Tuesday, December 23, 2008

M3A1 - Mathematical expressions (1)

M3A1. Students will use mathematical expressions to represent relationships between quantities and interpret given expressions.

One of the things that some people are surprised (or even get upset) about is the fact that algebra is included as a content strand for elementary school students (grades 3-5). Unfortunately, there are some even well-educated people who mistakenly think that this means we will be teaching algebra as they experienced in high schools in elementary schools. Clearly, that's absurd. What is being expected, however, is that children begin developing some foundational ideas about algebra and algebraic reasoning. Of course, that raises the question, "What is algebra in elementary school?"

Even though the GPS is heavily influenced by the 1989 Japanese Course of Study, interestingly enough, there is no "algebra" strand in the Japanese standards. Instead, they have a strand titled, Quantitative Relations, in which student learn much of what we would typically include in Algebra and also Statistics (Data Analysis). In the elaboration document the Ministry of Education publishes, they state that two important "themes" in Quantitative Relations are learning about mathematical expressions and studying functional relationships. The current standard (M3A1) is clearly about mathematical expressions. In fact, this standard really needs to be considered as soon as we start teaching addition operation in Grade 1. Students should learn that 5+3=8 is a representation of a situation like where Johnny had 5 apples and his mom gave him 3 more to make the total number of apples to be 8. Mathematical expressions aren't about computation problems to be completed. They represent situations/physical phenomena/one's thinking, concisely and precisely.

Because they are representations of situations etc., it is also perfectly possible to write something like 8=3+5 to represent decomposition of 8 into 3 and 5, for example. Teachers should include this type of expressions from early on to help students understand that "=" means the two quantities on both sides are equal in magnitude. It does not mean "do something" to get the answer to be written on the right side. By understanding mathematical expressions as representations of situations etc., students can think about writing missing number situations using some place holders like a little box, for example 3+[ ]=8.

When you consider mathematical expressions as representations, and also tools for communications, there are some implications. One such implication is how you write multiplication expressions - I wrote about this in June, 2007 (M2N3a). Another implication is the last part of this standard, "interpret given expressions." If mathematical expressions are the language of mathematics, as I believe they are, then we have to not only worry about "writing" but also "reading." Ability to read/interpret given expressions must become an explicit focus of mathematics instruction, starting in Grade 1. Possible instructional activities may include having students tell stories (or write word problems, when students are old enough) that will match the given expressions and interpreting other students' thinking processes when they present their solutions using mathematical expressions.

Moreover, just as we sometimes "read in between the lines," mathematical expressions can be interpreted in different ways. For example, if we are given 5+3=8, we can simply interpret this statement to mean, "If you add 3 to 5, you get 8." However, we can interpret this statement even further. For example, 5 must be 3 less than 8 since you need to add 3 to 5 to get 8. This means that the difference between 8 and 5 is 3, or 8-5=3. Now, the original addition sentence can also be interpreted as "if you add 5 to 3 you get 8," or 3+5=8. Then, using the similar argument, we can also say that 3 is 5 less than 8, or the difference between 8 and 3 is 5, i.e., 8-3=5. In many US textbooks, students learn about "fact families." I have never heard of such a phrase while growing up in Japan. Instead of simply memorizing how numbers can be shifted around and the operation signs manipulated, it would be much better if our students can "read" math sentences like "5+3=8" and interpret all the relationships that are expressed by so-called fact families, wouldn't it?

Thursday, December 18, 2008

MKM1, M1M1, & M2M1: Teaching Measurement in Primary Grades

MKM1. Students will group objects according to common properties such as longer/shorter, more/less, taller/shorter, and heavier/lighter.
Students will compare and/or order the length, weight, or capacity of two or more objects by using direct comparison or a nonstandard unit.
Students will know the standard units of inch, foot, yard, and metric units of centimeter and meter and measure length to the nearest inch or centimeter.

When discussing teaching and learning of measurement, we need to keep in mind there are three different (yet clearly related) aspects that students must learn. They are,
  • understanding the attribute being measured
  • process of measurement
  • how to use measuring instruments.
The three standards above reflects the four-stage general sequence of teaching measurement:
  1. Direct comparison
  2. Indirect comparison
  3. Measuring with non-standard units
  4. Measuring with standard units
MKM1 is clearly in the first stage, and M2M1 is in the fourth and final stage. M1M1 explicitly indicate stages 1 and 3, but there is no mention of stage 2. Considering the general sequence of instruction, what should be happening is more in stages 2 and 3.

So, why is it important to follow these four stages as we begin our instruction on measurement? The major focus of the first two stages is to help students understand attributes that are being measured. After all, before we can measure anything, we really need to understand what it is that we want to measure. Thus, before we can measure length, we need to understand what length is. By putting two objects next to each other (direct comparison), students can determine which is longer/shorter. Through such experiences, students gain the understanding that length is about the amount of space between the two ends of an object. [Although we may use different words, "height" is not really an attribute. It is really length in the vertical orientation.] Of course, through direct comparison activities, students are gaining some fundamental understanding about how to measure an object as well. For example, when comparing the lengths of two objects, it is important that one end of the objects must be lined up. You cannot say the segment on top in the figure below is longer just because it "sticks out" farther to the right.

Students will also learn that the "amount of space" we are interested in is along a straight pat. Thus, we cannot simply compare the positions of the end points as shown in the figure below.

It should be obvious that these understanding play an important role in the process of measurement later on.

Unfortunately, not every two objects may be directly compared. In those situations, it is sometimes useful to use a third object that can be compared directly to each of the two objects that are being compared. Thus, if a door way is wider than your arm span but a second door way is narrower than your arm span, then you know that the first door way is wider than the second one. Indirect comparison provides more flexibility as you compare two objects. It also provides opportunities for children to experience an important mathematical property of relationships called transitivity, that is, if a > b and b > c, then a > c. Of course, the formal study of such property will not take place until much later. Perhaps more important reason for indirect comparison is that it sets the stage for the most fundamental idea about the measurement process - the use of a unit. When using a third object, it may not be in between the two objects - for example, a wooden stick may be much shorter than two door ways. In those cases, however, it may be possible to determine that one door way is taller than the three (of the same) wood sticks put end to end while the other one is shorter than three wood sticks. Now, we can say that the first door way is taller than the second one.

You can easily see that such experiences become the foundation for the idea of expressing an attribute in terms of the number of a third object, unit, necessary to "cover" it. When we move into this stage, we are now indeed "measuring" in the sense that we are assigning a number to an object in terms of how much of the attribute it has. There are many merits for expressing the amount of an attribute using numbers. Clearly, it simplifies the process of comparison as we no longer need to find different object to use as the reference. Comparison of multiple objects can be easily done by simply comparing numbers. Moreover, once we assign numbers, we can answer not only "which is longer?" but also "by how much?" In general, once we express the amount of attributes with numbers, arithmetic operations may be used to answer some questions. Although the GPS does not explicitly state those merits, I hope teachers help students experience and understand those merits.

Some people may argue that, once we get to this stage, we should just use standard units. This argument perhaps makes sense later in the elementary grades after students have learned about measuring three or four different attributes. However, at the primary grade level, it is also important to keep in mind that students are still learning about the process of measurement - pick a unit, then determine how many of the unit is necessary to equal the object you are measuring. For us, this is so obvious, but not so with children. Introducing standard units at this stage will require children to deal with two new ideas simultaneously - new units and new process. There are also other considerations. First, some units may be too small or too large so that the size of the resulting numbers may not be appropriate for children at this particular time. By using non-standard units, teachers can control the range of numbers students might obtain. Also, it is important to note that measuring with standard unit typically means measuring with various instruments. For example, if you are measuring with inches, you are most likely to be measuring with a ruler. However, learning to use a ruler is also a challenging task - this might be a third new idea students have to deal with if we are to introduce standard unit at this stage.

Although it may sound a bit paradoxical, the use of non-standard units is a useful experience for children to understand the need for having standard units. For example, if two students measure the width of the same door way using their pencils, they may get different results. They will soon realize that they cannot compare numbers unless their units are the same. This is when we can introduce standard units such as inches, feet, centimeters and meters.

Finally, learning how to measure with common instruments such as rulers is not as simple as adults might think. For that purpose, it may be useful if children had some experiences using their own measurement tools. For example, during the third stage (measuring with non-standard unit), students can tape together index cards to form their own measuring "tape." Initially, students may actually count the number of index cards, but eventually they may realize simply labeling the cards 1, 2, 3,... will make it simpler. Such experiences will allow them to understand that what we are counting on a measurement tape is the number of spaces between the tick marks, and the numerical label at a given tick mark indicates the total number of units up to that mark. Furthermore, as we learned in the first stage, the end (actually the starting point) of the measuring tape must be lined up with an end of the object, not the tick mark labeled "1." A variety of home-made measuring instruments can be made to measure length, capacity/volume, weight, and even angles. Making and measuring with home-made instrument may be a very fruitful experiences as students learn to measure with standard units.

Finally, it should be noted that weight is not formally studied until Grade 4. Thus, children's experiences in Grades K and 1 should be viewed within the context of teaching children more about the existence of different attributes. Weight is a difficult concept for children because we cannot "see" it - that is, some objects that look big may be light while others that look small may be quite heavy. Thus, direct and indirect comparison activities may be what we should focus on in Grades K and 1 with respect to weight.

Sunday, November 30, 2008

M5N4(d) - Modeling 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.

In the last three posts, I discussed multiplication and division of decimal numbers that do not depend on the knowledge of multiplication and division of fractions. That was necessary because in the GPS decimal multiplication and division are discussed prior to fraction multiplication and division. In this post, I would like to discuss multiplication and division of fractions. I have previously discussed this topic (November, 2007). In the post, I briefly discussed how the area model may be used to represent multiplication of fractions, as well as the double number line representation that can be used for both multiplication and division. So, in today's post, I want to focus on how to model division of fractions.

As students are introduced to division operation in Grade 3, they are expected to understand that "division m many equal parts of a given size or amount may be taken away from the who 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" (M3N4b). We discussed how these interpretations must be extended as the number of "groups" become decimal numbers - whether as the divisor in a fair sharing problem or as the quotient in a measurement division problem.

The situation is basically the same with fraction multiplication and division. If the divisor is a whole number, we can use the fair sharing interpretation. When the divisor becomes a fraction, we must use either the extended meaning of fair sharing, that is, an operation that determines the per-one quantity, or the measurement interpretation. The measurement interpretation is much more easily modeled using manipulatives such as pattern blocks. We can model 3/4 divided by 1/6 this way: First, let's represent a whole using two hexagon pieces together. Then, 3 trapezoids will represent 3/4 and a blue rhombus will represent 1/6. The division question is asking how many blue pieces will fit in the 3 red trapezoids together. You can easily show that 4 blue pieces will fit completely inside the 3 red trapezoids. The remaining section is 1/2 of a blue rhombus. Therefore, the quotient is 4 1/2.

How else can we model division of fractions? In particular, how can we model fraction division that may also reflect the inverse relationship between multiplication and division? In my previous post on this standard, I mentioned how the area model of multiplication may be used to represent multiplication of fractions. In this model, the two dimensions of a rectangle represent the two factors and the product is represented by the area of the rectangle (in relationship to the unit rectangle). Thus, the figure below represents 1/3 x 2/3:

So, is there a way to represent division using the area model? For example, how can we model 3/4÷2/5? [I encourage you to think about how you may be able to represent this division using pattern blocks. You may find it a bit cumbersome.]

Since division is the inverse operation of multiplication, 3/4 must be the area of the rectangle, and the divisior, 3/4, is one of the two dimensions. Thus, we are trying to determine the other dimension of the rectangle so that the area will be 3/4. So, how can we model this? I'm sure that there are different ways, but here is one possibility.

Let's start by first representing 3/4:

Of course, this fraction has the dimension of 1 unit (vertically) by 3/4 units (horizontally). What we want is a rectangle that has the same area as the yellow rectangle but has 2/5 as one of the dimensions. If we say that the vertical dimension is 2/5 units, then we are looking for the horizontal dimension, as shown in the figure below:

So, how can we find the horizontal dimension? First, let's first draw in the segments showing the fifths in the yellow rectangle.

Now, we can see that another set of 2/5 by 3/4 rectangle (shown in green below) can be shifted to fit inside the rectangle whose vertical dimension is 2/5.

Now, two of the remaining 3 small rectangles (shown in blue) can be shifted.

Finally, the remaining small rectangle has to be split into two equal parts (shown in red).

So, how long is the horizontal dimension, which will be the quotient? Each of the small rectangle has the horizontal dimension of 1/4 unit. Clearly, we have 7 1/4-units. Finally, the red segment is a half of the small rectangle, or a half of 1/4. Thus, we have 7/4 and 1/8, or 15/8 altogether.

The figure below shows 1 2/3 ÷ 3/4.

Now, in this situation, the 2 small rectangles (in blue) had to be split into 3 equal parts, so the horizontal segment of the blue segment of the quotient is 2/3 of 1/3-unit, or 2/9. Thus, the quotient is 2 2/9.

Now, in this model, each small rectangle you obtain has the horizontal dimension which is the unit fraction with the denominator for the dividend (4 in the first example and 3 in the second). The total number of the small rectangles in the dividend is the product of the numerator of the dividend and the denominator of the divisor. The number of horizontal column of the unit fraction can be calculated by dividing the total number of the small rectangles by the numerator of the divisor. Thus, the quotient can be expressed as:

In other words, to divide a fraction by another fraction, you simply multiply the dividend by the reciprocal of the divisor. Of course, this generalization may be straightforward for us, but it is extremely important that we analyze what mathematical ideas are involved in making that generalization. Then, we can decide whether or not this generalization is accessible to our students.

In any event, it does raise some questions about why the GPS asks students to model division (and multiplication) of fractions in Grade 5 without specifying the development of the algorithm in the same grade level. As I stated earlier, I believe the appropriate interpretation of the current GPS is that the algorithms are to be developed (and mastered) in Grade 6. However, it seems rather strange to separate modeling from the algorithm development, which is the generalization based on the models.

Finally, I would like to emphasize that the area models are useful when we know the operation involved. The area model cannot help students determine which operation to use. For that purpose, models like double number line are much more suited.

Friday, November 28, 2008

M5N3 Multiplication & Division of Decimal Numbers (3)

M5N3. Students will further develop their understanding of the meaning of multiplication and division with decimal fractions and use them.

OK, this is the third (and hopefully the last) in the series of posts discussing multiplication and division of decimal numbers. In the last two posts, we discussed multiplying and dividing decimal numbers by whole numbers and multiplying by decimal numbers. We are developing these ideas using only our understanding of whole number multiplication and a powerful idea about our numeration system, relative size of numbers. What is left for us now is dividing by decimal numbers. Let's go back to our problem:

Problem 4
A wire that is 2.4 meters long weighs 3.6 grams. How much will the same wire weigh if it is 1 meter long?

This problem requires us to divide 3.6 by 2.4. We already looked at dividing decimal numbers by whole numbers, but we have yet to consider division by decimal numbers. In some curricula, fraction arithmetic is discussed first, so we can change this division to division of fractions. However, that line of reasoning is not available if we follow the GPS. So, what can students do?

Whenever students encounter a new problem, we would like them to ask, "What do I know that I can use?" or "How is this problem similar to what I have studied previously?" Such a habit is an example of what the authors of Adding It Up (National Research Council, 200?) call productive disposition. Again, a diagram might help us think about this problem.

One possibility is to think about 2.4 as 24 0.1's as we did before. But, what do we get if we divide 3.6 by 24? Let's see what the diagram will show us:

We can tell from this diagram that the result of dividing 3.6 by 24 is the weight of a 0.1-meter wire. So, how can we find the weight of a 1-meter wire if we know that a 0.1-meter wire weighs 0.15 grams? Since 1 meter is 10 times as long as 0.1 meter, the weight should also be 10 times as much. So, to find the weight of a 1-meter wire, we just need to multiply the weight of a 0.1-meter wire by 10. So, a 1-meter wire will weigh 1.5 grams.

With Problem 3, we also had another approach that considered 10 times of the multiplier. What would a parallel reasoning in Problem 4 be like? If we make the divisor (2.4) into a whole number, what does it mean? That means we are looking at a 24-meter wire, instead of a 2.4-meter wire. Again, it's 10 times as long, therefore, it should weigh 10 times as much, i.e., 36 grams. But if we know that a 24-meter wire weighs 36 grams, we can find the weight of a 1-meter wire by simply dividing 36 by 24. We don't have to do anything with the result since we haven't changed the weight of 1-meter of wire when we considered the weight of the 24-meter wire. A diagram might show this approach clearly:

This second approach may be more useful to generalize a paper-and-pencil algorithm. Basically what we did was to multiply the divisor by a power of 10 to make it into a whole number. Then, the dividend must be multiplied by the same power of 10 - since the length of the wire is now that many times as long, it should weigh also that many times as much. Then, we can simply divide the new weight by the new length, we can find the weight for 1 meter. Therefore,

Another way of describing this process is to move the decimal point of the divisor (the number outside of the long division symbol) as many places as necessary to the right to make it into a whole number. Then, move the decimal point of the dividend the same number of places to the right as well - annexing 0's if necessary. Then, we can perform long division as we have done previously - either a whole number divided by a whole number or a decimal number divided by a whole number. Again, this is the familiar algorithm, isn't it?

As we saw in the three recent posts, the familiar multiplication and division algorithms can be meaningfully derived using only our knowledge of whole numbers and the idea of relative size of numbers. In the Japanese standards, they discuss decimal multiplication and division first because the algorithms are essentially the same as those of whole number multiplication and division. Thus, when students study multiplication and division, they can focus more on extending the meaning of multiplication and division. Then, when students study multiplication and division of fractions, they do not have to worry about dealing with the new meaning of operations AND the new algorithms. It is not clear if the GPS writers had the same intent, but I hope you see how students can develop multiplication and division algorithm for decimal numbers without knowing multiplication and division of fractions.

Sunday, November 9, 2008

M5N3 Multiplication & Division of Decimal Numbers (2)

M5N3. Students will further develop their understanding of the meaning of
multiplication and division with decimal fractions and use them.

In the last post, I discussed how the idea of relative size can be used to think about multiplying and dividing decimal numbers by whole numbers - M4N5(d). In this post, I want to continue to the next step, multiplying and dividing by decimal numbers. As I discussed in October, 2007, when the multiplier and the divisor is something other than a whole number, we must extend the meaning of division from an equal-group perspective to a more proportional one. Let's look at the two problems I left as "homework" last time.

Problem 3
One meter of wire weighs 2.4 grams. How much will 1.8 meters of the same wire weigh?

Problem 4
A wire that is 2.4 meters long weighs 3.6 grams. How much will the same wire weigh if it is 1 meter long?

Clearly, in Problem 3, we must multiply 2.4 by 1.8, while in Problem 4, we must divide 3.6 by 2.4. Since these situations involve a decimal multiplier and a decimal divisor, we can no longer use the equal group interpretation of multiplication and division - what does 1.5 or 2.4 groups mean? Rather, we must look at these situations more proportionally. In Problem 3, we are asking, if 2.4 is to 1, how much is to 1.8, and in Problem 4, if 3.6 is to 2.4, what is to 1? Alternately, if you use multiple comparison idea, Problem 3 asks how much is 1.5 times as much as 2.4, while Problem 4 asks 3.6 is 2.4 times as much as what?

Let's now think about how students can solve these problems using only what they have learned so far, which does not include how to multiply or divide by decimal numbers.

Problem 3
One meter of wire weighs 2.4 grams. How much will 1.8 meters of the same wire weigh?

One possible idea that students might use is to consider the multiplier, 1.8, in terms of the decimal unit using the idea of relative size. That is, 1.8 means there are 18 pieces of 0.1's. But what does that mean? A diagram might be helpful. Using a double number line (November, 2007), we can represent the problem like this:

When we say 1.8 is made up of 18 pieces of 0.1's, the diagram may look like this:

In other words, 1.8 meters can be thought of as a collection of 18 0.1 meter pieces. But, how does that help us find the missing number. We are not multiplying 2.4 by 18 - we don't have 18 groups of 2.4. What do we have 18 groups of on the top number line?

From this diagram, we can tell that what we have 18 of on the top number line is actually the weight of 0.1 meter wire. In other words, if we know how much a 0.1-meter wire weighs, then, we can find the answer. But, it's easy to see that the weight of a 0.1-meter wire can be determined by simply dividing 2.4 by 10, which is what students learned in Grade 4. Once we determine the weight of a 0.1-meter wire, i.e., 0.24 grams, then, we can multiply that by 18, which is also a Grade 4 idea. 0.24 x 18 = 4.32, so the weight of a 1.8-meter wire is 4.32 grams.

Here is another idea that students might come up with. Although we are looking for the weight of a 1.8-meter wire, let's first think about the weight of 18-meter wire, which is easy enough - simply multiply 2.4 by 18, a Grade 4 idea. However, since a 18-meter wire is 10 times as long a 1.8-meter wire, it should also weigh 10 times as much, too. So, in order to determine the weight of a 1.8-meter wire, we can simply divide that by 10 to find its weight. Since we already know how to divide decimal numbers by whole numbers, this last step should not be a problem. This line of reasoning may be represented on a number line like this:

Different students will feel more comfortable with different approaches. However, this second approach may be more useful to generalize into a written computation algorithm. In general, what we do in the first step is to make the multiplier into a whole number by multiplying it by an appropriate power of 10. Now, if the multiplicand is a decimal number, we end up multiplying it by a power of 10 to make it into a whole number as well (that's another way of thinking about the use of relative size). Now that we have two whole numbers, we can multiply them easily. However, this product is too big, and it must be divided by those powers of 10. For example,

Since multiplying by 10 means that the decimal point will move to the right one place while dividing by 10 means moving the decimal point to the left one place, we can describe what happened above this way: when we think of 3.7x4.26 as 37x426, we moved the decimal point 3 places to the right altogether, therefore, we have to move the decimal point to the left 3 places in the product of 37x246 to get the product for 3.7x4.26. And, this is (to us) the familiar multiplication algorithm for decimal numbers, isn't it?

Well, this has gotten a bit too long - of course, with actual 5th graders, you may need several lessons to get this much discussion done. Anyway, I think I must postpone the discussion of dividing by decimal number until next time. However, if you can think about how we solved Problem 3, you may find that Problem 4 can be solved in similar ways.

Monday, November 3, 2008

Revisiting M4N5(d) & M5N3 - Multiplication and Division of Decimal Numbers (1)

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.

I have discussed these standards previously (October, 2007). However, I had an interesting conversation with a colleague from another university in Georgia recently. She also teaches mathematics courses for prospective elementary school teachers at her school. We were discussing how I incorporate Japanese curriculum materials to discuss decimal multiplication and division. She then shared with me that, when she teaches this topic, she relates it to fraction multiplication and division. Her approach is perfectly valid and viable; however, in the GPS, decimal multiplication and division are discussed before fraction multiplication and division. So, I feel it is important that our future elementary school teachers experience how decimal multiplication and division may be developed WITHOUT the knowledge of fraction multiplication and division. So, I want to discuss how multiplication and division of decimal numbers can be approached using the knowledge of whole numbers only.

Let's start with multiplying and dividing decimal numbers by whole numbers.

Problem 1
    One meter of wire weighs 5.7 grams. How much will 3 meters of the same wire weigh?

In the last post, I discussed the idea of relative size. If you use that idea, 5.7 grams can be considered as a collection of 57 0.1 g pieces. Thus, in 3 meters, we have 3 sets of 57 0.1 g's. That means we can use the calculation 57x3 to find out the total number of 0.1 g's in 3 meters. If there are 171 pieces of 0.1 g's, again using the idea of relative size, we know that is the same as 17.1 grams.

In general, when you are multiplying a decimal number by a whole number, you can just consider the given decimal number in terms of its smallest decimal place value as the unit. For example, 0.37 is 37 0.01's, 0.824 is 824 0.001's, etc.. You will then have a whole number as the multiplicand (in terms of a decimal unit). Since the multiplier is a whole number, we can use multiplication to find the total number of pieces of the decimal unit. We can then convert the final result into a decimal number by using the idea of relative size. Using the familiar paper-and-pencil algorithm, it basically means that the decimal point for the multiplicand (the number on top) and the product are in the same place.

Now, let's look at division of decimal numbers by whole numbers.

Problem 2
    A wire that is 3 meters long weighs 5.7 grams. How much will the same wire weigh if it is 1 meter long?

Again, using the idea of relative size, we can think of 5.7 grams as a collection of 57 pieces of 0.1 gram. Since 3 meters of this wire include these 57 pieces, and you want to know how much 1 meter will weigh, you simply need to divide 57 by 3, which is just a whole number division problem. The quotient, 19, tells us the number of 0.1 gram pieces in each group. Again, using the idea of relative size, we can conclude that 1 meter of this wire will weigh 1.9 grams.

In general, just as in the case of multiplication of decimal numbers by whole numbers, when you are dividing a decimal number by a whole number, you can consider the decimal dividend in terms of its smallest decimal place value as the unit. Then, we can simply use whole number division to find out how many pieces of the decimal unit will be in each group. The final quotient can be found by using the idea of relative size. Using the familiar long division notation, this suggests that the decimal point for the quotient and the dividend will be in the same place, i.e., the place values for the dividend and the quotient should line up, just as they did with whole numbers.

One point of complication we need to pay attention is what if the dividend (whole numbers pieces of a decimal unit) is not evenly divisible by the divisor. So, for example, what can we do if the weight of the 3-meter wire was 5.8 grams. When we divide 58 by 3, we have the remainder of 1. But, this is also the number of 0.1 gram piece. So, we can say that the answer to the division of 5.8 by 3 is 1.9 with the remainder of 0.1. In other words, the remainder is the number of the decimal unit. Thus, in the long division notation, the decimal point of the dividend and the decimal point for the remainder must also line up.

Of course, another option is to divide on - the remainder of 1 can be thought of as a collection of 10 0.1's (actually, the remainder is 0.1, so we are really talking about 10 0.01's), then we can keep dividing. Sometimes, this will result in a terminating decimal, while in other cases, you will have a repeating decimal. With this understanding of dividing on, when students learn about the quotient meaning of fractions they can then understand that every fraction can be re-written as a decimal number by simply dividing its numerator by the denominator.

OK, this post is already rather long. So, I will have to wait till the next time to discuss multiplication and division by decimal numbers. In the meantime, I encourage you to think about how students can reason about to solve the following problems, which involve multiplication and division by decimal numbers. Keep in mind that they have not learned how to multiply or divide by decimal numbers or fractions.

Problem 3
    One meter of wire weighs 2.4 grams. How much will 1.8 meters of the same wire weigh?

Problem 4
    A wire that is 2.4 meters long weighs 3.6 grams. How much will the same wire weigh if it is 1 meter long?

Friday, October 31, 2008

M2N1(b) & M3N1(b): Relative Magnitudes & Relative Sizes

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

M3N1 Students will further develop their understanding of whole numbers and ways of representing them.
b. Understand the relative sizes of digits in place value notation (10 times, 100 times, 1/10 of a single digit whole number) and ways to represent them.

In these two standards, you see phrases, "relative magnitudes" and "relative sizes." These standards actually elaborate what these phrases mean further - using 10, 100 or 1000 as a unit, and 10, 100 times or 1/10 of a single digit whole number. These statements seem to suggest that these phrases may be related but different.

As you know, the GPS was heavily influenced by the 1989 Japanese Course of Study (COS). Interestingly, in the COS, they use the same words, which can be translated either "relative magnitude" or "relative size." The Japanese Ministry of Education produces a document that explains the COS, and in this document, they explain what they meant by "relative size/magnitude":
"To understand the relative size of numbers" mean to grasp numbers' size by units of tens and hundreds." (Grade 2)
"In this grade, broaden the range of numbers up to unit of ten-thousands, and help children deepen their understanding of the relative size of numbers." (Grade 3)

Thus, it appears that, in the original Japanese COS, the authors' focus was on the meaning that is consistent with the meaning suggested by M2N1(b). So, what does this mean? Let's look at an example, 38291. Teachers and students are familiar with the question, "What numeral is in the hundreds place?" However, the idea of "relative magnitude/size" suggests another question: How many hundreds does this number include? The answer is 382. We can also say that this number also include 3829 tens. With our numeration system, therefore, telling the relative size of numbers is rather easy. Whatever the unit you want to use to consider the given number, think of that place as the "ones" place and consider the number made up of the numerals to the left. So, in 15076821, there are 1507 ten-thousands, 150768 hundreds, etc.. Actually, the way we read number in English take advantage of this idea. The number 38291 is read as "38 thousands 291," not "3 ten-thousands 8 thousands, ...," and 15076821 is "15 millions..."

This idea can also be extended to decimal numbers (and Japanese textbooks emphasizes this way of looking at numbers). For example, consider the number 0.873. You can say this number has 8 0.1's, 87 0.01's, or 873 0.001's. You can even say this number includes 8730 0.0001's. Moreover, the idea of considering a number using units other than 1 is an important foundation for fraction learning as well. It is very useful to consider non-unit fractions as collections of unit fractions. For example, 3/4 is 3 one-fourth's. When you consider numbers from this perspective, 30+40, 300+400, 0.3+0.4, and 3/5+4/5 can all be related to "3+4." The only difference is the unit, 3 and 4 of what (tens, hundreds, 0.1's or one-fifth's) we are combining.

By the way, there is actually a Grade 3 standard in the 1989 Japanese COS that states, "(Students are) To know about the size of 10 times, 100 times, 1/100 of a whole number and how to represent them." The elaboration document goes on to explain this standard by saying:
When teaching 10 times bigger, 100 times bigger, or 1/10 of a whole number, it is necessary to help children pay attention to the fact that the order of numerals does not change and that the size of corresponding numerals is 10 times, 100 times, or 1/10 of the original numbers.

Finally, the elaboration document states that this idea is related to the idea of "relative size/magnitude" discussed in another Grade 3 standard. So, it appears that M3N1(b) is more about this standard. However, as I noted above, the way we read large numbers in English take advantage of this idea, it is nevertheless important that Grade 3 mathematics instruction revisits and extends this idea further.

Tuesday, October 21, 2008

M3G1 - Geometry in Primary Grades (3)

M3G1. Students will further develop their understanding of geometric figures by drawing them. They will also state and explain their properties.

In Kindergarten, students are expected to "recognize" certain geometric figures. In Grade 1, students are expected to classify geometric figures by comparing their structures. In M3G1, the GPS mentions "properties" of geometric figures for the first time. By the end of Grade 3, students are expected to state and explain properties of geometric figures. The GPS does not clearly spell out what properties are to be explored. However, based on the types of figures students have previously explored, properties such as the equality of the base angles of isosceles triangles are within the reach for 3rd graders. Of course, we need to keep in mind that equality of angles at this point is based on the fact that two angles can be made to overlap each other completely as measuring angles is a Grade 4 expectation.

The general flow of the geometry standards in the GPS is very much consistent with the developmental levels of geometric thinking identified by Dina and Pierre van Hiele. According to the van Hiele's model, children can only treat geometric figures as wholes. Their thinking is based on how figures appear. In the second stage, children develop the ability to identify and use various components of geometric figures as objects of their thought. It is in this stage, they can start identifying figures based on relationships among their component parts. As children move into the 3rd level, typically called Informal Deduction, they can now start focusing on those relationships as the objects of their thought. One characteristic of students in this stage is that they can now start using some logical statements, like if ... then .... In order for students to be successful a typical high school geometry class, students must be at this stage at the beginning of the course. Since geometry is discussed throughout Math 1 ~ 4 in the GPS, and since geometry proof happens in Math 1, this means students must be at this stage by Grade 9 the latest.

As children begin to explore properties of geometric figures in Grade 3, it is essential that teachers distinguish properties from definitions. The van Hiele theory suggests that children in the second stage can identify relationships among various components of a given figure. However, as children move toward the third stage, they can identify the minimum amount of relationships that will be sufficient to define a shape. Those relationships become the definition of the shape, while other relationships are now treated as properties. For example, children in the second stage might be able to identify the following relationships in isosceles triangles:
  • two sides are equal in length
  • two angles are equal
  • has a line of symmetry (you can fold it in half - symmetry is a Grade 6 topic)
  • two angles that are equal are both acute angles
  • third angles can be acute, right or obtuse
  • etc.
As children move to the third stage, they can go from this laundry list of relationships to the realization that only the first relationship is necessary to define an isosceles triangle. A useful activity to help children go from a laundry list to the minimum set of defining characteristics is to have children actually draw shapes. As children are asked to draw isosceles triangle, they will realize that they don't need to use all of those characteristics to draw one - indeed if you draw a triangle with two equal length sides that's enough.

Although most 3rd graders are still in the second van Hiele stage, it is important that teachers' communication (with students and with parents) clearly distinguish definitions and properties. The parent letter for the third grade geometry unit have the following "terminology":
  • Parallelogram: A quadrilateral with opposite sides that are parallel and of equal length and with opposite angles that are of equal measure.
  • Rectangle: A quadrilateral with four right angles and two pairs of opposite, equal parallel sides.
  • Rhombus: A parallelogram with four equal sides and equal opposite angles.
All of these are "laundry lists" and not definitions. Although at times it is perfectly fine to be less rigorous about our language use, when we do use language loosely, we should do so with clear understanding that we are indeed being less rigorous.

Saturday, September 20, 2008

M2G1 - Geometry in Primary Grades (2)

M2G1. Students will describe and classify plane figures (triangles, square, rectangle, trapezoid, quadrilateral, pentagon, hexagon, and irregular polygonal shapes) according to the number of edges and vertices and the sizes of angles (right angle, obtuse, acute).

Well, it has been a long time since I wrote an entry here. I apologize for those who posted their comments for not responding. I will try to keep updating this blog a bit more frequently.

Last time, I started writing on the geometry standards, and I will continue the discussion of geometry in this entry. In M2G1, students are now expected to classify plane figures "according to the number of edges and vertices and the sizes of angles (right angle, obtuse, acute)." The GPS includes the following figures: triangles, square, rectangle, trapezoid, quadrilateral, pentagon, hexagon, and irregular polygonal shapes. Last time, I questioned the appropriateness of the expectation that kindergarteners to distinguish squares and rectangles. In Grade 2, students are learning about different types of angles, right, acute and obtuse. Therefore, it is in Grade 2 when it is appropriate for students to learn that rectangles are quadrilaterals with 4 right angles, and squares are rectangles with all sides equal.

However, let's think about how students can understand right angles. When I ask my adults, including some teachers and teacher candidates, what right angles are, they almost always respond by saying "90-degree angles." Although it is true that right angles measure 90 degrees, measuring angles using "degree" as the unit is a Grade 4 standard. Thus, how are second grade students to understand what right angles are? A Japanese elementary math textbook by Hironaka and Sugiyama has an interesting approach to this topic. They define a right angle to be the angle you obtain when you fold a piece of paper as shown in the figures below:

Note that the piece of paper can be any shape to start with. The second fold is made in such a way that the first fold line will be folded onto itself. Although it might also be helpful to point out to children that the corners of note papers, index cards, etc. are right angles, we cannot always be sure that corners of any piece of paper are right angles.

Interestingly, this definition of a right angle is very much comparable to Euclid's definition of right angles in his book The Elements. He defines that the angles you obtain by equally dividing a straight angle are right angles. When the second fold is made so that the first fold line will be folded onto itself, we are indeed dividing the straight line (the first fold line) into two equal angles.

I want to end this entry by raising another issue with the standard, however. This standard expects students to describe and classify trapezoid. However, in order to describe trapezoids, children need to concept of parallelism. The Grade 2 Geometry unit of Math Frameworks define trapezoids as "quadrilaterals with two parallel sides." Unfortunately, parallelism is a Grade 4 topic. Therefore, it is very strange that we should expect students in Grade 2 to describe and classify trapezoids.

To make the matter even worse, Grade 4 Geometry unit of Math Frameworks defines trapezoids as quadrilaterals "with only one pair of parallel sides." This definition is different from the Grade 2 definition, which does not say anything about the other two sides of quadrilaterals. In a recent publication, Zalman Usiskin and his collaborators document how these two definitions of trapezoids has existed in US mathematics textbooks. Therefore, the fact that the definitions are different isn't too surprising. However, it is rather unfortunate that a document that emphasizes coherence will not try to be consistent in their definition of a geometric figure.

Thursday, January 17, 2008

MKG1 - Geometry in Primary Grades (1)

MKG1. Students will correctly name simple two and three-dimensional figures, and recognize them in the environment.
a. Recognize and name the following basic two-dimensional figures: triangles, rectangles, squares, and circles.

All of my posts so far have been on the Number and Operations strand. This semester, I am teaching a geometry course for prospective elementary school teachers, so I want to start 2008 with a post on a geometry standard, MKG1.

As I read this standard, I wonder how students are to recognize figures, in particular rectangles and squares. Two Dutch mathematics educators, Dina and Pierr van Hiele, showed that children’s geometric thinking develops sequentially along 5 different levels. In the first stage, called Visualization, children recognize shapes by looking at them as a whole, not their component parts. Thus, they would say a shape is a square because it looks like a square. As they move to the next level, Analysis, they can start paying attention to the characteristics of shapes. However, it is not until the third level that children can actually think about interrelationship among classes of shapes, for example, rectangles are parallelograms. A typical HS geometry class requires students to be in the fourth level, called Deduction. In the fifth level of geometric thinking, a person can now think about different types of geometry, e.g., Euclidean, Spherical, etc. Although the works of van Hiele showed that these levels are not so much age dependent, it is probably safe to assume that most Kindergarteners are still operating in the first level. From this perspective, the inclusion of rectangles and squares in MKG1 is a bit problematic.

There are many different ways we can classify figures, and as we create classes of geometric figures, we sometimes give them specific names. Some classes of figures are mutually exclusive, while others are overlapping. All four types of figures named in this standard are similar in that they are all closed figures. Therefore, we are clearly expecting students to recognize closed figures from non-closed figures. Triangles, rectangles, and squares are members of a class of figures called polygons, while circles are not. Thus, recognizing circles (and distinguishing them from other shapes such as triangles, rectangles and squares) seems to be a slightly simpler task for children, appropriate for those children who are in the Visualization level. Distinguishing triangles from rectangles and squares also seems to be reasonable for such children. Although they may not be explicitly pay attention the number of sides as characteristics of these figures, triangles will definitely “look differently” from rectangles and squares – or any other four (or more) sided figures.

However, recognizing rectangles and squares seem to require much more sophistication that recognizing circles or triangles. For one thing, rectangles and squares are members of a class of figures called quadrilaterals (4-sided figures). Thus, simply noting the number of sides is insufficient to recognize them. What makes them special is the fact that all of them are 4-sided figures with 4 right angles. Moreover, squares and rectangles are not mutually exclusive, as circles and triangles are. In fact, squares are special type of rectangles. I doubt that the writers of the GPS expected Kindergarten children to understand this relationship – that would be developmentally inappropriate. What would have been a more appropriate standard is “to recognize triangles, quadrilaterals (or quadrangles), and circles,” but the term “quadrilateral” seems to be a bit too much for Kindergarteners. Thus, in order to avoid (perhaps) developmentally inappropriate words, we may have unintentionally introduced (perhaps) developmentally inappropriate mathematics expectations.

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.