Students' Creative Computations: My Way or Your Way?

Students' Creative Computations: My Way or Your Way?Author(s): Eleanor J. HamicSource: The Arithmetic Teacher, Vol. 34, No. 1 (September 1986), pp. 39-41Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41194197 .

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Students' Creative Computations: My Way or Your Way?

By Eleanor J. Hamic

My fifth-grade students had finished working through the concepts in- volved in single-digit division and had been struggling with double-digit division for several days when I selected students to work at the chalkboard.

I dictated the first problem to be solved, "6480 divided by 16."

Jenny's Way All the students correctly wrote

16)6480

and then began to work the problem by trying to determine how many 16s are contained in 64. Rounding and estimating give 3 as the digit for the hundreds place in the quotient, but a 4 is needed. As students thought and wrote, erased and rewrote, I noticed that Jenny was finished with her work. Her quotient was correctly written as 405. I smiled and nodded, but just as Jenny was about to erase her work, I spotted something unusual. This was Jenny's work:

405 16)6480 8}3240 32

4 0 40 40 0

Eleanor Hamic is a mathematics teacher at Fallen Timbers Middle School in Whitehouse, OH 43571.

September 1986

I asked Jenny not to erase just yet so that I might look at her work to see what she had done.

She had "split" the given numbers in half and, of course, gotten the same answer as the children who worked much longer on the "twice as big" problem. 1 asked Jenny why she did it, and she answered, "It's easier to divide by one number." (She meant a one-digit number.)

When I asked Jenny how she knew her way would work, she said, "It always worked on the little numbers

Children are creative in vary- ing the techniques that we teach them.

so I tried it on the big ones." I asked her to show me what she meant, and she wrote out pairs of facts that had equal quotients:

2 2 8ÏÏ6 and 4)8

3 3 6ÏÏ8 and 3)9

4 4 2Î8 and 1Î4

She explained that the second problem in each pair was "half as big" as the first but the answer was the same.

Jenny's way of knowing is not at all remarkable. In fact, children frequently notice patterns and apply them to similar or even different situ- ations. I call this process schematizing. Although the outcome can be

successful, as was Jenny's way, schematizing can result in incorrect responses as well. Consider the tod- dler who says "buyed," "runned," and "eated." The ed pattern for past- tense verbs has been noticed and ap- plied. The schema works for the majority of verbs and allows the child to speak many words correctly even though they were not specifically learned. Likewise, schematizing can lead a child to think that subtraction is commutative. The student who sub- tracts "backward" instead of renam- ing has likely noticed that 7 + 5 and 5 + 7 are equal for addition and as- sumes that 7-5 equals 5-7.

Although Jenny's way of dividing certain numbers was surprising, her way of knowing was not very unusual. The remarkable thing in Jenny's case is that she was able to verbalize her thinking. Seldom are students able to explain how they know, and so the teacher must try to find and understand the student's way of knowing. Students should be encouraged to show their work, but much creative computation is done mentally and the student may not have work to show. The teacher must watch the child as the problem is being solved, not just check the answer. We shouldn't pe- nalize them if they have their own ways that are difficult to show or explain:

John's Way John, another fifth grader, was aver- age in other subject areas but very good in mathematics. He was always the first to finish assignments. His answers were quick and accurate. As

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we reviewed addition and subtraction, I had no idea that John's way was unusual. Then we began multiplying by two-digit numbers. John finished first, as usual, but surprisingly his paper showed no work-only answers. John had not copied the problems, nor had he shown any partial products. (I don't require students to copy problems for computation. My only rule is, "If you need to do 'work,' then do it on the paper that you turn in.") I know fifth-grade students can mentally add, subtract, and multiply by one digit, but I knew of no way to multiply by two digits without doing some work, so even though he had missed only one out of the fifteen problems, I called John up to my desk to ask him about his paper.

John trudged up and said resign- edly, "I know; do it over and show the work. That's what my teacher said last year."

"Did you do the work on scrap paper?" I asked.

"No," he sighed. "How do you know what the an-

swers are?" I asked. "I look at the numbers," he an-

swered somewhat less dejectedly. Hoping to discover his way of

knowing, I wrote a similar problem on a piece of paper and asked him to solve it. He looked at the problem (for no longer than five seconds) and wrote the correct answer as anyone might, from right to left. I praised him; he looked surprised that I was pleased.

"Do I have to do the page over the long way?" he asked.

"No," I assured him, "but I do want you to tell me what you are thinking when you look at the numbers. Can you tell me what is going on inside your head?"

"I'll try," he said, but he solved two more problems correctly without saying anything. "I don't know," he said apologetically. I thanked him for trying, and he returned to his work.

It was apparent that whatever process John used, he had not yet formu- lated it into words. I had, however, seen enough to know that he had a workable way and felt that I could help him to verbalize his thought pro- cesses. Recalling that he had written the answers from right to left each

40

Fig. 1 John used these images in his method.

r f тип) r I mm îrb с i MM-MI) ( j^ r'

/TjTk «r/7jj> /Tli» ¿=ь5ШЪ

Ä. /rntfoi. ,.Bündle,. "Big bundles" (100) (10)

Fig. 2 John describes his procedure

4{p' Y ^ Я / R 1 Do this first' ̂6 x 2) = 121 ^ Я

(j^/ / R 1 "Make a bundle and write a 2."

2

S T. "Do these next." (6 x 4) + (3 x 2) = 30

У A' гЛ "Make 3 big bundles plus the ten from the first

у?У and write a 1 ." step makes 31 tens.

1 2

(4) 2 "Last, do this." r- V ' 4 x 3 = 12 big bundles and

X( 3 1 6 the 3 *rom the seconcl steP } -Л 1 - gives 1 5 all together.

15 12 "Write a 15."

time, I decided to use that as a begin- ning the next time we talked. In the meantime, I jotted down things to be considered in the solving of the multiplication algorithm: Place value is necessary. A knowledge of the basic multiplication facts is important, and it was evident in the speed of John's solutions. I thought it unlikely that a

ten-year-old would have memorized facts beyond 10 x 10 = 100, but I did not rule out that possibility. I also considered the possibility that he had used an addition process of some sort, either by itself or in combination with multiplication.

The next day I began as planned. "Do you always write the answers

Arithmetic Teacher



from right to left?" "No," he answered, "not always."

I wrote another problem. He considered it for a moment, but this time wrote the correct answer from left to right. "It's not much different," he said.

"Which numbers do you look at first?" I persisted.

"All of them," was his reply. Hoping for another clue, I wrote a

three-digit multiplication problem on the paper:

623 x 574

"Can you solve this problem?" I asked.

He looked at it for a few seconds and wrote the answer slowly from right to left. I worked the problem out in the usual way: multiplying to get each of the three partial products and adding to get the final product. I got the same answer he had written. He smiled broadly and confided, "This one is easier if you do it right to left." I still had no idea what he was doing, but I was sure he had created a new method to find products and I was determined to understand it.

During another of our sessions together, I said to John, 'Til hold the pencil and you tell me what to do." In the telling he used the word bundles.

"What bundles?" I wanted to know more about bundles!

Finally John explained the "pictures," or images, that he was working with. He was combining "bundles of sticks." We've all seen those bundles of sticks that are pictured in text- books and printed on worksheets to help students understand place value. He was using pictures (fig. 1) in his head to bundle up tens and hundreds. His procedure is described in figure 2.

As you have probably noticed, John's way, which I call visualizing, is not unlike the customary algorithm, but it requires that one remember how many tens and hundreds to total. Vi- sualizing is a powerful memory aid. Stacked bundles of sticks, being more concrete, are much easier to keep in mind than the digits that represent them.

September 1986

Visualizing is frequently a child's way of knowing, but it is very hard to verbalize. Often children, when asked how they know the answer, respond, "I don't know." But what they are really saying is, "I can't explain it to you." I've had students who calcu- lated by visualizing bags of jelly beans, marbles, seashells, and money. They all had trouble explain- ing their thinking, but visualizing worked well for them. Teachers can

VII show you my way, then you show me your way.

and should help these students to verbalize and share their ways of knowing.

More Than One Way Usually more than one way can be found to get where you're going. When you plan a trip (once you know where you're going), you choose the roads you want to take. I might rec- ommend the expressway, thinking to save you time, but you might prefer the scenic route. Unfortunately, our mathematics students do not always know where they are going, and we must direct them. We must show them where they are headed and rec- ommend a way to get there. The problem that we demonstrate on the chalkboard or the one at the top of the textbook page is illustrating a way, not the way. It is a suggestion, not an edict.

No doubt, the method shown in the sample problem was selected because it is effective for the majority of students and for most problems of the type being demonstrated. It may not, however, be the best way for some students or certain problems. John's way was better for him, and Jenny's way was better for her.

Every school, perhaps every classroom, has students like John and Jenny. They become less conspicuous as they get older because they "give up" in their efforts to find their own ways. Some join the crowd and do mathematics the "proper" way, but

others become discouraged and frus- trated by teachers who fail to recog- nize their need for independence and originality.

The school records of both John and Jenny had teachers' notations of "lazy" and "careless." John's fourth-grade teacher had written "seldom turns in math assignments" on his grade card as the reason for his low mathematics grade. I understand why John stopped turning in papers to a teacher who wrote on them "Do over!" "Show work!" or "Did not follow directions."

In my classroom students are encouraged to discuss alternative ways of solving various problems. I may start a review lesson with the com- ment "I'll show you my way, then you show me your way." (I usually show them a seldom-used algorithm.) I may ask them to show two ways to solve each problem and circle the one they like best. I often ask questions like these: • When is this way easier? • How many ways can you solve this

problem? • How else might you get the

answer? • Does anyone see another way? • Can you solve this problem without

using division?

You, as the mathematics teacher, should know many ways of solving problems, be able to present alternative methods when appropriate, and be receptive to new or different ways of thinking. Above all, you must allow the child to choose his or her own way. We want our students to con- tinue to explore and discover mathematics. We want them to have a wide range of problem-solving strategies from which they can readily choose an efficient method.

Explore with your students! Be ready to share with them not only the expressways but the back roads and scenic routes, too. Encourage them to be pathfinders and to blaze their own trails. Your students will be eager to find out where they are going next. And you'll notice new and interesting approaches even though you've been there many times, w

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Students' Creative Computations: My Way or Your Way?