Bridging the divide: Comparing the efficiency and transparency of two division algorithms.

Teaching students to use mathematical algorithms is a major part of standard math curricula, but there is long-standing debate about how-or even if-algorithms should be taught.Division is a fundamental operation for which students regularly use algorithms, the most popular being traditional long division (TLD).Although regarded as highly efficient, TLD is conceptually opaque and challenging to learn.The partial quotients (PQ) algorithm has been proposed as a more understandable alternative to TLD, but no study has directly compared their efficiency or transparency.In this study, we investigated how well fourth-and fifth-grade students performed and understood the rationale for TLD and PQ.Students (n 5 65) received a brief lesson on either TLD or PQ and then solved seven problems using that algorithm.Problems varied by dividend size (small 5 three digits, large 5 four to five digits).Afterward, students explained the conceptual rationale behind their assigned algorithm during a guided interview.Problem-solving performance was high overall.Regression analyses controlling for pretest performance did not show differences in accuracy between algorithms, but showed an advantage in speed for large-dividend problems favoring TLD.Interview scores indicated that PQ was more easily understood than TLD, t(25) 5 2.97, p 5 .005,Cohen's d 5 0.85.Accordingly, students made more algorithm-based errors and more extreme errors when using TLD, whereas students using PQ made more computational and less extreme errors.These results suggest that incorporating PQ into elementary mathematics instruction may enhance students' understanding of division. Educational Impact and Implications StatementThis study suggests that elementary students heavily struggle to understand why the traditional long division algorithm works, but they are better able to understand why the partial quotients algorithm (an alternative division procedure) works.When solving division problems, students are comparably accurate in using either algorithm.These results suggest that teaching partial quotients may allow students to more deeply understand a division algorithm without sacrificing efficiency.