanks for post, realy like'd it.
— JJ, Mon, 21 Sep 2009 10:31:23 0400 (EDT)
I was born in 1963 and went to a parochial school in CT. We learnd CDM, but like others heard about the "new math" rumblings. And like others, I never heard of PQM until this post. In fact, I never heard of the show MadMen. Going to have to look that up, although I don't need another reminder of how long ago 1963 was... :) Thanks for the great post!
— Jeff, Mon, 21 Sep 2009 12:36:24 0400 (EDT)
I learned the Column Division method in private school in Texas during the 1970s.
The Partial Quotient method is how I actually do division in my head without a calculator. But I didn't know it was a "method" people were actually taught. I thought I invented it!
— Matthew J Sullivan, Mon, 21 Sep 2009 16:26:48 0400 (EDT)
I remember being taught something similar in grade school in Miami, Florida in the early 70's and getting into trouble because my father had already taught me division. I don't remember what it was called, but because short division and the column method of long division don't "show all your work" I would actually have my answers marked wrong and couldn't really understand why. Fortunately, later that school year we transitioned to the column method.
But reading your blog has brought back bad memories. Not too awful, but you're right, a large portion of the audience, especially Baby Boomers, can relate more to Sally & Bobby than to the adults. Even the oldest of the Boomers, those born in 1946 would only be 17 in 1963.
As to the usefulness of the PQM past the decimal point, it wasn't an issue. When long division is (was) first taught a number smaller than the divisor was represented as a remainder, not as a fraction or decimal.
— Ileana, Thu, 24 Sep 2009 23:41:30 0400 (EDT)
I am teaching a community college course of mathematics for elementary school teachers, and will be providing them with many different approaches to teaching arithmetic basics including long and short division and the partial products method
Also, Mad Men is an awesome show. They nail every detail
— Rebecca, Sun, 11 Oct 2009 15:49:50 0400
It's the way most kids are taught division in places like Sylvan and other tutoring programs. Kids don't have to be able to multiply numbers by more than two or three in order to do it so those who can't/won't memorize the multiplication tables can be successful in their division. Sometimes it's called the 7 method because the divisor sign and the long tail look like a seven:






Obviously the better one knows ones multiplication tables, the faster one can do division. That's known as the carrot approach to getting kids to memorize their multiplication tables.
— ValorieO, Tue, 10 Nov 2009 14:45:15 0500
I teach PQM to my 4th graders and love it. It just makes more sense!!
— Amy, Thu, 7 Jan 2010 16:06:14 0500
My 4th grade daughter first brought home lattice multiplication & then the partial quotient division homework. She hadn't even learned all her multiplication tables yet. And I was clueless on how to help her with it. My fustration increased when talking with her teacher I was told; it's the concept that matters not the actual mastery. I have now checked out enough online sites to figure it out so I can help her. But still have doubts about how this can be built on as a foundation for more complex math & what to do with decimals. Now that my daughter has learned these tecniques; she totally shuts down to me trying to teach her traditional methods. But on a lighter note love the fashion on Mad Men.
— chris, Thu, 18 Feb 2010 14:20:51 0500
I find the Partial Quotient Method more imperfect in instructing mathemathics to children. It may have value at a point where a child is learning division. This type of learning if continued can lead to additional mathematical issues for the student. As a math major, I learned by multiplication tables and was able to quickly divide multiple digit number in my head. Without a firm base a student stands no chance and the PQM does not give it. It takes the easy way out.
— Mort, Thu, 25 Mar 2010 10:17:48 0400