Finally, the teacher should fully understand the information on the topics he is going to discuss. For example, he should know the classes of problems or problem types and the mathematical operations and rules governing these operations. If the teacher is very well familiar with the topic he will be teaching, he will gain more confidence. Background knowledge also provides teachers confidence that their curriculum is mathematically correct.

Teachers may use textbooks but they should not fully rely on these materials since many modern textbooks do not provide enough background information for the teacher or interesting opportunities for the children. Teachers should be encouraged to attend classes or workshops to develop their ability in developing challenging problems for the students. Among teachers, there is a common misperception that there are only two classes of problems. In one problem, the students add two numbers to find the solution. In the other, the students subtract to solve the problem.

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However, problem types cannot just be classified with these two. Math scholars break down the problem types in many ways. One textbook describes only one real-world use for addition, the joining of two sets in a problem (Kennedy & Tipps, 1994). Smith (2006) gives an example: “Dewey had 6 goldfish. His uncle brought him 4 more. How many goldfish does Dewey have? ” The solution to this problem is to add 6 and 4 together. Subtraction, on the other hand, occurs in four situations. The most common type is the take away problem.

For example: “Mario had 7 baseball cards. He gave 1 to his brother. How many baseball cards does he have left? ” (Smith, 2006). To solve this, just simply subtract 1 from 7. Another type of subtraction is a comparison between the sizes of two sets. For example: “Twelve people like chocolate ice cream. Nine people like vanilla ice cream. How many more people like chocolate ice cream? ” (Smith, 2006). A third type is a completion problem using subtraction to find a missing number. This type of problem is often called the ‘missing addend’ problem.

For example: “Zaneta had 3 berets. Her sister gave her some more berets. Now she has 7 berets. How many did her sister give her? ” (Smith, 2006). Finally, there is a whole-part-part problem used in finding the size of subgroup. For example: “Aunt Edna planted 12 tulips. Five were red. The rest were yellow. How many tulips were yellow? ” (Smith, 2006). Another textbook describes a fifth kind of addition and subtraction problems called incremental problems. These problems involve measuring continuous quantities such as sand, water, or temperature.

Troutman and Lichtenberg (1991) stress that these problems are important in training students not to rely on counting discrete objects only. There have been extensive researches dealing with how children think about addition and subtraction. Problem types form the basis for conducting researches. At the University of Wisconsin, Madison, research was conducted focusing on using problem types. These problem types are based on a different philosophy from the classification methods described earlier (Carpenter, Carey, & Kouba, 1990). Adults may use subtraction to solve many problems.

Children, on the other hand, use the method of counting from 1, or counting-on, to solve a problem. In effect, the children are adding. Bebout (1990) state that the using the eleven problem types encourages to children to learn the standard or canonical number sentences that mathematicians might use. Teachers are encouraged to study all eleven types to be able to teach mathematics with confidence. Furthermore, studies show that there is an increase in students’ academic achievement using the eleven problem types (Peterson, Fennema, & Carpenter, 1989).

According to Smith (2006), this method uses four basic classes of problems, namely, the join (three types), separate (three types), part-part-whole (two types), and comparison (three types). Part-part-whole and comparison problems are discussed earlier. Join problems involve combining or adding numbers while separate problems involve removing objects. According to Smith (2006), “The difference between the three kinds of problems has to do with where the unknown number of the box symbol occurs. ” The following table will show this difference.