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7MATHEMATICS TEACHERS' DEVELOPMENT IN CALIFORNIA (USA) AND UKRAINE. BRIEF COMPARATIVE ANALYSIS (СОВЕРШЕНСТВОВАНИЕ УЧИТЕЛЯ МАТЕМАТИКИ В КАЛИФОРНИИ (США) И УКРАИНЕ. КРАТКИЙ СРАВНИТЕЛЬНЫЙ АНАЛИЗ)
I.Subbotin, Professor,
National University, Los Angeles, USA,
N.N.Bilotskii, Assocoate Professor, National Pedagogic University, Kiev, UKRAIN Milla Hill, Mathematics Coordinator, Yavneh Academy, Los Angeles, USA
Ця стаття перша з циклу, присвяченому процесу тдготовки викладачгв математики в Кал1форнп в пор1внянт з тдготовкою викладач1в математики в Украш.
The main goal of this article is to highlight main stages of the Mathematics teachers' preparation process in California, USA and compare it with the Mathematics teachers' development in Ukraine. We truly believe that educators of the both countries could be significantly benefited sharing the best practices and experience in this and other professional areas. We consider this article as the first step towards an open discussion concerned with main issues of Mathematics teachers' preparation.
We will begin with the quote from the Mathematics Equals Opportunity (White Paper -- October 20, 1997) -the document issued by the United States Department of Education: "Shortages in workers skilled in mathematics and science could affect U.S. performance in global markets. According to a recent report, America's New Deficit: The Shortage of Information Technology Workers, from the Office of Technology Policy at the U.S. Department of Commerce, as computer and data processing become more important to the economy, more and more workers skilled in mathematics- and science-related disciplines will be needed to
maintain the U.S.'s international competitiveness. The report cites a survey by the Information Technology Association of America indicating that 50 percent of company executives in information technology report a lack of skilled workers as "the most significant barrier" to their companies' growth during the next year."
It is well known that Mathematics education becomes one of the most important factors of economics development. There is a strong society demand on well skilled in mathematics high school graduates. Here is another citation from the mentioned above Mathematics Equals Opportunity (White Paper -- October 20, 1997): "Mathematics ability will be even more important for well-paying jobs in the future. Some major firms already require job applicants to pass standardized mathematics and reading tests. For example, Diamond-Star Motors, a joint venture of Chrysler and Mitsubishi, tests all applicants for production and maintenance positions on their ability to do high school level mathematics."
However the situation with Mathematics in the American secondary schools is very far from being satisfactory. For example, according to the 2005 National Assessment of Educational Progress only 22% of eight-graders in California tested at or above proficiency in Mathematics. In 2005 an estimated 88 % of high school graduates who took the California High School Exit Exam (the class of 2006 is the first class in California required to pass the high school exit exam in order to receive a high school diploma) have passed the math portion of the test (see. Schools chief Jack O'connell releases 2004-05 California High School Exit Exam results, http://www.cde. ca.gov/nr/ne/ yr05/ yr05rel87. asp). A well trained, knowledgeable, methodically armed, dedicated Mathematics teacher is a main figure who enables to find the solution for all problems related to above issues, who enables to make a difference. There is a significant shortage of such teachers in California (not only in public, but also in private) schools. "California's public universities plan to more than double the number of science and math teachers they graduate to overcome a shortage of trained and credential instructors in those fields.... Both the University of California (UC) and the California State University (CSU) systems, which together now graduate about 1,000 math and science teachers, will use a combination of incentives to reach their goal of 2,500 teachers in four years. The initiative is vital to maintaining the state's competitiveness in a global economy, said CSU Chancellor Charles Reed. "Math and science is tied to California's economic future. Nothing we can do could be more important than preparing math and science teachers for California students." (California Major push to mint math, science teachers UC, CSU announce incentives to increase number of graduates - Tanya Schevitz, Chronicle Staff Writer Wednesday, June 1, 2005).
We are not going to talk here about private school teachers since each of these schools, even accredited with the most prestigious accreditation agency WASC (Western
Association of School and Colleges) has their own rules and requirements (usually much weaker than in public system) for its staff. We will focus on public school teacher training. There are few distinct ways that one could choose in order to become a Mathematics public school teacher. The main traditional way is to get a Bachelor Degree in Mathematics from an accredited by WASC university, to pass than a special mathematics subject exam (called CSET), and to take some additional general credential classes giving the applicant the main training in general education. This way could be significantly shorter if the applicant would choose an approved by CCTC (California Commission on Teacher Credentialing) university program. Such a program will give the applicant an opportunity to complete her/is teaching credentials together with the bachelor degree. In this case the applicant does not need to pass the CSET exam or takes additional courses. The restrictions on such approved programs have significantly straightening lately. To get such approval from CCTC is a very complicated process during which the university should prove that its Mathematics program satisfied by all CCTC high standards. Another way of achieving of mathematics teaching credential for the person having a Bachelor Degree in some majors using significant mathematics background (20 semester units, which equivalent of 300 lectures hours plus laboratory and homework hours, awarded from a mathematics department to the holder of this degree in such areas as engineering, accounting or finance, and so on) is the passing the mentioned above CSET exam and taking some general education credentialing classes. Few years ago besides regular single subject credentialing another form of Mathematics credentialing has been endorsed; namely The Single Subject Teaching Credential in Foundational-Level Mathematics "authorizes the holder to teach the content areas taught to the vast majority of California's K-12 public school math students: general Mathematics, Algebra, Geometry, Probability and Statistics, and Consumer Mathematics. Instruction is permitted in grades
twelve and below (CCTC, http://www.ctc.ca. gov/notices/coded/030010/03 0010.html). In other words, a teacher holding this type of credential can teach all mathematics secondary school courses except of Calculus based. The person who wants to get this credential needs to pass only two parts of the CSET, but not the third part dedicated to the Calculus based Mathematics. This opportunity has been given lately to applicants in order to satisfy an increasing demand for mathematics teachers in California public schools. It really makes sense since more than 90% of school courses are at foundational level and do not use any Calculus materials. We will talk more specifically about all three parts of CSET later in our next article.
Now we will expose an example of a typical CCTC approved Single Subject Credential in Mathematics program - the National University Mathematics Bachelor in Sciences Degree program. We will try to compare this program with approved by the Department of Education and Sciences of Ukraine program in Bachelor of Sciences in Mathematics implemented by Kiev National Pedagogic University named by M.P.Dragomanov. We will talk only Mathematics and natural sciences content courses. According to the National University 2006 Catalog description the Single Subject Preparation program in Mathematics at National University includes 945 lecture hours of mathematics coursework, 90 hours of computer science related coursework, and 45 lecture hours of physical sciences survey. Since the Ukrainian educators may not be familiar with some American titles for the course we supply each course with the brief description. Please also take in account that for each lecture hour the student needs to spend additional three hours doing his homework and projects.
Preparation to the Major Introduction to Probability and
Statistics (45 lecture hours)
This course offers an introduction to Probability Theory and Statistics, simple probability distributions, conditional probability (Bayes Rule), independence, expected value, binomial distributions, the
Central Limit Theorem, hypothesis testing, sampling, and analysis of variance.
Coupled with the described below Statistical Analysis course this course covered almost the same content as Ukrainian Probability Theory and Mathematical Statistics course (34 lecture hours and 51 seminar hours) does.
College Algebra and Trigonometry (45 lecture hours) This course examines higher degree polynomials, rational functions, trigonometry, and matrix algebra as needed for more specialized study in mathematics, computer science, engineering, and other related fields.
The content of this course are a part of the content of the Ukrainian Elementary Mathematics course (68 lecture hours and 128 seminar hours).
Introduction to Programming Concepts and Methods (45 lecture hours) An introduction to modern programming design techniques, and examines problem decomposition, modern programming paradigms and methods.
The corresponding course in the Ukrainian program is much wider and deeper course of Informatics (106 lecture hours, and 176 laboratory hours). Such a fundamental course included in the Ukrainian program as the base for the an additional minor in Informatics.
As a related course we can also mention Computer Technology in the Mathematics Classroom (45 lecture hours) course.
Survey of Physical Sciences (45 lecture hours) This course is an introduction to the basic principles and general concepts of the physical sciences. Topics include: the scientific method; laws of motion; energy; electricity and magnetism; heat; waves (sound and light); the atom; chemical bonds in molecules, solids, and liquids; chemical reactions; organic and inorganic chemistry; space and time; evolution of the universe.
In the Ukrainian Program we can list the Theoretical Physics and Mechanics courses as the corresponding area courses for this one (68 lecture hours and 68 laboratory hours
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total). We can count also the elective course of Experimental Physics (18 lecture hours plus 36 seminar hours). Definitely this area of the Ukrainian program is much stronger than American analog.
Calculus I, II, III, IV (180 lecture hours) This course is just an analog of the Ukrainian Mathematical Analysis course (210 lecture hours and 174 seminar hours) examines differentiation and integration concepts with applications to related rates, curve sketching, engineering optimization problems, and business applications, including the study of functions of several variables. Coupled with College Algebra and Trigonometry these courses also cover Analytic Geometry. We would say that these Ukrainian and American courses are very similar by contents.
Upper Division Requirements for the Major Topics from Geometry (45 lecture hours) This class is a survey of main concepts of Euclidean geometry with the emphasis on the axiomatic approach, constructions, logic of proof, and some ideas from non-Euclidean geometry including historical aspects.
The main content could be considered as a part of the Elementary Mathematics Ukrainian course content.
Mathematical Modeling I (45 lecture hours) An introductory course in mathematical modeling, utilizing a variety of interesting, useful, and diverse applications from physical, biological, business, social, and computer sciences.
A very attractive and useful course introduces prospective mathematicians to apply research.
Discrete Mathematics (45 lecture hours) This course studies combinatory and graph theory. It analyzes algorithms, logic, circuits, number bases, and proofs. Ample applications (graphs, counting problems, Turing Machines, codes) examine the ideas of Euler, Boole, Floyd, Warshall, Dijkstra,
Church and Turing, Shannon, Bernoulli. Graphing calculator is required.
The Ukrainian program includes a course under the same title (17 lecture hours plus 34 seminar hours).
Linear Algebra (45 lecture hours) This course examines systems of linear equations and matrices, elementary vector-space concepts, and geometric interpretations. Discusses finite dimensional vector spaces, linear functions and their matrix representations, determinants, similarity of matrices, inner product, rank, eigenvalues and eigenvectors, canonical form, and GramSchmidt process.
The Ukrainian program includes the same course but covered with 70 lecture and 70 seminar hours. Differential Equations (45 lecture hours) This course is a study of ordinary differential equations with emphasis on linear equations and systems of linear equations. An analysis of the existence and uniqueness of solutions of ordinary differential equations with initial conditions, so called Cauchy problem. Examines linear differential equations of first, second and higher orders, and linear systems of ordinary differential equations. It includes infinite series, Laplace transform and matrix methods of solution. It stresses application to engineering problems.
In the Ukrainian program this course covered by the Differential Equations (36 lectures hours and 36 seminar hours). Number Theory (45 lecture hours) An examination of fundamental concepts of numbers, including divisibility, congruencies, the distribution of Primes, Pythagorean triples, the Euclidean Algorithm, the Fundamental Theorem of Arithmetic, Diophantine equations and Goldbach's conjecture and other unsolved problems of number theory.
Algebraic Structures (45 lecture hours) A look at groups, rings, and fields, as well as applications of these systems. Discusses equivalence relations, Lagrange's theorem, homomorphisms, isomorphisms, Cayley's theorem, and quaternions. Also examines error correcting codes, and issues of
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cryptography. Graphing calculator may be required.
Together with Abstract Algebra with Applications and Number Theory this course would compete with the Ukrainian Algebra and Number Theory course (70 lecture hours and 70 seminar hours).
Foundation of Geometry (45 lecture hours) A discussion of fundamental ideas and processes common to Euclidean and Non-Euclidean geometries: projective, affine and metric geometry. Examines the interplay between inductive and deductive reasoning, and formal and informal proof. Addresses uses in areas such as science (transformations, scaling), art (Escher-type tessellations, projections), architecture (three-dimensional figures), and computer science (fractals, computer-aided design.)
Related Ukrainian program course is an elective Projective Geometry course (34 lecture hours and 34 seminar hours). However in this program we have such an advance course as Differential Geometry and Topology (36 lecture hours and 36 seminar hours) the partial content of which is very briefly highlighted in the Advanced Calculus American course.
Statistical Analysis (45 lecture hours) An examination of statistical applications to business, computer science, psychology, education, social sciences, and mathematics with fundamental concepts of probability distribution, mathematical models relating independent and dependent random variables, hypothesis testing and experimental design. Study includes fundamental analysis of variance, various distributions and methods of regression, analysis and scaling.
Advanced Calculus (45 lecture hours) This course has no analog in the Ukrainian Program. However a big part of it is covered by the Mathematical Analysis course. A look at sets, functions, and the real numbers as an ordered set. Topics include the Completeness axiom, cardinality, and Cantor's theorem, Limsup, and Liminf; the of R1 and R2, open sets, and limit points as well as compactness and the Heine-Borel theorem;
the properties of continuous functions, uniform continuity, the Mean-Value theorem, inverse functions and differentiability; the Riemann integral, and Lebesgue measure.
History of Mathematics (45 lecture hours) Throughout history, mathematics has changed the way people view the world. This course examines currents in the development of mathematics and throughout ancient Egypt, Babylon, China, and the Middle East. It studies math's influence on society through the major events of Europe, contemporary developments, and some projections into the future, including the women and men who played key roles in evolution. Readings and problems are taken from original as well as secondary sources.
We think that such a course is very important for the teacher development and would be a great asset to the Ukrainian program.
Applied Mathematical Modeling
(capstone course) (45 lecture hours) A capstone course for BS in Mathematics, this course is intended to culminate the mathematics major studies and should be taken at or near the end of the program. Addresses important problem areas such as political science, ecology, psychology, sociology, economics, anthropology, business, and institutional planning using mathematical techniques from areas such as calculus, geometry, probability and statistics, linear and matrix algebra, and linear programming. Discusses principles and methods of constructing, analyzing, interpreting, evaluating, and refining models. Compares mathematical models including analytic and simulation, discrete and continuous, and deterministic and stochastic.
Additional Requirement for Single Subject Preparation students only Mathematics Practicum and Portfolio (Should be taken as early in the student's program as possible.
This pedagogical field experience course has two objectives. First, it provides an opportunity for students to observe and reflect on the actual work of public middle/secondary school mathematics teachers. Students observe
at least 28 hours in public middle or secondary school mathematics classrooms and at least 3.5 hours of mathematics-related student activities and administrative meetings. The second objective of the course is to familiarize students with the requirements of the assessment portfolio they must submit as part of the Single Subject Matter Preparation program. To meet this objective, students begin planning the production of their assessment portfolio and write a brief essay related to one they will have to submit with that portfolio.
In the Ukrainian program one can see three practicum courses with significantly bigger amount of hours, including real classroom teaching experience.
Upper-Division Concentration Requirements Concentration in Mathematics and
Applications Numerical Analysis (45 lecture hours) An introduction to numerical computation employed widely in industry and research. Discusses errors in numerical computation, truncation and discretization, and machine storage restrictions as well as function approximation, roots of nonlinear equations, systems of linear equations, algebraic eigenvalue problems, polynomial interpolation, and cubic spline interpolations, quadratures, numerical differentiation, initial and boundary-value problems. Programmed algorithms may be utilized. Corresponds to the Ukrainian Methods of Computations (34 lecture hours and 34 laboratory hours).
Abstract Algebra with Applications
(45 lecture hours) This course continues and advances the work done in Algebraic Structures, discussing selected fundamental algebraic structures and their applications to computations. The main concepts of Sylow Theory of finite groups, Galois Theory, Lattices Theory, Coding Theory and Cryptography, Boolean Algebra and Switching Theory are developed. Finite permutation groups (Cayley's Theorem) and their applications in science and arts are studied.
Functions of Complex Variables and its Applications (45 lecture hours) This course is an analog to Ukrainian Complex Analysis (34 lecture hours and 34 seminar hours) and includes study of functions of complex variables and their applications to other mathematics branches, sciences, and engineering. The following topics will be examined: the complex plane, analytic functions, integration and Cauchy's Theorem, sequences and series, residue calculus, Fourier and Laplace transforms, and applications.
Mathematics Project Course
The project courses are not independent study. They are directed student team project or internships in mathematics. Utilization of previously acquired skills knowledge is required to complete the project. Students can select project topics from industry, government, business, education, or research.
Upper-Division Concentration Requirements Single Subject Teaching Concentration Computer Technology in the Mathematics Classroom (45 lecture hours) An overview of the use of computer-based technology in mathematics educational environments. Evaluates graphing calculators, and computer software such as Maple, Scientific Workplace, Geometer's Sketchpad, MiniTab, SPSS, and others.
Same kind of course one can see among the elective courses of the Ukrainian program (11 lecture hours and 22 seminar hours).
Problem Solving Strategy (45 lecture hours) The course will develop student's abilities to solve mathematics problems. The aim in the course is not to impart any specific body of knowledge, but rather to foster the students' understanding that mathematics is a science of identifying, solving problems and generalizing. The course helps to prospective mathematics teachers to acquire their professional skills in the teaching of mathematics in secondary school, teach and assess problem solving. A survey of most famous math problems will be
given. Most popular problems from the secondary school mathematics courses will be considered. The course includes the description of main approaches to solving standard and challenge math problems. Students will learn strategies most widely used: pattern recognition, working backwards, guess and test, experimentation or simulation, reduction expansion, organized listing and exhaustive listing, logical deduction, mathematics induction, divide and conquer, writing equations, producing fruitful sketches.
Methods of Teaching of Mathematics (45 lecture hours) This course is designed as a critical inquiry into present-day tendencies in teaching mathematics in order to help prospective mathematics teachers to acquire their professional skills in the teaching of mathematics in secondary school. Fundamental concepts of mathematics teaching, main teaching strategies, methods and forms of organization of students'' learning, survey of main concepts of basics mathematics, algebra, geometry, trigonometry, functions, discrete mathematics, probability, statistics, beginning calculus will be studied. Effective approaches to the teaching of main mathematics ideas will be discussed. Graphics calculators, computer mathematics learning and tutorial software,
different kinds of manipulatives and their uses in the classroom will also be considered.
These two courses above form the analog of the Ukrainian Methods of Teaching Mathematics Course (64 lectures and 74 seminar hours).
Mathematics Project Course
(See above)
Upon completion of the program the student submits a Portfolio reflecting the main stages of her/is development and takes an exit exam highlighting main topics of the program. The primary instrument of summative assessment of programs outcomes is the required Portfolio, which students submit at the end of the Program. This Portfolio requires students to present samples of work from all required classes and electives and to write an essay reflecting on their experiences in the program and how those experiences have prepared them for their future work as teachers.
In general, we can observe that the American and the Ukrainian Mathematics teachers training have a lot of in common. This fact is a simple consequence of the international character of Mathematics. However, we can also note some dissimilarity reflecting the tradition of Mathematics educations in these two countries. We are going to continue the cycle of articles devoted to this subject.
Резюме. Subbotin I., Bilotskii N.N., Milla Hill. СОВЕРШЕНСТВОВАНИЕ УЧИТЕЛЯ МАТЕМАТИКИ В КАЛИФОРНИИ (США) И УКРАИНЕ. КРАТКИЙ СРАВНИТЕЛЬНЫЙ АНАЛИЗ.
Эта статья первая в цикле, посвященная процессу подготовки преподавателей математики в Калифорнии в сравнении с подготовкой преподавателей математики в Украине.
Summary. Subbotin I., Bilotskii N.N., Milla Hill. MATHEMATICS TEACHERS' DEVELOPMENT IN CALIFORNIA (USA) AND UKRAINE. BRIEF COMPARATIVE ANALYSIS. This article is the first in the cycle dedicated to the Mathematics teachers' development process in California comparing with the Mathematics teachers' preparation in Ukraine.
Надшшла до редакци 11.05.2006р.
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