# CHARATERIZING THE DEVELOPMENT OF A SCHEMA FOR REPRESENTING AND SOLVING ALGEBRA WORD PROBLEMS BY PRE-ALGEBRAIC STUDENTS ENGAGED IN A STRUCTURED DIAGRAMMATIC ENVIRONMENT

Open Access

- Graduate Program:
- Curriculum and Instruction
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- August 26, 2009
- Committee Members:
- Rose Mary Zbiek, Dissertation Advisor
- Mary Kathleen Heid, Committee Chair
- Rose Mary Zbiek, Committee Member
- Glendon Wilbur Blume, Committee Member
- Richard Alan Carlson, Committee Member

- Keywords:
- algebra
- diagram
- problem solving
- quantitative reasoning
- part-whole
- procept

- Abstract:
- In recent years, the learning of algebra by all students has become a high national priority (Moses & Cobb, 2001; National Council of Teachers of Mathematics, 2000). Algebra is considered to be a foundational topic in mathematics (Usiskin, 1988) and some have argued that an understanding of algebra is fundamental to success in today’s technological society (Moses & Cobb 2001; Nathan & Koellner, 2007). Whereas the National Council of Teachers of Mathematics [NCTM] (2000) advocates that students should “develop their skill in…solving linear equations in the middle grades” (p. 39), many students in middle school and high school algebra have been reported to experience difficulty when attempting to represent and solve algebra word problems. Students have also been found to demonstrate a lack of understanding when solving equations. The purpose of this study is to explore the nature of the algebraic understandings developed by students who learn to solve algebra word problems within the context of a structured diagrammatic environment. Three sixth grade students who had never before taken algebra took part in a twelve-week teaching experiment in which they learned to represent and solve algebra word problems. Students were given tasks which were similar in structure to those utilized by the Singaporean model method. However, the use of these tasks differed from the Singaporean curriculum in that, following their use of diagrams to model and solve algebra word problems, students were introduced to the use of letters and numbers to accomplish the same. The tenets of Realistic Mathematics Education’s theory of lesson design were employed in choosing the scope and sequencing of the tasks. Using APOS theory (Dubinsky, 1991), Simon and colleagues’ (2004) activity-effect theory, and Gray and Tall’s (2007) procept theory, the nature and development of students’ algebraic understandings were examined. It was found that students developed four overarching understandings. Three of these understandings are prominent in the literature as being prerequisite to a students’ algebraic development. Underlying and supporting students’ development of these understandings were the following conceptions: an additive quantity is simultaneously a process and an object (a procept), a quantitative whole is decomposable, and the parts of a quantitative whole are commutative.