Characteristics of student statistical
reasoning in mathematical problem solving
Cite as: AIP Conference Proceedings 2215, 030028 (2020); https://doi.org/10.1063/5.0003653 Published Online: 01 April 2020
Erry Hidayanto, and Desi Rahmatina
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Characteristics of Student Statistical Reasoning in Mathematical Problem Solving
Erry Hidayanto
1,a)and Desi Rahmatina
21Universitas Negeri Malang, Indonesia
2Universitas Maritim Raja Ali Haji, Tanjungpinang, Indonesia
a)Corresponding author: [email protected].
Abstract. The purpose of this study is to describe the characteristics of statistical reasoning in mathematical problem- solving. The characteristics of statistical reasoning are relational, synthesis, understanding, explain, inference, and interpretation. Respondents in this study were 2 MIPA high school students in Malang. The instrument of this research was a written test question about mathematical problem solving and semi-structural interviews conducted to reveal in-depth regarding the students’ statistical reasoning in mathematical problem-solving. The results showed that the students’
statistical reasoning characteristics were as follows: 1) relational: students could connect several concepts such as the concepts of space-building volume, comparison of numbers, and system concepts of one-variable equations in mathematical problem solving 2) synthesis: students were able to combine various ideas in mathematical problem solving, 3) understanding: students understood the information contained in the problem, understood the steps to solve the problem, 4) explain: students were able to explain the process of presenting data, the process of analyzing data until the decision making process of which shipping service to choose, 5) inference: students were able to choose the “DEF Express” shipping service to send their book order, and 6) interpretation: students were able to make decision regarding the service costs in which the cheapest shipping service was chosen.
INTRODUCTION
Reasoning is a process of logical thinking to solve a problem. Leighton & Sternberg [1] stated that reasoning is a process of drawing conclusions to achieve goals, these conclusions inform efforts to solve problems and make decisions because goals drive humans, and conclusions are taken to help someone achieve and fulfil his goals.
Sometimes, in order to achieve goals, a person firstly encounters a problem. The problem, according to Dunbar &
Fugelsang [2], is a situation with no clear, standardized, or routine way to achieve a goal. With the problem that arises, we need a way to solve it so that the goal can be achieved. Furthermore, Dunbar & Fugelsang [2] also argue that problem-solving is the process of overcoming obstacles to achieve goals. In dealing with the problems, reasoning skill is necessary because, without thinking or reasoning, the problem would be difficult to solve. This is in line with the opinion of Dunbar & Fugelsang [2], who said that problem solving and reasoning are the two main keys in thinking.
Furthermore, Dunbar & Fugelsang [2] said that solving problems include a series of cognitive procedures and thought processes to achieve goals. If someone experiences an obstacle in achieving a goal, then that person must overcome these obstacles. In reasoning, cognitive procedures are used to draw conclusions from knowledge. Making conclusions is one form of the construct in statistics.
Garfield & Ben-Zvi [3] stated that statistical reasoning is a way of reasoning by using various statistical ideas and understanding statistical information. Statistical reasoning involves the relationship of one concept with another concept (for example, concentration and dissemination) or combining ideas about data and opportunities. According to Kalobo [4], statistical reasoning means understanding and being able to explain statistical processes and interpret various statistical results. Statistical reasoning is a topic that is widely used in various fields of work, including
The 3rd International Conference on Mathematics and Sciences Education (ICoMSE) 2019 AIP Conf. Proc. 2215, 030028-1–030028-10; https://doi.org/10.1063/5.0003653
Published by AIP Publishing. 978-0-7354-1968-1/$30.00
psychologists, doctors, journalists, politicians and teachers ([5]). Research on statistical reasoning on non-statistical topics was also carried out outside of statistics such as statistical reasoning in biology.
Research on the level or characteristics of statistical reasoning has been carried out by several experts, both in making various levels, as well as revising the levels of pre-existing statistical reasoning, including ([5], [6], and [7]).
Garfield [5] examined the topic of sampling distribution in statistics. Garfield [5] divided the level of statistical reasoning into 5 levels, namely: idiosyncratic reasoning, verbal reasoning, transitional, procedural, and integrated processes. Jones et. al [6] created a framework to characterize students’ statistical thinking in 4 levels of statistical reasoning, namely: 1) idiosyncratic, 2) transitional, 3) quantitative, and 4) analytical, and divided them into 4 constructs, namely: 1) describe data, 2) collecting and reducing data, 3) representing data, and 4) analyzing and interpreting data. Leavy [7] divided students’ statistical reasoning into three levels of statistical reasoning, namely:
idiosyncratic, transitional, and quantitative.
In addition, many researchers also examined the level of statistical reasoning by using a framework or level of previous researchers such as Bansilal [8], Chan & Ismail [9], Chan et. al [10], and Oslington et. al [11]. Bansilal [8]
used the APOS framework (Action, Process, Object, Schema) to assess the teacher’s response to the tasks given about normal distribution. Chan & Ismail [9] used technology-based statistical reasoning to assess and characterize students’
statistical reasoning on a topic of 4 constructs and 5 levels of reasoning and also improve and validate the statistical reasoning framework. Furthermore, Chan et. al [10] made an assessment on the framework of students’ statistical reasoning on descriptive statistical topics about the size of concentration and the size of the spread. They combine 5 levels of statistical reasoning made by Garfield [5] and used 4 statistical constructs made by Jones et. al [6]. So that the appearance of this model is almost similar to that made by Jones et. al [6]. Where student’s statistical reasoning levels were determined by calculating the average value of each code made, from the results of the task-based interviews provided, the researchers have revised the sub-process of the initial framework. Oslington et. al [11]
investigated the statistical reasoning of grade 1 students in the early years of school, by allowing children to be involved in data modelling activities. Researchers connected student responses based on the level of statistical reasoning by Leavy [7].
Based on the description of level research or statistical reasoning characteristics above, no researcher or expert has discussed the characteristics of statistical reasoning in solving mathematical problems. Although the application of statistical reasoning in non-statistical problems is widely used in various fields such as medicine, education, psychology, and neuroscience, business fields ([12]). The purpose of this study was to describe the characteristics of students’ statistical reasoning processes on mathematical problems based on the definition of statistical reasoning made by Garfield & Ben-Zvi [3] and Kalobo [4]. There were six statistical reasoning characteristics used in this study, namely: 1) relational, 2) synthesis, 3) understanding, 4) explain, 5) inference and 6) interpretation.
METHOD
This research was qualitative research to describe and investigate in-depth on the students’ statistical reasoning characteristics in solving mathematical problems. Respondents in this study were two eleventh grade students of SMAN MIPA in Malang, Indonesia. Data were retrieved through tests and interviews. The instrument in this research was as follows:
The bookseller will send the book you ordered, the book is made using beam-shaped boxes and the boxes are filled with books, all books are the same size as shown in Figure 1. The weight of 1 book is 0.5 kg. The seller provides two alternative shipping services to the buyer, namely “ABC Express” and “DEF Express.” “ABC Express” charges a principal shipping fee of Rp.25.000,- plus Rp.22,000,- /kg, while “DEF Express” charges a principal shipping cost of Rp.19.000,- plus Rp.22,500,- /kg, while the box fee is not charged to the buyer. Which shipping service should you choose?
FIGURE 1. The book and the box
RESULTS AND DISCUSSION
Based on students’ answers and the results of interviews, the characteristics of students’ statistical reasoning in solving mathematical problems will be described. Both of the students’ answers (Student 1 and Student 2) showed similarities in using mathematical concepts, but after clarification through interviews, Student 2’s statistical reasoning process was more varied than Student 1 because student 2 had metacognition in solving mathematical problems.
Initially, Student 2 was able to complete the statistical reasoning process in making a decision, but after cross- checking, it turned out Student 2used an incorrect concept. Student 1 was able to use statistical reasoning process and consistent in using mathematical concepts. The answers from Student 1 and Student 2 are shown in Fig. 2 and Fig. 3 respectively.
V book = 6 x 20 x 15 = 1800 cm2 V box = 30 x 36 x 20 = 21600 cm3 V book = V box 21600 = 1800 12 books . 12 x 0,5 = 6
I chose the shipping service DEF Express more save Rp. 3000
FIGURE 2. Answers from Student 1
The number of books in a box = Vcarton/box
Vbook Vbox = 30 x 36 x 20
= 21600 cm3
The number of books = 21600
1800 =12 books 12 x 0,5 = 6 kg/box
So, I chose the "DEF Express" shipping service because it was cheaper than the "ABC Express" shipping service. DEF Express "3000 cheaper than" ABC Express ".
FIGURE 3. Answers from Student 2
Based on students’ answers to the test given, followed by interviews on each of the students, characteristics of students’ statistical reasoning processes in solving mathematical problems were discerned.
ABC Express 6 x 22.000 + 25.000
= 132.000 + 25.000
=157.000 DEF Express 6 x 22.500 + 19.000
= 135.000 + 19.000
=154.000
Vbook = 6 x 20 x 15 =1800 cm3
1. Relational
Answers from Student 1 and Student 2 have similarities as shown in Table 1.
TABLE 1. Relational related student answers
Student 1 Student 2
The concept of building volume
V book = 6 x 20 x 15 = 1800 cm3 V box = 30 x 36 x 20 = 21600 cm3
The concept of comparison of numbers
V box : V book 21600 : 1800 12 books . 12 x 0,5 = 6 The concept of linear equations
ABC Express
ݔʹʹǤͲͲͲ ʹͷǤͲͲͲ = 132.000 + 25.000 = 157.000
DEF Express
ݔʹʹǤͷͲͲ ͳͻǤͲͲͲ = 135.000 + 19.000 = 154.000
The concept of comparison of numbers
The number of books in a box = Vcarton/box Vbook
The concept of building volume
Vbox = 30 x 36 x 20 = 21600 cm3
The concept of linear equations
In Table 1 appears that Student 1 started the answer by determining the volume of the box and volume of the book, then used the relationship of the concept of volume building space with the concept of comparative numbers to determine the amount of numbers with the volume of space and finally connected the concept of the comparison of
Vbook = 6 x 20 x 15 =1800 cm3
numbers with the concept of one variable linear equation. To clarify the students’ answers, researchers conducted interviews as follows.
P: What is the process of completing the above question?
S1: Reading and understanding the contents of the questions, calculating by associating each number of books in the compartment, entering into the count of “ABC Express” and “DEF Express” and determining the shipping service.
S2: Understanding the meaning of the question, count the number of books per box, multiply the number of books per box by weight per book, if I have found the weight of books, entered it into the formula “ABC Express” or “DEF Express”
From the interviews of Researcher (P) with Student 1 (S1) and Student 2 (S2), it was found that they could connect or associate several mathematical concepts to make a decision on which delivery service to choose. From the answers to questions and interviews above, it appears that both Student 1 and Student 2 were able to determine the relationship of one concept to another, as Student 1 said that “associating the number of books per box.” Student 1 already knew the use of box volume and book volume. Student 2 also understood the relationship between mathematical concepts, the number of books per box obtained from determining the volume of books and boxes. Understanding the relationship between mathematical concepts is in accordance with the opinion of Skemp [13] which stated that a relational understanding of someone using a mathematical procedure comes from the results of connecting various relevant mathematical concepts in solving a problem and knowing why the procedure can be used.
2. Synthesis
The synthesis process of Student 1 and Student 2 in solving mathematical problems can be seen in Fig. 4 and Fig.
5.
FIGURE 4. The synthesis process of Student 1
V box : V book
12 books
DEF Express
ABC Express
I chose the shipping service DEF Express more save Rp. 3000 V book
V box
From Fig. 4, the calculation of the volume of the room is presented, which is 1800 cm3 and 21.600 cm3.
FIGURE 5. The synthesis process of Student 2
From Fig. 5, it appears that Student 2 started by determining the number of books in the box by determining the volume of the box and volume of the book and then calculating the volume of the box divided by the volume of books and found that the number of books in the box is 12 books and the total weight of books is 6 kg.
From Fig. 4 and Fig. 5, and the results of the interview, Student 1 and Student 2 carried out a synthesis by combining various mathematical ideas or concepts such as combining the concept of a number comparison, one- variable linear equation and the concept of volume building space in solving problems. As a result, the concept of data distribution emerged as a fundamental concept.
3. Understanding
Both Student 1 and Student 2 understood how to solve mathematical problems by starting from identifying problems, what are the steps to solve the problems, reasoning about what data is in the problem starting from gathering information in the form of book size and box size so that the number of books in a box were obtained, then presented several possibilities for the number of boxes ordered then analyzed it using a linear equation of one variable to draw conclusions and interpret which shipping services to choose.
4. Explain
In the interview, Student 1 and Student 2 were able to explain the process of presenting data, analyzing data as shown in Table 2 below.
The number of books in a box = Vcarton/box
Vbook Vbox = 30 x 36 x 20 = 21600 cm3
The number of books = 21600
1800 =12 books 12 x 0,5 = 6 kg/box
Vbook = 6 x 20 x 15 =1800 cm3
So, I chose the "DEF Express" shipping service because it was cheaper than the "ABC Express"
shipping service. DEF Express "3000 cheaper than" ABC Express ".
TABLE 2. Students ability to explain mathematical problem solving
Student 1 Student 2
P : What if you ordered 2 more boxes?
Which would you choose?
S1: Stay the same "DEF Express"
P : What if you ordered 4 boxes?
S1 : 4 boxes means 48 books weighing 24 kg while still using the "DEF Express" shipping service
P : Has it been checked correctly?
S1 : (checking) 4 boxes Æ 48 books Æ 24 kg = 4 x 12 books 1 shipping box DEF = 154.000 x 4 = 616.000
1 shipping box ABC = 157.000 x 4 =628.000.
Thrifty 628.000 – 616.000 =Rp. 12.000
P : Where did you get the number 154,000 for DEF Express and 157,000 for ABC Express?
S1: From the results of the initial calculation, ma’am
P: Does the cost depend on the number of boxes? Does it not depend on kg weight?
S1 : Yes, I think ...
P : What if you ordered 2 more boxes? Which would you choose?
S2: Still ”DEF Express”
P : What if you ordered 3 boxes?
S2 : Wait ma’am, (calculation process), if 1 box = 12 books, 3 boxes =36 books, weight 3 boxes = 36 x 0,5 kg = 18 kg
ABC Express = 25.000+22.000 x 18 = 421.000 DEF Express = 19.000 + 22500 x 18 = 424.000 The cheaper ABC Express, sorry, sorry, I made the
wrong prediction beforehand, may I revise the answer earlier.
P : Wait, try if 2 boxes, which one will be selected?
S2 : if 2 boxes = 24 books, weight of 2 boxes = 24 x 0,5 kg = 12 kg
ABC Express = 25.000+22.000 x 12 = 289.000 DEF Express = 19.000 + 22500 x 12 = 289.000 Ma’am, can I use both?
From Table 2 above, Student 2 was able to explain the same amount of costs for both ABC Express and DEF Express shipping services when buyers order as many as 2 boxes at a shipping cost of Rp289.000,- for both ABC Express and DEF Express. While Student 1 said that the cheapest shipping cost was DEF Express, regardless of the weight of the book sent. The ability of the students to explain ideas in solving mathematical problems is in accordance with the research conducted by Penna et. al [14] which analyzed how the ability of students to solve problems was influenced by how statistical problems are presented.
5. Inference
Inference or conclusions made by Student 1 and Student 2 were revealed in the answer sheet of the written tests and interviews, as shown in Table 3 below. Based on the answers on the written test and interview in Table 3 above, it appears that both Student 1 and Student 2 were able to make conclusions. This is by Erickson [15] which said that inference is very difficult that even professional researchers use it inappropriately; this is due to the existence of counterfactual ideas. Also, in accordance with Garfield & Ben-Zvi [3] which stated that inference can be taken based on the type of data after the type of data or information is determined, then one can present data, construct and or read data to interpret data and recognize characteristics.
TABLE 3. Student responses in drawing conclusions
Student 1 Student 2
P: So what's the conclusion?
S1: "DEF Express" shipping services are more efficient than "ABC Express" even though the number of shipping boxes is doubled
P : So what's the conclusion?
S2 : For the 1 box I choose DEF Express if 2 boxes I can choose any one of them because the series or price is the same. If 3 boxes I choose ABC Express because it's cheaper
P : How about 4 boxes?
S2: I choose ABC Express, ma'am P : So what's the conclusion?
S2 : (Calculation Process), wait ma’am, I think I have made a mistake
If 2 boxes:
ABC = 157.000 x 2 = 314.000 DEF =154.000 x 2 = 304.000 If 3 boxes:
ABC = 157.000 x 3 = 471.000 DEF = 154.000 x 3 = 462.000 If 4 boxes:
ABC = 157.000 x 4 = 628.000 DEF = 154.000 x 4 = 616.000
1 shipping box ABC = 157.000 x 4 = 628.000.
DEF = 154.000 x 4 = 616.000
So the conclusion is that although a lot of boxes (2 / more boxes) which are cheaper are still DEF Express
P : Even 1 box?
S2 : Yes ma'am
P: Has it been checked for 1 box?
S2 : Already ma'am, for ABC 1 box 157,000, if DEF is 154,000
So in my opinion, even if 2,3,4 boxes are the same as the price of 1 box multiplied by the many boxes ordered because it can only hold 12 books.
I chose the shipping service DEF Express more save
So, I chose the "DEF Express" shipping service because it was cheaper than the "ABC Express"
hi i i DEF E "3000 h
6. Interpretation
Both Student 1 and Student 2 were able to interpret the conclusions drawn, as shown in Fig. 6 and Fig. 7.
FIGURE 6. Interpretation of Student 1 answers
FIGURE 7. Interpretation of Student 2 answers
From Fig. 6 and Fig. 7, it appears that both Student 1 and Student 2 chose DEF Express shipping service on the grounds that shipping costs imposed by DEF Express were Rp. 3000 from ABC Express, choosing a cheaper shipping service. The ability of students’ interpretation can be used to determine the level of students’ understanding of mathematical concepts. This is in line with Parke [16] which stated that the interpretation of the conclusions obtained can differ between students and they are aware of the many ways to describe the results and do not have a correct answer. This result illustrates students’ creativity ([17]; [18]).
CONCLUSIONS
The characteristics of statistical reasoning used in solving mathematical problems in this study consisted of relational, synthesis, understanding, explain, inference, and interpretation. Relational was shown in the form of students’ ability to connect several concepts to solve mathematical problems. In synthesis, students were able to combine several of these concepts, including combining the concepts of linear one-variable equations, the volume of space build and the concept of number comparisons. Students’ understanding is show when they could identify problems, understand steps to solve problems, understand data or any information within the problem, although the students' understanding was initially only one box containing 12 books that were ordered and did not think there was a possibility that many books were ordered for more than one box. In explain, students were able to explain the process from presenting data to making decisions about which delivery service to choose to send books ordered, an explanation of the decision-making process was more detailed when students were interviewed, as students were able to explain decision making when given several alternatives of the number of books ordered, which is not only one box containing 12 books but students were able to explain the shipping costs on each shipping service if many books were ordered in more than one box. Inference, students concluded from the results of their calculations either through written tests or interviews that students chose either ABC Express or DEF Express freight forwarding services to send the books they ordered, even though during interviews students were able to conclude three alternatives to ship the books. First, if one box of books were ordered, then DEF Express shipping service would be selected, second, if two boxes of books were ordered, then one of the two shipping services can be used, third, if more than two boxes of books were ordered, then ABC Express shipping service would be used. Finally, Interpretation, students were able to interpret from the conclusions made that the delivery service that provides the lowest cost would be chosen to send the books ordered.
I chose the shipping service DEF Express more save Rp. 3000
So, I chose the "DEF Express" shipping service because it was cheaper than the "ABC Express" shipping service.
DEF Express "3000 cheaper than" ABC Express ".
So, I chhhososoooosossssssseeeeeeeeeeethttthtthththtthhhhhhhe eeeee DDDDDDDDDDDDDEFEEFEEEFEEEExxpxxpxprereressssss shshshiphhhippipppipiiiiiiingnnnnngnnnnggggggggssssssssserereeereeeerervirvivvvivicececcececeeeebbbbbbecause i DE
DE D D D DE DE D D D D D D D D D D D D D D D D D D D D D D D
D F Express "3000 cheaper than" ABC Express ".
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