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1. Construction of Arithmetic-Algebraic Thinking in a Socio-Cultural Instructional Approach = Construction d'une pensée arithémico-algébrique dans une approche socioculturelle de l'enseignement

Author

Hitt, Fernando

Abstract

We present the results of a research project on arithmetic-algebraic thinking that was carried out jointly by a team in Mexico and another in Quebec. The project deals with the concepts of variable and covariation between variables in the sixth grade at the elementary level and the first, second, and third years of secondary school--namely, children from 11 to 14 years old. We target secondary students (first year or K7) in this article. Our objective relates to the development of a gradual generalization in arithmetic-algebraic thinking in a socio-cultural approach to the learning of mathematics. We experimented with investigative situations using a paper-and-pencil approach and technology. We analyze the emergence, in this context, of a visual abstraction, the production of institutional and non-institutional representations, a sensitivity to contradiction, and, finally, the concepts of variable and of covariation between variables. [For the complete proceedings, see ED629884.]

Published
2020

2. The Spontaneous Emergence of Elementary Number-Theoretic Concepts and Techniques in Interaction with Computing Technology

Author

Kieran, Carolyn and Guzman, Jose

Abstract

We present in this paper five case studies of instrumental genesis of number-theoretic concepts and techniques involving multiples, divisors, and numerical decomposition. The pupils, who were 12 to 15 years old, used the multi-line screen display of the graphing calculator to explore numerical tasks related to the "Five steps to zero" problem. The paper begins with a mathematical task analysis of various techniques that could be used to determine divisibility by 9. Findings show that the use of the technological tool permits students of this age to spontaneously develop variants of these techniques within tasks that simultaneously afford the co-emergence of related conceptual notions. (Contains 3 footnotes and 6 figures.) [Additional support for the research described in this paper was provided by CONACYT of Mexico. For complete proceedings, see ED500858.]

Published
2003

3. Coordinating Invisible and Visible Sameness within Equivalence Transformations of Numerical Equalities by 10- to 12-Year-Olds in Their Movement from Computational to Structural Approaches

Author

Kieran, Carolyn and Martínez-Hernández, Cesar

Abstract

"They are the same" is a phrase that teachers often hear from their students in arithmetic and algebra. But what do students mean when they say this? The present paper researches the notion of sameness within algebraic thinking in the context of generating equivalent numerical equalities. A group of Grade 6 Mexican students (10- to 12-year-olds) was presented with tasks that required transforming the given numerical equalities in such a way as to show their truth-value. The students initially indicated this by calculating and demonstrating that the total was the same on both sides. When asked not to calculate, their approaches evolved into more structural transformations involving decomposition so as to arrive at an equality with the same expression on each side. Students used the language of sameness--both visible and invisible--to describe the truth-value of their transformed expressions and equalities. The visible sameness referred to the resulting identical form of each side of the equality and the invisible sameness to the top-down equivalences that they had generated by their decomposing transformations--both types of sameness being characterized by their hidden numerical values. These findings suggest implications for transitioning to the algebraic domain of equations with their similarly hidden values.

Published
2022
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4. Negative Numbers in the Teaching of Arithmetic. Repercussions in Elementary Algebra.

Author

Gallardo, Aurora

Abstract

This article reports the results of a questionnaire applied to secondary school students (n=35) to explore efficiency in the resolution of equations in the domain of whole numbers and the spontaneous responses to problems leading to negative solutions. The most significant results obtained with the questionnaire are the lack of knowledge of the double use of brackets in arithmetic expressions, the partial comprehension of the operation of subtraction, and the difficulty in the operativity of expressions with double signs. The conclusions of this research suggest recommendations for the teaching of whole numbers. Contains 13 references. (Author/MKR)

Published
1995

5. Exploring Mathematical Connections of Pre-University Students through Tasks Involving Rates of Change

Author

Dolores-Flores, Crisólogo, Rivera-López, Martha Iris, and García-García, Javier

Abstract

This paper reports the results of a research exploring the mathematical connections of pre-university students while they solving tasks which involving rates of change. We assume mathematical connections as a cognitive process through which a person finds real relationships between two or more ideas, concepts, definitions, theorems, procedures, representations or meanings or with other disciplines or the real-world. Four tasks were proposed to the 33 pre-university students that participated in this research; the central concept of the first task is the slope, the last three tasks contain concepts like velocity, speed and acceleration. Task-based interviews were conducted to collect data and later analysed with thematic analysis. Results showed most of the students made mathematical connections of the procedural type, the mathematical connections of the common features type are made in smaller quantities and the mathematical connection of the generalization type is scarcely made. Furthermore, students considered slope as a concept disconnected from velocity, speed and acceleration.

Published
2019
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6. Arithmetic/Algebraic Problem-Solving and the Representation of Two Unknown Quantities

Author

Filloy, Eugenio, Rojano, Teresa, and Solares, Armando

Abstract

We deal with the study of the senses and the meanings generated in the representation of the unknowns in the resolution of word problems involving two unknown quantities. The discussed cases show the difficulties that the students beginning the algebra learning have to deal with when using the equality between "unknown things". For them, applying the equality transitivity property between different (but equivalent) algebraic expressions, or replacing an unknown quantity with its representation in terms of another one is not derived from an extension of the transitivity between numerical equalities or from the numerical substitution. This may have important implications in the algebra problem-solving teaching domain, in which it is usual to take for granted that students spontaneously transfer these numerical issues to the algebraic realm. [For complete proceedings, see ED489632.]

Published
2004

7. Uso del CAS (Computer Algebra Systems) Como Manipulador Simbólico en la Resolución de Problemas Verbales Aritmético-Algebraicos.

Author

Esparza, Miguel Benito Huesca

Subjects
ALGEBRA, ARITHMETIC, COMPUTERS, SCHOOLS
Abstract

Este documento reporta observaciones al utilizar el CAS como manipulador simbólico, para resolver problemas verbales aritmético-algebraicos. El CAS se utiliza como resolutor sintáctico que hace que el alumno se concentre en la parte semántica del problema verbal aritmético-algebraico y reflexione sobre lo que tiene en la pantalla de la calculadora con CAS, para explorar, descubrir y manipular cosas que desconocía. ..PAT.-Unpublished Manuscript [ABSTRACT FROM AUTHOR]

Published
2006