Crossing the cognitive gap between arithmetic and algebra: Operating on the unknown in the context of equations

Abstract

The objective of the teaching experiment reported in this article was to overcome the "cognitive gap", that is, students' inability to spontaneously operate with or on the unknown. Following an analysis of the cognitive obstacles involved, this paper reports the results of an alternative approach. We designed an individualized teaching experiment which was tested in six case studies. In the first part the students' natural tendency to group singletons in the unknown within the equations was expanded to a process of grouping like terms. In the second part we introduced a reverse process to grouping like terms, that of decomposition of a term into a sum. This process, combined with the cancellation of identical terms, provides a procedure for the solution of first degree equations with the unknown on both sides of the equality sign. The last part of the teaching experiment involved the decomposition of an additive term into a difference. The first two parts proved very successful and the students developed procedures on their own that were more efficient than the initial ones. The results of the third part, however, revealed the lirnits of this approach. The students experienced difficulties in choosing the required decomposition. It seems that some of these obstacles are rather robust and perhaps should not be dealt with incidentally but should be addressed as part of a pre-algebra course. PREFACE

Grade 9 Learners’ Structural and Operational Conceptions of the Equal Sign: A Case Study of a Secondary School in Soshanguve

“…Much thought and research have gone into exploring learners' understanding of the equal sign in elementary and middle schools (Essien & Setati, 2006Hattikudur & Alibali, 2010Hohensee, 2017, Jones, Inglis, Gilmore, & Dowens, 2012Stephens, Knuth, Blanton, Isler, Gardiner, & Marum, 2013). Linchevski and Herscovics (1996) argue that teachers and learners should be made aware of the demarcation between arithmetic and algebra. In other words, teachers should be able to understand the gap between arithmetic and algebra to enhance learners' readiness to learn secondary school mathematics.…”

“…In other words, teachers should be able to understand the gap between arithmetic and algebra to enhance learners' readiness to learn secondary school mathematics. The gap between arithmetic and algebra may explain learners' lack of readiness and the poor results achieved in algebra (Linchevski & Herscovics, 1996).…”

Designation, Implementation and Evaluation of Activities to Ensure Transition from Arithmetic to Algebra

“…Aritmetikten cebire geçişin tam olarak sağlanması durumunda, öğrencilerin değişken ve denklem kavramını anlamada (Arcavi & Schaenfeld, 1988, Ursini & Trigueros, 2001, denklemlerin çözüm kümelerini bulabilmede (Hersovics & Linchevski, 1994;MacGragor & Stacey, 1997) ve cebirsel sözel problemleri öğrenmede (Kieran, 1991;Londholz, 1993, Linchevski, 1995 (Akkan, 2009;Linchevski & Herscovics, 1996;Cortes & Pfaff, 2000;Lee, 2002;Dede, Yalın ve Argün, 2002;Van Dooren, Verschafel ve Onghena, 2003;Vlasis, 2004;Dede & Peker, 2007). Genelleme yapma sürecinde örüntü ve dört işlem özelliklerinde (değişme özelliği, birleşme özelliği gibi), sembollerin kullanımı sürecinde (eşittir işareti, parantez kullanımı vb.…”

La comprensión del signo de igual en la entrada al álgebra: el diseño de tareas y la conversación en la clase de matemática

“…Varios estudios reportan ciertas dificultades de los estudiantes en el álgebra relacionadas con la comprensión del signo de igual (Kieran, 1981;Rojano y Gallardo, 1988;Sfard y Linchevski, 1994;Linchevski y Herscovics, 1996;Knuth, Alibali, McNeil, Weinberg y Stephens, 2011;Burgell y Ochoviet, 2015). En la mayoría de estos trabajos se aborda la problemática en un contexto aritmético e incipientemente en un contexto algebraico de ecuaciones.…”