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.

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.

M3N4d - Meaning of a remainder in division

M3N4. Students will understand the meaning of division and develop the ability to apply it in problem solving.
d. Explain the meaning of a remainder in division in different circumstances.
In an earlier post (August, 2007), I discussed the relationship between the two division situations, fair sharing and measurement (discussed in element b) and multiplication situations. When students are first introduced to division, they must understand that the division is an operation needed when you are making equal groups with the given amount. We can use the division to find the number of groups given the number in each group (measurement division), or we can use the division to find the number in each group given the number of groups (fair sharing). We want students to develop a unified understanding of these two situations as "division."

In the introductory stage of division instruction, students focus on those division problems that are the inverse of the basic multiplication facts. Thus, they develop the strategy to find the quotient of division by looking for the related multiplication facts. For example, to solve 48 ÷ 6, students think about 6 and what multiplied together will equal 48. Since 6x8=48 (or 8x6=48, depending on the type of division), we can say the answer is 8. Thus, if we are giving each person 6 candies, 48 candies can be shared by 8 people.

What if we had 50 candies? There is no multiplication fact with 6 as a factor that will give the product of 50. Most children can solve this problem if they are allowed to use concrete materials to manipulate. As division with remainders is introduced, it is important that students initially use concrete materials to model the problem situation. They should compare and contrast the situation with earlier division situations (without remainder) to realize that these situations are also creating equal groups, thus it is appropriate to represent it with division sentences. Furthermore, implicit in division problem is to maximize the quotient (either the number of people sharing or the number of items for each person) - that is, the point isn't just to make equal groups but to use up as many of the given amount. The "remainder" is the amount left over when the maximum amount of the given amount is used up.

When students simply rely on computation (multiplication) to find the remainder, sometimes you see mistakes like 50÷6=9 remainder 4. This occurs because when you look for the quotient by checking the 6's multiplication facts in order, you recognize the quotient only after the product exceeds the total. Thus, some children mistakenly think that the quotient is 9.

If students do not understand the meaning of the remainder clearly, the opposite can also happen. Some students may say that 50÷6=7 remainder 8. Those students do not understand that we have to use up as many of the given amount. Since this is not a computation error (in the sense of getting an incorrect product or incorrect difference), the answer checking algorithm (Dividend = Divisor x Quotient + Remainder - Gr. 4 GPS) will not detect it. To avoid this error, students must clearly understand the meaning of remainders. Furthermore, since we are using up as many of the given amount, the remainder cannot be greater than the divisor - in either type of division situation. If the left over amount is greater than the divisor, that means we can give (at least) one more person his/her share (measurement division) or (at least) one more item to each person (fair sharing). But, this relationship, Remainder is less than Divisor, is the result of the often implicit requirement of division that we must use up as many of the given amount as possible.

By the way, in many Japanese elementary mathematics textbooks, the long division notation is introduced when students are learning division with remainders - after students understand the meaning of division, remainder, and the relationship between the remainder and the divisor. This notation is introduced to ease the mental demand involved in division with remainders. For example, in the case of 50÷6, children must identify the first multiplication facts with 6 as a factor that exceeds 50, 9, then subtract 1 from it to make 8 as the quotient, find the product of 6x8, then subtract the product, 48, from 50 to find the remainder. The long division notation can provide a way for children to record the intermediate steps and procedures more explicitly. However, it is important to remember that we are not really teaching children the long division algorithm here. The notation is simply introduced as a way to deal with simple division with remainders (that is, those division that requires the application of the multiplication facts only once). I will discuss the development of the long division algorithm, element e, in a separate post.

M1N2 - Understanding place value notation

M1N2. Understand place value notation for the numbers between 1 and 100. (Discussions may allude to 3-digit numbers to assist in understanding place value.)

Most, if not all, elementary school teachers know that understanding of place value is the key to success in elementary school mathematics. But what does it mean to understand place value? How did we get stuck with such a complex notation system? What benefits are there that this particular system offers that other system didn't?

Probably the simplest number notation system is the tally system. It may have started with people carrying around some pebbles (or acorns or whatever), but eventually became a written system. If you see "|||||||||||||" you can actually "see" the number. Unfortunately, when the number gets large, it becomes difficult to distinguish numbers. Soon, people started to come up with simplified system, such as noting 5 as ||||. A further extension is a system like the Egyptian system where they used a new notation for 10, 100, etc. With those notations, it became easier to distinguish large numbers, but numbers themselves are no longer "visible."

One of the shortcomings of a system like the Egyptian system is that you need, in theory, infinite number of symbols in order to express very large numbers. Other people, like the Babylonians and the Mayans, came up with a system in which where a numeral is written also contributed in the way the number is written. The picture below shows how the Mayan system worked:

Each group of symbols actually represents a number between 0 and 19 inclusively. However, where it is written makes difference - so, "17" in the second from the bottom is in the 20's place, so it actually represents 340. Therefore, this system is like ours in that they used place values.

So, why didn't this system survive? It used only 3 symbols: 1 (dot), 5 (horizontal bar), and 0 (sea shell). Our system requires 10 numerals (0 through 9). One of the reason why this system did not survive is probably because of this economy of the symbols. When you have 5 in one place and another 5 in the adjacent place, it is difficult to distinguish that from 10 in one place.

So, it appears that our history can be characterized as the search for a balance between simplicity and complexity. Our current numeration system, typically called the Hindu-Arabic system, eliminated the confusion of the Mayan (and the Babylonian) system by using more symbols but a smaller exchange rate for adjacent places. So, here are the major "rules" of our number system:
1. Where a numeral is written matters, i.e., "1" in 31 and 15 represents different numbers.
2. Any pair of adjacent places have 10-to-1 relationship, i.e., you need 10 of the smaller place value to exchange with 1 of the larger value - therefore, the place values are all powers of 10.
3. The total value of a number is determined by multiplying the numeral by the place value and finding their sum.
4. There must be one and only one numeral in each place.
5. Because of (4), we must use "0" as a place holder - except for the leading 0's (and trailing 0's in decimal numbers).
(4) and (5) are often learned in the process of learning how to record addition/subtraction using written algorithms. Most teachers have seen children who tried to write "12" in the ones place when they add 35 + 17. Up till that point, (4) is not made explicit so children do what is most natural thing to do. In fact, (4) is the reason why we have to worry about re-grouping. And now we seemed to have introduced another complexity to our numeration system.

So, what are the merits of our number system? Probably the biggest merit of our number system is that calculation is simple. Wait a minute! we just said because of a rule of our system, we had to worry about re-grouping, a difficult idea for many children.

Yes, it is true that re-grouping is difficult, but with our system, if we know the basic addition and multiplication facts, we can do any calculation. Just think about this for a minute. If you know 100 addition facts (0+0 through 9+9) and 100 multiplication facts (0x0 through 9x9), you can do ANY calculation, no matter how large or how small numbers are. Just imagine how you would calculate 34x72 using the tally system, the Mayan system, the Roman numerals, etc.. So, one important, and often implicit, goal of teaching children computational algorithms is to help students understand this merit of our number system.