Do Your Kids Need a Dawning Division Moment?

After sitting in a week’s worth of meetings about Common Core instruction and best math practices, I feel as if my way of instructing students is gradually morphing into a new beast, which I am not sure I fully can describe yet.  In the midst of all of this, I went back to the fifth graders I have been working with on the day after these meetings.  I decided to try out some of the teacher practices I had been learning.  Sidenote:  This set of fifth graders isn’t yet proficient at long division or ladder division for that matter.  I posed a long division problem to the students, but instead of giving it to them in “naked numbers” alone, I gave it to them in word problem form as well.  In reflecting, I should have started out with the word problem alone and let the students figure out that they were supposed to divide.  After watching the students struggle with solving using ladder division or the traditional long division algorithm, I encouraged the  students make equal groups to see if that helped them.  One student in particular that I worked closely with (whose work is pictured below) managed to find a correct answer using equal grouping, however he could not find his mistake when he used ladder division.  Another student…one I might add who typically doesn’t have much perseverance to solve problems…related completely to the contextual situation and persevered through to solve the problem to the end.  To get to my MAJOR DIVISION REVELATION, let’s look at this student work below:

One student's incorrect ladder division and solving with equal grouping correctly.

The same student's equal grouping and the relationship to the correct ladder division which I showed him. Notice how 20, 30, 10, and 5 are the same in the circles and to the side of the division problem.

So, you ask, what is the big AH-HA?  If you notice that the numbers in the student’s circles (equal groups) are the same numbers to the side of the division problem (or the ladder).  The numbers to the side of the ladder are actually the chunks that students would naturally break off if they were naturally dividing cookies among people etc.!  Now I know according to my PD classes that I shouldn’t have just told and shown the student this relationship, but I should have allowed him to figure it out.  I was just a little too excited to hold in my own personal discovery I suppose.

Another discovery I had in experimenting with having students discover their own strategies, I learned that students actually do better and persevere more when the problem is in a context familiar to them and it isn’t just a naked numbers problem.  Having a contextual situation actually gives the students an entry point into the problem, and so they don’t give up as quickly.  I have always thought backwards.  Students MUST UNDERSTAND THE NAKED NUMBERS FIRST and THEN they can SOLVE HARD WORD PROBLEMS.  But in reality after my experiment with these fifth graders, they were much better at solving this division problem in their own way when they had a context.  I have had a complete paradigm shift, and you were all here to see it  above– that is.