High School Student Representational Adaptivity and Transfer in Multiplying Polynomials
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While there is an extensive amount of research on representations for solving problems involving functions, there are few studies on high school student use of multiple representations for multiplying polynomials. This study contributes to current mathematics education literature by focusing on the appropriateness of high school student choices of representations for multiplying polynomials and the extent of their transfer of knowledge from multiplication of integers. Study participants are 85 students enrolled in four high school algebra classes in which multiplication of polynomials is covered and the teacher has been observed to encourage students to use multiple methods for problem solving. Choice/no-choice assessments were administered to determine student representational fluency and adaptivity with standard distribution, lattice and place value multiplication of polynomials. Semi-structured task-based interviews were also conducted with ten students from the study to examine student choices of representation for multiplying polynomials and components transferred from integer multiplication. The results of generalized estimating equations for ordinal logistic regression reveal that students are more likely to accurately use lattice than standard distribution to obtain accurate solutions for polynomial multiplication tasks. Students also tended to transition from choosing standard distribution to the lattice as the number of terms in the polynomials to be multiplied increased, stating that standard distribution was more efficient for solving simple multiplication tasks while the lattice made it easier to organize multiplication of polynomials with many terms. Students selected place value for fewer choice assessment polynomial multiplication tasks than standard distribution, and though they used it to obtain nearly as many accurate solutions on the no-choice assessment, the place value representation was used less accurately as many students forgot to align place values in the factors. However, more students interviewed recognized a larger number of symbolic components transferred from integer multiplication to place value multiplication of polynomials than the other two representations. The value of teaching polynomial multiplication with multiple representations is then introducing students to adaptive choices for differing tasks and preferences as well as relating the representations to familiar integer multiplication.