Partial quotient method of division: An alternate to traditional long division
If a typical 6th grade student is given the same simple fraction computation problem over and over over, one that only involves just a few simple steps, such as a simple fraction multiplication problem, eventually after a few trials most will remember the steps, be able to mirror them during direct teaching, do it on their own during guided practice, and are quickly ready for independent practice. Even if a typical student does not fully understand why each step is required, most can at least match, mirror, notice a pattern, and do a series of steps independently. Not my most at-risk class for which I am being challenged to teach 6th grade curriculum when only a few can perform 3rd grade division, even during guided practice. Independent practice? What a joke! Solving word problems? Heartburn!
Even though the steps are posted, with each step a different color, the ability of most of these students to follow a simple multi-step procedure breaks down after about the second or third step, leaving many of these students smiling obtusely, sitting like bumps on a log. After multiple repetitions of the same problems, the majority can at lest perform steps one through four and are able to get to the fifth step when they must change an improper fraction to a mixed number or reduce a fraction to simplest form, a step that involves division. I give students credit for 80% of an "exit ticket" problem. I give students unlimited opportunities to do retakes throughout a quarter, but only a few take advantage of my unlimited opportunities policy. Only a few care enough about their learning to come for help after school or during lunch, so I am constantly trying to chase squirrely students down. The lack of independent learning skills, lack of stamina, and overall listlessness is maddening! 6th grade math is a life skill, but these kids in self-contained classrooms just don't seem to get it!
Long division has been problematic largely because most of these students do not know their multiplication facts. Most have no automaticity with their addition and subtraction fact families to 20, which makes computation without a calculator virtually unbearable. The root of the problem with division seems to be a general inability to follow multi-step procedures, even if they sort of get the concept of equal grouping. I noticed Lena getting stumped by the concept of how many 7's go into 63, guessing 6 instead of 9, looking puzzled as usual. After observing Grant successfully solve a fraction multiplication problem using what was a combination of guessing and repeated subtraction, it suddenly dawned on me that my "division problem" is not going away unless something changes drastically. Division is becoming a black hole, a destroyer of motivation for frustrated learners.
Time to try an alternative strategy. What I like about the partial quotients strategy is the way it allows students to chunk a larger number down even if they cannot quickly find appropriate multiples of a divisor to find "the right" quotient. Close can be good enough, Even not close can be good enough. All students need to remember is that, whatever multiple of the divisor they choose, the product cannot be larger than larger than what remains of the dividend, (or the amount that is being divided). Having studied partial quotients for a few hours, having found a few well-written notes, and having found a nice video courtesy of Khan Academy, I am ready to teach the partial quotients strategy, and praying that this strategy will prove to be a magic bullet!
On Monday, even if students can take advantage of the visual scaffolding and explicit strategy cues I have provided on their quiz, which I spent over two hours after school yesterday modifying and perfecting with feedback from a master teacher who works with intellectually disabled students (students with IQ's below 70), I expect most to still fail because they cannot perform the computations. My value added chart for my self-contained students remains mired in the red, but hopefully I can pull a few of these students up a little.
