Educational Sciences: Theory & Practice • 14(1) • 411-421
©
2014 Educational Consultancy and Research Center
www.edam.com.tr/estp
DOI: 10.12738/estp.2014.1.1688
A Comparative Analysis of Questions in American,
Singaporean, and Turkish Mathematics Textbooks Based
on the Topics Covered in 8th Grade in Turkey
a
b
Eren ÖZER
Renan SEZER
Ankara University
Ankara University
Abstract
This study offers a comparative analysis of questions found in Turkish, Singaporean, and American mathematics
textbooks and workbooks based on topics covered in the 8th grade mathematics curriculum in Turkey. To this
end, the study utilizes the 3-dimensional framework developed by Li. When the questions in the textbooks and
workbooks representative of those used in the United States, Singapore, and Turkey were analyzed with respect
to their mathematical features, the percentage of questions that require multiple computation procedures were
found to be 90%, 96%, and 85% respectively. When the questions were analyzed with respect to their contextual
features, it was observed that questions categorized as purely mathematical in context made up 72%, 76%, and
61% of the questions in the books. When the questions were compared with respect to their response type, a
sub-category of performance requirements, it was found that 83%, 85%, and 66% of the questions respectively
required only numerical answers. In the representative books from the US, when questions were categorized
with respect to their cognitive requirements, a sub-category of performance requirements, it was determined
that conceptual understanding, procedural practice, problem solving, and special requirements constituted 9%,
81%, 9%, and 1% of the questions. These percentages were determined to be 7%, 83%, 9%, and 1% for the
Singaporean books, and 21%, 67%, 11%, and 1% for the Turkish books. Even though the percentage of questions
that required problem solving was higher in the Turkish books than in those of the other two nations, the number
of such problems in the books was less (US 259, Singapore 246, Turkey 144).
Key Words
Li’s Dimensions of Problem Requirements, Mathematics Textbooks, Middle School Mathematics, National
Comparisons, 8th Grade.
As a result of reform movements in mathematics
education around the world, Turkey felt the need
to revise its educational goals and prepare new
curricula (Toptaş, Elkatmış, & Karaca, 2012). In
the academic year 2006-07, a new mathematics
curriculum for the 6th, 7th, and 8th grades
was prepared, and starting with the 6th grade
mathematics textbook in 2006, a new textbook for
each grade was written and put to implementation.
The project was finalized by the academic year
2008-09 (Eğitim Reformu Girişimi [ERG], 2005).
Textbooks are considered to be the most important
a Eren ÖZER is a mathematics teacher who got his masters’ degree from Ankara University in Elementary
Mathematics Education. Email: [email protected]
b Renan SEZER, Ph.D., is currently a professor of Elementary and Secondary School Mathematics Education.
Her research interests include international comparisons in mathematics education, problem solving and
real-life mathematics problems. Correspondence: Ankara University, Faculty of Education, Department of
Elementary Education, Cebeci, Ankara, Turkey. Email: [email protected]
EDUCATIONAL SCIENCES: THEORY & PRACTICE
component of a reformed curriculum, because they
are a reflection of the curriculum for the teachers,
students, and parents (Valverde, Bianchi, Wolfe,
Schmidt, & Houang, 2002). Which textbooks
are used is a good indicator of a curriculum, and
analysis of the textbooks explores how well the
intended curriculum is implemented (Remillard,
2000).
Of all learning materials, textbooks offer the most
learning opportunities (Garner, 1992). In order
to best measure students’ learning opportunities,
Törnroos (2005) assessed the mathematical topics
included in Trends in Mathematics and Science
Study (TIMSS 1999) using three categories. First,
the learning opportunity in the content of each
chapter was assessed using the content titles in the
textbooks. Second, learning opportunities offered
by teachers, and third, the content of the textbooks
were analyzed with regard to how the material
was presented. The result of this study indicated
that textbooks are the best source for providing
learning opportunities, as well as the best indicator
for measuring the learning opportunities provided.
Learning opportunities is an important factor in
explaining differences in students’ performances
(Schimdt et al., 2001).
The finding that textbooks are a good source in
establishing learning opportunities resulted in
an increase in the number of studies focusing
on textbook analyses of different nations after
the 1990’s. Foxman’s (1999) research indicated
that students using the mathematics textbook in
class were more successful in TIMSS than their
counterparts who had not. Similarly, Yeap’s (2005)
study established that textbooks played a vital role
in the mathematics achievement of Singaporean
students. For this reason, many nations strove to
obtain minimum standards in the quality of their
textbooks. In order to determine the potential
effect of textbooks and pedagogical approaches
on the mathematics achievement of students,
the textbooks of many nations such as the US,
Singapore, China, and Japan were analyzed (Fuson,
Stigler, & Bartsch, 1988; Li, 2000; Mayer, Sims, &
Tajika, 1995; Schmidt, McKnight, & Raizen, 1997).
For instance, in his study comparing American and
Japanese 1st and 2nd grade mathematics textbooks,
Stevenson (1985) stated that the questions in the
American textbooks were less challenging than
their Japanese counterparts. In another study
where Stevenson collaborated with Bartsch (1992),
the content of mathematics textbooks used in the
US and Japan was analyzed, and it was found that
412
in the Japanese textbooks, many concepts were
introduced several years earlier than those in the
American ones. Stevenson, Stigler, Lee, and Lucker
(1982) analyzed mathematics textbooks utilized in
Japan, Taiwan, and the US to find when a concept
was introduced, and the level of skills expected in a
particular grade in different countries. In a similar
study, Lo, Cai, and Watanabe (2001) compared
mathematics textbooks used in China, Taiwan,
Japan, and the US with respect to the overall
appearance of the textbooks, as well as the questions
and solutions they contained. The result of this
comparison indicated that questions in the Chinese,
Taiwanese, and Japanese textbooks had much more
complex and difficult questions than those in the
American books. A study comparing the treatment
of addition and subtraction of fractions in the
textbooks used in Southern Cyprus, Ireland, and
Taiwan was published by Charalambous, Delaney,
Seán, Hsu, and Mesa (2010). The authors argued
that in order to understand the differences in the
teaching and achievement of different nations,
their textbooks should be analyzed. In a study
completed by Mayer et al. in 1995, Japanese and
American textbooks were compared based on their
approaches to teaching problem-solving, and it was
found that Japanese books emphasize problemsolving more than their American counterparts.
Beckmann’s 2004 study showed that Singaporean
textbooks were the reason for the success of 8th
grade Singaporean students in TIMSS.
In a paper published in 2000, Li developed
a framework called “dimensions of problem
requirements” in order to analyze mathematics
questions in textbooks. This framework analyzes
questions from three dimensions and is explained
below (Li, 2000, p. 237):
Li’s (2000) framework was the main methodology
used in numerous studies, such as Arnold and
Son’s (2011), where response types and cognitive
demands were categorized using Li’s (2000) as well
as Son and Senk’s (2010) methods. In 2003, Cheung
utilized Li’s framework to compare questions in
ÖZER, SEZER / A Comparative Analysis of Questions in American, Singaporean, and Turkish Mathematics...
the algebra chapters of two mathematics textbooks
used in Hong Kong, one which was written for
mathematics program of 1985 and the other for
that of 2001. İncikabı and Tjoe’s (2012) study
compared questions related to ratio and proportion
in American and Turkish mathematics textbooks
with respect to mathematical features, contextual
features, and performance requirements. The
method used was similar to Li’s (2000) in many
respects, with the addition of a technology
component for those questions that required its
use. Conklin (2004) compared German, American,
and Japanese mathematics textbooks with respect
to their size, weight, structural organization, page
length, and question characteristics. Li’s framework
was utilized for the analyses of the contextual
features, response types, and cognitive demands.
Hu’s (2011) analysis of the response types and
performance requirements of questions also
employed Li’s dimensions of problem requirements.
As is the case in other countries, studies focusing on
mathematics textbooks in Turkey have increased
over the years; however, these studies were mostly
at the national level, based on the feedback of
teachers, students (Arslan & Özpınar, 2009; A.
Çakır, 2006; İ. Çakır, 2009; Işık, 2008). In a 2006
study, Delil analyzed geometry questions in the 6th,
7th, and 8th grade textbooks based on the cognitive
requirements used in TIMSS 1999. The study
found that although applying a known procedure
constituted only 24% of TIMSS geometry
questions, 50% of the questions in the textbook
fell into this category, and whereas 30% of TIMSS
questions required reasoning skills, only 10% of the
textbook questions required it. Another study by
Erbaş and Alacacı (2009) compared mathematics
textbooks and workbooks used in the US,
Singapore, and Turkey. According to the findings
of this study, in the Turkish books, there was one
type of question limited to the understanding of a
concept, and multiple solutions were not offered.
In contrast, Singaporean and American textbooks
tried to develop various mathematical skills related
to the concept. Moreover, in the Singaporean
textbooks, problems of a wide range of difficulty
and increasing complexity were offered, helping
students internalize concepts.
There is no study in the literature comparing the
8th grade mathematics textbooks of Turkey with
those in nations that take the lead in TIMSS and
Programme for International Student Assessment
(PISA); however, numerous such studies exist for
other countries (Charalambous et al., 2010; Fuson
et al., 1988; Haggarty & Pepin, 2002; Li, 2000; Mayer
et al., 1995; Schmidt et al., 1997). The current study
aims to be a continuation of the Erbaş and Alacacı
(2009) study, thus its purpose is to analyze the
overall picture of how middle school mathematics
textbooks compared to those of nations that score
higher than Turkey in global comparisons such as
TIMSS and PISA (Eğitimi Araştırma Geliştirme
Dairesi Başkanlığı [EARGED], 2003). One country
that was chosen for this comparison was Singapore,
a country that ranks in the top five in TIMSS and
PISA. The other was the US, a country whose
reform in education has affected many others, and
which ranks about average in global comparisons.
Authors of the present study believe that such an
evaluation will help in the writing of mathematics
textbooks that are up to par in the future. Li’s (2000)
three-dimensional framework will be used to
compare the questions found in Turkey’s 8th grade
textbook and workbook with questions in books
from the US and Singapore, in order to answer the
following research questions:
1. a) Which topics are found across the 8th grade
text books of the US, Singapore, and Turkey?
1. b) When are topics that are covered in the
Turkish 8th grade textbook but not in the
American or Singaporean textbooks taught in
these countries?
Taking the topics covered in the Turkish 8th
grade mathematics textbook approved by the
Turkish Ministry of Education as a basis:
2. a) How do the questions pertaining to these
topics compare in the American, Singaporean,
and Turkish books with respect to their
mathematical features?
2. b) How do the questions pertaining to these
topics in each content area compare in the
American, Singaporean, and Turkish books with
respect to their mathematical features?
3. a) How do the questions pertaining to these
topics compare in the American, Singaporean,
and Turkish books with respect to their
contextual features?
3. b) How do the questions pertaining to these
topics in each content area compare in the
American, Singaporean, and Turkish books with
respect to their contextual features?
4. a) How do the questions pertaining to these
topics compare in the American, Singaporean,
and Turkish books with respect to their response
type, a sub-category of performance features?
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EDUCATIONAL SCIENCES: THEORY & PRACTICE
4. b) How do the questions pertaining to these
topics in each content area compare in the
American, Singaporean, and Turkish books with
respect to their response type, a sub-category of
performance features?
5. a) How do the questions pertaining to these
topics compare in the American, Singaporean,
and Turkish books with respect to their cognitive
requirements, a sub-category of performance
features?
5. b) How do the questions pertaining to these
topics in each content area compare in the
American, Singaporean, and Turkish books with
respect to their cognitive requirements, a subcategory of performance features?
Method
This is a qualitative study, where the questions in
the textbooks and accompanying workbooks (if
available) that are representative of those used
in the US, Singapore, and Turkey are analyzed
according to Li’s framework. The reason Li’s (2000)
framework was used is because it was designed
specifically to analyze mathematics questions in
textbooks.
1. Mathematical features:
that the 8th grade mathematics textbook most
widely used was Mathematics: Applications &
Concepts by Rhonda Bailey, Roger Day, Patricia
Frey, Arthur Howard, Deborah Hutchens, Kay
McClain, Beatrice Moore, Jack Ott, Ronald Pelfrey,
Jack Price, Kathleen Vielhaber, and Teri Willard
(Glencoe). For this reason, the present study used
the same book. In Singapore, all textbooks need to
be approved by Singapore’s Ministry of Education
before they can be adopted. As a representative of
math textbooks and workbooks in Singapore, the
New Syllabus series was chosen. For Turkey, the
2010 reprint of the textbook and workbook first
published and approved in 2008 by the Ministry of
Education was chosen. For this study, all questions
in the textbooks and workbooks’ learning activities
and worked-out examples, except those used at the
introduction of the topic, were coded. For those
topics found in the 8th grade books in Turkey,
but not in those of the US and Singapore, 6th to
9th grade textbooks were surveyed. The books
surveyed are given in Table 1.
Table 1.
Text Books Used in the Study
Grade US
6
Mathematics:
Applications &
Concepts
Glencoe, 2006,
Mcgraw-Hill
7
Mathematics:
Applications &
Concepts
Glencoe, 2006,
Mcgraw-Hill
8
Mathematics:
Applications &
Concepts
Glencoe, 2006,
Mcgraw-Hill
9
Algebra 1
Glencoe, 2006,
Mcgraw-Hill
· Single procedure (S)
· Multiple procedure (M)
2. Contextual features:
· Purely mathematical context in numerical or
word form (PM)
· Illustrative
context
with
representation or story (IS)
pictorial
3. Performance requirements:
a) Response type:
· Numerical answer only (NA)
· Numerical expression only (NE)
· Explanation or solution required (ES)
b) Cognitive requirements:
· Procedural practice (PP)
· Conceptual understanding (CU)
· Problem solving (PS)
· Special requirements (SR)
A study conducted by Adaptive Curriculum in
2010 in 100 school districts in the US indicated
414
Singapore
Turkey
New Syllabus
1, 2010,
Shinglee
New Syllabus
2, 2010,
Shinglee
New Syllabus
3, 2010,
Shinglee
Ministry
of
Education,
2010
New Syllabus
4, 2010,
Shinglee
To answer the second part of the fifth research
question, the questions in the textbooks and
workbooks were coded according to Li’s framework.
Before the coding was started, a workshop was given
to the two coders, both of whom are mathematics
teachers, by a faculty member working in this
area. During this workshop, questions were coded
and the codings were discussed. Later, the two
coders coded all the questions independently, and
results were compared. In any category where the
reliability was less than 90%, the questions that
were coded differently were discussed and coded
again. Reliability was calculated based on the last
coding. The reliability calculated for each category
ÖZER, SEZER / A Comparative Analysis of Questions in American, Singaporean, and Turkish Mathematics...
was found to be 94% for mathematical features,
98% for contextual features, 96% for response type,
a sub-category of performance requirements, and
97% for cognitive requirements, a sub-category of
performance requirements.
Results
The first part of the first research question asked
about topics that were found across the 8th grade
textbooks of all three countries. These topics were
“Exponents and Operations with Exponents,”
“Relationships Formed by the Sides of a Triangle,”
“Slope,” “Inequalities,” and “Ratios in a Right
Triangle;” however, the coverage of some of these
topics ranged between 7th to 9th grades in the US
and 6th to 8th grades in Singapore (Table 2).
Table 2.
Topics Found in the American, Singaporean, and Turkish 8th
Grade Mathematics Text Books
Topics
US
Exponents and Operations
7-8-9
with Exponents
Singapore
Turkey
8
8
Relationships Formed by the
Sides of a Triangle
8
6, 8
8
Slope
8
8
8
Inequalities
8
8
8
Ratios in a Right Triangle
8
8
8
The second part of the first research question asked
when the topics covered in Turkey’s 8th grade
textbooks are taught with the same depth in the
US and Singapore. To answer this question, the
objectives of the mathematics program of these
countries were analyzed, and the results are given
in Table 3.
The first part of the second research question
investigates the mathematical features of questions
in the textbooks and workbooks of the US,
Singapore, and Turkey, based on the topics covered
in the Turkish 8th grade textbooks. Questions
were categorized into two groups, namely, those
requiring a single procedure in their solution and
those requiring multiple procedures. There are a
total of 2,736 questions in the American books,
2,669 questions in the Singaporean books, and
1,367 questions in the Turkish books. Of those,
2,454 (90%) of the questions in the American books,
2,560 (96%) of the questions in the Singaporean
books, and 1,163 (85%) of the questions in the
Turkish books require multiple procedures. The
distribution of these questions in the textbooks and
workbooks is given in Table 4.
Table 3.
The Grades At Which Topics Covered in the 8th Grade Textbook
in Turkey Are Covered in the US and Singapore
Topics
US
Singapore Turkey
Numbers
Exponents and Operations
7-8-9
8
8
with Exponents
Square Roots and Operation
7-8-9
6
8
with Square Roots
Real Numbers
8
6
8
Geometry and Measurement
Reflection, Translation, and
8
8
Rotation
Fractals
8
Relationships Formed by the
8
6, 8
8
Sides of a Triangle
Congruence and Similarity of
7-9
8
8
Triangles
Prisms, Surface Area, and
8
6
8
Volume of Prisms
Surface Area and Volume of
8
7
8
Pyramids, Cones, and Spheres
Projection and Polyhedra
8
Perspective Drawing
8
Polyhedra and Its Cross
8
Sections
Geometric
Objects
and
8
8
Symmetry
Ratios in a Right Triangle
8
8
8
Slope
8
8
8
Algebra
Number Sequences
7-8
6
8
Pythagorean Theorem
7-8-9
7
8
Identities and Factors
8
7
8
Rational Expressions
9
7
8
Inequalities
8
8
8
System of Equations
9
7
8
Statistics, Probability, and
Data Analysis
Combination
8
8
Probability and Event Types
8
9
8
Histogram
8
8
Standard Deviation
9
8
*Topics written in bold are those mutually included in the 8th
grade text books of the three countries.
Table 4.
Mathematical Features (Number of Procedures Required) of
Questions in the Representative Textbooks and Workbooks
from the US, Singapore, and Turkey
Country
US Textbooks
Number of Procedures
Required (Percentage)
Single
Multiple
217 (%10)
1882 (%90)
US Workbooks
65 (%10)
572 (%90)
US Total
282 (%10)
2454 (%90)
Singapore Textbooks
73 (%4)
1758 (%96)
Singapore Workbooks
36 (%4)
802 (%96)
Singapore Total
109 (%4)
2560 (%96)
Turkish Textbook
95 (%15)
544 (%85)
Turkish Workbook
109 (%15)
619 (%85)
Turkish Total
204 (%15)
1163 (%85)
The second part of the second research question
investigates how the mathematical requirements
415
EDUCATIONAL SCIENCES: THEORY & PRACTICE
of the questions in (2a) were distributed across
content in the American, Singaporean, and Turkish
books. The questions were categorized into four
content areas, namely, numbers, geometry and
measurement, algebra, and statistics, probability,
and data analysis. A summary of these results is
given in Table 5.
The second part of the third research question
investigates how the contextual requirements of
the questions in (3a) are distributed across the four
content areas in the American, Singaporean, and
Turkish books. A summary of these results is given
in Table 7.
Table 5.
Mathematical Features (Number of Procedures Required) of Questions in the Representative Textbooks and Workbooks from the US,
Singapore, and Turkey With Respect to Content Areas
Countries
Geometry and
Measurement
Numbers
Algebra
Statistics and
Probability
Number of Procedures Required
US Textbooks
Single
Multi
Single
Multi
Single
Multi
Single
Multi
94 (%18)
430 (%82)
51 (%8)
579 (%92)
34 (%5)
651 (%95)
38 (%15)
222 (%85)
0
185(%100)
39 (%19)
167 (%81)
16 (%9)
167 (%91)
10 (%16)
53 (%84)
94 (%13)
615 (%87)
90 (%11)
746 (%89)
50 (%6)
818 (%94)
48 (%15)
275 (%85)
Singapore Textbooks
14 (%6)
221 (%94)
43 (%8)
478 (%92)
11 (%1)
978 (%99)
5 (%6)
81 (%94)
Singapore Workbooks
7 (%2)
419 (%98)
0
64 (%100)
29 (%10)
275 (%90)
0
44 (%100)
Singapore Total
21 (%3)
640 (%97)
43 (%7)
542 (%93)
40 (%3)
1253(%97)
5 (%4)
125 (%96)
Turkish Textbook
62 (%35)
117 (%65)
21 (%9)
225 (%91)
10 (%6)
148 (%94)
2 (%4)
54 (%96)
Turkish Workbook
58 (%27)
154 (%73)
25 (%9)
256 (%91)
25 (%9)
151 (%91)
1 (%2)
120 (%31)
271 (%69)
481 (%91)
35 (%10)
299 (%90)
US Workbooks
US Total
Turkish Total
46 (%9)
The first part of the third research question investigates
the contextual features of questions in the textbooks
and workbooks from the US, Singapore, and Turkey
based on the topics covered in the Turkish 8th grade
textbooks. Questions were categorized into two
groups, namely, those having a purely mathematical
context and those having an illustrative or story
context. It was found that 1,963 of the 2,736 questions
(72%) in the American books, 2,403 of the 2,669
questions (77%) in the Singaporean books, and 834 of
the 1,367 questions (61%) in the Turkish books had a
purely mathematical context. The distribution of these
questions in the textbooks and workbooks is given in
Table 6.
Table 6.
Contextual Features (Pure/Illustration or Story) of Questions
in the Representative Textbooks and Workbooks from the US,
Singapore, and Turkey
Country
Contextual Features (Percentages)
Pure
Mathematics
Illustrative Context
or Story
US Textbooks
1456 (%69)
643 (%31)
US Workbooks
507 (%80)
130 (%20)
US Total
1963 (%72)
773 (%28)
Singapore Textbooks
1348 (%74)
483 (%26)
Singapore Workbooks
695 (%83)
143 (%17)
Singapore Total
2043 (%77)
626 (%23)
Turkish Textbook
381 (%60)
258 (%40)
Turkish Workbook
453 (%62)
275 (%38)
Turkish Total
834 (%61)
533 (%39)
416
3 (%3)
58 (%98)
112 (%97)
The first part of the fourth research question
investigates the response type, a sub-category of
performance requirements, of questions in the
textbooks and workbooks from the US, Singapore,
and Turkey based on the topics covered in the
Turkish 8th grade text books. Questions were
categorized into four groups, namely, those
requiring a numerical answer only, those requiring
a numerical expression only, and those requiring
either an explanation or a solution. It was found
that of the 2,736 questions in the American books,
2,264 (83%) of them required numerical answers
only, 471(17%) required an explanation or solution,
and two required a numerical expression only. In
the Singaporean books, of the 2,669 questions,
2,165 (81%) required a numerical answer only,
504 (19%) required an explanation or solution,
and two required a numerical expression only. In
the Turkish books, there were 1,367 questions, of
which 897 (65%) required a numerical answer only,
448 (33%) required an explanation or solution, and
22 (2%) required numerical expressions. As can
be seen from the example below, although Turkey
had a higher percentage of questions requiring an
explanation or solution, these explanations did not
focus on higher-level thinking skills used towards
the solution, but merely on the answer itself. The
distribution of these questions in the textbooks and
workbooks is given in Table 8.
ÖZER, SEZER / A Comparative Analysis of Questions in American, Singaporean, and Turkish Mathematics...
Table 7.
Contextual Features (Pure/Illustration or Story) in the Representative Textbooks and Workbooks from the US, Singapore, and Turkey
With Respect to Content Areas
Countries
Numbers
Geometry
Algebra
Statistics and
Probability
Contextual Features
Pure Math.
Illustrative
or Story
Pure Math.
Illustrative
or Story
Pure Math.
Illustrative
or Story
Pure Math.
Illustrative
or Story
US Textbooks
481 (%92)
US Workbooks
176 (%95)
43 (%8)
280 (%44)
350(%56)
546 (%80)
139(%20)
149 (%57)
111(%43)
9 (%5)
115 (%56)
91 (%44)
176 (%96)
7 (%4)
40 (%63)
US Total
23 (%37)
657 (%93)
52 (%7)
395 (%47)
441(%53)
722 (%83)
146(%17)
189 (%59)
134(%41)
Singapore
Textbooks
222 (%94)
13 (%6)
185 (%36)
336(%64)
900 (%91)
89 (%9)
41 (%48)
45 (%52)
Singapore
Workbooks
380 (%89)
46 (%11)
29 (%45)
35 (%55)
282 (%93)
22 (%7)
4 (%9)
40 (%91)
Singapore Total
602 (%91)
59 (%9)
214 (%37)
371(%63)
1182(%91)
111 (%9)
45 (%35)
85 (%65)
Turkish Textbook
170 (%95)
9 (%5)
98 (%40)
148(%60)
96 (%61)
62 (%39)
17 (%30)
39 (%70)
Turkish Workbook
192 (%91)
20 (%9)
97 (%35)
184(%65)
134 (%76)
42 (%24)
30 (%51)
29 (%49)
Turkish Total
362 (%93)
29 (%7)
195 (%37)
332(%63)
230 (%69)
104(%31)
47 (%41)
68 (%59)
Table 8.
Response Type (a Sub-Category of Performance Requirements)
of Questions in the Representative Textbooks and Workbooks
from the US, Singapore, and Turkey
Countries
Number of Questions According to
Response Type (Percentages)
Numerical
Answer
Only
Explanation
or Solution
Numerical
Expression
Only
US Textbooks
1702 (%81)
396 (%19)
1 (%0)
US Workbooks
562 (%88)
75 (%12)
1 (%0)
US Total
2264 (%83)
471 (%17)
1 (%0)
Singapore
Textbooks
1452 (%79)
379 (%21)
1 (%0)
Singapore
Workbooks
713 (%85)
125 (%15)
1 (%0)
Singapore
Total
2165 (%81)
504 (%19)
1 (%0)
Turkish
Textbook
418 (%66)
213 (%33)
8 (%1)
Turkish
Workbook
479 (%66)
235 (%32)
14 (%2)
Turkish Total
897 (%65)
448 (%33)
22 (%2)
Example
Explain if the numbers given below are a perfect
square.
a) 230
b) 156
c) 196
d) 0
The second part of the fourth research question
investigates how the response type (a sub-category
of performance requirements) of the questions in
(4a) is distributed across the four content areas in
the American, Singaporean, and Turkish books. A
summary of these results is given in Table 9.
from the US, Singapore, and Turkey, based on the
topics covered in the 8th grade Turkish textbooks.
Questions were categorized into four groups,
namely, those requiring conceptual understanding,
those requiring procedural practice, those
requiring problem solving, and those having special
requirements.
It was found that of the 2,736 questions in the
American books, 241 (9%) of them required
conceptual understanding, 2,215 (81%) required
procedural practice, 260 (9%) required problem
solving, and 20 (1%) had special requirements.
In the Singaporean books, of the 2,669 questions,
187 (7%) required conceptual understanding,
2,200 (83%) required procedural practice, 246
(9%) required problem solving, and 36 (1%) had
special requirements. In the Turkish books, there
were 1,367 questions, of which 291 (21%) required
conceptual understanding, 918 (67%) required
procedural practice, 144 (11%) required problem
solving, and 14 (1%) had special requirements. The
distribution of these questions in the textbooks and
workbooks is given in Table 10.
The second part of the fifth research question
investigates how the cognitive requirements (a subcategory of the performance requirements) of the
questions in (5a) were distributed across the four
content areas in the American, Singaporean, and
Turkish books. A summary of these results is given
in Table 11.
The first part of the fifth research question
investigates the cognitive requirements (a subcategory of the performance requirements) of
the questions in the textbooks and workbooks
417
EDUCATIONAL SCIENCES: THEORY & PRACTICE
Table 9.
Response Type (a Sub-Category of Performance Requirements) of Questions in Representative Textbooks and Workbooks from the US,
Singapore, and Turkey With Respect to Content Areas
Countries
Numbers
Geometry
Algebra
Statistics and probability
Response Types
Num.
Answ.
Explain/
Solution.
Num.
Exp.
Num.
Answ.
Explain
Soln.
Num.
Exp.
Num.
Answ.
Explain./
Solution
Num.
Exp.
Num.
Answ.
Explain/
Solution
Num
Exp.
US Textb.
467
(%89)
57
(%11)
0
467
(%74)
162
(%26)
1
(%0)
558
(%81)
127
(%19)
0
210
(%81)
50
(%19)
0
US Workb.
185
(%100)
0
0
181
(%88)
25
(%12)
0
139
(%76)
44
(%24)
0
57
(%90)
6 (%10)
0
US Total
652
(%92)
57
(%8)
0
648
(%78)
187
(%22)
1
(%0)
697
(%80)
171
(%20)
0
267
(%83)
56
(%17)
0
Singapore
Textb.
196
(%83)
39
(%17)
0
373
(%72)
148
(%28)
0
823
(%83)
166
(%17)
0
60
(%70)
26
(%30)
0
Singapore
Workb.
365
(%86)
61
(%14)
0
44
(%69)
20
(%31)
0
261
(%86)
43
(%14)
0
43
(%98)
1 (%2)
0
Singapore
Total
561
(%85)
100
(%15)
0
417
(%71)
168
(%29)
0
1084
(%84)
209
(%16)
0
103
(%79)
27
(%21)
0
Turkish
Textb.
131
(%73)
45
(%25)
3
(%2)
132
(%54)
113
(%46)
1
(%0)
116
(%73)
38
(%24)
4
(%3)
39
(%70)
17
(%30)
0
Turkish
Workb.
195
(%92)
11
(%5)
6
(%3)
148
(%53)
132
(%47)
1
(%0)
103
(%58)
66
(%38)
7
(%4)
33
(%56)
26
(%44)
0
Turkish
Total
326
(%84)
56
(%14)
9
(%2)
280
(%54)
245
(%46)
2
(%0)
219
(%66)
104
(%31)
11
(%3)
72
(%63)
43
(%37)
0
Table 10.
Cognitive Requirements (a Sub-Category of Performance
Requirements) of Questions in the Representative Textbooks and
Workbooks from the US, Singapore, and Turkey
Country Number of Questions with Respect to Cognitive
Requirements
Procedural Contextual Problem
Special
Practice Understanding Solving Requirements
US
1643
216
220
20
Textbooks
(%79)
(%10)
(%10)
(%1)
US
572
25
40
0
Workbooks (%90)
(%4)
(%6)
2215
241
260
20
US Total
(%81)
(%9)
(%9)
(%1)
Singapore
1478
142
195
16
Textbooks
(%80)
(%8)
(%11)
(%1)
Singapore
722
45
51
20
Workbooks (%87)
(%5)
(%6)
(%2)
Singapore
2200
187
246
36
Total
(%83)
(%7)
(%9)
(%1)
Turkish
405
157
72
5
Textbook
(%63)
(%25)
(%11)
(%1)
Turkish
513
134
72
9
Workbook
(%71)
(%18)
(%10)
(%1)
Turkish
918
291
144
14
Total
(%67)
(%21)
(%11)
(%1)
Discussion and Recommendations
This study analyzing the 8th grade textbooks and
workbooks from the US, Singapore, and Turkey,
found that the topics in the Turkish textbook
overlapped more with those in the American
books (85%) than those in the Singaporean books
(30%), and that the overlap between the American
and Singaporean books was even less (20%).
Although national curricula are considered as
one of the most influential factors contributing to
a country’s success in global comparisons such as
TIMSS (Peak, 1996), this alone does not explain the
418
ranking of these three countries in recent TIMSS.
Both in TIMSS 2007 and PISA 2009, the US
ranked above Turkey, even though its curriculum
had less of an overlap with Singapore’s. This study
indicated that Singaporean students are introduced
to mathematical concepts a year or two earlier than
Turkish students. Stevenson and Bartsch (1992)
attributed Japanese students’ success over their
American counterparts to having been introduced
to topics in earlier grades. This feature seems to be
an important characteristic of the education system
of countries in the Far East such as Singapore and
Japan.
In the American and Singaporean books, there
were 2,736 and 2,669 questions respectively, though
in Turkish books the number was considerably
less (1,367). Fewer questions may result in fewer
question types, especially a fewer number of highlevel questions. Numerous studies on Turkish
mathematics textbooks recommend increasing the
number and difficulty level of the questions in the
textbooks (Altun, Arslan, & Yazgan, 2004; Aydoğdu
İskenderoğlu & Baki, 2011; Erbaş & Alacacı, 2009;
Yüksel & Artut, 2010).
An analysis of the books revealed that in all three
countries and in all content areas, the percentage of
questions requiring multiple procedures was higher
than those requiring single procedures, although
the number of questions requiring a single
procedure in their solutions was highest in Turkey’s
books. These findings are parallel to those found
ÖZER, SEZER / A Comparative Analysis of Questions in American, Singaporean, and Turkish Mathematics...
Table 11.
Cognitive Requirements (a Sub-Category of Performance Requirements) of Questions in the Representative Textbooks and Workbooks
from the US, Singapore, and Turkey With Respect to Content Areas
Statistics and
Country
Numbers
Geometry
Algebra
Probability
Cognitive Requirements
PP
CU
PS
SR
PP
CU
PS
SR
PP
CU
PS
SR
PP
CU
PS
SR
469
31
24
438 117
69
6
578
37
62
8
158
31
65
6
US Textb.
0
%89 %6
%5
%69 %19 %11 %1 %85 %5
%9
%1 %61 %12 %25 %2
178
7
172
6
28
167
16
55
3
5
US Workb.
0
0
0
0
0
0
%96
%4
%83 %3 %14
%91 %9
%87 %5
%8
647
31
31
610 123
97
6
745
53
62
8
213
34
70
6
US Total
0
%92 %4
%4
%73 %15 %11 %1 %86 %6
%7
%1 %65 %11 %22 %2
190
36
7
2
380
64
65
12
873
36
79
1
35
6
44
1
Singap. Textb.
%81 %15 %3
%1 %76 %12 %12 %2 %88 %4
%8
%0 %41 %7 %51 %1
394
21
2
9
39
13
5
7
284
10
6
4
5
1
38
Singap. Workb.
0
%93 %5
%0
%2 %61 %20 %8 %11 %94 %3
%2
%1 %11 %2 %87
584
57
9
11
419
77
70
19 1157 46
85
5
40
7
82
1
Singap. Total
%88 %9
%1
%2 %72 %13 %12 %3 %89 %4
%7
%0 %31 %5 %63 %1
136
39
1
3
129
85
30
2
121
23
14
19
10
27
Turkish Textb.
0
0
%75 %22 %1
%2 %52 %35 %12 %1 %76 %15 %9
%34 %18 %48
195
9
2
6
159
97
25
142
14
20
17
14
25
3
Turkish Workb.
0
0
%92 %4
%1
%3 %56 %35 %9
%81 %8 %11
%29 %24 %42 %5
331
48
3
9
288 182
55
2
263
37
34
36
24
52
3
Turkish Total
0
%85 %12 %1
%2 %55 %35 %10 %0 %79 %11 %10
%31 %21 %45 %3
Note: PP: Procedural Practice, CU: Conceptual Understanding, PS: Problem Solving, SR: Special Requirements
by Erbaş and Alacacı (2009). When questions were
analyzed with respect to their contextual features,
the percentage of pure math problems was 76% in
the Singaporean, 72% in the American, and 61% in
the Turkish books. These results also support Erbaş
and Alacacı’s (2009) findings. Olkun and Aydoğdu
(2003) cited insufficient quality of the visuals in the
geometry content area in the books published by
the Ministry of Education in Turkey as a reason for
Turkish students doing poorly on these questions
in TIMSS. Ng and Lee’s (2009) research indicated
that the visuals in the Singaporean books aided
students in their solutions and strengthened their
conceptual understanding. When questions were
analyzed with respect to their required response
type, it was observed that in all three countries’
books, questions requiring a numerical answer only
constituted the largest percentage of the questions.
In the Turkish books, even though the percentage
of questions requiring an explanation or solution
was higher than in the books of the other two
countries, the responses focused on the explanation
of the answer rather than on the solution method.
Erbaş and Alacacı (2009), Cai (2003), and Soylu
and Aydın’s (2006) results confirm this finding
and contrast it to the focus of the questions in the
Singaporean books on solution methods. When the
questions were compared with respect to cognitive
demands, it was found that in all the countries
investigated, procedural practice took the lead.
Li, Chen, and An (2009) indicated that in the Far
Eastern countries, conceptual understanding is
attained during the presentation of a topic, and as
Hiebert and Carpenter (1992) argue, one needs to
have a conceptual understanding, as well as the
necessary skills for procedural practice, in order to
be successful in mathematics.
The recommendations of this study are to increase
both the number and variety of questions in the
Turkish mathematics books, especially those
with a high level of cognitive demand. In the
American and Singaporean books, the questions
were labeled as easy, mediocre, and difficult; such
a categorization in the teacher’s resource book
used in Turkey would increase teachers’ awareness
as to the kind of questions they are assigning.
In the Turkish textbooks, both the number and
percentage of pure mathematics questions were
less than those of the other two countries’ books.
Increasing the pure mathematics questions in
Turkish books is also recommended. Alternative
solution strategies that enhance students’ repertoire
and help them internalize concepts should be
offered in Turkish textbooks. Finally, this study can
be repeated utilizing the cognitive levels used in
TIMSS or PISA.
419
EDUCATIONAL SCIENCES: THEORY & PRACTICE
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421
ERRATUM
Sosyal Bilimlerde Alternatif Regresyon Yöntemi Kullanımı ve En Küçük Kareler ile M Tahmin Yöntemlerinin Belirleyicilik Katsayısı Açısından Karşılaştırılması
Orkun COŞKUNTUNCEL
Kuram ve Uygulamada Eğitim Bilimleri
13(4) • 2139-2158
©2013 Eğitim Danışmanlığı ve Araştırmaları İletişim Hizmetleri Tic. Ltd. Şti.
DOI: 10.12738/estp.2013.4.1867
* Yukarıda künye bilgisi verilen makalenin 2140., 2141., 2142. ve 2143. sayfalarında yer alan bazı istatistiki
formüller Online ortamda hatasız görünmekle birlikte basılı nüshada hatalı çıkmıştır. İlgili formüllerin
yer aldığı sayfalar müteakip olarak yer almaktadır.
Çalışmanın Online versiyonuna şu adresten ulaşılabilir.
http://dx.doi.org/10.12738/estp.2013.4.1867
Yrd. Doç. Dr. Orkun COŞKUNTUNCEL
Mersin Üniversitesi, Eğitim Fakültesi
İlköğretim Bölümü, 33110 Yenişehir, Mersin
Elektronik posta: [email protected]
422
ERRATUM / BASIM HATASI
Regresyon analizinde ayrıca değişkenler arasındaki
ilişkinin biçimi de çok önemlidir. Eğer değişkenler
arasındaki ilişki doğrusal ise probleme doğrusal
veya lineer regresyon analizi, aksi takdirde doğrusal
olmayan veya eğrisel (curvilinear) regresyon analizi denir. Uygulamada sıklıkla doğrusal regresyon
analizi problemleriyle karşılaşılmaktadır (Büyüköztürk, Çokluk ve Köklü, 2011).
Regresyon analizi değişkenlerin bağımlı ve bağımsız değişkenler olarak ayrımı ile başlar. Bağımlı değişken genellikle “y” ve bağımsız (açıklayıcı) değişken ise “x” ile gösterilir. Genel olarak y değişkeni, x
değişkenin bir fonksiyonudur ve y değişkeni üzerindeki değişim, araştırmacının kontrolünde ihmal
edilebilir hata ile ölçülen, x değişkeninin değerine
bağlıdır. Bu durum matematiksel olarak Y = f(X)
biçiminde bir eşitlikle yazılır (Arıcı, 1991). Basit
doğrusal regresyon modeli genel olarak,
y = β0 + β1x + ε (1)
(4)
olmak üzere
y = Xβ + ε olarak yazılabilir. Burada y, n x 1 tipinde gözlemlerin
vektörü, X matrisi, n x p tipinde bağımsız değişken
matrisi, β, p x 1 tipinde tahmin edilecek regresyon katsayılarının vektörü ve ε, n x 1 tipinde rastgele hataların
vektörüdür. Burada k bağımsız değişken sayısı olmak
üzere p = k + 1’dir ve 1’ler sabit terim içindir. Ayrıca k
= 1 alınırsa basit doğrusal regresyon modeli elde edilir.
Regresyon analizi aşağıda verilen önemli varsayımlar/
sayıltılar altında gerçekleşir:
i) y bağımlı değişkeni ile bağımsız değişken/değişkenler arasındaki ilişki lineer ya da yaklaşık
lineerdir.
veya (y1, x1), … , (yn, xn) şeklinde n birimlik veri
çiftine sahip olduğumuzu varsayarsak
ii) ε hataları sıfır ortalamalıdır.
yi = β0 + β1xi + εi, i = 1, … , n iv) Hatalar ilişkisizdir.
(2)
şeklinde yazılabilir. Burada β0 kesim noktasını gösteren sabit değer, β1 regresyon doğrusunun eğimi ve
ε rastgele hata bileşenidir. Hataların sıfır ortalamalı,
bilinmeyen σ2 varyanslı ve ilişkisiz olduğu varsayılır.
Yani i-inci hata miktarı geriye kalan başka bir hata değerine bağlı değildir. Tahmin edilmesi gereken β0 ve
β1 katsayılarına regresyon katsayıları denir. β1 eğimi,
x’deki bir birim değişimin, y’de meydan getirdiği ortalama değişim miktarını gösterir (Büyüköztürk, 2005).
Benzer şekilde çoklu doğrusal regresyon modeli
genel olarak, y bağımlı değişkeni, k tane bağımsız
değişken ile ilişkili olmak üzere,
y = β0 + β1x1 + β2x2 + … + βkxk + ε
(3)
şeklindedir. Bu model xj bağımsız değişkenlerinin
oluşturduğu k boyutlu uzaydaki alt düzlemi gösterir. Tahmin edilecek βj bilinmeyen regresyon katsayısı, xj bağımlı değişkeni dışındaki tüm bağımlı
değişkenleri sabit tutulmak kaydıyla, xj’deki bir birim değişmenin y’de yol açması beklenen değişimi
gösterir. Bundan dolayı βj’ye kısmi regresyon katsayısı da denir (Draper ve Smith, 1998).
Genel olarak regresyon analizinde modelleri matris
formunda göstermek model, veri ve sonuçlar açısından daha etkili ve pratiktir (Montgomery, Peck
ve Vining, 2001). Bu gösterim gerek en küçük kareler gerekse M-tahmin edicileri ile yapılacak tahminlerin teorik olarak daha anlaşılabilir olmasını
sağlamaktadır. (3)’te verilen çoklu doğrusal regresyon modeli matris gösterimi ile
(5)
iii)ε hataları σ2 sabit varyanslıdır.
v)Hatalar normal dağılıma sahiptir.
Normallik varsayımı hipotez testleri ve aralık tahminleri için gereklidir. Son iki varsayımın bir arada
düşünülmesi hataların bağımsız rastgele değişkenler olmaları anlamına gelir (Draper ve Smith, 1998;
Montgomery ve ark., 2001).
Regresyon analizinde bilinmeyen regresyon katsayılarını tahmin etmek için en sık kullanılan yöntem
en küçük kareler yöntemidir. (5) ile verilen matris
formundaki doğrusal regresyon modelinden hareketle en küçük kareler tahmin edicileri ε = y – Xβ
olmak üzere ∑ ε2i hata kareleri toplamını minimize
eder. Böylece b’nın en küçük kareler tahmini ,
,
,
= (X X)-1X y (6)
şeklinde elde edilir. Böylece tahmin edilmiş regresyon modeli
=X =
0
+ 1x1 + 2x2 + … + kxk (7)
şeklinde formüle edilebilir. En küçük kareler için
standartlaştırılmış rezidüler ei = yi – i olmak üzere,
(8)
(9)
ile verilir. Burada,
dir ve hatalar bağımsız, sıfır ortalamalı, σ standart sapmalı, özdeş dağılıma sahip olduğunda
σ2, σ’nin yansız tahmin edicisidir (Birkes ve Dodge,
423
EDUCATIONAL SCIENCES: THEORY & PRACTICE
1993; Chatterjee, Hadi ve Price, 2000; Draper ve
Smith, 1998; Montgomery ve ark., 2001).
En küçük kareler yönteminin klasikleşmiş olmasının temel nedeni hesaplanmasının kolaylığıdır.
Tahmin veriler yardımıyla herhangi bir iteratif yönteme ihtiyaç olmaksızın direk ve kolaylıkla hesaplanmaktadır. Ayrıca diğer tüm yansız tahmin ediciler içinde en iyi lineer yansız tahmin edicidir ve eğer
hatalar normal dağılıyorsa, Maksimum Likelihood
tahmin edicisine benzediği gibi bu durumda diğer
yansız tahmin edicilere göre minimum varyanslı
tahmin verir (Arslan, 1992; Arslan ve Billor, 1996;
Coşkuntuncel, 2005; Montgomery ve ark., 2001).
En küçük kareler yöntemi, veriler normal dağılıma sahipken çoğu zaman uygun yöntemdir ve iyi
istatistiksel özelliklere sahip tahminler verir. Ancak
normallikten sapmalar olduğu durumlarda uygun
yöntem olmaktan çok uzaktadır. Bu tip durumlarda
alternatif tahmin edici olarak robust tahmin ediciler düşünülebilir (Arslan ve Billor, 2000).
Birçok robust tahmin yöntemi olmasına karşın,
bunlar arasında en çok kullanılanlar klasik robust
tahmin yöntemi olarak da bilinen M-tahmin edicisi
(maksimum likelihood tipi tahmin edici), L-tahmin
edicisi (sıra istatistiklerinin lineer kombinasyonları), R-tahmin edicisi (ranka dayalı veya rank dönüşümüne dayalı tahmin edici), RM-tahmin edicisi
(repeated median-tekrarlı medyan tahmin edicileri), LMS-tahmin edicisi (medyan karelerinin en
küçüğünü kullanan tahmin edici)’dir (Arslan, 1992,
2004a, 2004b; Arslan, Edlund ve Ekblom, 2001;
Belsley, Kuh ve Welsch, 1980; Hample, Ronchetti,
Rousseeuw ve Stahel, 1986; Huber, 1981; Rousseeuw, 1984; Rousseeuw ve Leroy, 1987; Rousseeuw ve
Yohai, 1984; Rousseeuw ve Zomeren, 1990).
(5)’te verilen model için en küçük kareler tahmini
(6)’da verilmişti. En genel halde (5)’deki b katsayısı
için M tahmin edicisi, r(e),
i) ρ(e) ≥ 0 ii) ρ(0) = 0 iii) ρ(e) = ρ(-e) iv) |ei| > |ej|, i
≠ j iken ρ(ei) ≥ ρ(ei), ei = yi – xʹiβ
İstatistiksel yöntemlerin sağlamlık teorisi olan robust
istatistik, verilerin normalden sapmalarının klasik yöntemlere etkilerini inceler ve gerekiyorsa daha uygun bir
yöntem geliştirir. İstatistiksel çalışmalarda, regresyon
tahmin edicisi için başlangıçta kabul edilen varsayımlar doğru olmazsa bile iyi sonuçlar verebilen yöntemler
elde edebilme uğruna önemli çabalar sarf edilmiştir.
En küçük karelerin normallik varsayımı altında bile en
uygun tahmin edici olduğu düşüncesi Tukey’in (1960)
“A survey of sampling from contaminated distributions” adlı çalışması ile son bulmuştur. Daha sonra bu
çalışmadan esinlenerek birbirine paralel dört önemli
robust teori, Huber (1964; 1965), Hample (1968), Rousseuw (1984) tarafından ortaya atılmıştır.
koşullarını sağlayan bilinen bir fonksiyon olmak üzere
Yukarıda ifade edildiği gibi, regresyon analizinde karşılaşılan en büyük problem verideki bir veya daha fazla
gözlemin diğer gözlemlerden farklı olması, yani ayrık
değer problemidir. Robust istatistiksel yöntemlerin esas
hedefi bu tip ayrık değer içeren verilerde bile tutarlı sonuçlar veren yöntemler geliştirmektir. Robust regresyon
tahmin yöntemleri, genellikle en küçük kareler yönteminden daha iyi istatistiksel sonuçlar üretmelerine
rağmen, literatür incelendiğinde, istatistiksel analizlere
gereksinim duyan ve istatistiksel yöntemler kullanan bilim dallarından sosyal bilimlerde, bağımlı değişkeni en
iyi yordayan bağımsız değişken/değişkenleri belirlemek
için en küçük kareler klasik yöntemi tercih edilmektedir (Aktaş, 2005; Aluçdibi ve Ekici, 2012; Gündüz ve
Çelikkaleli, 2009; Güneş ve Tulçal, 2002; İnandı, 2009;
Rahman ve Amri, 2011; Şahin ve Anıl, 2012).
ρ(e) =
fonksiyonunu minimum yapar. En çok kullanılan
iki ρ fonksiyonu, Huber ve Tukey ρ fonksiyonlarıdır. Huber ρ fonksiyonu k = 1,345 olmak üzere
ρ(e) =
şeklindedir. Tukey ρ fonksiyonu c = 5 veya 6 olmak
üzere
şeklindedir (Huber, 1981). Her iki fonksiyonda birbirlerine yakın tahminler üretmesine rağmen Huber r
fonksiyonunun bir sakıncası bazen ayrık değerler için
küçük ama sıfıra uzak ağırlıklar üretebilecek olmasıdır. Tukey ρ fonksiyonu matematiksel özelliklerinden
dolayı bu sakıncaları ortadan kaldırmaktadır (Hample ve ark., 1986; Maronna, Martin ve Yohai, 2006; Rousseeuw ve Leroy, 1987). Bu çalışmada Tukey ρ fonksiyonu kullanılacaktır. Fonksiyonu minimum yapacak
β değerini elde etmek için (10)’daki fonksiyonun β’ye
göre türevi sıfıra eşitlenirse,
elde edilir. (13)’teki fonksiyonu
424
(10)
(11)
ERRATUM / BASIM HATASI
, ei ≠ 0
(12)
olarak yazabiliriz. (14)’teki fonksiyonda wi = ρʹ(ei)/
ei dersek fonksiyon,
(13)
şeklinde yazılır ve buradan
(14)
normal denklemleri elde edilir. Buna göre b için M
tahmin edicisi,
M
=
(15)
dir. Matris formunda, (15)’deki normal denklemler,
W matrisi köşegeninde wi ağırlıkları bulunan köşegen bir matris olmak üzere,
XʹWXβ = XʹWy ve
M
M
(16)
tahmini
= (XʹWX)-1XʹWy (17)
olarak elde edilir. Buradaki wi ağırlıkları her bir gözlem için ele edilecek ve eğer gözlem veya gözlemlerin bir gurubu geriye kalanlardan farklı olan, ayrık
değerlerse, sıfıra yakın ağırlıklara sahip olacaktır.
Buna karşılık kalan gözlemler bire yakın ağırlıklara sahip olacaktır. Böylece ayrık değerlerin modele
yaptıkları olumsuz katkı minimuma indirilmiş olacaktır (Coşkuntuncel, 2009). Yöntem (11)’de verilen
minimizasyon problemi için bir algoritmaya gerek
vardır. Algoritmaya başlamak için genellikle en
küçük kareler tahmini başlangıç değeri olarak alınarak iterasyonlar yaptırılır ve ayrık değerlere karşı
en uygun wi ağırlıkları elde edilir (Birkes ve Dodge,
1993). Gerek bağımlı, gerekse bağımsız değişkenler
ayrık değerlere sahip olabilirler. Bu çalışmada sadece bağımlı değişkenlerin ayrık değer veya değerlere
sahip olması durumu ele alınacaktır.
Regresyon analizi kullanılan bir araştırmada tahmin edilen modelin kalitesini değerlendirmek gerekir. Başka bir ifadeyle, uydurulmuş modelin y’deki
değişimleri ne kadar açıklayabildiği bilinmelidir.
Bunun için en yaygın kullanılan yöntem R2 belirleyicilik (determinasyon) katsayısına bakmaktır. Belirleyicilik katsayısı x bağımlı değişken/değişkenleri
ile y’de açıklanabilen değişimin oranıdır. En küçük
kareler için R2 değeri Tablo 1’de verilen regresyonun
anlamlılığı için varyans analizi (ANOVA) tablosun-
dan hesaplanabilir. ANOVA’yı oluşturmak için y –
hata kareleri toplamından modelin varyansının
tahmin edilmesi ile başlayan bir süreci yürütmek
gerekir (Montgomery ve ark., 2001). Buna göre, en
küçük kareler için R2 belirleyicilik katsayısı (klasik
belirleyicilik katsayısı)
R2 =
(18)
ile hesaplanır. KTT, x bağımsız değişkenlerinin etkileri göz önüne alınmadan y’deki değişebilirliğin ölçüsü
ve KTH ise x bağımsız değişkenleri kabul edildikten
sonra y’deki değişebilirliğin ölçüsü olduğundan R2’ye
x ile açıklanabilen değişim oranı da denir. KTH, sıfır ve
KTT değerlerine eşit ya da aralarında bir değer olması
nedeniyle 0 ≤ R2 ≤ 1’dir. R2’nin 1’e yakın değerleri regresyon modelinin y’deki değişebilirliği büyük oranda
açıklayabildiği anlamına gelir.
En küçük kareler için verilen R2 klasik belirleyicilik
katsayısı, çok dikkatle kullanılması gereken bir istatistiktir. Çünkü veriye yeni gözlemlerin eklenmesi
ve modeldeki bağımsız değişken sayısının arttırılması belirleyicilik katsayısını yükseltir. Ancak bu
artış yeni modelin daha iyi olduğu anlamına gelmemektedir. Ayrıca bağımsız değişkenin değişim genişliğinin çok küçük olması durumunda küçük bir
belirleyicilik katsayısı elde edilebilir. Benzer şekilde
genişlik büyük ise büyük bir belirleyicilik katsayısı
elde edilebilir. Ek olarak başlangıç varsayımlarının yanlışlığı ve özellikle ayrık değerlere karşı çok
hassas olan belirleyicilik katsayısı değişkenler arasındaki lineer ilişkinin zayıf olması durumunda da
büyük elde edilebilir (Montgomery ve ark., 2001).
Klasik yaklaşımdaki bu sorunu gidermek için Renaud ve Feser (2010), y bağımlı veya amaç değişkeni ile onun robust tahmini arasındaki korelasyona
2
dayalı robust belirleyicilik katsayısı Rw’yi önermiş2
lerdir. Rw’yi, yw = (1/Σwi)Σwiyi, = (1/Σwi)Σwi i ve
wi ile i robust M tahmini ile elde edilen ağırlıklar
ve uydurulmuş değerler olmak üzere
(19)
şeklinde verilmiştir. Aynı ağırlık ve uydurulan değer2
ler için Rw’nin toplam kareler toplamı (KTT) formunu
(20)
425
EDUCATIONAL SCIENCES: THEORY & PRACTICE
Tablo 1.
Regresyonun Anlamlılığı İçin Varyans Analizi Tablosu
Değişimin
Kaynağı
Kareler Toplamı
(KT)
Serbestlik
Derecesi
Kareler Ortalaması
(KO)
F
KOR / KOH
Regresyon
KTR =
k
KOR = KTR / k
Hata
KTH =
n–k–1
KOH = KTH / (n – k – 1)
Toplam
KTT =
n–1
şeklinde vermişler ve bu iki belirleyicilik katsayısının
eşit olduğunu göstermişlerdir. Diğer yandan, robust
belirleyicilik katsayısı verideki ayrık değerlerden daha
az etkilenir ve hataların normal dağılması durumunda daha verimlidir. Ayrıca açıklayıcı değişkenlerin
dağılımları üzerinde herhangi bir varsayım yapmaz ve
dolayısıyla amaç değişkenin koşulsuz dağılımı içinde
varsayıma ihtiyaç duymaz (Renaud ve Feser, 2010).
Başta sosyal bilimler olmak üzere birçok alanda yukarıda ayrık değerlerle ilgili söz edilen bu dezavantaja
rağmen regresyon analizinde en küçük kareler yöntemi kullanılmaya devam edilmektedir. Ülkemiz açısından bunun en önemli nedeni, robust yöntemlerin
en küçük karelerin aksine iteratif yöntemlere ihtiyaç
duymalarından dolayı hesaplanmasının zor olmasının yanında (Rousseeuw ve Leroy, 1987), regresyon
analizinde en küçük kareler yöntemi dışında başka
yöntemlerin yeterince ele alınmamış olmasıdır. Bu
çalışmada hipotetik ve gerçek veriler çerçevesinde en
küçük kareler ve M tahmini karşılaştırılarak söz konusu değer sapmalarının etkisini minimuma indirecek en uygun yöntem belirlenmeye çalışılmıştır. Coşkuntuncel (2009) en küçük kareler tahmininin ayrık
değerlere karşı çok hassas olduğunu M tahmininin ise
bu ayrık değerlere karşı daha dirençli olduğunu göstermiştir. Bu çalışmada hem gerçek hem de hipotetik
veriler üzerinde ayrık değerler belirlenmiş ve R2 belirleyicilik katsayısının robust formu yardımıyla tahminler karşılaştırılmıştır. Bu çalışma ile bilimsel araştırmalarda sıklıkla kullanılan tahmin yöntemlerinde
en uygun sonuçları verebilecek alternatif tahmin yöntemini ortaya koymak ve tanıtmak amaçlanmaktadır.
Yöntem
Bu çalışmada basit ve çoklu lineer regresyon yöntemleri uygulanan veriler için klasik ve robust M regresyon analiz yöntemleri karşılaştırılmıştır. Ayrık değerlerin en küçük kareler klasik yöntemi üzerindeki etkileri ve robust M yönteminin bu ayrık değerlere karşı
426
gösterdiği direnç, belirleyicilik katsayısının klasik ve
robust formu yardımıyla gösterilmiştir.
Araştırma Verileri
Bu çalışmada 1 hipotetik ve 2 gerçek olmak üzere toplam
3 veri seti üzerinde çalışılmıştır. Hipotetik veride rastgele
seçilen 15 öğrencinin Genel Matematik başarısının Lineer Cebir dersi başarısı üzerindeki yordayıcı etkisine bakılmıştır. Basit Lineer regresyon modelinin kullanıldığı
bu veri seti 2011-2012 eğitim öğretim döneminde Mersin Üniversitesi Eğitim Fakültesi İlköğretim Matematik
Öğretmenliği Bölümündeki rastgele seçilen 15 öğrencinin derslere ilişkin başarı notlarından elde edilmiştir.
Çoklu regresyon modelinin kullanıldığı gerçek veriler daha önce en küçük kareler yöntemi ile analizi
yapılmış ve yayınlanmış çalışmalardan elde edilmiştir. Bunlardan ilki Gündüz ve Çelikkaleli’nin (2009)
akademik yetkinlik inancı, akran baskısı ve kaygı
değişkenlerinin kız ve erkek ergenlerin saldırganlık
düzeylerini yordamadaki katkılarının incelendiği saldırganlık düzeyi verisidir. Veri, 2008–2009 öğretim
yılı bahar yarıyılında, Mersin il merkezinde bulunan
genel, Anadolu ve meslek liselerinde okumakta olan
ve rastgele örnekleme yöntemiyle ulaşılan 129 kız, 102
erkek toplam 231 lise öğrencisi ergenden oluşmaktadır. Veri toplama aracı olarak; Saldırganlık Ölçeği,
Akademik Yetkinlik Beklentisi Ölçeği, Akran Baskısı
Ölçeği ve Sürekli Kaygı Envanteri kullanılan araştırmada Akademik Yetkinlik İnancı (AYİ), Akran Baskısı (AB) ve Sürekli Kaygı (SK) değişkenlerinin kız ve
erkek ergenlerin saldırganlık düzeylerini yordamadaki katkıları incelenmiştir.
İkinci veri seti öğrenci tükenmişliğini yordamada
stresle başa çıkma, sınav kaygısı, akademik yetkinlik
ve anne-baba tutumları değişkenlerinin incelendiği
Çapulcuoğlu (2012) tarafından Mersin merkez ilçelerinde bulunan lise kurumlarından küme örnekleme
yöntemi kullanılarak belirlenen 1385 öğrenci ile yapılan yüksek lisans tezinden elde edilmiştir.
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