Promoting insight into algebraic formulas through graphing by hand

Abstract

Student insight into algebraic formulas, including the ability to identify the structure of a formula and its components and to reason with and about formulas, is an issue in mathematics education. In this study, we investigated how 16-and 17-year-old pre-university students' insight into algebraic formulas can be promoted through graphing formulas by hand. In an intervention of five 90-min lessons, 21 grade 11 students were taught to graph formulas by hand. The intervention's design was based on experts' strategies in graphing formulas, that is, using a combination of recognition and qualitative reasoning, and on principles of teaching complex skills. To assess the effect of this intervention, pre-, post-, and retention tests were administered, as well as a post-intervention questionnaire. Six students were asked to think aloud during the pre-and posttests. The results show that all students improved their abilities to graph formulas by hand. The think-aloud data suggest that the students improved both on recognition and reasoning, and give a detailed picture of how students used recognition and qualitative reasoning in combination. We conclude that graphing formulas by hand, based on the interplay of recognition and qualitative reasoning, might be a means to promote students' insight into algebraic formulas.

The relation between graphing formulas by hand and students’ symbol sense

“…In discussing the findings, we note that the results of this study seem to confirm earlier research about the problems Dutch students have with algebra: students have problems graphing formulas by hand (Kop et al, 2020) and with identifying and using the structure of algebraic expressions (Van Stiphout, Drijvers, & Gravemeijer, 2013). Regular teaching of algebra does not seem to develop these aspects of symbol sense.…”

“…This suggests that teaching symbol sense in the domain of graphing formulas by hand might be an effective means to teach essential aspects of symbol sense involved in solving non-routine algebra tasks. In a previous study, we showed how to teach graphing formulas by hand, using these essential aspects of symbol sense (Kop et al, 2020).…”

“…However, it is not clear how to teach symbol sense appropriately (Arcavi, 2005;Hoch & Dreyfus, 2005). In a previous study, we showed how teaching graphing formulas by hand, defined as graphing using recognition and reasoning, without technology, to grade 11 students improved their insight into algebraic formulas (Kop, Janssen, Drijvers, & Van Driel, 2020). Insight into algebraic formulas is an aspect of symbol sense and involves recognizing structure and key features of a formula and qualitative reasoning with and about a formula.…”

Pre-Service Teachers’ Strategies in Solving Absolute Value Equations and Inequalities

“…In line with other studies [13,27,28], the difficulties include drawing absolute value function graphs without using graphing software and determining x values intervals from the graphs that are satisfying the absolute value inequalities. The difficulties dealing with absolute value function graphs show a lack of symbol sense, i.e., a lack of ability in recognizing function families and graph features from the structure of functions [22,29]. In other studies [30,31], the ability to switch from symbolic algebra to geometrical ideas, showing versatile thinking of different representations, which also shows symbol sense behavior, concerns the ability to intertwine between mathematical topics.…”

Das Fach Mathematik in der gymnasialen Oberstufe

“…Although school textbooks aim to pay attention to mathematical thinking, there is not much explicit instruction on symbol sense. In order to systematically teach symbol sense, Kop (2020) and Kop et al (2021) have suggested teaching students in grades 11 and 12 to graph formulas based on recognition and reasoning.…”