The objective of this study was to evaluate and characterize the argumentation patterns used by seventh-grade students in the context of algebraic addition and subtraction problems. A qualitative case study was conducted using the Claim-Evidence-Reasoning (CER) model (McNeill & Krajcik, 2008) and argument quality framework. To describe the…

This study explores learning vector addition and demonstrating the understanding of vector addition. Vector addition is a key skill in mathematics and science education, but there is a limited amount of research that focuses on this topic. My study investigates how an instructional design framework "concreteness fading" can be used for…

Teacher-directed and self-directed learning have been compared across various contexts. Depending on the settings and the presentation of material, mixed benefits are found; the specific circumstances under which either condition is advantageous are unclear. We combined and reanalyzed data from two experimental studies investigating the effects of…

This paper presents an analysis of fractions errors displayed by learners due to deficient mastery of prerequisite concepts. Fractions continue to pose a critical challenge for learners. Fractions can be a tricky concept for learners although they often use the concept of sharing in their daily lives. 30 purposefully sampled learners participated…

In the present research, we provide empirical evidence for the process of symbolic integration of number associations, focusing on the development of simple addition (e.g., 5 + 3 = 8), subtraction (e.g., 5 - 3 = 2), and multiplication (e.g., 5 × 3 = 15). Canadian children were assessed twice, in Grade 2 and Grade 3 (N = 244; 55% girls). All…

Knowledge representations that result from practicing problem solving can be expected to differ from knowledge representations that emerge from explicit verbalizing of principles and rules. We examined the degree to which the two types of learning improve problem-solving knowledge and verbal explanation knowledge in classroom instruction. We…

Whole number arithmetic is the foundation of higher mathematics and a core part of elementary mathematics. Awareness of pattern and underlying problem structure promote the learning of whole number arithmetic. A growing consensus has emerged on the necessity to provide students with the opportunity to engage in algebraic reasoning earlier in their…

This is the second of a two-part article that presents a theory of unit construction and coordination that underlies radical constructivist empirical studies of student learning ranging from young students' counting strategies to high school students' algebraic reasoning. In Part I, I discussed the formation of arithmetical units and composite…

In the domain of "Operations & Algebraic Thinking," Common Core State Standards indicate that in kindergarten, first grade, and second grade, children should demonstrate and expand their ability to understand, represent, and solve problems using the operations of addition and subtraction, laying the foundation for operations using…

The attainment and retention of later algebra skills in high school has been identified as a factor significantly impacting the postsecondary success of students majoring in STEM fields. Researchers maintain that learners develop meaning for algebraic procedures by forming connections to the basic number system properties. The present study…

This interactive electronic report provides an overview of an innovative new instrument developed by CPRE researchers to authentically measure teachers' formative assessment practices in mathematics. The Teacher Analysis of Student Knowledge, or TASK, instrument assesses mathematics teachers' knowledge of formative assessment and learning…

High academic standards are an important first step in ensuring that all Alaska's students have the tools they need for success. These standards reflect the collaborative work of Alaskan educators and national experts from the nonprofit National Center for the Improvement of Educational Assessment. Further, they are informed by public comments.…