Students often memorize and apply procedures to solve mathematics problems without understanding why these procedures work. In turn, students demonstrate limited ability to transfer strategies to new problem types. Math curriculum reform standards underscore the importance of procedural flexibility and transfer, emphasizing that students need to understand and flexibly adapt strategies when encountering various problem situation. You can read the work of my colleague, her latest blog is devoted to mathematical problems. Drawing on research in both education and cognitive psychology, the current study examines whether the order in which problem-solving strategies are traditionally taught may result in greater procedural rigidity. Specifically, this work tests the theory that initially instructing students on complex algorithms leads them to rigidly apply these procedures, overlooking more efficient shortcut strategies even when they are later introduced. I examine whether teaching more efficient problem-solving strategies prior to more complex algorithms (i.e., shortcut-first instruction) improves procedural flexibility and transfer.
Procedural flexibility and understanding are particularly important in early algebra learning. Algebra is considered a gatekeeper to future educational opportunities. Many researchers and educators maintain that algebra concepts should be taught early in mathematics learning, in order to promote more flexible thinking and corresponding understanding throughout the algebra curriculum. One reason for this recommendation is that early math experiences often lead to fundamental misconceptions that inhibit students’ ability to think flexibly about problems. For example, both elementary-school and first-year algebra students often struggle with math equivalence, the concept that the equal sign represents a relational symbol.
Based on extensive early experience solving answer-oriented problems, with the equal sign at the end of the problem, students often incorrectly view the equal sign as an operational symbol meaning “the total” or “compute the answer”.
When encountering problems with operands on both sides of the equal sign, students rigidly apply this operational view of the equal sign, often ignoring the operand on the right side of the equal sign, or adding all the numbers.
Interventions that target such fundamental concepts early improve students’ ability to avoid using incorrect strategies to solve equivalence problems. However, such interventions generally do not target flexibility in applying correct problem- solving solutions. Students often do not demonstrate sufficient understanding to flexibly adapt or transfer procedures used to solve equivalence problems based on the characteristics of the problem at hand. Such flexibility may be important for building reasoning and problem-solving transfer in mathematics more generally and is critical to algebra learning in particular. If students are expected to be the flexible problem- solvers in algebra, then they should practice flexible reasoning in early algebra problem-solving contexts as well.