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:

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:

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.

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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