M3N3d - Developing multiplication algorithms (6)

M3N3. Students will further develop their understanding of multiplication of whole numbers and develop the ability to apply it in problem solving.
d.Understand the effect on the product when multiplying by multiples of 10.

This standard talks about multiplying by multiples of 10, for example 37x30. This situation is different from multiplying multiples of 10, 100, etc. (which we have discussed in a previous post) because we now have 30 groups of 37. Now, if we study this idea after students have already developed a paper-and-pencil algorithm, these problems can be considered as a special case where there will be a 0 in the product. So, procedurally, there are different ways to deal with these problems. Some will carry out the calculation exactly in the same manner as they do with other multipliers:

After students get used to this calculation, they might try to combine the steps to make it more efficient:


From this perspective, this multiplication isn't much different from something like 35x18. The important idea is that we have to write a 0 in the ones place as a place holder.

However, M3N3d states that students must understand "the effect on the product when multiplying by multiples of 10." Moreover, according to the GPS, students do not study how to multiply by 2-digit number until Grade 4 (next post). So, it seems rather odd to talk about multiplying by multiples of 10, which are 2-digit number, at this point. If students' don't know how to multiply by 2-digit number, then we can't focus on the procedural aspect discussed above. Rather, we want students to understand what is going on when we multiply by multiples of 10. Although we cannot use the idea of 10 as a unit in the same way as we did when we were multiplying multiples of 10, we can still use the idea of 10 as a unit when the multipliers are multiples of 10. For example, you can think of 37x30 as 37x3x10. Alternately, you can think of 37x30 as 37x10x3. Either way, multiplying a 2- or 3-digit number by 3 is something students have already learned. What students may not have studied is multiplying 2- (or 3-) digit number by 10. So, that seems to be the primary focus of this standard.

As we explore multiplying 2- and 3-digit numbers by 10, we may again want to go back to the area model of multiplication. For example, if students are to model 17x10 using base-10 blocks, they might at first construct something like this by simply extending what they have done previously:

At this point, some might notice that we can actually use a flat on the left side since there are 10 longs. Moreover, on the right side, since there are 10 rows of units, we can replace each column by a long, resulting in an arrangement like this:

Students can also record the process more abstractly like this, too:

They can also consider cases like 40x10 by extending their thinking of 40 as four 10's. If you have 10 groups of four 10's, you can think of that as 4 groups of ten 10's as well, or 100x4.

From these exploration, students may notice that when you multiply 2- and 3-digit numbers by 10, the product will contain the same set of numerals in the same order but every numeral is moved one place to the left - and there is a 0 in the ones place as a place holder.

Although we may be able to consider multiplying by multiples of 10 as a special case of multiplying by 2-digit numbers, students still need to learn the effect of multiplying by 10 before they can explore multiplying by 2-digit numbers. Moreover, once you study the effect of multiplying by 10, extending it to multiplication by multiples of 10 may be useful to help students deepen their understanding of multiplication operation. Although the formal study of properties of multiplication is done in Grade 4, Grade 3 students can use the associative property to reason about the effect of multiplying by multiples of 10. I believe that's the point behind this standard, not just the procedural fluency.

M3N3c - Developing multiplication algorithms (5)

M3N3. Students will further develop their understanding of multiplication of whole numbers and develop the ability to apply it in problem solving. c. Use arrays and area models to develop understanding of the distributive property and to determine partial products for multiplication of 2- or 3-digit numbers by a 1-digit number.
In the previous post, I discussed how students can develop a paper-and-pencil algorithm for multiplying 2-digit numbers by 1-digit numbers. Let's consider how we can help students extend the procedure to multiplication of 3-digit numbers by 1-digit number.

How can we multiply 312 x 3? How can students use what they have learned so far to calculate this? One possibility is to think of 312 as 300+12. Then, we can multiply 300x3 and 12x3. Both of these are already learned ideas. If students have already understood how to multiply a 2-digit number by a 1-digit number using a paper-and-pencil method, they can then combine their learning and record this multiplication something like this:

When extending the multiplicand from 2-digit to 3-digit, therefore, there isn't really any new concept involved. Even the idea of looking at 312x3 as 300x3+12x3 is really the same idea as looking at 12x3 as 10x3+2x3, i.e., the distributive property of multiplication, which will be formally studied in Grade 4.

One important thing to think about when we study multiplying 3-digit numbers by 1-digit numbers is different situations where re-grouping must take place, or when there is a 0 (or more) in either the multiplicand or the product. The example we just saw, 312x3, does not involve re-grouping and there is no 0 in the multiplicand nor the product. So, in a way, it is a "general" case of multiplying 3-digit numbers by 1-digit numbers. But, here are some of other cases:
Re-grouping is involved
• 227x3
• 227x5
• 162x3
• etc.

0 is involved
• 406x7
• 365x4
• 527x4
• etc.
I encourage you to think about other cases. As teachers, we must also think about how we want to deal with them. We can carefully sequence those cases and have students think about how they can adapt the written procedure they developed those situations. As you do, it will be helpful if you explicitly ask students what is different about each case compared to the most general one that we start with.

As we look at those special cases, it is important that students understand what is actually happening when we are multiplying 3-digit numbers by 1-digit numbers. For that, it might be useful to go back to the notation system that we used when we developed when we were multiplying 2-digit numbers by 1-digit numbers. For example, let's think about 427x4. Since we can think of 427x4 as 400x4+27x4, and we can use a pictorial notation like this:

Or, we can use more symbolic notation like this (with the previous agreement that we start recording with the partial product of the ones digits first):

We can combine some of the steps involved in this notation and develop a notation like this:

No matter how you approach this topic, what we cannot do is to start with the standard algorithm, which is the most sophisticated way of recording the processes. Help students extend what they have previously learned, which may be the standard algorithm for multiplying 2-digit numbers by 1-digit number by thinking about the structure of numbers and the meaning of operations. If necessary, go back to the intermediate notations that were used while developing the algorithm for multiplying 2-digit number by 1-digit numbers. By experiencing this extension, students can then think about how they can extend the algorithm for multiplying 3-digit numbers by 1-digit numbers to multiplying 4-digit (or even longer) numbers by 1-digit numbers. They have not only the experiences of multiplying two numbers but also the experience of "extending" their procedure from one case to another. So they can ask themselves not just "How did I multiply 2- or 3-digit numbers by 1-digit numbers?" but also "How did I extend the algorithm for multiplying 2-digit numbers to 3-digit numbers?" Therefore, when teaching multiplication of 3-digit numbers by 1-digit numbers, what is important is not the procedure but the idea of how to extend the previously learned procedure (2-digit multiplicands) to a new situation (3-digit multiplicands).

M3N3c - Developing multiplication algorithms (4)

M3N3. Students will further develop their understanding of multiplication of whole numbers and develop the ability to apply it in problem solving. c. Use arrays and area models to develop understanding of the distributive property and to determine partial products for multiplication of 2- or 3-digit numbers by a 1-digit number.
This is the fourth in a series of posts in which I am discussing the development of multiplication algorithms. Up to this point, students were calculating mentally. The focus has been more on consolidating students' understanding of our number system and the meaning of multiplication by using those understanding to figure out multiplication beyond the basic facts. Today's standard is the first step toward developing paper-and-pencil algorithms. As I begin my post, let me emphasize that teaching of an algorithm for any operation should focus on helping students develop the algorithm on their own. In other words, we need to move away from the show-and-tell approach where teachers show students how to multiply using the multiplication algorithm and then have them practice over and over. Practice is important, but students should first develop the algorithm themselves. Of course, that does NOT mean that we just leave students on their own. Rather, teachers must plan carefully to guide students' thinking.

One useful idea in developing a multiplication algorithm is the area model of multiplication. In Grade 3, students learn about area of rectangles and squares. When students cover a rectangle with unit squares, they notice that they are arranged rows and columns of equal sizes. Because all rows (or columns) are equal, we can use multiplication to efficiently determine the area. This idea can be used to model multiplication where the two factors are represented by the two dimensions of a rectangle and the product is represented by the area. So, for example, 4x6=24 can be modeled as shown below.

Notice that since you can turn the rectangles around without changing the area, this is also a useful model to show why the commutative property of multiplication is true. It is also useful to model the distributive property of multiplication.

When you model multiplication problems like 14x7 using base-10 blocks, you can certainly try to make 7 groups of 14 (1 long and 4 units). However, we want to encourage students to organize the model more systematically using the area model. The area model representation will make it much easier to determine the product by observation if the same type of blocks are grouped together.

Eventually, we want to help students move beyond modeling with actual base-10 blocks. One useful approach to do so is to have students draw what they would have done with base-10 blocks. Thus, drawing the picture like the one above. Grid papers can be very helpful in that process. However, as they become comfortable with drawing pictures, they realize that drawing can be rather tedious. Given our goal is to determine the product, what we want to know is how many longs and how many units we have. Thus, we can model the multiplication explicitly showing only the information we need. Here is an example for 14x7.

Once students become comfortable with modeling multiplying 2-digit number by 1-digit number this way, we can ask if they can think of a way to represent this model using a vertical notation like we did with addition and subtraction. Here are two possibilities:

Students can see that 70+28 and 28+70 are the same. Thus, we can write it either way. At this point, it is ok to suggest that we agree to write the product of the ones digit first. Again, after students practice this notation, they might notice that the ones digit for the partial product of the tens digit on the multiplicand and the multiplier is always 0. Therefore, the ones' digit of the product is always the ones digit of the partial product of the ones digit of the multiplicand and the multiplier, 8 in the example above. Then, we have to add the tens digits of the partial products to find the product. This process can be combined if you use a notation like this:

This may be a slightly different notation than some of us are used to, where the tens digit of the partial product above the tens digit of the multiplicand. That notation sometimes causes students to add the re-grouped digit and the tens digit of the multiplicand before multiplying by the multiplier - that is students end up doing (2+1)x7 instead of 1x7+2. Writing the re-grouped digit below the horizontal bar (the equal sign) might minimize that error.

In the next post, I will discuss how this procedure may be extended to multiplying 3-digit numbers

M3N3 Developing multiplication algorithms (3)

M3N3. Students will further develop their understanding of multiplication of whole numbers and develop the ability to apply it in problem solving.

This is the third in a series of posts in which I discuss the development of multiplication algorithms in Grades 3 and 4. If you haven't read the first two, I encourage you to do so, either before or after reading this post.

One idea that is important as children continue to develop multiplication algorithms yet not explicitly mentioned in the GPS is the idea of multiplying multiples of 10 and 100, such as 40x7 and 600x3. Note, multiplying by multiples of 10 and 100 is a Grade 4 standard.

So, how can students make sense of multiplying multiples of 10 and 100? So, let's think about 40x7. Again, just a reminder that I am following the Japanese convention of writing the multiplicand (number in a group) first. Therefore, this problem is asking us to find the total amount when there are 40 groups in each and there are 4 groups.

An important idea here is the understanding of 10 and 100 (and 1000) as unit. When students first learned simple addition and subtraction of 2-digit numbers such as 30+40 and 70-20 in Grade 1, they used the idea of 10 as a unit. Since 30 and 40 are made up of 3 and 4 tens, putting those two numbers together meant there are seven 10's, or 70. In a similar way, we can think of 40x7 using 10 as a unit. Since there are 7 groups of 40, or 7 groups of four 10's, we see that there are 7x4=28 tens altogether. Therefore, the product is 280. Similarly, you can think of 600x3 as 3 groups of six 100's, or 3x6=18 hundreds, i.e., 1800.

The idea of 10 and 100 (and 1000) as units was the focus of M2N1(b):
Understand the relative magnitudes of numbers using 10 as a unit, 100 as a unit, or 1000 as a unit. Represent 2-digit numbers with drawings of tens and ones and 3-digit numbers with drawings of hundreds, tens, and ones.
Although the second part of the statement emphasizes looking at 3-digit numbers as composed of hundreds, tens, and ones, it is also important for children to understand numbers like 280 as twenty-eight 10's. This notion of relative magnitude ("relative size" in M3N1(b)) plays an important role in students' mathematics learning in the future. Therefore, it is important to help them deepen and consolidate their understanding as we teach multiplication of multiples of tens and hundreds.

M3N3(b) - Developing multiplication algorithms (2)

M3N3. Students will further develop their understanding of multiplication of whole numbers and develop the ability to apply it in problem solving. b. Know the multiplication facts with understanding and fluency to 10 x 10.
The new idea here is multiplication with 10 as a factor, either the multiplicand or the multiplier. Let's first look at the cases where 10 is used as the multiplicand, i.e., 10x1, 10x2, 10x3, .... These can be interpreted as one 10, two 10's, three 10's, ... respectively. In Grade 1, when students studied the numbers up to 100, this is something they should have encountered. Thus, they can use that particular prior knowledge to figure out what these facts will be, i.e., 10x1=10, 10x2=20, 10x3=30, ....

What about 10 as the multiplier, i.e., 1x10, 2x10, 3x10, .... These means, respectively, ten 1's, ten 2's, ten 3's, .... For adults, these are obvious and we might think they should be obvious to children, too. However, although young students can answer the problem on the left very quickly, many of the same students have much more difficult time with the problem on the right.

So, how can students think about problems like 3x10? Hopefully, when they were constructing their multiplication table, they have used the idea that when the multiplier increases by 1, the product increases by the multiplicand. For example, the answer for 3x6 should be 3 (the multiplicand) more than 3x5. This idea, then, can be used to think about 3x10. The answer to 3x10 should be 3 more than 3x9, which is a part of the basic fact they have learned in Grade 2. This idea is really a particular case of the distributive property, which students will formally study in Grade 4. However, the distributive property plays an important role as students think about how to multiply by larger numbers. Therefore, it may be useful if this idea is discussed explicitly in classrooms.

Some of us grew up memorizing the multiplication table up to 12x12. Even today, some teachers/schools/districts still make their students consider the multiplication table up to 12x12. Although it may have some usefulness in everyday situations to know the multiplication facts of 11's and 12's, there is really no particular mathematical reason for expanding the multiplication table to 12x12. Once students develop an algorithm for multiplying by 2-digit numbers, they can calculate anything beyond 10x10 using the algorithm. On the other hand, students can also use the property of multiplication and think of 11's and 12's as simply 10's and 1's or 10's and 2's. Thus, 7x12 can be thought of as 7x10+7x2 and 12x8 can be thought of as 10x8+2x8. Perhaps it is much more important for students to develop that form of flexible thinking than simply memorizing those facts.

M3N3/M4N3 -- developing multiplication algorithms

M3N3. Students will further develop their understanding of multiplication of whole numbers and develop the ability to apply it in problem solving.
M4N3. Students will solve problems involving multiplication of 2-3 digit numbers by 1 or 2 digit numbers.

In the previous 2 posts, I discussed division algorithms. So, in the next few posts, I would like to discuss multiplication algorithms. Today, as the first entry on multiplication algorithm, I want to discuss an overview of teaching and learning multiplication algorithms.

Students are introduced to multiplication in Grade 2. The GPS (M2N3) states that students should construct the multiplication table and correctly multiply 1-digit numbers. What is not quite clear is where multiplication involving 0 as a factor (either the multiplicand or the multiplier) should be discussed. Many US textbooks introduce multiplication with 0 and 1 as factors fairly early on in their discussion of multiplication. In contrast, in the Japanese textbooks, multiplication with 1 as the multiplicand, i.e., 1x1, 1x2, 1x3,..., are discussed AFTER students study the 9's facts. [Note: in the Japanese notation, the first number is the multiplicand, i.e., the number in a group.] They do not discuss 0 as a factor until the 3rd grade. They do this because the emphasis in Grade 2 is developing the meaning of multiplication first. For children, considering 1, or even 0, item as a "group" may be strange. From the equal group perspective of multiplication, therefore, 0 and 1 as the multiplicand are special cases. Therefore, they start with more general cases first (2's through 9's), then discuss the special cases (1's and 0's). Textbooks often treat 0's and 1's early because getting the answers is easy. However, if our focus is on the meaning of multiplication, that may not be a wise choice.

Anyway, after students study 1x1 through 9x9 in Grade 2 (and possibly 0's), students are expected to learn to multiply larger numbers in Grades 3 and 4 (M3N3 and M4N3). So, by the end of the 4th grade, we want students to be able to calculate problems like 512 x 43. Using the conventional algorithm, we can calculate this problem as shown below:



With this algorithm, we can calculate this problem by performing 6 basic multiplication and 5 basic addition. In fact, with our base-10 numeration system, once we learn the basic addition and multiplication facts, we can perform the basic 4 operations with any size numbers. Although this is not explicitly spelled out in the GPS, we would like students to understand this merit of our number system as a result of learning the computational algorithms.

Of course, this is by the end of Grade 4, and we have to think about how to help students go from knowing only the 1-digit multiplication facts to that point. So, how should we organize our instruction? What are some important mile markers in this endeavor?

Here are some important understandings students need.
* We can think of 512x43 as 512x40+512x3.
* 512x3 can be thought of as 500x3+10x3+2x3.
* 512x40 can be thought of as 512x10x4.

The first idea involves the use of the distributive property. Although the formal study of the properties of operations is in Grade 4, students use the distributive property as they construct the multiplication table. For example, they might have thought of 7x6 as7x5+7. Or, they thought of 8x7 as 8x5+8x2. So, this is not a completely new idea. However, multiplying 512x3 certainly is. So, they need to learn how to multiply 2- and 3-digit numbers by 1-digit number. The third idea uses the associative property of multiplication. Students my have used it to find something like 7x4 as 7x2x2. So, the use of property itself may not be new, but 512x10 certainly is. Students must learn how to multiply numbers by 10 before they think about multiplying a number by multiples of 10.

So, from this example, we can see 5 important mile markers of multiplication instruction in Grades 3 and 4.
a. Expand the basic multiplication up to 10x10. [M3N3b]
b. Understand how to multiply multiples of 10 and 100 by 1-digit number (e.g., 30x8, 400x6, etc.).
c. Understand how to multiply 2- and 3-digit numbers (but not multiples of 10 and 100) by 1-digit numbers. [M3N3c]
d. Understand how to multiply by multiples of 10. [M3N3d]
e. Understand how to multiply be general 2-digit numbers [M4N3]

As you can see, all but one mile marker is explicitly noted by the GPS. Starting next entry, I will discuss these 5 mile markers in more details.

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.

M5N3c - Multiplying and dividing by numbers less than one

M5N3. Students will further develop their understanding of the meaning of multiplication and division with decimals and use them. c. Multiply and divide with decimal fractions including decimal fractions less than one and greater than one.
Consider a problem like the following:
A ribbon costs $ 1.80 for one meter. If you want to buy 0.8 meter of the ribbon, how much will it cost?
When children are asked what operation they would use to solve this problem, many will pick division. Some might think those children simply do not understand multiplication and division. However, that is not the case. Those children pick division as the operation because they know that "multiplication make bigger and division makes smaller." Research has shown that many children, and adults, hold this misconception.

Actually, calling it a "misconception" may be inappropriate. Rather, it is an overgeneralization children make based on their experiences. While students are working only with whole numbers, the only exception to this generalization is multiplication by 0 and 1. However, once they go beyond the "basic facts" stage of multiplication learning, practically all experiences involve multiplying by a number greater than one. The same can be said of division. The only time division does not result in a number less than the dividend is when it is divided by one. However, once again, practically all children's experiences before this point are division by a number greater than one.

Once the range of numbers is expanded to include decimal numbers and fractions, however, there are many cases where we do multiply or divide by numbers less than one. Therefore, it is an important goal of mathematics teaching that our students overcome this overgeneralization. A potentially powerful tool for this purpose is double number lines. If we represent the ribbon problem, it will look like this:

From this diagram, you can easily see that if the multiplier (represented on the bottom number line) is less than 1, the product (? mark) will be on the left of the multiplicand (1.80, i.e., amount corresponding to 1). On the other hand, if the multiplier is greater than 1, the product will be to the right of the multiplicand. Therefore, we can generalize:
If multiplier is greater than 1, multiplicand < product.
If multiplier is less than 1, product < multiplicand.
Similarly, you can use double number lines to contrast the situations when the divisors are less than 1 and those cases where the divisors are greater than 1.

However, the most difficult part for students (thus for teachers) is to help them understand that these situations are indeed situations where multiplication is the appropriate operation. For some students, double number line may not be sufficient. Another possible tool is to write mathematical expressions using words to describe the relationship among the quantities involved. In the ribbon problem, there are three quantities: cost of 1 meter of ribbon, total length of ribbon, and the price. The relationship among these three quantities can be expressed as
Price = [Cost of 1 meter] x [Total Length].Thus, for this problem, ? = 1.80 x 0.8.

An implicit, yet very important, goal of teaching multiplication and division of fractions and decimal numbers is to expand students' understanding of these operations. In early elementary grades, these operations are considered in equal group situations. Thus, when the multiplier or the divisor (in the case of fair-sharing division, the quotient in the case of measurement division) becomes something other than whole numbers, students have difficulty interpreting what it means. Through teaching of multiplication and division of decimal numbers and fractions, we want students to develop more proportional understanding of these operations. For example, A x B = C, should be interpreted as "A is to 1, C is to B," or "C is B times as much as A." Although this is not an explicitly stated goal in the GPS, it is something all teachers must keep in mind.

MKN1 h, i, j - Coins

MKN1. Students will connect numerals to the quantities they represent.
h. Identify coins by name and value (penny, nickel, dime, and quarter).

i. Count out pennies to buy items that together cost less than 30 cents.

j. Make fair trades using combinations involving pennies and nickels and pennies and dimes.

Quite frankly, I really don't understand why money and clock reading are in the mathematics curriculum. Those ideas should be learned in everyday contexts in which they mean something. The names and the values of each coin aren't mathematical concepts. However, whether or not I like these topics in the mathematics curriculum really matters much as teachers are expected to teach them. I heard many teachers say that money is a difficult idea for children. There may be a number of reasons for children to have difficulty with money. For one thing, being able to exchange a merchandise with a piece of metal or paper cannot be really "natural" to children. Another major source of difficulty for young children is the notion of exchanging coins. It's not quite logical why 5 shiny pennies can be exchanged with one, slightly larger coin, for example. Neither of these is mathematical ideas, and I'm not really sure how you can help young children with these ideas.

However, I do want to say something about exchanging coins. A part of the difficulty for young children is, I believe, because they have yet to develop a very sophisticated understanding of numbers. Kindergarten teachers are familiar with an exchange like the following:
Teacher: How many blue counters do you see?

Child: (counting to herself) One, two, three, four. Four !

Teacher: How many red counters do you see?

Child: (counting to herself) One, two, three. Three!

Teacher: So, how many counters are there altogether?

Child: One, two, three,...

Teacher: (interrupting). Wait a second. How many blue counters?

Child: One, two, three, four. Four.

Teacher: How many red counters?

Child: One, two, three. Three.

Teacher: So, how many counters altogether?

Child: One, two, three, four, five, six, seven. Seven!
Many adults are simply puzzled why a child will have to count all the counters when they know that there are four blue counters and three more red ones. However, for many young children, numbers exist only after they count - "four" cannot exist by itself. Furthermore, for many of them, four simply means four ones. Children must develop an understanding that four can be considered as an entity, or a unit, in itself before they can count on. In the previous post, I discussed the idea of five and ten as a benchmark. Children cannot think of five as a benchmark unless they can think of five as five ones and, simultaneously, one five. For adults, this is so obvious, and it is difficult to even fathom anyone (including children) not understanding it. However, research clearly shows that children don't automatically understand this idea.

So, if a child is still "counting all" stage, it is probably not reasonable to expect him/her to be able to make an exchange of five pennies with one nickel with understanding. One way to help children overcome difficulty with money is to help them develop good number sense - the ability to see numbers flexibly. Without number sense, money is much more difficulty to make sense.

MKN1f - Five and ten as benchmarks

MKN1. Students will connect numerals to the quantities they represent.
f. Estimate quantities using five and ten as a benchmark. (e.g. 9 is one five and four more. It is closer to 10, which can be represented as one ten or two fives, than it is to five.)

Although this indicator includes the words "estimate," what it is talking about isn't really estimation in the sense of "about how big" a number is. Rather, it is more about looking at a number from different perspectives. Thus, 8 isn't just eight ones, but rather, it is three more than 5 and two less than 10 as well. From that perspective, this standard relates very closely to MKN2 b, "Build number combinations up to 10 (e.g., 4 and 1, 2 and 3, 3 and 2, 4 and 1 for five) and for doubles to 10 (3 and 3 for six)." Using five and ten as a benchmark is in a way a special case of this indicator. Furthermore, being able to look at numbers from multiple perspectives is something that is continuously emphasized in the elementary GPS. According to Elementary School Teaching Guide for the Japanese Course of Study, the ability to see a number as a sum, a difference, a product, or a quotient of other numbers is an important foundation for algebraic thinking.

One common tool that is often used to help students develop this idea of five and ten as a benchmark is a ten frame:

It is just a table with 2 rows of 5 cells. Different numbers can be represented by filling up these cells with counters. However, when you represent numbers 6, 7, and 8, it is important that a row is filled up completely so that those numbers are represented as 5 and some more,

not like

The latter representation is useful if we want children to develop the understanding that 8 can be represented as 4 and 4 (MKN2b). Ten frames are very versatile tools, but that means we, as teachers, must be very intentional about how we use them to help students develop a specific understanding.

M4N2 a - Rounding

M4N2. Students will understand and apply the concept of rounding numbers.

a. Round numbers to the nearest ten, hundred, or thousand.

Rounding is a specific technique to approximate numbers. Some teachers in primary grades actually teach their students rounding when they want students to "estimate." However, "estimation" and "rounding" aren't the same idea. In fact, as an approximation technique, it is probably better to teach rounding when students are working with larger numbers.

As you know, rounding a number to the nearest designated place means to look at the numeral to the right of the place to which we are rounding. If the numeral is 4 or less, we will round down (i.e., simply change all places to the right of the designated place 0's) and if it is 5 or above, we round up (i.e., increase the numeral in the designated place by 1 and change all numerals to the right 0's). So, when 45,542 is rounded to the nearest thousands place, it will be 46,000, and when it is rounded to the nearest hundreds, it will be 45,500.

One question students sometime ask is why we round up with a "5" even though 5 is right in the middle (of 0, 1, ..., 9). Some teachers will simply say it's just a rule. But, is it?

Let's consider 45,542. If we want to round this number to the nearest thousands place, we are really asking is it closer to 45,000 or 46,000. According to the procedure, we will be checking the numeral in the hundreds place. So, what numbers between 45,000 and 46,000 have a 5 in the hundreds place? Well, 45,500 is definitely one. But there are a lot more: 45,501, 45,502, 45,503, ... 45, 598, 45,599. Altogether there are actually 100 numbers in this range with a 5 in the hundreds place? So, which of these numbers are closer to 45,000? 46,000? Right in the middle? Well, it's obvious that all but one of these numbers are actually closer to 46,000, and the one exception is right in the middle. If that's the case, in general, does it make sense to round a number with a 5 in the hundreds place up or down?

The problem with "5 is right in the middle" comes up only when you are rounding to the nearest tens (and only if we are looking at whole numbers). Since approximate numbers are used when we have very large numbers of very small numbers, perhaps trying to teach rounding, a specific approximation procedure, with such small numbers may not make any sense.

M7n1 a - Meaning of 0

M7N1. Students will understand the meaning of positive and negative rational numbers and use them in computation.
a. Find the absolute value of a number and understand it as the distance from zero on a number line.

I usually don't get many comments on my blog entry (and I would be happy to hear from more of you), but on May 30, PJGould said that he had come across a child who started his counting with zero. Of course, he noted, that made his counting always off by one. After all, zero is not a counting (natural) number. But what does zero mean?

In elementary (K-5) curriculum, there are 3 meanings of zero - perhaps it is more accurate to say 3 ways zero is used. First, zero indicates the cardinality of an empty set - that is, zero means 'nothing.' This is probably the most commonly used meaning of zero in elementary school. Another place zero is used is as a place holder in a written numbers, such as 3042. Of course, this is a slight extension of the first meaning in that there is no unit of one-hundred in this written number. So, it is still pretty close to the first meaning.

The third usage of zero in elementary schools is the starting point of a number line. In some textbooks, a number line actually starts with zero as shown below.

In other textbooks, the tick mark for zero is not at the end of a number line, implying that there may be something to the left of zero as well.

As students study positive and negative number, one of the important understanding students have to make is the meaning of zero as a referent point, or the origin, on the number line. As long as students are stuck with the idea that zero means 'nothing,' some will have difficulty making sense of numbers that is 'less than nothing.' Rather, students must look at zero as a referent point on a number line, and those number to the right of zero are positive and those on the left are negative. The distance from zero, whether on the right or left is the absolute value of the number.

For those of us who already understand positive and negative numbers, this way of looking at zero is not a major issue. However, we should be aware that this meaning of zero isn't something students are familiar with. In most elementary curriculum, very little explicit discussion takes place about the role of zero on a number line. Thus, when we introduce positive and negative numbers, we do have to keep this shift in understanding of zero in our mind.

M3N5 a - Modeling decimal numbers

M3N5. Students will understand the meaning of decimal fractions and common fractions in simple cases and apply them in problem-solving situations.
a. Understand a decimal fraction (i.e., 0.1) and a common fraction (i.e., 1/10 represent parts of a whole.

According to the GPS, decimal numbers are introduced in Grade 3. In Grade 3, though, students only consider decimal numbers only in the first decimal place (or the tenths place, if students have already learned fraction terminology). Decimal numbers to the 2nd decimal place and beyond are studied in Grade 4 and above.

I have heard many teachers who say they use money as the model for decimal numbers. However, if we are limiting decimal numbers to be discussed in Grade 3 to the first decimal place, we cannot use money as an appropriate model as money amounts are shown to the second decimal place. Thus, money as a model for decimal numbers is appropriate only starting in Grade 4, and only with decimal numbers with 2 decimal places. Some people argue, and I agree, that money is not a good model for decimal numbers.

So, why isn't money a good model for decimal numbers? First, as we saw above, it is a very limited utility as a model - only in Grade 4 (and above) and only when we are dealing with decimal numbers with 2 decimal places. Although it is true that a 0 may be annexed to a decimal number with only 1 decimal place, e.g., 0.4 = 0.40, looking at all decimal number as 2-digit decimal numbers may not be the most helpful habit to develop.

Perhaps more serious problem with money as a model is that students (and adults) don't really have to think about money amounts as decimal numbers. Rather, they are really combinations of two monetary units, dollars and cents. By using two different units, we can simply work with two whole numbers. For example, we don't consider $2.35 as two and 45 hundredths dollars. Rather, it is TWO dollars and THIRTY-FIVE cents. If we get additional $3.18, we simply add TWO and THREE dollars and THIRTY-FIVE and EIGHTEEN cents. Therefore, we are not really considering those numbers as decimal numbers - they only use notations similar to decimal numbers. Mathematically speaking, there isn't really that much difference between monetary amounts and durations expressed in hours and minutes. If you spend 2 hours and 35 minutes watching TV and 3 hours and 18 minutes playing computer games, then you wasted 2+3=5 hours and 35+18=53 minutes!

So, if money isn't a good model, what other models can we use? Base-10 blocks are always an option - we just have to designate something other than unit cubes as "1." We can also use paper strips, too, just as you might use them to model fractions. No matter what model you decide to use, an important idea we want children to develop is the unitary perspective of decimal numbers. For example, 0.4 is made up of 4 0.1-units. This is very similar to the unitary perspective of fractions I discussed in the last post. This way of looking at decimal numbers will allow students to bridge decimal numbers to whole numbers. So, it is very important for us to think about models to use, but we should also keep in mind the goal understanding we want our students to develop.

M3N5 - Simple cases of fraction addition/subtraction

M3N5. Students will understand the meaning of decimal fractions and common fractions in simple cases and apply them in problem-solving situations.e. Understand the concept of addition and subtraction of decimal fractions and common fractions with like denominators.
Addition and subtraction of fractions are discussed in three different grades (M3N5e; M4N6b; M5N4g). Both this current standard and M4N6b involve fractions with like denominators. For M4N6b, there is a note stating that denominators should not exceed 12. So, what is the difference between M3N5e and M4N6b? If one of the reasons for developing the GPS was to minimize repetitions, why is this topic repeated in Grade 4?

One of the differences is that in Grade 3, the sum or the minuend must be less than or equal to 1 as students will not be studying improper fractions and mixed numbers until Grade 4. Thus, 2/5 + 1/5 is appropriate in Grade 3 but not 4/5 + 2/5. However, the most important reason for discussing simple addition and subtraction in Grade 3 is to help students understand fractions as numbers, just like whole numbers.

Fractions are often introduced as parts of a whole. Although this way of looking at fractions is relatively easy for students to grasp, research also shows that this is a very limiting view of fractions. In other words, if students can consider fractions only as parts of a whole, they will have difficulty dealing with fraction arithmetic. Part of a whole is a relationship, and we cannot perform arithmetic operations on relationships. We can only add, subtract, multiply, and divide numbers. Thus, students must understand fractions as numbers in order to make sense of fraction arithmetic. So, how do we help students to see fractions as numbers? Well, one way is to help students experience situations where fractions are added or subtracted. From those experiences, students can realize that fractions are numbers because they can be added or subtracted. It sounds like a circular argument, and it probably is. However, I would like to think this relationship more of reflexive, i.e., neither one is a prerequisite for the other, and an understanding of one can actually promote and deepen the understanding of the other.

What is important, though, is that experiences students will encounter are something that they can determine as addition/subtraction situations. For example, we can ask students what is the total length of a tape if a 2/5-meter segment and 1/5-meter segment are put together end to end. They can see that this situation is an addition situation - you would use addition if the lengths of the segments were 2 meters and 1 meter, respectively.

Another key idea is the unitary view of fractions. In other words, students should understand 2/5-meters as made up of 2 1/5-meter segments. Then, 2/5 + 1/5 is really 2 1/5-units and 1 1/5-unit put together, or 2+1 1/5-units. By recognizing that fractions may be added or subtracted, and having a way to reason through to find the answers, students can develop the understanding of fractions as numbers. With this knowledge as the starting point, students in Grade 4 can explore fraction addition and subtraction more formally.

M2N1 - Number lines

M2N1. Students will use multiple representation of numbers to connect symbols to quantities.

When I visit primary grade classrooms, I often see a large number line posted above the whiteboard in the front of the room. Sometimes I also see number lines taped on students' desks. A variety of experts, including the National Math Panel, state that number lines are powerful representation tools and mathematics instruction should develop students' proficiency with number lines. Singapore elementary mathematics textbooks are famous, in part, because of their use of "tape diagrams" to help students deal with complicated mathematical problems. As I have discussed previously, double number lines can be powerful thinking tools to support students' comprehension of multiplication and division of rational numbers. So, on the surface, the display of number lines in the primary grades (K-2) seems to be a sound teaching practice. But, is it?

When you examine Japanese elementary mathematics textbooks, the formal term, "number line" does not appear until Grade 3. However, that does not mean number lines are not used in Grades 1 and 2 (there is no Kindergarten in Japanese elementary schools). As usual, Japanese textbooks carefully and gradually develop number line representations. Thus, students' first encounter with something like number line is simply placing number cards 1 through 10 in order going from left to right. They will be asked to fill in the missing number in a sequence like 3 - 4 - [ ], or 7 - [ ] - 9. A little later on, once the range of numbers has been extended up to 20, there is a question which asks how far a space alien character hopped along a number line, starting at 0. Missing number problems may also involve number cards sequenced in backward (from large to small). When students are studying numbers up to 100, students are asked to locate given numbers on a number line, and similar questions are asked in Grade 2 when the range of numbers is extended to 1000.

What is conspicuously absent in the Japanese primary mathematics textbooks is the use of number lines to deal with addition and subtraction. Rather, number lines are used to represent visually relative sizes of numbers. I recently heard that some people distinguish number paths and number lines. Number paths, as I understand it, simply string together numbers, 1, 2, 3, ... On a number path, numbers are represented more by their positions (or orders) whereas on a number line, a number is represented by the distance of the tick mark from the origin, i.e., 0. So, the way the Japanese textbooks introduce and use number lines are much more along the line of number paths.

So, why don't Japanese textbooks use number lines to represent addition and subtraction, as is often done in some US textbooks? There are at least a couple of reasons. The idea that a number is represented by the distance of the tick mark from the origin is a difficult one for students in primary grades. This is difficult, in part, because those students are still learning about measuring length. So, they really don't have the prerequisite knowledge to interpret number lines in that manner. What they tend to do is to simply count the tick marks. However, when students count, they start with "1," and this is another reason number lines are complicated for young children. I have yet to meet a child who started his/her counting by saying, "zero... one, two, three, ..." For many young children the role of 0 (the origin) on a number line is mysterious. So, when they have to use number line to solve 5+3, they will start with the tick mark labeled "5," some will point to "5" and say, "one." Most, if not all, teachers of primary grades have seen young children line up their rulers starting at "1." It's the same problem.

Some people suggest that number lines are inappropriate for primary students, and we should not use number lines. However, I do think it is important that number lines are introduced in primary grades. However, we should be careful about how we use them. We can use them to think about relative sizes of numbers. However, it is probably a good idea to wait to use number lines as a tool for arithmetic. We can use something like tape diagram for that purpose. But, developing the idea that numbers can be represented on number lines is an idea that should start in primary grades, and we should guide students to understand how numbers are represented (as distance from the origin) on number lines, perhaps connecting to the study of linear measurement.

P1a - Teaching THROUGH problem solving

P1. Students will solve problems (using appropriate technology). a. Build new mathematical knowledge through problem solving.

I'm going to write about the same process standard as the last entry; however, this time, I want to focus on the actual indicator, "build new mathematical knowledge through problem solving."

Teaching through problem solving has been a major emphasis in mathematics education over the last (at least) 2 decades - the emphasis on problem solving was there in the 1980 NCTM document. So, it is not necessarily a new idea, but it's not quite clear what this might actually look like in a real classroom. Some people have discussed the three related ideas:
* teaching for problem solving
* teaching about problem solving
* teaching through problem solving

Teaching for problem solving is exemplified by the common textbook organization where students are taught various rules and formulas in a unit, and at the end of the unit is the lesson(s) titled "applications." Students are taught necessary tools, so to speak, and they are given numerous problems for which those tools may be useful.

Teaching about problem solving typically means teaching various problem solving strategies such as guess and check, draw a diagram, look for a simpler problem, make a table, etc. Some textbooks will include a mini-unit on these strategies throughout their textbook, and students are asked to solve problems using the specified strategy.

However, neither approach really produces new mathematical knowledge by solving problems. Teaching through problem solving means students will solve a problem, using only what they have previously learned. Then, by examining their solution strategies, they will generate a new idea/rule/formula. Let's take a look at an example.

In the GPS, students are expected to learn how to determine the area of rectangles and squares by multiplying their dimensions in Grade 3 (M3M4c). Then in Grade 5, students are expected to derive the formulas for calculating the area of parallelograms and triangles (M5M1 b & c). Somewhere in between, students are often asked to find the area of L-shape like the one shown below.

If you ask students to find the area of this shape in many different ways, they may come up with solutions like the ones shown below.


All of these methods will determine the area of the L-shape. However, if you are teaching through problem solving, your real lesson starts once these solution strategies are shared because the goal of the lesson is NOT to determine the area of the L-shape. Rather, you may ask students, "What is common about ALL of these strategies?" One conclusion students may reach is that all of the strategies are somehow using rectangles and squares, shapes for which they already know how to calculate the area. Thus, by discussing that question, students may reach a new understanding that "when we are given an unfamiliar shape, we may still be able to calculate its area by somehow making a familiar shape (or a collection of familiar shapes)."

Your lesson may not stop there. You may want to ask students to sort these strategies - "which strategies are alike?" Often times, students will come up with the following three categories:
* divide the given shape up into several familiar shapes
* cut and re-arrange to make a familiar shape
* make-it-bigger
Thus, students can learn some specific strategies for creating familiar shapes by critically analyzing these strategies.

So, what can we say about teaching through problem solving? One important idea is that the discussion after various solution approaches are shared is the meat of the lesson. That means we must make sure that we leave sufficient amount of time for such discussion. Too often, we see lessons where very little time is left after the last solution is shared. Sometimes this happens because teachers lost track of time as they circulate around the classroom. Other times teachers feel that students need more time to solve the problem. However, I think it is very important for us to remember that the goal is not the answer to the problem. Rather, even if students have not completed their solution, perhaps their incomplete answer may still be sufficient for conducting productive discussion.

Teaching through problem solving is extremely challenging. It requires teachers to have deep understanding of mathematics they are teaching. It also requires teachers to understand their students' mathematical knowledge so that they can anticipate various solution strategies might come up. Furthermore, teachers must have a plan on how to orchestrate the discussion once strategies are shared. Few teachers, if any, can naturally do this; however, it is something teachers can learn, too. Japanese teachers continuously sharpen their craft of mathematics teaching through a process called lesson study. You can learn more about lesson study and also watch some interesting lessons by clicking here.