Mathematical Creative Model: Theory Framework and Application in Mathematics Learning Activities

Abstract

One of the most crucial abilities in the face of global issues is creativity. Because of this, research on creativity is always intriguing, especially in the context of mathematics education. Two investigations pertaining to original mathematical models are presented in this paper. The first study had 72 primary school instructors who took part in enhancing mathematical learning. Participants given open challenges with cube net material. Out of 137 high school students that consistently responded, 124 were included in the second study. Two descriptive statistical issues affect secondary school students. After a thorough interview to understand the subject’s thought process, the results of the problem-solving process were used to support the subject’s stance using an original mathematical model. Impersonation, adaptation, and building make up the creative model’s structure. The findings demonstrate the consistency of mathematical creativity model properties. The subject only imitates a successful routine method due to the level of imitation. According to the characteristics of the level of modification, the subject is able to alter the methods discovered in order to meet the difficulties at hand. The hallmarks of the creation level are that they develop novel, rational, and methodical approaches to problem-solving. Based on the findings of this study, a new method of learning mathematics was devised. The open-ended activities that have been packed into the form of a mathematical tree are utilized, together with the strategic approach to problem-posing.

1. Introduction

In order to prepare future generations for the challenges of a globalized society, creativity is crucial. Pioneers and entrants to change and progress can only be creative people. The foundation of artistic, technological, and scientific advancement is creativity. The advancement of information technology as a creative human endeavor that can make solving problems easier. Numerous academics have looked into the value of creativity, notably in the teaching of mathematics. This is consistent with the essence of mathematics education, which entails learning by fostering critical thinking and reasoning in order to become an effective problem-solver. Consequently, it is applied in mathematics to develop higher-order thinking skills (HOTS). One of HOTS’s most important components is creativity. The greatest degree of cognition, in accordance with the revised Bloom’s taxonomy, is creativity. The ability to be creative makes one an excellent problem solver and innovator.

Studying creativity is always fascinating, particularly when it comes to the realm of math instruction. The importance of creative research in mathematics education and the expansion of the scope of study were emphasized by Joklitschke et al. [1]. This supports a study by Schindler & Lilienthal [2] that emphasizes the importance of student creativity processes in mathematics education research. The development of creative mathematics is followed through open challenges to show how creative research progresses from creative outputs to creative processes. Multiple Solution Tasks (MSTs) were employed by Schindler & Lilienthal [2] in order to foster and assess students’ mathematical creativity, particularly the creative process.

In addition to the usage of open assignments, problem solving can be used to stimulate creativity. Problem-based research (PPI) is a method used by Leikin & Elgrably [3] to examine how creative solutions to mathematical issues are produced. According to their research, a stronger creative process does not always result in a stronger creative product and a higher degree of strategic originality. But it relates to the end product of creativity. Here, the two distinct aspects of cognitive processing connected to creative problem-solving are the creative outcome and the creative process. Three aspects of creativity—fluidity, flexibility, and creativity—are the foundation of their research. Their findings indicate that problem generation and problem solving cannot be separated when someone completes PPI creativity challenges.

Schoevers et al. [4] investigated into the connection between mathematical problem-solving in elementary schools and creative thinking. There are four closed routine issues, six closed non-routine problems, and four open non-routine problems using geometry tests (multiple solution problems). The findings indicate that creativity plays a significant role in predicting how well children will succeed on various geometry problems. However, it was more closely connected with how well students performed on open tasks that were not routine, which shows that kids with more creativity were more successful at solving geometry problems.

Numerous research, spanning from the classification of creative thinking components to the level of creativity, have been conducted as a result of the significance of creativity. The classification of creativity in this study is not based on the cognitive process by which it is formed, though. The mathematical creative model is a development of Subanji et al. [5] and is based on the mental process of creative formation. The context of issues involving mathematical thinking is examined in their research. Having unique qualities like reason and logic, mathematics is a fundamental science. The rationale behind using the framework of mathematical reasoning is that it calls for higher-order thinking, which is necessary for creative thinking. On the basis of the properties of the mathematical content, a study of argumentation in the field of mathematics was conducted. It is known as covariational reasoning in the context of building graphs, algebraic reasoning in the context of solving comparative problems, proportional reasoning in the context of solving analogy problems, analogical reasoning in the context of solving controversial problems, and controversial reasoning in the context of solving controversies. In this study, a problematic mathematical problem—one that defies the preexisting framework of thought—is the focus of the mathematical problem that is being used. The controversial question was picked because it was good at causing cognitive conflicts, or dis equilibration in Piaget’s terminology, which pushes for higher order thinking, particularly creative thinking. The development of mathematical creative models is based on the cognitive process of the formation of creative thought in solving mathematical problems. This creative model is important for research because it can be used to develop mathematical thinking in the correct way based on cognitive processes. The learning of mathematics is the learning of reason and logic, especially the formation of HOTS, which includes: analysis, assessment, and creation (creative). According to Bloom’s taxonomy, creativity is the highest level of thinking. Whoever can reach the level of creative thought will become a good problem-solver, initiator and innovator. The importance of creativity has led to various studies, from the classification of elements of creative thinking to creative thinking levels. Researchers have assessed the originality, fluidity, and adaptability of pupils’ problem-solving to assess their creativity [6, 7]; Additionally, Kattou et al. [6] discovered a link between creativity and mathematical aptitude. Four stages—preparation, incubation, illumination, and verification—were identified by Sriraman [8] after studying the creative processes of five mathematicians.

Learning mathematics requires a lot of creative thinking. Numerous academics looked at the elements of fluidity, flexibility, and invention in addressing open questions to determine how creatively kids thought [6, 7]. The number of alternative solutions to a given problem is considered in the assessment of fluency. The capacity to modify several concepts to generate various means of completion is related to flexibility. Originality is the generation of fresh approaches to challenges. Additionally, the structure of the connection between mathematical prowess and creativity was examined by Kattou et al. [6]. According to their research, pupils fall into one of three categories based on their mathematical prowess: those with low, medium, or high mathematical prowess. Sriraman [8] conducted a study of five mathematicians to ascertain the characteristics of the creative process, and the results showed that the creative process of mathematicians followed the four stages of the Gestalt model, namely the preparation-incubation-illumination-verification. However, the mathematical creativity of the three categories also varied, so it was found that students with the highest scores on the math test were also the most creative. In this situation, learning mathematics, especially statistics, can involve the creative process.

The basic and secondary mathematics curricula include statistical content on data analysis and probability, according to the National Council of Teachers of Mathematics [9]. This demonstrates the significance of statistics in mathematics. The mathematics curriculum in Indonesia similarly includes statistical components from primary to higher education. This is because statistics have a wide range of applications in the fields of law, medicine, agriculture, and economics. The sciences of data collection, analysis, presentation, interpretation, and decision-making are known as statistics [10]. There are two types of statistics: descriptive statistics and inference [11]. Measurements of concentrations and dispersion fall within the purview of descriptive statistics, whereas hypothesis tests that can extrapolate from samples and make generalizations about population features go under inference statistics. In this study, descriptive statistics were utilized to present data visually. The kind of qualitative or quantitative data also affects the type of diagram [12]. For instance, histograms, line graphs, stem and leaf charts, and pie charts are frequently used to depict the distribution of qualitative data (on a nominal or ordinal scale) (interval or ratio scale). In this situation, understanding information requires the capacity to read graphs and diagrams.

Sharma [13] examines and discusses how pupils comprehend the information in graphical forms such as tables and bar graphs. According to his research, a lot of pupils employ instinctive and experience-based tactics. Further research by Aoyama [14] into the hierarchy of students’ interpretations of graphs revealed several challenges younger students face when considering open-ended questions due to their lack of prior learning. According to Mann & Lacke [10], descriptive statistics include techniques for gathering data, showing it, and summarizing it using tables, graphs, and summary measures of concentration. Since there are so many graphs used to depict data in written and electronic media, it is important to comprehend how the graph might be used to interpret numbers. In order to solve the measure of concentration problem, students used the descriptive statistics they had learned in this course to portray data as line graphs and bar charts. This study’s problem is an open one that has to be solved with original thought.

To address common difficulties, creative thinking is necessary. Mathematical exercises can foster the development of creative thinking [8, 13, 15, 16, 17, 18, 19]. Logic-based concepts, structures, and interactions are a foundational part of mathematics, according to this theory. Using logical and methodical justifications, the truth of mathematics is established. Numerous mathematical tasks are performed through logical and methodical thought processes, including as formulating and testing hypotheses, seeking parallels, drawing connections, creating representations, creating generalizations, proving theorems, and ultimately solving problems. High-level thinking calls for pupils to exercise both critical and creative thought when completing these mathematical assignments.

Creative thinking occurs in mathematical activities, called mathematical creative thinking, and is often associated with problem solving. [9, 15, 20, 21]. The National Council of Teachers of Mathematics [9] proposes to give students difficult problems that can promote their mathematical creativity. This can be done because problem solving enables students to improve their creativity skills through various solutions. Baran et al. [15] discovered that mathematical creativity can be seen in problem-solving abilities, especially in open mathematical situations. Chamberlin & Moon [20] discovered creativity in the thinking processes of mathematicians related to the solution of non-routine problems.

Mathematical creative thinking, which happens when performing mathematical tasks, is frequently linked to problem-solving [9, 15, 20, 21]. Giving pupils challenging tasks that can foster their mathematical creativity is a suggestion made by The National Council of Teachers of Mathematics [9]. This is possible because problem solving gives kids the chance to develop their creativity through a variety of solutions. Mathematical creativity can be evident in one’s ability to solve problems, particularly in scenarios involving open-ended mathematics, according to research Baran et al. [15]. Chamberlin & Moon [20] found that the way mathematicians solve non-routine problems involves creativity. Therefore, one part of mathematical creativity is problem solving. Additionally, Beghetto & Karwowski [22] contend that teachers might accomplish this balance by turning some normal tasks into nonroutine difficulties. Routine practice must be matched with innovative and creative approaches. Can be achieved by teachers by changing some routine tasks into nonroutine problems.

Researchers have investigated the use of mathematics to foster creativity [15, 19, 23, 24]. To predict students’ creativity in solving mathematical problems, Lin & Cho [24] created a model of creative problem-solving abilities. According to gender, Baran et al. [15] discovered a correlation between creativity and mathematical aptitude. Voica & Singer [19] looked at the creativity of math-gifted children and discovered that those who had a strong grasp of the subject had good inventiveness. Using Model-Eliciting-Activities, Coxbill et al. [23] developed and tracked students’ mathematical inventiveness (MEASs). Sheffield [25, 26] has conducted research on the value of fostering and enhancing pupils’ creativity as they study mathematics. Understanding mathematics can aid in the development of students’ creativity [27]. Some of these studies highlight how crucial it is to research students’ mathematical creativity.

The other aspect of the study of creative thinking is examined in this article, specifically the creative model, which is founded on the cognitive process of creative formation.

2. Creative model framework

General creativity includes creative mathematical thinking. It is possible to think of creativity as a process of thought that involves original concepts and ideas [28, 29]. Numerous methods, such as studies of creative outputs and creative processes, can be used to examine creative mathematical thinking [3]. The cognitive process that results in creative thinking is referred to as the creative process. A type of creativity whose focus is mathematics is known as mathematical creativity. Therefore, the development of mathematical creative frameworks is founded on general creative frameworks that consider the properties of mathematical structures. In general, creativity occurs in daily life and can be divided into three categories: creation, modification, and imitation.

When someone wishes to create a product that replicates an already existing product, they start at the lowest level, which is later referred to as a creative model of imitation. The process of invention in this instance is restricted to product imitation. Even if they just copy, imitators nevertheless engage in the creative process since they consider ways to make their creations more affordable than the originals. Simple cognitive processes are used in the creative model of imitation. This imitation creativity model in mathematics is influenced by the learning process, which only prioritizes procedures. Students can solve a problem if the procedures are known.

The Creative Model of “modification” refers to the second level, which is a change. This level is reached when a product is transformed into a new one by looking at its “functions, advantages, and forms.” The highest level is creation, which is referred to as the “creation” model. This occurs when someone creates a new product without first considering the associated items that already exist.

A mathematical creative model is one that can be created from a general creative model in the context of mathematics. The general creative, mathematical creative models categorize creativity into three levels (three): imitation (imitation), modification (change), and creation. Table 1 below provides the conceptual framework for the construction of general creativity-based mathematical creativity models.

Creative models	General creativity	Mathematical creativity
Imitation	Imitate a product with a simpler process or lower cost Imitate a production process to produce another product that is better	Just imitate similar forms of settlement to solve the problems at hand
Modification	Changing the function/benefit/form of a product so that it becomes a new product	Changing the problem/data/solution procedure to obtain a more efficient solution
Creation	Create new works that are more interesting, more practical, and have more functions	Develop new settlement procedures according to the demands of the problem

Table 1.

Mathematical creative model.

The core of the imitation level creative model is the principle of making products with the same design and functionality but at a cheaper cost. You must cut or substitute other less expensive materials in order to lower the cost of the product. A creative imitation model produces a varied grade of goods as its outcome. The definitions of “original product,” “super replica product” (quality 1), “medium replica” (quality 2), and “low replica” (quality 3). Models for creative modification are based on the idea of modifying items to make them more cozy, lovely, alluring, and practical. The process of modifying a product’s function, benefit, or form results in a new product with additional functions, one that is more useful, and one with a more appealing shape. This process is the basis for the creative modification model. Finding new ideas, thinking, and solutions to issues is the foundation of creative creation models.

In the context of mathematics, imitation levels, i.e., imitate only the facts or techniques obtained when completing problems/tasks. The methods he was able to find in this situation varies from the most basic to the most sophisticated. The technique of generating new challenges or tasks by adjusting the data, graphics, and procedures obtained is known as modification level mathematical creative modeling. The process of constructing or resolving issues by gathering data, making graphs, or coming up with novel problem-solving techniques constitutes the creation level of the mathematical creative model.

3. Method

The research develops a model of creativity based on the idea that creativity is seen as a cognitive process of creative formation. The first research was carried out on teachers of elementary schools attending cube net materials training, and on students of secondary schools undergoing descriptive statistics training. The paper describes only part of the two studies of the creative model. The first study was a creative model for 72 primary school teachers who participated in mathematics learning courses jointly held by Universitas Negeri Malang and the PT. Pertamina. This teacher-powering cooperation lasted 6 years, but the data sources used in the study were only last year. The training is conducted every year at three strengthening stages and in two classroom exercises. The first and second strengthening phases took 10 days (80 hours) to complete. The third phase of strengthening was 5 days (40 hours) long. Learning practice is conducted in intervals between strengthening stages. There are materials to solve mathematical problems at each stage of the reinforcement. Among the problems-solving materials, the most interesting are cube net materials. This material is always chosen by the participants as the most interesting material, because it is related to real life, different from what is usually presented to students, the problem is open, makes sense, and is very challenging. Then, the process of solving the problem of cube net material and conducting an in-depth interview was used as a data tool to investigate creative models.

The second study was the creative model of 137 secondary students to solve problems in descriptive statistical material. Students received two open problems relating to descriptive statistical materials, the tea sale problem and the math test results problem. To assess the consistency of the level of student creative models, two problems were presented. Of the 137 students, 124 were re-examined (59 boys and 65 girls). From the process of solving statistical descriptive problems, followed by tracing the process of cognition through in-depth interviews, and used to justify the position of the subject based on the creative model.

4. Results

The results of the research are presented on the basis of two studies, namely the mathematical creative model for solving the problem of cube nets and creative mathematical model to solve descriptive statistical problems.

4.1 Creative model of cube net problem solving activities

This study included 72 primary school teachers participating in mathematics education training. One of the materials presented is a mathematical problem solved with a cube net. The topic was given the following problem about applying cube networks.

Cake box company

Pak Romli is the head of the company “MAKMUR” which produces cake boxes. Pak Romli always thinks how to get big profits. Several attempts have been made: first, to make various models of boxes that may be liked by buyers and secondly to make savings, namely to produce as many boxes as possible from the same material. Sir. Regarding the first attempt, Pak Romli has conducted a survey of 200 cake sellers, the result is that 70% of the respondents liked the cube-shaped cake box on the grounds that it could contain more. To save money, Pak Romli conducted an experiment by making a box and opening it with each side still intertwined (not separated from each other), which are often called cube nets, as shown inFigure 1.

Figure 1.

Net cube.

To make an efficient cube design, Pak Romli prepared a “cube material” (Figure 2) in the form of a sheet of paper that has been completed with unit squares. Then Pak Romli thought about how to use a piece of paper to make as many cubes as possible (and as little paper wasted as possible). You are asked to help Pak Romli efficiently design cube nets fromFigure 2material (the cubes produced are the most and the paper is wasted the least)! Determine the maximum number of cubes that can be made!

Figure 2.

Cube material.

Research subjects participating in mathematics learning training had the characteristics of 11 early-career teachers (1–3) and 61 high-career teachers (4–6). Of the 72 subjects, 9 (13.50%) were imitation levels, 34 (47.2%) were modifications levels, and 29 (40.28%) were creativity levels. Each level’s description is given as follows.

4.2 Imitation level

At the level of imitation, the object only imitates Pak Romli’s cube nets and immediately places the cube nets on the cube material sheet.

One can see from the subject’s response on the level of imitation that the subject only copies the information provided in the question and keeps the relevant cube materials structured. The subject does not take into account various cube net configurations; rather, what is noteworthy is the subject’s cognition, which copies the provided material and simply modifies its position to produce consequences. To further understand the subject’s position at imitation level, the researcher conducted an interview.

Q: What is your comment on this matter?

S: The problem is contextual and very challenging.

Q: Have you ever had a question like this?

S: Never.

Q: What do you think when you face a problem like this?

S: Since there are already nets of the cube, then I make the same nets with different positions so that the solution is immediately obtained.

Q: In your opinion, are there other forms of nets?

S: There should be, but I do not remember because all this time I have been teaching the lower class (grade 2), so I do not know the cube nets by heart.

From the interviews, the subject seems to “imitate” the net only in different horizontal and vertical positions. This may occur because the subject is taught in a lower class and the net material of the cube is given in 5’ class.

4.3 Modification level

At the level of modification, the topic changes the shape of the problem’s cube net by changing the position of one unit square. In addition to (1), you can change one step of the left square to (2). Modification from (1) to (3) is done by shifting the topmost square to the bottommost position. Next is the position adjustment.

To further explore the creative model of the subject in modification level, a task-based interview was conducted.

Q: What is your comment on this matter?

S: This problem makes sense but is not easy to solve.

Q: Have you ever had a question like this?

S: Never. Usually the problem of cube nets is limited to their type.

Q: What do you think when you face a problem like this?

S: I am curious. At first I thought it was simple, but after thinking about it, it’s not easy, even though I’ve got this answer, I am still curious.

Q: How do you get the nets of cubes (2) and (3)?

S: I changed it at the same time wondering whether the result of the conversion is still a net of a cube. And after I imagined the results of the conversion were cube nets. Next, I arrange like that position and get three closed cubes.

From the interview, it appears that in solving the problem, the subject begins by “changing” the cube nets that are already known in the problem into several different cube nets and then arranging them in the cube material provided.

4.4 Creation level

At the level of creation, integrative thought focuses on the shape of the 7x5 cube. The subject is “How to cover a cube material as many cube nets as possible and ignore the nets in the problem. The topic was explored and was successfully created by making a “new” cube net that allowed for more cube material configuration.

To further investigate the characteristics of creativity models at the creation level, task-based interviews were conducted. The interview focused on the process of constructing answers for the subject.

Q: What is your comment on this matter?

S: I never thought of a problem like this, because when teaching cube nets, we only showed several alternatives for cube nets.

Q: Have you ever had a question like this?

S: Never. This problem inspired me to connect math material with life.

Q: What do you think when you face a problem like this?

S: I realized that to teach mathematics it is necessary to think about the use of mathematics in life. Therefore, so far, I teach cube nets only procedures, so I feel guilty.

Q: How would you construct an answer like this?

S: At first, I thought this problem was simple, but it turned out to be complex. I started with a 7x5 cube material, while one cube only needed six unit squares. I suspect that five nets of the cube can be made.

Q: Why do not you use the cube nets that have been given?

S: I suspect that the exemplified web cannot produce many cubes. It was a hindrance, so I ignored it. And I created new cube nets alternatives.

Q: What are your next steps?

S: I arrange cube nets into cube materials with various alternatives. It turns out that the maximum number of cubes that can be made is four. Then I was still thinking again, the rest, it turns out that there are some that cannot be made into cubes anymore, but there are also some that can be made into open cubes. I think that the open cube is still of some use, too. Finally, I conclude that the most efficient arrangement is four closed cubes and one open cube, there are still six unit squares left but randomized, no more cubes can be made. I am also thinking further, what if the size of the cube material is different (not 7x5), but I have not tried.

The results of the interviews show that the creation level topics continue to think about the creation of the most effective cube. In fact, the topic does not stop with the solution he is given, but thinks he wants to change the size of the cube material so that it’s a little wasted.

5. Mathematical creative models on descriptive statistical problem-solving activities

This section provides students with the level of creative model to solve statistical descriptive problems [5]. The study was conducted with 137 secondary students and provided two open problems with descriptive statistical materials, namely problem 1 (tea sales) and problem 2 (mathematical test scores). To see the consistency of the level of creative models of students, two problems have been given. Of the 137 students, 124 (59 boys and 65 girls) were consistently re-recognized.

Problem 1. Tea Sales

A shop sells two soft drinks “tea bottle” and “tea box” from 2000 to 2009. The graph below shows the sales of the drink “Teh Botol” for 5 years. Make a line graph of the sales results of “Teh Botol” and “Teh Kotak” in one graph so that in 2009 the sales of the two soft drinks were the same.

Problem 2. Math Exam Score

A Mathematics teacher teaches in two classes, namely classes A and B.Table 1below is the result of the Mathematics score for class A. The number of students in class B is 35 people. Make a bar chart of the Grade B Mathematics grades (Scores 0–100) if the mean, median, and class B mode scores are greater than the class A mode.

One Hundred and Twenty four people responded consistently to questions based on creative model levels. 25 subjects (20.16%) are classified as imitation levels, and their problem solving processes are characterized by imitating data display, graphics, or strategy. Fifty six subjects (45.16%) are characterized by modifying data and graphics display processes to solve problems.; and 43 creative subjects (34.68%) are characterized by problems solved by the construction of new data and graphics. Figure 3(a) shows the distribution of the creative model for solving the open problem.

Figure 3.

(a) Distribution of creative models; (b) creative model by gender.

Figure 3(b) shows that the distribution of the creative model of male subjects: level of imitation 12 (9.7%), level of modification 26 (21%), and level of creation 21 (16.9%). Creative models of female subjects: level of imitation 13 (10.5%), level of modification 30 (24.2%), and level of creation 22 (17.7%). In carrying out open-ended problem solving activities, both male and female students are mostly on the modified-level creative model. The following presents a creative model of the subject in solving open-ended problems for each level.

5.1 Imitation level

The creative model of imitation level is characterized by imitation of the context of problem solving strategies. In the first problem, the creation model at the imitation level is reflected in the process of imitating graphic shapes and numbers to make them similar to existing graphics and numbers. In the second problem, the creativity model at the level of imitation is reflected in the process of imitation only of the available data. This shows that the creative models of the level of imitation of the subjects in problems 1 and 2 are consistent. The student’s activities in the mathematical creative model of imitation level are presented in the following.

In problem 1, the subject only imitated the shape of the graph and the trend of the sales of bottled tea so that the two soft drinks had similarities both in the form of graphs and the increase in sales results which experienced the same increase every year. The results of interviews with researchers (P) and students (S) are as follows:

Q: What is your process for completing problem 1?

S: I saw the difference in the increase in tea bottles by six, then I followed it until 2009, it went up by six until I got 64. Then for the tea box graph, the value must be the same in 2009, so I started to make it from the back first from 2009. From my score of 64 less 4 continues every year until 2000, sales results 28

Q: Why did you choose the difference 4, not any other number?

S: because you are free to choose ma’am

Q: Why not make six to make it the same as tea bottle

S: Therefore, that the graphs do not overlap, mam

In problem 1, the subject who is at the imitation level in determining the sales of tea bottles in 2005–2009 by “imitating” is exactly the same as the increase in sales in 2000–2004, which is Rp. 6 million every year. The subject used the Rp. 6 million to continue the pattern of increasing sales of tea bottle in 2005–2009 so that the sales of tea bottles in 2009 were Rp. 64 Million. Furthermore, the subject made a graph of the sales results of tea boxes starting in 2009 and imitated the pattern of increasing sales of tea bottle which the increase was the same every year. This subject makes an increase in the sales of tea boxes by Rp. 4 million every year. From the sales in 2009 of Rp. 64 million, the subject made a downward trend by counting backward from Rp. 4 million every year so that the sales of tea box in 2000 amounted to Rp. 28 million. In this case, the subject makes a pattern for the sale of tea bottles in the form of an arithmetic sequence with a difference of Rp. 6 million. By using the concept of the arithmetic sequence, the subject imitated it to determine the sales of tea boxes with a difference of Rp. 4 million. The imitation level creative model that the subject did in problem 1 is reflected in the process of imitating graphic shapes and imitating number patterns.

In problem 2, the creative model of the imitation level is reflected in the subject’s activity in determining the score of mathematics in class B which “imitates” the values of mathematics in class A. Although in Problem 2 it is stated that the subject is free to determine the range of mathematics values for class B (values 0–100), but the subject chose to imitate all grade A math scores. To create a bar chart, the subject made the horizontal axis the same as the math score of class A. In other words, the subject used all the scores of the class A, i.e., grades 40, 45, 60, 70, and 85 to make a bar chart of grade class B math scores. Next, the subject made the frequency on the vertical axis so that the mean, median, and mode of class B were greater than class A. The subject’s position at the imitation level was strengthened by the following interview results.

Q: Why did you choose the value on the horizontal axis like that? (pointing to the bar chart).

S: I made the value the same as the score of the class A, mam.

Q: What is your reason for making the values the same and not choosing another value?

S: I will just make it the same, ma’am.

Q: In the question from being free to choose of class B math score from 0 to 100, did you not read.

question?

S: Read ma’am,

The question reveals that although he read the question, he can freely choose from 0 to 100 the Class B score. Furthermore, the subject has made the class B score equal to the class A score. Subjects can determine class A average values of 65, median values of 60, and mode values. Therefore, if the subject determines 70 as the highest frequency of 11 students, the pattern and average of class B are larger than class A. The subject determines that 26 students got a score of more than 70 and as many as nine students got a score of less than 70, so the subject believed that the average value of class B was greater than class A.

The descriptive statistical activities for problems 1 and 2 depend only on the information shown in the task. This is in line with the opinion of Mecca & Mumford [30] which states that imitation occurs if there is an object to be imitated or imitated so that imitation depends on how people work with examples. For example, in problem 1, students create a positive trend in the sales of tea box because there is an example of a graph in the form of a positive trend in the sales results of tea bottle which is displayed in problem 1. Likewise, in Problem 2, students scored mathematics in class B because students observed that class A contained mathematics scores. The math scores in class A can also be used for students in class B. In this case, students can get information about something that they experience directly and from what is around them. Buttelmann et al. [31] state that children are more likely to imitate reliable models than unreliable ones. While Okada & Ishibashi [32] state that imitation is the core of the learning process, someone imitates the attitudes of others not only superficially but also at the level of deep cognitive processes.

5.2 Modification level

The Creative Model of the Modification Level occurs when the subject changes and combines components, data, and strategies to complete the task. In problem 1, subject changes some or all of the existing graphics and numerical models to construct the graph. This is consistent with problem 2 completion, which is used to modify the scores of the A class students and construct the completion of the B class data. Examples of mathematical creative models at the level of modification are given below.

In the creative model of the modified level, subject changes the pattern of increasing the sale of tea bottles so that they are different from tea bottles and that the graphics are different from tea bottles. The results of research interviews with students confirm the position of modification level as follow.

Q: How did you complete problem 1?

S: In making the tea box from 2005 to 2009 ma’am, I followed the multiples of six ma’am like.

year 2000–2004,

Q: Why not make another multiple?

S: I think from 2000 to 2004 the increase was regular, so I also made the increase regularly for 2005-.

2009

Q: Why not make up and down like a tea box chart?

S: I just continued the previous one, mam.

Q: Why do the lines on the tea box chart go up and down?

S: Because I made it different from tea bottle mam.

In the first issue, the subject made a series on tea bottles sales from 2005 to 2009, continuing the previous year’s series. Between 2000 and 2004, tea bottles were sold by Rp. 6 million every year, so the students added Rp. 6 million to the sales of the last year. In this case, the subject made modifications by changing the trend of the sales of tea bottles so that the sales of tea box experienced an up and down trend. The student made the sales of tea box that are different from the tea bottle, so that the shape of the trend line on the graphs of the two soft drinks is also different.

In problem 2, the subject uses some of the score of class A; namely scores 45, 60 and 70 and modifies them to 50, 65, and 90. These values are used on the horizontal axis on the bar chart of grade B math scores. While on the vertical axis, the subject determines the number of students so that the mean, median, and mode of class B are greater than class A. The subject uses a score of 85 in class A and modifies the other values to 90, 95, and 100. These values are used by the student to make a bar chart for class B. The position of the subject at this modified level is supported by the following interview results:

Q: Why did you make a bar chart like this (while pointing to the student’s answers).

S: Because judging from the problem, mam, from the questions obtained in class A there are 30 students, from various grades from 40 to 85, right, the order is to make a bar chart for class B whose value is up to us, sir, from 0 to 100. The average value, the median and mode of class B must be greater than class A. The first score is recorded for class A, in class A the average is 65 books, the mode is 60 and the median is also 60. So, how do I find the value for class B, I choose First the numbers are high so that the mean, median and mode of class B are greater than class A.

Q: Why were the scores 85, 90.95 and 100 chosen for class B score, why not other grades?

S: I choose the largest value in class A, which is 85, then I choose a number greater than 85 so that the average, median and mode exceed class A.

Q: Are you sure?

S: Sure, because the value is bigger than the grade A class bu.

The interview showed that the subject merged the highest score of class A 85 and other values of more than 85 and modified the data. Students do so without mathematical calculation, so that the average, average, and mode score of class B is higher than class A.

The creative modification level model is an electronic data modification by merging and synthesizing several objects and concepts to generate new objects and concepts. This is in line with Batanero et al. [33] which states that through synthesis, such as combining the concept of a measure of concentration with the concept of a measure of dispersion, a new concept emerges, namely the distribution of data as a fundamental concept.

5.3 Creation level

The level of creation of creative models is an activity performed by a person in the development of new information. The level of creativity in solving descriptive statistical problems is marked by the subject developing the existing line graph into a new line graph to solve the problem of presenting data in graphical form. The subject creates a new bar chart based on the table to solve the centering size problem. Subject activities on problem 1 and problem 2 on the creative level of creative model are presented in below.

To further study the status of the subject in the creation phase, the researcher conducted the following interviews.

Q: How is your process to complete problem 1?

S: I made a graph of tea box and tea bottles from 2005 to 2009, some of which went up and some went down, mam, because I think that every sale does not always increase or decrease, there are times when sales also increase and decrease, that’s why I made a graph, some are up and some are down.

Q: What are the initial steps taken?

S: First, I made the sales of tea bottle from 2005 to 2009 go up and down, mam. Then I made the sales of tea box starting in 2000, the value was 17, then in 2001 it decreased to 10, and in 2002 it increased again to 26, in 2003 it increased again to 23, in 2004 it became 43. decreased to 25, in 2006 it rose again slowly to 29, in 2007 it rose to 35, in 2008 it rose to 40, and finally in 2009 the sales of tea box and tea bottle were equal to 50. So in the middle of the year the sales had decreased drastic.

Q: Is there a number pattern that you make in determining it?

S: No mam, I’ll take anything.

Q: What is the reason for taking any?

S: No, mam, just create your own.

In completing problem 1, the subject of the creative level of model creative is not affected by the shape of the pattern or the shape of the trend in the graph in the problem, the subject makes or creates his own pattern and shape of the trend on the graph. It is believed that sales results should not continue to rise, and sometimes they must also rise and decrease. In this case, the subject is able to connect the problem in the problem with real life, the subject is not fixated on what is shown in the problem, but students think realistically and logically to complete tasks associated with everyday life. The pattern of numbers made by the subject is irregular, meaning that the amount of increase in sales results does not form a line that has a regular pattern so that it results in a different graphic form from the graph shown in the assignment.

In problem 2, the creation-level, subject creates a class B mathematical score bar chart by designing its own class A mathematics score. In problem 2, students can use values in the range 0–100, the range of values used by the subject to create new score of class B. The activity carried out by the subject in the creative level creative model is designing new scores so that the math scores of students in class B are different from those of students in class A. The student chooses scores of B student’s score by 30, 35, 50, 65, 90, 95 and 100 where all these values are different from the scores of students in class A. These values are used by students to make a bar chart, where the horizontal axis contains student scores and the vertical axis contains the number of students who got these scores. The subject determined that more students scored 90, 95 and 100, i.e. there were 27 students and eight students scored lower than 90. The results of the following interviews reflect the creative process of a creative subject.

Q: How is the process of answering problem 2? (while pointing to the student’s answer paper).

S: The first step is to determine the mean, median and mode of class A. The average of class A is 65, the median and mode are 60. To make it bigger than class A, I make high marks in class B, there are 90, 95 and 100, I made the stems high, class B has a mode of 95 because there are 10 people, while in class A there are low scores and high scores, but the score is not up to 100. Because in the question class B has to be bigger in average, median, and mode, so I make a few few who score low and many who score high.

Q: Did you use formulas to solve for mean, median, and class B mode?

S: No, mam.

Q: How can you be sure that the mean, median and class B mode are greater?

S: I made a lot of students get high marks from class A mam.

Q: Why did you choose the score in class B like this? (pointing to the horizontal axis of the bar chart).

S: Because in a class usually there are those who get low scores, there are also moderate, and some are high and very high, I made it so that the grades of classes A and B are different.

Q: Any other reasons?

S: No, mam, so that the values are different.

The subject first determines the mean, medium and model of Class A before selecting the mathematical score of Class B so that the average, medium and model of Class B are greater than Class A. The average, average and class A mode values are 65, 60, and 60. Subjects can ensure that the mean, median and mode of score of class B are greater than class A without performing mathematical calculations by placing more students with scores higher than 65. Activities of these topics are classified as creative. This is in accordance with Sheffield [34] opinion that student solutions are considered creative if the student can produce something unique and new to what is in their environment.

6. Creative models in mathematics learning assisted by mathematics tree media

The mathematical problem that can promote students’ mathematical creative thinking is usually open-ended, allowing students to create new ideas and ideas freely. Students should participate in activities that allow them to explore the problems, ideas, and ideas necessary to solve the problems [17, 18]. Problems that stimulate creativity are usually addressed in several ways [8] or open-ended problems. In addition, creative thinking can be built by problem posing. In the development of the level of creative models, the learning of mathematics must use open-ended or problem posing [35].

The importance of open ended in learning mathematics activities has been studied by several researchers [36, 37]. Hitt & Dufour [37] examined students’ mathematical activities when they completed an open-ended task related to speed and found that students used different representations in the process of modeling the situation when they were solving open-ended problems. Chan & Clarke [36] provide an open-ended problem in collaborative group work activities. Mathematics learning activities using open ended are able to develop problem solving skills and negotiation skills in collaborative learning groups. This shows that open-ended assignments can challenge students to think higher and ultimately be able to improve problem-solving skills.

Many experts believe that creativity can be developed by open-ended activities and problem posing. The problem is how to package open ended and problem posing in learning mathematics. This paper offers mathematics learning with open ended activities and problem posing which is packaged in the form of a mathematical tree. Mathematics learning is assisted by mathematical tree media, hereinafter referred to as the Mathematical Tree Learning Model. The mathematical tree learning model is intended as a learning model that facilitates students to: (1) pose a problem whose answers are known or (2) answer questions from open-ended problems.

Learning with the mathematical tree media is a form of learning with the following syntax: (1) the teacher models/explains the material, (2) the teacher presents problems and students solve them in groups, (3a) the teacher gives answers on the twigs & students construct the appropriate questions on the leaves. Or (3b) the teacher gives open ended questions on the twig & students determine all possible answers on the leaf, (4) the teacher asks students to exchange and correct other students’ answers, (5) the teacher asks students to rate other students’ answers, and (6) the teacher provide reinforcement to the problems or answers made by students. In this case the teacher has prepared media in the form of a mathematical tree, which consists of stems, twigs, and leaves. The stem contains the subject matter, the twig contains open-ended problems or answers, the leaves contain answers to open-ended problems or problems whose answers are already known. Students make leaves (compose problems or determine answers) as much as possible. The more leaves produced, the more fertile the tree is. On the other hand, if the leaves are made incorrectly, they will become INSTRUCTIONS. Therefore, in determining the assessment, the leaf (the correct answer/problem) is scored 3 (three). When the parasite (answer/problem made) is wrong, then the value is –1 (negative 1).

For example, in teaching students about the application of definite integrals, the questions usually given by the teacher are determining the area of a region bounded by some curves or determining the volume of a curve that is rotated around an axis. While learning with the mathematical tree media is done by determining the tree, namely, the integral and the branch is the area or volume whose value has been given. Next, students are asked to make leaves (find as many problems as possible) whose answers are on the branch. The integral tree can be made as shown in Figure 4 below.

Figure 4.

Integral tree.

In the integral tree, students are asked to construct a leaf (i.e., a problem whose answer is already known) - the definite integral will result in 12. The definite integral form whose result is 12 is very large, therefore students can arrange as many definite integrals as possible, the important thing is that the result is 12. In this case, it is not enough for students to just remember the procedures exemplified by the teacher, but students must be creative in determining as many alternatives as possible. Therefore, learning with this mathematical tree media can develop students’ creative models.

Another example, learning straight line equations. The tree is the equation of a straight line. The stick is to determine as many equations as possible the line that passes through the point (1.2) and determine as many equations as possible the line parallel to $y=2x–3$. The line equation tree is presented in Figure 5 below.

Figure 5.

Line equation tree.

The thinking process of students who are built in learning with the media tree of mathematics can be described as follows.

In determining the equation of a straight line that passes through $\left(-2.1\right)$, the students’ thinking process is built by determining any line equation. For example $y=2x$, then students will be able to think that if $x=-2$, then y must be worth one. Whereas when $x=-2$, the value of y is $2\left(-2\right)=-4$. To get a value of 1, you must add five. Therefore, so that$y=2x$ through $\left(-2.1\right)$, it must be changed to $y=2x+5$. This thought process will be able to empower students’ reasoning. If the students’ reasoning is well developed, then when students are asked to solve other problems, for example the equation of a line through $\left(-2.1\right)$ with a gradient of $–3$, students will easily determine by thinking, if the gradient is–3, it means the equation of the line is $y=–3x$ . If $x=–2$, then y must have a value of one. Whereas when we substitute $x=-2$, we get $y=-3\left(-2\right)=6$. Therefore, in order to fulfill 1, the equation of the line is $y=-3x–5$. Therefore the equation of the line through $\left(-2.1\right)$ with a gradient of$-3$ is $y=-3x–5$.

Mathematical trees can also be developed in elementary schools in various materials: integer operations, fraction operations, perimeter and area. Here’s an example of a math tree in elementary school about the number operation tree (Figure 6).

Figure 6.

The number operation tree.

When students are asked to make pairs of numbers that meet $3-a=b+10$, students will think, one of which is if $a=2$, then the left side is equal to 1, and so on the right side the value is 1, then$b=-9$. If $a=3$, then the left-hand side is zero and so that the right-hand side is zero, b must be $-10$. In this case, students are doing creative thinking activities, where students make many answers and finally students can find a pattern if an increases by 1, then b decreases by one. Mathematical tree learning will build HOTS and creative thinking.

7. Conclusion

Mathematical creative models are based on cognitive processes of creative thinking in mathematical activities and are divided into three levels: imitation, modification, and creation. Levels of imitation are characterized by cognitive processes in which people can only imitate provided strategies/processes problem solving methods. The level of modification is characterized by a cognitive process in which one hopes to change strategy/procedure, and the problem-solving process is more effective or simple. The creation level is characterized by the cognitive process, i.e. the ability to construct new strategies/procedures for solving problems.

Mathematics learning emphasizes only procedures and only forms creative models at imitation levels. Students who can only remember procedures can apply them only to problems similar to solved problems. Students face problems if they face new problems or modified problems. Therefore, mathematics must be learned to stimulate creative thinking, such as using the mathematical tree media.

Mathematical tree learning was developed based on problem posing and open ended. The stimulus is given in the form of open ended, requiring students to determine several alternative correct answers. Problem posing in a mathematical tree has a special characteristic, namely the stimulus given is in the form of an answer and students are asked to construct a problem with the answers already provided.