“Can you show me another way?”
Multiple representations show students there is more than one correct way to do the math. This is an important message in itself.
Multiple representations also allow students to learn new mathematical concepts and procedures.
For example, division can be thought of as sharing or grouping.
8 ÷ 2 = 4 can be thought of as:
I have 8 items. I share them equally between 2 people. Each person gets 4 items.- I have 8 items. I put them in groups of 2. I can make 4 groups.
I prefer the adjective flexible over multiple. Adaptability of, not number of, is what is important.
To learn how to divide integers and fractions, students must be able to visualize both representations.
For example, -8 ÷ 2 can be thought of as sharing equally between 2 groups. Each group contains four negative counters. However, -8 cannot be put in groups of +2.
Alternatively, having a negative number of groups does not make sense. However, -8 can be put into groups of -2.
Think about why 6 ÷ ½ is 12 (without simply applying the invert and multiply rule). Having a fraction for the number of groups doesn’t make sense. However, students can explore how many halves there are in 6 using pattern blocks or number lines.
3 × 2 is more than simply 3 groups of 2. An understanding of an area model of multiplication helps students to learn two-digit multiplication.
An understanding of this model will help students make connections between multiplying binomials and multiplying two-digit numbers.
As a secondary department head pointed out a meeting last year, teaching how to multiply binomials may be easier than teaching how to multiply two-digit numbers – in algebra, there isn’t the added complication of place value.
Now if only we would stop using the term FOIL…
Chris Hunter 
Nice. Not everyone agrees that this distinction matters, but I think it’s a fundamental one and I’m glad to see you spreading the good word.
I’m gonna take you to task on this, though:
I’ll agree that it’s harder for most of us to imagine a fractional group than it is to imagine a fractional group size. But that it doesn’t make sense is too strong a claim.
Here are some quickies that can lead to grouping division with fractional divisor (the first involves a whole number divisor, but sets up the division structure to follow):
1. Kate walked 5 miles in 2 hours at a steady pace. How far did she walk in one hour?
2. Sean walked 3/4 of a mile in 15 minutes. At that pace, how far can he walk in one hour?
3. 1 cup of whole milk has 8 grams of fat. How many grams of fat are in a gallon of whole milk?
Sure, you might not think of (2) as 3/4 divided by 1/4, nor of (3) as 8 divided by 1/16. But it makes sense to do so.
Oh, and sorry about the imperial units. Metric conversions available for a fee.
Great post! I also think the distinction of grouping and sharing in division is important. But, it can make sense to have 1/2 a group. What about this situation? Mrs. G had 6 cookies that she didn’t want, so she decided to give them to some students. 1/2 of the students in the room got a cookie. How many students were in the room? This way of thinking is not as natural, but it can make sense.
It’s also helpful for students if they act out these various situations before drawing them.