Friday, December 21, 2007

M6N1f - Percents (2)

M6N1. Students will understand the meaning of the four arithmetic operations as related to positive rational numbers and will use these concepts to solve problems.
f. Use fractions, decimals, and percents interchangeably.

To help us understand the nature of percents, let’s look at how percents are introduced in Japan. A very typical introduction to percents will involve problems like this one:
Here are the records of the basketball teams of three city schools. Which team has the best record?
East 15 7
West 10 7
North 15 10
Keep in mind that students do not know anything about percents at this point. Thus, students are initially encouraged to think about which two teams’ records are easier to compare. Students can easily see that the comparison of East and North is easy as they have played the same number of games – thus North has a better record with more wins. Some students will also notice that the comparison of East and West is also easy as they have won the same number of games – West needed fewer games to win the same number of games, so they have a better record than West. From this investigation, they generalize that if one of the two quantities (number of games played or the number of games won), the comparison is easy.

Students will then asked to think about how they can compare West and North. In this case, neither the number of games played nor the number of games won is the same. However, some students may realize that we can make the number of games same by pretending these two teams keep playing at the same pace. If West plays 30 games at this pace, they will win 21 games, while North will win 20 games out of 30 games. Thus, West has a slightly better record than North.

Thus, they can summarize what they have discovered so far:
* if one of the quantities are the same, the comparison is easy.
* if neither quantity is the same, then make one the same
Now suppose the team from South has won 12 games out of 17 games they played. We can compare West and South by making them play the same number of games (or make their numbers of wins the same). However, if we have to compare two teams at a time, the comparison may be a bit too tedious. So, what can we do?

One possibility is to think about what fractions of games played each team won. This can also be interpreted as pretending each team playing 1 game. However, it does not really make sense to think about a team winning a fraction of a game. But, the idea of making all teams play the same number is useful. An easy way to do this is to use a common multiple of games for the teams involved. However, that will mean we will use different number for different set of teams – or different points during the season. Instead, we can standardize the process by pretending each team played 100 games. The number of games each team won out of 100 games is a relative value – not an actual number of wins. However, because we standardized the total number of games, we can compare these relative values.

Note that in this situation, we are comparing the number of games won to the number of games played. When we are comparing two like quantities, we can calculate a relative value by considering the base of the comparison (in this case, the number of games played) as 100. That relative value is called percents. It is very important that we remember that percents are relative values.

Because we can also express relative values in terms of fractions or decimals, we can use fractions, decimals, and percents interchangeably WHEN WE ARE DEALING WITH RELATIVE VALUES. However, when we are looking at fractions or decimals as numbers, we cannot use percents. Thus, the GPS statement M6N1f should have included a phrase, “to express relative values.”

Tuesday, December 18, 2007

M6N1 f - Percents (1)

M6N1. Students will understand the meaning of the four arithmetic operations as related to positive rational numbers and will use these concepts to solve problems.
f. Use fractions, decimals, and percents interchangeably.

Since I discussed ideas related to fractions and decimals, let me discuss another standard that deals with those ideas, M6N1f. As most of you are aware, the Georgia Performance Standards were heavily influenced by the 1989 Japanese Course of Study. There are many GPS statements that are identical to what you would find in the Japanese standards. However, this is one statement you will NOT find in the Japanese standards. Since a standard like this is probably very familiar to many of us, that might come as a surprise. Why doesn’t the Japanese standards expect students to be able to use “fractions, decimals and percents interchangeably”?

Before answering that question, let’s consider why fractions, decimals and percents might be used interchangeably. Some people claim that percent is just another representation of rational numbers. After all, “percents” means “out of 100.” Therefore, just like decimal numbers are based on powers of 10, percents are based on a particular power of 10, namely 100. Thus, 32% means 32 out of 100. M5N5a does state that students should be able to “model percent on 10 by 10 grids.” Thus, 32% may be represented by shading in 32 small squares on a 10 by 10 grids. Of course, the same model may be used to model a decimal number, 0.32, or a fraction 32/100. So, since percent is another way to represent rational numbers, decimals, fractions and percents are interchangeable.

However, if percents, fractions, and decimals are interchangeable, then we should be able to make statements like,
• I bought a 50% gallon of milk.
• The next exit is 75% mile away.
• My son is 350% year old.
These statements sound rather absurd, but why is it? If fractions, decimals, and percents are truly interchangeable, shouldn’t these statements make perfect sense? Or, consider these cases:

Jenny made 7 out of 10 (70%) free throws during the first half of a game. In the second half, she made 8 out of 10 (80%) free throws.
So, did Jenny make 70+80 = 150% of free throws in the game?
Can we express this situation as 7/10 + 8/10 = 15/20?

These situations suggest that there are some significant differences between percents and fractions/decimals, and they are not interchangeable all the time. So, when can we use these three interchangeably and why can we not? To answer this question, we must first understand what percents mean – they are not just another representation of rational numbers.

Wednesday, November 7, 2007

M5N4d & M6N1e - Multiplication & Division of Fractions

M5N4. Students will continue to develop their understanding of the meaning of common fractions and compute with them.
d. Model the multiplication and division of common fractions.

M6N1. Students will understand the meaning of the four arithmetic operations as related to positive rational numbers and will use these concepts to solve problems.
e. Multiply and divide fractions and mixed numbers.

In the previous post, I discussed similar statements about decimal number multiplication and division. These statements are very similar, but there is a significant difference. In M4N5, the GPS makes it clear that the focus is on multiplying and dividing decimal numbers BY WHOLE NUMBERS. In contrast, in M5N4, the GPS simply states “multiplication and division of common fractions.” Does this mean that students should understand the meaning of multiplication and division BY FRACTIONS?

As I discussed in the previous post, when we multiply or divide by decimal numbers, the meaning of multiplication and division must be expanded. However, the same meaning can be applied whether or not we are multiplying/dividing by decimal numbers or fractions. Thus, it is perfectly possible to discuss the situations involving multiplication/division by fractions in Grade 5 – as long as we are aware of this subtle yet significant difference. However, it would have been helpful had the GPS made the boundary of M5N4 a little more explicit.

There is another problem with M5N4 and M6N1. One of the goals of establishing the GPS was to reduce the repetition in the curriculum. That raises a question: what is the difference between these two statements regarding the multiplication and division of fractions? One obvious difference we can see is the use of the word, “model” in M5N4. This raises 2 questions for me. The first question is, how do we model the multiplication and division of fractions? As long as the multiplier is a whole number, we can (relatively) easily model the multiplication by making so many sets of a specific fraction (the multiplicand). So, 3x2/3 can be modeled by making 3 sets of 2/3 – with Pattern Blocks, Fraction Bars, Fraction Pieces, drawings, etc. How would you model if the multiplier becomes a fraction (or both the multiplier and the multiplicand are fractions), like 1/3 x 2/3? One possibility is to use the area model of multiplication, that is, 1/3 x 2/3 may be represented by the area of a rectangle with the dimensions, 1/3 unit by 2/3 units. We can do this by folding paper or by drawing picture like this:

What about the division? An easier way of representing division of fractions is to interpret the division as the measurement division – how many groups of (the divisor) can we make with (the dividend)? We can model using various fraction manipulatives, or drawing pictures. However, the difficulty arises when there is a left over piece, or the dividend is less than the divisor. In those situations, students must understand that the division is asking, “What part of the divisor is the dividend?” This shift is not always so simple for students. However, not all division situations involve the measurement division. How might we model those situations?

The second question is when should students develop the algorithms for multiplying and dividing fractions? Are the algorithms types of “models”? Many of you are aware that the GPS was heavily influenced by the 1989 Japanese national course of study. In the Japanese standards, the multiplication and division of fractions are Grade 6 topics. They do not touch on these topics in Grades 5 at all, though since they study adding and subtraction fractions in Grade 5, they can easily consider the situation with whole number multipliers. In any event, it is in Grade 6, the Japanese curriculum develops the algorithms. Did the GPS also mean that the algorithms were to be developed in Grade 6 – that is, “modeling” excludes the algorithms? This is another important point that the GPS could have made much more explicit.

By the way, the Japanese textbooks often use a model of multiplication and division called double number line. So, for a problem like, “If 5 1/3 feet of wire weights 2/3 ounce, how much will 1 foot of the same wire weigh?” they will model the problem like this:

This model is used not necessarily to find the answer but to help students understand the way the quantities relate to each other. By understanding the relationship, students can then decide what operation must be performed to find the missing quantity. What is important about this model is that this same model may be used to model both the multiplication and the division situation. So, if you have a problem like, “if you can paint 3 1/3 square feet with 1 pint of paint, how much area can you paint with 4 3/5 pints?” may be modeled like this:

Finally, it can also be used to represent the measurement division situation like, “if you car can travel 18 3/5 miles on 1 gallon of gasoline, how many gallons of gasoline do you need to go 180 miles?” will be modeled like this:

The strength of this particular model is that the same model may be used for both multiplication and division, thus potentially helping students to understand the relationship between those two operations. Furthermore, this model is also a powerful tool for representing proportional situations in general. One weakness of this model, though, is that it cannot be used to model area situations as those situations will not involve “per-one” quantities.

Friday, October 26, 2007

M4N5(d) & M5N3 - Multiplication & Division of Decimal Numbers

M4N5. Students will further develop their understanding of the meaning of decimal fractions and use them in computations.
d. Model multiplication and division of decimal fractions by whole numbers.
M5N3. Students will further develop their understanding of the meaning of multiplication and division with decimal fractions and use them.

So, what is the difference between these two statements (M4N5d and M5N3)? There are two aspects about these standards I would like to discuss, but I will focus on one in this post. Then, in the next post, I will address the other issue.

The primary difference between M4N5d and M5N3 is that, in Grade 4, students are to deal with problem situations where decimal numbers are used as the multiplicand or the dividend, but not as the multiplier or the divisor. As I discussed earlier (in a June 2007 post on M2N3a), the multiplier is the number of groups when there are equal sized groups, while the multiplicand is the number in a (or each) group. Thus, according to M4N5d, in Grade 4, students should be dealing with problems like:
With 1 gallon of gasoline, Steve’s car can travel 17.4 miles. How far can Steve’s car go with 8 gallons of gasoline?
But not problems like:
With 1 gallon of gasoline, Steve’s car can travel 18 miles. How far can Steve’s car go with 6.5 gallons of gasoline?

Some may ask, what’s the difference, they both involve multiplying a whole number and a decimal number. The big (mathematically) difference is that in the first problem we can think of it as 17.4+17.4+17.4+17.4+17.4+17.4+17.4+17.4, but not in the second problem. The second problem is not 6.5+6.5+6.5+6.5+…+6.5 (6.5 18 times). Why not? Think about what 6.5 represents in this problem. 6.5 is the amount of gasoline. So, why do we have 6.5 gallons 18 times? Actually, the problem involves 18+18+…+18, but we have to use 18 6.5 times? What does that mean? Helping students to deal with that situation, therefore, is the major focus of Grade 5 – and that’s why the GPS says, “the meaning of multiplication and division.”

So, what about division? In division, the dividend is the total amount. The divisor, however, may be the amount in one group or the number of groups. If the divisor is the amount in one group, then the quotient will tell us the number of groups (this is called measurement division), while, if the divisor is the number of groups, the quotient tells us the amount in each group (this is usually called fair sharing division). When the divisor is a decimal number, it may be a bit easier to conceive of the division situation as the measurement division. How many groups of 3.6 can you make with 18.7, for example. The fair sharing division situation will be something like this:
With 3.6 pints of paint, we can paint 18.7 square feet of board. How much area can we paint with 1 pint?
In this question, we are asked to determine the per-one unit amount. Again, this requires an expansion of the meaning of division.

Some may argue that, if it is easier to perceive division by a decimal number as the measurement division, why not focus on the measurement division. If we use the measurement division and if there is no whole number quotient, we will have to help students interpret the meaning of the quotient. Thus, either way, we have to help students expand their understanding of the meaning of division.

As Grades 4 and 5 teachers deal with multiplication and division of decimal numbers, they must pay very close attention to these points. In Grade 4, we are still using the same meaning of multiplication and division, and the focus is on helping students develop procedures for multiplying or dividing decimal numbers by whole numbers. In Grade 5, the first focus is helping students expand the meaning of multiplication and division, then we must help students develop computational procedures when the multiplier and the divisor become decimal numbers.

Saturday, October 13, 2007

Grade 4 tasks

It has been a rather hectic month, and I haven't been able to write anything for this blog for a few weeks.

I just wanted to share that I uploaded Grade 4 tasks that I wrote for the GA Math Frameworks. None of these tasks has been published yet, and I am not sure if any will be. However, I believe these are all grade appropriate tasks, some are definitely more challenging than others.

Here is the URL:

Scroll down to the bottom of the page to see the list of the tasks.

Monday, September 17, 2007

In pursuit of a focused and rigorous curriculum

This entry will be a bit different from others. Instead of providing my commentary on specific GPS statements, I want to talk about the challenge of creating a focused and rigorous school mathematics curriculum. According to the Executive Summary of the GPS online, the state Board of Education asked the Department of Education (DoE) to write a new standards (GPS) with four specific charges. Those charges were,
(1) The curriculum needs to be rigorous…
(2) The curriculum needs to be focused…
(3) The curriculum needs to be clearly understandable by teachers…
(4) Instruction needs to be student-centered…

So, the GPS is supposed to be standards for a focused and rigorous curriculum that is easily understood by teachers and implementable (is this a word?) through student-centered instruction. There are many questions we can ask about the GPS based on these charges, but today, I want to focus on a “focused and rigorous curriculum.” In fact, much of my attention will be on a “rigorous” curriculum and how difficult it is to actually create one.

What does a rigorous curriculum look like? In our everyday language, the word “rigorous” is often used interchangeably with the words like “difficult” or “challenging.” Clearly, we want a curriculum that is (appropriately) challenging to our students. However, speaking of mathematics and mathematics education, the word “rigor” or “rigorous” should also mean something else. When mathematicians use the word “rigor,” what they typically refer to is logical cohesiveness of their arguments. Thus, a rigorous mathematics curriculum must be logically cohesive and coherent. There must be logical sequencing of topics.

This idea seems to be so obvious and easy to accomplish, but it is not that simple because it requires us to know the standards very well. It is very easy to overlook something while creating a curriculum. A good example can be found in the Mathematics Framework, a document that was created by the DoE “to be models for articulating desired results, assessment processes, and teaching-learning activities that can maximize student achievement relative to the Georgia Performance Standards.” So, these documents are supposed to be illustrative of a rigorous curriculum. Unfortunately, there are some oversights.

For example, look at the “Quotient if Greater Than One” task found in the Grade 3 multiplication and division unit. The task gives the following expression: []/^ > 1 (it's supposed to be a box over a triangle), and ask students questions such as “If the dividend is 10, what is the largest number the divisor can be? Why is that true?” and “If the divisor is 6, what is the smallest number the dividend can be? Tell why.” In the discussion section, the Framework states the following:

“This activity requires that students prove their answers to be true. Just a simple numerical answer is not enough. Help students articulate their thinking with correct math vocabulary and make sure they understand the connection between division and fractions. Students should be able to write any division problem as a fraction and for any division problem that has a remainder they should be able to write the remainder as a fraction also.”

Although these goals are very important goals for our students, looking at these goals in Grade 3 is totally inappropriate within the GPS. In the GPS, understanding that a fraction indicates the division of the numerator by the denominator is a Grade 5 topic (M5N4a). Although it is not quite clear in the GPS when students should learn to express remainders as fractions, certainly we can’t expect them to understand this idea until they have made this connection between division and fraction. Thus, this task, though it may be perfectly appropriate and useful task in Grade 5, is inappropriate for Grade 3 because it requires students to use something they have yet to study. It is a good example how difficult it is to create a rigorous curriculum, that is not only challenging but also logically cohesive.

Friday, September 7, 2007

M5N4 (b) - Equivalent Fractions (2)

M5N4. Students will continue to develop their understanding of the meaning of common fractions and compute with them.
a. Understand division of whole numbers can be represented as a fraction (a/b= a ÷ b).
b. Understand the value of a fraction is not changed when both its numerator and denominator are multiplied or divided by the same number because it is the same as multiplying or dividing by one.

In the previous post, we looked at the idea of equivalent fractions. According to the GPS, students are expected to become aware that two fractions that look different may represent the same number, i.e., the concept of equivalent fractions, in Grade 4, while they are expected to understand how to create equivalent fractions by multiplying or dividing both the numerator and the denominator by the same number in Grade 5 (M5N4b). In this post, I would like to look at the reason why this procedure works.

In M5N4b, it is stated that this procedure works “because it is the same as multiplying or dividing by one.” This explanation makes perfect sense for us since we already know how to multiply or divide by fractions. For example,

However, this explanation is inappropriate for Grade 5 students because they have not studied how to multiply fractions by fractions. In Grade 5, students are expected to “model the multiplication and division of common fractions” (M5N4d). However, learning the procedures/algorithms for multiplying and dividing fractions is a Grade 6 expectation. Furthermore, even if students were to study the algorithms for multiplication and division of fractions in Grade 5, most likely they will be studying the procedure of creating equivalent fractions before they study multiplying and dividing fractions because they need to have this procedure to add and subtract fractions with unlike denominators. So, how can students understand that why this procedure for creating equivalent fractions work using only what they have already studied? Or, do they simply have to accept this procedure for now and justify it later?

Although there may be some topics in school mathematics where students may have to accept a formula or an algorithm just because teachers tell them it works, this is NOT such an occasion. Students do have something they have studied to justify this procedure. They key for understanding this procedure is M5N4a, which expands the meaning of fractions from simply a part of a whole or a collection of unit fractions (see my earlier post on the meaning of fractions). According to M5N4a, 5th grade students are expected to understand a fraction represents the answer for division of a whole number by another whole number. For example,

Moreover, in Grade 4, students are expected to “understand and explain the effect on the quotient of multiplying or dividing both the divisor and dividend by the same number” (M4N4d). Therefore, if you multiply or divide both 3 and 5 in the above example by the same number, the quotient does not change. For example,

But, this last expression, according to M5N4a, is the same as because a fraction represents the quotient of a whole number divided by another whole number. Or, in general,
, for any whole number .

The Georgia Performance Standards emphasizes that students should construct new mathematics understanding based on what they have previously studied. However, this is much easier said than done. Because we are already very familiar with mathematics students are still learning, we can easily overlook the fact that we slipped in something students have not studied yet. M5N4b is a very good illustration how difficult it is to create a rigorous, i.e., logically cohesive, curriculum.

Sunday, September 2, 2007

M4N6 (a); M5N4 (b) & (c) - Equivalent Fractions (1)

M4N6. Students will further develop their understanding of the meaning of common fractions and use them in computations.
a. Understand representations of simple equivalent fractions.
M5N4. Students will continue to develop their understanding of the meaning of common fractions and compute with them.
b. Understand the value of a fraction is not changed when both its numerator and denominator are multiplied or divided by the same number because it is the same as multiplying or dividing by one.
c. Find equivalent fractions and simplify fractions.

In an earlier post, I discussed different meaning of fractions. The fact that fractions may be interpreted in many different ways is a major reason why fraction is so difficult to teach and learn. Another major difference between fractions and whole numbers is the fact that fractions that look different can represent the same number – we call those fractions “equivalent fractions.” In the Georgia Performance Standards for school mathematics, the idea of equivalent fractions first appear in Grade 4, M4N6 (a). However, it should be noted that the goal in Grade 4 is that students become aware of the fact that two fractions that look different may represent the same number. Students are to understand this idea and be able to demonstrate their understanding by representing those fractions to show their equivalence. Understanding how to create equivalent fractions by multiplying (or dividing) both the numerator and the denominator by the same number is a Grade 5 standard, M5N4 (b) and (c). Consequently, answering problems involving fractions with the simplest form should NOT be a focus in Grade 4. Thus, for example, if students calculate 3/4 – 1/4 in Grade 4, the answer should be written as 2/4, not 1/2. Clearly, some students will understand that 2/4 and 1/2 are the same based on their study of equivalent fractions. Therefore, if they do present their answers in the simplest form, that is ok. However, we should not penalize students who do not present their solutions in the simplest form. That emphasis should begin only after students understand the procedure for creating equivalent fractions.

Monday, August 13, 2007

M3N4 a & b - Meaning of Division

M3N4. Students will understand the meaning of division and develop the ability to apply it in problem solving.
a. Understand the relationship between division and multiplication and between division and subtraction.
b. Recognize that division m ay be two situations: the first is determining how many equal parts of a given size or amount may be taken away from the whole as in repeated subtraction, and the second is determining the size of the parts when the whole is separated into a given number of equal parts as in a sharing model.

If you are asked to write a simple word problem for which 12 ÷ 4 is the appropriate computation, what will you do? Here are two possibilities:

* Cathy has 12 apples and she wants to give them to her 4 friends equally. How many apples will each friend receive?
* James’ mom baked 12 cookies for James and his 3 brothers. If they share the cookies equally, how many will each receive?

In both of these problems, we are sharing the total amount (12) among 4 people. Some people call these division situations “fair sharing” division situation. In a fair sharing situation, you know how many total you have and the number of groups the total is being shared equally.

There are other situations in which division is appropriate. For example,

* Cathy has 12 apples. If she puts 4 apples in a bag, how many bags will she need to put all apples away?
* James’ mom baked 12 cookies. If she puts 3 cookies on a plate, how many plates will she need?

In these situations, you are trying to find out how many groups there are. Thus, you are given the total amount and how many in each group. These situations are often called “measurement” or “repeated subtraction” division situations.

So, where do these differences come? Actually, the difference is closely related to the way we define multiplication. Recall that multiplication is an arithmetic operation that is appropriate when you have equal groups. The number of groups is called multiplier and the number of items in a group is called multiplicand. Division, just like multiplication, is also used in equal groups are involved. The number you are dividing, called dividend, is the total number of items you have. The divisor, however, may be the number of groups as in a fair sharing situation or the number in a group as in a measurement situation. Here is the summary:

multiplicand multiplier product
(Number in a group) x (Number of groups) = (Total)

dividend divisor quotient
Total ÷ (Number of groups) = (Number in a group): Fair sharing division
Total ÷ (Number in a group) = (Number of groups): Measurement division

We often say division is the inverse operation of multiplication, and it is. However, if we pay attention to the multiplier-multiplicand distinction, we get two different situations for division. In one situation, you are trying to figure out the multiplier (number of groups) and in the other, you are trying to determine the multiplicand (number in a group).

Thus, to help students understand these standards, it is very important that they understand the multiplier-multiplicand distinction when they are first introduced to multiplication.

Saturday, July 21, 2007

M1N3 - Diagrams for Addition & Subtraction

[I just want to apologize for the sizes of some of the pictures - I am still trying to figure out how to do this better.]

M1N3. Students will add and subtract numbers less than 100 as well as understand and use the inverse relationship between addition and subtraction.

Problem 1
Cathy had some candies. She gave 5 to her brother, and she now has 7 candies left. How many candies did Cathy have at first?

Problems like this one is a very difficult one for some children. In the previous post, I discussed different meanings of addition and subtraction. According to the GPS, there are 3 situations in which addition and subtraction are used: combine, separate, and compare. This particular problem is a separate situation – a set (subtrahend) is separated from another set (minuend) to result in another set (difference). The take-away subtraction is used to find the number in the resulting set given the numbers for the first 2 sets, that is, (minuend) – (subtrahend) = (difference). However, in the problem above, what is not known is the minuend. To find the minuend, we have to add the subtrahend and the difference.

Similarly, a combine problems like the ones below require subtraction to find the answer.

Problem 2
Juan had some marbles. Salvador gave him 7 more and Juan now has 12 marbles. How many marbles did Juan have at first?

Problem 3
Kim bought 7 books with her allowance. Her grandmother gave her some more books for her birthday, and Kim now has 12 books altogether. How many books did Kim receive from her grandmother?

One thing you might notice is that these problems show that the “key words” do not work – in fact, those children who depend on key words are much more likely to miss these problems. Therefore, our instruction should not emphasize key words in word problems. Rather, what we want children to think about is how the quantities in a problem relate to each other.

One potentially useful tool to help children visualize the relationship among the quantities in a problem is to use a diagram. In many Japanese elementary mathematics textbooks, a linear model called “tape diagram” is often used. Using this diagram, the relationship among the quantities in Problem 3 may be represented like this:

From this diagram, we can tell that, in order to find the number of books Kim received from her grandmother, we will have to subtract 7 from 12. Can you draw tape diagrams for the other 2 problems?

So, how can we help our students make this diagram as their tool? We need keep in mind that it will take some time before children can make these representations as their thinking tools. However, it has to start some time toward the end of students’ initial study of addition. As we have students model addition (most likely combine problems) using manipulatives (counters), we can intentionally arrange them in a straight line. For example, suppose we are working on the following word problem: Cameron had 5 apples. His mom gave him 7 more. How many apples does Cameron have now? We can model this problem (5 + 7) this way,

instead of,

At some point, we might even want to place boxes around the counters like this,

You may even want to label “Apples Cameron had at first” and “Apples Cameron’s mom gave him.” We can introduce similar diagrams as students study subtraction. In the context of take away subtraction, students will be introduced to the idea that an empty “tape” may stand for a number, which is not a trivial idea.

We should introduce a new diagram only after students are reasonably comfortable with the operation the diagram is supposed to represent. Japanese teachers believe that we teach a new concept using a familiar diagram and a new diagram with a familiar concept. We should not try to teach both a new concept and a new diagram simultaneously.

Here is a tape diagram template for a comparison problem:

I encourage you to represent problems you find in your textbooks until you feel comfortable with these tape diagram representations. Then, start thinking about how you might introduce the diagram to your students.

Click here for the discussion of other uses of tape diagram.

Tuesday, July 17, 2007

MKN2(a) & M1N3(d) - Meaning of Addition & Subtraction

MKN2. Students will use representations to model addition and subtraction.
a. Use counting strategies to find out how many items are in two sets when they are combined, separated, or compared.
M1N3. Students will add and subtract numbers less than 100 as well as understand and use the inverse relationship between addition and subtraction.
d. Understand a variety of situations to which subtraction may apply: taking away from a set, comparing two sets, and determining how many more or how many less.

In the previous two posts, we discussed the meanings of multiplication. In this post, I want to go back even further to discuss the meanings of addition and subtraction. Although the GPS does not explicitly state what the meanings of addition and subtraction are, MKN2(a) suggests the meanings that the GPS writers had in mind. According to MKN2(a), and also K1N3(d), addition and subtraction are operations that describe situations in which items in two sets are “combined, separated, or compared.” It is absolutely critical that we keep in mind that human beings give meaning to a new idea in context. Therefore, it is very important that the initial instruction on these operations utilize word problems in which two sets are “combined, separated, or compared.”

Addition is an arithmetic operation used to determine the total amount when two sets are combined. There are two slightly different situations when two sets are combined. In one case, the two sets are actually combined to make one set – that is, the number of items in a set increases. In the other case, two sets are “combined” in our mind, but not necessarily physically. For example, when we ask: Tom has 4 marbles and Carey has 3 marbles. How many marbles do they have in all? we are simply changing our perspective and looking at those marbles irregardless of who they belong to. However, neither Tom nor Carey is losing or gaining any marble. The Cognitively Guided Instruction (CGI) calls the first case as “Join” (or “Combine”) while the second as “Part-Part-Whole.”

There are two major situations for subtraction. The first is when you are trying to find out how many items will be left when a set is removed from another set – this is the take-away situation. Many people equate subtraction as “take away,” but this is not the only case where subtraction is used. Another important meaning of subtraction is to find out the difference between two sets. In fact, the formal term for the answer to subtraction is “difference,” suggesting that this “comparison” meaning of subtraction is perhaps more significant mathematically.

Although we use addition and subtraction when two sets are “combined, separated, or compared,” when we use which operation in each situation depends on what quantity is unknown. That is, even though we have a combining situation, if we don’t know the starting amount or the added amount, we must use subtraction to find the answer. Some people may call this as the missing-addend meaning of subtraction. In the same manner, if we don’t know the starting amount in a take-away situation, we use addition. The important idea we want children to understand, therefore, is how addition and subtraction are related to each other, and this is suggested by M1N3 statement – “the inverse relationship between addition and subtraction.”

So, here is the summary of the important ideas we want Kindergarteners and 1st graders to understand:

* meanings of addition
- join/combine
- part-part-whole
* meaning of subtraction
- take-away
- compare (to find the difference)
* the inverse relationship between addition and subtraction

In Kindergarten, our emphasis should be more on the meanings of these operations (using numbers to 10). As 1st graders continue their study of addition and subtraction, with increasingly larger numbers, they will also have to understand the relationship between addition and subtraction. However, I believe it is important for us to focus on one idea at a time. That is, if you want students to understand addition with sums greater than 10, let’s focus on that aspect, using only the familiar situations (combining two sets). However, if you want students to understand the relationship between two operations, let’s use numbers with which children are comfortable. A focused lesson should not expect students to learn two new ideas simultaneously.

Sunday, July 15, 2007

M2N3 b - Multiplication Table

M2N3. Students will understand multiplication, multiply numbers, and verify results.
b. Use repeated addition, arrays, and counting by multiples (skip counting) to correctly multiply 1-digit numbers and construct the multiplication table.

In the previous post, I discussed that multiplication is not the same thing as repeated addition. Rather, repeated addition is a way to obtain the product (the answer to a multiplication problem). In the same way, skip counting is a method of getting the product. Therefore, repeated addition and skip counting are two strategies to obtain the product.

Arrays, on the other hand, is a slightly different object. An array may be used to model multiplication situation. An array with 3 rows and 4 columns possesses the equal group structure that can be described by multiplication. It is a very powerful model to help students construct their understanding of various properties of multiplication, such as commutative and distributive properties. However, if we are emphasizing the equal group meaning of multiplication, we need to be careful how we use arrays to model multiplication as students are first being introduced to the concept. That is because the distinction between the multiplier and the multiplicand is not clear in an array. There is not logical reason that a row or a column is to be considered as a group. Thus, an array with 3 rows and 4 columns may be 3x4 or 4x3. Thus, we should use the array model with caution.

Perhaps the most important part of this particular standard is that students are to “construct the multiplication table.” Thus, we should look at the strategies such as repeated addition and skip counting and the model like arrays as tools for students to construct the multiplication table. As students construct the multiplication table, one very powerful pattern that can be useful for them to explicitly think about is that when the multiplier increases by 1, the product increases by the multiplicand. In other words, if you add one more set or skip count one more time, the product will become greater by the size of the group. With an array model, you can see this that when you reveal one more column, the total number is increased by the number of rows.

So, how can we help students construct the multiplication table on their own? Here are a couple of suggestions. First, organize your lessons according to the size of multiplicand, that is the number in a group. So, when you are looking at the multiplication facts of 6’s, you are looking at 1x6, 2x6, … 9x6 (1 group of 6, 2 groups of 6, … 9 groups of 6). Second, start with the facts of 2’s and 5’s, then 3, 4, 6, 7, 8, and 9. Wait till the end to treat the multiplication facts of 1. You might find it strange to wait to discuss 1’s until the end, but if the focus of the initial treatment of multiplication is helping students understand the meaning of operation, not just get the answers, then you might notice that “group of 1” seems to be a rather strange idea – particularly for children. A “group” usually consists of more than 1 person or item. So, as far as multiplication is concerned, 1’s are special cases, and it is probably not a wise move to start with a special case.

One particular teaching strategy teachers may want to consider is asking students questions like the following:

• How much greater is the answer to 4 x 7 compared to 3 x 7?
• What do you need to add to the answer of 3 x 7 to get the answer to 4 x 7?
• How much greater will the answer be when the multiplier (i.e., the number of groups) increases by 1 in the 6’s facts (i.e., the multiplicand is 6)?

There are many strategies children can develop to master multiplication facts, but it is probably a good idea for all children to have at least one common strategy they can rely on. Don’t discourage other strategies (such as 4’s are doubles of 2’s), but make sure everyone understand this particular strategy, which is a special instance of the distributive property they will study later.

Saturday, June 2, 2007

M2N3a - Meaning of Multiplication

M2N3. Students will understand multiplication, multiply numbers, and verify results.
a. Understand multiplication as repeated addition.

In my last post, I discussed a particular property of division. I am going backward this time to talk a bit more about multiplication.

So, let’s start from the very beginning. What is multiplication? M2N3 (a) states that children are to understand multiplication as repeated addition, but is it? If multiplication is just another form of addition, why do we need a separate operation?

Multiplication is an operation that can be used when you have equal size groups to determine the total number of objects in those groups. For example, if you have 4 plates and on each plate there are 6 strawberries, we can express the situation with an equation, 6 x 4 = 24 (or 24 = 6 x 4). On the other hand, if one of the plates has 5 strawberries and another has 7, then, we can no longer use a multiplication sentence to express this relationship.

What is important to remember here is that mathematical operations and mathematical sentences are used to represent relationships among quantities in different situations. Clearly, the example above could have be written as 6+6+6+6=24 as well. However, repeated addition may be a way to calculate the product (the answer for a multiplication problem), but not necessarily the same thing as multiplication as an operation. This becomes a problem later when students start studying multiplication of fractions and decimals.

How else is multiplication different from addition? Well, you are probably familiar with the cliché, “you can’t add apples and oranges.” Addition (and subtraction) requires two numbers that are referring to the same thing (4 apples and 5 apples, 4 oranges and 5 oranges, but not 4 apples and 5 oranges – unless we change our referents to 4 fruits and 5 fruits). On the other hand, multiplication can be performed between two numbers that are referring to two different things – 6 strawberries and 4 plates.

So, if we can find the total amount by repeated addition, what is the advantage of using multiplication? One advantage is that when you write 6 x 4, it is much easier to tell that there are 4 plates. If you write 6+6+6+6, you have to count the number of 6’s to get that information. Clearly if there are only four 6’s, it may not be too difficult but imagine if you had the situation that can be written as 24 x 31.

Finally, because the two numbers in a multiplication sentence tells us specific information about the situation, we should be consistent in the way we write multiplication sentences. Let’s go back to the original example of plates and strawberries. Could we have written 4 x 6 = 24 as well as 6 x 4 = 24? The answers are clearly the same, but do they mean the same thing? Well, that depends if we have any agreement on how we are to write multiplication sentences. If we agree that a multiplication sentence is to be written
{number in a group } x {number of groups} = total number of objects
then, the situation must be written as 6 x 4 = 24 as we don’t have 4 strawberries each on 6 plates.

Some people may argue that since the answers are the same, then it really doesn’t matter which way you write. One thing to keep in mind is that 6 groups of 4 things and 4 groups of 6 things are two different situations. If we want multiplication sentences to be useful in communication, we need to feel confident that when we write “6 x 4,” it can be interpreted only one way. Otherwise, the sentence becomes a useless as a tool for communication.

Mathematically speaking, the first number is the number in a group (called “multiplicand”) and the second is the number of groups (called “multiplier”). The sentence “6 x 4 = 24” should be read as “6 multiplied by 4 equals 24.” However, because the language of “times” (which is not really a mathematical term) is much more common, some may prefer to say the first number is the number of groups. After all, if you read “6 times 4,” it feels much more natural to thing of “6 times of 4.” My strong preference is to write the multiplicand first, but the important thing in classrooms is that teachers (hopefully within the same system – across grades) stay consistent one way or the other.

Thursday, May 17, 2007

M4N4(d) - Property of Division

M4N4. Students will further develop their understanding of division of whole numbers and divide in problem solving situations without calculators.
d. Understand and explain the effect on the quotient of multiplying or dividing both the divisor and dividend by the same number. (2050 ÷ 50 yields the same answer as 205 ÷ 5).

When we study properties of operations, we tend to focus on properties such as commutativity (a+b=b+a, or ab=ba), associativity [a+(b+c)=(a+b)+c, or a(bc)=(ab)c], identity [0+a=a+0=a, or 1xa=ax1=a], etc., which are typically associated with either addition or multiplication. Another important property is the distributive property of multiplication over addition/subtraction [(a+b)c=ac+bc and a(b+c)=ab+ac]. Not much about subtraction nor division is typically discussed. However, there is a useful relationship for both subtraction and division operations. That relationship concerning division is what is discussed in M4N4(d), which I have been calling the Equal Multiplication Principle (although you can multiply OR divide both the dividend and the divisor by the same number). The parallel relationship for subtraction is the Equal Addition Principle, which states that if the same number is added to (or subtracted from) both the minuend and the subtrahend, the difference remains the same.

So, when/where/how is the Equal Multiplication Principle useful? The example included in M4N4(d) is a familiar rule that says we can omit the same number of 0’s from both the dividend and the divisor. This is the specific instance of dividing both numbers by a particular power of 10. Clearly, this is a useful mental computation strategy. Moreover, for problems like 480 ÷ 15, doubling both (that is, multiplying both the dividend and the divisor by 2) will change the problem to 960 ÷ 30, which then can be changed to 96 ÷ 3 (which is the same thing as dividing the original numbers by 5).

In addition to being a useful mental computation strategy, the Equal Multiplication Principle plays a central role in division of decimal numbers. When you have a problem like 5.38 ÷ 1.6, we can multiply both the dividend (5.38) and the divisor (1.6) by 10 to change to problem into 53.8 ÷ 16. Once the divisor becomes a whole number, then we can simply use the long division to calculate the answer. Alternately, we could multiply both numbers by 100 to make both of them into whole numbers. Either way, the Equal Multiplication Principle says that the quotient we obtain is equal to the quotient for the original problem.

Furthermore, the Equal Multiplication Principle may also be useful while dividing a number by a fraction. For example, let’s look at 2/3 ÷ 3/4. We know that if multiplying both the dividend and the divisor by the same number, we will not change the quotient. So, how can we use the Principle here? Another familiar pattern of division is that the quotient of any number divided by 1 is the number itself. So, if we can change the divisor (3/4) to 1, then, we know that the quotient is the same as the dividend. To make the divisor 1, we have to multiply it by its reciprocal (that’s the definition of the reciprocal, isn’t it?). So,

2/3 ÷ 3/4 = (2/3 x 4/3) ÷ (3/4 ÷ 3/4) = (2/3 x 4/3) ÷ 1 = 2/3 x 4/3

So, we see that the division by a fraction is the same as the multiplication by the reciprocal of the divisor.

[By the way, you can also use a very similar argument to show that subtraction of a negative is the same as addition of its opposite.]

Anyway, this is a property of division that is not always emphasized in an elementary school mathematics. However, I hope you have a better sense of its potential usefulness.

So, why is the Equal Multiplication Principle true? One way you can see this is to use the measurement meaning of division (that is, how many groups of {divisor} are there in the {dividend}). So, if we have 2400 ÷ 400, we are asking how many groups of 400 can we make with 2400. If we look at both of these numbers using a hundred as a unit, we have “how many groups of 4 hundreds can we make with 24 hundreds?” Thus, the answer should be the same as 24 ÷ 4. It is important that we look at both the dividend and the divisor with respect to the same unit. You could think about 2400 ÷ 400 in terms of units of 5 if you want to. To do so, you need to know how many 5’s are in 2400 and how many 5’s are in 400. Of course, to find those answers, you have to divide 2400 and 400 by 5 – that’s the Principle.

One word of caution. The Equal Multiplication Principle says that the quotient does not change. However, it does not say anything about the remainder. So, for example, if you have 2500 ÷ 400, a common error is to say the answer of 6 remainder 1 because 25 ÷ 4 = 6 remainder 1. However, we need to keep in mind that we are using 100 as a unit. So, the remainder of 1 actually is telling us we have 1 unit of 100 remainder. So, the answer should be 6 remainder 100.

Sunday, May 6, 2007

M3N5(b) - Meaning of Fractions

M3N5. Students will understand the meaning of decimal fractions and common fractions in simple cases and apply them in problem-solving situations.
b. Understand the fraction a/b represents a equal sized parts of a whole that is divided into b equal sized parts.

Teaching and learning of fractions continue to be a major challenge for both teachers and students. Many children (and teachers) think of fractions as parts of a whole [M3N5(a)]. However, M3N5(b) suggests that we look at fractions in a slightly different way as well. For example, according to M3N5(b), the fraction 2/3 means there are 2 pieces of 1/3. So, why is it important that children understand fractions in this manner?

In "Elementary School Teaching Guide for the Japanese Course of Study: Arithmetic (Grades 1 – 6)," the authors suggest that there are 5 meanings of fractions. For example, the fraction 2/3 may mean:

1. two parts of a whole that is partitioned into three equal parts,
2. representation of measured quantities such as 2/3 liter or 2/3 meter,
3. two times of the unit obtained by partitioning 1 into 3 equal parts,
4. quotient fraction (2 ÷ 3), and
5. A is 2/3 of B – if we consider B as 1 (a unit), then the relative size of A is 2/3.

Thus, M3N5(b) corresponds to the third meaning above. So, why is it not sufficient to think of 2/3 as 2 out of three equal parts (of a whole)? What advantages are there to think of 2/3 as two 1/3’s?

The most important reason for going beyond the part-whole view of fractions is that we want students to understand fractions as numbers. The part-whole interpretation of fractions is more about relationships, and it does not necessarily signify a quantity/number. When someone makes 6 out of 8 free throw attempts, the fraction 6/8 doesn’t signify a number. In fact, if he makes 8 of 10 attempts in the next game, we can say he was successful at 14/18 of attempts in those two games combined. This combination is NOT addition of numbers 6/8 and 8/10, in that case, we have to find a common denominator to find the sum. Rather, 6/8 and 8/10 are both ratios. The part-whole interpretation will signify a number if the whole we are considering is the number 1.

The part-whole interpretation is important, and may be a prerequisite, before students can consider 2/3 as 2 pieces of 1/3’s. For this interpretation to be truly useful, students must first understand 1/3 as a number – it is a number such that if you have 3 of them together, you will make the number 1. In other words, 1/3 is a number that is equal to the number in a group when 1 is divided into three equal sized numbers – 1 out of 3 of the number 1.

There are many places in the elementary school curriculum the interpretation of a/b as a copies of 1/b’s. For example, if students’ view of fractions is limited to the part-whole interpretation, they will have a hard time making sense of an improper fraction. After all, what does 4 out of 3 mean? On the other hand, if you consider 4/3 as 4 pieces of 1/3-units, then there is nothing different about 2/3 and 4/3. Or, consider the simple addition/subtraction of fractions with like denominators. For example, 3/5 + 4/5 means putting together 3 pieces of 1/5’s and 4 pieces of 1/5’s, giving us 7 pieces of 1/5’s all together, or 7/5. This reasoning is, in principle, the same as thinking of 30 + 40 as adding 3 tens and 4 tens, thus 7 tens.

The importance of looking at non-unit fractions as collections of unit fractions is not a Japanese idea. Thompson and Saldanha indicated in their chapter on fractions in the Research Companion to the Principles and Standards for School Mathematics, that this is a very important view of fractions. Unfortunately, they also note that this idea is rarely seen in US mathematics textbooks. As we begin implementing the GPS, therefore, it is important for us to remember this perhaps unfamiliar way of looking at fractions.

By the way, the fourth meaning is discussed in M5N4(a). The fifth meaning of fraction, i.e., fractions as ratios, are not treated until Grade 6 when students are introduced to the idea of ratios (makes sense, doesn’t it?). Moreover, in the Japanese elementary textbooks, the idea of a fraction of a set (or discrete model of fractions) does not appear until Grade 6 because they believe that the meaning of fractions in that context is much closer to the ratio meaning of fraction. [In fact, the part-whole meaning of fractions is very close to the ratio meaning of fractions.] This is an interesting contrast to GPS M2N4(a) and something Georgia educators must think about carefully.

I will be discussing models of fractions in another post.

Saturday, April 28, 2007

MKN1a, c, d & e - Numbers

MKN1. Students will connect numerals to the quantities they represent.
a. Count a number of objects up to 30.
c. Write numerals through 20 to label sets.
d. Sequence and identify using ordinal numbers (1st-10th).
e. Compare two or more sets of objects (1-10) and identify which set is equal to, more than, or less than the other.
Helping children understand numbers is one of the major focus of primary grades mathematics instruction. However, what does it mean to understand numbers? Careful reading of this GPS standard suggest different aspects of coming to “understand numbers.”

1. Numbers are used to represent quantities.
Children encounter “numbers” everyday in many different contexts. Here are some examples: I weigh 53 pounds; and I have 24 baseball cards. In each of these examples, “numbers” indicate “how many” or “how much” of something. Because some quantities are more than, less than, or equal to other quantities, numbers may be more than, less than, or equal to others (indicator e).

2. Understanding numbers is more than counting, but counting is important.
The previous discussion suggests that understanding numbers is much more than counting. However, counting plays an important role in understanding numbers as well (indicator a). After all, unless you can count, how would you know that you have “24” baseball cards? So, what are important ideas related to learning to count? First, children need to be able to recite the number words sequence in the correct order, i.e., “one, two, three, four, …”.

Second, children must make one-to-one correspondence between the number words and the objects they are counting. Young children often point/touch objects as they count, but some children point/touch in one rhythm while saying number words in a completely different rhythm. So, sometimes they point/touch more than one object while saying only one number word.

Third, children must understand that the result of counting indicates how many objects there are. To do so, each object must be counted once and exactly once. Therefore, this understanding helps children realize the necessity of making one-to-one correspondence. This understanding also creates the need to keep track of objects that are already counted so that no object is skipped or counted more than once.

Moreover, children must understand that the result of counting, i.e., the last number word produced, indicates how many objects in the whole set. Children sometimes think that a particular number word is a “name” of a particular object. So, after they counted five objects, they will point to the last object when asked to show “five.” They must understand “five” is the amount of objects in the whole group, not the name for a particular objects. Without this understanding, children will not be able to truly understand, for example, 7 is 5 and 2 more, or 9 is 1 less than 10.

3. Numbers are expressed using numerals.
We saw that numbers represent quantities, and counting plays an important role in determining quantities as well as connecting quantities with numbers, or more accurately, number words. In addition to number words, we can also represent numbers using numerals (indicator c). Learning to write numerals is like learning to write letters of alphabets. Moreover, learning to write 2-digit numbers at this level is like learning to spell simple words. Only when children understand our number system, they can make sense of the logic of writing “thirteen” as “13,” but that goes beyond the Kindergarten GPS.

4. Because numbers may be more than, less than, or equal to others, we can use numbers to represent an order (indicator d).
Another way children might see numerals used in their everyday life includes examples like these: My classroom is room 14; My parents watch news on channel 11, and My soccer jersey number is 7. In each of these cases, the numerals do not indicate any particular quantities. Rather, because numbers can be sequenced (1, 2, 3, …) based on the size relationship of the quantities they represent, we can use then to represent order and to label objects uniquely. Unlike a number, which indicates how many objects are in a whole group, an ordinal number is a label for a particular object.

In closing...
I hope this short discussion gave you a better sense of what this GPS is trying to address. Kindergarteners need a lot of experiences in counting objects, making groups, splitting groups, ordering groups, etc. These are activities they encounter throughout a day, not just during the math lesson. Because much, if not all, of elementary mathematics derives out of everyday phenomena, we should try to take advantage of those everyday situations where numbers play important roles. We cannot teach children mathematics by simply giving them worksheets after worksheets.

Monday, April 23, 2007

As I begin...

The new Georgia Performance Standards for mathematics is modeled after (based on?) the 1989 Japanese Course of Study. One of my research interests as a mathematics educator is mathematics education in Japan, where I am originally from.

From time to time, I plan to post my own interpretation of the GPS based on my study of the Japanese curriculum (and my interactions with Japanese mathematics educators at all levels).

I was involved in the translation of a Japanese elementary mathematics textbook series. My colleagues and I have also translated an elaboration document of the Japanese standards that the Ministry of Education published. Both of those materials may be obtained from Global Education Resources (

Creative Commons

Creative Commons License
Elaboration of Georgia Performance Standards by Tad Watanabe is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.