This study focuses on proportional division, a specific problem type of proportions and proportional relationships, which embeds a part-part-whole relationship. We investigated how students in grades 6 to 9 solved and perceived three proportional division problems. Analysis of the students' performance on the tasks revealed three types of…

The purpose of this study was to gain insight into 30, first year calculus students' understanding of the relationship between the concept of vertex of a quadratic function and the concept of the derivative. APOS (action-process-object-schema) theory was applied as a guiding framework of analysis on student written work, think-aloud and follow up…

We have used "GeoGebra," a dynamic geometry software environment, to explore how Year 4 students understand definitions of angles. Seven students defined angle and then completed several activities adapted for the dynamic environment. Afterward, students again shared their definitions of angles. We found that even a short investigation…

Instruction promoting reasoning and proof should be a part of K-12 mathematics teaching. Preservice teachers (PSTs) need mathematics content instruction that helps them construct valid justifications for mathematical notions and develop positive perceptions of proof. This case study describes instruction with proof-related tasks and manipulatives…

Two studies were conducted to examine and compare the construct validity of scores on the Junior Metacognition Awareness Inventory (JMAI) and problem-solving interview protocols. Participants consisted of 183 middle and high school students attending a university summer program for academically talented youth. Study 1 results indicated that JMAI…

Considerable research suggests that standardizing curricular materials can undermine teachers' ability to facilitate learning. Nonetheless, U.S. school districts' curriculum guides have become increasingly standardized, with teachers being required to follow predetermined pacing calendars indicating which standards to teach during specific months,…

Within the current test-centered educational reform movement, considerable emphasis is placed on using assessment results to make instructional decisions for individual students. Test scores are used to estimate a student's current level of skill, monitor his or her progress during instruction, and identify whether the student has gained the…

Existing research from Western countries has consistently shown that the learning environment is an important factor for academic motivation. Self-determination theory, which serves as a theoretical foundation of the present study, suggests that teachers' involvement is an important element of students' learning, regardless of the cultural…

In the generative classroom, teachers provide well-designed learning environments that result in the combination, recombination, and reorganization of repertoires such that new untaught repertoires are likely to occur. One component that can contribute to such generativity is Precision Teaching (PT), a frequency building instructional…