During our geometry unit,

bright colored pattern blocks were spread out across our "groovy grape" rug, small groups delved into computer activities on our interactive whiteboard, students were scattered about the room drawing, building, designing, measuring, exploring and analyzing. We laughed together as we watched entertaining Euclid videos. We marveled at the real world ties to architecture and space travel. Math was interesting. Math was fun!

bright colored pattern blocks were spread out across our "groovy grape" rug, small groups delved into computer activities on our interactive whiteboard, students were scattered about the room drawing, building, designing, measuring, exploring and analyzing. We laughed together as we watched entertaining Euclid videos. We marveled at the real world ties to architecture and space travel. Math was interesting. Math was fun!

But now we were moving on to long division...ugh!

This unit would entail rote memorization of steps (divide, multiply, subtract, bring down) and endless drill and practice sheets taking us step by step through the process (2-digit by one digit without remainder, 2-digit by one digit with

remainder, 3-digit by one digit without remainder, 3-digit by one digit with a zero in the quotient, and on and on). Then, just to spice things up, we throw in the multiplication algorithm to check our work.

You get the picture.

There's really no way of getting around it, the long division algorithm is the most efficient and effective method for finding a quotient in the absence of a calculator. Here's the problem: the constant drill and practice may improve our ability to solve

for a quotient with precision and accuracy, but it takes us further and further from the context of real world math.

Students stop thinking about what the numbers represent.

They know what to do, but not

So how do we make sure students understand more than just the process?

Give them a problem they can relate to.

We began our division unit with this problem:

This unit would entail rote memorization of steps (divide, multiply, subtract, bring down) and endless drill and practice sheets taking us step by step through the process (2-digit by one digit without remainder, 2-digit by one digit with

remainder, 3-digit by one digit without remainder, 3-digit by one digit with a zero in the quotient, and on and on). Then, just to spice things up, we throw in the multiplication algorithm to check our work.

You get the picture.

There's really no way of getting around it, the long division algorithm is the most efficient and effective method for finding a quotient in the absence of a calculator. Here's the problem: the constant drill and practice may improve our ability to solve

for a quotient with precision and accuracy, but it takes us further and further from the context of real world math.

Students stop thinking about what the numbers represent.

They know what to do, but not

*why*they are doing it.So how do we make sure students understand more than just the process?

Give them a problem they can relate to.

We began our division unit with this problem:

Of course there were confident students who already knew the long division algorithm,

Most groups started by taking apart the number. Some began with the hundreds and worked backwards. First they divided the hundreds into equal groups, then the tens, then the ones. They had to take the leftover hundreds and convert into tens and the leftover tens into ones. Some groups started small

and worked their way up. Others used a "Guess and Check" method. Each group found a creative way to represent their thinking, keeping in mind that their audience should be able to

understand their strategy.

Most groups were thrown off by the remainder. Because there was a number leftover, they felt frustrated and defeated

believing they had failed the challenge. This demonstrates how quickly students forget about real world context when working with numbers. As I moved around the room, I asked, "What does the '1' remainder represent? One pickle? One alligator?" students giggled as they reminded themselves they were working with money. Many groups still needed more prompting. "How can one dollar be shared among four friends?" One-by-one, light bulbs flashed on. A dollar can be divided up into quarters.

In the end, all groups were successful. Students shared their posters in small groups describing their strategies and explaining their thinking. Yes, we did have to move on to the algorithm and a great deal of practice, but now we had a frame of reference to fall back on and students had a better understanding of division in the real world.

Now onto decimals...

*Piece of cake!*Except they had to explain not just "what" they did, but "why" they did it. Now things got a lot more complicated. Students puzzled over the steps to the process. Why do we divide these numbers, multiply these numbers, and subtract these numbers? Interestingly enough, while the more practiced mathematicians mulled over the steps of the algorithm, those students who were yet unfamiliar with the traditional method were quicker to dive into the problem. Working in small groups, students created posters to solve the problem and "make their thinking visible."Most groups started by taking apart the number. Some began with the hundreds and worked backwards. First they divided the hundreds into equal groups, then the tens, then the ones. They had to take the leftover hundreds and convert into tens and the leftover tens into ones. Some groups started small

and worked their way up. Others used a "Guess and Check" method. Each group found a creative way to represent their thinking, keeping in mind that their audience should be able to

understand their strategy.

Most groups were thrown off by the remainder. Because there was a number leftover, they felt frustrated and defeated

believing they had failed the challenge. This demonstrates how quickly students forget about real world context when working with numbers. As I moved around the room, I asked, "What does the '1' remainder represent? One pickle? One alligator?" students giggled as they reminded themselves they were working with money. Many groups still needed more prompting. "How can one dollar be shared among four friends?" One-by-one, light bulbs flashed on. A dollar can be divided up into quarters.

*Aha!*

In the end, all groups were successful. Students shared their posters in small groups describing their strategies and explaining their thinking. Yes, we did have to move on to the algorithm and a great deal of practice, but now we had a frame of reference to fall back on and students had a better understanding of division in the real world.

Now onto decimals...