Guest Post by John Mighton
Many educators now believe that algorithms such as long division are simply a series of rote rules that do not involve any “concepts,” so students should invent their own algorithms instead of learning the ones people have already devised. I have developed a way to teach long division that enables kids to discover the steps of the algorithm and understand the underlying concepts while learning to perform the algorithm proficiently. I connect it with a problem involving money.
I tell students that the notation:
can be interpreted to mean: 3 friends wish to share 7 dimes and 2 pennies (72 cents) as equally as possible. I then write three or four questions like the one in my example on the board and ask students to tell me how many friends, dimes and pennies are indicated in each case. Students answer orally or in a notebook.
NOTE: I give an assessment, consisting of three or four questions similar to whatever example I have just worked on, in each of the steps below. Because the steps are easy, students generally get a perfect score, and that success makes them engaged and attentive.
I ask students to draw a picture to show how, for the division statement,
they would divide the dimes among the friends. If students use a circle for each friend and an X for each dime, the diagram would look like this:
I ask students to tell me the meaning of their diagram: each friend gets two dimes and there is one dime left over.
I tell students that if they happened to see someone carrying out the first few steps of the long division algorithm, this is what they would see:
I challenge students to figure out what the steps in the algorithm mean by identifying where they see each number in their diagram. Students readily make the following connections between their diagram and the algorithm.
(If necessary I proceed in smaller steps, only working on one number at a time.)
I ask students to complete their diagram from step two to show me how much money still has to be divided among the friends. If students use a circle to represent a penny, their diagram looks like this:
I invite three students to come to the front of the class so I can demonstrate how I would divide the remaining coins among the three friends. I give two students a penny each, and one student a dime. The students always protest that my way of dividing up the coins isn’t fair: they tell me they would exchange the dime for ten pennies and divide the twelve pennies among the friends. I inform students that this process of “regrouping” the tens (dimes) as ones (pennies) is actually a step in the long division algorithm. Most adults call this the “bring down” step, but very few understand it.
column, you implicitly change the number in the tens (dimes) column into the smaller unit (pennies). Then you combine all of your smaller units to give twelve pennies altogether.
I ask students to show me, in their diagrams, how they would divide the (twelve) remaining pennies among the friends. I also ask them to connect the numbers in their diagram with the remaining steps of the algorithm:
For more about John Mighton and his methods, please visit: "New Techniques Make Math Fun for All" and "The Making of a Mathematical Mind: One Step at a Time"