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4MATHEMATICS TEACHERS' DEVELOPMENT IN CALIFORNIA (USA): CALIFORNIA SUBJECT EXAMINATIONS FOR TEACHERS
(СОВЕРШЕНСТВОВАНИЕ УЧИТЕЛЕЙ МАТЕМАТИКИ В КАЛИФОРНИИ (США): ЭКЗАМЕН ПО ПРЕДМЕТУ ДЛЯ УЧИТЕЛЕЙ В КАЛИФОРНИИ)
I.Subbotin, Professor,
National University, Los Angeles, USA,
N.N.Bilotskii, Assocoate Professor, National Pedagogic University, Kiev, UKRAIN M.Hill,
Mathematics Coordinator, Yavneh Academy, Los Angeles, USA
Ця стаття e другою в циклi, присвяченому процесу удосконалювання вчителгв математики в Калiфорнii (США) i Украт. Вона стосуеться екзамену з предмету для вчителiв математики в Калiфорнii.
The main goal of this article is to observe and discuss the California Subject Examinations for Teachers (in Mathematics). This very important part of the Mathematics teachers' preparation process in California, USA, ensures the minimum level of knowledge in the subject that the prospective mathematics teacher supposes to demonstrate in order to be able effectively teach Mathematics in the secondary school. We believe that this information will be interesting and useful for Ukrainian mathematics educators.
A well trained, knowledgeable, methodically armed, dedicated mathematics teacher is one of the key figures in the secondary school education. California State University System Chancellor Charles Reed wrote: "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).
In our previous article (Subbotin,I., Bilotsckii,N, Hill,M., Mathematics teachers' development in California (USA) and Ukraine. Brief comparative analysis. Didactics of Mathematics: Problems and Investigations. -Doneck: Company TEAN, 26, 2006) we discussed and compared Californian and Ukrainian Mathematics teacher's preparation programs. In the current article we will briefly discuss the California Subject Examinations for Teachers in Mathematics (SCET). In order to achieve the mathematics teaching credential (license to work in California Public School System) a 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) supposes to pass the CSET exam and takes some education credentialing classes. Few years, ago besides the 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/030010/0 30010.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 examinees 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.
The content of the exams has been determined by the following document: Mathematics Teacher Preparation in California: Standards of Quality and Effectiveness for Subject Matter Programs, Created and Recommended by the Mathematics Subject Matter Advisory Panel (2001-2003), State of California 1900 Capitol Avenue Sacramento, California 95814 2003, p 48-54. According to Mathematics Content Standards for California Public Schools (1997), as outlined in the Mathematics Framework for California Public Schools: Kindergarten Through Grade Twelve (1999) the prospective mathematics teacher needs to know from an advanced standpoint the following mathematics domains:
Domain 1. Algebra
1.1. Algebraic Structures: a. Know why the real and complex numbers are each a field, and that particular rings are not fields (e.g., integers, polynomial rings, matrix rings);
b. Apply basic properties of real and complex numbers in constructing mathematical arguments (e.g., if a<b and c<0, then ac>bc);
c. Know that the rational numbers and real numbers can be ordered and that the complex
numbers cannot be ordered, but that any polynomial equation with real coefficients can be solved in the complex field.
1.2. Polynomial Equations and Inequalities.
a. Know why graphs of linear inequalities are half planes and be able to apply this fact (e.g., linear programming); b. Prove and use the following: The Rational Root Theorem for polynomials with integer coefficients; The Factor Theorem; The Conjugate Roots Theorem for polynomial equations with real coefficients; The Quadratic Formula for real and complex quadratic polynomials; The Binomial Theorem; c. Analyze and solve polynomial equations with real coefficients using the Fundamental Theorem of Algebra.
1.3. Functions. a. Analyze and prove general properties of functions (i.e., domain and range, one-to-one, onto, inverses, composition, and differences between relations and functions); b. Analyze properties of polynomial, rational, radical, and absolute value functions in a variety of ways (e.g., graphing, solving problems); c. Analyze properties of exponential and logarithmic functions in a variety of ways (e.g., graphing, solving problems).
1.4. Linear Algebra. a. Understand and apply the geometric interpretation and basic operations of vectors in two and three dimensions, including their scalar multiples and scalar (dot) and cross products; b. Prove the basic properties of vectors (e.g., perpendicular vectors have zero dot product); c. Understand and apply the basic properties and operations of matrices and determinants (e.g., to determine the solvability of linear systems of equations).
Domain 2. Geometry
2.1. Parallelism. a. Know the Parallel Postulate and its implications, and justify its equivalents (e.g., the Alternate Interior Angle Theorem, the angle sum of every triangle is 180 degrees); b. Know that variants of the Parallel Postulate produce non-Euclidean geometries (e.g., spherical, hyperbolic).
2.2. Plane Euclidean Geometry. a. Prove theorems and solve problems involving similarity and congruence; b. Understand, apply, and justify properties of triangles (e.g., the Exterior Angle Theorem, concurrence theorems,
trigonometric ratios, Triangle Inequality, Law of Sines, Law of Cosines, the Pythagorean Theorem and its converse); c. Understand, apply, and justify properties of polygons and circles from an advanced standpoint (e.g., derive the area formulas for regular polygons and circles from the area of a triangle); d. Justify and perform the classical constructions (e.g., angle bisector, perpendicular bisector, replicating shapes, regular n-gons for n equal to 3, 4, 5, 6, and 8); e. Use techniques in coordinate geometry to prove geometric theorems.
2.3. Three-Dimensional Geometry. a. Demonstrate an understanding of parallelism and perpendicularity of lines and planes in three dimensions; b. Understand, apply, and justify properties of three-dimensional objects from an advanced standpoint (e.g., derive the volume and surface area formulas for prisms, pyramids, cones, cylinders, and spheres).
2.4. Transformational Geometry.
a. Demonstrate an understanding of the basic properties of isometries in two- and three-dimensional space (e.g., rotation, translation, reflection); b. Understand and prove the basic properties of dilations (e.g., similarity transformations or change of scale).
Domain 3. Number Theory
3.1. Natural Numbers. a. Prove and use basic properties of natural numbers (e.g., properties of divisibility); b. Use the Principle of Mathematical Induction to prove results in number theory; c. Know and apply the Euclidean Algorithm; d. Apply the Fundamental Theorem of Arithmetic (e.g., find the greatest common factor and the least common multiple, show that every fraction is equivalent to a unique fraction where the numerator and denominator are relatively prime, prove that the square root of any number, not a perfect square number, is irrational).
Domain 4. Probability and Statistics
4.1. Probability. a. Prove and apply basic principles of permutations and combinations;
b. Illustrate finite probability using a variety of examples and models (e.g., the fundamental counting principles); c. Use and explain the concept of conditional probability; d. Interpret the probability of an outcome; e. Use normal,
binomial, and exponential distributions to solve and interpret probability problems.
4.2. Statistics. a. Compute and interpret the mean, median, and mode of both discrete and continuous distributions; b. Compute and interpret quartiles, range, variance, and standard deviation of both discrete and continuous distributions; c. Select and evaluate sampling methods appropriate to a task (e.g., random, systematic, cluster, convenience sampling) and display the results; d. Know the method of least squares and apply it to linear regression and correlation; e. Know and apply the chi-square test.
Domain 5. Calculus
5.1. Trigonometry. a. Prove that the
Pythagorean Theorem is equivalent to the
2 2
trigonometric identity sinx+cosx=1 and that
this identity leads to 1+tan2x=sec2x and
22
1+cot x=csc x; b. Prove the sine, cosine, and tangent sum formulas for all real values, and derive special applications of the sum formulas (e.g., double angle, half angle); c. Analyze properties of trigonometric functions in a variety of ways (e.g., graphing and solving problems); d. Know and apply the definitions and properties of inverse trigonometric functions (i.e., arcsin, arccos, and arctan); e.Understand and apply polar representations of complex numbers (e.g., DeMoivre's Theorem).
5.2. Limits and Continuity. a. Derive basic properties of limits and continuity, including the Sum, Difference, Product, Constant Multiple, and Quotient Rules, using the formal definition of a limit; b. Show that a polynomial function is continuous at a point; c. Know and apply the Intermediate Value Theorem, using the geometric implications of continuity.
5.3. Derivatives and Applications. a.
Derive the rules of differentiation for polynomial, trigonometric, and logarithmic functions using the formal definition of derivative; b. Interpret the concept of derivative geometrically, numerically, and analytically (i.e., slope of the tangent, limit of difference quotients, extrema, Newton's method, and instantaneous rate of change); c. Interpret both continuous and differentiable functions geometrically and analytically and apply Rolle's Theorem, the Mean Value Theorem, and L'Hopital's rule; d. Use the derivative to solve
dD
rectilinear motion, related rate, and optimization problems; e. Use the derivative to analyze functions and planar curves (e.g., maxima, minima, inflection points, concavity); f. Solve separable first-order differential equations and apply them to growth and decay problems.
5.4. Integrals and Applications. a. Derive definite integrals of standard algebraic functions using the formal definition of integral; b. Interpret the concept of a definite integral geometrically, numerically, and analytically (e.g., limit of Riemann sums); c. Prove the Fundamental Theorem of Calculus, and use it to interpret definite integrals as antiderivatives; d. Apply the concept of integrals to compute the length of curves and the areas and volumes of geometric figures.
5.5. Sequences and Series. a. Derive and apply the formulas for the sums of finite arithmetic series and finite and infinite geometric series (e.g., express repeating decimals as a rational number); b. Determine convergence of a given sequence or series using standard techniques (e.g., Ratio, Comparison, Integral Tests); c. Calculate Taylor series and Taylor polynomials of basic functions.
Domain 6. History of Mathematics
6.1. Chronological and Topical Development of Mathematics. a. Demonstrate understanding of the development of mathematics, its cultural connections, and its contributions to society; b. Demonstrate understanding of the historical development of mathematics, including the contributions of diverse populations as determined by race, ethnicity, culture, geography, and gender.
Based on this document National Evaluation System, Inc (NES) has developed SCET for prospective mathematics teachers. This exam consists from three subtests: Subtest I: Algebra and Number Theory, Subtest II: Geometry, Probability and Statistics, and Subtest III: Calculus and History of Mathematics. A graphing calculator is allowed for Subtest II only. The examinee can take all three subtests in one day or each subtest per a time. The exam duration for each subtest is five hours. The exams are conducted six times per year. In average, each subtest consist from 30 multiply answers questions and four open answer problems. To pass the each part of subtest the examinee must answers correctly on
75% of questions. A partial solution of an open answer problem is also counted as a partial grade. The main idea of the test is checking not only basic skills and knowledge but mostly the level of mathematical culture. All problems in general are moderate; some of them even easy, but it allows checking effectively the examinee knowledge and skills level.
Below we comment some examples of such problems and their solutions retrieved from the NES website www.cset.nesinc.com. We begin with the following simple questions from Subtest I.
The problem below checks only some basic concept of algebraic structures, but requires knowledge in matrixes and operations in GL3(3).
1. Which of the following statements refutes the claim that GLR(3), the set of 3 by 3 invertible matrices over the real numbers, is a field?
A. There exist elements A and B of GLR(3) such that AB * BA.
B. There exist elements A and B of GLR(3) such that det(AB) = det(A)det(B).
C. If A is an element of GLR(3), then there exists a matrix A-1 such that A-1 A = I.
D. If A is an element of GLR(3), then there exists a matrix A such that det(A) .
The answer is A. A field is a commutative ring with identity in which every nonzero element has an inverse. The set of 3 by 3 invertible matrices is not a commutative ring because there exist matrices A and B such that AB * BA.
The following simple problem checks understanding of main concepts of number systems with its connections to algebraic structures.
2. Which of the following sets is an ordered field?
A. the complex numbers
B. the rational numbers
C. the integers
D. the natural numbers
The correct answer is B. Of the sets given in the response choices, only the complex numbers and the rational numbers are fields. Since the complex numbers are not ordered, response choice B is the correct response.
The next example confirms understanding of main notions of polynomials.
3. Use the graph of a polynomial function below to answer the question that follows.
©
y= pW
Which of the following statements about p(x) must be true?
A. p(x) has at least one complex root
B. p(x) is divisible by (x - 2)
C. p(x) is an odd function
D. p(x) is divisible by x2 - 6x + 9
The correct answer is D. The polynomial p(x) has x-intercepts at 3 and -2, so (x-3) and (x+2) are factors. In fact, x = 3 is a double root, since the function is tangent to the x-axis at x=3. This means that (x-3)2, or (x2-6x+9), is a factor of p(x), so p(x) is divisible by x2-6x+9.
The following problem uses some basic knowledge in Combinatorial Analysis and Number Theory.
4. If x, y, and z are nonnegative integers, what is the total number of factors of 2x3y5z?
A. (2 + 3 + 5) (x + y + z)
B. xyz
C. (x +3 1) (y + 1) (z + 1)
D. x2yV
The correct answer is D: Each factor of 2x3y5z is the product of between 0 and x twos, between 0 and y threes, and between 0 and z fives. Since this yields (x + 1) possible products of twos, (y + 1 ) possible products of threes and (z + 1) possible products of fives, there are (x + 1)(y + 1)(z + 1) factors.
The next question requires the knowing of
the Euclid proof of the irrationality of V2.
5. Use the informal proof below to answer the question that follows.
Let x be a positive integer that is not a
perfect square. Assume that Vx =y/z, and that y and z are relatively prime integers. It follows that x = (yz)2 = y2/z2. Since x is an integer, z2 is equal to 1, and x is the square of y. But this is not possible.
Which of the following is shown by this informal proof?
A. The square root of any integer that is not a perfect square is an algebraic number.
B. The square of any rational number is also a rational number.
C. The square root of any positive integer that is not a perfect square is irrational.
D. The square of any real number is also a real number.
The correct answer is C: The proof begins by assuming that the square root of a positive integer that is not a perfect square is rational, by rewriting it as a fraction. This assertion leads to a contradiction, which shows that the initial assumption is not correct.
Here is an example of open answer problem (which means that the examinee should complete and carefully write the solution) pertaining to the Linear Algebra.
6. If vectors a = (a1, a2) and b = (b1, b2) ! cos0 - sin0^
are perpendicular, A =
, and
v sin 0 cos 0 y if we identify vectors with column matrices in the usual manner, then show that the vectors
A a and A b are perpendicular for all values of H.
The following two problems are from the Subtest II and give you an idea about Geometry and Statistics involved in the exam. It worthy to mention, that this subtest includes a significant amount of simple geometric and probability problems. However some problems require the base knowledge on non-Euclidian Geometry and far from basic knowledge on Probability and Statistics.
7. A hexagon undergoes a coordinate transformation given by T:(x, y) ^ (3x, 2y). What is the ratio of the area of the original polygon to the area of the transformed hexagon?
The following solution demonstrates a significant level of mathematical culture and experience.
Since T is a linear transformation, one need only check how T transforms the area of a square with vertex coordinates (0, 0), (0, 1), (1, 1), and (1, 0). This square has area 1. The image of these points is (0, 0), (0, 2), (3, 2), and (3, 0). This is a rectangle of area 2. Hence, T maps each unit square to a rectangle of area 2. It follows that if A is the area of the original hexagon, 3A is the area of the transformed
<ü)
hexagon, and the ratio of the original hexagon to the transformed hexagon is 2.
8. The volume of liquid in soda cans is normally distributed with a mean of 12 fl. oz. and a standard deviation of 0.05 fl. oz. What is the approximate percentage of cans of this brand of soda that contain less than 11.9 fl. oz.?
A. 0.5% B. 1 C. 2.5% D. 5%
The correct answer is C.: If the mean of a normally distributed data set is 12 fl. oz. and the standard deviation is 0.05 fl. oz., then approximately 67% of the points in the data set have values within 0.05 fl. oz. of the mean and approximately 95% of the data points are within 0.1 fl. oz. (i.e., two standard deviations) of the mean, that is, between 11.9 fl. oz. and 12.1 fl. oz. This means that 2.5% of the data points are below 11.9 fl. oz., and 2.5% are above 12.1 fl. oz.
The Subtest III checks the examinee skills in Calculus and History of Mathematics. The problems from Calculus are pretty basic. For instance:
9. Which of the following represents the area under the curve the function h(x) = 4/3x + 2 on the interval [0, 2]?
1 0. For what values of x does the infinite series 1-(x-2)+(x-2)2-(x-2)3+... converge?
The following two problems require some expertise in History of Mathematics.
11. A Babylonian tablet from circa 300 B.C. contains the following problem:
There are two fields whose total area is 1800 square yards. One produces grain at the rate of 3 of a bushel per square yard while the other produces grain at the rate of 2 of a bushel per square yard. if the total yield is 1100 bushels, what is the size of each field?
The above problem indicates the beginning of which of the following contemporary topics of mathematics?
A. systems of linear equations
B. calculus
C. quadratic equations
D. Euclidian geometry.
1 2. In the eleventh century, the mathematician Omar Khayyam presented the generalized geometric solution to some cubic equations using intersecting conic sections. This concept of applying algebra to geometry, which centuries later developed into the topic of analytical geometry, is more notably attributed to which of the following mathematicians?
A. David Hilbert
B. Blaise Pascal
C. Rene Descartes
As an open answer question example we can show the following problem.
13. Use the Fundamental Theorem of Calculus stated below to complete the exercise that follows.
Let f be continuous on a closed interval [a, b] and let x be any point in [a, b]. If F is
defined by F (x) = J f (t )dt, then F'(x)=f(x) at
a
each point x in the interval [a, b].
Using the formal definition of the derivative, prove the above theorem.
From this small pool of examples we can see, the SCET test in Mathematics satisfactory checks the required minimum skills and knowledge of a prospective Mathematics teacher. It does not include any problem above the basic level, but quite positively insures that the examinee who successfully passed all three parts of this test has enough skills, knowledge and mathematical culture for the future profession.
Резюме. Subbotin I., Bilotskii N., Hill M. СОВЕРШЕНСТВОВАНИЕ УЧИТЕЛЕЙ МАТЕМАТИКИ В КАЛИФОРНИИ (США): ЭКЗАМЕН ПО ПРЕДМЕТУ ДЛЯ УЧИТЕЛЕЙ В КАЛИФОРНИИ. Эта статья является второй в цикле, посвященном процессу совершенствования учителей математики в Калифорнии (США) и Украине. Она касается экзамена по предмету для учителей математики в Калифорнии.
Summary. Subbotin I., Bilotskii N., Hill M. MATHEMATICS TEACHERS' DEVELOPMENT IN CALIFORNIA (USA): CALIFORNIA SUBJECT EXAMINATIONS FOR TEACHERS. This article is the second part in the cycle dedicated to the Mathematics teachers' development process in California (USA) and Ukraine. It dedicated to the California Subject Examinations for Mathematics Teachers.
Надшшла доредакцп 28.12.2006р.
