Exploring Student-Centred Teaching, Open-Ended Tasks, and Real Data Analysis to Promote Students’ Reasoning about Variation Thesis submitted for the MSc in Mathematics and Science Education Dian Kusumawati Research Supervisor: drs. André Heck AMSTEL Instituut Universiteit van Amsterdam Science Park 904 1098 XH Amsterdam The Netherlands July 2010
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Exploring Student-Centred Teaching,
Open-Ended Tasks, and Real Data Analysis
to Promote Students’ Reasoning about Variation
Thesis submitted for the MSc in Mathematics and Science Education
Dian Kusumawati
Research Supervisor: drs. André Heck
AMSTEL Instituut
Universiteit van Amsterdam
Science Park 904
1098 XH Amsterdam
The Netherlands
July 2010
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iii
Abstract
In this master research thesis I report about a study in which I explored the influence of a
specific approach in teaching variation on the progress and development of students‟
statistical reasoning about variation. A socio-constructivist teaching and learning approach
was designed and tried out in a pretest-posttest experimental-control-group research design.
This was done with students of a social science stream in a secondary school in a rural area of
Indonesia. The teaching approach contained three new key elements, namely, the use of
(1) real data within a context instead of the use of artificial data; (2) open-ended tasks; and
(3) group work. The research results indicated that the experimental teaching approach pro-
vided students a more conducive learning environment for developing statistical reasoning.
Although students from both experimental and control groups were mostly at a low level of
reasoning, the quantitative and qualitative analysis of their response indicated that there were
more students in the experimental group that improved regarding the level of statistical
reasoning. Qualitatively, students in the experimental group began to use central measures in
making their conclusions. Regarding the procedural knowledge, there was no statistically
significant difference in the performance between the two groups. These results and the fact
that the cooperating teacher was ready to adopt the teaching approach have encouraged me to
conclude that the chosen teaching approach has potential to help students develop and
progress with statistical reasoning about variation. Based on the analysis of the teaching
experiment, recommendations for adopting the teaching approach in future practice are given.
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Acknowledgement
I would like to express my deep gratitude to my supervisor, drs. André Heck, whose valuable
advice, supervision, flexibility, and motivational support have enabled me to complete this
thesis. Many thanks are directed to dr. Mary Beth Key, who has helped me go through this
master program without much troubles.
I thank the cooperating teachers who have accommodated me to do my teaching experiment
in an unusual time in their school plan. I also would like to thank dr. Willem Jan Gerver from
Maastricht University who gave me raw data of a recent Indonesian growth survey for use in
the teaching and learning activities in my study.
Not least importantly, I also thank my fellow students at the AMSTEL Institute and my
friends in Amsterdam for all support they have given me, especially Clea Matson, Lilia
Ekimova, and Budi Mulyono.
Finally, by nature I tend to be easily worried and self-negative. This trait played tricks on me
when finishing the master‟s thesis. I wish to thank my parents, my brother, my sisters, and my
dear friend Dharma, for listening to my worries when the self-negativity engulfed me and
cheering me up all the way. To my mother and father, this thesis is simply dedicated to you.
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Table of Contents
ABSTRACT III
ACKNOWLEDGEMENT V
1. INTRODUCTION 1
1.1. Statement of the Problem 1
1.2. The Indonesian Education System 2
1.3. Statistics in the Indonesian Mathematics Curriculum 4
1.4. The Purpose of the Study 6
2. THEORETICAL BACKGROUND 9
2.1. Statistical Literacy, Statistical Reasoning and Statistical Thinking 9
2.2. Assessment of Statistical Reasoning 10 2.2.1. The Structure of the Observed Learning Outcome (SOLO) Taxonomy 10
2.2.2. Statistical Thinking 13
2.2.3. Statistical Reasoning about Variation 15
2.3. Teaching Variation 17 2.3.1. Conceptions about Variability 17
2.3.2. Suggestions from Research Studies about Contexts 18
2.3.3. Principles Underpinning the Design of My Lesson Activities 19
3. RESEARCH DESIGN AND METHODOLOGY 21
3.1. Research Question 21
3.2. Research Setting and Research Methodology 22 3.2.1. The School Setting 22
3.2.2. Research Methodology 23
3.2.3. The Teaching Materials 24
3.3. Research Instruments 26 3.3.1. The Pretest and Posttest 26
3.3.2. The Questionnaire 32
3.3.3. The Interview 33
4. RESULTS AND ANALYSIS OF THE TEACHING EXPERIMENT 35
4.1. Classroom Observations Prior to the Teaching Experiment 35
4.2. The Teaching Experiment 36 Lesson 1: Activity 1 37
Lesson 2: Activity 1 revisited 40
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Lesson 3: Activity 2 42
Lesson 4: Activity 2 42
4.3. The Teaching in the Control Group 44
4.4. The Questionnaire 45 The Use of Real Data 45
Group Work 46
The Teaching Approach 46
Students‟ Free Feedback 47
4.5. The Interview with the teacher of the experimental group 48
4.6. Analysis of the Teaching Experiment 50
5. RESULTS AND ANALYSIS OF THE PRETEST AND POSTTEST 53
5.1. Subtest A: Question 1 and 2 53
5.2. Subtest B: Question 3 and 4 60
5.3. Subtest C: Question 5-10 61
5.4. Result of the Interview 82
5.5. Summary and Analysis of the Findings from Pretest and Posttest 84 5.5.1. Subtest A 84
5.5.2. Subtest B 85
5.5.3. Subtest C 85
6. CONCLUSIONS AND DISCUSSIONS 87
6.1. Conclusions 87
6.2. Limitations of my Study and Suggestions for Future Research 91
REFERENCES 93
EXTENDED BIBLIOGRAPHY 95
LIST OF APPENDICES 97
Appendix A. Garfield and Ben-Zvi’s Framework for Assessing Reasoning about
Variability 99
Appendix B. Students’ Activity Sheets 101
Appendix C. Pretest/Posttest 111
Appendix D. The Questionnaire 117
1
1. Introduction In this chapter, I explain the background of my research, including my motivation. Firstly, I
state the background problem in general. Secondly, I describe the Indonesian education
system. Thirdly, I give an overview of the Indonesian statistics curriculum at primary and
secondary level. Finally, I explain the purpose of my research. I hope that my personal aims
and motivation for doing this research can be grasped from the contents of this chapter.
1.1. Statement of the Problem
Statistics has become part of primary and secondary mathematics education curriculum across
the world, although the breadth and depth of its content differ from country to country.
Statistics being considered by many people as a part of mathematics, it is no surprise that
statistics teaching in school practice does not differ much from mathematics teaching.
Therefore, the recent reform efforts in mathematics education based on a constructivist view
of education have also influenced statistics education (cf., Moore, 1997).
Recent reforms in statistics education promote the idea that the focus of teaching and
learning statistics should be on the understanding of statistical concepts, rather than on the
procedural knowledge and skills. As Moore & Cabe (2005, p. xxxi) wrote: “The goal of
statistics is to gain understanding from data.” Thus, students should not merely be able to
compute statistical measures. It is recommended that the focus of the statistical content to be
learned by students is the understanding of statistical ideas and concepts. To this end,
statistics should be less taught by lectures, but more through real data investigation carried out
by students. Cobb & Moore (1997, p. 801) pointed out that the role of context in statistics and
mathematics are different: “Statistics requires a different kind of thinking, because data are
not just numbers, they are number with context”. This requires the introduction of a real
world context in any interesting statistics problem.
What can be said about constructivist approaches in Indonesian mathematics
education? Since 2000, the Indonesian Ministry of Education has enforced a new curriculum
that promotes a constructivist view of education (Badan Litbang Puskur, 2002). Student-
centred teaching and learning is endorsed and the use of ICT in education is promoted.
However, the implementation of the top-down reform has not been successful yet. Teachers
still rely heavily on textbooks, and their teaching and learning still tends to emphasize
formulas and procedures (Sembiring et al., 2008). In other words, rote learning and teacher-
centred activities are still the dominant ways of knowledge transfer. In particular, the in-
service teacher training for this new curriculum has not been effective.
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If current mathematics teaching in Indonesia still gives students the impression that
mathematics is only about plugging numbers into formulas to get the correct answer of a
problem, then I believe that the impression is even more strongly felt by students toward
statistics, as statistics teaching is still mostly based on textbooks which mainly deal with
formulas, computation and closed problems with artificial data. My belief is in line with Ben-
Zvi & Garfield (2004, p. 4) who wrote: “Students equate statistics with mathematics and
expect the focus to be in numbers, computations, formulas, and one right answer.”
In fact, I can still remember that statistics meant just number plugging to me when I
was a high school student. I first got interested in statistics when I did a course in Regression
Analysis during my bachelor study. Only then, when I got in contact with data analysis, I
started to see that statistics is useful in making decisions and drawing conclusions.
The above problem in statistics education in Indonesia at secondary school level
motivated me in my master‟s study to investigate a different approach of statistics
teaching that helps students improve their understanding of and reasoning about statistical
concepts and ideas, and not just learn how to do statistical computations. The usual teaching
sequence of (1) explaining the formulas, (2) working out examples, and (3) giving procedural
problems has not been a sufficiently successful approach to enable students to reason statisti-
cally at a proficient level. Exploratory data analysis by students seems to me more promising.
My personal experience as an Indonesian secondary school student and as a teacher-
student in mathematics education also motivated me to try out a student-centred approach in
which students would analyze real data. I saw and still see no reason why the students could
not learn how to draw conclusions based on real data and simple descriptive statistics which
they had learned before. The depth of the data analysis can be adjusted to the content of the
Indonesian curriculum.
In my study, I conducted an experiment in a secondary school class in which I tried
out a constructivist approach to learning about variation. I compared the results of the
experimental group with that of a control group, who received traditional teaching. I hoped
and expected that the results of my study could lead to recommendations for teachers and
future teachers and give them ideas and/or suggestions about better ways of teaching the
subject of variation.
1.2. The Indonesian Education System
Based on Law Number 20 [UU no. 20 year 2003] about the Indonesian education system, the
national education system consists of formal, non-formal and informal education. The system
of formal education consists of primary education (Grade 1-9), secondary education
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(Grade 10-12) and higher education (see Figure 1). The primary education consists of 6 years
elementary school or Sekolah Dasar (SD) and 3 years lower (literally, first) secondary school
or Sekolah Menengah Pertama (SMP). It is free of charge and compulsory for every child of
age between 7 to 15 years. There are two types of secondary education: General Secondary
School or Sekolah Menengah Atas (SMA) and vocational secondary school or Sekolah
Menengah Kejuruan (SMK). In SMA, there are three streams:
Natural science or Ilmu Pengetahuan Alam (IPA)
Social science or Ilmu Pengetahuan Sosial (IPS)
Language or Bahasa.
A student graduates from secondary education through a nationwide examination.
Figure 1. The education system in Indonesia.
Regarding the curriculum, as mentioned in Section 1.1., the government introduced in
2000 a new curriculum, which is competency-based and promotes a constructivist student-
centred approach. In 2005, the government introduced another curriculum called Kurikulum
Tingkat Satuan Pendidikan (KTSP) or Curriculum of an Education Unit Level (Naskah
Akademik KTSP Pendidikan Dasar dan Menengah, Puskur, 2005). KTSP is basically an
extension and diversification of curriculum 2000 in the spirit of school autonomy and local
government autonomy. The curriculum is still competence-based but the central government
gives freedom to every school to develop its own implementation of the curriculum, based
Higher Education
Secondary Education (3 years)
SMA/SMK
Lower Secondary (3 years)
SMP
Elementary School (6 years)
SD
National Examination
National Examination
Primary Education
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upon the potential of its own students, the social characteristics and the potential of the local
community. The KTSP implementation must conform to the basic structure of the formal
curriculum and the competency standards of graduates dictated by the Ministry of Education.
Personally, I really agree with this curricular scheme and really like the fact that this
means that teachers have freedom to develop their own subject curriculum, which is then later
together with all other subjects compiled into the KTSP implementation of the school. I also
agree that education should be tailored according to the potential of the students. However, in
practice, the teachers and the school still have trouble with design and implementation of their
own curriculum. As mentioned in the first section, the student-centred approach of curriculum
2000 has not yet been adopted by many teachers and in the end, the syllabus and KTSP of
many schools is produced from copying other school‟s KTSP or from an in-service teacher
training event held by the government (Kajian Kebijakan Kurikulum Matematika, Puskur,
2007). Usually only better-facilitated schools in the bigger cities are able to produce their own
curriculum. Efforts are still needed to improve teacher professionalism so that the goal of
accommodating each student‟s needs can be realized. With this research study I hope to
contribute to such standard of education regarding statistical notion of variation.
1.3. Statistics in the Indonesian Mathematics Curriculum
The Indonesian mathematics curriculum is somewhat modular in the sense that each big
mathematical concept is taught in a separate chapter of the textbook. Once it has been
completed, students are not likely to touch upon the subject again for a while, except for
reviewing or refreshing purposes when needed as prerequisites of subsequent book chapters.
The Indonesian curriculum is also a spiral curriculum in the sense that at every higher level of
education, the breadth and depth of a big concept are increased.
At each level of schooling, elementary, lower and upper secondary level, there is a
book chapter about statistics (see Table 1). In elementary school (SD), statistics is taught in
grade 6, in the first semester, under the topic of „Data Analysis‟. In this grade, students mainly
learn to analyze data in simple ways, to present data in simple graphs and tables and to
interpret them. I reviewed one book from the government (Sumanto et al., 2008) and in this
textbook; the measure of centre is indeed not present. However, my personal communication
with a primary school teacher revealed that the common measures of centre, namely mode,
median and mean, are taught in reality because they usually appear in the final school
examination.
In lower secondary school (SMP), statistics is taught in Grade 9, first semester, under
the topic „Statistics and Probability‟. The students learn more ways to represent data and, in
addition, they learn about central measures. Moreover, probability is introduced.
5
Finally, in upper secondary school (SMA/SMK), statistics is taught in grade 11, under
the topic of „Statistics and Probability‟. In Table 1, it is shown that measures of dispersion are
included in the contents of statistics in SMA. Furthermore, the probability content is more
advanced compared to that in SMP. Regarding the standard contents of statistics, the three
streams have the same statistics contents, but the contents of probability differ. Students in
natural science stream (IPA) learn more about probability. Another difference is the time allo-
cation for learning this topic. For IPA students, the topic „Statistics and Probability‟ is only
one out of three topics in the first semester. On the other hand, students in the social science
stream (IPS) only learn this topic for the whole semester and students in the language stream
(Bahasa) have the whole year to learn Statistics and Probability. I believe that the underlying
idea is to adjust the pace of mathematical learning of students from stream to stream.
Level Standard Competency Explanation of Standard Competency
SD Grade 6
Semester 1
Data Analysis
Collecting and analyzing
data
Collect and read data
Analyze and present data in table form
Interpret data representations
SMP Grade 9
Semester 1
Statistics and
Probability
Analyzing and presenting
data
Understanding the
probability of simple events
Determine the mean, median, and mode
Present data in tabular forms, bar chart,
line graph and pie chart
Determine the sample space of an
experiment
Determine the probability of a simple
event
SMA IPA
Grade 11
Semester 1
Statistics and
Probability
Using the rules of statistics,
counting rules, and
properties of probability in
problem solving
Read data in tabular forms, bar chart, line
graph, pie chart, and ogive
Present data in forms of table, bar chart,
line graph, pie chart, and ogive, and
interpret them
Compute measures of centre, location
and dispersion, and interpret them.
Use the rules of multiplication,
permutation and combination in problem
solving
Determine the sample space of an
experiment
Determine the probability of an event
and interpret it (the meaning)
Table 1. The standard curriculum of statistics and probability.
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SMA IPS
Grade 11
Semester 1
Statistics and
Probability
Using rules of statistics,
counting rules, and properties
of probability in problem
solving
Read data in forms of table, bar chart,
line graph, pie chart, and ogive
Present data in forms of table, bar
chart, line graph, pie chart, and ogive
and interpret them
Compute measures of centre, location
and dispersion, and interpret them.
Use the rules of multiplication,
permutation and combination in
problem solving
Determine the sample space of an
experiment
Determine the probability of an event
and interpret it (the meaning)
SMA Bahasa
Grade 11
Statistics and
Probability
Semester 1
Analyzing, presenting,
and interpreting data
Read data in forms of table, bar chart,
line graph, pie chart, and ogive
Present data in forms of table, bar
chart, line graph, pie chart, and ogive
and interpret them
Compute measures of centre, location
and dispersion, and interpret them.
Semester 2
Using counting rules to
determine the probability of
an event and interpret it.
Use the rules of multiplication,
permutation and combination in
problem solving
Determine the sample space of an
experiment
Determine the probability of an event
and interpret it (the meaning)
Table 1. (continued).
1.4. The Purpose of the Study
In my research, I chose one main topic in the secondary school curriculum: measure of
dispersion/variation. Unlike central measures such as mean, research on the notion of measure
of variation is rather limited. This is unfortunate since variation can be considered as one of
the basic concept of statistical thinking (Cobb & Moore, 1997, p. 801).
I conducted a small teaching experiment about the notion of variation and the measure
of dispersion (specifically standard deviation) in one school in a class of IPS students.
Another class of IPS students at the same school acted as a control group in which the usual
teacher-centred approach was applied. My research was in essence a case study, the results of
which depended much on the characteristics of the students in this particular school where
this experiment took place. This implies that at this stage of the research no easy
generalization of the results could be obtained. I designed the research study such that the
teaching experiment can be repeated in other regular classes of SMA in Indonesian schools in
order to obtain in future more general results.
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In my research, I investigated and compared students‟ statistical reasoning about
variation prior and after teaching the topic of variation and measure of dispersion. One of my
other objectives was that the results would lead to recommendations to mathematics teachers
regarding statistics teaching and learning, and hopefully would have a positive effect on
statistics teaching in Indonesia.
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2. Theoretical background In my study, I wanted to investigate to what extent my teaching and learning approach
affected students‟ statistical reasoning, especially reasoning about variation. In this section, I
present a definition and assessment framework of statistical thinking and reasoning taken
from research literature and that I used for my research. I also summarize suggestions from
research literature regarding teaching statistics, in particular teaching about variation, that
were taken into account in my study.
2.1. Statistical Literacy, Statistical Reasoning and Statistical Thinking
Research in statistics education in the last two decades has changed direction from research
on misconceptions about statistical concepts or ideas into research focusing on how students
learn and reason about statistical concepts (Shaughnessy, 2007). Many researchers in the field
of statistics education classify this research into three big ideas: statistical literacy, statistical
reasoning, and statistical thinking. There is no formally agreed definition yet of these three
ideas. Ben-Zvi & Garfield (2004, p. 7) defined them as follows:
“Statistical literacy includes basic and important skills that may be used in under-
standing statistical information or research results. These skills include being able to
organize data, construct and display tables, and work with different representations of
data. Statistical literacy also includes an understanding of the concepts, vocabulary,
and symbols, and includes an understanding of probability as a measure of uncer-
tainty.
Statistical reasoning may be defined as the way people reason with statistical
ideas and make sense of statistical information. This involves making interpretation
based on sets of data, representation of data, or statistical summary of data. Statistical
reasoning may involve connecting one concept and another (e.g., center and spread),
or it may combine ideas about data and chance. Reasoning means understanding and
being able to explain statistical processes and being able to fully interpret statistical
results.
Statistical thinking involves an understanding of why and how statistical investiga-
tions are conducted and the „big ideas‟ that underlie statistical investigations. These
ideas include the omnipresence nature of variation and when and how to use appro-
priate methods of data analysis such as numerical summaries and visual display of
data. Statistical thinking involves an understanding of the natures of sampling, how
we make inferences from samples to populations, and why design experiments are
needed in order to establish causation…”
Chance (2002) reviewed the literature on the definition of statistical thinking and
concluded that: “Perhaps what is unique to statistical thinking, beyond reasoning and literacy, is the
ability to see the process as a whole (with iteration), including „why,‟ to understand
the relationship and meaning of variation in this process, to have the ability to
explore data in ways beyond what has been prescribed in texts, and to generate new
questions beyond those asked by the principal investigator. While literacy can be
10
narrowly viewed as understanding and interpreting statistical information presented,
for example in the media, and reasoning can be narrowly viewed as working through
the tools and concepts learned in the course, the statistical thinker is able to move
beyond what is taught in the course, to spontaneously question and investigate the
issues and data involved in a specific context.”
Mooney (2002) adopted the definition of statistical thinking from (Shaughnessy,
Garfield, & Greer, 1996) where it means the cognitive actions that students engage in during
the data-handling processes of describing, organizing and reducing, representing, and ana-
lyzing and interpreting data.
From the above definitions, it seemed to me that the definition of statistical literacy,
reasoning and thinking are not mutually exclusive. By statistical reasoning about variation, I
mean the way people reason with variation (as Ben-Zvi and Garfield defined this) and the
way people make use of the concept of variation to investigate issues and data (as Chance
defined statistical thinking). However, to be honest, taking the review of Chance (2002) into
deeper account and carefully reading of Ben-Zvi & Garfield‟s (2004) definition of statistical
thinking, led me to the conclusion that the students in my research project were not much
involved in such type of activity, that is, they did not have to learn about nor carry out a
statistical inquiry. In my teaching experiment I engaged students in data exploration activities
that mainly involved statistical literacy and statistical reasoning in the sense of Ben-Zvi and
Garfield (2004), the two ideas which in my point of views constitute Mooney‟s (2002)
definition of statistical thinking. To avoid confusion in terminology, I prefer in this thesis to
adopt the distinction between literacy and reasoning. When I refer to Mooney‟s statistical
thinking I understand it mostly as statistical reasoning.
2.2. Assessment of Statistical Reasoning
Because I wanted to investigate in my research whether my teaching and learning
approach would improve students‟ reasoning, I searched the literature to find an appropriate
assessment tool, preferably one suitable for classroom practice (i.e., not only suitable in a
small group setting or laboratory setting, and furthermore easy to use by teachers in their
practice). The following subsections are about the assessment framework of statistical
thinking and reasoning from existing literature that I selected for use.
2.2.1. The Structure of the Observed Learning Outcome (SOLO) Taxonomy
The SOLO taxonomy is a neo-Piagetian framework proposed by Biggs and Collis (1982) to
analyze the complexity level at which students carry out tasks and answer questions. The
SOLO taxonomy is not specifically designed for statistics or mathematics, but I discovered
11
that many researchers in statistics education use this taxonomy for characterizing and
assessing the students‟ statistical thinking and reasoning. For this reason I briefly review the
SOLO taxonomy; further details about the general model can be found in Biggs & Collis
(1982) and for its application in statistics education I refer to (Jones et al., 2004; Shaughnessy
et al., 2007) and references therein.
The SOLO taxonomy posits five modes of functioning (similar to Piaget‟s
development stages: preoperational, early concrete, middle concrete, concrete generalization,
and formal operation) and five hierarchical levels of complexity at which tasks can be carried
out in principle (prestructural, unistructural, multistructural, relational, and extended abstract).
The development stage sets mainly the upper limit for the cognitive level that can be reached,
but this does not mean that at thus stage of functioning lower levels of complexity cannot be
observed anymore.
The five levels of complexity of students‟ responses to tasks, usually referred to as the
SOLO levels, are as follows: at the prestructural level, a student avoids the question (denial),
repeats the question (tautology), or engages in the task but is distracted or misled by irrelevant
aspects belonging to an earlier mode of functioning. For the unistructural level, the student
focuses on the relevant domain and picks up on one relevant aspect of the task, running in this
way the risk to come to a limited conclusion or a dogmatic answer only. At the multistructural
level, the student picks up several disjoint and relevant aspects of the task but does not
integrate them and ignores inconsistencies or conflicts in the provided information. At the
relational level, the student integrates the various aspects and produces a more coherent
understanding of the task. At the extended abstract level, the student recognizes that a given
example is an instance of a more general case, that is, (s)he generalizes the structure to take in
new and more abstract features that represent thinking in a higher mode of functioning. As
noted already by Biggs and Collis, only the first four cognitive levels are encountered up to
and including secondary education; one hardly notices the extended abstract level in
classroom practice.
Biggs & Collis (1982) described certain crucial characteristics of each SOLO level in
terms of the dimensions of capacity (the required amount of working memory or attention
span), relating operation (between cue and response), and consistency and closure (no contra-
dictions in the final conclusion). See Table 2, taken from Biggs & Collis (1982, p. 24-25).
12
Developmental
Base Stage with
Minimal Age
SOLO Description
Capacity Relating Operation Consistency and Closure
Extended Abstract
(16+ years)
Extended
Abstract
Maximal: cue +
relevant data +
interrelations +
hypotheses
Deduction and Induction. Can
generalize to situations not
experienced
Inconsistencies resolved. No felt need to give
closed decisions-conclusions held open, or
qualified to allow logically possible
alternatives.
Concrete
generalization
(13-15 years)
Relational High: cue + relevant
data + interrelations
Induction. Can generalize
within given or experienced
context using related aspects
NO inconsistency within the given system, but
since closure is unique so inconsistencies may
occur when he goes outside the system.
Middle Concrete
(10-12 years) Multistructural
Medium: cue +
isolated relevant data
Can “generalize” only in terms
of a few limited and
independent aspects
Although has feeling for consistency can be
inconsistent because closes too soon on basis of
isolated fixations on data, and so can come to
different conclusions with the same data
Early Concrete
(7-9 years) Unistructural
Low: cue + one
relevant datum
Can “generalize” only in terms
of one aspect.
No felt need for consistency, thus closes too
quickly; jumps to conclusions on one aspect,
and so can be very inconsistent.
Pre-operational
(4-6 years) Prestructural
Minimal: cue and
response confused
Denial, Tautology,
transduction. Bound to specifics
No felt need for consistency. Closes without
even seeing the problem.
Table 2. Base stage of cognitive development and response description according to the SOLO Taxonomy (note that the SOLO description in the 2nd
column refers to the
maximum level at the given developmental stage in the corresponding entry in the 1st column).
13
2.2.2. Statistical Thinking
There are several studies in which the main goal was to develop a framework for characteriz-
ing and assessing statistical thinking (in the sense of Mooney, 2002). Below, I will discuss
three of them.
Jones et al. (2000) developed a framework for characterizing elementary children‟s
statistical thinking situated in the SOLO taxonomy. They focused on data handling and used
the following four constructs in their framework were: (1) describing; (2) organizing and
reducing; (3) representing; and (4) analyzing and interpreting data. To characterize students‟
statistical thinking in each of these constructs, they used four levels corresponding with the
first four levels in the SOLO taxonomy:
1) Idiosyncratic: idiosyncratic students are engaged in a task but they are easily distracted or
misled by irrelevant aspects;
2) Transitional : students focus on a single relevant aspect of a task;
3) Quantitative: students can focus on multiple relevant aspects of the task but have
problems in integrating them;
4) Analytical: students are able to make links between different aspects of the task
(demonstrate relational level of thinking).
Jones et al. (2000) conducted their study by analyzing the interviews of sixteen students
(grade 2-5) who responded to several data handling tasks with questions for every construct.
Statistics concepts like average, spread were probed at elementary level and the way children
would work with basic data displays like bar graphs.
Mooney (2002) used the framework of Jones et al. (2000) as his initial framework to
assess middle school students‟ statistical thinking in data handling tasks and extended it with
another level of statistical thinking: Extended Analytical, meaning that students can examine
data from more than one perspective. However, Mooney did not find data that support the
existence of the fifth level in middle school students and thus also used the four levels of
statistical thinking above in his result. The final framework Mooney (2002) is reproduced in
Figure 2. It was actually used for statistics concepts of measures of centre and spread. In this
study, I took an eclectic approach and selected suitable parts of Mooney‟s framework to
assess my students‟ statistical reasoning (see Chapter 4).
14
Figure 2. Mooney‟s framework of middle school students‟statistical thinking.
15
Groth (2003) sought a framework for describing high school students‟ statistical
thinking, when it comes to describing, organizing and reducing, representing, analyzing and
collecting data. Groth conducted a qualitative study to find out characteristics or patterns for
the four constructs that were used by Mooney (2002) and Jones et al. (2000). He developed a
set of statistical thinking tasks and used it in structured, task-based clinical interview sessions
with high school students and recent high school graduates. Students were asked to solve
these statistical thinking tasks and then the students‟ responses were analyzed to define
patterns of responses to questions regarding processes of data handling, applying the SOLO
taxonomy. In my study, I used a modification of Question 8 from his fifth task (Question 7 in
my pretest, see p.25), which was part of a set of questions Groth used to probe students‟
understanding about summarizing data through a measure of centre and measure of spread.
The pattern descriptors for using measures of centre and spread that Groth (2003, p.85, 90)
identified are listed in Table 3.
Four Pattern Descriptors for Using Measures of Centre
A student uses:
1. reasonable formal measures to locate centres of data sets
2. a combination of reasonable formal and visual measures to locate centres of data sets.
3. a combination of formal and visual measures to find centres of data sets, only some of which are reasonable for the given set of data
4. only visual approaches to find centres of data sets, only some of which are reasonable for
the given sets of data.
Three Pattern Descriptors for Using Measure of Spread
A student gives:
1. quantifications and subjective verbal descriptions of spread that are suitable for given sets of data.
2. quantifications and subjective verbal descriptions of spread. Some descriptions or quantifications are not suitable for given sets of data
3. subjective verbal descriptions of spread rather than quantifications
Table 3. Groth‟s pattern descriptors for using measures of centre and spread.
2.2.3. Statistical Reasoning about Variation
The term „variation‟ and „variability‟ are often used interchangeably in research literature (cf.,
Shaughnessy, 2007; Reading & Shaughnessy, 2004). However, Reading & Shaughnessy
(2004, p. 202) made the following distinction between the two terms:
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“The term variability will be taken to mean the characteristics of the entity that is
observable, and the term variation to mean the describing or measuring that
characteristics. Consequently, … „reasoning about variation‟ will deal with the
cognitive process involved in describing the observed phenomena in situations that
exhibit variability, or the propensity to change. ”
In my study, I adopted the above definition of Reading & Shaughnessy (2004) of statistical
reasoning about variation. But it is noted that in the Indonesian language there is no word for
variation in this sense; only words such as variability, diversity and variety exist in every day
speech. In mathematics textbooks the term variability is also used for variation (in the sense
of Reading & Shaughnessy). Therefore it may happen that I use the words terms inter-
changeably.
In this section I briefly discuss literatures on reasoning about variation. To begin with,
Watson et al. (2003) conducted a study to measure understanding of variation in a chance
setting. They gave questionnaires to students in grades 3, 5, 7, and 9, and from the results of
their analysis initially based on the SOLO taxonomy, they defined four ability levels of
students‟ understanding of variation:
1) prerequisites for variation: working out the environment, table/graph reading,
intuitive reasoning for chance;
2) partial recognition of variation: putting ideas in context, tendency to focus on single
aspects and neglect others;
3) applications of variations: consolidating and using ideas in context, inconsistent in
picking salient features
4) critical aspects of variation: employing complex justification or critical reasoning.
Watson et al. (2003) did not give clear descriptors for each of these levels, but it seemed to
me that these four levels are equivalent to Mooney‟s four levels of statistical thinking (for
example, compare Mooney‟s transitional level with Watson et al.‟s level of partial recogni-
tion of variation). As mentioned in Subsection 2.2.2, I used part of Mooney‟s framework for
assessing students‟ reasoning about variation (see Chapter 4). My teaching design was also
about understanding measures of variation (range, interquartile range, average deviation, and
standard deviation), but I wanted to investigate students‟ reasoning about variation without
specific connection to chance processes or sampling. Nevertheless the above descriptors of
students‟ understanding of variation are of interest to me, but alas too general to apply fruit-
fully in my research. As mentioned in Subsection 2.2.2, I used part of Mooney‟s framework
for assessing students‟ reasoning about variation (see Chapter 4).
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In addition to the above results of Watson and coworkers, Garfield & Ben-Zvi (2005,
pp. 93-95) identified seven areas of knowledge of variability and seven corresponding assess-
ment areas. The seven areas of knowledge are:
1) developing intuitive ideas of variability;
2) describing and representing variability;
3) using variability to make comparisons;
4) recognizing variability in special type of distributions;
5) identifying patterns of variability in fitting models;
6) using variability to predict random samples or outcomes;
7) considering variability as part of statistical thinking.
I used these areas of knowledge as a guidance to develop my assessment test. However, Areas
4-7 are not relevant to my study, which was situated in Indonesian curriculum. Thus I only
used the first three areas in this framework. For reasons of completeness and possible interest
of readers, the assessment items for all areas are presented in Appendix A.
2.3. Teaching Variation
In this section I present what I consider as the most relevant research literature about teaching
variation. This includes a summary of suggestions made in various studies about teaching
variation in various contexts. At the end of this section I list the principles that I incorporated
in my designed activities.
2.3.1. Conceptions about Variability
Shaughnessy (2007) summarized students‟ conceptions of variability into eight types:
(1) variability in particular values, including extremes or outliers;
(2) variability as change over time;
(3) variability as whole range (the spread) of all possible values;
(4) variability as the likely range of a sample;
(5) variability as distance or difference from some fixed point;
(6) variability as the sum of residuals;
(7) variation as covariation or association;
(8) variation as distribution.
These conceptions resemble the areas of knowledge about variability in the framework of
Garfield & Ben-Zvi (2005), which was discussed in Subsection 2.2.2. In my study I focused
on getting the students so far to develop the conception of variability as type 1, 3, 5, and 8.
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2.3.2. Suggestions from Research Studies about Contexts
In the statistics education literature one can find several contexts to teach variability. Garfield
(2008, pp. 205-206) reviewed contexts used in mostly exploratory and qualitative research.
Below are several contexts that Garfield (2008, pp. 205-206) found:
measurement and natural context: students investigate the variety in height of plants
by measurement activities and comparing distribution of plants;
a „growing sample‟: students reason what happens to graphs if the sample size gets
larger;
measurement of minutes per day spent on various activities, e.g., time spent on study-
ing or talking in the phone: students make conjectures; students reason informally
about the distribution;
variability in data;
bivariate relationships;
comparing variability within and between data sets;
standard deviation and histogram: students explore the concept of standard deviation
through comparing histograms;
probability contexts;
sampling contexts.
As I have mentioned before, I decided not to use the context of probability and sampling for
my study. The focus in the statistics unit selected in my study was descriptive statistics and
thus the context „bivariate relationships‟, for example, was not suitable. The context
„comparing data‟ is a context I would like to have used, however not as the first introduction
to variability. In comparing data sets, there are at least two types of comparisons:
1. reading between data sets;
2. reading beyond data sets.
Curcio (1989, p. 384) defined that in reading between data, one makes comparison and use
mathematical concepts and skills. In reading beyond data, one makes extension, prediction
and inferences. As reasoning about variation between data needs higher level of statistical
thinking then of variation within groups (Mooney 2002; Jones, 2000), I decided to use the
context „comparing variability within data sets‟, meaning analyzing one data set, to introduce
measures of dispersion and hereafter to use „comparing variability between data sets‟.
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2.3.3. Principles Underpinning the Design of My Lesson Activities
I originally planned to use the following principles in designing my activities:
1. Use ICT to change students‟ computational efforts into reasoning efforts.
The use of ICT enables the use of real and large data set (Reading & Shaugnessy, 2004,
p. 223) and helps students to visualize and explore data (Garvield & Ben-Zvi, 2007). I
considered a statistical software package or simply a calculator to ease the computational
efforts of my students. However, it is noted that the use of ICT in statistics education is
not the main focus of my research.
2. Foster the student‟s integration of concepts of central measures and variability during
data exploration (Reading & Shaughnessy, 2004, p. 223; Shaugnessy, 2007, p. 1002).
This is particularly important in my setting because textbooks separate four concepts of
statistics in Indonesian curriculum. Furthermore, shape of a distribution should also be in
a lesson package with centre and spread because “a brief description of a distribution
should include its shape and numbers describing its center and spread” (Moore &
McCabe, 2005, p. 40).
3. Discuss variability in various different contexts and using different questions. (Reading &
Shaugnessy, 2004, p. 223).
4. Use real data comparison to reason about variability (See, for example, Garfield & Ben-
Appendix A. Garfield and Ben-Zvi‟s Framework for Assessing Reasoning about Variability.
Appendix B. Students‟ Activity Sheets.
Appendix C. Pretest/Posttest.
Appendix D. The Questionnaire
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Appendix A. Garfield and Ben-Zvi’s Framework for Assessing Reasoning about Variability
Garfield, J., & Ben-Zvi, D. (2005). A Framework for teaching and Assessing Reasoning about
Variability, Statistics Education Research Journal, 4(1), 92-99.
1. Assessment - Developing intuitive ideas of variability
Items that provide descriptions of variables or raw data sets (e.g., the ages of children in a grade school, or the height of these children) and asking students to describe variability or shape of distribution.
Items that ask students to make predictions about data sets that are not provided (e.g., if the students in this class were given a very easy test, what would you predict for the expected graph and expected variability of the test scores?).
Given a context, students are asked to think of ways to decrease the variability of a variable (e.g., measurements of one students’ jump).
Items that ask students to compare two or more graphs and reason about which one would have larger or smaller measures of variability (e.g., Range or Standard Deviation).
2. Assessment - Describing and representing variability
Items that provide a graph and summary measures, and ask students to interpret it and write a description of the variability for each variable.
Items that ask students to choose appropriate measures of variability for particular distributions (e.g., IQR for skewed distribution) and select measure of center that are appropriate (e.g., median with IQR, mean with SD).
Items that provide a data set with an outlier that ask students to analyze the effect of different measures of spread if the outlier is removed. Or, given a data set without an outlier, asking students what effect adding an outlier will have on measures of variability.
Items that ask students to draw graphs of distributions for data sets with given center and spread.
3. Assessment - Using variability to make comparisons
Items that present two or more graphs and ask students to make a comparison either to see if an intervention has made a difference or to see if intact groups differ. For example, asking students to compare two graphs to determine which one of two medicines is more effective in treating a disease, or whether there is a difference in length of first names for boys and girls in a class.
Items that ask students which graph shows less (or more) variability, where they have to coordinate shape, center, and different measures of spread.
4. Assessment - Recognizing variability in special types of distributions
Items that provide the mean and standard deviation for a data set that has a normal distribution and students are asked to use these to draw graphs showing the spread of the data.
Items that provide a scatterplot for a specific bivariate data set and students have to consider if values are outliers for either the x or y variables or for both.
Items that provide graphs of bivariate data sets where students are asked to determine if the variability in one variable (y) can be explained by the variability in the other variable (x).
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5. Assessment - Identifying patterns of variability in fitting models
Items that ask students to determine if a set of data appear normal, or if a bivariate plot suggests a linear relationship, based on scatter from a fitted line.
6. Assessment - Using variability to predict random samples or outcomes
Items that provide students choices of sample statistics (e.g., proportions) from a specified population (e.g., colored candies) for a given sample size and ask which sequence of statistics is most plausible.
Items that ask students to predict one or more samples of data from a given population.
Items that ask students which outcome is most likely as a result of a random experiment when all outcomes are equally likely (e.g., different sequences of colors of candies)
Items that ask students to make conjectures about a sample statistic given the variability of possible sample means.
7. Assessment - Considering variability as part of statistical thinking
Items that give students a problem to investigate along with a data set, that requires them to graph, describe, and explain the variability in solving the problem.
Items that allow students to carry out the steps of a statistical investigation, revealing if and how the students consider
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Appendix B. Students’ Activity Sheets
Activity 1 Am I Normal?
As we have discussed, it is a common practice to check a child‟s height, weight, head
circumference, etc to see whether his or her growth is normal.
In this activity, you will use the following data to create an easy rule to decide whether a boy
or girl of your age is growing normally, based on his or her height and weight.
Below is the data of height and weight of 16-year-old boys and girls in Jakarta. These data
were collected in 2005 by a PhD student of VU University, Amsterdam, for the making of
Indonesian growth chart.
Boys Girls Height (cm) Weight (kg) Height (cm) Weight (kg)
a. Make a histogram of the boys height and weight data. From that histogram, what can
you say about boys‟ height and weight? How is the data spread out?
b. Make a rule that allows you to determine whether a 16-year-old boy or girl has a
height that is :
- Very common;
- Still normal, but needs attention;
- Abnormal, does not mean there is a health problem, only need to be checked
up.
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Explain how your rule works, why it might be a good one, and how it could work in
practice.
If your rule uses numbers, you must explain how you compute that numbers.
c. Make an easy visual aid (for example, a table or a diagram) that allows you to:
- Quickly apply your rule;
- Explain it to others, for example, to your classmates or your parents.
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Activity 2a Who is Taller?
In 2005, a PhD student conducted a study about the growth of Indonesian children and he
created a growth chart. Below is the histogram of boys height data collected in Jakarta for this
study.
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Histogram of Boys' Height in Jakarta
2.a. Compute the mean and standard deviation of the boys height in Jakarta, based on the
histogram above. Show your work/computation.
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Activity 2b Who is Taller?
In 2007, the Ministry of Health had a social survey carried out in all provinces of Indonesia.
This survey covered many topics in public health. The histogram below shows the height data
of boys in Bengkulu obtained from this survey.
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Histogram of Boys' Height in Bengkulu
The mean of the raw data of boys’ height in Bengkulu is 154.7.
On the next page are shown two histograms of boys height in Bengkulu and Jakarta. Use
these histograms to answer the following questions.
2b.1. Without doing any computation, based only two histograms in the next page, is
the standard deviation of Bengkulu’s data higher than the standard deviation of
Jakarta’s data (6.1)? Give your reasons.
2b.2. Now check your answer for (a) by computing: What is the standard deviation of the
boys’ height in Bengkulu?
2b.3. Can you conclude that boys in Jakarta are taller that boys in Bengkulu? Explain your
reason.
2b.4. What makes the histogram of boys’ height in Bengkulu looked like the above
histogram?
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Where does the time fly? Homework
I have collected from your questionnaires the data on the time that you spend for various activities.
Analyze the data (on the next page) and work in groups to answer the questions below:
a) On which activity do students in your class spend most time per week? Give your reasons.
b) Which activity is the most popular, that is, the one the most students participate in? Is this the same activity as the one identified in a)?
c) On which activity do students in your class spend the least amount time per week? Give your reasons.
d) Which activity is the least popular? Is this also the one on which students in your class spend the least amount of time in a week?
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Activity 3
Where does the time fly?
Rural Vs Urban?
In this activity you are to analyze similar data (on the next page) which are collected from SMAN No.5 Bengkulu.
There are 9 activities on the data sheet. Choose just one activity to be analyzed, for example doing homework.
We name the activity you have chosen as activity X.
Compare this data (of activity X) with the data of activity X from your class and answer the following question:
“Who spends more time on activity X: students from Bengkulu city or Lebong?”
Explain your reasons!
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No Activities Number of Hours Per
Week
Frequency
1. Doing Homework 0-2
3-4
5-6
6-8
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2. Reading (not school work) 0-2
3-4
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3. Playing computer, Play Station or video
games
0-2
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13-14
4. Watching TV, videos or movies 0-2
3-4
5-6
6-8
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11-12
13-14
5. Playing or listening to music 0-2
3-4
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8-10
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6. Doing jobs at home 0-2
3-4
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6-8
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11-12
13-14
7. Working for pay outside the home 0-2
3-4
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6-8
8-10
11-12
13-14
8. Participating in sports 0-2
3-4
5-6
6-8
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11-12
13-14
9. Hanging out with friends 0-2
3-4
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6-8
8-10
11-12
13-14
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Questionnaire Name : __________________ School: _________________________ Age :__________________ Grade : ___________________ In the last week, approximately how much time did you spend on each of the following activities?
No Activities Number of Hours Per Week
1. Doing Homework
2. Reading (not school work)
3. Playing computer, Play Station or video games
4. Watching TV, videos or movies
5. Playing or listening to music
6. Doing jobs at home
7. Working for pay outside the home
8. Participating in sports
9. Hanging out with friends
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Appendix C. Pretest/Posttest Name: __________________________
Class : __________________________
Do the following problems as carefully and as best as you can.
You may use a calculator if needed.
1. Based on your experience, what does variation mean to you? Give an explanation and/or
an example.
2. For each of the following cases, answer the following question:
“Which is more desirable: high variation or low variation?”
Add your reason.
a. Age of trees in a national forest.
b. Diameter of new tires coming off one production line.
c. Scores on an aptitude test given to a large number of job applicants.
d. Weight of a box milk of the same brand.
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3. Given the data: 11, 32, 17, 34, 24, 15, 28
For the data above, fill in the table below.
Range
Mean
Median
Standard Deviation
Interquartile Range
4. Below are the data of monthly income.
Monthly Income
( in thousand Euros )
Number
of People
3 – 5
6 – 8
9 – 11
12 – 14
15 – 17
3
4
9
6
2
The mean of the data above is ________.
The standard deviation is _______.
5. Four histograms and two descriptions of data are displayed below.
i. A data set of Mathematics test scores where the test was very easy
ii. A data set of wrist circumferences of newborn female babies (measured in
centimeters).
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a. Which histogram best matches the data in description (i)? Give your reason.
b. Which histogram best matches the data in description (ii)? Give your reason.
6. Two students who took mathematics tests received the following scores (out of 100):
Student A: 60, 90, 80, 60, 80
Student B: 40, 100, 100, 40, 90
If you had an upcoming mathematics test next week, who would you prefer to be your
study partner, A or B? Why?
7. One day Jeroen caught a very big catfish. He wanted to be sure of the weight of the fish
and therefore he weighed it 7 times on the same scale/balance. Below are the
measurements (in kilogram) that he found:
2.9; 2.7; 5.1; 3.1; 3.0; 2.8; 3.0 kg.
e. How spread out are the measurements he obtained?
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f. How many kilograms do you think the true weight of the catfish was? Give your
reason.
8. The histogram below shows the number of hours of exercising per week by marketing
staffs of a bank.
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Histogram of Number of Exercising Hours Per Week
g. Compute the median. _______________
h. Compute the mean. ________________
i. Based on the histogram, what is the typical number of hours of exercising per
week of the staffs in this company? Give your reason.
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9. Forty college students participated in a study of the effect of sleep on test scores. Twenty
of the students studied all night before the tests in the following morning (no-sleep group)
while the other 20 students (the control group) went to bed by 11.00 pm on the evening.
The test scores for each group are shown in the diagrams below. Each dot on the diagram
represents a particular student‟s score. For example, the two dots above the 80 in the
bottom diagram indicate that two students in the sleep group scored 80 on the test.
• •••
•••
•••
•••
•••
• • • •
30 40 50 60 70 80 90 100
Test Scores: No-Sleep Group
• • • • •••
•••
•••
••
••
••
•
30 40 50 60 70 80 90 100
Test Scores: Sleep Group
Examine the two diagrams carefully.
Which group is better: the sleep group or the no-sleep group? Explain your reasons.
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Then circle one from the 6 possible conclusions listed below the one you most agree with.
a. The no-sleep group did better because none of these students scored below 35 and the
highest score was achieved by a student in this group
b. The no-sleep group did better because its average appears to be a little higher than the
average of the sleep group.
c. There is no difference between the two groups because there is considerable overlap in
the scores of the two groups.
d. There is no difference between the two groups because the difference between their
averages is small compared to the amount of variation in the scores.
e. The sleep ground did better because more students in this group scored 80 or above.
f. The sleep group did better because its average appears to be a little higher than the
average of the no-sleep group.
10. Below is the histogram of the scores of Mathematics test in two classes.
Scores
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b. Comparing the two histograms, one could infer
i. Variation of scores in Class A is higher variation than in class B. (The scores
in class A vary more than the scores in class B)
ii. Variation of scores in Class B is higher than in class A (The scores in class A
vary more than the scores in class B)
iii. Class A and class B have equal variation.
iv. I don‟t know.
c. Why? Give your reason.
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Appendix D. The Questionnaire
For statements no1-6, choose one answer that is suitable to your opinion.
1. The use of real data makes the learning of Statistics more interesting and fun.
a. I strongly agree b. I agree c. Neutral d. I disagree e. I strongly disagree
2. Analyzing real data makes it easier for me to understand statistical concepts; for example
standard deviation.
a. I strongly agree b. I agree c. Neutral d. I disagree e. I strongly disagree
3. The use of real data shows me the application of Statistics in real life.
a. I strongly agree b. I agree c. Neutral d. I disagree e. I strongly disagree
4. I am used to like working in groups.
a. I strongly agree b. I agree c. Neutral d. I disagree e. I strongly disagree
5. I like working in groups.
a. I strongly agree b. I agree c. Neutral d. I disagree e. I strongly disagree
6. The group‟s discussion helps me understanding statistical concepts.
a. I strongly agree b. I agree c. Neutral d. I disagree e. I strongly disagree
7. I actively participate in contributing ideas and in group discussions.
a. I strongly agree b. I agree c. Neutral d. I disagree e. I strongly disagree
For statement no. 8, give a tick mark √ in the box besides the statement that you agree
with. If you have something to add, please write it in the provided box.
8. I think that the way of learning and teaching in the last two lessons differs from the way of
learning mathematics I usually experience in the following sense:
Using real data, not only artificial numbers.
Using real data, not only artificial numbers.
Demanding students to develop their ideas and then defend those ideas through
correct correct arguments.
Giving students the chance to try solving problems, not directly “telling” the correct
ways. Ways.
Using calculator is allowed.
Others (please write it in the box below)
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For question 9-11, please fill in your answers in the provided boxes.
9. What are according to you the strengths and/or weaknesses of the last two lessons?
10. What suggestions do you have for future improvement of the last two lessons?
11. Do you have any other comment about the last two lessons? If so, please write it down.