Here is the interaction that I had with my 5th grade son yesterday:

Me: “What does 6 ÷ 2 mean?”

Son: “Six split into two equal groups.”

Me: “What else could 6 ÷ 2 mean?”

Son: [crickets]

Me: “…how many groups of two are in six… so…what does 6 ÷ ½ mean?”

Son: “…how many groups of one-half are in six…12.”

Me: “What is 6 ÷ ¼?”

Son: “…how many groups of one-fourth are in six…24. I remember this every time you remind me…but I always forget.”

My son is very strong with fractions, and he’ll go straight to invert and multiply if I let him. I often engage him in this same conversation…working on the understanding of dividing a whole number by a unit fraction because I know how difficult division of fractions can be for students.

My hypothesis is that it has to do with the two interpretations of division:

(a) How many groups? and (b) How many in each group?

****At this point…I will hope that elementary people will tell me I’m wrong if I’m wrong…*

I began teaching Math for Teachers at the local college (St. Mary’s College of Maryland~my alma mater) two years ago. I use Sybilla Beckmann’s Math for Elementary Teachers fourth edition for this course.

The first activity for the division unit is this:

*Write a simple word problem and make a math drawing that you could use to help children understand what 10 ÷ 2 means.*

Each year I’ve taught this course EVERY single student wrote a **how many in each group** problem. This is called the **sharing model** of division.

I think it’s less natural for students to consider the **how many groups** (or **measurement model**) interpretation of division. But it seems that this is the model that makes the most sense for the division of fractions.

I think this less familiar interpretation of division can also impact a student’s ability to be successful with long division.

A student needs to be able to think, “How many groups of 30 are in 1429??”

I think it’s important to be purposeful with the language we’re using with students, and how we are exposing them to different interpretations of division specifically. Secondary math teachers can learn A LOT by digging into the elementary material.

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As a sixth grade math teacher, this insight into teaching different ways of understanding division was a big breakthrough for me, too.

Your post is nice and clear and covers most of the same ground already, but you might also like these:

https://christopherdanielson.wordpress.com/2011/06/30/more-on-fraction-division-you-know-you-love-it/

https://lanier180.wordpress.com/2013/02/20/day-93-three-eighths-of-a-pile/

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The other way I’ve looked at division is not just “How many of” but the alternative “how much of” for a problem like 1/4 divided by 3/4. 1/4 is how much of 3/4? Is is 1/3. 5 divided by 10. 5 is how much of 10, it is 1/2 of 10.

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