M4N2. Students will understand and apply the concept of rounding numbers.
a. Round numbers to the nearest ten, hundred, or thousand.
Rounding is a specific technique to approximate numbers. Some teachers in primary grades actually teach their students rounding when they want students to "estimate." However, "estimation" and "rounding" aren't the same idea. In fact, as an approximation technique, it is probably better to teach rounding when students are working with larger numbers.
As you know, rounding a number to the nearest designated place means to look at the numeral to the right of the place to which we are rounding. If the numeral is 4 or less, we will round down (i.e., simply change all places to the right of the designated place 0's) and if it is 5 or above, we round up (i.e., increase the numeral in the designated place by 1 and change all numerals to the right 0's). So, when 45,542 is rounded to the nearest thousands place, it will be 46,000, and when it is rounded to the nearest hundreds, it will be 45,500.
One question students sometime ask is why we round up with a "5" even though 5 is right in the middle (of 0, 1, ..., 9). Some teachers will simply say it's just a rule. But, is it?
Let's consider 45,542. If we want to round this number to the nearest thousands place, we are really asking is it closer to 45,000 or 46,000. According to the procedure, we will be checking the numeral in the hundreds place. So, what numbers between 45,000 and 46,000 have a 5 in the hundreds place? Well, 45,500 is definitely one. But there are a lot more: 45,501, 45,502, 45,503, ... 45, 598, 45,599. Altogether there are actually 100 numbers in this range with a 5 in the hundreds place? So, which of these numbers are closer to 45,000? 46,000? Right in the middle? Well, it's obvious that all but one of these numbers are actually closer to 46,000, and the one exception is right in the middle. If that's the case, in general, does it make sense to round a number with a 5 in the hundreds place up or down?
The problem with "5 is right in the middle" comes up only when you are rounding to the nearest tens (and only if we are looking at whole numbers). Since approximate numbers are used when we have very large numbers of very small numbers, perhaps trying to teach rounding, a specific approximation procedure, with such small numbers may not make any sense.
Friday, June 19, 2009
Tuesday, June 9, 2009
M7n1 a - Meaning of 0
M7N1. Students will understand the meaning of positive and negative rational numbers and use them in computation.
a. Find the absolute value of a number and understand it as the distance from zero on a number line.
I usually don't get many comments on my blog entry (and I would be happy to hear from more of you), but on May 30, PJGould said that he had come across a child who started his counting with zero. Of course, he noted, that made his counting always off by one. After all, zero is not a counting (natural) number. But what does zero mean?
In elementary (K-5) curriculum, there are 3 meanings of zero - perhaps it is more accurate to say 3 ways zero is used. First, zero indicates the cardinality of an empty set - that is, zero means 'nothing.' This is probably the most commonly used meaning of zero in elementary school. Another place zero is used is as a place holder in a written numbers, such as 3042. Of course, this is a slight extension of the first meaning in that there is no unit of one-hundred in this written number. So, it is still pretty close to the first meaning.
The third usage of zero in elementary schools is the starting point of a number line. In some textbooks, a number line actually starts with zero as shown below.
In other textbooks, the tick mark for zero is not at the end of a number line, implying that there may be something to the left of zero as well.
As students study positive and negative number, one of the important understanding students have to make is the meaning of zero as a referent point, or the origin, on the number line. As long as students are stuck with the idea that zero means 'nothing,' some will have difficulty making sense of numbers that is 'less than nothing.' Rather, students must look at zero as a referent point on a number line, and those number to the right of zero are positive and those on the left are negative. The distance from zero, whether on the right or left is the absolute value of the number.
For those of us who already understand positive and negative numbers, this way of looking at zero is not a major issue. However, we should be aware that this meaning of zero isn't something students are familiar with. In most elementary curriculum, very little explicit discussion takes place about the role of zero on a number line. Thus, when we introduce positive and negative numbers, we do have to keep this shift in understanding of zero in our mind.
a. Find the absolute value of a number and understand it as the distance from zero on a number line.
I usually don't get many comments on my blog entry (and I would be happy to hear from more of you), but on May 30, PJGould said that he had come across a child who started his counting with zero. Of course, he noted, that made his counting always off by one. After all, zero is not a counting (natural) number. But what does zero mean?
In elementary (K-5) curriculum, there are 3 meanings of zero - perhaps it is more accurate to say 3 ways zero is used. First, zero indicates the cardinality of an empty set - that is, zero means 'nothing.' This is probably the most commonly used meaning of zero in elementary school. Another place zero is used is as a place holder in a written numbers, such as 3042. Of course, this is a slight extension of the first meaning in that there is no unit of one-hundred in this written number. So, it is still pretty close to the first meaning.
The third usage of zero in elementary schools is the starting point of a number line. In some textbooks, a number line actually starts with zero as shown below.
In other textbooks, the tick mark for zero is not at the end of a number line, implying that there may be something to the left of zero as well.
As students study positive and negative number, one of the important understanding students have to make is the meaning of zero as a referent point, or the origin, on the number line. As long as students are stuck with the idea that zero means 'nothing,' some will have difficulty making sense of numbers that is 'less than nothing.' Rather, students must look at zero as a referent point on a number line, and those number to the right of zero are positive and those on the left are negative. The distance from zero, whether on the right or left is the absolute value of the number.
For those of us who already understand positive and negative numbers, this way of looking at zero is not a major issue. However, we should be aware that this meaning of zero isn't something students are familiar with. In most elementary curriculum, very little explicit discussion takes place about the role of zero on a number line. Thus, when we introduce positive and negative numbers, we do have to keep this shift in understanding of zero in our mind.
Tuesday, June 2, 2009
M3N5 a - Modeling decimal numbers
M3N5. Students will understand the meaning of decimal fractions and common fractions in simple cases and apply them in problem-solving situations.
a. Understand a decimal fraction (i.e., 0.1) and a common fraction (i.e., 1/10 represent parts of a whole.
According to the GPS, decimal numbers are introduced in Grade 3. In Grade 3, though, students only consider decimal numbers only in the first decimal place (or the tenths place, if students have already learned fraction terminology). Decimal numbers to the 2nd decimal place and beyond are studied in Grade 4 and above.
I have heard many teachers who say they use money as the model for decimal numbers. However, if we are limiting decimal numbers to be discussed in Grade 3 to the first decimal place, we cannot use money as an appropriate model as money amounts are shown to the second decimal place. Thus, money as a model for decimal numbers is appropriate only starting in Grade 4, and only with decimal numbers with 2 decimal places. Some people argue, and I agree, that money is not a good model for decimal numbers.
So, why isn't money a good model for decimal numbers? First, as we saw above, it is a very limited utility as a model - only in Grade 4 (and above) and only when we are dealing with decimal numbers with 2 decimal places. Although it is true that a 0 may be annexed to a decimal number with only 1 decimal place, e.g., 0.4 = 0.40, looking at all decimal number as 2-digit decimal numbers may not be the most helpful habit to develop.
Perhaps more serious problem with money as a model is that students (and adults) don't really have to think about money amounts as decimal numbers. Rather, they are really combinations of two monetary units, dollars and cents. By using two different units, we can simply work with two whole numbers. For example, we don't consider $2.35 as two and 45 hundredths dollars. Rather, it is TWO dollars and THIRTY-FIVE cents. If we get additional $3.18, we simply add TWO and THREE dollars and THIRTY-FIVE and EIGHTEEN cents. Therefore, we are not really considering those numbers as decimal numbers - they only use notations similar to decimal numbers. Mathematically speaking, there isn't really that much difference between monetary amounts and durations expressed in hours and minutes. If you spend 2 hours and 35 minutes watching TV and 3 hours and 18 minutes playing computer games, then you wasted 2+3=5 hours and 35+18=53 minutes!
So, if money isn't a good model, what other models can we use? Base-10 blocks are always an option - we just have to designate something other than unit cubes as "1." We can also use paper strips, too, just as you might use them to model fractions. No matter what model you decide to use, an important idea we want children to develop is the unitary perspective of decimal numbers. For example, 0.4 is made up of 4 0.1-units. This is very similar to the unitary perspective of fractions I discussed in the last post. This way of looking at decimal numbers will allow students to bridge decimal numbers to whole numbers. So, it is very important for us to think about models to use, but we should also keep in mind the goal understanding we want our students to develop.
a. Understand a decimal fraction (i.e., 0.1) and a common fraction (i.e., 1/10 represent parts of a whole.
According to the GPS, decimal numbers are introduced in Grade 3. In Grade 3, though, students only consider decimal numbers only in the first decimal place (or the tenths place, if students have already learned fraction terminology). Decimal numbers to the 2nd decimal place and beyond are studied in Grade 4 and above.
I have heard many teachers who say they use money as the model for decimal numbers. However, if we are limiting decimal numbers to be discussed in Grade 3 to the first decimal place, we cannot use money as an appropriate model as money amounts are shown to the second decimal place. Thus, money as a model for decimal numbers is appropriate only starting in Grade 4, and only with decimal numbers with 2 decimal places. Some people argue, and I agree, that money is not a good model for decimal numbers.
So, why isn't money a good model for decimal numbers? First, as we saw above, it is a very limited utility as a model - only in Grade 4 (and above) and only when we are dealing with decimal numbers with 2 decimal places. Although it is true that a 0 may be annexed to a decimal number with only 1 decimal place, e.g., 0.4 = 0.40, looking at all decimal number as 2-digit decimal numbers may not be the most helpful habit to develop.
Perhaps more serious problem with money as a model is that students (and adults) don't really have to think about money amounts as decimal numbers. Rather, they are really combinations of two monetary units, dollars and cents. By using two different units, we can simply work with two whole numbers. For example, we don't consider $2.35 as two and 45 hundredths dollars. Rather, it is TWO dollars and THIRTY-FIVE cents. If we get additional $3.18, we simply add TWO and THREE dollars and THIRTY-FIVE and EIGHTEEN cents. Therefore, we are not really considering those numbers as decimal numbers - they only use notations similar to decimal numbers. Mathematically speaking, there isn't really that much difference between monetary amounts and durations expressed in hours and minutes. If you spend 2 hours and 35 minutes watching TV and 3 hours and 18 minutes playing computer games, then you wasted 2+3=5 hours and 35+18=53 minutes!
So, if money isn't a good model, what other models can we use? Base-10 blocks are always an option - we just have to designate something other than unit cubes as "1." We can also use paper strips, too, just as you might use them to model fractions. No matter what model you decide to use, an important idea we want children to develop is the unitary perspective of decimal numbers. For example, 0.4 is made up of 4 0.1-units. This is very similar to the unitary perspective of fractions I discussed in the last post. This way of looking at decimal numbers will allow students to bridge decimal numbers to whole numbers. So, it is very important for us to think about models to use, but we should also keep in mind the goal understanding we want our students to develop.
Thursday, May 21, 2009
M3N5 - Simple cases of fraction addition/subtraction
M3N5. Students will understand the meaning of decimal fractions and common fractions in simple cases and apply them in problem-solving situations.e. Understand the concept of addition and subtraction of decimal fractions and common fractions with like denominators.
Addition and subtraction of fractions are discussed in three different grades (M3N5e; M4N6b; M5N4g). Both this current standard and M4N6b involve fractions with like denominators. For M4N6b, there is a note stating that denominators should not exceed 12. So, what is the difference between M3N5e and M4N6b? If one of the reasons for developing the GPS was to minimize repetitions, why is this topic repeated in Grade 4?
One of the differences is that in Grade 3, the sum or the minuend must be less than or equal to 1 as students will not be studying improper fractions and mixed numbers until Grade 4. Thus, 2/5 + 1/5 is appropriate in Grade 3 but not 4/5 + 2/5. However, the most important reason for discussing simple addition and subtraction in Grade 3 is to help students understand fractions as numbers, just like whole numbers.
Fractions are often introduced as parts of a whole. Although this way of looking at fractions is relatively easy for students to grasp, research also shows that this is a very limiting view of fractions. In other words, if students can consider fractions only as parts of a whole, they will have difficulty dealing with fraction arithmetic. Part of a whole is a relationship, and we cannot perform arithmetic operations on relationships. We can only add, subtract, multiply, and divide numbers. Thus, students must understand fractions as numbers in order to make sense of fraction arithmetic. So, how do we help students to see fractions as numbers? Well, one way is to help students experience situations where fractions are added or subtracted. From those experiences, students can realize that fractions are numbers because they can be added or subtracted. It sounds like a circular argument, and it probably is. However, I would like to think this relationship more of reflexive, i.e., neither one is a prerequisite for the other, and an understanding of one can actually promote and deepen the understanding of the other.
What is important, though, is that experiences students will encounter are something that they can determine as addition/subtraction situations. For example, we can ask students what is the total length of a tape if a 2/5-meter segment and 1/5-meter segment are put together end to end. They can see that this situation is an addition situation - you would use addition if the lengths of the segments were 2 meters and 1 meter, respectively.
Another key idea is the unitary view of fractions. In other words, students should understand 2/5-meters as made up of 2 1/5-meter segments. Then, 2/5 + 1/5 is really 2 1/5-units and 1 1/5-unit put together, or 2+1 1/5-units. By recognizing that fractions may be added or subtracted, and having a way to reason through to find the answers, students can develop the understanding of fractions as numbers. With this knowledge as the starting point, students in Grade 4 can explore fraction addition and subtraction more formally.
Addition and subtraction of fractions are discussed in three different grades (M3N5e; M4N6b; M5N4g). Both this current standard and M4N6b involve fractions with like denominators. For M4N6b, there is a note stating that denominators should not exceed 12. So, what is the difference between M3N5e and M4N6b? If one of the reasons for developing the GPS was to minimize repetitions, why is this topic repeated in Grade 4?
One of the differences is that in Grade 3, the sum or the minuend must be less than or equal to 1 as students will not be studying improper fractions and mixed numbers until Grade 4. Thus, 2/5 + 1/5 is appropriate in Grade 3 but not 4/5 + 2/5. However, the most important reason for discussing simple addition and subtraction in Grade 3 is to help students understand fractions as numbers, just like whole numbers.
Fractions are often introduced as parts of a whole. Although this way of looking at fractions is relatively easy for students to grasp, research also shows that this is a very limiting view of fractions. In other words, if students can consider fractions only as parts of a whole, they will have difficulty dealing with fraction arithmetic. Part of a whole is a relationship, and we cannot perform arithmetic operations on relationships. We can only add, subtract, multiply, and divide numbers. Thus, students must understand fractions as numbers in order to make sense of fraction arithmetic. So, how do we help students to see fractions as numbers? Well, one way is to help students experience situations where fractions are added or subtracted. From those experiences, students can realize that fractions are numbers because they can be added or subtracted. It sounds like a circular argument, and it probably is. However, I would like to think this relationship more of reflexive, i.e., neither one is a prerequisite for the other, and an understanding of one can actually promote and deepen the understanding of the other.
What is important, though, is that experiences students will encounter are something that they can determine as addition/subtraction situations. For example, we can ask students what is the total length of a tape if a 2/5-meter segment and 1/5-meter segment are put together end to end. They can see that this situation is an addition situation - you would use addition if the lengths of the segments were 2 meters and 1 meter, respectively.
Another key idea is the unitary view of fractions. In other words, students should understand 2/5-meters as made up of 2 1/5-meter segments. Then, 2/5 + 1/5 is really 2 1/5-units and 1 1/5-unit put together, or 2+1 1/5-units. By recognizing that fractions may be added or subtracted, and having a way to reason through to find the answers, students can develop the understanding of fractions as numbers. With this knowledge as the starting point, students in Grade 4 can explore fraction addition and subtraction more formally.
Wednesday, May 6, 2009
M2N1 - Number lines
M2N1. Students will use multiple representation of numbers to connect symbols to quantities.
When I visit primary grade classrooms, I often see a large number line posted above the whiteboard in the front of the room. Sometimes I also see number lines taped on students' desks. A variety of experts, including the National Math Panel, state that number lines are powerful representation tools and mathematics instruction should develop students' proficiency with number lines. Singapore elementary mathematics textbooks are famous, in part, because of their use of "tape diagrams" to help students deal with complicated mathematical problems. As I have discussed previously, double number lines can be powerful thinking tools to support students' comprehension of multiplication and division of rational numbers. So, on the surface, the display of number lines in the primary grades (K-2) seems to be a sound teaching practice. But, is it?
When you examine Japanese elementary mathematics textbooks, the formal term, "number line" does not appear until Grade 3. However, that does not mean number lines are not used in Grades 1 and 2 (there is no Kindergarten in Japanese elementary schools). As usual, Japanese textbooks carefully and gradually develop number line representations. Thus, students' first encounter with something like number line is simply placing number cards 1 through 10 in order going from left to right. They will be asked to fill in the missing number in a sequence like 3 - 4 - [ ], or 7 - [ ] - 9. A little later on, once the range of numbers has been extended up to 20, there is a question which asks how far a space alien character hopped along a number line, starting at 0. Missing number problems may also involve number cards sequenced in backward (from large to small). When students are studying numbers up to 100, students are asked to locate given numbers on a number line, and similar questions are asked in Grade 2 when the range of numbers is extended to 1000.
What is conspicuously absent in the Japanese primary mathematics textbooks is the use of number lines to deal with addition and subtraction. Rather, number lines are used to represent visually relative sizes of numbers. I recently heard that some people distinguish number paths and number lines. Number paths, as I understand it, simply string together numbers, 1, 2, 3, ... On a number path, numbers are represented more by their positions (or orders) whereas on a number line, a number is represented by the distance of the tick mark from the origin, i.e., 0. So, the way the Japanese textbooks introduce and use number lines are much more along the line of number paths.
So, why don't Japanese textbooks use number lines to represent addition and subtraction, as is often done in some US textbooks? There are at least a couple of reasons. The idea that a number is represented by the distance of the tick mark from the origin is a difficult one for students in primary grades. This is difficult, in part, because those students are still learning about measuring length. So, they really don't have the prerequisite knowledge to interpret number lines in that manner. What they tend to do is to simply count the tick marks. However, when students count, they start with "1," and this is another reason number lines are complicated for young children. I have yet to meet a child who started his/her counting by saying, "zero... one, two, three, ..." For many young children the role of 0 (the origin) on a number line is mysterious. So, when they have to use number line to solve 5+3, they will start with the tick mark labeled "5," some will point to "5" and say, "one." Most, if not all, teachers of primary grades have seen young children line up their rulers starting at "1." It's the same problem.
Some people suggest that number lines are inappropriate for primary students, and we should not use number lines. However, I do think it is important that number lines are introduced in primary grades. However, we should be careful about how we use them. We can use them to think about relative sizes of numbers. However, it is probably a good idea to wait to use number lines as a tool for arithmetic. We can use something like tape diagram for that purpose. But, developing the idea that numbers can be represented on number lines is an idea that should start in primary grades, and we should guide students to understand how numbers are represented (as distance from the origin) on number lines, perhaps connecting to the study of linear measurement.
When I visit primary grade classrooms, I often see a large number line posted above the whiteboard in the front of the room. Sometimes I also see number lines taped on students' desks. A variety of experts, including the National Math Panel, state that number lines are powerful representation tools and mathematics instruction should develop students' proficiency with number lines. Singapore elementary mathematics textbooks are famous, in part, because of their use of "tape diagrams" to help students deal with complicated mathematical problems. As I have discussed previously, double number lines can be powerful thinking tools to support students' comprehension of multiplication and division of rational numbers. So, on the surface, the display of number lines in the primary grades (K-2) seems to be a sound teaching practice. But, is it?
When you examine Japanese elementary mathematics textbooks, the formal term, "number line" does not appear until Grade 3. However, that does not mean number lines are not used in Grades 1 and 2 (there is no Kindergarten in Japanese elementary schools). As usual, Japanese textbooks carefully and gradually develop number line representations. Thus, students' first encounter with something like number line is simply placing number cards 1 through 10 in order going from left to right. They will be asked to fill in the missing number in a sequence like 3 - 4 - [ ], or 7 - [ ] - 9. A little later on, once the range of numbers has been extended up to 20, there is a question which asks how far a space alien character hopped along a number line, starting at 0. Missing number problems may also involve number cards sequenced in backward (from large to small). When students are studying numbers up to 100, students are asked to locate given numbers on a number line, and similar questions are asked in Grade 2 when the range of numbers is extended to 1000.
What is conspicuously absent in the Japanese primary mathematics textbooks is the use of number lines to deal with addition and subtraction. Rather, number lines are used to represent visually relative sizes of numbers. I recently heard that some people distinguish number paths and number lines. Number paths, as I understand it, simply string together numbers, 1, 2, 3, ... On a number path, numbers are represented more by their positions (or orders) whereas on a number line, a number is represented by the distance of the tick mark from the origin, i.e., 0. So, the way the Japanese textbooks introduce and use number lines are much more along the line of number paths.
So, why don't Japanese textbooks use number lines to represent addition and subtraction, as is often done in some US textbooks? There are at least a couple of reasons. The idea that a number is represented by the distance of the tick mark from the origin is a difficult one for students in primary grades. This is difficult, in part, because those students are still learning about measuring length. So, they really don't have the prerequisite knowledge to interpret number lines in that manner. What they tend to do is to simply count the tick marks. However, when students count, they start with "1," and this is another reason number lines are complicated for young children. I have yet to meet a child who started his/her counting by saying, "zero... one, two, three, ..." For many young children the role of 0 (the origin) on a number line is mysterious. So, when they have to use number line to solve 5+3, they will start with the tick mark labeled "5," some will point to "5" and say, "one." Most, if not all, teachers of primary grades have seen young children line up their rulers starting at "1." It's the same problem.
Some people suggest that number lines are inappropriate for primary students, and we should not use number lines. However, I do think it is important that number lines are introduced in primary grades. However, we should be careful about how we use them. We can use them to think about relative sizes of numbers. However, it is probably a good idea to wait to use number lines as a tool for arithmetic. We can use something like tape diagram for that purpose. But, developing the idea that numbers can be represented on number lines is an idea that should start in primary grades, and we should guide students to understand how numbers are represented (as distance from the origin) on number lines, perhaps connecting to the study of linear measurement.
Friday, April 17, 2009
P1a - Teaching THROUGH problem solving
P1. Students will solve problems (using appropriate technology). a. Build new mathematical knowledge through problem solving.
I'm going to write about the same process standard as the last entry; however, this time, I want to focus on the actual indicator, "build new mathematical knowledge through problem solving."
Teaching through problem solving has been a major emphasis in mathematics education over the last (at least) 2 decades - the emphasis on problem solving was there in the 1980 NCTM document. So, it is not necessarily a new idea, but it's not quite clear what this might actually look like in a real classroom. Some people have discussed the three related ideas:
* teaching for problem solving
* teaching about problem solving
* teaching through problem solving
Teaching for problem solving is exemplified by the common textbook organization where students are taught various rules and formulas in a unit, and at the end of the unit is the lesson(s) titled "applications." Students are taught necessary tools, so to speak, and they are given numerous problems for which those tools may be useful.
Teaching about problem solving typically means teaching various problem solving strategies such as guess and check, draw a diagram, look for a simpler problem, make a table, etc. Some textbooks will include a mini-unit on these strategies throughout their textbook, and students are asked to solve problems using the specified strategy.
However, neither approach really produces new mathematical knowledge by solving problems. Teaching through problem solving means students will solve a problem, using only what they have previously learned. Then, by examining their solution strategies, they will generate a new idea/rule/formula. Let's take a look at an example.
In the GPS, students are expected to learn how to determine the area of rectangles and squares by multiplying their dimensions in Grade 3 (M3M4c). Then in Grade 5, students are expected to derive the formulas for calculating the area of parallelograms and triangles (M5M1 b & c). Somewhere in between, students are often asked to find the area of L-shape like the one shown below.
If you ask students to find the area of this shape in many different ways, they may come up with solutions like the ones shown below.
All of these methods will determine the area of the L-shape. However, if you are teaching through problem solving, your real lesson starts once these solution strategies are shared because the goal of the lesson is NOT to determine the area of the L-shape. Rather, you may ask students, "What is common about ALL of these strategies?" One conclusion students may reach is that all of the strategies are somehow using rectangles and squares, shapes for which they already know how to calculate the area. Thus, by discussing that question, students may reach a new understanding that "when we are given an unfamiliar shape, we may still be able to calculate its area by somehow making a familiar shape (or a collection of familiar shapes)."
Your lesson may not stop there. You may want to ask students to sort these strategies - "which strategies are alike?" Often times, students will come up with the following three categories:
* divide the given shape up into several familiar shapes
* cut and re-arrange to make a familiar shape
* make-it-bigger
Thus, students can learn some specific strategies for creating familiar shapes by critically analyzing these strategies.
So, what can we say about teaching through problem solving? One important idea is that the discussion after various solution approaches are shared is the meat of the lesson. That means we must make sure that we leave sufficient amount of time for such discussion. Too often, we see lessons where very little time is left after the last solution is shared. Sometimes this happens because teachers lost track of time as they circulate around the classroom. Other times teachers feel that students need more time to solve the problem. However, I think it is very important for us to remember that the goal is not the answer to the problem. Rather, even if students have not completed their solution, perhaps their incomplete answer may still be sufficient for conducting productive discussion.
Teaching through problem solving is extremely challenging. It requires teachers to have deep understanding of mathematics they are teaching. It also requires teachers to understand their students' mathematical knowledge so that they can anticipate various solution strategies might come up. Furthermore, teachers must have a plan on how to orchestrate the discussion once strategies are shared. Few teachers, if any, can naturally do this; however, it is something teachers can learn, too. Japanese teachers continuously sharpen their craft of mathematics teaching through a process called lesson study. You can learn more about lesson study and also watch some interesting lessons by clicking here.
I'm going to write about the same process standard as the last entry; however, this time, I want to focus on the actual indicator, "build new mathematical knowledge through problem solving."
Teaching through problem solving has been a major emphasis in mathematics education over the last (at least) 2 decades - the emphasis on problem solving was there in the 1980 NCTM document. So, it is not necessarily a new idea, but it's not quite clear what this might actually look like in a real classroom. Some people have discussed the three related ideas:
* teaching for problem solving
* teaching about problem solving
* teaching through problem solving
Teaching for problem solving is exemplified by the common textbook organization where students are taught various rules and formulas in a unit, and at the end of the unit is the lesson(s) titled "applications." Students are taught necessary tools, so to speak, and they are given numerous problems for which those tools may be useful.
Teaching about problem solving typically means teaching various problem solving strategies such as guess and check, draw a diagram, look for a simpler problem, make a table, etc. Some textbooks will include a mini-unit on these strategies throughout their textbook, and students are asked to solve problems using the specified strategy.
However, neither approach really produces new mathematical knowledge by solving problems. Teaching through problem solving means students will solve a problem, using only what they have previously learned. Then, by examining their solution strategies, they will generate a new idea/rule/formula. Let's take a look at an example.
In the GPS, students are expected to learn how to determine the area of rectangles and squares by multiplying their dimensions in Grade 3 (M3M4c). Then in Grade 5, students are expected to derive the formulas for calculating the area of parallelograms and triangles (M5M1 b & c). Somewhere in between, students are often asked to find the area of L-shape like the one shown below.
If you ask students to find the area of this shape in many different ways, they may come up with solutions like the ones shown below.
All of these methods will determine the area of the L-shape. However, if you are teaching through problem solving, your real lesson starts once these solution strategies are shared because the goal of the lesson is NOT to determine the area of the L-shape. Rather, you may ask students, "What is common about ALL of these strategies?" One conclusion students may reach is that all of the strategies are somehow using rectangles and squares, shapes for which they already know how to calculate the area. Thus, by discussing that question, students may reach a new understanding that "when we are given an unfamiliar shape, we may still be able to calculate its area by somehow making a familiar shape (or a collection of familiar shapes)."
Your lesson may not stop there. You may want to ask students to sort these strategies - "which strategies are alike?" Often times, students will come up with the following three categories:
* divide the given shape up into several familiar shapes
* cut and re-arrange to make a familiar shape
* make-it-bigger
Thus, students can learn some specific strategies for creating familiar shapes by critically analyzing these strategies.
So, what can we say about teaching through problem solving? One important idea is that the discussion after various solution approaches are shared is the meat of the lesson. That means we must make sure that we leave sufficient amount of time for such discussion. Too often, we see lessons where very little time is left after the last solution is shared. Sometimes this happens because teachers lost track of time as they circulate around the classroom. Other times teachers feel that students need more time to solve the problem. However, I think it is very important for us to remember that the goal is not the answer to the problem. Rather, even if students have not completed their solution, perhaps their incomplete answer may still be sufficient for conducting productive discussion.
Teaching through problem solving is extremely challenging. It requires teachers to have deep understanding of mathematics they are teaching. It also requires teachers to understand their students' mathematical knowledge so that they can anticipate various solution strategies might come up. Furthermore, teachers must have a plan on how to orchestrate the discussion once strategies are shared. Few teachers, if any, can naturally do this; however, it is something teachers can learn, too. Japanese teachers continuously sharpen their craft of mathematics teaching through a process called lesson study. You can learn more about lesson study and also watch some interesting lessons by clicking here.
Monday, April 6, 2009
P1 - Technology and manipulatives
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.
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.
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.
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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.