Saturday, July 21, 2007

M1N3 - Diagrams for Addition & Subtraction

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

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

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

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

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

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

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

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

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

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

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

instead of,

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

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

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

Here is a tape diagram template for a comparison problem:

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

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

Tuesday, July 17, 2007

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

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

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

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

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

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

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

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

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

Sunday, July 15, 2007

M2N3 b - Multiplication Table

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

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

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

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

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

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

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

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

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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.