Sequential development of algebra knowledge: A cognitive analysis

Abstract

Learning to operate algebraically is a complex process that is dependent upon extending arithmetic knowledge to the more complex concepts of algebra. Current research has shown a gap between arithmetic and algebraic knowledge and suggests a pre-algebraic level as a step between the two knowledge types. This paper examines arithmetic and algebraic knowledge from a cognitive perspective in an effort to determine what constitutes a pre-algebraic level of understanding. Results of a longitudinal study designed to investigate students' readiness for algebra are presented. Thirty-three students in Grades 7, 8, and 9 participated. A model for the transition from arithmetic to pre-algebra to algebra is proposed and students' understanding of relevant knowledge is discussed.The transition from concrete and numerical foundations of arithmetic to the symbolic and abstract reasoning of algebra has been conceptualised as a didactic cut by Filloy and Rojano (1989) and as a cognitive gap by Herscovics and Linchevski (1994). A consequence of this cut/gap has been recognition of the need for an operational level of pre-algebraic knowledge between arithmetic and algebra. This study proposes a model that depicts the transition from arithmetic to algebra and includes an operational level of pre-algebra. However, instead of explaining this transition as a cut/gap, it is explained in terms of a sequence of development from arithmetic to algebraic knowledge.

Difficulties in initial algebra learning in Indonesia

“…This seems to be caused by students' lack of understanding of the meaning of the equal sign as algebraic equivalence (Herscovics and Linchevski 1994;Ketterlin-Geller et al 2007;Kieran 1981;Linchevski 1995;Pillay et al 1998).…”

“…Therefore, in this study, we focus on linear equations in one variable, and the related linear inequalities. As many researchers (e.g., Herscovics and Linchevski 1994;Linchevski and Herscovics 1996;Pillay et al 1998) have addressed linear equations in one variable to comprehend students' learning and thinking in the transition from arithmetic to algebra and students' capability to understand and use variables in particular, this seems an appropriate topic to further elaborate on.…”

“…In arithmetic, the equal sign often invites carrying out a calculation and writing down a numerical answer, whereas in algebra, it usually means "is algebraically equivalent to" (Filloy and Rojano 1989;Herscovics and Linchevski 1994;Ketterlin-Geller et al 2007;Kieran 1981;Linchevski 1995;Pillay et al 1998). With the former insight, students may interpret 2+3 =.... as adding 2 and 3 to get the specific answer 5 and may not view 2+3=3+2; 2+3=1+4; or 5=2+3 as possible solutions to the same task.…”

“…Also, students (11-14 year olds) misapply commutative as well as associative properties when carrying out subtractions or divisions (Booth 1988;Pillay et al 1998;Warren 2003), and fail to use the distributive property of a multiplication over an addition (Booth 1988;Pillay et al 1998). In our view, these difficulties reveal students' limited mastery of addition, subtraction, multiplication, and division; of applying the priority rules of arithmetic operations in calculations; and of using properties of numerical operations.…”

El Álgebra Elemental en la Escuela Secundaria. Un análisis desde la Teoría Antropológica de lo Didáctico

“…Los procesos de enseñanza y de aprendizaje del álgebra en distintos niveles educativos han sido motivo de numerosas investigaciones en didáctica de la matemática [10,11,14,13,7,9]. En general, los investigadores han tratado de buscar respuestas a los principales interrogantes en torno a la naturaleza del álgebra y a los procesos de pensamiento implicados, que permitan a los alumnos construir significados para los símbolos algebraicos y para su manipulación.…”