Mathematical Communication, Conceptual Understanding, and Students' Attitudes Toward Mathematics

This action research study of my 8th grade classroom investigated the use of mathematical communication, through oral homework presentations and written journals entries, and its impact on conceptual understanding of mathematics. This change in expectation and its impact on students’ attitudes towards mathematics was also investigated. Challenging my students to communicate mathematics both orally and in writing deepened the students’ understanding of the mathematics. Levels of understanding deepened when a variety of instructional methods were presented and discussed where students could comprehend the ideas that best suited their learning styles. Increased understanding occurred through probing questions causing students to reflect on their learning and reevaluate their reasoning. This transpired when students were expected to write more than one draft to math journals. By making students aware of their understanding through communicating orally and in writing, students realized that true understanding did not come from mere homework completion, but from evaluating and assessing their own and other’s ideas and reasoning. I discovered that when students were challenged to communicate their reasoning both orally and in writing, students enjoyed math more and thought math was more fun. As a result of this research, I will continue to require students to communicate their thinking and reasoning both orally and in writing.

Mathematical Communication through Written and Oral Expression

In this action research study of my classroom of sixth grade mathematics, I investigated the use of communication of mathematics through both written and oral expression. Giving my students the opportunity to communicate mathematics both in writing and orally helped to deepen the students’ understanding of mathematics. The students’ levels of comprehension were increased when they were presented with a variety of instructional methods. Through discussion and reflection the students were able to find methods that worked best for them and their learning ability. Students’ understanding increased from probing questions that made the students reflect and re-evaluate their solutions. This learning took place when students were made aware of different solutions or ways of doing things from the class discussions that were held. I discovered that when students are challenged to express their thinking both in writing and orally, the students found that they could communicate their thinking in a new way. Some of my students were only comfortable expressing their thoughts in one of the two ways but by the time the project was completed, they all expressed that they enjoyed both ways, and maybe changed the original way they preferred doing mathematics. As a result of this research, I will continue to require students to communicate their thinking and reasoning both in writing and orally.

Rubric Assessment of Mathematical Processes in Homework

In this action research study of my classroom of 11th grade geometry, I investigated the use of rubrics to help me assess my students during homework presentations. I wanted to know more about the processes students went through as they did their homework problems, so homework presentations were implemented with the rubrics being the main form of assessment. I discovered that students are willing to speak about mathematics and can gain more understanding of mathematical processes as a result of homework presentations. The scores of the class improved after they talked about the homework assignments with each other. As a result of this research, I plan to keep on using homework presentations in my classroom to talk about homework, but discontinue the use of rubrics in assessment of students in mathematics. I also found students going to the board to solve problems in small groups are another helpful way to use presentations prior to assessment to help me understand where the students are with a new concept prior to assigning homework or giving an assessment.

Written Solutions of Mathematical Word Problems

In this action research study of my classroom of ten ninth grade algebra students, I investigated how my students expressed written solutions of mathematical word problems. I discovered that my students writing and performance improved as they experienced different strategies to attack problem solving. These experiences helped improve the confidence of my students in their problem solving skills and in their mathematical writing. I also discovered that my teaching style changed, as my students took on more responsibility for their learning. As a result of this research, I plan to implement problem solving activities in all my classrooms next year. I also plan to have my students develop their written communication skills by presenting their solutions to their problem solving activities in writing.

Understanding the Mathematical Language

In this action research study of my calculus classroom consisting of only 12th grade students, I investigated activities that would affect a student’s understanding of mathematical language. The goal in examining these activities in a systematic way was to see if a student’s deeper understanding of math terms and symbols resulted in a better understanding of the mathematical concepts being taught. I discovered that some students will rise to the challenge of understanding mathematics more deeply, and some will not. In the process of expecting more from students, the frustration level of both the students and the teacher increased. As a result of this research, I plan to see what other activities will enhance the understanding of mathematical language.

Exploring the Influence of Vocabulary Instruction on Students’ Understanding of Mathematical Concepts

In this action research study of my classroom of 8th grade mathematics, I investigated the influence of vocabulary instruction on students’ understanding of the mathematics concepts. I discovered that knowing the meaning of the vocabulary did play a major role in the students’ understanding of the daily lessons and the ability to take tests. Understanding the vocabulary and the concepts allowed the students to be successful on their daily assignments, chapter tests, and standardized achievement tests. I also discovered that using different vocabulary teaching strategies enhanced equity in my classroom among diverse learners. The knowledge of the math vocabulary increased my students’ confidence levels, which in turn increased their daily and test scores. As a result of this research, I plan to find ways to incorporate the vocabulary teaching strategies I have used into current math curriculum. I will start this process at the beginning of the next school year, and will continue looking for new strategies that will promote math vocabulary retention.

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.

Daily Problem-Solving Warm-Ups: Harboring Mathematical Thinking In The Middle School Classroom

In this action research study of my classroom of 8th grade mathematics, I investigated the use of daily warm-ups written in problem-solving format. Data was collected to determine if use of such warm-ups would have an effect on students’ abilities to problem solve, their overall attitudes regarding problem solving and whether such an activity could also enhance their readiness each day to learn new mathematics concepts. It was also my hope that this practice would have some positive impact on maximizing the amount of time I have with my students for math instruction. I discovered that daily exposure to problem-solving practices did impact the students’ overall abilities and achievement (though sometimes not positively) and similarly the students’ attitudes showed slight changes as well. It certainly seemed to improve their readiness for the day’s lesson as class started in a more timely manner and students were more actively involved in learning mathematics (or perhaps working on mathematics) than other classes not involved in the research. As a result of this study, I plan to continue using daily warm-ups and problem-solving (perhaps on a less formal or regimented level) and continue gathering data to further determine if this methodology can be useful in improving students’ overall mathematical skills, abilities and achievement.

Effects of arterial oxygen tension and cardiac output on venous saturation: a mathematical modeling approach

OBJECTIVES: Hemodynamic support is aimed at providing adequate O-2 delivery to the tissues; most interventions target O-2 delivery increase. Mixed venous O-2 saturation is a frequently used parameter to evaluate the adequacy of O-2 delivery. METHODS: We describe a mathematical model to compare the effects of increasing O-2 delivery on venous oxygen saturation through increases in the inspired O-2 fraction versus increases in cardiac output. The model was created based on the lungs, which were divided into shunted and non-shunted areas, and on seven peripheral compartments, each with normal values of perfusion, optimal oxygen consumption, and critical O-2 extraction rate. O-2 delivery was increased by changing the inspired fraction of oxygen from 0.21 to 1.0 in steps of 0.1 under conditions of low (2.0 L.min(-1)) or normal (6.5 L.min(-1)) cardiac output. The same O-2 delivery values were also obtained by maintaining a fixed O-2 inspired fraction value of 0.21 while changing cardiac output. RESULTS: Venous oxygen saturation was higher when produced through increases in inspired O-2 fraction versus increases in cardiac output, even at the same O-2 delivery and consumption values. Specifically, at high inspired O-2 fractions, the measured O-2 saturation values failed to detect conditions of low oxygen supply. CONCLUSIONS: The mode of O-2 delivery optimization, specifically increases in the fraction of inspired oxygen versus increases in cardiac output, can compromise the capability of the "venous O-2 saturation" parameter to measure the adequacy of oxygen supply. Consequently, venous saturation at high inspired O-2 fractions should be interpreted with caution.

ACID-BASE REACTIONS: CONCEPT, REPRESENTATION AND GENERALIZATION FROM THE ENERGY INVOLVED IN TRANSFORMATIONS

ACID-BASE REACTIONS: CONCEPT, REPRESENTATION AND GENERALIZATION FROM THE ENERGY INVOLVED IN TRANSFORMATIONS. Undergraduate students on the first year of Chemistry Courses are unfamiliar with the representation of acid-base reactions using the ionic equation H+ + OH- -> H2O. A chemistry class was proposed about acid-base reactions using theory and experimental evaluation of neutralization heat to discuss the energy involved when water is formed from H+ and OH- ions. The experiment is suggested using different strong acids and strong base pairs. The presentation of the theme within a chemistry class for high school teachers increased the number of individuals that saw the acid-base reaction from this perspective.

MATHEMATICAL ANALYSIS OF MAXIMUM POWER GENERATED BY PHOTOVOLTAIC SYSTEMS AND FITTING CURVES FOR STANDARD TEST CONDITIONS

The rural electrification is characterized by geographical dispersion of the population, low consumption, high investment by consumers and high cost. Moreover, solar radiation constitutes an inexhaustible source of energy and in its conversion into electricity photovoltaic panels are used. In this study, equations were adjusted to field conditions presented by the manufacturer for current and power of small photovoltaic systems. The mathematical analysis was performed on the photovoltaic rural system I- 100 from ISOFOTON, with power 300 Wp, located at the Experimental Farm Lageado of FCA/UNESP. For the development of such equations, the circuitry of photovoltaic cells has been studied to apply iterative numerical methods for the determination of electrical parameters and possible errors in the appropriate equations in the literature to reality. Therefore, a simulation of a photovoltaic panel was proposed through mathematical equations that were adjusted according to the data of local radiation. The results have presented equations that provide real answers to the user and may assist in the design of these systems, once calculated that the maximum power limit ensures a supply of energy generated. This real sizing helps establishing the possible applications of solar energy to the rural producer and informing the real possibilities of generating electricity from the sun.

Progress in the mathematical theory of quantum disordered systems

We review recent progress in the mathematical theory of quantum disordered systems: the Anderson transition, including some joint work with Marchetti, the (quantum and classical) Edwards-Anderson (EA) spin-glass model and return to equilibrium for a class of spin-glass models, which includes the EA model initially in a very large transverse magnetic field. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4770066]

Mathematical modeling and analysis of the particle interactions in the direct filtration using the colloidal theory

This work used the colloidal theory to describe forces and energy interactions of colloidal complexes in the water and those formed during filtration run in direct filtration. Many interactions of particle energy profiles between colloidal surfaces for three geometries are presented here in: spherical, plate and cylindrical; and four surface interactions arrangements: two cylinders, two spheres, two plates and a sphere and a plate. Two different situations were analyzed, before and after electrostatic destabilization by action of the alum sulfate as coagulant in water studies samples prepared with kaolin. In the case were used mathematical modeling by extended DLVO theory (from the names: Derjarguin-Landau-Verwey-Overbeek) or XDLVO, which include traditional approach of the electric double layer (EDL), surfaces attraction forces or London-van der Waals (LvdW), esteric forces and hydrophobic forces, additionally considering another forces in colloidal system, like molecular repulsion or Born Repulsion and Acid-Base (AB) chemical function forces from Lewis.

Gauge and integrable theories in loop spaces

We propose an integral formulation of the equations of motion of a large class of field theories which leads in a quite natural and direct way to the construction of conservation laws. The approach is based on generalized non-abelian Stokes theorems for p-form connections, and its appropriate mathematical language is that of loop spaces. The equations of motion are written as the equality of a hyper-volume ordered integral to a hyper-surface ordered integral on the border of that hyper-volume. The approach applies to integrable field theories in (1 + 1) dimensions, Chern-Simons theories in (2 + 1) dimensions, and non-abelian gauge theories in (2 + 1) and (3 + 1) dimensions. The results presented in this paper are relevant for the understanding of global properties of those theories. As a special byproduct we solve a long standing problem in (3 + 1)-dimensional Yang-Mills theory, namely the construction of conserved charges, valid for any solution, which are invariant under arbitrary gauge transformations. (C) 2012 Elsevier B.V. All rights reserved.

Freeze-drying microscopy in mathematical modeling of a biomaterial freeze-drying

Transplantation brings hope for many patients. A multidisciplinary approach on this field aims at creating biologically functional tissues to be used as implants and prostheses. The freeze-drying process allows the fundamental properties of these materials to be preserved, making future manipulation and storage easier. Optimizing a freeze-drying cycle is of great importance since it aims at reducing process costs while increasing product quality of this time-and-energy-consuming process. Mathematical modeling comes as a tool to help a better understanding of the process variables behavior and consequently it helps optimization studies. Freeze-drying microscopy is a technique usually applied to determine critical temperatures of liquid formulations. It has been used in this work to determine the sublimation rates of a biological tissue freeze-drying. The sublimation rates were measured from the speed of the moving interface between the dried and the frozen layer under 21.33, 42.66 and 63.99 Pa. The studied variables were used in a theoretical model to simulate various temperature profiles of the freeze-drying process. Good agreement between the experimental and the simulated results was found.