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Early algebraic reasoning can be viewed as developing a bridge between arithmetic and algebra. Accordingly, this research examines how middle-grades students' arithmetic reasoning, classified by their number sequences, can be used to model their algebraic reasoning as it pertains to generalizing, writing, and solving linear equations and systems of equations. In the quantitative phase of research, 326 students in grades six through nine completed a survey to assess their number sequence construction. In the qualitative phase, 18 students participated in clinical interviews, the purpose of which was to elicit their algebraic reasoning. Results show that the numbers of students who had constructed the two least sophisticated number sequences did not change significantly across grades six through nine. In contrast, the numbers of students who had constructed the three most sophisticated number sequences did change significantly from grades six and seven to grades eight and nine. Furthermore, students did not consistently reason algebraically unless they had constructed at least the fourth number sequence. Thus, it is concluded that students with the two least sophisticated number sequences are no more prepared to reason algebraically in ninth grade than they were in sixth. / Doctor of Philosophy / Early algebraic reasoning can be viewed as developing a bridge between arithmetic and algebra. This study examines how students in grades six through nine reason about numbers, and whether their reasoning about numbers can be used to explain how they reason on algebra tasks. Particularly, the students were asked to extend numerical patterns by writing algebraic expressions, and were asked to read contextualized word problems and write algebraic equations and systems of equations to represent the problems. In the first phase of research, 326 students completed a survey to assess their understanding of numbers and their ability to reason about numbers. In the second phase, 18 students participated in interviews, the purpose of which was to elicit their algebraic reasoning. Results show that the numbers of students who had constructed a more sophisticated understanding of number did not change significantly across grades six through nine. In contrast, the numbers of students who had constructed a less sophisticated understanding of number did change significantly from grades six and seven to grades eight and nine. Furthermore, students were not consistently successful on algebraic tasks unless they had constructed a more sophisticated understanding of number. Thus, it is concluded that students with an unsophisticated understanding of number are no more prepared to reason algebraically in ninth grade than they were in sixth.

An exploratory study on instructional design for classroom activities that encourage algebraic reasoning at the elementary school level. Assistance with the activities was provided as students needed further scaffolding, and multiple solutions were encouraged. An analysis of student responses to the activities is discussed.

Student difficulties learning algebra are well documented. Many mathematics education researchers (e.g., Bednarz&Janvier, 1996; Davis, 1985, 1989; Vergnaud, 1988) argued that the difficulties students encounter in algebra arose when students were expected to shift suddenly from arithmetic to algebraic reasoning and that the solution to the problem was to integrate opportunities for elementary school students to simultaneously develop both arithmetic and algebraic reasoning. The process of generalization, or describing the overall pattern underlying a set of mathematical data, emerged as a focal point for extending beyond arithmetic reasoning to algebraic reasoning (Kaput, 1998; Mason, 1996). Given the critical importance for students to have opportunities to develop understanding of the fundamental algebraic concepts of variable and relationship, one could argue that providing opportunities to explore linear functions, the first function studied in depth in a formal algebra course, should be a priority for elementary students in grades 4-5. This study informs this debate by providing data about connections between different representations of constant growth functions and student algebraic reasoning in a context open to individual construction of representations and reasoning approaches. Participants included 9 fifth graders from the same elementary class. Data shows that students can generate representations which are effective reasoning tools for finding particular cases of the function and generalizing the function but that this depends on features of the problem representation, most importantly the representation of the additive constant. I identified four categories of algebraic reasoning on the task to find the tenth term and found that only students who used reasoning approaches with the additive constant separate and functional reasoning to find the variable component were able to generalize the function. These instances occurred on a story problem and two geometric pattern problems. None of the students used such a reasoning approach or were able to generalize on the numeric sequence problem which did not represent the additive constant separately. Implications for future research and for teaching for conceptual understanding of variable and relationship are discussed.

Magister Educationis - MEd / This research has employed an intentional teaching model to investigate the algebraic reasoning of grade 9 learners from a low socio-economic background. It has also sought to study how learners engage with algebra to make generalizations and to scrutinize any misconceptions deriving from the experience. They looked for patterns, paid attention to aspects of the patterns that are important and then generalized from familiar to unfamiliar situations. Algebraic reasoning underpins all mathematical thinking including arithmetic because it allows us to explore the structure of mathematics. This study is based on the Curriculum and Assessment Policy Statement which states that learners are expected to investigate patterns to establish the relationships between variables, as well as represent and analyse the change of patterns. The study also had a huge emphasis on algebra. According to Mphuthi & Machaba (2016): “Algebraic expressions form part of the senior phase CAPS curriculum in South Africa. A substantial amount of time is allocated to this section on evaluating expressions and simplifications of algebraic expressions in grade 7-9.” The study is premised on a qualitative research paradigm and a design-based research methodology for data collection. A set of tasks based on algebraic patterns and generalizations was given to an opportunistic sample of 20 grade 9 learners in a school in Delft, a low socio-economic suburb about 30 kilometres from Cape Town in the Western Cape Province of South Africa. Three weeks after completing the tasks, learners were interviewed to identify their reasoning and how they felt about the tasks. The results of the study show that the majority of the learners struggled with tasks especially when asked what the rules they could derive from the patterns. Learners did not seem to understand what they were doing because they were unable to articulate the given tasks in words and did not have knowledge of concepts like the perimeter.

abstract: The purpose of this study was to identify the algebraic reasoning abilities of young students prior to instruction. The goals of the study were to determine the influence of problem, problem type, question, grade level, and gender on: (a) young children’s abilities to predict the number of shapes in near and far positions in a “growing” pattern without assistance; (b) the nature and amount of assistance needed to solve the problems; and (c) reasoning methods employed by children. The 8-problem Growing Patterns and Functions Assessment (GPFA), with an accompanying interview protocol, were developed to respond to these goals. Each problem presents sequences of figures of geometric shapes that differ in complexity and can be represented by the function, y = mf +b: in Type 1 problems (1 - 4), m = 1, and in Type 2 problems (5 - 8), m = 2. The two questions in each problem require participants to first, name the number of shapes in the pattern in a near position, and then to identify the number of shapes in a far position. To clarify reasoning methods, participants were asked how they solved the problems. The GPFA was administered, one-on-one, to 60 students in Grades 1, 2, and 3 with an equal number of males and females from the same elementary school. Problem solution scores without and with assistance, along with reasoning method(s) employed, were tabulated. Results of data analyses showed that when no assistance was required, scores varied significantly by problem, problem type, and question, but not grade level or gender. With assistance, problem scores varied significantly by problem, problem type, question, and grade level, but not gender. / Dissertation/Thesis / Doctoral Dissertation Curriculum and Instruction 2016

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