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…

Walk into any primary grade music class in the U.S., and you will likely hear teacher and students singing a musical greeting, such as "Good morning boys and girls" (sol-mi-mi-sol-sol-mi) and the response "Good morning Miss Purdy" (sol-mi-mi-sol-mi-mi). Since about the 1970s, teachers have been beginning and ending music class for young children…