First off…if you’re an elementary school teacher and you’re not reading Joe Schwartz‘s blog, stop reading this, go subscribe and come back! Great stuff! It seems we are always diving into the same conversations with our teachers which is awesome! Last week, the beast known as long division reared its ugly head again as teachers scramble to prepare for state testing. A time when procedural understanding supersedes the conceptual! Joe’s latest post perfectly explains the type of numerical reasoning that should be taking place in our elementary schools when in comes to division (before, during and after testing season).

Language plays an enormous role in thinking conceptually about standard division algorithm. Most teachers and students are accustomed to saying *“GUZINTA”* which is hard to let go. Traditionally if we were to do a problem such as 583÷4, we might say “4 guzinta 5, one time.” Initially, this is mysterious to students. How can you just ignore the “83” and keep changing the problem? ~~Preferably we want~~ We NEED students to think of 583 as 5 hundreds, 8 tens, and 3 ones… **NOT AS INDEPENDENT DIGITS 5, 8, AND 3! **

This Kahn Academy video uses “guzinta” at least ten times and talks about bringing down another place value (ugh!). We all know the rule:

*Divide-Multiply-Subtract-Bring down-Repeat*(DMSBR)**D**oing-**M**y-**S**illy-**B**rainless-**R***ule*.

I came across this context from John Van de Walle last week and tried it with a 4th grade class on Monday. The context served the math beautifully because it contains the place-value underpinnings of the standard algorithm for division. My hope is that MS teachers jump on board this context.

A Context that Serves Long Division (here’s the task sheet we used)

- He started by drawing 5 circles and partitioning out the 5 pallets to each class and did the same with the cartons.
- He had 1 carton leftover which he decomposed into 10 boxes leaving him with 14 boxes.
- He gave 2 boxes to each of the 5 classes leaving him with 4 leftover boxes and he broke them down into individual donuts (40).
- He shared the 48 donuts with the 5 classes and each class got 9 (with 3 extra donuts)
- Each class received 1 pallet, 1 carton, 2 boxes, and 9 donuts

Base-ten blocks were made available to students but only a few chose to use them. (1000s block=pallet, 100s flat=carton, 10s rod=boxes, and 1s cube=a donut). To introduce the task we had to clarify some vocabulary terms but after that …students hit the ground running.

Whether it was at school or at home, a different student had already been exposed to traditional algorithm but it was 100% procedural. I asked him to make sense of the algorithm using the aforementioned context. My typing would not serve his thinking justice so please take a second and watch this video.

The context of the task asks that students disseminate the donuts in the most efficient way possible. The most efficient way is to keep as many pallets, cartons and boxes intact as possible. The underlying mathematics of this task mimics the standard algorithm and this student made that connection. I was dancing on the inside!!!!!!!

With CCSS, students are not expected to solve problems using the standard division algorithm until 6th grade (6.NS.2). We’re hoping that when a student from our school “guzinta” middle school they’ll conceptually understanding what they are doing!

Here is Joe Schwartz’s tape diagram lesson. It was so good that I had to type it up! 4th Grade-SCAFFOLDING DIVISION THROUGH STRIP MODEL DIAGRAMMING

Thanks Graham! My old supervisor talked about “guzinta” and called the long division bracket (or whatever it is) the “guzinta box”. Love the pallet problem. Also much the same could be accomplished dividing with money (thousand, hundred, ten, and one dollar bills, though there are no thousand dollar bills anymore). How much different does this problem look in the equal sharing context, where the shipment gets dropped off at, say, a high school, and each kid in the school gets 5 donuts, and we need to figure out how many kids there are? In this case every pallet, box, and carton has to be opened and donuts distributed individually. The algorithm is the same, but does it still mimic the underlying mathematics of the task? Does it matter?

I don’t think it matters at all Joe! Ideally we want every 4th and 5th grade student to explore division through equal sharing models like you mentioned in you blog… but unfortunately that’s not the case. I love the fact that students can partition the dividend into partial quotients, but in quantities that is manageable and makes sense to them. Whether students are modeling measurement or partitioning division the math is the same. What matters is that students are able to make mathematical connections and relationships across grade levels. Unfortunately many students will not be given the opportunity to explore partial quotients which leads to them never making sense of the traditional algorithm.