Sunday, August 16, 2009

M3N4e - Developing division algorithms (1)

M3N4. Students will understand the meaning of division and develop the ability to apply it in problem solving.
e. Divide a 2 and 3-digit number by a 1-digit divisor.A lot of people seem to have a very negative feeling toward "long division," the division algorithm which is commonly used in the US today. Some people even call for eliminating long division since we can use calculators. Clearly, the writers of the GPS did realize the foolishness of such a recommendation. Thus, in M4N4, they specifically state that students can divide without calculators. So, why is "long division" so disliked by many?

One of the reasons is probably the multiple steps involved in the procedure. There are many different mnemonics that is supposedly help children remember the sequence of those steps. How else is the long division algorithm different from other algorithms? One difference is that the long division is the only common algorithm that goes left to right. With addition, subtraction and multiplication, we are taught to start with the ones place - of course, it is perfectly possible to go left to right, but that's a different story. Constance Kamii and other researchers have pointed out that children, when they are asked to think about numbers, naturally start with the largest places. So, many first graders, when asked to find the sum of 23 and 31, they would think like, "20 and 30 make 50, and 3 and 1 make 4, so the answer is 54." Many adults, when they are estimating the sum or difference of 3- or 4-digit numbers mentally, they find it much easier to go from left to right. For example, with 584+279, you would think, "500 and 200 is 700, 80 and 70 is 150, so 850 altogether, 4 and 9 is 13, so the answer is 863." So, in a way, we can argue that the long division algorithm is the only common algorithm that aligns with our natural way of thinking.

So, how can we help our students more naturally develop the long division algorithm? One of the keys is how we organize our instruction. First, let's think about the meaning of division. We know that students are introduced to two types of division situations (M3N4b): partitive (fair sharing) and quotitive (repeated subtraction). Which should we use when teaching the long division algorithm? Some might say it would not make any difference, but I argue that it is much easier to work with partitive situations if you want students to develop the long division algorithm. Thus, we should start with Problem 1, instead of Problem 2:
Problem 1: There are 72 sheets of construction paper. If you share them among 4 students, how many sheets will each student receive?

Problem 2: There are 72 sheets of construction paper. If you make bundles of 4 sheets, how many bundles can you make?

Second, let's think about what learning tools students should use. For teaching and learning of the long division algorithm, I think base-10 blocks are very useful. Of course, if base-10 blocks are to serve as students' thinking tools, they have to be comfortable with them before they start using to work on problems like Problem 1. If students are familiar with base-10 blocks, how might they solve this problem? It is not unreasonable to think that they will first make 72 by using 7 rods and 2 unit cubes. They will then give one rod to each of the 4 groups. At that point, they will trade in the remaining 3 rods to get 30 unit cubes, and they now have 32 unit cubes altogether to share among 4 groups. They will then distribute 8 unit cubes to each group, with no remainder. Thus, the quotient is 18.

Once students gotten used to solving division problems with base-10 blocks, it's time to help them move beyond the blocks. You can have them draw what they would have done with the blocks, instead of actually using blocks. So, with Problem 1, students will draw 7 rods and 2 units, and 4 circles for the groups.

You can cross out 4 rods and give 1 rod to each group.

Then, you have to cross out the remaining 3 rods and draw 30 units.

Now, you can give 8 units to each.

After while, students will feel this is too much drawing, and that's when you can suggest a couple of things. First, you can suggest that you really don't need all four groups since the final answer is how much is in each group. The second suggestion is to write numerals instead of pictures of blocks using a place value mat. So, for Problem 1, you would write something like this:

Now, when you give 1 rod to each group, you used 4 rods, so you have to take away 4 of the 7.

Now you have to exchange those 3 remaining rods with 30 units, but since you already had 2 units, you now have 32 units.

After giving each group 8 units, you used 32 units and 32-32=0.

You see how similar these notations are to the actual long division algorithm. Once students get used to using this notation, you can probably show the long division algorithm and ask students if they can explain what is happening at each step.

Finally, you want to consider what kinds of numbers you use. When thinking about a 2-digit number divided by a 1-digit number, you want to think about each of the numerals in the 2-digit number in relationship to the divisor. Do you want that to be greater than, equal to, or less than the divisor? If it is less than, that means the tens place in the quotient will be empty - which may be a bit too much for the opening problem. If it is equal, that means there is no left over after all rods are shared. Basically, you can divide each numeral by the divisor. Again, that is a special case, and you may wonder about whether or not starting with a special case is good Something like 72 and 4 as in Problem 1 where the tens digit is greater than the divisor may be a good starting point.

After students develop the division algorithm, that's when we might want to think about those special cases. In addition to the case when the leading digit is equal to the divisor, we must think about those situations when there is an empty place in the quotient - the case when the leading digit is less than the divisor is one such case, i.e., 0 in the tens place (or the leading place). Other cases are when 0 is in the ones place and 0 is in the middle, when dividing 3- or longer digit numbers divided by 1-digit numbers.

With these ideas in mind, the long division algorithm can be learned more naturally. As stated earlier, the long division algorithm may be more "natural" of the four algorithms. However, that doesn't mean that students will automatically develop the algorithm. It takes careful planning by teachers.

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Elaboration of Georgia Performance Standards by Tad Watanabe is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.