The project focuses on the development of conceptual and procedural knowledge for two core algebra topics in elementary school: mathematical equivalence and patterns. First, I will examine and improve the reliability and validity of measures of conceptual and procedural knowledge for each topic. Then, I will evaluate how self- and instructional explanations support growth of each type of knowledge. The overall goals are to develop an information-processing framework for the iterative development of conceptual and procedural knowledge and to provide guidance on how to use explanations as instructional tools. I will also teach an undergraduate course on mathematical cognition.
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). Inculcating algebraic thinking in the early grades is thought to reduce students' difficulty with algebra in middle and high school. We investigated knowledge development in three early algebra topics: mathematical equivalence (Grades 2 - 6), functions (Grades 2 - 6) and patterns (preK). First, we developed an assessment for each topic so we could accurately test children’s knowledge and how it changed, and we rigorously tested the quality of each. These high-quality assessments are valuable tools for researchers and teachers to better gauge children’s early algebraic thinking. Next, we evaluated several instructional approaches for helping children learn about early algebra, based on evidence from basic cognitive science. These studies revealed that: 1) Prompting children to generate their own explanations can improve both understanding and problem-solving success, but a substantial number of children struggle to provide relevant explanations and do not learn more when prompted to explain; 2) Delaying instructional explanations until after students have an opportunity to explore related problems can lead to greater learning because learners are better prepared to learn from the explanations. However, delaying instruction was not always beneficial, especially if students missed opportunities to generate their own explanations after receiving instructional explanations; 3) Including instruction on solution procedures in conjunction with instructional explanations harmed learning relative to instructional explanations alone (that focused only on concepts); 4) Using abstract language in explanations can support better problem-solving than concrete language; 5) Accuracy feedback during problem solving aids learning for low-knowledge learners, but accuracy feedback can harm learning for students with modest prior knowledge. 5) Across these studies, we have found three characteristics of the learner that impact learning under different conditions – prior knowledge, working memory capacity (i.e., the amount of information children can actively think about at one time) and achievement motivation (e.g,, motivation to master mathematics content). Attention to prior knowledge is particularly important because instructional approaches that are most effective with students with little prior knowledge may be ineffective or even harmful for students in the same grade-level with moderate prior knowledge. The intellectual merit of this work includes advancements in theories of learning, including how direct instruction and self-generated discoveries work together and how characteristics of the learner impact the learning process. It has also led to assessments that can be used in future research on instructional intervention and experimental evaluations of learning processes. The broader impact of this work includes materials and methods for helping preschool and elementary school children develop their algebraic thinking, which is critical for their success in mathematics as well as in life outcomes, such as college attendance and salary. We helped close to a thousand children in our local community understand important mathematics topics and provided high-quality instructional ideas and materials to many local teachers. A course in how children learn math was also developed and implemented on two occasions with prospective teachers and researchers.
