## Tuesday, December 23, 2008

### M3A1 - Mathematical expressions (1)

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

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

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

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

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

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

## Thursday, December 18, 2008

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

MKM1. Students will group objects according to common properties such as longer/shorter, more/less, taller/shorter, and heavier/lighter.
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.
The three standards above reflects the four-stage general sequence of teaching measurement:
1. Direct comparison
2. Indirect comparison
3. Measuring with non-standard units
4. Measuring with standard units
MKM1 is clearly in the first stage, and M2M1 is in the fourth and final stage. M1M1 explicitly indicate stages 1 and 3, but there is no mention of stage 2. Considering the general sequence of instruction, what should be happening is more in stages 2 and 3.

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

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

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

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

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

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

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

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

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