Aiming to enhance understanding of visual obstacles inherent in two-dimensional (2-D) sketches used in high school spatial geometry instruction, we propose a measure of visual difficulty based on two attributes of the sketches: potentially misleading geometrical information (PMI) and potentially helpful geometrical information (PHI). The…

This article presents results of a research study. Sixty-one first graders' responses to interview questions about negative integer values and order and directed magnitudes were examined to characterize the students' mental models. The models reveal that initially, students overrelied on various combinations of whole-number principles as…

College students (n=32) in their first abstract algebra course were asked to list elements and find the product of 2 elements in the dihedral group D4. Most students found solutions using a combination of visual approaches and analytic strategies. Proposes an alternative model, the Visualizer/Analyzer model, which assumes visualization and…

Demonstrates how research-based knowledge of students' incompatible solutions to various representations of the same problem could be used to raise their awareness of inconsistencies in their reasoning. Reports that students' decisions as to whether two given infinite sets have the same number of elements depend on the specific representation of…

Defines spatial structuring as the mental operation of constructing an organization or form for an object/set of objects. Examines in detail students' structuring and enumeration of two-dimensional rectangular arrays of squares. Concludes that many students do not see row-by-column structure. Describes various levels of sophistication in students'…

Explores the role of spatial orientation skill in the solution of mathematics problems. Reports the role of estimating the approximate magnitude of a figure; demonstrating the flexibility to change; adding marks to show relationships; moving or assessing the size and shape of figure; and getting the correct answer. (YP)

Focuses on the strategies young children (N=115) use to solve rectangular covering tasks before they have been taught area measurement. Children's solution strategies were classified into five developmental levels. Emphasizes the importance of understanding the relationship between the size of a unit and the dimensions of the rectangle in learning…

Mathematics educators frequently recommend that students use strategies for measurement estimation, such as the reference point or benchmark strategy; however, little is known about the effects of using this strategy on estimation accuracy or representations of standard measurement units. One reason for the paucity of research in this area is that…

One group of elementary education majors received an algebraic instructional treatment of a unit on linear absolute value equations while another received a geometric treatment. The hypothesis of an aptitude-by-treatment interaction was statistically confirmed. (MN)

Presents a study in which children in four grade 2 classrooms learned about transformational geometry and symmetry by designing quilts, while their four teachers participated in professional development focused on understanding children's thinking in arithmetic, and two of the teachers participated in additional workshops on students' thinking…

Examines preschool children's understanding of the mathematical significance of unit size through problems that involve making judgments about the number of larger objects versus the number of smaller objects. Elicits children's judgments about the effects of object size both before and after the presentation of a series of demonstration trials.…

In the study reported here, I examined the diagrams that mathematically capable high school students produced in solving applied calculus problems in which a diagram was provided in the problem statement. Analyses of the diagrams contained in written solutions to selected free-response problems from the 1996 BC level Advanced Placement Calculus…

Utilizes the psychological and sociocultural components of a constructivist paradigm to provide a detailed analysis of how the cognitive constructions students make as they enumerate 3D arrays of cubes develop and change in an inquiry-based, problem-centered mathematics classroom. Describes the classroom work of three pairs of 5th-grade students…

Investigates gender differences and the role of spatial visualization in problem solving in high school geometry. Reports that, whereas males and females differed in spatial visualization and in their performance, they did not differ in logical reasoning ability or in their use of geometric problem-solving strategies. (Author)