Distinguishing multiplication principle problems and permutation problems in the classroom helps us examine common student errors.
Mary C. Caddle and Bárbara M. Brizuela
David W. Carraher, Analúcia D. Schliemann, Bárbara M. Brizuela and Darrell Earnest
Algebra instruction has traditionally been postponed until adolescence because of historical reasons (algebra emerged relatively recently), assumptions about psychological development (“developmental constraints” and “developmental readiness”), and data documenting the difficulties that adolescents have with algebra. Here we provide evidence that young students, aged 9–10 years, can make use of algebraic ideas and representations typically absent from the early mathematics curriculum and thought to be beyond students' reach. The data come from a 30-month longitudinal classroom study of four classrooms in a public school in Massachusetts, with students between Grades 2–4. The data help clarify the conditions under which young students can integrate algebraic concepts and representations into their thinking. It is hoped that the present findings, along with those emerging from other research groups, will provide a research basis for integrating algebra into early mathematics education.
Maria Blanton, Bárbara M. Brizuela, Angela Murphy Gardiner, Katie Sawrey and Ashley Newman-Owens
The study of functions is a critical route into teaching and learning algebra in the elementary grades, yet important questions remain regarding the nature of young children's understanding of functions. This article reports an empirically developed learning trajectory in first-grade children's (6-year-olds') thinking about generalizing functional relationships. We employed design research and analyzed data qualitatively to characterize the levels of sophistication in children's thinking about functional relationships. Findings suggest that children can learn to think in quite sophisticated and generalized ways about relationships in function data, thus challenging the typical curricular approach in the lower elementary grades in which children consider only variation in a single sequence of values.