Problem solving

This thesis presents some results in triangle geometry discovered using dynamic
software, namely, Geometer’s Sketchpad, and confirmed with computations using
Mathematica 9.0. Using barycentric coordinates, we study geometric problems associated
with the triangle of reflections T of a given triangle T, yielding interesting triangle
centers and simple loci such as circles and conics. These lead to some new triangle
centers with reasonably simple coordinates, and also new properties of some known,
classical centers. Particularly, we show that the Parry reflection point is the common
point of two triads of circles, one associated with the tangential triangle, and another with
the excentral triangle. More interestingly, we show that a certain rectangular hyperbola
through the vertices of T appears as the locus of the perspector of a family of triangles
perspective with T, and in a different context as the locus of the orthology center of T
with another family of triangles.

The present study evaluated three instructional methods for solving algebra word problems. The instructions consisted of either an example, a set of procedures, or both the example and procedures. The procedures were rules designed to aid in solving word problems. The example was a worked-out solution to a word problem that was representative of the test problems. The subject's task was to study the instructional material and then generate the equations needed to solve the subsequent test problems. The test problems differed from the example by either 0, 1, 2, or 3 transformations. The results showed that subjects in the example and example-plus-procedure groups performed the best but the difference between these two groups was not significant. The procedures alone were relatively ineffective. A mathematical model is proposed to explain how the performance was influenced by the three types of instructional material and four levels of transformation.

The present study investigated how students in a college
algebra class select potentially useful solutions for
solving algebra word problems. Several variables, including
similarity, complexity, and inclusiveness were assessed to
determine how they influence subjects' selections. Results
indicated that neither complexity nor inclusiveness had a
significant impact on students' selections and that students
select solutions primarily on the basis of similarity.
Students' inability to select potentially useful solutions
was revealed by their failure to select the more inclusive
solution, even though their performance greatly improved
through the use of these solutions. The influence of
similarity on the selection of solutions and inclusiveness
on the effective use of solutions occurred when the
effective solutions had the same story context (Experiment
1) and when the effective solutions had a different story
context (Experiment 2).

Previous research on algebra word problems revealed that students
usually could not use a solution of a problem to solve a related
problem that required a slight modification of the solution. In an
attempt to improve performance on related problems, students were given
practice on either filling in tables or using completed tables.
Practice on entering values in tables did not improve performance
because students lacked the domain specific knowledge to correctly
enter the values. Students' use of completed tables resulted in
improved performance when tables were provided, but did not result in
an improvement on generalization problems in which no tables were
given. These findings were discussed within the context of a
hierarchical classification scheme which describes the relationship
between two related problems. In addition, the instructional
implications of the study were considered and suggestions are offered
for improving the instructional procedures.

This was an exploratory study of factors that predict individual differences in chemistry learning. Consistent with cognitive learning theory, working memory was assumed to be an important predictor of chemistry knowledge. Measures of chemistry affect, cognitive ability, demographics and mathematical ability were examined in relation to visual/schematic and algebra-like stoichiometry chemistry word problem solving ability and strategy use. 139 undergraduate students (91 females, 48 males) at a major Southeastern university participated in this study (Age ranged from 18 to 39 years (M = 20.70 years of age)). Perceived usefulness of the chemistry material, mathematical ability, GPA, and SGPA uniquely predicted conceptual stoichiometry problem solving ability.

The purpose of this study was to investigate which of two different instructional formats of a geometric construction activity was best suited to enhance student task performance. The subjects (N=50) were selected from community college students enrolled in introductory college mathematics courses and randomly assigned to one of two treatment groups. The groups differed in that group A received a different format of the procedural steps to reconstruct a given angle than that of group B. Form A consisted of adjoining text and diagrams where a diagram for each of the five steps of the task was pictured. Form B consisted of only two diagrams for the five-step process where it was necessary for subjects to determine which of the two diagrams was being referred to while proceeding through the steps. Both formats are representative of instructional materials currently in use in classrooms from the middle school level to the post-secondary level. During a personal interview session, each subject received one of two sets of directions describing the procedure of reproducing a given angle using a compass and straightedge. Each subject was asked to study the assigned set of directions with materials provided to practice the task. With the directions removed, each subject was then asked to reconstruct another, slightly different angle. Students given Form B, the reduced diagram format, significantly outperformed those students given Form A, x^2 (1, $N=50)=5.19, with no significant differences in the two groups with respect to the time spent studying the directions, t(48)=0.04, or completing the testing phase, t(48)=1.58. Other than use of the assigned form, mathematics ability was a significant factor in the subjects' ability to successfully complete the construction task, F(1,46)=7.79, p<.01. Verbal ability was not a significant factor, F(1,46)=1.81, p=.19, in the subjects' ability to successfully complete the construction task. Gender alone was not a significant factor, F1,46 =0.31, p = .58; however, when examined in combination with mathematics ability a significant interaction resulted, F1,46)=6.41, p=.02. Overall, success was significantly related to format, mathematics ability, and gender relative to mathematics ability.