Edward’s current tutoring subjects are listed at the left. You
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Edward’s qualifications in specific subjects below.
Algebra introduces the concept of variables. Students of algebra apply previously learned rules like the distributive law and PEMDAS to expressions and equations which now include variables. In addition, they learn rules like the addition principle of equality and the multiplication principle of equality. The goal is to manipulate the equation so a value of the variable can be calculated. Sorting out all these rules and deciding what should be done next can be confusing. I have taught classes in Algebra 1 as well as tutored the subject to students of all ages, backgrounds and abilities. I strive to cut through the confusion by introducing structure to the process and having my students view it from the perspective of "isolate and normalize" the variable. I also am effective at helping students struggling with word problems by stressing the concepts of identifying keywords and drawing diagrams. Later in an Algebra I course, equations and expressions with two variables are introduced. There, problems can be solved using multiple techniques (e.g. algebraic, geometric and numeric). Each student learns math differently, I strive to continuously adapt my tutoring style to each student's learning style.
Algebra 2 is a continuation of algebra 1. In algebra 2 students are introduced to polynomials. The rules for basic arithmetic operations on polynomials (addition, subtraction, multiplication, division and exponentiation) are examined. Rational expressions (quotients of two polynomials) are studied along with the rules for basic arithmetic operations on rational expression. At the core of rational expressions is factorization. I simplify the learning of these concepts by showing how easy they can be if students factor the polynomials and look for opportunities to cancel terms before multiplying or dividing. Often times, this requires addressing student weaknesses in factorization. Algebra 2 also covers quadratic functions in detail. Students learn multiple ways for finding roots of quadratics. Finally, algebra 2 typically ends with complex numbers, logarithmic and exponential functions. When I teach and tutor algebra 2, I stress mathematical reasoning and multiple approaches (algebraic, geometric and numeric). For example, many students are unaware that arithmetic on complex numbers can be done on the TI83/84 calculator. Teaching them this skill gives them a way to check answers.
Biostatistics is the application of statistics to biology. It typically involves the design of a biological experiment from which data is gathered. From that point, the tools of descriptive and inferential statistics are brought to bear. The data are first analyzed and second summarized by numbers, tables and graphs. Finally, inferences are drawn from the results. At the core of an introductory biostatistics class, are the principles and practices taught in the introductory statistics class I have taught. I have experience helping bright undergraduate and graduate biology students get over the math hurdle they face when they take this course.
The C computer language is one of the most popular computer languages of all time. C is a general-purpose computer language which is flexible enough for implementing system software (through its low level memory access capabilities) as well as portable application software. Long before multicore processors hit the consumer marketplace, I had been developing software for massively parallel computers with as many as 65,536 processors using C. I have several decades of software development experience using the C programming language. In addition, I taught a class called "Introduction to the C Programming Language" at Daley College, one of the City Colleges of Chicago, for several years.
C++ is one of the most popular programming languages. The language was developed by Bjarne Stroustrup as an enhancement to the C language. Consequently, C++ inherits most of C's syntax. C++ added classes, virtual functions, operator overloading, multiple-inheritance, templates and exception handling among other features.
It is a multi-paradigm language enabling programmers to blend functional, generic, modular, procedural and object-oriented styles. I have several decades experience developing engineering simulation software during which time I have used each of the above programming styles. Long before shuttered glasses and 3D televisions became mainstream, I was developing C++ software for the 3D visualization of engineering simulations using shuttered glasses for specialized graphics devices such as the CAVE.
Calculus is the mathematical study of change. It has two branches. Differential calculus is the branch of calculus focusing on rates of change; for example, the slopes of curves and surfaces. Integral calculus is the branch of calculus focusing on accumulations; for example, areas under curves and volumes enclosed by surfaces. The two branches are connected by the Fundamental Theorem of Calculus discovered independently by Isaac Newton and Gottfried Leibnitz. My first exposure to calculus was in high school. I was fortunate enough to have had a teacher who was a true scholar. He placed equal emphasis on having his students understand the principles and theorems as he did on having his students master procedures. Consequently, I started the undergraduate calculus sequence with a significant advantage over my classmates. I completed the entire undergraduate calculus sequence through differential equations with straight A's.
I have taught business calculus at Harper College. I have tutored high school students (honors and AP) as well as college students. My tutoring approach is to solve problems alongside my students and supplement what they are learning in the classroom (hows) with simple examples and sketches illustrating the underlying principles (whys). I believe students will apply the rules of calculus confidently and correctly after they develop an intuition for the underlying principles.
I graduated from Illinois Institute of Technology in 1981 with a bachelor's degree in Civil Engineering. I have worked in the water distribution department for the City of Chicago, a piping stress analyst for the nuclear industry and engineering simulation software developer for a national laboratory.
Computer programming involves designing, writing, testing and refining source code. The purpose of computer programming is to build a set of instructions computers follow to accomplish various tasks such as numerical computation, information storage, information retrieval and display of images and information. I have almost four decades of computer programming experience. I have developed programs in Basic, Fortran, Pascal, C, C++ and Java. Most of these have involved engineering simulation. In addition, I have used both low-level graphics libraries such as OpenGL and high level graphics libraries like vtk to display the results of engineering simulations. I have experience developing user interfaces in Visual Basic and Java swing. I have developed shell scripts in Perl. I can help you in crafting new programs as well as deciphering the inner workings of legacy code. I have experience developing programs under a variety of paradigms (e.g. structured programming, object based programming and object oriented programming) as well as target platforms ranging from desktop PC to networked workstations ans supercomputers.
Differential equations are equations involving a function and one or more of its derivatives. Traditionally, they have frequently appeared in the mathematical models constructed by natural scientists and engineers. However, in recent times, their use by social scientists has increased dramatically. This is especially true in the area of economics. Only the simplest differential equations admit solutions given by explicit formulas. Beyond this, numerical methods using computers are called upon to approximately solve differential equations. Determining the explicit formulas for exact solutions to basic differential equations is covered in the traditional "diffy q" class at the end of the undergraduate calculus course sequence. My experience encompasses both the classical methods of solving differential equations taught in the traditional undergraduate "diffy q" course required for an engineering or science undergraduate degree as well as over two decades of experience solving systems of partial differential equations numerically on computers. I have completed graduate coursework on numerical methods such as the finite difference and finite element method for the solution of differential equations. In addition, my PhD thesis discussed solving the system of partial differential equations arising from structural dynamics on high performance parallel computer architectures using software I had developed.
Discrete math is a catch-all term encompassing many diverse areas of mathematics. There is no universal agreement as to what constitutes discrete math. Discrete math is defined less by what topics are included than by what is excluded. Excluded are notions of continuity upon which calculus is built. Consequently, discrete math is described as "non-calculus" math. Finite math is an introductory course in discrete math. A typical finite math course is a survey course consisting of: linear functions, matrices, linear inequalities, linear programming, the Simplex Method, counting (combinatorics), and probability. I have taught finite math within the university and community college setting for the past nine years. I am aware of the errors most frequently made by students. I diligently will point these out to you and make sure you won't fall into these traps when the exam comes around.
I have extensive experience in computer programming. One programming language I specialize in is Fortran. The name Fortran comes from the phrase "formula translation". It is the programming language of choice for scientists and engineers. Most computer programmers are unfamiliar with Fortran. I wrote my first Fortran program in 1976. I have kept up with the changes to the language through Fortran 2008.
Linear algebra is the study of sets of linear equations and their transformation properties. Combined with calculus, linear algebra allows the solution of systems of linear differential equations. Linear systems of equations arise in a diverse range of applications including agriculture, business, economics, finance, sociology, demography, political science, biology, chemistry, ecology, genetics, astronomy, electronics, engineering and physics. For this reason, linear algebra is often viewed as the work horse of modern applied mathematics. Linear algebra employs matrices as a tool for representing systems of equations. It goes further by establishing efficient methodologies for solving systems of equations which can be done by hand using pencil and paper (for small systems) or which can be readily automated and executed on computers and calculators. The main objective is finding values of variables satisfying all equations in the system.
I have studied linear algebra from both a pure and applied perspective. I have extensive experience developing computer programs for Gaussian decomposition, LU decomposition and eigenvectors. I have experience helping students learn the basics of linear algebra and the matrix solution of systems of equations in College Algebra and Finite Mathematics courses I have taught. In addition, I have tutored engineering students and shown them how to apply what they had learned in their math classes to systems of equations representing electrical circuits and civil engineering structures. In both the classroom and tutoring settings, I have helped students learn how to apply the linear algebra capabilities built into the TI83 and TI84 calculators to solve systems of linear equations.
I graduated from Northwestern University in 1990 with a doctorate in Theoretical and Applied Mechanics. Applied mechanics bridges the gap between physical theory and its application to technology. It is fundamental to many branches of engineering, especially mechanical and civil engineering. In these disciplines it is commonly referred to as engineering mechanics. Engineering mechanics can be subdivided into statics, dynamics, mechanics of materials, fluid mechanics and continuum mechanics. I have taken advanced courses in these areas as part of my graduate studies. In addition, my PhD thesis outlined the development of novel computational tools for engineering mechanics simulations on high performance parallel computer architectures.
Pascal is a computer language designed from scratch by Professor Niklaus Wirth in the 1960s and implemented in 1970. Pascal was originally targeted towards academia. The objective was the teaching of good programming style. It embraced structured programming and stepwise refinement (top-down programming design). The original Macintosh OS was written in Pascal. Like many Fortran programmers of the 1980s, Pascal captured my interest because it was a way to transfer engineering simulation programs from expensive mainframes to minicomputers and desktop computers like the Macintosh 512K I had just purchased. I went on to develop several computer programs for the classes I took at Northwestern University on that computer. For me, Pascal provided a stepping stone, first to C and then later to C++, Java and object oriented programming.
Prealgebra focuses primarily on arithmetic. The most fundamental skills are reading and writing of whole numbers. From there, basic arithmetic operations of addition, subtraction, multiplication and division are defined for the whole numbers. Wrapping up the study of arithmetic on whole numbers, exponentiation and order of operations (PEMDAS) are introduced. Knowing how to work with whole numbers well is the key to the bulk of a prealgebra course. The same skills (reading and writing, basic arithmetic operations, exponentiation and PEMDAS) are introduced for fractions, mixed numbers and decimals. After these skills are mastered for all number types, they are brought to bear in solving problems involving ratios, proportions, percents, measurement and basic geometry. Signed numbers are introduced at the end. I have helped students of all ages, backgrounds and abilities with prealgebra. Among them, have been adults returning to school whose skills have gotten rusty through lack of use or who perhaps felt discouraged by previous bad experiences with math and have since avoided it.
Precalculus explores topics that will be applied when a student studies calculus in the future. A strong mastery of precalculus will streamline a student's study of calculus. Conversely, having a weak background in precalculus makes the study of calculus more difficult since the student will feel he has an overwhelming amount of things to learn simultaneously. Precalculus involves memorization. For many students, it is the first math class they have taken where memorization plays such a profound role. The objective must be long-term memorization so that the definitions and skills learned can be recalled and applied throughout the student's study of calculus. My tutoring approach focuses on making sure a student learns the fundamentals and realizes how often the fundamentals appear in the solution of problems. I train students to break a problem into smaller problems. I have them repeat that process until each problem involves applying a single fact they have previously learned. Every time they encounter and apply a previously learned skill, they are memorizing it. Repetition is key to memorization.
The study of randomness is at the heart of probability. While individual occurrences for the outcome of a random event (e.g. the toss of a fair coin) are impossible to predict in advance, if repeated many times the sequence of outcomes will exhibit patterns. In probability, formulas are devised for the predicted of these patterns (e.g. 50% heads, 50% tails for a sequence of coin tosses). Probability forms the mathematical foundation in inferential statistics. It is traditionally placed in between descriptive and inferential statistics in an introductory statistics class. When teaching and tutoring probability, I try to tie back the theory to everyday random phenomena (e.g. coin tosses, playing cards, dice) which students can experiment with to see how formulas can be used to predict the outcomes they are observing. I also stress a visual (Venn diagram or tabular) depiction of sample spaces and outcomes which often can be used in lieu of formulas to compute probabilities.
Statistics is the science of collecting, organizing, summarizing and analyzing information in order to draw conclusions. A typical statistics class begins with an overview of descriptive statistics. Descriptive statistics consists of organizing and summarizing the information collected. This is usually done through charts, graphs and tables as well as computing numerical summaries such as the mean and standard deviation. Most students have seen these types of summaries in the media and have no problem interpreting, using and constructing them. The second phase of a statistics course (inferential statistics) is much more difficult for students. Inferential statistics uses methods that generalize results obtained from a sample to the population. When guiding students towards the solution of a problem in inferential statistics, I use a visual approach to diagram the interrelationship between sample and population. Specifically, that the sample being analyzed is one specific member of a much larger sampling distribution. I emphasize the central limit theorem and its ramifications on the shape of the distributions (population and sampling distribution) in the diagram. Finally, all numbers given in the problem are transferred to the appropriate locations on the diagram. By being able to visualize the underlying logic and reasoning behind inferential statistics, my students are able to comprehend inferential statistics, choose the correct test to apply in a given situation and interpret the results of the statistical tests they conduct confidently and correctly. I also teach my students how to use the built-in capabilities of the TI83/84 calculator to do statistics.
Trigonometry has been largely replaced by Precalculus in many schools. Like Precalculus, trigonometry involves a lot of long term memorization. A trigonometry class focuses on the six fundamental trigonometric functions and the relationships between them. Students learn laws that allow them to solve triangles (determine angles and sides) for oblique (non-right) triangles. Trigonometry has many applications. The word problems in a trigonometry course challenge students by drawing from applications such as navigation, surveying, constructing models of periodic behavior and simple harmonic motion. My tutoring approach focuses on making sure a student learns the fundamentals and realizes how often the fundamentals appear in the solution of problems. The first step in a trigonometry problem often involves looking at a problems and devising a strategy to solve it. I train students to start problems by trying to identify previously learned facts that could be brought to bear on its solution. Every time they encounter and apply a previously learned skill, they are memorizing it. Repetition is key to memorization.
I took my first class in Basic in 1977. Back then, in keeping with the acronym Beginners All Purpose Symbolic Instruction Code, it was simple, small and uncomplicated. My first internship was to maintain and extend a Basic computer program designed to calculate pressures and flows in a piping network. Over the years, the art of computer programming evolved. Unlike many early computer languages, Basic has continued to be relevant. This is due, to a large extent, to Microsoft and its Visual Basic product. Object-oriented and event-driven programming as well as many other developments in computer languages have been incorporated into Visual Basic. Visual Basic also has tight ties to Microsoft Office. This has transformed the language from something a "hobbyist" might use for toy applications to a language which can deliver full-fledged custom business applications. I have several decades of computer software experience and witnessed this evolution of the Basic language. I can help you untangle the features of the language and develop your software program whether it be a toy hobbyist application or a custom business application tied into Microsoft Office.