Variables versus numbers: Effects of symbols and algebraic knowledge on students’ problem-solving strategies

Abstract

To efficiently solve mathematical expressions and equations, students need to notice the systemic structure of mathematical expressions (e.g., inverse relation between 3 and − 3 in 3 + 5 – 3). We examined how symbols—specifically variables versus numbers—and students’ algebraic knowledge impacted seventh graders’ problem-solving strategies and use of systemic structures within an online algebra game where students could dynamically transform expressions. We found that on simple problems, symbols did not impact students’ strategy efficiency, although students with higher vs lower algebraic knowledge were more efficient at solving these problems. On complex problems, students with higher algebraic knowledge were more efficient at solving variable vs numerical problems, whereas students with lower algebraic knowledge were less efficient at solving variable vs numerical problems. The visualizations and examination of first steps further revealed that whether students leveraged systemic structures during their problem-solving varied across problems. The findings have implications for research on cognitive processes of symbols and practices on teaching mathematics.