For many students, algebraic formulas are abracadabra: they lack symbol sense. Symbol sense includes identifying the structure of algebraic formulas, giving meaning to them, and reasoning with and... Show moreFor many students, algebraic formulas are abracadabra: they lack symbol sense. Symbol sense includes identifying the structure of algebraic formulas, giving meaning to them, and reasoning with and about formulas. Besides basic skills, symbol sense is needed to solve algebraic problems and it is not known how to teach symbol sense systematically.In this research we have studied how teaching graphing formulas by hand (i.e. sketching a graph of a formula) could promote grade 11- and 12-students’ symbol sense. Our GQR-design (Graphing formulas through recognition and qualitative reasoning) is based on expert research and pays explicit attention to recognition of basic function families and features, and to qualitative reasoning, that focuses on the global shape of a graph, with global descriptions and ignoring what is not relevant. These aspects get little attention in regular education, that often focuses on algebraic manipulation.We found that students improved their insight into formulas and found a positive correlation between students’ abilities to graph formulas and their abilities to solve non-routine algebra problems with symbol sense. The students were able to use essential aspects of symbol sense, learned in the context of graphing formulas, such as taking a global view for recognition, qualitative reasoning, and questioning a formula, when solving algebra problems.In the future, when technology will take over the manipulation of algebraic formulas, symbol sense will become even more important. Graphing formulas could promote students’ symbol sense in upper secondary school, and therefore deserves a prominent place in mathematics curriculum. Show less
Kop, P.M.G.M.; Janssen, F.J.J.M.; Drijvers, P.H.M.; Driel, J.H. van 2020
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.... Show moreStudent 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. Show less
