Students’ Ability of Statistical Reasoning in Descriptive Statistics

Problem Solving

Nila Kesumawati

Universitas PGRI Palembang, Jl.Jend. A. Yani, Lrg. Gotong Royong 9/10 Ulu, Palembang, Indonesia

Department of Mathematics Education, Universitas PGRI Palembang, Indonesia

Keywords: Students’ Statistical Reasoning; Descriptive Statistics; Problem Solving

Abstract: The problem of this research was some students unable to interpret the results of calculations in solving the

problem of descriptive statistics associated with the statistical reasoning. Students were able to solve the

problem procedurally only without understanding the meaning of the calculated results. Students knew the

concept, but they could not identify the use of the concept. This descriptive quantitative and qualitative

research aimed at describing the students' statistical reasoning ability in solving the problem of descriptive

statistics. The subjects of this research were 134 students of mathematics education and sports education,

University of PGRI Palembang. They were chosen using purposive sampling because they enrolled in Basic

Statistics Course. Reasoning test and interview was used to collect the data. The results stated that students'

statistical reasoning ability is in the multi-structural level category. This can be known from the ability of

statistical reasoning based on the measurement of Reading and Reid. In Prestructural Level there were 2

students who have no clear conceptual foundation; and in the Unistructural Level there were 25 students

focusing on one concept of statistics; at multi-structural level there were 104 students who focus on more

than one statistical concept; and at Relational Level there were 3 students who develop relationships with

various other statistics concept.

1 INTRODUCTION

Basic statistics is one of the important courses in

curriculum 2016 which based on KKNI, so the

students are expected to understand and implement

it. Statistics as knowledge provides a means to solve

the phenomenon or problems of life, in the working

environment, and in the science itself, as well as a

tool for evaluation of what has happened in

determining future policy (Moore, 1997; Kadir,

2015; Kesumawati, et.al. 2017). According to

Sundayana (2012), basic statistics course has four

aspects to be achieved, such as: (1) to provide

theoretical knowledge to the students; (2) to provide

practical skills in the form of statistical calculations;

(3) to provide an overview and experience of how to

solve problems with daily life related to problems;

and (4) to train students how to communicate the

results of the study, both in written and oral form.

The ability of students to communicate the

results of his studies by explaining or answering

questions about interpreting the results of

calculations is part of writing in communicating

understanding. NCTM (2000) stated that writing is

the way of communicating mathematics and

reinforcing students’ thinking, as this may affect

their thinking about ideas and concepts that writing

as the most important element in learning

mathematics. In addition, writing also can support

mathematical reasoning with problem-solving and as

a tool to internalize the characteristics of effective

communication. In line with, Bosse and Faulconer

(2008) stated outlined procedures that can be

employed in mathematics assessment to create

experiences that promote reading and writing as

tools for expressing mathematics understanding.

The researcher often sees students have difficulty

in interpreting the results that have been done. Based

on Lanani’s research (2015), some of the

weaknesses for students in the course of statistics

during this time are these materials: classification of

statistical data types, representation of statistical

data, measurement of statistical data, sample as a

representation of the population, and hypothesis

testing.

530

Kesumawati, N.

Students’ Ability of Statistical Reasoning in Descriptive Statistics Problem Solving.

DOI: 10.5220/0008524905300536

In Proceedings of the International Conference on Mathematics and Islam (ICMIs 2018), pages 530-536

ISBN: 978-989-758-407-7

Associated with the weakness for students in

statistics during this time, it is often encountered that

many students have not been able to interpret the

results for problem-solving of descriptive statistics.

Besides the weakness for statistical ability is also

experienced by the teacher. This is evidenced based

on the results of research that has been done by

Martadiputra (2010). It was found that statistical

reasoning ability of mathematics teachers, in both

junior and high school, on the material statistics

descriptive median material, population, and sample

are categorized very less; material representation

data and central tendency are categorized enough.

Overall, the average statistical reasoning ability of

them was in the medium category. In 2012, research

of Martadiputra was continued and still about the

statistical reasoning ability of undergraduate student

majoring in mathematics education at a State

University in Bandung. The results show that

students’ statistical reasoning ability is in a low

category. Other evidence is from community service

activity conducted by lecturer majoring in

mathematics education UPI in Kabupaten Subang

(Avip, 2010). It was found that the mean statistical

reasoning ability of teachers in the junior high

school and the senior high school reached 46% in

the medium category.

Statistical reasoning is a reasoning activity on

statistical materials which is the developed statistical

ideas form skills in using statistical concepts

(Lanani, 2015; Gal and Garfield, 1997; Garfield and

Chance, 2000). The important understanding of

statistical concepts in statistical ideas, such as

central tendency, deviation, various data

presentation, and correlation is part of the statistical

reasoning.

All this time, statistics is considered only as a

science for the solution to problems that are

mechanistic. However, it has a lot of usefulness in

everyday life related to principles if it is studied

further. This statement is supported by Dasari (2009:

40), that statistics are still considered only as a series

of the thinking process. Teachers and students

emphasize more on the particular rather than the

principle, the emphasis on mechanistic rather than

the main methodology, and the emphasis on a

special formula rather than a common one.

Specifically, the use of statistics is to describe

and predict based on phenomena as the collection of

results from the study. The ability of statistical

reasoning is needed to interpret and represent data

onto determining the correct decision on the data.

According to Dasari (2009), the ability of statistical

reasoning is the ability to make conclusions and

provide explanations based on data orientation with

respect to structured procedures, unstructured

procedures, and statistical concepts, and interpret the

process and statistical result.

Furthermore, Del Mas (2002) interprets

statistical reasoning as the ability to explain why and

how a result is processed and why and how to draw

conclusions. To find out how far the students’

statistical reasoning abilities, it is surely needed

measurement tools. The required measuring

instruments are the indicator of statistical reasoning

ability. According to La Nani (2014), indicators of

statistical reasoning ability are that students capable

of: (1) summarizing and explaining based on data

orientation; (2) understanding and interpreting

process and statistical results.

The aim of this research is to assess students’

reasoning statistical ability on central tendency,

deviation and data representation through essay

examinations particular, the question posed in this

research is how the student’s level of statistical

reasoning in basic statistics courses seen from the

way students answer/explain: (1) Prestructural; (2)

Unistructural; (3) Multistructural; and (4) Relational.

This research attempts to be different, yet

complementary, the emphasis on describes students’

statistical reasoning ability in descriptive statistics

problem solving based on the levels developed by

Reading and Reid (2006), i.e. pre-structural,

unistructural, multi-structural, and relational. It is a

little information about how students have studied

basic statistics courses to interpret the results

obtained, and such research is rare in colleges in

South Sumatera. This research contributed to (1) the

basis of knowledge for students, and (2) as the basis

for students in the course.

2 RESEARCH METHOD

This research used descriptive quantitative. The

instrument of this research was essay test that had

been validated by expert and had been declared

valid. The test made based on mathematical

statistical reasoning which consist of 5 questions.

This research was conducted at PGRI University of

Palembang included 134 students of academic year

2017/2018. It covers undergraduate students of the

second semester in mathematics education program

and the third semester in sports education program.

The reasons for choosing the students are that: (1)

the students are following the basic statistics course;

(2) the students are more easily managed by the

researcher to follow the planned research procedure.

Students’ Ability of Statistical Reasoning in Descriptive Statistics Problem Solving

531

Data was collected through essay test and interview.

In collecting the data, researcher giving an essay test

material consists of central tendency, data deviation,

and data representation at the middle of semester.

Students explain or interpret the results of the

research in descriptive statistics, which are necessary

to determine the level of students’ statistical

reasoning. The development of essay tests based on

reasoning statistical ability indicators has been

validated to determine student responses to research

instruments. Interview was conducted to find out

more student’s reasoning of their answer sheets.

The level of statistical reasoning with this

research is guided by Reading and Reid (2016) that

has four phases and arranged hierarchically in Table

Table 1. Phase of Statistical Reasoning

Reasoning

Level

Descriptions of Statistical

Reasoning

Prestructural

No obvious concept based.

Unistructural

Only focuses on one statistics

concept.

Multistructural

Focusing on more than one

statistics concept.

Relational

Developing connection to

another statistics connection.

Furthermore, after analyzing the results of

student answers, the aspects used and the level of

statistical reasoning is adapted from Yusuf’s which

has been modified (2017), can be seen in Table 2.

Table 2. Phase of Statistical Reasoning for

Statistic Data Measuring

Phase 1

Phase 2

Phase 3

Phase 4

Prestruct

ural

0 ≤ score

≤ 20

Unistructu

ral

20 ≤

score ≤

Multistructu

ral

50 ≤ score

≤ 90

Relation

90 ≤ score

≤ 100

3 RESULT AND DISCUSSION

From the results of examination in quantitative and

qualitative, completion of the students’ essay

resulted four identification level. These four levels

are adopted from the description of Reading and

Reid (2006) modified SOLO taxonomic. The

description of each level is as follows: (1) the

student does not have obvious conceptual

foundation. It means that they are only able to

determine the data of the problem but cannot

continue the solution onto the problem. It

categorized at prestructural level; (2) the students are

only focusing on one statistical concept. It means

that they can use only one information such as the

problem of determining standard deviation value and

variance and not yet know the relationship between

them as well as the relationship between mean,

median and mode of data. It is categorized at level

Unistructural; (3) the students focus on more than

one concept of statistics. It means that they can use

some information but do not connect. It is

categorized multistructural level; and (4) the

students can develop relation to other statistical

concepts. It means that they have completed

understanding of the process, the relation of rules

and the use of statistics, and students can conclude

in their own words. It is categorized relational level.

Results of students' overall statistical reasoning

appraisal (average = 57,2 and SD = 12,6) included in

the multistructural level category. This result is in

line with other findings about the average ability of

statistical reasoning ability (Dasari 2009;

Martadiputra, 2012; Avip, 2010). The following are

four main findings of the study based on Reading

and Reid (2006).

1. Statistical reasoning ability at prestructural

level is only reached by 2 students (1.5% of all

students) who have no obvious conceptual

base. At this level, students cannot start an

essay because they do not try to focus on the

problem. They are distracted by irrelevant

things. Students still think the more dominant

concrete-symbolic way so that students cannot

answer the problem to choose and give reason.

From 134 students that became the subject of

the research, there are 2 students at the

Prestructural level.

2. Statistical reasoning ability at unistructural

levels is reached by 25 students (18.7% of all

students) focusing only on one concept of

statistics. At this level, the student can focus on

the problem, generally only focus on one

aspect. They can answer issues related to

central tendency and dispersion tendency of

data given. In addition, they can solve the

Pearson slope problem and the coefficient of

tangles. They are also able to show that a curve

is normal, oblique to left, or right oblique.

They have a higher tendency to support their

answers by completing the ordered data. From

134 students who became the subject of

research, only 25 students are at the level of

Unistructural. Figure 1 is an example of the

ICMIs 2018 - International Conference on Mathematics and Islam

532

results of student’s work that can determine the

results of calculations and can determine that

the data is oblique to the left. The student also

understands that the more pointed a curve then

the smaller standard deviation so that the data

homogen.

Figure 1. Student's Answers Material Deviation

3. Statistical reasoning abilities at multistructural

level are reached by 104 students (77.6% of all

students) focusing on more than one statistics

concept. At this level, students display the

ability to think quantitatively and know more

than one aspect of data exploration. Generally,

there is a concrete-symbolic way of thinking

and has a concept that is more than one

relevant aspect. They can interpret histogram

and frequency polygon images into the table

frequency distribution. In the histogram and the

known frequency polygon, only middle scores

and the frequency is showed, but 30 students

cannot determine the lower and upper limits for

each interval. At this level, students can change

the value of z to standard numbers in order to

determine which candidates should be accepted

and give reasons. From 134 students that

became the subject of research, there are 104

students are at the Multistructural level. Figure

2 is an example of a student's answer in

interpreting the results after the calculation.

Figure 2. Student's Answer on Material of Data

Presentation

Student was interviewed about their reasoning

to solve the problem. The students observed the

bar chart from the problem then remember how

to create a frequency distribution table that the

interval should be the same. After creating a

frequency distribution table, they analyze data

from the table to determine the central

tendency. It means that the student knows more

than one concept, i.e. presenting data from bar

chart to frequency distribution table, determine

central tendency of it. Another example of

student answers can be seen in figure 3. It can

be showed that the result is wrong in

determining the length of the interval. Only

first interval class has right length of interval.

Otherwise, the next interval class is wrong. It is

because carelessness so that there is an error in

making the table distribution of frequency.

Figure 3. Student’s Inaccuracy on Material Creating

Table Frequency Distribution

4. Relational level of statistical reasoning ability

is reached by 3 students (2.2% of all students)

who can develop relation with various other

Students’ Ability of Statistical Reasoning in Descriptive Statistics Problem Solving

533

statistical concepts. At this level, students

display the ability to think analytically and

quantitatively about the data and be able to

explain various perceptive based on the data

obtained. From 134 students who became the

subject of research, there are 3 students who

are at the level of Relational. Figure 4 is an

example of student answers.

Figure 4. Student’s answer to the material of z-score

Student was interviewed about their reasoning

to conclude who is the best candidate to be

accepted. The student using z-score to

determine the best candidate because there

were given value of x, mean, and deviation

standard of each candidate. It means that the

student can relate various statistical concepts.

In this research there are two important

limitations. First, the source of assessment provided

for the analysis was an essay and the second was the

three basic statistical topics i.e. central tendency,

data deviation, and data presentation. This research

provides rich analysis through response comparisons

showing various levels of students' statistical

reasoning. It then offers insight into how the

students' statistical reasoning can influence the

lecture strategy in order to interpret the data.

The level of statistical reasoning can be as a

guide in identifying the weaknesses and strengths of

students in statistical reasoning (Joseph, 2017).

According to Rumsey (2002) the purpose of learning

statistics is that students understand statistics well in

order to make decisions and can develop research

skills. In addition, Lanani (2015) argues that

statistical reasoning plays a role in shaping students’

skills using concepts, rules and statistical processes.

Olani, et al (2011) states that the ability of statistical

reasoning refers to the ability to understand and

integrate statistical and ideas to interpret data and

make decisions based on statistical concepts.

Lovett (2001) found that to understand and

improve students’ statistical reasoning, needs

integrating three approaches, literature study

approach (theoretical), empirical studies, and

classroom-based research. Chan and Ismail (2014)

stated that there are five statistical reasoning tasks

in the assessment instrument which have been

created based on the initial statistical reasoning

framework, each item is associated with the sub-

processes of four key constructs, i.e. describing

data, organizing and reducing data, representing

data, and analyzing and interpreting data. The

dynamic spreadsheet of GeoGebra software is used

as a technological tool in solving tasks. This

statistical reasoning assessment instrument can be

utilized by instructors and researchers for further

investigation in future studies.

Agus et. al. (2013) concluded to develop a

measurement instrument to assess undergraduate

students’ statistical reasoning on uncertainty and on

association, in relation to methods of proof

presentation. By the construction of paired items

in two forms, we could compare the reasoning

applied to problem resolution, regarding the specific

problem structure. Furthermore, P

ta (2006)

found

new technologies involve a reformulation of

contents and methodology used

or teaching

statistics. Developing students’ statistical reasoning

becomes an important task

or teachers of applied

statistics. This is particularly true in the field of

health sciences. In this wor

the statistical

reasoning ability acquired by health sciences

students was evaluated in the

text

of their final

undergraduate pro

ject

4 CONCLUSIONS

Based on the research result, the conclusion of this

research is that the students' statistical reasoning

ability in problem solving of descriptive statistics is

in the multistructural level category. In detail, the

statistical reasoning abilities obtained based on

Reading and Reid measurements, i.e. in Prestructural

level categories there are 2 students has no clear

conceptual foundation; in unistructural level

category, there are 25 students only focus on one

concept of statistics; in the multistructural level

category, there are 104 students focusing on more

than one concept of statistics; and in the relational

level category, there are 3 students developed in

connection with various other statistical concepts.

According to the limitation of this research, for

further researcher who will conduct similar research,

ICMIs 2018 - International Conference on Mathematics and Islam

534

it is suggested that: (1) the instrument is not only

using essay but also completed with interview and

observation; (2) the topic should be extended,

because the more topics is used the more it will be

known the whole ability of students' statistical

reasoning.

ACKNOWLEDGEMENTS

I would like to express my special thanks and

gratitude to my rector (Bukman Lian) who gave me

the golden opportunity to do this wonderful project

on the topic “Students’ Ability of Statistical

Reasoning in Descriptive Statistics Problem

Solving”. Secondly, I would also like to thank my

dean (Dessy Wardiah) and friends (Muhammad

Kristiawan, Ansari Saleh Ahmar, Indah Rahayu, Ali

Akbarzam, and Novita Sari) who helped me a lot in

finalizing this project within the limited time frame.

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