I closed my recent post, So Long, Long Division, by asking you to either join me in acknowledging that long division is a meaningless math procedure or to share reasons students should learn long division that I had overlooked. And I appreciate all the responses from both sides of the debate--in the comments section of this blog and in discussion groups on LinkedIn (I’ve included links to these groups, but I think they’ll only work with a LinkedIn account open in another window). I’ll share and respond to some of those comments shortly, but first let’s lighten things up with this video that retired math teacher Danny Seaver shared on the Math, Math Education, Math Culture group on LinkedIn:
As for the debate, the most common content-focused rebuttal to my case against long division relates to the connection between long division and polynomial division, as expressed in a comment on this blog by Mark32092:
I'd be more open to this suggestion if someone had a way and could show me a better way to divide polynomials or do synthetic division that did not require the prerequisite skill using the standard division algorithm.
But Peter Smyth, a math teacher and administrator in South Carolina, downplayed the long division and polynomial division connection on the CCSS - Common Core State Standards Network group on LinkedIn:
I can't think of anytime I've done long division by hand in 30 years, nor has it helped me understand something. Nor have I found any use for long division with polynomials. My kids have done fine in AP Calculus.
I agree with Smyth. Yet even if we were to agree that long division with polynomials is an essential skill, would this justify the time and energy we (and students) spend on long division in elementary school? How often in algebra will students encounter coefficients larger than small two-digit numbers? What’s the benefit of having elementary students divide six-digit dividends using long division? Let’s help students acquire skills if and when they need them.
On the NCSM: Mathematics Education Leadership group on LinkedIn, John Madden, a college math teacher in Australia, wrote, “I like students to master long division, not because they can’t get through without it, but because it helps build persistence and a striving for precision which are vital components of resilient mathematicians.” I agree with Madden that persistence and precision are important, as did the Common Core authors, per Standards for Mathematical Practice #s 1 and 6. But does long division really build persistence or precision for most students? Not from what I’ve seen. Besides, there’s no shortage of rich, challenging problems through which we can cultivate persistence and precision in students.
Several more readers joined Peter Smyth and me in criticizing the procedural approach toward division and, more important, shared alternative approaches that help students understand division rather than just do division. In the Stem Connections for K-12 Education group on LinkedIn, University of Massachusetts Professor Emeritus of Mathematics Walter Rosenkrantz wrote, “The long division algorithm (LDA) is an example of what a colleague of mine referred to as ‘mindless, formal computation.’” He then wrote:
A true understanding of LDA requires an understanding of the "division algorithm" (DA) which is not to be confused with the LDA itself. A good discussion of the DA is given on page 35 of the book by Judith Sally and Paul Sally Jr., Integers, Fractions, and Arithmetic: A Guide for Teachers. The LDA is just an implementation of the DA, which asserts that if a and b are positive integers, with b < a, then a = bq + r, where q and r are unique non-negative integers, a is the dividend, b is the divisor, q is the quotient, and r is the remainder. An example from the Sallys' book: Divide a=39 by b=7. Using the division algorithm, we have 39=7x5+4. Here the dividend is 39, the divisor is 7, the quotient q=5 and the remainder r= 4 and 0<4<7
Walter Rosencrantz also mentioned that, on pages 161-178 of their book, the Sallys apply the DA to compute the decimal equivalent of a fraction a/b, and he concluded by writing, “Lessons learned: Teach the DA and derive the LDA from it.”
On the CCSS - Common Core State Standards Network group on LinkedIn, Georgia teacher Evelyn Hines wrote:
...we do teach students to make sense of the problem. For example, 3,657 divided by 35 - a student looks for a 'friendly number,' so he/she might say that 35 time 100 equals 3500 which leaves 657. Then, the student might say they know that 35 times 1 equals 35, so 35 times 10 equals 350, leaving 307. It goes on... With this method, students look at the whole number rather than separating it into parts, which fits nicely into the estimation of the ballpark idea of where they want to be. By the time they get to 5th grade, they can choose between this strategy or use the standard algorithm after they understand WHY the algorithm works.
Evelyn Hines also shared the link to an anti-long division blog post in which Owen Elton wrote, "... this archaic algorithm is completely and utterly obsolete. We have calculators built into our smartphones that will do the division for us. You might as well require all students to learn how to use a washboard and a mangle to do their laundry; at least in this case they might learn something useful about social history.” Like me, Elton suggests replacing long division with an estimation-based approach.
Finally, in a comment on this blog, Markov Chaney wrote:
I believe that division should start by having students do repeated subtraction of the divisor from the dividend... As students start to see the relationship among the constituent parts - divisor, dividend, quotient (and remainder, if there is one), and when they are "comfortable" with this approach, the dividend should be replace with a large enough number that students should start complaining that the process is time-consuming and boring. And that's where the teacher can ask if students have other ideas about how to find the quotient. Another look at how multiplication works and how it can be used to replace repeated addition calculations might be helpful here. Eventually, methods that employ partial quotients in a non-systematic manner can be introduced, with an emphasis on why students using different combinations of partial quotients can still all arrive at the same correct final result. With sufficient grounding and a careful examination of the role of place value in this process, it should then be possible to explore the standard algorithm with much deeper understanding as to what's going on, why it makes sense, what each step in the repeated cycle means, and why the final quotient + remainder, if any, makes sense as the answer to the initial question.
So, how has this discussion and debate affected me? Well, for one thing, I’ll be encouraging teachers and students to use the above alternatives to long division in addition to those I shared in my previous post on this topic. I also plan to look into how countries like Singapore and Japan “use long division to deepen understanding of the place value system,” as MrsMath mentioned in her comment on this blog. And I look forward to checking out Judith and Paul Sally’s book, Integers, Fractions, and Arithmetic: A Guide for Teachers. (Several years ago I attended a seminar for teachers that Paul Sally led at the University of Chicago, and it was outstanding. Sally passed away last year, but his work will definitely live on!)
One thing I won’t be doing is welcoming back long division. Instead, I’m with UK Math Consultant Ruth Sharpe who wrote on the NCSM: Mathematics Education Leadership group on LinkedIn that long division “does nothing for mathematical thinking AND has no discernible purpose except perhaps to be one of the nails (and quite a big one by all accounts) in the coffin of the enjoyment of mathematics.”