Saturday, August 29, 2009

M4N4b - Developing division algorithms (2)

M4N4. Students will further develop their understanding of division of whole numbers and divide in problem solving situations without calculators.
b. Solve problems involving division by 1 or 2-digit numbers (including those that generate a remainder).In the previous post about the long division algorithm, I mentioned that the partitive (fair sharing) division may be more useful to develop that algorithm. What students might do to model a partitive problem with concrete materials like base-10 blocks will match up very nicely with the paper-and-pencil algorithm you are trying to help students develop. What will happen if we have a quotitive (measurement, or repeated subtraction) division problem? Let's look at Problem 2 from the last post:
Problem 2: There are 72 sheets of construction paper. If you make bundles of 4 sheets, how many bundles can you make?To solve this problem using concrete materials, students will make groups of 4. Often times, adults (or students) will describe the first step of the long division by saying, "how many times can 4 go into 7?" However, since we are dealing with 72, "7" is actually 7 rods. To match up the algorithm, what we are asking ourselves is, "how many groups of 4 rods can we make with 7 rods?" The fact that you can make 1 group of 4 rods, actually suggests that we can make 10 groups of 4 units. So, if you already know the long division algorithm, you can make the process match the algorithm. However, for children who are learning the algorithm for the first time, that task isn't as straightforward as it will be with the partitive division.

However, this alternative way of looking at division may be useful when you are actually dividing by large numbers - like when you have to divide by a 2-digit number in Grade 4. Suppose you have the following problem:
Problem 3: There are 1950 sheets of construction paper. If you make bundles of 38 sheets, how many bundles can you make?So, you will ask, "how many times can 38 go into 195?" To estimate this partial quotient, you may round up 38 and think about, "how many times can 40 go into 195?" 40x5 is 200, and that's too big. So you estimate the tens digit of the quotient is 4. You multiply 38x4 and subtract it from 195 and get the difference of 43! So, the tens digit of the quotient must be 5, not 4. So, you have to re-calculate.

Instead of doing this, you can think about the problem differently. The question is to determine how many groups of 38 you can make with 1950. If you have a reasonable number sense, you can see that the double of 38 will be 76. So, you can easily make 20 groups, or use 760 sheets. 1950 - 760 = 1190, so you can make another set of 20 groups. 1190 - 760 = 430, so we can make 10 more groups. 430 - 380 =50, and that's one more group. 50 - 38 = 12, so we can't make any more group. Therefore, we made 20 + 20 + 10 + 1 = 51 groups, with 12 sheets left over.

This process can be made into a written process like this:


This algorithm is sometimes called the Scaffold algorithm. Others may call it a "forgiving method," as it doesn't require the best estimate of the partial quotient. It is useful in some situations, like when the divisor becomes large. Should all students know this algorithm? I am not so sure. One of the important ideas of teaching students computational algorithms is that students understand that with our numeration system, we can look carry out calculation by focusing on one place value at a time. This algorithm treats the numbers (divisors) as a whole.

On the other hand, it does have some usefulness as we can see. For me, teaching of an algorithm means helping children make their own procedures (with concrete materials or thinking strategies) into a written procedure. So, if children aren't thinking this way, then imposing a method doesn't seem to be too productive. Of course, by asking students to think about quotitive (measurement) division problems, you can increase the likelihood of students thinking this way, too.

If we are to teach this algorithm, I think it is important for students to realize when this method might be more useful than the long division algorithm. We want students to make intelligent decisions about how to calculate - which algorithm to use, whether or not an estimation is good enough, etc. So, if this algorithm is included in your curriculum, I encourage you to help your students understand its merits so that they can use different methods flexibly.

2 comments:

Blank said...

I wonder is worksheets that do a little hand-holding would help to improve confidence and get familiar with the algorithm... something like these long division worksheets.

Tad said...

I don't see anything wrong with the worksheet. It is important to provide opportunities to practice skills, but we should keep in mind that practices should be short, focused, and consistent. We should not be using up a half of class period just to practice, for example. It is probably better to spend 5 minutes everyday - and in elementary schools, this can happen outside of designated math periods in some cases. Also, it is important to focus practices on what children are having difficulty with - yes, maintenance of skills is important, but we don't need so many problems to do that. Practice time may be the time where "differentiation" is particularly important.

As for developing confidence, we should think about praising children not only for their correct answers but their thinking. If we want students to be better thinkers, then we should communicate that to our students by praising their thinking. As we do, however, whenever we can, we should point out the mathematical value of the thinking process instead of a blanket praise like "that was a really good thinking." Instead say something like, "You tried to do ____, and that was a good idea."

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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.