P1. Students will solve problems (using appropriate technology).
This process standard includes the parenthetical statement, "using appropriate technology." Although the use of computers and other technologies are generally accepted by teachers and general public, the use of hand-held calculators, particularly in elementary classrooms, continue to be controversial. On the other hand, most people seem to endorse the use of concrete materials (manipulatives) in elementary school classrooms. So, what's the difference between technology (not just calculators) and manipulatives? Although some people may say both technology and manipulatives are both simply learning tools, I believe there is a fundamental difference in their nature.
Let's consider how long division algorithm may be taught using a very commonly found manipulatives, base-10 blocks. A typical instructional sequence will start with problems where students are asked to solve sharing problems using base-10 blocks. As students continue to solve these problems using base-10 blocks, teachers may encourage students to start drawing the picture of blocks and modify the picture as blocks are manipulated. Eventually, teachers will ask students to simply draw pictures of what they would do with base-10 blocks without actually working with the blocks. As students continue solving problems by drawing pictures, teachers will encourage students to use numerals to record the process - instead of drawing 4 flats (hundreds), students can simply write "4" under the heading of "flats (or hundreds)." Eventually students can organize the record using the familiar long-division notation. [See, for example, two activities Sharing Base-10 Blocks and Doing and Recording on my university web page: http://science.kennesaw.edu/~twatanab/.]
Now, here is a calculator game that may help students to develop mathematical thinking. It is called "NIM with Calculator." It is a 2-player game. Clear the calculator so that "0" is shown in the display. Players take turn adding 1, 2, or 3. The winner is the player who gets the sum of 21 after his/her turn. It is a very simple game and children do not have problem remembering the rules. When students become comfortable with the game, you may want to ask if there is a "winning strategy" for either player - the player who goes first or the player who goes second. It turns out there is a winning strategy for the player who goes first - that is, if you know the strategy, you can be 100 % sure that you will win if you go first. I encourage you to figure out the strategy.
Once you figure out the strategy, an interesting extension question is how you may be able to figure out the winning strategy if you change the goal number - for example, you can make the player who gets the sum of 24 to be the winner. You can then determine the relationship between the goal number and the winning strategy (in particular, the first number you must enter).
All of these activities - base-10 block division activities and NIM with Calculator - may be appropriate in elementary classrooms at the appropriate time. However, the roles these tools (base-10 blocks and calculators) play are very different in nature. With base-10 blocks, teachers ultimate goal is to help their students go beyond base-10 blocks. That's the reason teachers will start asking children to draw base-10 blocks or imagine what they might do with base-10 blocks. It is possible with base-10 blocks, and other manipulatives, for children to imagine what they would do and what the results of their actions might look like. Thus, they can examine the effects of their actions without having to use manipulatives. Of course, it is essential that children have opportunities to physically manipulate the blocks BEFORE they start imagining what they might do or what the results of their actions might look like.
On the other hand, calculators and other technological tools are often suited for such an instructional step. Children might be able to imagine which calculator keys to push, but it isn't always possible for students to imagine what the results of their actions may look like - in other words, they don't always know what they will see after they hit the "=" key. Similar point can be made with graphing calculators, dynamic geometry software, or productivity software like spreadsheet.
Teachers should be aware of this difference in the nature of these tools. Manipulatives are useful tools for students to make sense of different processes so that they don't have to use manipulatives to figure out the results. On the other hand, technological tools are to be used to help students think about mathematical relationships that might exist among different numbers and shapes in the problem context. We are not interested in "weaning" students from technology - we want our students to become better at using technology. Judicious use of technology requires us to pay attention to this difference.
Monday, April 6, 2009
Thursday, March 26, 2009
M6G1 a - Symmetry
M6G1. Students will further develop their understanding of plane figures.
a. Determine and use lines of symmetry.
In the last entry, I mentioned that the topic of odd/even numbers is one of the topics some teachers are surprised to see discussed so much later than they used to. Another topic that some teachers have expressed their surprise because of the lateness of the treatment is the idea of symmetry. Many teachers of primary grades have children explore (reflective) symmetry through paper folding. They will have students make symmetrical shape by cutting a folded papers, or have them fold symmetric figures so that the two sides will coincide.
Clearly, young children can explore, and enjoy exploring, symmetries through such activities. However, as valuable as such informal experiences may be, they are still "informal" explorations. It is important for children to consider and understand symmetry as a mathematical idea, too. Such study of symmetry is the focus of this particular standard.
According to this standards, students are supposed to "further develop their understanding of plane figures" by studying symmetry. Thus, the purpose of studying symmetry isn't just about learning symmetry. Rather, using symmetry as a new perspective to review those shapes that have been previously studied. So, for example, what kinds of triangles are symmetric? From this perspective, isosceles triangles and equilateral triangles are in one group, symmetric triangles.
Students can also explore which types quadrilaterals have reflective (line) symmetry. Parallelograms, with the exception of those which are also rectangles, do not possess reflective symmetry. Many children (and adults) think that the line that are parallel and in between a pair of sides will serve as the line of symmetry. When they actually fold a parallelogram, they are surprised that the two sides do not match up. That experience, in turn, can help students understand that the line of reflection must be the perpendicular bisector of the segments connecting corresponding points. [The notion of "corresponding points" follows from their study of congruent figures in Grade 5 -- because the two sides of a symmetric figures are congruent, there are corresponding points. As a result, the formal study of symmetry must follow the study of congruence.] With that understanding, children can now determine the line of symmetry (M6G1a) without having to actually fold the paper, or simply eye balling it. This knowledge will also allow them to complete a figure when one side of the figure and the line of symmetry are given.
Many Japanese mathematics teachers consider the study of symmetry in Grade 6 as the culminating point of the study of geometry in elementary schools (in Japan, elementary schools cover grades 1 through 6). Children not only learn about symmetry, but they also learn to use symmetry as a perspective to re-analyze shapes they have learned. Most Grade 6 classrooms in Georgia are in middle schools. So, perhaps we can position the study of symmetry as an entry point into a more formal study of geometry in secondary schools.
a. Determine and use lines of symmetry.
In the last entry, I mentioned that the topic of odd/even numbers is one of the topics some teachers are surprised to see discussed so much later than they used to. Another topic that some teachers have expressed their surprise because of the lateness of the treatment is the idea of symmetry. Many teachers of primary grades have children explore (reflective) symmetry through paper folding. They will have students make symmetrical shape by cutting a folded papers, or have them fold symmetric figures so that the two sides will coincide.
Clearly, young children can explore, and enjoy exploring, symmetries through such activities. However, as valuable as such informal experiences may be, they are still "informal" explorations. It is important for children to consider and understand symmetry as a mathematical idea, too. Such study of symmetry is the focus of this particular standard.
According to this standards, students are supposed to "further develop their understanding of plane figures" by studying symmetry. Thus, the purpose of studying symmetry isn't just about learning symmetry. Rather, using symmetry as a new perspective to review those shapes that have been previously studied. So, for example, what kinds of triangles are symmetric? From this perspective, isosceles triangles and equilateral triangles are in one group, symmetric triangles.
Students can also explore which types quadrilaterals have reflective (line) symmetry. Parallelograms, with the exception of those which are also rectangles, do not possess reflective symmetry. Many children (and adults) think that the line that are parallel and in between a pair of sides will serve as the line of symmetry. When they actually fold a parallelogram, they are surprised that the two sides do not match up. That experience, in turn, can help students understand that the line of reflection must be the perpendicular bisector of the segments connecting corresponding points. [The notion of "corresponding points" follows from their study of congruent figures in Grade 5 -- because the two sides of a symmetric figures are congruent, there are corresponding points. As a result, the formal study of symmetry must follow the study of congruence.] With that understanding, children can now determine the line of symmetry (M6G1a) without having to actually fold the paper, or simply eye balling it. This knowledge will also allow them to complete a figure when one side of the figure and the line of symmetry are given.
Many Japanese mathematics teachers consider the study of symmetry in Grade 6 as the culminating point of the study of geometry in elementary schools (in Japan, elementary schools cover grades 1 through 6). Children not only learn about symmetry, but they also learn to use symmetry as a perspective to re-analyze shapes they have learned. Most Grade 6 classrooms in Georgia are in middle schools. So, perhaps we can position the study of symmetry as an entry point into a more formal study of geometry in secondary schools.
Tuesday, March 10, 2009
M5N1 a - Even and odd numbers
M5N1. Students will further develop their understanding of whole numbers.
a. Classify the set of counting numbers into subsets with distinguishing characteristics (odd/even, prime/composite).
When teachers examine the GPS, there are (at least) a couple of topics they are surprised to see so much later in elementary schools than they are used to. The topic of odd/even numbers is one of those topics. Teachers are surprised that a topic that they used to discuss in the second grade (or even in the first grade) is now delayed until Grade 5. Some may be tempted to include this topic in an earlier grade. So, why does the GPS wait to discuss this topic until Grade 5? Although I cannot speak for the committee who developed the GPS, I can share with you the Japanese perspective.
There is no question that we can teach second grade children to distinguish odd/even numbers. We can connect to skip counting or simply tell students that all numbers in the sequence, "2, 4, 6, 8, 10, ..." are called even numbers and the rest are odd numbers. For larger numbers, they can simply use the rule, "if a number ends with a 0, 2, 4, 6, or 8, it is an even number." However, the point is NOT identifying even/odd numbers. Let's look at the GPS statement:
Classify the set of counting numbers into subsets with distinguishing characteristics (odd/even, prime/composite).
It is important to note that what we want students to understand is that "counting numbers" (whole numbers?) can be classified into different subsets by paying attention to various distinguishing characteristics. So, what is the distinguishing characteristics for even/odd numbers? It is the divisibility by 2. Even numbers are those numbers that are divisible by 2, while odd numbers are those that cannot be divided (with a whole number quotient) by 2. So, the emphasis is not about identifying even/odd numbers, but understanding ways to sort whole numbers. Even/odd numbers are just an example of one such classification schemes.
To help students focus more on ways to classify whole numbers, teachers may want to engage students with a task that require students to sort whole numbers in a similar way - by focusing on the remainder when divided by a number. You can watch a videotaped lesson from Japan, in which the teacher posed an interesting question involving the idea of classifying numbers by looking at the remainders when they are divided by 4.
http://tiny.cc/rkB1X
When teachers examine the GPS, there are (at least) a couple of topics they are surprised to see so much later in elementary schools than they are used to. The topic of odd/even numbers is one of those topics. Teachers are surprised that a topic that they used to discuss in the second grade (or even in the first grade) is now delayed until Grade 5. Some may be tempted to include this topic in an earlier grade. So, why does the GPS wait to discuss this topic until Grade 5? Although I cannot speak for the committee who developed the GPS, I can share with you the Japanese perspective.
There is no question that we can teach second grade children to distinguish odd/even numbers. We can connect to skip counting or simply tell students that all numbers in the sequence, "2, 4, 6, 8, 10, ..." are called even numbers and the rest are odd numbers. For larger numbers, they can simply use the rule, "if a number ends with a 0, 2, 4, 6, or 8, it is an even number." However, the point is NOT identifying even/odd numbers. Let's look at the GPS statement:
It is important to note that what we want students to understand is that "counting numbers" (whole numbers?) can be classified into different subsets by paying attention to various distinguishing characteristics. So, what is the distinguishing characteristics for even/odd numbers? It is the divisibility by 2. Even numbers are those numbers that are divisible by 2, while odd numbers are those that cannot be divided (with a whole number quotient) by 2. So, the emphasis is not about identifying even/odd numbers, but understanding ways to sort whole numbers. Even/odd numbers are just an example of one such classification schemes.
To help students focus more on ways to classify whole numbers, teachers may want to engage students with a task that require students to sort whole numbers in a similar way - by focusing on the remainder when divided by a number. You can watch a videotaped lesson from Japan, in which the teacher posed an interesting question involving the idea of classifying numbers by looking at the remainders when they are divided by 4.
http://tiny.cc/rkB1X
Monday, February 23, 2009
M4N7 - Order of operations and writing math sentences
M4N7. Students will explain and use properties of the four arithmetic operations to solve and check problems.
b. Compute using the order of operations, including parentheses.
As I look back on my own school experiences in Japan, I really don't remember explicitly learning about the order of operations. As I examine the current Japanese elementary school mathematics textbooks, there is really no unit titled "Order of Operations." Of course, that does not mean that Japanese children do not learn the order of operations. What is important to notice is that how they study order of operations is more from the perspective of writing mathematical expressions.
For example, in Grade 4, students are given the following problem:
Makoto had a 1000-yen bill. He bought a 460-yen notebook and a 140-yen pair of scissors. How much change did he get back. However, the focus here is not on solving this problem as the solution is actually given on the textbook page. In fact, there are two possible solutions given. Here is Naoko's method:
1000 - 140 = 860
860 - 460 = 400 Makoto, on the other hand, solved the problem this way:
140 + 460 = 600
1000 - 600 = 400 The task given to students is to think about how these sentences may be combined into one math sentence. They are also given a math sentence with words:
[Money Paid] - [Total Price] = Change From here, students are expected to understand that Makoto's 140 + 160 is one quantity, namely [Total Price] in the math sentence with words. Therefore, they learn that the two sentences may be combined into: 1000 - (140 + 460). Since (140 + 460) represents one quantity, it has to be calculated first.
A little later in the unit, students are given the following problem:
Let's make one math sentence for each of the following problems, then find the answers.
a) You have a 100-yen coin. You buy 3 sheets of paper, and each sheet costs 25 yen. How much change will you get?
b) You buy a 500-yen pencil case and a half dozen pencils. A dozen of pencils cost 480 yen. How much will you pay? The textbook simply tells students that, in math sentences, multiplication and division (i.e., products and quotients) can be considered as one quantity and no parentheses is needed. Thus (a) can be represented by the math sentence, 100 - 25 x 3, while (b) can be represented as 500 + 480 ÷ 2.
Finally, after a few practice problems, the textbook summarizes the rules about the order of operations:
b. Compute using the order of operations, including parentheses.
As I look back on my own school experiences in Japan, I really don't remember explicitly learning about the order of operations. As I examine the current Japanese elementary school mathematics textbooks, there is really no unit titled "Order of Operations." Of course, that does not mean that Japanese children do not learn the order of operations. What is important to notice is that how they study order of operations is more from the perspective of writing mathematical expressions.
For example, in Grade 4, students are given the following problem:
860 - 460 = 400
1000 - 600 = 400
A little later in the unit, students are given the following problem:
a) You have a 100-yen coin. You buy 3 sheets of paper, and each sheet costs 25 yen. How much change will you get?
b) You buy a 500-yen pencil case and a half dozen pencils. A dozen of pencils cost 480 yen. How much will you pay?
Finally, after a few practice problems, the textbook summarizes the rules about the order of operations:
- Generally goes from left to right.
- If there are any parentheses, we calculate inside the parentheses first.
- Multiplication and division are performed before addition and subtraction.
Saturday, January 31, 2009
M4A1 - To use tables or not?
M4A1. Students will represent and interpret mathematical relationships in quantitative expressions. a. Understand and apply patterns and rules to describe relationships and solve problems.
One of the important aspect of the study of algebra at the elementary school level is the idea of patterns. As a result, many curricula include problems like the following.
A square table can seat 1 person on each side. For an outside picnic, we are going to make a long "train" of tables like the picture below.
a) How many seats will there be if we make a train of 8 tables?
b) How many seats will there be if we make a train of 24 tables?
c) How can we easily calculate the number of seats if you know the number of tables?
Often times, children are encouraged to make a table and find out how many seats there will be for 1, 2, 3, ... tables. Then, they are encouraged to find patterns so that they can answer questions (b) and (c). Most children will have no problem coming up with the table like the following. [I'm having a little problem formatting this page -- please scroll down further.]
Children can easily notice that every time we add a table, we increase the number of seat by 2. So, some will try to extend this pattern to find the number of seats when there are 24 tables. Others may notice that 24 is 3 times as many as 8, so they think that the number of seats must also be 3 times as many as 18, the number of seats with 8 tables. Of course, the answers will be different, and there could be a very productive discussion about what is going on here. Such a discussion may be particularly important in Grade 6 when students are studying proportional relationships. It is only in proportional situations where the latter reasoning process will work. In fact, in the Japanese curriculum, a proportional relationship is defined as the following. When one quantity becomes 2, 3, 4, ... times as much, the other quantity also becomes 2, 3, 4, ... times as much. Then, we say those two quantities are in proportion. So, this table problem is an example of relationship something other than proportional.
In order to answer (c), some may suggest students modify the way the number of seats are expressed slightly.
From this table, we can see that: Seats = 4 + 2 x (Tables - 1). Other children may notice that the number of seats is always 2 more than the double of the number of tables. Therefore, they will come up with Seats = 2 x Tables + 2. However, many of these students will not be able to explain why we multiply by 2 (in both cases) or add 2 (in the second case). Some may say that +2 in the second case signifies the fact that the number of seats increases by 2 every time we add a table. But, is it?
Let's think about how children might approach this problem if we changed the way we pose this problem slightly.
A square table can seat 1 person on each side. For an outside picnic, we are going to make a long "train" of tables.
a) Think about different ways you can count (or calculate) the total number of seats when we make a train of 8 tables as shown below.
b) How many seats will there be if we make a train of 24 tables?
c) How can we easily calculate the number of seats if you know the number of tables?
Clearly, some will count seats going around this train one by one. However, there are many other possibilities. Here are four ways children might determine the number of seats.
Method 1
There are 2 seats on each table, top and bottom in the picture above (in alternating colors), and 2 more on the ends of the train. Therefore, with 8 tables, 2x8+2=18, or 18 seats.
In general, Seats = 2 x Tables + 2
Method 2
There are 3 seats on each of the tables on the end, and all other tables have 2 seats. So, 6+2x(8-2)=18, or 18 seats.
In general, Seats = 6 + 2 x (Tables -2).
Method 3
Each table can seat 4 people by itself. However, every time we put a table together with another, we loose 2 seats for each "joint." With 8 tables, there are 7 joints. So, 4x8-2x7=18, or 18 seats.
In general, Seats = 4 x Tables - 2 x (Tables - 1).
Method 4
There are 4 seats on the first table. Every time we add a table, one seat at the joint is shifted to the end of the train, but that seat really came from the first table. Then, each new table will add 2 additional seats, top and bottom in the picture above. So, 4+2x7=18, or 18 seats.
In general, Seats = 4 + 2 x (Tables - 1).
Although the final generalizations may be all different, it is not unreasonable to expect these students to be able to understand what each number and operation means in other students' equations.
From Method 1, we can tell that "+2" in Seats = 2 x Tables + 2 really comes from those 2 seats on the ends of the train. In other words, "2" in "+2" is actually the constant in this situation, not the 2 seats that will be added every time a table is added to the train. The 2 additional seats are actually represented by the coefficient of Tables. From an algebraic perspective, this makes sense because 2 new seats for each new table is actually the rate of change, or the slope of the line. Thus, it should be the coefficient of the independent variable, in this case Tables.
So, what was different about these two situations? Obviously the way the problems were posed was different, but how did that difference influence the outcomes? In the original problem, students created the table and try to find number patterns in the table. However, in the second situation, students were asked to focus on the way they came up with the number of seats. The students in the first situations might have counted the number of seats in many different ways. However, I suspect most children will simply count the number of seats around the train. Furthermore, once they notice "+2" relationship in the number of seats, some may even skip the counting step and simply fill in the table. On the other hand, in the second situation, students' focus was on their actions. What they were doing was to mathematically express, or represent, their actions. Particularly in elementary schools, mathematical expressions should represent the way quantities relate to each other, and that relationship often becomes explicit in students' actions. Thus, we need to encourage them to reflect on their actions. This is not to say looking for patterns in a table is unimportant. We want students to develop their number sense and mathematical reasoning with numbers abstractly as well. Such an ability may be particularly critical in science. However, in the fourth grade when students are first learning about expressing quantitative relationships using mathematical notations, perhaps we may want to emphasize students' reflection on their own actions so that they can understand the meaning behind each part of the mathematical expressions.
One of the important aspect of the study of algebra at the elementary school level is the idea of patterns. As a result, many curricula include problems like the following.
A square table can seat 1 person on each side. For an outside picnic, we are going to make a long "train" of tables like the picture below.
b) How many seats will there be if we make a train of 24 tables?
c) How can we easily calculate the number of seats if you know the number of tables?
Often times, children are encouraged to make a table and find out how many seats there will be for 1, 2, 3, ... tables. Then, they are encouraged to find patterns so that they can answer questions (b) and (c). Most children will have no problem coming up with the table like the following. [I'm having a little problem formatting this page -- please scroll down further.]
| Tables | Number of Seats |
| 1 | 4 |
| 2 | 6 |
| 3 | 8 |
| 4 | 10 |
| 5 | 12 |
| 6 | 14 |
| 7 | 16 |
| 8 | 18 |
Children can easily notice that every time we add a table, we increase the number of seat by 2. So, some will try to extend this pattern to find the number of seats when there are 24 tables. Others may notice that 24 is 3 times as many as 8, so they think that the number of seats must also be 3 times as many as 18, the number of seats with 8 tables. Of course, the answers will be different, and there could be a very productive discussion about what is going on here. Such a discussion may be particularly important in Grade 6 when students are studying proportional relationships. It is only in proportional situations where the latter reasoning process will work. In fact, in the Japanese curriculum, a proportional relationship is defined as the following. When one quantity becomes 2, 3, 4, ... times as much, the other quantity also becomes 2, 3, 4, ... times as much. Then, we say those two quantities are in proportion. So, this table problem is an example of relationship something other than proportional.
In order to answer (c), some may suggest students modify the way the number of seats are expressed slightly.
| Tables | Number of Seats | details | summarized |
| 1 | 4 | 4 | 4+2x0 |
| 2 | 6 | 4+2 | 4+2x1 |
| 3 | 8 | 4+2+2 | 4+2x2 |
| 4 | 10 | 4+2+2+2 | 4+2x3 |
| 5 | 12 | 4+2+2+2+2 | 4+2x4 |
| 6 | 14 | 4+2+2+2+2+2 | 4+2x5 |
| 7 | 16 | 4+2+2+2+2+2+2 | 4+2x6 |
| 8 | 18 | 4+2+2+2+2+2+2+2 | 4+2x7 |
From this table, we can see that: Seats = 4 + 2 x (Tables - 1). Other children may notice that the number of seats is always 2 more than the double of the number of tables. Therefore, they will come up with Seats = 2 x Tables + 2. However, many of these students will not be able to explain why we multiply by 2 (in both cases) or add 2 (in the second case). Some may say that +2 in the second case signifies the fact that the number of seats increases by 2 every time we add a table. But, is it?
Let's think about how children might approach this problem if we changed the way we pose this problem slightly.
A square table can seat 1 person on each side. For an outside picnic, we are going to make a long "train" of tables.
a) Think about different ways you can count (or calculate) the total number of seats when we make a train of 8 tables as shown below.
b) How many seats will there be if we make a train of 24 tables?
c) How can we easily calculate the number of seats if you know the number of tables?
Clearly, some will count seats going around this train one by one. However, there are many other possibilities. Here are four ways children might determine the number of seats.
Method 1
In general, Seats = 2 x Tables + 2
Method 2
In general, Seats = 6 + 2 x (Tables -2).
Method 3
In general, Seats = 4 x Tables - 2 x (Tables - 1).
Method 4
In general, Seats = 4 + 2 x (Tables - 1).
Although the final generalizations may be all different, it is not unreasonable to expect these students to be able to understand what each number and operation means in other students' equations.
From Method 1, we can tell that "+2" in Seats = 2 x Tables + 2 really comes from those 2 seats on the ends of the train. In other words, "2" in "+2" is actually the constant in this situation, not the 2 seats that will be added every time a table is added to the train. The 2 additional seats are actually represented by the coefficient of Tables. From an algebraic perspective, this makes sense because 2 new seats for each new table is actually the rate of change, or the slope of the line. Thus, it should be the coefficient of the independent variable, in this case Tables.
So, what was different about these two situations? Obviously the way the problems were posed was different, but how did that difference influence the outcomes? In the original problem, students created the table and try to find number patterns in the table. However, in the second situation, students were asked to focus on the way they came up with the number of seats. The students in the first situations might have counted the number of seats in many different ways. However, I suspect most children will simply count the number of seats around the train. Furthermore, once they notice "+2" relationship in the number of seats, some may even skip the counting step and simply fill in the table. On the other hand, in the second situation, students' focus was on their actions. What they were doing was to mathematically express, or represent, their actions. Particularly in elementary schools, mathematical expressions should represent the way quantities relate to each other, and that relationship often becomes explicit in students' actions. Thus, we need to encourage them to reflect on their actions. This is not to say looking for patterns in a table is unimportant. We want students to develop their number sense and mathematical reasoning with numbers abstractly as well. Such an ability may be particularly critical in science. However, in the fourth grade when students are first learning about expressing quantitative relationships using mathematical notations, perhaps we may want to emphasize students' reflection on their own actions so that they can understand the meaning behind each part of the mathematical expressions.
Saturday, January 17, 2009
M4A1bc & M5A1a - symbols such as □ and Δ
M4A1. Students will represent and interpret mathematical relationships in quantitative expressions.
b. Represent unknowns using symbols, such as □ and Δ.
c. Write and evaluate mathematical expressions using symbols and different
values.
M5A1. Students will represent and interpret the relationships between quantities algebraically.
a. Use variables, such as n or x, for unknown quantities in algebraic
expressions.
In the last entry, I discussed mathematical expressions as the language of mathematics and how important it is for students to learn to write and read mathematical expressions, starting when students start studying addition formally in Grade 1. I also discussed it may be possible for students to represent situations involving missing addend situations using mathematical expressions using a box, like 5+[ ]=8 (or 5+?=8, 5+__=8, etc.). In grades 1-3, those symbols are used as place holders for particular values. However, in Grade 4, students begin the next phase of using symbols to represent numbers and quantities, that is, the concept of variables.
As we consider teaching of this complex idea (variables), it may be worth noting a progression across grades:
Grades 1 - 3: writing math sentences with numbers, and occasionally with place holders
Grade 4: writing math sentences with symbols like □ and Δ.
Grade 5: writing math sentences using letters as symbols Some people may wonder what's the point of using symbols like □ and Δ in Grade 4. Why not just use letters since that's what is typically done in higher math? Although there are probably many reasons for using symbols like □ and Δ, one possible reason is the principle I have observed in many Japanese curriculum materials: do not introduce a new representation and a new concept simultaneously. Although the idea of using letters to stand for numbers may be straightforward to those of us who already learned the concept, I'm also sure that you have heard people say how they were confused by the idea of using letters in math sentences. This suggests that use of letters to represent numbers and quantities isn't that simple. So, it may not be a good idea to introduce both letters as representations and the concept of variables at the same time. However, since we do need symbols to talk about variables, the natural choice seems to be to use something familiar, symbols like □ and Δ.
Now, there are a couple of implications from the previous paragraph. First, it is important that symbols like □ and Δ are familiar to the 4th grade students - that means they should be introduced to the use of those symbols in math sentences before Grade 4. The other implication is that the primary focus on Grade 4 is, then, on developing the concept of variables, not necessarily about using the symbols like □ and Δ. In fact, to develop the concept of variables, in some cases, you may not want to use symbols like □ and Δ. Instead, you may want to write mathematical expressions using words. For example, in the third grade, children learn about calculating the area of rectangles and squares. The GPS (M3M4) isn't quite clear whether or not the formulas should be developed in Grade 3. However, it may not be a bad idea to develop the formulas in the context of studying the concept of variables. In Grade 3, students learned that the area of rectangles and squares can be calculated by multiplying the lengths and the widths. Thus, we can express the relationship with a mathematical expression using words like this: Area = Length x Width. [Moreover, it is important to note that we cannot write the formula as A = lw yet since the use of letters is a fifth grade standards!] You can probably think of many other situations that will be appropriate for Grade 4 students, for example, Change = Amount Paid - Price, Number of Children = Boys + Girls, etc.. In fact, as students explore different patterns and rules to describe relationships (M4A1a), they can use mathematical expressions with appropriate words to represent the patterns and rules.
When students are comfortable with mathematical expressions with words, you may want to suggest using symbols like □ and Δ in some cases. In fact, having some experiences with mathematical expressions with words, may help students' transition to the use of letters as variables in Grade 5. In those situations, instead of using letters like x, y, a, b, etc., you may want to start with the initial of the words used in the expressions (thus A = lw).
Evaluating mathematical expressions (perhaps derived by students) by substituting different values (M4A1c) is also an important activity to help students understand the concept of variables. Again, it is important that we keep in mind that the main focus here is the concept of variables. By substituting different values, students are learning that the variables (words, symbols, or letters) stand for quantities that can vary.
b. Represent unknowns using symbols, such as □ and Δ.
c. Write and evaluate mathematical expressions using symbols and different
values.
M5A1. Students will represent and interpret the relationships between quantities algebraically.
a. Use variables, such as n or x, for unknown quantities in algebraic
expressions.
In the last entry, I discussed mathematical expressions as the language of mathematics and how important it is for students to learn to write and read mathematical expressions, starting when students start studying addition formally in Grade 1. I also discussed it may be possible for students to represent situations involving missing addend situations using mathematical expressions using a box, like 5+[ ]=8 (or 5+?=8, 5+__=8, etc.). In grades 1-3, those symbols are used as place holders for particular values. However, in Grade 4, students begin the next phase of using symbols to represent numbers and quantities, that is, the concept of variables.
As we consider teaching of this complex idea (variables), it may be worth noting a progression across grades:
Grade 4: writing math sentences with symbols like □ and Δ.
Grade 5: writing math sentences using letters as symbols
Now, there are a couple of implications from the previous paragraph. First, it is important that symbols like □ and Δ are familiar to the 4th grade students - that means they should be introduced to the use of those symbols in math sentences before Grade 4. The other implication is that the primary focus on Grade 4 is, then, on developing the concept of variables, not necessarily about using the symbols like □ and Δ. In fact, to develop the concept of variables, in some cases, you may not want to use symbols like □ and Δ. Instead, you may want to write mathematical expressions using words. For example, in the third grade, children learn about calculating the area of rectangles and squares. The GPS (M3M4) isn't quite clear whether or not the formulas should be developed in Grade 3. However, it may not be a bad idea to develop the formulas in the context of studying the concept of variables. In Grade 3, students learned that the area of rectangles and squares can be calculated by multiplying the lengths and the widths. Thus, we can express the relationship with a mathematical expression using words like this: Area = Length x Width. [Moreover, it is important to note that we cannot write the formula as A = lw yet since the use of letters is a fifth grade standards!] You can probably think of many other situations that will be appropriate for Grade 4 students, for example, Change = Amount Paid - Price, Number of Children = Boys + Girls, etc.. In fact, as students explore different patterns and rules to describe relationships (M4A1a), they can use mathematical expressions with appropriate words to represent the patterns and rules.
When students are comfortable with mathematical expressions with words, you may want to suggest using symbols like □ and Δ in some cases. In fact, having some experiences with mathematical expressions with words, may help students' transition to the use of letters as variables in Grade 5. In those situations, instead of using letters like x, y, a, b, etc., you may want to start with the initial of the words used in the expressions (thus A = lw).
Evaluating mathematical expressions (perhaps derived by students) by substituting different values (M4A1c) is also an important activity to help students understand the concept of variables. Again, it is important that we keep in mind that the main focus here is the concept of variables. By substituting different values, students are learning that the variables (words, symbols, or letters) stand for quantities that can vary.
Saturday, January 3, 2009
M4A1 - Mathematical expressions (2)
M4A1. Students will represent and interpret mathematical relationships in quantitative expressions.
b. Represent unknowns using symbols, such as □ and Δ.
c. Write and evaluate mathematical expressions using symbols and different values.
M5A1. Students will represent and interpret the relationships between quantities algebraically.
a. Use variables, such as n or x, for unknown quantities in algebraic expressions.
In the last entry, I discussed mathematical expressions as the language of mathematics and how important it is for students to learn to write and read mathematical expressions, starting when students start studying addition formally in Grade 1. I also discussed it may be possible for students to represent situations involving missing addend situations using mathematical expressions using a box, like 5+[ ]=8 (or 5+?=8, 5+__=8, etc.). In grades 1-3, those symbols are used as place holders for particular values. However, in Grade 4, students begin the next phase of using symbols to represent numbers and quantities, that is, the concept of variables.
As we consider teaching of this complex idea (variables), it may be worth noting a progression across grades:
Grades 1 - 3: writing math sentences with numbers, and occasionally with place holders
Grade 4: writing math sentences with symbols like □ and Δ.
Grade 5: writing math sentences using letters as symbols
Some people may wonder what's the point of using symbols like □ and Δ in Grade 4. Why not just use letters since that's what is typically done in higher math? Although there are probably many reasons for using symbols like □ and Δ, one possible reason is the principle I have observed in many Japanese curriculum materials: do not introduce a new representation and a new concept simultaneously. Although the idea of using letters to stand for numbers may be straightforward to those of us who already learned the concept, I'm also sure that you have heard people say how they were confused by the idea of using letters in math sentences. This suggests that use of letters to represent numbers and quantities isn't that simple. So, it may not be a good idea to introduce both letters as representations and the concept of variables at the same time. However, since we do need symbols to talk about variables, the natural choice seems to be to use something familiar, symbols like □ and Δ.
Now, there are a couple of implications from the previous paragraph. First, it is important that symbols like □ and Δ are familiar to the 4th grade students - that means they should be introduced to the use of those symbols in math sentences before Grade 4. The other implication is that the primary focus on Grade 4 is, then, on developing the concept of variables, not necessarily about using the symbols like □ and Δ. In fact, to develop the concept of variables, in some cases, you may not want to use symbols like □ and Δ. Instead, you may want to write mathematical expressions using words. For example, in the third grade, children learn about calculating the area of rectangles and squares. The GPS (M3M4) isn't quite clear whether or not the formulas should be developed in Grade 3. However, it may not be a bad idea to develop the formulas in the context of studying the concept of variables. In Grade 3, students learned that the area of rectangles and squares can be calculated by multiplying the lengths and the widths. Thus, we can express the relationship with a mathematical expression using words like this: Area = Length x Width. [Moreover, it is important to note that we cannot write the formula as A = lw yet since the use of letters is a fifth grade standards!] You can probably think of many other situations that will be appropriate for Grade 4 students, for example, Change = Amount Paid - Price, Number of Children = Boys + Girls, etc.. In fact, as students explore different patterns and rules to describe relationships (M4A1a), they can use mathematical expressions with appropriate words to represent the patterns and rules.
When students are comfortable with mathematical expressions with words, you may want to suggest using symbols like □ and Δ in some cases. In fact, having some experiences with mathematical expressions with words, may help students' transition to the use of letters as variables in Grade 5. In those situations, instead of using letters like x, y, a, b, etc., you may want to start with the initial of the words used in the expressions (thus A = lw).
Evaluating mathematical expressions (perhaps derived by students) by substituting different values (M4A1c) is also an important activity to help students understand the concept of variables. Again, it is important that we keep in mind that the main focus here is the concept of variables. By substituting different values, students are learning that the variables (words, symbols, or letters) stand for quantities that can vary.
c. Write and evaluate mathematical expressions using symbols and different values.
M5A1. Students will represent and interpret the relationships between quantities algebraically.
In the last entry, I discussed mathematical expressions as the language of mathematics and how important it is for students to learn to write and read mathematical expressions, starting when students start studying addition formally in Grade 1. I also discussed it may be possible for students to represent situations involving missing addend situations using mathematical expressions using a box, like 5+[ ]=8 (or 5+?=8, 5+__=8, etc.). In grades 1-3, those symbols are used as place holders for particular values. However, in Grade 4, students begin the next phase of using symbols to represent numbers and quantities, that is, the concept of variables.
As we consider teaching of this complex idea (variables), it may be worth noting a progression across grades:
Grade 4: writing math sentences with symbols like □ and Δ.
Grade 5: writing math sentences using letters as symbols
Some people may wonder what's the point of using symbols like □ and Δ in Grade 4. Why not just use letters since that's what is typically done in higher math? Although there are probably many reasons for using symbols like □ and Δ, one possible reason is the principle I have observed in many Japanese curriculum materials: do not introduce a new representation and a new concept simultaneously. Although the idea of using letters to stand for numbers may be straightforward to those of us who already learned the concept, I'm also sure that you have heard people say how they were confused by the idea of using letters in math sentences. This suggests that use of letters to represent numbers and quantities isn't that simple. So, it may not be a good idea to introduce both letters as representations and the concept of variables at the same time. However, since we do need symbols to talk about variables, the natural choice seems to be to use something familiar, symbols like □ and Δ.
Now, there are a couple of implications from the previous paragraph. First, it is important that symbols like □ and Δ are familiar to the 4th grade students - that means they should be introduced to the use of those symbols in math sentences before Grade 4. The other implication is that the primary focus on Grade 4 is, then, on developing the concept of variables, not necessarily about using the symbols like □ and Δ. In fact, to develop the concept of variables, in some cases, you may not want to use symbols like □ and Δ. Instead, you may want to write mathematical expressions using words. For example, in the third grade, children learn about calculating the area of rectangles and squares. The GPS (M3M4) isn't quite clear whether or not the formulas should be developed in Grade 3. However, it may not be a bad idea to develop the formulas in the context of studying the concept of variables. In Grade 3, students learned that the area of rectangles and squares can be calculated by multiplying the lengths and the widths. Thus, we can express the relationship with a mathematical expression using words like this: Area = Length x Width. [Moreover, it is important to note that we cannot write the formula as A = lw yet since the use of letters is a fifth grade standards!] You can probably think of many other situations that will be appropriate for Grade 4 students, for example, Change = Amount Paid - Price, Number of Children = Boys + Girls, etc.. In fact, as students explore different patterns and rules to describe relationships (M4A1a), they can use mathematical expressions with appropriate words to represent the patterns and rules.
When students are comfortable with mathematical expressions with words, you may want to suggest using symbols like □ and Δ in some cases. In fact, having some experiences with mathematical expressions with words, may help students' transition to the use of letters as variables in Grade 5. In those situations, instead of using letters like x, y, a, b, etc., you may want to start with the initial of the words used in the expressions (thus A = lw).
Evaluating mathematical expressions (perhaps derived by students) by substituting different values (M4A1c) is also an important activity to help students understand the concept of variables. Again, it is important that we keep in mind that the main focus here is the concept of variables. By substituting different values, students are learning that the variables (words, symbols, or letters) stand for quantities that can vary.
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?
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.
M1M1. Students will compare and/or order the length, weight, or capacity of two or more objects by using direct comparison or a nonstandard unit.
M2M1. 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,
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.
M1M1. Students will compare and/or order the length, weight, or capacity of two or more objects by using direct comparison or a nonstandard unit.
M2M1. 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.
- Direct comparison
- Indirect comparison
- Measuring with non-standard units
- Measuring with standard units
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.
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.
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.
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
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
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
Problem 4
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.
M3N1 Students will further develop their understanding of whole numbers and ways of representing 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":
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.
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:
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":
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.
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.
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.
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.
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Elaboration of Georgia Performance Standards by Tad Watanabe is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.