Early algebra and mathematical generalization

Abstract

We examine issues that arise in students' making of generalizations about geometrical figures as they are introduced to linear functions. We focus on the concepts of patterns, function, and generalization in mathematics education in examining how 15 third grade students (9 years old) come to produce and represent generalizations during the implementation of two lessons from a longitudinal study of early algebra. Many students scan output values of f(n) as n increases, conceptualizing the function as a recursive sequence. If this instructional route is pursued, educators need to recognize how students' conceptualizations of functions depart from the closed form expressions ultimately aimed for. Even more fundamentally, it is important to nurture a transition from empirical generalizations, based on conjectures regarding cases at hand, to theoretical generalizations that follow from operations on explicit statements about mathematical relations.

Mathematics Intervention Supporting Allen, a Latino EL: A Case Study

“…These findings, suggesting language features as a means to provide Allen access to these algebraic tasks, echo findings from Carraher et al's [22] study. Essentially, Carraher et al [22] found that student developed conjectures as a mean to provide students explicit statements of mathematical relationships.…”

“…Recent findings suggest that young students are capable of constructing algebraic generalizations, paying attention to the explicit language and use of conjectures in solving operation problems [19][20][21][22]. Students as young as preschool-age have been found capable of transitioning their everyday experiences to mathematical experiences when developing an additive-subtractive inversion principle [19].…”

“…These studies show that educators need to more explicitly bridge empirical generalizations around young students' use of conjectures towards abstract generalizations [22]. Therefore, ELs lacking "typical" experiences and necessary English language levels of proficiency in which to access school Algebra may be limited when developing abstract generalizations about mathematics relationships.…”

Um modelo teórico de Matemática para o Ensino do Conceito de Função a partir de realizações em livros didáticos <br> A theoretical model of Mathematics for Teaching of the concept of function from realizations in textbooks

“…Compõem esse panorama as realizações que comunicam o conceito de função como um texto que descreve uma regra (funcional) para determinar o valor de um elemento de uma posição arbitrária em uma sequência, com base no conhecimento dos seus elementos iniciais (CARRAHER; MARTINEZ; SCHLIEMANN, 2008). A construção e a validação dessa regra não são baseadas em uma inferência formal, ou seja, não são fundamentadas na realização de uma prova (demonstração), trata-se de processo indutivo "informal" que é legitimado como uma forma de argumentação no contexto da Escola Básica.…”

Neurodevelopment of relational reasoning: Implications for mathematical pedagogy

“…A different approach to early algebra, called the 'functional approach', uses numeric or geometric patterns to explore the relationships between quantities, variables and representations. One classic example is figuring out how many people could sit at restaurant tables made up of one, two, three or nine square tables pushed together [50]. Students readily understand the function and can iteratively calculate the number of people in each case.…”