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21CHALLENGES OF EXAMINATIONS FOR MATHEMATICS TEACHERS IN CALIFORNIA
(ТРУДНОСТИ ПРИ СДАЧЕ ЛИЦЕНЗИОННЫХ ЭКЗАМЕНОВ УЧИТЕЛЕЙ МАТЕМАТИКИ В КАЛИФОРНИИ)
Nina V. Stankous, Ph.D.
National University 11255 N. Torrey Pines Road La Jolla, CA 92037 USA
У св1тл1 того, що школярг в США показують низькг результати в математицг, а вчител1в математики недостатньо, питания тдготовки останнх стае критичним. У дант робот1 розглядаеться кшька важливих питань, пов'язаних з тдготовкою вчите-л1в математики, особливо в Кал1форнП Визначен основы проблеми при тдготовщ до здач1 спещально тесту для вчител1в математики в старших класах (CSET). Також в робот1 пропонуються шляхи вир1шення цих проблем I розроблена низка рекомендаций для полтшення якост1 тдготовки майбутшх учителгв математики в Кал1форнП
Ключовi слова: тдготовка вчителя математики, концептуальный тдхгд, логгчна аргуме-нтацгя.
Introduction: Analysis of Mathematics Teachers Preparation in US and California
The first indicator of mathematics education in any country is the student performance. Let's take a look at how achievement of American students compare to students in other countries.
The Program for International Student Assessment (PISA) is a system of international assessments that measures 15-year-olds' capabilities in reading, mathematics, and science every 3 years. PISA was first implemented in 2000 and is carried out by the Organization for Economic Cooperation and Development (OECD), an intergovernmental organization of industrialized countries.
On the 2006 PISA, the average score of U.S. 15-year-olds in mathematics literacy was 474, which was lower than the OECD average of 498. (Possible scores on PISA assessments range from 0 to 1,000.) The average mathematics literacy score in the United States was lower than the average score in 23 of the other 29 OECD countries for which comparable PISA results were reported, higher than the average score in 4 of the other OECD countries, and not measurably different from the
average score in 2 of the OECD countries. Comparable mathematics literacy results were also reported for 27 non-OECD jurisdictions, 8 of which had higher average scores than did the United States (U.S. Department of Education, National Center for Education Statistics. Digest of Education Statistics, 2009, Chapter 6, (NCES 2010-013)).
Those results show that mathematics education in US is in need of radical improvement, as is acknowledged by politicians, educators and society in general. To meet these expectations we need to prepare better teachers for schools, yet this problems is quite complicated. In his article :"Strengthen Teacher Quality", Dr. Whitehurst says: "We would not tolerate a system in which airline pilots varied appreciably in their ability to accomplish their tasks successfully, for who would want to be a passenger on the plane with the pilot who is in the 10th percentile of safe landing", and yet, that's unfortunately true about mathematics teachers.
A new report from the Teacher Education Study in Mathematics posted on April 19, 2010, states that U.S. mathematics teachers are not as prepared as their international counterparts. Based on a survey of more than 23,000
future teachers in sixteen countries, including about 3,300 in US, the research found that the top-achieving countries allocated half of their teacher preparation courses for future middle school teachers to the study of formal mathematics, compared to 40 percent in the United States. In the top-performing countries, at least 90 percent of future mathematics teachers took both linear algebra and a basic year-long sequence in calculus - considered core courses in the study of formal mathematics - compared to the 66 percent of teachers who took linear algebra and 55 percent of teachers who took calculus in the U.S.
To address these disappointing results, the authors of the report recommend recruiting teachers with stronger mathematics backgrounds, establishing more rigorous state certification requirements for mathematics teachers, and requiring more demanding mathematics courses in all teacher preparation programs.
The history of research on mathematics teachers' preparation is long and contradictory. In 1966, sociologist James Coleman in his work "Equality of Educational Opportunity" suggested that differences in teachers such as their scores on a vocabulary test, level of education, years of experience etc. did not matter much for students' achievement. The study involved 60,000 teachers in over 3,000 schools.
More recent studies (Brewer & Goldhaber, 2000; Monk, 1994; Monk & King, 1994; Rowan, Chiang, & Miller, 1997) have shown much higher influence of teachers on student academic achievement than reported by Coleman. One of the major characteristics of effective teachers is subject matter knowledge. The effects of teacher preparation on student academic achievement become clearer when the focus of research moved to subject matter knowledge. The research is generally consistent in indicating that high school mathematics and science teachers with a major in their field of instruction have higher achieving students than teachers who are teaching out-of-field (this term means that the teachers may be fully prepared, but are assigned to teach science and mathematics which are out of their area of expertise). These effects be-
come stronger in advanced mathematics and science courses in which the teacher's content knowledge is presumably more critical (Monk, 1994; Chiang, 1996).
The situation with mathematics teachers in California is just a reflection of the national picture. The California Council on Science and Technology issued a report "Critical Path Analysis of California's Science and Mathematics Teacher Preparation System" (2007). The report revealed that more than ten percent of all mathematics and science teachers are underprepared, and lack the training and experience necessary for a teaching credential in the subject they teach. More than one third of novice teachers (those in their first or second year) teaching mathematics or science are un-derprepared. They call teachers "fully prepared" when they are credentialed and are teaching courses in which they are certified, while the rest of the teachers are considered underprepared. If one takes a look more carefully at the numbers and break them down into the two categories of middle and high schools you will see that the most critical situation is in high school.
According to this report, at the middle school level, 10% of science and mathematics teachers are underprepared, and nearly 30% of novice science and mathematics teachers are underprepared. About 9% of science teachers and 12% of mathematics teachers teach out-of-field.
At the high school level, 9% and 12 % of science and mathematics teachers respectfully, are underprepared; and an even larger percentage (35% and 40% of novice science and mathematics teachers respectfully) are under-prepared.
The conclusion of the report is troubling: despite efforts in California to boost the quantity and quality of fully prepared and effective teachers a shortage of these teachers in science and mathematics persists. If the current trends continue, California will remain in the distressing condition of leaving students with under-prepared science and math teachers, and out-of-field teachers throughout the entire state school system.
After getting acquainted with the analysis
of mathematics teacher preparation one may be tempted to ask a question: What is so difficult about the preparation of mathematics teachers? That is the title of one of the publications devoted to this problem (Wu, 2002). One can also find discussion of this topic in many other publications (Ball, 1991; Cuoco, 2001; MET, 2001) starting in late 1980's through recent times. As stated by H.Wu, "it is generally recognized that the absence of logical reasoning from mathematics classrooms is a main culprit in bringing about the present mathematics education crisis"(Wu, 2002, p.4). At the end of his article, he says that his basic conviction is "In mathematics, content guides pedagogy". It does not mean that the content knowledge is all it takes to be a good teacher, it just means that a solid knowledge of mathematics is crucial to competent mathematical teaching. One cannot discuss pedagogy without reference to mathematical content; pedagogical considerations make sense only after the teachers are comfortable with the content. One of the problems of mathematics teacher preparation is that those perspective teachers already graduated from schools with the existing mathematics education troubles, and bring their fear and anxiety of mathematics to the higher education classroom. To break that vicious circle is the main challenge of mathematics teacher preparation.
Structure and requirements of CSET Single Subject Mathematics examination
There are two alternatives (established by the Ryan Act) for prospective teachers to satisfy the subject matter requirements:
• complete an approved subject matter program, or
• achieve a passing score on an adopted examination.
The California Subject Examinations for Teachers® (CSET®) have been developed by the California Commission on Teacher Cre-dentialing (CTC) for prospective teachers who choose to or are required to meet specific requirements for certification by taking examinations (http://www.cset.nesinc.com). Completion of a Commission-approved program in mathematics, as an alternative way to meet the subject matter requirements, takes on average
3 to 4 years of study in an accredited College or University. This option is designed to provide the opportunity to be exempt from the CSET testing, and only for 50% - 60% of all prospective teachers enroll in such programs. The remaining 40% - 50% of prospective mathematics teachers dare to challenge the CSET examination in mathematics.
The California Commission on Teacher Credentialing (CCTC) has approved the content area examination for Single Subject Areas. Successfully passing the examination is one of the statutory requirements for the California Single Subject Teaching Credential. The tests of the CSET are criterion referenced and based on CCTC- approved subject matter requirements (also called content specifications). A criterion-referenced test is designed to measure a candidate's knowledge and skills in relation to an established standard. It is designed to measure domains of subject-matter content knowledge. All subject matter requirements are developed for the CSET program by committees of California educators and approved by the CCTC.
The test questions matching to the subject matter requirements were developed using, in part, textbooks, California curriculum syllabi, teacher education curricula, and teacher cre-dentialing standards. The questions were developed in consultation with and approved by committees of educators, teacher educators, and other content and assessment specialists in California.
Among recommendations given in the report (Critical Path Analysis of California's Science and Mathematics Teacher Preparation System", 2007) regarding of how to improve and fix the situation, there are a few related to the development of programs designed to encourage experienced people from business and industry to enter teaching, especially in the areas of science and mathematics. One of the ways for them to become mathematics teachers is to pass CSET Single Subject Mathematics exam.
Usually, CSET takers are:
• Current California high school teachers.
• Future teachers.
• Teachers who move to California.
• Teachers who want to change their subject and want to teach mathematics.
• People who change their occupations from other industries like engineers, high tech specialists.
• Educators from other countries.
CSET Single Subject Mathematics Preparation classes at National University is one of the initiatives to assist future mathematics teachers in gaining conceptual knowledge in mathematics and problem solving skills which they need to pass the test and, what is more essential, to become good mathematics teach-
ers.
Before describing specific problems related to math teachers preparation using CSET it is instructive to consider the structure of CSET Single Subject Math exam. CSET: Mathematics consists of three separate subtests, each composed of both multiple-choice and constructed-response questions. Each subtest is scored separately. The structure of the examination is shown in the table below (Table 1).
Table 1
CSET Mathematics
Subtest Domain Number of Multiple- Number of Con-
Choice Questions structed- Response
Questions
I Algebra 24 3
Number Theory 6 1
Subtest Total 30 4
II Geometry 22 3
Probability and Statistics 8 1
Subtest Total 30 4
III Calculus 26 3
History of Mathematics 4 1
Subtest Total 30 4
Candidates verifying subject matter competence by examination for a credential in Foundational-Level Mathematics are required to take and pass Subtests I and II only.
CSET Single Subject Mathematics Preparation Class Practice
Now when an instructor can see which areas of mathematics are required to be covered, he/she can design how to prepare students to demonstrate masteries in them. Students who come to CSET Single Subject Mathematics Preparation class have practically the same problems as high school students. Even with some math education of a higher level they still have trouble passing the test. Some students attempt CSET Single Subject Mathematics exams 6-7 times without success.
Results of this study are based on observation of 17 groups of students (about 200 peo-
ple) who came to National University CSET mathematics preparation classes during 20042008. The purpose of the research was to determine why CSET exam is so difficult, what are the most critical issues of the test and how to dramatically improve student performance, increase the quantity of students with successful scores on the exam, and overcome math anxiety among examinees.
During this research, the following main concerns of the students have been recognized:
1. They don't understand the questions
2. They hate constructed response questions
3. They don't know how to present proofs with reasoning
4. They don't think they need to know all the things which are asked in the test
5. They have to memorize too much
Let's consider a few typical problems from CSET sample and main difficulties related to them.
Students always say they are frustrated with the way how CSET asks the questions. For example: If f(x) = -2x2 + 8x +16, then which of the following is the absolute value of the difference between the zeros of f(x)? (and four choices as usual in the test).
Many students don't understand the question. If they were asked to solve a quadratic equation it would not be a problem, so, what is the difference?
A general form of a quadratic equation is ax2 + bx + c = 0. At the same time, we can write that ax2 + bx + c = f (x), and if f (x) = 0 it means that we can find zeros of the function f (x), and it is the same thing as solving the quadratic equation
f (x) = ax2 + bx + c = 0.
Students cannot make the link between the zeros of a quadratic function and the roots of a quadratic equation. The reason for that is that don't see the whole picture of functions, equations and their relationships, they don't see the concept.
Very often, students don't understand the whole concept, they just remember how to solve some problems using a rule. For exam-
ple, If f (x) =
e5 x + 6 2
and g(f(x)) = x, then
which of the following is equivalent to g(x)?
In the most of the cases, students believe that finding an inverse function just means switching variables. They don't understand that it is not enough, and they should also show that for two functions f (x) and g(x) to be inverses of each other they need to satisfy the following conditions: f (g (x)) = x and g(f (x)) = x . In that particular problem, it is not said that g (x) is the inverse of f (x), it is just said g(f (x)) = x, and if students don't understand the whole concept of inverse functions they don't understand the question because nothing is said about switching variables in the question.
Many students feel very uncomfortable
with trigonometry. Even a simple question about Cos J behavior makes them confused, and questions in CSET about vectors and dot products are among the most complicated for them. One of the first questions they ask when they come to CSET prep class is "Will we cover dot product?"
Also, students don't understand the value of definitions. When they are asked to give a definition of some basic mathematical terms they usually give some features, or examples related to those terms. For example, given any
two vectors a and b such that a =
= 1.
which of the following statements about the inner product, a»b , must be true?
A. a»b = 1
B. -1 < =a»b < 1
C. 1 <d'b < 2
D. 1 < a»b < 2
To solve this problem, they just need to know the definition of the inner product and behavior of main trigonometric functions.
Also, if they need to find an angle between two vectors they need to know two definitions of the dot product,
1. u »v = |u|»|v| »Cos J
2. u»v = UjVj + u2v2
and at this point, they are completely lost.
Very often, students mix up definitions and properties. For example, a student may say that "parallel sides of a parallelogram are congruent by definition (!) of a parallelogram.
Another thing the students struggle with is logical reasoning. It's difficult for them to see that there are cornerstone theorems, and their corollaries which help to solve problems. For example, many students don't know the theorem about conjugate roots: if one root is a complex number then, its conjugate must be the second root. Some of them don't know that polynomial of degree n always has n roots , but all those things come from the Main Theorem of Algebra which should be considered as a main result of algebra.
The issues diseased above are even more critical when it comes to proofs, especially in geometry. Sometimes, they are asked to solve
b
a word problem which requires to sketch a picture first, and then to prove some statement. For example, prove: if two circles are tangent externally, their common internal tangent bisects a common external tangent. The most challenging thing is to create a picture based on words. They are confused about "external" and "internal" tangents, they try to assume that both circles have equal radii, and if the radii are different that it cannot be true.
It is surprising how many students don't know how to prove that the sum of angles in a triangle is 180°. For example, given: If a transversal intersects two parallel lines, then the alternate interior angles are congruent. Questions: If the above statement is false, which of the following is also false?
A. If two angles are supplements of congruent angles (or the same angle), then the two angles are congruent.
B. Vertical angles are congruent.
C. The base angles of an isosceles triangle are congruent.
D. The angle sum of every triangle is 180°.
Many students cannot see any connection between the Euclid postulate and the fact that the sum of the angles in a triangle is 180°.
The hardest part of CSET math exam is the one related to Abstract Algebra. Every time when students come to the prep class they ask: "Will we cover Fields and Rings? And Vectors?" Those topics of Abstract Algebra are the most challenging and confusing for them.
The example below shows how several issues related to definitions and proofs come together just within one problem and make it so complicated:
Which of the following statements refutes the claim that GLR(3), the set of 3 x 3 invert-ible 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"11 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) ^ 1.
Again, the way how the question is asked
is unusual. They say: "Which statement refutes..." Students cannot distinguish which properties are not related to the definition of a field at all like det(AB) = det(A)det(B) or det(A) ^ 1. Also, they try to prove that the set of 3x3 invertible matrices is a field, and don't see that to disprove a statement one needs just to find one contradiction, instead, they check all the properties.
The list of the problems can go on, but the main thing those examples illustrate is that practicing a conceptual approach and logical reasoning, students and future teachers can overcome their fear of math and CSET exam in particular.
The students in CSET mathematics preparation classes complain that mathematics requires memorizing too much. It's very common myth related to the fact that they don't understand a concept and don't know how to apply it. It's getting critical for an advance level of mathematics where the volume of information is practically impossible to memorize, and students' skills in logical reasoning including definitions and proofs determine their success in problem solving.
Some observations from CSET preparation classes show that even when students can see answers, in 90% cases, they cannot do the problems on their own with explanations as a constructed response, but it is required in CSET exam. The reason for that is that if they don't understand concepts they cannot understand the answers even when they see them.
Although the CSET Single Subject Mathematics preparation classes are short, intensive and require quite strong background in mathematics, the results of that approach are impressive. Over 200 students took those classes during two years 2004-2008, and about 80% of the students passed CSET after the classes( based on their E-mail communication with the author). When they come to the class CSET looks like a puzzle for them with no connection between pieces. Post classes, they say they can finally see the whole picture, and they gain much better understanding of mathematical concepts and reasoning.
Conclusions and Recommendations
Based on our CSET Single Subject
Mathematics preparation classes experience the following critical issues have been recognized:
• Examinees don't understand questions.
• They hate constructed response questions.
• They don't know how to present proofs with reasoning.
• They don't think they need to know all the things which are asked in the test.
• They have to memorize too much.
While practicing the conceptual approach
and logical reasoning in CSET preparation classes almost all the students' concerns went away. The idea of understanding mathematical concepts and reasoning is not new, but what is really important for this study that highlighting specific areas of concerns and resolving the most of issues blocking successful passing of CSET examination by that approach demonstrates the effectiveness of critical analysis of typical problems and practical application of the approach.
Main recommendations for CSET Single Subject Mathematics takers can be formulated as following:
• Students need to see the whole picture instead of unrelated fragments.
• Understanding of concepts and reasoning is the key to be successful in CSET examinations and teaching.
• Students should know definitions very well.
• Knowing the difference between definitions and properties is a must , i.e. the student must understand what exactly is to be proven, and what reasoning is needed for this. Recognize the basic postulates and proofs of main theorems, and not to memorize everything.
If examinees follow those recommendations their chance of success in CSET Single Subject Mathematics examination and, more essential, becoming a good mathematics teacher increases dramatically. This is also good recommendations for instructors who want to get involved in mathematics teachers preparation, i.e. college and university professors, professional development experts etc.
Sometimes students who come to the
CSET preparation class don't have enough background to capture concepts from those short prep classes. National University offers very high quality Mathematics major program which gives students a waiver for CSET Single Subject Mathematics exams. The program covers all SMR (Subject Matter Requirements) for CSET Single Subject Mathematics exam. In addition, the program gives very solid knowledge of mathematical principles and applications.
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Резюме. Stankous N. ТРУДНОСТИ ПРИ СДАЧЕ ЛИЦЕНЗИОННЫХ ЭКЗАМЕНОВ УЧИТЕЛЕЙ МАТЕМАТИКИ В КАЛИФОРНИИ. В свете того, что школьники в США показывают низкие результаты в математике, а учителей математики недостаточно, вопрос подготовки последних становится критическим. В данной работе рассматривается несколько важных вопросов, связанных с подготовкой учителей математики, особенно в Калифорнии. Определены основные проблемы при подготовке к сдаче специально теста для учителей математики в старших классах (CSET). Также в работе предлагаются пути решения этих проблем и разработан ряд рекомендаций для улучшения качества подготовки будущих учителей математики в Калифорнии.
Ключевые слова: подготовка учителя математики, концептуальный подход, логическая аргументация.
Abstract. Stankous N. CHALLENGES OF EXAMINATIONS FOR MATHEMATICS TEACHERS IN CALIFORNIA. In the light of low student performance and mathematics teacher shortages, the issue of mathematics teacher preparation becomes critical. This paper addresses a few important issues focused on mathematics teacher preparation, specifically on preparation for California schools. This study identifies problems related to mathematics teacher preparation, and describes the main concerns of CSET Single Subject Math exam takers. The study also offers an effective way to evaluate and address those concerns, and provides a list of recommendations to improve the quality offuture mathematics teachers.
Key words: mathematics teacher preparation, CSET Single Subject Math, CSET exam challenges, conceptual approach, logical reasoning.
Стаття представлена професором 1.Я.Суббот1ним.
Надшшла доредакцп 12.09.2010р.
