Long division that sticks

I recently saw a division problem that a middle school student attempted and her solution looked like this:

Ouch!  She got the format down-- divide, multiply, subtract, bring the next digit down-- and used a remainder; and technically, it is true that 16 x 39 +1023 = 1647 (points for bringing down the 1000 after the fact!), but she missed the point of the process entirely, which is to find the number of times that 39 can go into 1647.

Whether long division, short division, division of fractions, or whatever, students need to know that division asks a question:  How many sets of one number can go into another?  It's easy to get so caught up in the algorithm that the purpose is completely lost.  That's true of a lot of math skills-- it's not enough for the student to be able to run through a procedure; they need to know what they're doing and why.

Many students have a hard time learning long division, simply because there are so many steps involved.  Years ago, my own hands-on girl needed the following method to help make things click:

Materials:

• Colored, 3D numbers: multicolored plastic or wooden magnetic numbers, or play dough, or modeling clay; 
• A markable base to put them on: cookie sheet lined with paper, or magnetic whiteboard; 
• Something to make the math symbols: clay, craft sticks, felt, paper, or pipe cleaners
• Something to mark lines with: dry erase markers, washable markers
• Index cards or other paper

Procedure:

1. Draw 6-8 columns lengthwise on the cookie sheet or whiteboard.
2. Set up the problem:  Construct the long division bracket out of felt, paper or a pipe cleaner, and line up the digits of the dividend in the columns, using a different color number for each digit in the dividend.  The last digit of the dividend should go in the far right column.
3. Write the divisor on an index card or piece of paper.  This illustrates the distinction between the dividend, which is the actual number we are working with, and the divisor, which is the number we're trying to fit into it.  Place the divisor outside of the bracket.
4. As the problem is solved, make a point to line up digits in the columns and match the color of all digits in each column.  I didn't have enough wooden numbers of the same color to complete the division, so I used matching modeling clay to make the numbers I needed.
5. Use popsicle sticks or other neutrally-colored 3D object for the subtraction base lines.
6. If your number set does not include operation signs, make your own subtraction symbols.  It is best to have the student be able to pick up and move a physical subtraction sign.  

By keeping the numbers lined up and color coordinated, place value is emphasized.  When talking with your student,  point out that all the numbers in the same column have the same place value-- for example, in the picture at left, the yellow 1's both mean 1000, the blue digits mean 600 and 500, the red digits indicate 40, 40, 60, 80, and 70, the green digits are 2, 7, 7, 8, and 9.

The benefit to using the moveable numbers is that they are easily picked up and moved around, and the student doesn't have to worry about writing neatly.  If the columns don't line up, a simple push of a finger will fix it.  If a mistake is made, it is corrected as easily as changing out numbers.  They should not be writing anything in this activity, just moving the 3D numbers and symbols.

Have the student practice this procedure over many days.  If she has a hard time remembering the steps of the algorithm, try chanting, "Divide, multiply, subtract, bring it down!" over and over.  You can add in dance moves or hand motions: for "divide," move your hands as if you were separating something into two parts; for "multiply," cross your arms to make an "X;" "subtract" might be pointing away with your thumb as in 'hit the road, Jack,' and "bring it down" could be pantomiming pulling down a window shade or something.  Your student can also come up with her own moves.

Once she's got the procedure down, have the student do the same procedure using colored pencils on notebook paper that is turned sideways.  The column lines are already drawn!

It is important to have the student get in the habit of checking her work.  In this case, point to the original 1647 and ask the student, What two numbers can be multiplied together to get this number? If she understands what she just did, she will know that 39 x 42 will get him close to 1647, but because the solution has a remainder, the remainder 9 must be added to the product to get the final answer.

Getting back to the original problem of the middle school student's division error,  why do you suppose she made the error?  It appears that she did not understand the purpose of the problem to begin with.  If she can learn to see the problem as, "How many 39's are in 1647," that will help her see that "16 remainder 1,023" is a useless answer.  Another way to help her would be to take the card with 39 on it and say, "What could you round this number to that would be easier to work with?"  If she can see that rounding the divisor to 40-- and perhaps also rounding 1647 to 1600-- can give her a good estimate, it will go along way toward getting to the correct solution.