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A Touch-Stone for a Few Teaching Principles
The teaching of long division is the focus of heated arguments in the world of mathematical education. Some claim it is too difficult and that the children do not understand it, but rather perform it mechanically. There are those who go so far as to compare it to calculating square roots, a subject that used to be included in the seventh and eighth grades many years ago, and has now been completely, and probably justifiably, abandoned. In Israel, failing to teach long division has led to its omission for a very long time: the 1988 curriculum gave it a low priority, interpreted by many teachers as permission to omit it entirely. It is now making its comeback. It appears as part of the Ministry of Educations new proposed curriculum, which contains no priorities, ensuring that all subjects are mandatory.
The return of long division to the curriculum is highly justified. This is a fundamental algorithm, containing several important principles. Mastering it requires a good understanding of the meaning of division, knowledge of the multiplication table and the ability to estimate. The algorithm is based on the idea of extracting the number in stages, that is, dividing one part of the number after the other. Its inductive structure is perhaps even more apparent than that of other calculation algorithms. And most importantly: the ability to calculate such a basic operation provides the child with a sense of mastery. Such an operation should not be left in the hands of a calculator, thus attaching to it a sense of mystery.
In this article, I will relate the methods of teaching the algorithm, and particularly my experience in teaching it. The purpose is not only to help teachers and authors of books, although this is most certainly an important cause. I would also like to demonstrate, through the teaching of long division, several principles that I believe to be fundamental to teaching elementary school mathematics, and teaching in general.
A Few Didcatic Principles
First Principle: Respect for the Complexity of Elementary Mathematics
The first element required for good teaching, and for writing textbooks in the proper order, is respect for the complexity of elementary mathematics. It is important to understand that it is not simple, and that processes that appear to consist of one piece do in fact consist of several stages. Those who do not acknowledge this fact may assume that their role as teachers or as authors of textbooks is to convey sophisticated concepts to the children or invent teaching aids. Those who are familiar with it will know that they must seek the secret of teaching a concept in its mathematical aspect, before anything else. This means, above all, breaking the concept down to its components. And it brings us to the second didactic principle:
Second Principle: Breaking Down
When I first entered elementary school, the teacher whom I accompanied, or, more precisely, accompanied me, would stop me from time to time: pacing, pacing, pacing, she would repeat again and again. This is the most important rule in mathematical education: good teaching depends, first and foremost, on knowing how to break the mathematical principles into the correct stages.
It is impossible to move from the ground to the top of a building in one leap. One must use steps, or a ladder. Teaching is similar to building such a ladder, and presenting its steps in the proper order. Some pupils have long legs, and can make do with large gaps between the steps; others have short legs. However, as a rule, it is best not to skip any stage, not even with the more gifted pupils. Skipping stages is the main cause of mathematics anxiety.
Third Principle: Divide and Conquer
Breaking down principles depends firstly on knowing the correct order of the elements taught: a layer cannot be constructed on a missing layer. However, in some cases, there is no particular order between two principles, and they are not dependent on each other. Even so they must not be taught simultaneously. Each principle should be taught separately divide and conquer. The effort to understand one abstract concept is enough. There is no need to burden the pupils with understanding two concepts at once.
Fourth Principle: Experimentation
This is a well known principle among teachers who work with young children, but it is also true in the higher grades, where teachers tend to forget it. Even in more advanced stages of learning, it is necessary to return to the concrete basis of abstract concepts.
Experimentation is important in the simplest of matters. There is nothing too simple to experiment. To establish the concept of the number, it is necessary to count objects again and again; to understand division, it is important to experiment with dividing groups of objects into equal sets again and again.
Fifth Principle: Raising the Principles from Within the Child
Teachers should guide experimentation so that the child arrives at the principles on his own. One understands concepts that arise from within in a completely different manner than something one is told. We do not truly understand what we are told, unless we experiment on our own. It is important to remember: arising from within the child does not mean abandoning the child to his own devices, to discover the principles alone. Rather, it means mediation. Namely, gentle guidance towards the principles, by means of questions and indication of phenomena. The most important thing is that the children experience the Aha! sense of understanding on their own.
Sixth Principle: Explicit and Accurate Phrasing
Of all the novelties that awaited me in elementary school, the most surprising was discovering the importance of explicit phrasing, and even more how much the pupils love them. Explicit phrasing enables communication relating to concepts, clarifies them, and stabilizes knowledge. Above all, explicit phrasing of a mathematical principle enables one to use it, at a later stage. Words are footholds for thought. Accurate wording is the stage before last in acquiring a mathematical principle, the last stage being knowing how to operate it.
Needless to say, the phrasing should also arise from within the children, with guidance. They should also understand the origin of the names of concepts. For example, why is the unknown factor in an equation called unknown? Why is the denominator called so?
The same rule mentioned when discussing experimentation is also true of phrasing: it should be applied even to the simplest of principles. Even the obvious must be stated explicitly.
Teaching Long Division in the Third Grade
The following is my experience with teaching long division in the third grade. In the past, if long division was taught at all in Israel, it was introduced in the fourth grade, at the earliest. As will we shall try to show, the subject isnt too difficult for third graders, either. This is true on two conditions: the first, that the children are prepared (the meaning of this preparation will be explained immediately), and the second that the textbook, or teacher, or preferably both, know how to break down the process. In this article, we will try to break the algorithm into the smallest possible stages.
A. Dividing with a Remainder
The long division algorithm is based, first and foremost, on dividing with a remainder. This is the first stage of the process, and thus, we must open with it.
The concept of the remainder should be taught from the start when teaching division. When first experimenting with division, it is better not to arrange for the result to be an integer. 2 children can be given 7 popsicle sticks to divide among themselves, or 5 children can divide 12 caps. The children will discover on their own that some of the items remain undivided, thus providing a chance to discuss what to do with these. One possibility is to divide each of the remaining items, and the other option is to leave them aside, as a remainder.
As part of the introduction to long division, a simple, but necessary, step must be taken: writing division with a remainder in the format of long division. This relates to division with a remainder within the limits of the multiplication table, that is, where the result is smaller than 10, and long division is not really necessary, as yet. For instance:
6 32 5 30 2
Notice that there is no true long division here. Long division becomes meaningful only when the result is a two-digit number. What we have here is only a way of writing. Nonetheless, this is an important step, since this method of calculting the remainder is a crucial stage in understanding the algorithm.
In one third grade class, I began the first lesson on long division with the concept of the remainder. I asked the children what is division, and following a short discussion, I asked them for a story that requires division. As expected, they told me a story of division without a remainder. Then I told them: I am going to tell you two division stories. It is your job to discover the difference between them. The first story: a mother had 5 sandwiches, and she divided them between her two children. How many did each child receive? The second story: a mother had 5 marbles, and she divided them between her two children. How many did each child receive? After a short discussion, the children arrived at the conclusion: in the first story each child received 2 sandwiches, and the one that was left over could be divided into two, leaving each child with two and a half sandwiches. In the second story, the remaining marble cannot be divided, and therefore we leave it aside a remainder.
I asked the children to tell me more division stories in which the remaining items cannot be divided after the equal division, requiring those items to be left as a remainder. Amongst others, we arrived at the example of there were seven children and they had to divide into two groups is it possible to divide the remaining child? We told the story of King Solomons judgment.
Now I told the children: you know, there are many markings for division. It can be written as 7 2, or as EMBED Equation.3 , and some use the marking:
7 2
The result is written on top, as follows:
3
7 2
Then I took 7 markers and asked one of the children to demonstrate how she divides them between two children. She divided the markers into 2 sets of 3, and left one aside. I asked her if you have 2 sets of 3, how many markers did you divide? 6, she answered. How do you know? I did 2 times 3. And how can you tell how many you didnt divide? You had 7 to begin with, and you divided 6, how many did you leave undivided? 1, she said. What exercise did you just do? 7 6, she said. At this point, I wrote the 6 under the 7, and added the subtraction:
3
7 2
6
1
Now I asked why did we write the 6 below the 7? With a little bit of guidance, the arrived at the answer: since we are used to writing subtraction vertically. Is it really easier to calculate 7 6 when written vertically? No, but when we get to more complicated exercises it will indeed be easier.
Lets stop here, and ask ourselves what principles we used thus far. First of all, we broke into stages we preceded the algorithm itself with an important stage. Second we made it concrete. We explained what a remainder is, and when it is necessary in real life, and then we accompanied the division of abstract numbers with the real division of a set of markers. Third we let matters arise from within the child. The children themselves discovered why the process is written vertically. Fourth we used explicit phrasing. We explicitly mentioned the fact that division can be written in different ways, and discussed why it is written vertically.
Lets return to the lesson. The next stage was to repeat similar exercises, in which only this part, the calculation of the remainder, was extracted from the long division process. I believe in an additional, slightly odd, teaching rule: repeat an idea ad nauseam. Namely until the pupils feel it is extremely simple, until they feel you are exaggerating. The best sign of saturation is when they, and you, begin to laugh. Satiety of a principle means it has been internalized.
I didnt continue teaching division in that class, but if I had continued, I would have asked the children in following lessons more and more about exercises of this type: 11 2, 12 2, 19 2, 23 3, 57 8, and so forth.
What we did here was to isolate the first stage of long division: through the calculation of the remainder, by multiplying the result by the divisor and subtracting the product from the dividend. What kind of preparation is required from the pupils for this stage? One requirement is the understanding of the meaning of the division, multiplication and subtraction operations. Another is the ability to tell arithmetic stories about these operations. And the third, the ability to phrase mathematical principles and analyze their reasons.
Is the stage described above too simple? Not at all. As mentioned before: nothing is too simple to be taught explicitly. Once the foundation is firmly laid, it is possible to continue building.
B. The Distributive Law
The second component of long division is the distributive law of division: 24 2 are 20 2 plus 4 2.
Here, too, it is important to begin with concrete experimentation. Give a child 6 markers and 8 popsicle sticks, and ask him to divide them between 2 children. How many will each child receive? The children will discover: 3 markers and 4 popsicle sticks. If so, how many are 6 markers and 8 popsicle sticks divided by 2? The answer: 3 markers and 4 popsicle sticks. Now tell them: this rule has a special name the distributive law. To distribute means to separate into parts.
C. Separating the Two Components of the Algorithm The General Structure and The Division at Each Stage
We now arrive at the algorithm itself. Long division consists of stages: each stage includes a division operation within the limits of the multiplication table (that is resulting in a number smaller than 10) and the calculation of the remainder what has not yet been divided. The pupil will face two difficulties: first understanding the general structure of the process, namely, where the various stages lead; second performing the division at each stage.
This brings us to the third didactic principle mentioned earlier: divide and conquer. These two components must be separated. This can be done, initially, by neutralizing one of the two the performance of division within the limits of the multiplication table. This enables concentrating on understanding the other component, which is the general structure of the process. To do so, we use a simple trick: begin with dividing by 2. When dividing by 2, each stage of the division is especially simple anyone can perform divisions such as 16 2, where the dividend is smaller than 10. Therefore, when teaching long division, it is important to dwell on dividing by 2 for a prolonged period of time.
Dividing by 2. First Case: All Digits Are Divisible by 2.
Sub-Case: Dividing an Even Number of Tens
Ask the pupils: what are 60 2? The will immediately answer: 30. How do you know? Because 60 are 6 tens. Draw 6 sets of 10 items on the blackboard and ask what happens when they are divided between 2 children. They will tell you each will receive 3 of them. Now move on to a slightly more abstract stage: draw 6 coins of 10 and ask what happens when they are divided between 2 children.
This should be repeated again and again, 40 2, and 160 2, 280 2. The mathematical principle taught here is: one ten is like one apple. Dividing whole tens is like dividing apples.
D. Combining the Distributive Law with the Principle of Dividing Whole Tens
The next stage is to examine an exercise such as 68 2. The children immediately know that the result is 34. Why? Because 60 2 are 30, and 8 2 are 4. Its best to return to drawings. Draw 68 objects on the blackboard, symbolically: 6 circles (coins) with the number 10 written inside, namely 6 tens, and 8 ones. Now lets divide the 6 tens between two children is this possible? Yes. Each child will receive 3 tens. And what about the ones? Each child will receive 4 of the ones. How many will each child receive altogether? 3 tens plus 4 ones.
Does this remind you of anything? Thats right, its very similar to what we did when we divided 6 markers and 8 popsicle sticks between two children. What was the name of the rule used to divide two objects of two different types? The distributive law.
More practice is required here: what are 46 2? 82 2?
The next stage we arrive at is phrasing the rule. Can you tell me how to calculate 68 2? Thats right, each digit is divided by 2. And what about 268 2?
Can you phrase a general rule? The teacher will most likely have to lead the class to the formulation: What happens when we divide by 2 a number whose digits all divide by 2? The children will find the answer the quotient is a result of dividing each digit by 2.
E. The General Case: Long Division Begins on the Left!
Now we have all the components. All that is left is to join them together.
Begin with dividing a two-digit number greater than 20 by 2. For example, 74 2. Write down the exercise, and next to it draw 7 coins of 10 alongside 4 lines, representing the 4 ones. The writing of long division should be accompanied concomitantly by this drawing.
Long division begins on the left. That is, we begin by dividing the tens. Why? Well explain this later. Lets try to divide the 7 tens. What happens if we divide 7 tens between 2 children? Each one will receive 3 tens. If so, 7 2 = 3. But not all 7 tens have been divided: only 3 x 2 tens have been divided, namely, 6 tens. How many are left undivided? 7 6 = 1. Lets write this like we learned earlier.
3
74 2
6
1
Why did we write the 3 above the 7? Because it represents the tens. Why did we write the 6 beneath the 7? Because it represents 6 tens. These are the 6 tens already divided. How many tens are left undivided? One ten, and what else? The 4 ones are also undivided. One ten cannot be divided into 2, at least not as a whole ten. What can be done? The ten can be changed for ones. How many ones are there together with the 4? 14. This is apparent as the 4 is moved downwards, thus:
3
74 2
6
14
We still havent divided the 14 ones by 2. Remember its best to show this through the drawing accompanying the exercise! The drawing shows one ten, left over as a remainder, plus 4 ones. When divided, the result is 7 ones. This is written in the ones digit place:
37
74 2
6
14
This should be repeated with more and more examples.
I assume that from here on any teacher familiar with the algorithm can continue on her own. Here is an additional example, including hundreds as well: 758 2. Again, it is best to accompany the process with a drawing, where 758 is represented by 7 coins of 100, 5 coins of 10 and 8 ones. Begin on the left, that is, by dividing the hundreds. 7 hundreds divided by 2 are 3. This leaves us with one undivided hundred. It cannot be divided as a whole hundred. Therefore, it is changed for tens. But 758 already contain 5 tens, represented by the second digit. Added to the 10 tens obtained when changing the hundred, there are 15 tens. When divided by 2, the result is 7 tens. This is the tens digit of the result.
In the formal writing of long division:
3 758 2 6 15
What are the 15? If accompanied by a drawing, the children will know immediately: 15 tens. 15 tens divided by 2 are 7 tens. This is written in the tens digit place of the result. Now only 14 tens have been divided. How do we know? Multiply 7 by 2. In the writing of the algorithm:
37 758 2 6 15 14 1
What is the remainder? It is the single undivided ten. Can it be divided? No. But it can be changed for 10 ones, and added to the 8 ones of the 758, which are also still undivided. There are now 18 ones, which, when divided by 2, result in 9 ones. This is written as the ones digit of the result, thus:
379 758 2 6 15 14 18
Thats it!
After mastering division by 2, the road is paved. Long division by 3 and 4 should also be practiced. Then, it is possible to introduce the general case. Since there are no new principles involved in teaching more complicated cases, we will make do with those mentioned thus far.
Why Do We Begin on the Left?
Finally, it is a good idea to discuss why the calculation of division begins on the left with the children. Children who discussed a similar issue when learning other operations why the calculation of addition, subtraction and multiplication begins on the right wont find this difficult. Addition, for example, begins with the rightmost digit, that is, the ones digit, since the ones digit of the result can be calculated immediately and written down, without concern of having to retrace our steps at a later stage. In division, the opposite is true: the digit that can be calculated immediately, without concern of having to retrace our steps, is the leftmost digit.
As an example, lets return to the exercise 758 2. How many ones will the result include? The answer is not obvious. It is not necessarily 8 2, namely, 4 for we have seen that the result actually includes 9 single ones! But the number of hundreds is obvious. There are 7 hundreds in the dividend, and when 7 are divided by 2 the answer is 3. Therefore, the hundreds digit of the result is 3. Nothing will change this further on: the division of the tens by 2 will not add hundreds to the result. Therefore, it is best to begin with the calculation of the hundreds digit.
Summary
Is it really possible to teach all this to children in the third grade? I believe so, on two conditions. First, that the children completely internalize the decimal system beforehand, by actively grouping tens, hundreds and thousands. Second, that the teacher herself be familiar with all the stages. This is not beyond reach of an ordinary teacher. Obviously, she must go through the same stages as a pupil, and the place to do so is at the Teachers Seminar. That is the place to teach long division systematically and in depth.
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