The Problem with Word Problems
A train leaves Santa Fe, New Mexico heading east at 8:00 am. It is traveling at a speed of 70 mph. Another train leaves Wilmington, Delaware at the same time but it is heading west. The train from Delaware is traveling at 60 mph. What time will both trains meet? Most adults at this time will start rolling their eyes or their eyes begin to glaze over as they stare off into space. The next words that utter from their lips usually are, "I hate word problems!" They don't usually say, "Ooh, this sounds like a cool problem! Let me get a pencil and paper and figure this out. This sounds like fun!" Somewhere along the way, word problems have become associated with pain and agony. This reaction is very different for a 1st grader as they begin their foray into the world of word problems. Using various types of manipulatives such as bears, cubes, popsicle sticks, etc., 1st graders have a wide-eyed wonderment as they seek to solve the problem. The book Children's Mathematics: Cognitively Guided Instruction by T. Carpenter, et al provides a wonderful diagram as to what encompasses addition and subtraction problems. The following explanation comes directly from this excellent book. The purpose is not for the terms to be used to teach to your students, but to give you the teacher a better insight into the construction of word problems. Addition and subtraction fall into four categories of problems. Those categories are Join, Separate, Part-Part-Whole and Compare.(4) These categories help educators identify the different type of word problems that students will encounter. When teachers are better able to identify the type of word problem that they are presenting to their student, the more beneficial to the student as the teacher builds and scaffolds their learning. This also beneficial to the teacher because they can make sure that their students see all the different types, so they will be able to deal with any addition and subtraction problem. Join problems are problems in which a set is increased by a specific amount. The set is increased due to a direct or implied action. In simpler terms, this represents an addition problem. However, sometimes the total is given and one of the summands is asked for. Then it must be found by subtraction. The following example is an example of a Join addition problem:
5 birds were sitting on a fence. 4 more birds flew on to the fence. How many birds were on the fence then?
The action in this problem takes place over time. There is a starting amount at Time 1 (5 birds sitting on the fence); a second (or change) amount is joined to the initial amount at Time 2 (4 more birds flew on to the fence); the result is a final amount at Time 3 (the 9 birds then on the fence). T. Carpenter, et al finds that this type of problem spawns three distinct sub-categories. The reason for this is although the resulting set of birds is composed of the birds initially sitting on the fence, another group of birds joined the first group of birds on the fence. The two sets of birds take on very different roles in the problem due to the action which happens in a sequence of time. What does this mean? Well, here is where Carpenter, et al, takes Join problems as well as Separate problems (which we will discuss next) and identifies the sub-categories. The sub-categories are Result Unknown, Change Unknown, and Start Unknown. This is important because children may not realize that 4 birds joining 5 birds is the same as 5 birds joining 4 birds. Let's take a word problem and manipulate the sequence that the numbers are given to illustrate the sub-categories of Join and Separate.
Result Unknown: Tanya has 2 dolls. Her parents give her 4 more dolls for her birthday. How many dolls did she have then?
Change Unknown: Tanya had 2 dolls. Her parents gave her some more dolls for her birthday. Then she had 6 dolls. How many dolls did Tanya's parents give her for her birthday?
Start Unknown: Tanya had some dolls. Her parents gave her 4 more dolls for her birthday. Then she had 6 dolls. How many dolls did Tanya have before her birthday?
Separate problems are similar to Join problems in many ways. Like Join problems, there is an action that takes place over time, but with Separate problems, the initial quantity is decreased rather than increased. There can also be three distinct sub-categories for Separate problems like Join problems. They also have a starting quantity, a change quantity (this is the amount that is removed), and the result (difference). Examples of the different sub-categories that fall under Separate problems are as follows:
Result Unknown: Valerie had 12 marbles. She gave 7 away to Nancy. How many marbles does Valerie have left?
Change Unknown: Valerie had 12 marbles. She gave some to Nancy. Now she has 5 marbles left. How many marbles did Valerie give Nancy?
Start Unknown: Valerie had some marbles. She gave 7 to Nancy. Now she has 5 marbles left. How many marbles did Valerie start with?
The next type of problems that Carpenter, et al have identified are the Part-Part-Whole problems. These problems are different from Join and Separate problems because there is no change over time. There is also no direct or implied action. This means that because one set is not being joined to the other, both sets assume equal roles in the problem. "Part-Part-Whole problems involve static relationships among its two disjoint subsets."(5) Due to the fact that Part-Part-Whole problems are static, there are only two types of sub-categories. The problem will either give the two parts and ask students to find the size of the whole, or, it gives one of the parts and the whole and asks students to find the size of the other part. These two sub-categories can be labeled as Whole Unknown, or Part Unknown. An example of a Whole Unknown is as follows:
Whole Unknown - 3 boys and 5 girls were playing basketball. How many children were playing basketball?
In this example, we know the two parts (3 boys and 5 girls playing basketball). What we don't know is how many children (boys and girls) were playing basketball all together (the Whole Unknown). Next we will examine the Part Unknown word problem.
Part Unknown - 8 children were playing basketball. 3 were boys and the rest were girls. How many girls were playing basketball?
As we look at the Part Unknown example we know one of the parts (3 boys playing basketball). We also know the Whole (8 boys and girls all together). What we don't know is the 2nd part (the number of girls playing basketball).
The final type of problem that Carpenter, et al, have distinguished is the class of Compare problems. Compare problems entail relationships between amounts rather than a joining (addition) or separating (subtraction) action. This is similar to Part-Part-Whole problems. Where they both differ is that Compare problems involve comparing two different sets rather than the relationship between a set and its subsets. Given that one set is compared to another set, one set is given a label of the referent set, and the other set is called the compared set. The third component of a compare problem is called the difference. This is the amount by which one set is greater than the other set. Carpenter, et al, does a good job illustrating this concept by using clear examples. Here is an example that illustrates the different elements of Compare problems.
- Nancy has 2 cats. - Referent set
- Karli has 6 cats. - Compared set
- Karli has 4 more cats than Karli. - Difference
With Compare problems, any one of the three components, the difference, the referent set or the compared set, can be the unknown. Using the same problem, let's examine how it is used in a word problem.
- Difference Unknown - Nancy has 2 cats. Karli has 6 cats. Karli has how many more cats than Nancy?
- Compared Set - Nancy has 2 cats. Karli has 4 more cats than Nancy. How many cats does Karli have?
- Referent Set - Karli has 6 cats. She has 4 more than Nancy. How many cats does Nancy have?
Each problem is slightly different but clearly shows the distinctions that Carpenter, et al, have identified as sub-categories of Compared problems.
We will next look at how these distinctions among problems translate into number sentences. Number sentences can be used to represent the Join and Separate problems.
Number sentences are another way to represent the subtleties and differences in certain problem types. Join and Separate problems are the easiest type of problems to be represented by number sentences as seen in Table 1
(table 07.06.02.01 available in print form)
Table 1
When teachers are making word problems or even selecting word problems for their children to solve, they should look to make the language as clear as possible. This can be done by making the action or relationship as clear as possible. When working with kindergartners or 1st graders, it is important to make sure that the units are consistent. What do I mean by this? Take this simple word problem:
Moses has 2 apples. Tyler has 3 oranges. How many fruit do the boys have all together?
As an adult the answer may seem obvious. To a young learner this may be confusing to sort in their mind the fact that each boy has different fruit. The fruit becomes the unit. Combining 2 apples and 3 oranges may be a very foreign concept because they look different, they smell different, they taste different and it may not seem to make sense in the mind of a very young person why they should put them together. The old saying of "comparing apples to oranges" comes to mind. The students also have to make the connection that although apples and oranges are different, they are both fruit. This type of problem involves an implicit agreement to ignore the differences.(6)It can be so easy to see all the ways that apples and oranges are different that a child misses how they are alike. This combining of two very different items can be very perplexing to a six or seven year old because they see each set as two very dissimilar things.
The less abstract a problem is the easier it is for children to make a one to one correspondence. It is important that problems represent easily identifiable, discrete sets of objects. Compare these two sets of problems:
- Angel has a puppy that weighed 3 pounds when her mom brought her home. The puppy now weighs 9 pounds. How many pounds has the puppy gained?
- Paula has 2 stickers to start her collection. She needs 10 to complete the page. How many more stickers does Paula need?
The pounds of puppy are not clearly identifiable objects or "things" that a child can see and touch. Using counters like bears or cubes to represent the puppy's pounds is an abstract representation that may be hard for young children to grasp. Weight is a continuous variable. It takes time to learn to segment it.(7) The second problem deals with items that a young student can easily identify with. It is much easier to use counters to represent something more tangible like stickers.
We will next look at how teachers can use reading strategies to help students with their math. Although the reading strategies are gong to be explored individually, it is important to know that the reading strategies interact and work in conjunction with one another and are not independent of each other. The focus is going to be using a KWC chart to help children solve word problems. The KWC will work in conjunction with the reading strategies to give students an excellent tool to use to solve any word problem.
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