Math 388: Continuous Mathematical Modeling

The mathematics 388 course at Claremont Graduate University goal was the construction, simplification, analysis, interpretation and evaluation of mathematical models that shed light on problems arising in the physical, biological, and social sciences. Furthermore, derivations and methods for solutions of model equations, heat conduction problems, simple random walk processes, simplification of model equations, dimensional analysis and scaling, perturbation theory, and a discussion of self-contained modular units that illustrate the principal modeling ideas were emphasized throughout the class. The major class assignment was to develop a modeling project.

The modeling project that I selected was the predator-prey model. The predator-prey model, also known as Lotka-Volterra, is used to understand the population dynamics involved in a simple system that involves a single predator specie and a single prey specie. By ignoring such things as the variability of individuals of each species, variation in the environment over time, and the effects of other species, the properties of the interactions between the two species are examined and are used to understand the mechanisms involved in the cyclical behavior of their population levels. In my project I examine a modified model that illustrates the fruitful interactions of algebraic and geometric methods, and the classic Lotka-Volterra model.

The class increased my mathematical knowledge in the areas of advance calculus, linear algebra, ordinary and partial differential equations, and numerical methods. Moreover, the project provided me with first hand experience of what mathematical modeling is and of all its elements. Through the class lectures, homework assignments, and project, I was able to increase my experience of mathematical modeling. I was provided with extensive practice of formulating real-world problems in mathematical terms, and to use appropriate analytical and numerical methods to solve simplified equations or approximate the solution for certain parameter regimes. In addition, the course professor emphasized the importance of interpreting solutions or results and to make reasonable predictions on the behavior of different systems as parameters were varied.

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Math 294: Advanced Applied Mathematics for Applications

The mathematics 294 course of Claremont Graduate University introduced the derivation of classical equations of applied mathematics. The equations emphasized were: quasilinear hyperbolic, Laplace, Poisson, Helmholtz, advection-diffusion, and wave equations. Furthermore, most of the equations were examined in various orthogonal curvilinear coordinate systems. For example, the gradient, divergence, curl, and Laplacian operators were studied in Cartesian, cylindrical and spherical coordinates, acting on scalar or vector fields. In addition, methods of solutions on bounded domains using eigenfunction expansions and the associated Sturm-Liouville eigenvalues problems, including Bessel functions and Legendre polynomials were covered.

The class was challenging, but rewarding. I was able to review my basic calculus knowledge and expand my mathematical content knowledge at the same time. The homework assignments were demanding and I frequently sought the help of my classmates, the class tutor, and the professor. The most challenging topic for me was to understand scalar and vector fields, as well as the gradient, divergence, curl and Laplacian operators acting on such fields.

In conclusion, the course increased mathematical content knowledge and made me a better teacher. I was provided with the opportunity to study the classical equations of applied mathematics. For example, we applied the method of characteristics to solve first-order quasi-linear partial differential equations and analyze the formation of shocks and expansion waves in such systems. Prior to the course I knew that differential equations were greatly used within the mathematics field, but was not able to point out several specific examples. In addition, the course increased and sharpened my calculus knowledge, which will make me a better calculus teacher.

Math 389: Introduction to Operations Research

The mathematics 389 course at Claremont Graduate University introduced operations research (OR), which is the use of mathematical models to guide decisions and improve the performance of systems. Although there are many branches to OR, the class emphasized the fundamental branch of optimization; which is maximizing an object function over a constrained solution space. The main goal of the class was not only to introduce us to basic techniques, but also to the thought process of operations research. Furthermore, through the major assignment of the course, we learned how to think like an operations researcher, by framing problems mathematically, by identifying underlying assumptions and by testing the sensitivity of the solutions to these assumptions.

The main assignment of the course was a case study, which was a group project in which we had the freedom to select the topic. My group decided to create a linear problem that would create the master schedule for the Mathematics Department of Pacific High School. The master schedule we created was intended to not only meet the needs of students, but also to maximize teacher’s happiness. In order to maximize teacher’s happiness we took into account our returning teachers preference of courses to be taught the following school year. In addition, the master schedule had to be created with the purpose that all courses for students, grades nine through twelve, were met without exceeding the limits on the number of unique course preparations and the total number of coursed taught.

By exploring the basic applications of operations research, I gained an appreciation for its omnipresence in modern life. Moreover, it increased my mathematical knowledge since I knew almost nothing of OR before the course, and of all of its applications and effects on modern life. The class made me a better teacher because it increased my knowledge of the applications of mathematics. When my students ask me how is mathematics used in the real world, now I have a vast number of ways to answer that question and go a step further by providing them with my personal experience in the case study. For example, I can mention how operations research lies in the interface of computer science, engineering, and business.

Math 251: Probability

The mathematics 251 course of Claremont Graduate University covered the main elements of probability theory at an intermediate level. The topics covered included combinatorial analysis, conditional probabilities, discrete and continuous random variables, probability distributions, central limit theorem, and numerous applications.

The topic that provided me with the most challenge was conditional probabilities. I remember spending numerous hours trying to complete and to understand the homework assignments of this topic. Although conditional probabilities was the most difficult topic, it was the one that contributed to my mathematical content knowledge growth the most. After attempting to complete the first homework assignment on the topic, I realized my lack of knowledge or experience of the topic. I struggled with the concept and always sought out for help from my peers, tutor, on line sources, and from the course professor.

In conclusion, completing the math 251 course; not only increased my mathematical content knowledge, but also positively affected by pedagogical knowledge. Teaching the probabilities unit of the high school algebra course was always the most challenging topic to teach, and one in which my students consistently scored the worse on state tests. I realized that it was because I did not fully understand the concept of conditional probabilities. However, all change due to my successful completion of the course and I feel ready and capable of efficiently teaching this concept the next time it comes around.

EDUC 363: Theories related to Modeling in Mathematics

The education 363 course at Claremont Graduate University developed my understanding of mathematical modeling. During the course we investigated the realities of incorporating a strong emphasis on the mathematical practice of modeling with mathematics. The main goal of the course was to develop, write, field test, and report on a modeling project in our classroom.

The modeling project that I conducted involved weigh loss. I provided my students with my current weight of 230 pounds and stated the desire to reach my ideal weight of 190 pounds. The challenge that I presented my students with was to develop a weight loss plan for me that could help me reach my desired goal. The project lasted five days, students were provided with the opportunity to research the topic, and to create a power point presentation and present their findings. A presentation scoring rubric was used to grade the student presentations, which was completed by their classmates. At the end of the project, students were asked to complete a reflection on their experience and opinion of the project.

The course increased my educational content knowledge since I became familiar with the literature regarding mathematical modeling, and experienced first hand the difficulties and benefits of modeling projects. Before completing this project, I was hesitant in the idea or benefit of incorporating mathematical modeling in the classroom since I like to structure the learning outcomes within my students. I felt that with modeling projects it was difficult to control the learning outcomes and that too much time was allocated in the research and very little grade level mathematics was accomplished. After completing this project, I can honestly say that my students proved me wrong. Throughout the project they engaged in meaningful mathematical conversations and I was impressed by their technological knowledge. My students and I enjoyed the project, and my perspective of mathematical modeling was positively affected as a result of this course assignment.

EDUC 342: Action Research

The education 432 course at Claremont Graduate University developed my skills in action research by reading key literature as well as conducting an action research project. Throughout the course, we analyzed, discussed, and reflected on mathematics teaching by examining classroom videos of our own, as well as our classmates.  The main course assignment was to engage in our own classroom action research project and to present our research, reflections, artifacts, and conclusion at the end of the course.

For my action research project, I decided to research the use of open-ended questions to provoke conceptual development in students. I selected this topic because according to educational experts, teachers seldom write down their questions while planning instead they generate them extemporaneously during the lesson. This can lead to vague questions that do not engage students in high-quality thinking. I felt that this topic would help me grow as a professional since I noticed that many of the questions I asked during a lesson, for the most part, asked students to generate single number, figure, or mathematical object as a response. My professional goal for the research project was to develop and to incorporate questions in my teaching practices that are directed toward evaluating students’ thinking. Furthermore, the questions must provide learners an opportunity to communicate their reasoning process. In addition, the questions I created allowed for the teacher to gather detail data on how students think and what they actually learn from instructions.

In conclusion, the action research course increased my personal educational knowledge for it provided me with a tool to improve and refine my teaching practices. Moreover, it empowered me to contribute my educational institution in a more powerful way. The action research I conducted involved several teachers at my school; and at the end of the project I conducted an action research presentation to the mathematics department. The presentation had a positive effect on my colleagues, it prompted further research of my topic within the department, and new ideas of action research projects were discussed.

EDUC 340-1: Algebra/Geometry assignments and reflections

The education course 340-1 provided me with many rich reading and writing assignments that increased my educational pedagogical knowledge and teaching practices. Within the course we analyzed educational research as it applied to the teaching of algebra and geometry. One of the course assignments was to analyze a strand of the new Common Core State Standards (CCSS) for high school mathematics. I selected to research and learn about the new high school functions standards.

The paper provides an overview of the CCSS, analyzes the major differences between the CCSS and the current California Standards, and finally it examines each of the functions standards. The main goal of the paper was to connect the function standards to educational research and thus validate their educational importance. Furthermore, the curricular implications and possible effects of the standards are examined throughout the paper.

This assignment improved awareness of the CCSS and made me an educational leader in my educational institution since I was one of the few that had some knowledge of the CCSS. At this point in time, most teachers had heard of them but few had taken the time to read about them. It also affected my teaching practices as I made it a point to start preparing my self for the change by teaching more concepts conceptually and to emphasize the study and importance of functions when appropriate.

EDUC 340: Learning Theories Related to Mathematics

The Education 340 course of Claremont Graduate University provided many rich theoretical reading materials and meaningful writing assignments. The assignment that improved the writers’ educational pedagogical knowledge the most, was a written assignment that involved analyzing the three major learning theories the in teaching mathematics.

The paper analyzed constructivism, sociocultural, and habits of mind, which are identified by educational experts as the three major learning theories. The similarities, differences, and areas of compatibility between the three theories previously mentioned were identified. Moreover, the paper attempted to demonstrate how the three learning theories support and compliment each other. Finally, the writer stated the implications of his findings in regards to his own teaching practices.

This specific assignment impacted the author tremendously for he realized how much his teaching credential program had underprepared him for the teaching profession. He realized that most of what he learned in the credential program dealt with classroom management and lesson preparation, which are all important elements of the teaching profession; but it was evident that a vast hole existed in his knowledge of educational theories.  After completing this assignment, the writer became more methodical when lesson planning. Furthermore, the three learning theories became a guide for his teaching practices from a curriculum, lesson delivery, and creation of lessons point of view. As a result of this assignment, the writer created the following professional goals for his teaching practices: First, he would strive to learn the mathematical knowledge of his students to adapt his teaching practices and plans based on the mathematics of his students. Second, he would always attempt to learn what is the best environment for his students to learn in. Finally, he will strive to develop lessons in which students get an opportunity to become experimenters, pattern sniffers, describers, inventors, conjecturers, and guessers.