Early Algebraic Thinking
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1. Dr. Rittle-Johnson is interested in evaluating the role of explanation in mathematics learning--specifically "why" explanations, or reasoning about why something is true. Such explanations can be generated by the student to explain new content or provided by the instructor. Although explanations are often used as a learning tool, many factors can determine whether explaining will actually benefit learning, such as the source and timing of these explanations and individual differences such as working memory capacity. Her research group is performing a series of studies to uncover these factors. In doing so, they hope not only to reveal the learning mechanisms that make explanation helpful, but they will also use this knowledge to inform mathematics instruction in the classroom.
Their studies on explanations are exploring a number of important questions:
- How much does self-explanation actually help?
- Which is a better use of time, explaining or practicing extra problems?
- Who should explain, and when?
- When should students generate their own explanations, and when should instructors explain mathematical concepts for them?
- Does formal instruction matter?
- Is the role of self-explanation similar in young children before formal mathematics instruction begins and in elementary school after formal instruction has begun (but before formal instruction on justification and proof)?
- Do some students benefit from explanations more than others?
- Do students with more or less prior knowledge or working memory capacity benefit more or less from explanation?
2. To explore these ideas, they are investigating knowledge development in three early algebra topics: mathematical equivalence (Grades 2 - 6), functions (Grades 2 - 6) and patterns (preK). Algebra is no longer a course saved for high school; rather, fundamental components of algebraic thinking have been identified as focal points for instruction beginning in preschool (NCTM, 2006). The hope is that inculcating algebraic thinking in the early grades will reduce students' difficulty with algebra in middle and high school.
Mathematical equivalence, often represented by "=," is the principle that two sides of an equation represent the same amount. However, elementary school children often fail to understand equivalence, often interpreting the equals sign as simply an operator signal that means "get the answer"(Baroody & Ginsburg, 1983; Kieran, 1981; Rittle-Johnson & Alibali, 1999; Sfard & Linchevski, 1994).
Functions represent the relation between two varying quantities, such as f(x) = 3x. Functional thinking is one of the core strands of Kaput's (2008) framework of algebraic reasoning and a core expectation for mathematics curriculum. Elementary-school students often focus on particular numbers as outcomes. Finding a functional relationship between two sets of numbers is a way to jump from thinking of particulars to sets (Carrahar et al. 2008).
Repeating patterns (e.g., ABBABB patterns) are often children's first experiences with patterns, and creating patterns is one of the most common mathematical activities in which preschoolers spontaneously engage (Seo & Ginsburg, 2004). The National Council of Teachers of Mathematics has recommended that identifying, duplicating, and extending sequential patterns be a focal point of instruction in pre-K and kindergarten (NCTM, 2006). In our studies, we focus on identifying the pattern unit (e.g., ABB) because this is an early form of generalization - it is the rule underlying the pattern.