Rationale
A key feature of this unit is that it requires students to spend a significant portion of time working on word problems. Through this intensive exposure to word problems the students should become more comfortable with them and less afraid of them. Problem solving is a very important aspect of many of the high stakes tests that have become so important to school districts throughout the United States. On high stakes tests, the concepts are often embedded within a word problem. Some students automatically give up when they see the words. Others that know the concept have a difficult time understanding the task and are not able show what they know. By developing the confidence and the skills to tackle word problems, students will not only improve their math skills with the four basic operations, but also will hopefully transfer their knowledge to solving problems involving decimals, fractions, and all other areas of math.
The Singapore bar models were chosen for several reasons. First, Singapore leads the world in math achievement so the strategies that the Singapore approach teaches are proven to be successful. Secondly, the bar pictures are a visual representation of the words. They may help children translate from words to symbols, particularly if the words and language are difficult for them.
Another reason why this unit is appropriate and necessary for fourth graders is because the four operations are a major component of elementary math. Students need to understand the connectedness of the four operations and this unit ties the different operations together. According to Liping Ma, (Ma, 1999), there are two relationships that connect the four basic operations. These two relationships are the "derived operation" and the "inverse operation". An example of the "derived operation" is, "Multiplication is an operation derived from the operation of addition. It solves certain kinds of addition problems in an easier way" (Ma, 1999, p. 113). The inverse relationship is the other connection that Ma refers to in her explanation of the connections between the four operations. Subtraction is the inverse of addition because subtraction of a fixed number from a variable number is the inverse of addition of (the same) fixed number to a variable number. Multiplication and division are also inverse operations. Ma's (1999) analogy describes the importance of connecting the operations with this analogy:
"These two kinds of relationships tightly connect the four operations. Because all of the topics of elementary mathematics are related to the four operations, understanding of the relationships among the four operations, then, becomes a road system that connects all of elementary mathematics. With this road system, one can go anywhere in the domain" (Ma, 1999, p.113).
By developing a connected road system, I seek to help my students to develop a better understanding of arithmetic.
As students solve math and word problems, they need to be flexible and know different ways to approach the problems. While this unit is focused on one strategy, the Singapore bar models, the goal is to teach the students this technique so that they can increase their repertoire of strategies. Students will be required to use the bar models as I teach this unit. However, once this unit is completed, the students may or may not select the bar models as their approach to solving a given problem. As with any other strategy, some students will find that it works well for them and others may prefer a different method.
Why did I choose to use the Singapore bar models? Singapore students were first in the 1995, 1999 and 2003 Trends in International Mathematics and Science Study (TIMSS), which is designed to measure trends in students' mathematics and science achievement in four-year cycles. The TIMSS 2003 results were released on December 14, 2004. (New AIR Study Compares the Quality of U.S. Math Instruction with Singapore Recognized World Leader U.S. Trails, But Both Nations Could Learn from Each Other 2/7/05 http://www.air.org/news/ documents/Singapore.htm.)
Based on these findings I thought it would be interesting to expose my students and myself to Singapore models. Word problems seem to be particularly difficult for students. These models seem to simplify the problems and provide a visual representation to help the students to think about the problem. According to the Cognitively Guided Instruction model, "It is not children's manipulations of materials that is important; it is their understanding of the principles involved in the manipulations" (Carpenter, et al 1999, p. 68).
Based on a study conducted by the American Institute for Research (Leinwand et al., 2005) it is important to look to Singapore and to learn from how this country teaches math to its students. This report states:
"It is unreasonable to assume that Singaporean students have mathematical abilities inherently superior to those of U.S. students; rather, there must be something about the system that Singapore has developed to teach mathematics that is better than the system we use in the United States. That's why it's important to take a closer look, and see how the U.S can learn and how the U.S can improve."
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