The easiest way to explain this is to give an example of approximately a week's work in my grade five class. I'm assuming here that I've never introduced the topic in question before to this group. We're going to go with distance and rate as an introduction to multiplication and algebraic patterning; we haven't done a multiplication unit at this time, though they're familiar with multiplication from the previous year, and my observations were that they had very little sense of division at all.
So, Monday morning. I put up a problem on chart paper at the front of the room: "A spider travels 19 cm per second. How long will it take the spider to travel the perimeter of a room 3m x 4m?" The kids get the standard instructions for group work: they can work with anyone in the class and they can use whatever manipulatives or supplies will help them figure out the problem. If they seem lost, we may discuss a few possible ways of approaching the problem, for example: "Well, if the spider travels 19 cm in one second, how far does it travel in two seconds?"
The kids then start working through the problem. While it seems obvious to an adult that this is a division problem - 3m x 4m = 12 m = 1200 cm, so the answer will be 1200/19 - this is not obvious to the kids and I don't point it out to them. Most will start by adding up, 19+19+19, over and over again. I circulate and ask questions like, "What do these nineteens mean? What part of the answer are you getting here?" and "What strategy are you using?" to get their metacognitive thinking going. Meanwhile, I'm taking notes. Who jumps right to using multiplication - for example, 19x10 - and then adds groups from there? Who uses repeated subtraction instead of repeated addition? Who can't seem to add the number of nineteens at the same time as they're adding nineteens? Who is using an estimation strategy, for example 20x5 = 100?
Over the course of the next hour or so, I'll get different students up to the board to explain their strategies to the other kids. The kids get to ask questions of the person presenting, and if they have a suggestion, they can make it at this point. Usually, a bunch of kids realize that they can use what the other kids said to refine their strategies, because they were doing something similar. I don't reserve this step only for the people getting it right; I'll often get someone up who is missing one key piece to make an otherwise-sound strategy work, in the hope that another student will point out what they've missed.
When we get to the end of the problem - and it might very well take two days - we go back and revisit the strategies we used. I give the kids a chance to point out that the repeated addition worked, but it wasn't very efficient, and could be made so if they added groups instead of individual nineteens.
This got long.
So, Monday morning. I put up a problem on chart paper at the front of the room: "A spider travels 19 cm per second. How long will it take the spider to travel the perimeter of a room 3m x 4m?" The kids get the standard instructions for group work: they can work with anyone in the class and they can use whatever manipulatives or supplies will help them figure out the problem. If they seem lost, we may discuss a few possible ways of approaching the problem, for example: "Well, if the spider travels 19 cm in one second, how far does it travel in two seconds?"
The kids then start working through the problem. While it seems obvious to an adult that this is a division problem - 3m x 4m = 12 m = 1200 cm, so the answer will be 1200/19 - this is not obvious to the kids and I don't point it out to them. Most will start by adding up, 19+19+19, over and over again. I circulate and ask questions like, "What do these nineteens mean? What part of the answer are you getting here?" and "What strategy are you using?" to get their metacognitive thinking going. Meanwhile, I'm taking notes. Who jumps right to using multiplication - for example, 19x10 - and then adds groups from there? Who uses repeated subtraction instead of repeated addition? Who can't seem to add the number of nineteens at the same time as they're adding nineteens? Who is using an estimation strategy, for example 20x5 = 100?
Over the course of the next hour or so, I'll get different students up to the board to explain their strategies to the other kids. The kids get to ask questions of the person presenting, and if they have a suggestion, they can make it at this point. Usually, a bunch of kids realize that they can use what the other kids said to refine their strategies, because they were doing something similar. I don't reserve this step only for the people getting it right; I'll often get someone up who is missing one key piece to make an otherwise-sound strategy work, in the hope that another student will point out what they've missed.
When we get to the end of the problem - and it might very well take two days - we go back and revisit the strategies we used. I give the kids a chance to point out that the repeated addition worked, but it wasn't very efficient, and could be made so if they added groups instead of individual nineteens.