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How to Teach Long Division through Guided Discovery

The views expressed are those of the author and are not necessarily those of Scientific American.


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Courtesy of puuikibeach via Flickr.

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.

STEP ONE
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.

STEP TWO
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.

STEP THREE
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.)

STEP FOUR
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.

STEP FIVE
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

Ingrid Wickelgren About the Author: Ingrid Wickelgren is an editor at Scientific American Mind, but this is her personal blog at which, at random intervals, she shares the latest reports, hearsay and speculation on the mind, brain and behavior. Follow on Twitter @iwickelgren.

The views expressed are those of the author and are not necessarily those of Scientific American.





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  1. 1. jolie600 4:26 pm 08/15/2013

    my co-worker’s wife makes $80 an hour on the laptop. She has been out of a job for seven months but last month her income was $was $16620 just working on the computer/laptop for a few hours….Heres what she do ==== Buzz55. ℂℴM====
    ♫ ✂♫ ✂To find a mountain path all by oneself gives a greater feeling of strength than to take a path that is shown.

    –>

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  2. 2. mjonck 9:16 am 08/16/2013

    Thank you! I’m teaching this topic for the first time the semester. I’ll use this!

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  3. 3. Bobstr 10:34 am 08/16/2013

    On one hand your approach has merit, but on the other hand it over complicates long division. I am more from the old school. Just teach the “rote rules” and make a less complicated guided discovery lesson by asking 3 students take a penny from a hat containing 72 pennies until the hat is empty. The student then discovers that each has 24. So 72 pennies divided among 3 students equals 24. This is simpler approach and quicker to discover. Just my 2 cents (or pennies :-)

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  4. 4. markovchaney 8:56 pm 08/23/2013

    “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.”

    Please. I know no mathematics educator who says that. You’ve managed to take two separate true statements and cobble them together in a way that is classic anti-progressive math propaganda.

    The problem is that the second statement doesn’t follow from the first. The first one is true: people have long been teaching things like division as a bunch of steps to be mindlessly followed. Some kids are capable of doing that (I was one of them), and blow through long division without difficulty, though few, if any, give any thought to why any of it works.

    Others struggle recalling the steps or getting them consistently in the right order, or carrying out specific aspects of one or more steps, and thus fail at division, struggle with it unsuccessfully for years and years, and STILL don’t know what’s going on or why, should at some point they finally “master” the rote process.

    That’s a flaw of traditional math education, weak knowledge of math by elementary school teachers, and a generally wrong-headed approach to education that has dominated in the US for over a century.

    The SECOND statement is also true. But it is based on the notion that kids build mathematical confidence and thinking skills by having a chance to, among many other things, take a crack at developing their own algorithms. By so doing, they aren’t expected to NEVER learn the standard ones currently in fashion, but rather to both improve their general mathematical “chops” AND gain some insights into what any reasonable algorithm in arithmetic is likely going to have to do. Once kids have looked at their own algorithms and/or those of peers, they can now have some intelligent basis to look at adult algorithms and them make informed choices about what they like and why. Imagine that!

    Putting the first and second statements into once sentence makes it sound as if there’s some simple causal argument when there isn’t. The second situation actually precludes the first, and is designed in fact to obviate the mass confusion and failure.

    And if you can’t get that right, I’m not sure how interested I am in your particular methods. Because you open with something that really smells awful.

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