A decision method for elementary algebra and geometry

"This report...was written while the project was a part of Douglas Aircraft Co., inc."

Elementary algebra ...

Mode of access: Internet.

Higher algebra, a sequel to Elementary algebra for schools.

Mode of access: Internet.

An easy algebra for beginners; being a simple, plain presentation of the essentials of elementary algebra, and also adapted to the use of those who can take only a brief course in this study.

"Answers": p. 143-157.

A key containing the statements and solutions of questions in Charles Davies' New elementary algebra : for the use of teachers only.

Mode of access: Internet.

An elementary treatise on algebra : for the use of students in high schools and colleges /

Spine title: Sherwin's elementary algebra.

An elementary treatise on algebra : for the use of students in high schools and colleges /

Spine title: Sherwin's elementary algebra.

Investigating functional thinking in the elementary classroom : foundations of early algebraic reasoning

This paper examines the development of student functional thinking during a teaching experiment that was conducted in two classrooms with a total of 45 children whose average age was nine years and six months. The teaching comprised four lessons taught by a researcher, with a second researcher and classroom teacher acting as participant observers. These lessons were designed to enable students to build mental representations in order to explore the use of function tables by focusing on the relationship between input and output numbers with the intention of extracting the algebraic nature of the arithmetic involved. All lessons were videotaped. The results indicate that elementary students are not only capable of developing functional thinking but also of communicating their thinking both verbally and symbolically.

Algebra of Classical Theoretical Term Reductions in the Sciences

An elementary algebra identifies conceptual and corresponding applicational limitations in John Kemeny and Paul Oppenheim’s (K-O) 1956 model of theoretical reduction in the sciences. The K-O model was once widely accepted, at least in spirit, but seems afterward to have been discredited, or in any event superceeded. Today, the K-O reduction model is seldom mentioned, except to clarify when a reduction in the Kemeny-Oppenheim sense is not intended. The present essay takes a fresh look at the basic mathematics of K-O comparative vocabulary theoretical term reductions, from historical and philosophical standpoints, as a contribution to the history of the philosophy of science. The K-O theoretical reduction model qualifies a theory replacement as a successful reduction when preconditions of explanatory adequacy and comparable systematicization are met, and there occur fewer numbers of theoretical terms identified as replicable syntax types in the most economical statement of a theory’s putative propositional truths, as compared with the theoretical term count for the theory it replaces. The challenge to the historical model developed here, to help explain its scope and limitations, involves the potential for equivocal theoretical meanings of multiple theoretical term tokens of the same syntactical type.

Crunch time : how did the students perform?

Two newspaper numbers games based on simple arithmetic relationships are discussed. One is rather trivial, but very useful as an introduction to the second, whose potential to give students of elementary algebra practice in semi ad-hoc reasoning and to build general arithmetic reasoning skills was explored theoretically in an earlier paper. Preliminary results on the effectiveness of this general approach are presented, with student performance and feedback on an assignment task and formal examination included, and recommendations for future work.

The Number Crunch game : a simple vehicle for building algebraic reasoning skills

A newspaper numbers game based on simple arithmetic relationships is discussed. Its potential to give students of elementary algebra practice in semi-ad hoc reasoning and to build general arithmetic reasoning skills is explored.

相同知识背景下技能之间迁移机制的研究

Since the middle of 1980's, the mechanisms of transfer of training between cognitive subskills rest on the same body of declarative knowledge has been highly concerned. The dominant theory is theory of common element (Singley & Anderson, 1989) which predict that there will be little or no transfer between subskills within the same domain when knowledge is used in different ways, even though the subskills might rest on a common body of declarative knowledge. This idea is termed as "principle of use specificity of knowledge" (Anderson, 1987). Although this principle has gained some empirical evidence from different domains such as elementary geometry (Neves & Anderson, 1981) and computer programming (McKendree & Anderson, 1987), it is challenged by some research (Pennington et al., 1991; 1995) in which substantially larger amounts of transfer of training was found between substills that rest on a shared declarative knowledge but share little procedures (production rules). Pennington et al. (1995) provided evidence that this larger amounts of transfer are due to the elaboration of declarative knowledge. Our research provide a test of these two different explanation, by considering transfer between two subskills within the domain of elementary geometry and elementary algebra respectively, and the inference of learning method ("learning from examples" and "learning from declarative-text") and subject ability (high, middle, low) on the amounts of transfer. Within the domain of elementary geometry, the two subskills of generating proofs" (GP) and "explaining proofs" (EP) which are rest on the declarative knowledge of "theorems on the characters of parallelogram" share little procedures. Within the domain of elementary algebra, the two subskills of "calculation" (C) and "simplification" (S) which are rest on the declarative knowledge of "multiplication of radical" share some more procedures. The results demonstrate that: 1. Within the domain of elementary geometry, although little transfer was found between the two subskills of GP and EP within the total subjects, different results occurred when considering the factor of subject's ability. Within the high level subjects, significant positive transfer was found from EP to GP, while little transfer was found on the opposite direction (i. e. from GP to EP). Within the low level subjects, significant positive transfer was found from EP to GP, while significant negative transfer was found on the opposite direction. For the middle level subject, little transfer was found between the two subskills. 2. Within the domain of elementary algebra, significant positive transfer was found from S to C, while significant negative transfer was found on the opposite direction (i. e. from C to S), when considering the total subjects. The same pattern of transfer occurred within the middle level subjects and low level subject. Within the high level subjects, no transfer was found between the two subskills. 3. Within theses two domains, different learning methods yield little influence on transfer of training between subskills. Apparently, these results can not be attributed to either common procedures or elaboration of declarative knowledge. A kind of synthetic inspection is essential to construct a reasonable explanation of these results which should take into account the following three elements: (1) relations between the procedures of subskills; (2) elaboration of declarative knowledge; (3) elaboration of procedural knowledge. 排Excluding the factor of subject, transfer of training between subskills can be predicted and explained by analyzing the relations between the procedures of two subskills. However, when considering some certain subjects, the explanation of transfer of training between subskills must include subjects' elaboration of declarative knowledge and procedural knowledge, especially the influence of the elaboration on performing the other subskill. The fact that different learning methods yield little influence on transfer of training between subskills can be explained by the fact that these two methods did not effect the level of declarative knowledge. Protocol analysis provided evidence to support these hypothesis. From this research, we conclude that in order to expound the mechanisms of transfer of training between cognitive subskills rest on the same body of declarative knowledge, three elements must be considered synthetically which include: (1) relations between the procedures of subskills; (2) elaboration of declarative knowledge; (3) elaboration of procedural knowledge.

A relação entre a álgebra acadêmica e a álgebra escolar em um curso de licenciatura em matemática: concepções de alunos e professores

Pós-graduação em Educação - FCT

A produção de significados durante e processo de enisno-aprendizagem-avaliação de equações polinomiais

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)