Student Problem Solving

The purpose of this study is to determine if students solve math problems using addition, subtraction, multiplication, and division consistently and whether students transfer these skills to other mathematical situations and solutions. In this action research study, a classroom of 6th grade mathematics students was used to investigate how students solve word problems and how they determine which mathematical approach to use to solve a problem. It was discovered that many of the students read and re-read a question before they try to find an answer. Most students will check their answer to determine if it is correct and makes sense. Most students agree that mastering basic math facts is very important for problem solving and prefer mathematics that does not focus on problem solving. As a result of this research, it will be emphasized to the building principal and staff the need for a unified and focused curriculum with a scope and sequence for delivery that is consistently followed. The importance of managing basic math skills and making sure each student is challenged to be a mathematical thinker will be stressed.

Using Source-Code Analysis to Help End-user Programmers Create Dependable Software

Not long ago, most software was written by professional programmers, who could be presumed to have an interest in software engineering methodologies and in tools and techniques for improving software dependability. Today, however, a great deal of software is written not by professionals but by end-users, who create applications such as multimedia simulations, dynamic web pages, and spreadsheets. Applications such as these are often used to guide important decisions or aid in important tasks, and it is important that they be sufficiently dependable, but evidence shows that they frequently are not. For example, studies have shown that a large percentage of the spreadsheets created by end-users contain faults. Despite such evidence, until recently, relatively little research had been done to help end-users create more dependable software. We have been working to address this problem by finding ways to provide at least some of the benefits of formal software engineering techniques to end-user programmers. In this talk, focusing on the spreadsheet application paradigm, I present several of our approaches, focusing on methodologies that utilize source-code-analysis techniques to help end-users build more dependable spreadsheets. Behind the scenes, our methodologies use static analyses such as dataflow analysis and slicing, together with dynamic analyses such as execution monitoring, to support user tasks such as validation and fault localization. I show how, to accommodate the user base of spreadsheet languages, an interface to these methodologies can be provided in a manner that does not require an understanding of the theory behind the analyses, yet supports the interactive, incremental process by which spreadsheets are created. Finally, I present empirical results gathered in the use of our methodologies that highlight several costs and benefits trade-offs, and many opportunities for future work.

Saving the Vaquita: Immediate Action, Not More Data

The recent likely extinction of the baiji (Chinese river dolphin [Lipotes vexillifer]) (Turvey et al. 2007) makes the vaquita (Gulf of California porpoise [Phocoena sinus]) the most endangered cetacean. The vaquita has the smallest range of any porpoise, dolphin, or whale and, like the baiji, has long been threatened primarily by accidental deaths in fishing gear (bycatch) (Rojas-Bracho et al. 2006). Despite repeated recommendations from scientific bodies and conservation organizations, no effective actions have been taken to remove nets from the vaquita’s environment. Here, we address three questions that are important to vaquita conservation: (1) How many vaquitas remain? (2) How much time is left to find a solution to the bycatch problem? and (3) Are further abundance surveys or bycatch estimates needed to justify the immediate removal of all entangling nets from the range of the vaquita? Our answers are, in short: (1) there are about 150 vaquitas left, (2) there are at most 2 years within which to find a solution, and (3) further abundance surveys or bycatch estimates are not needed. The answers to the first two questions make clear that action is needed now, whereas the answer to the last question removes the excuse of uncertainty as a delay tactic. Herein we explain our reasoning.

Designing Software Product Lines for Testability

Software product line (SPL) engineering offers several advantages in the development of families of software products such as reduced costs, high quality and a short time to market. A software product line is a set of software intensive systems, each of which shares a common core set of functionalities, but also differs from the other products through customization tailored to fit the needs of individual groups of customers. The differences between products within the family are well-understood and organized into a feature model that represents the variability of the SPL. Products can then be built by generating and composing features described in the feature model. Testing of software product lines has become a bottleneck in the SPL development lifecycle, since many of the techniques used in their testing have been borrowed from traditional software testing and do not directly take advantage of the similarities between products. This limits the overall gains that can be achieved in SPL engineering. Recent work proposed by both industry and the research community for improving SPL testing has begun to consider this problem, but there is still a need for better testing techniques that are tailored to SPL development. In this thesis, I make two primary contributions to software product line testing. First I propose a new definition for testability of SPLs that is based on the ability to re-use test cases between products without a loss of fault detection effectiveness. I build on this idea to identify elements of the feature model that contribute positively and/or negatively towards SPL testability. Second, I provide a graph based testing approach called the FIG Basis Path method that selects products and features for testing based on a feature dependency graph. This method should increase our ability to re-use results of test cases across successive products in the family and reduce testing effort. I report the results of a case study involving several non-trivial SPLs and show that for these objects, the FIG Basis Path method is as effective as testing all products, but requires us to test no more than 24% of the products in the SPL.

Does Decoding Increase Word Problem Solving Skills?

In this action research study of my classroom of 7th grade mathematics, I investigated whether the use of decoding would increase the students’ ability to problem solve. I discovered that knowing how to decode a word problem is only one facet of being a successful problem solver. I also discovered that confidence, effective instruction, and practice have an impact on improving problem solving skills. Because of this research, I plan to alter my problem solving guide that will enable it to be used by any classroom teacher. I also plan to keep adding to my math problem solving clue words and share with others. My hope is that I will be able to explain my project to math teachers in my district to make them aware of the importance of knowing the steps to solve a word problem.

Improving Students’ Story Problem Solving Abilities

In this action research study of my classroom of 8th grade mathematics students, I investigated if learning different problem solving strategies helped students successfully solve problems. I also investigated if students’ knowledge of the topics involved in story problems had an impact on students’ success rates. I discovered that students were more successful after learning different problem solving strategies and when given problems with which they have experience. I also discovered that students put forth a greater effort when they approach the story problem like a game, instead of just being another math problem that they have to solve. An unexpected result was that the students’ degree of effort had a major impact on their success rate. As a result of this research, I plan to continue to focus on problem solving strategies in my classes. I also plan to improve my methods on getting students’ full effort in class.

Reasonable or Not? A Study of the Use of Teacher Questioning to Promote Reasonable Mathematical Answers from Sixth Grade Students

In this action research study of my sixth grade mathematics class, I investigated the influence a change in my questioning tactics would have on students’ ability to determine answer reasonability to mathematics problems. During the course of my research, students were asked to explain their problem solving and solutions. Students, amongst themselves, discussed solutions given by their peers and the reasonability of those solutions. They also completed daily questionnaires that inquired about my questioning practices, and 10 students were randomly chosen to be interviewed regarding their problem solving strategies. I discovered that by placing more emphasis on the process rather than the product, students became used to questioning problem solving strategies and explaining their reasoning. I plan to maintain this practice in the future while incorporating more visual and textual explanations to support verbal explanations.