Rationale
"The true virtue of mathematics (and not many know this) is that it saves efforts". This sadly underappreciated truth, stated so eloquently by Ron Aharoni, is the main premise of this unit. According to Aharoni, mathematics has three ways to economize: order, generalization, and concise representation. He refers to these as mathematical economy. Order is discerning patterns to make orientation easier; generalization is the idea that an idea discovered in one area can be applied to another area; and concise representation refers to mathematical formulas that represent propositions in a brief and clear manner. In this unit, all three of the economies will be addressed; however, concise representation will be the primary focus. The basis of concise representation allows us to translate our verbal and written words into a mathematical representation. As many teachers and students are very aware - "mathematics is a language of itself" that uses numbers as well as words. Aharoni defined this as a "mathematical proposition" - the equivalent of a sentence in spoken language. We are very familiar with formulas which are precise mathematical propositions that were condensed. By understanding and perfecting the art of translating English sentences to algebraic expressions, and vice versa, students will be better equipped to deal with the "infamous" word problems.
As discussed among my colleagues within my school as well as in the seminar, many students enjoy working with numeric mathematics problems as opposed to word problems for a number of reasons. Early in their education students become familiar with numerical expressions. Often those expressions are limited to addition and subtraction. Generally, they are very concrete. By the middle grades, students begin to study multiplication and division expressions with limited abstraction attached. Unfortunately, the tendency is to study computation only, to the neglect of applications through word problems. Therefore, by the time students arrive in high school, specifically Algebra, they have little experience in interpreting expressions or applying them in word problems. Needless to say, students prefer to avoid word problems and are uncertain of what is being asked and how to approach the word problem. To help students to reach a level of understanding of solving word problems, students must be introduced to word problem by a systematic approach. This unit will explore approaches to, and ultimately the solution of word problems through the use of numerical relationships and variables. Students will investigate, describe, and generalize a variety of patterns and represent situations using variables. They will also simplify algebraic expressions involving like terms and when given values are provided. Students will also review properties of numbers and operations using variables.
To reduce frustration and allow for "success", many teachers tend to stay away from word problems; however, word problems must not be ignored. Word problems must be an integral part of the curriculum. I generally try to use curricula and lessons that are relevant as well as familiar, as I will do in this unit. This will increase students' interest and achievement. In fact, Aharoni states, "one of the fundamental principles of teaching is to begin with the familiar." For example, students are familiar with reading; they may not be proficient with it, but the structure already exists in their mind. Therefore, adding word problems is an extension of reading curricula. Additionally, one principle for consistent success in solving word problems is that it is absolutely necessary to read the problem carefully, if necessary, read it again. A third reading may also be needed. Additionally, embedded into reading is the use of common language. Similarly, in math, along with all other subjects, it is imperative that there is a common language. However, the challenge is that there are many words that are used interchangeably. If their equivalence is not recognized, this will cause confusion. For the purpose of this unit, some of the words that present this challenge and may need clarification are listed in appendix 1.
Although, many teachers feel that teaching word problems is a "separate" unit, the integration of word problems with numerical problems will deepen student understanding as well as problem solving skills. The problem that I have with this integration is that many of my ninth grade Algebra students enter my class unequipped with the essential mathematic skills necessary to begin the algebra curriculum, or better yet, to be able to tackle word problems. In past years, students that were below the 48th percentile of the State's test were automatically enrolled in pre-algebra to develop weak skills. Today, the district now requires that all freshman students take Algebra as their first high school mathematics course - regardless of developmental level. This poses a tremendous challenge to all algebra teachers in my district as well as the students. In order to lessen the anxiety and frustration of my students, I constantly look for new ways to help my students achieve by staying abreast with the best instructional strategies and incorporating them whenever I feel they will help my students.
In addition to the above, my school is the second lowest high school in the city in regard to the Prairie State Achievement Examination (PSAE) scores. This summer our reform-minded principal who was hand-picked by school officials was transferred out the building. Prior to his principalship, the school was in physically bad shape; a third of the computers worked, showers were inoperable, the band had no uniforms, and more than a third of the students received special education services. Under the new principal's guidance, many strategies were implemented and renovation was made: the schools received new computers and books as well as a renovated swimming pool. Unfortunately, the improvements did not result in significant student academic gains. In fact, performance and attendance dropped in the last few years. In 2003, 5.9 percent students were passing the PSAE tests. In 2005, the number had dropped to 3.8 percent. Since 2003, the schools' four-year graduation rate has fallen almost 10 percentage points to 48.7. The attendance rate has also dropped. One area of slight improvement occurred with the ACT. The ACT scores have improved, rising from 13.9 in 2003 to 14.4 in 2006 on a 36 point-scale.
The Algebra curriculum involves an approach that encourages students to work in groups. Most of the activities are discovery-based cooperative learning. The idea is that students will make connections and take ownership of the lessons. The problem is that most of my students are reluctant to work in groups and prefer that the "rules" or shortcuts be given to them. A second problem is that the curriculum does not provide opportunity for the students to master the skills before beginning the next unit. In addition, the curriculum provides only limited opportunity to work with "traditional" word problems. Although the curriculum supports my belief that students must make connections as well as discoveries, I also strongly believe they need more exposure to reading, analyzing, and solving word problems.
Algebra I is the study of linear equations and functions. Throughout this course, students will examine linear equations, graphs, and tables. They will apply the skills that were developed to solve real life situations that usually come in word problems. It will be imperative that students know how to translate verbal expressions to algebraic expressions - the problem territory - and ultimately solve word problems. There are three dimensions of the problem territory: understand the relationship of the variable and the operation symbols, represent the variables and operations correctly, and evaluate expressions given a value of the variable. Despite the many challenges my students and I face, my goal for the completion of this unit is to have student master the state's standard of formulating and evaluating linear expressions algebraically. This goal is important because these are the fundamental skills for future progress.
For most students, math means that they will have to work with numbers, variables, operations and expressions. The general principle of the unit problem sets will enable the students to solve set of word problems that require writing expressions. However, the problems are carefully graduated in their complexity as well as the complexity of the required operation(s). Initially, as students explore the problem territory, they will use arithmetic that will extend to algebraic representation through the concrete, pictorial, and abstraction methods. They will begin to develop a deeper understanding in writing expressions and evaluating word problems. Students will begin to recognize that there are many approaches to solving mathematics problems. The opportunity for reinforcement and enhancement will be provided. By using problem solving strategies, students will organize information and solve novel as well as unique problems.
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