Dignify the Remainder

Long division is by far the coldest, most frightening and sinister of the four arithmetical operations. It's the one that traditionally is the last one that students have to master, the final and highest hurdle in the traditional elementary school arithmetic program, It looks completely different from the other three algorithms as the line is on top of the numbers when you start and and not below them. It also works from left to right which, again is completely the opposite to the other three.

It also has its own language like "guzinta' in that  '5 guzinta 10 two times" as well as "bring down". And then there's that mysterious r3 or R3 or 3/5 or .6. It always seemed to me that the remainder was a nuisance because math is 'supposed' to be exact and tidy. There are no remainders in addition, subtraction or multiplication so why does there have to be one in division, I remember thinking when I was a fifth grader.  I can even remember using the remainder idea to decide which operation to use to solve a problem. For example, you always knew that if a word problem contained the numbers 4 and 11 it couldn't be a divison problem because there would be a remainder which would be untidy. Sadly, I suspect these disabling strategies are still taught in some places.

What we need to do is to dignify the remainder by giving the numbers in a problem referents. By giving a number a referent we turn it into a magnitude which enables us to know what to do with the remainder. Let's think about problems involving 4 and 11 as in;

Problem 1.  How many cars do we need to take 11 people to the movies assuming each car holds 4 people and the drivers are part of the group of 11 going to the movies. It would not be 2 r3 unless three people were left behind. It could also not be 2 3/4 or 2.75, for obvious reasons.
     
The answer would have to be rounded up to 4.

Problem 2  How much orange juice would there be in each of 4 glasses if you poured 11 pints of OJ 
equally into the 4 glasses. Again, it couldn't be 2 r3 but it could be 2 3/4 or 2.75.

What would the 2.75 refer to; glasses or pints of OJ?  It would have to be pints of OJ .

Giving numbers referents allows students to dignify the remainder so that they know what to do with it. It also helps demystify the concepts of division. Yes, there is a different division concepts in each of the two problems. The first is an example of the repeated subtraction where you know the size of each group (a 4-seater car) but not how many groups. The second is an example of the fair sharing concept when you know the number of groups (4 glasses) but not how much in each 'group' (glass). The first problem also involves a discrete variable (people) which requires the use of the term "many" while the second involves a continuous variable (OJ) which requires the use of the term 'much'..