Zekiye Morkoyunlu & Alper Cihan Konyalıoğlu

In this study, mathematical creativity in a social cultural atmosphere and mathematical creativity in an individual atmosphere will be investigated. In other words, how the mathematical creativity differs from an individually study to a sociocultural study? The importance of mathematical creativity is on the research arena. However, how the the atmosphere affects the mathematical creativity of a child can be a significant factor in terms of improvement of mathematical creativity. The study is going to be a mixed design study. It is going to be an experimental case study. In order to conduct the study, 10 sixth grade students will be selected from a randomly selected primary school in Kırşehir, Turkey. This class will be selected as an average performed class in the school. Five of the students will be study with their peers, teachers and parents, and the others will be study individually on the same activities. Five mathematical activities will be designed based on the units in the Turkish curriculum. Various instruments will be used in the study. For example, pretest, posttest, focus group discussions, and diaries for parents. The study will be conducted during five week days. It will be required that each activity will be handled in a week day. The pretest will be applied to both groups at the beginning of the week. The posttest will be applied to both groups at the end of the handling activities. After gathering the data, all the information will be analyzed both quantitatively and qualitatively. At first, descriptive analysis will be used to reach the main results. After this analysis, content analysis will be used to reach deeper analysis.

**INTRODUCTION**

Parental involvement in mathematics education one of the important issue in the mathematics education arena. Many researchers have contributed to this arena by conducting several research and projects. Marta Civil and her colleagues are the significant researchers in this domain. They conduct different projects about parent involvement; such as, The Project MAPPS and the BRIDGE (Marta Civil webpage). Especially, the MAPPS emphasizes the interaction of parents and students. In the study, parents took mathematics courses (algebra,geometry, numbers) and with their knowledge and pedagogy they studied with their children. The results were positive in terms of the students’ achievements (Civil, 2001). In another research of Civil and others (1998), they aimed to develop the students’ and their parents’ understanding of mathematics based on daily life experiences. In order to do this the researchers provided everyday examples which encompasses mathematical uses in them. The results of the study shows that the students who were comfortable with doing everyday activities were found more unsuccessful compared to others. Additionally, the researchers report that the connection between in school and out of school mathematics should be reinforced in terms of curriculum (Civil, 1998). Thus, it seems crucial to support parents involvement in mathematics and make strong ties between school mathematics and out of school mathematics. These two factors can be considered important in terms of the development of students’ mathematical creativity.

On the other hand, mathematical creativity is another important research arena in the field of mathematics education. At first, it is crucial to address the creative nature of mathematics as a domain. Lewis and Aiken (1977) mention the creative nature of mathematics in their study called “Mathematics as a creative language”. They point out the creation of a mathematical concept by emphasizing the use of components of the mathematical terms; such as, symbols, rules, concept etc. (Lewis and Aiken, 1977). Today, students are required not only to develop intellectually in mathematics but they are also required to develop mathematical creativity skills in order to be fulfil their potential (Andzans and Koichu, undated).

In the Mann (2006)’s article called “Creativity: The Essence of Mathematics” various definitions of creativity were presented. Two of them as follows:

“divergent and convergent thinking, problem finding and problem solving, self expression, intrinsic motivation, a questioning attitude, and self confidence (Runco, 1993)”

“creativity includes the ability to see new relationships between techniques and areas of application to make associations between possibly unrelated ideas (Tummadge, as cited in Haylock, 1987, p.60)

Based on the different views of the definitions of mathematical creativity; fluency, flexibility, originality and elaboration were found as the components of mathematical creativity (Mann, 2006; Singh, 1985).

Also Singh (1985) simply states the the mathematical creativity as “the process of generating significant ideas, making theoretical ideas practical, converting innovative ideas from other fields into mathematics.”

An important issue has been existed in the literature; Mathematical creativity: For mathematicians and for school children (Mann, 2006). Many of the researchers think that Mathematical Creativity for Mathematicians should be put aside now and Mathematical Creativity for School children should be thought as significant (Sheffield 2009 as cited Lev and Leikin, undated).

In order to reinforce the students’ mathematical creativity, several of approaches some mathematical learning theories should be taken into account. For example, radical constructivist and socio-constructivist approach can be considered two of the important approaches in the field.

According to Mann (2009), the development of creativity mostly depend on the environment. In other words, the environment one of the most crucial factor of creativity of students in mathematics, too (Mann, 2009). Similarly, Münz (2014) states the importance of socio-constructivist environment in terms of emergent of various interaction yielding creative products. She based her idea on the idea of Sriraman (2004) that “…it is impossible for an individual acquire knowledge of the external world without social interaction”.

As with the other level of schooling, middle school students’ mathematical creativity can be examined in terms of the learning environment. For example, learning individually and learning in a social environment. The aim of this study is to compare the mathematical creativity of 4, 6th grade students studying individually with 6, 6th grade students studying with their parents. In order to reach this aim, the following research questions have been tried to solve:

- How do the 5 students studying with their parents perform during the study?
- How do the 5 students studying individually perform during the study?
- What were the results of parents’ ideas indicated on the diaries?

**METHOD**

*Research design*

In this study, mixed design study has been used as a research design. It is an experimental case study. It is an experimental study because the students’ mathematical creativity who studied with their parents has been compared with the students’ mathematical creativity who studied individually. The experimental group consists of the students studying with their parents. The control group consists of the students studying individually. On the other hand, the study is a case study because a phenomena –mathematical creativity of the students- wanted to be investigated in real situations (Cohen, Manion and Morrison, 2011).

*Study group*

10 sixth grade students have been selected from a middle school which was selected randomly in the city of Kırşehir, Turkey. These 10 students have also been selected based on their mathematics grades. All the students’ mathematics grade are 5 based on the fifth system. During the study, 6 of the students studied with their parents on the five activities prepared by the researchers based on both Turkish mathematics curriculum and TIMSS questions. 4 of the student studied individually on these five activities.

*Data collection*

Data were collected through pre test, post test, mathematics activities and parent diaries, and focus group discussion with students.

The study lasted five days. At the beginning of the study, the students were given a pretest consisting of 15 multiple choice questions. Then, each of the week day, 6 students studied on the activities with their parents. The other 4 students studied on the same activities on their own each of the week days, too.

After studying on the activities, the students were given post test, similar to the pre test, consisting of 15 multiple choice questions.

Focus group discussions were implemented with both the students and the parents after the study on the activities.

*Data analysis*

All the information will be analyzed both quantitatively and qualitatively.

The pretest and post tests have been analyzed with SPSS packet programme. Two way analysis of variance and t- test are used to analyze the results of pretest and posttest.

The main components of creativity; originality, fluency and flexibility, have been used to evaluate the creativity of students on their performance of the activities.

Descriptive analysis will be used to reach the main results of the activities. After this analysis, content analysis will be used to reach deeper analysis of the activities.

The results of the parent diaries and the focus group discussions of both the students and the parents are also analyzed based on the descriptive analysis methods. The content analysis is also used to reach deeper information.

**Findings and Results**

Analysis of pretest and posttest

SPSS 15 programme was used to reach the following results;

**Table 1.**

**Descriptive Statistics**

Group | Mean | Std. Deviation | N | |

Pretest | S | 59,2000 | 20,26573 | 5 |

SP | 46,0000 | 15,79557 | 5 | |

Total | 52,6000 | 18,48843 | 10 | |

Posttest | S | 83,6000 | 12,81796 | 5 |

SP | 89,4000 | 6,94982 | 5 | |

Total | 86,5000 | 10,18986 | 10 |

**Table 2.**

**Tests of Within-Subjects Effects**

Source | Type III Sum of Squares | df | Mean Square | F | Sig. | |

Measurement | Sphericity Assumed | 5746,050 | 1 | 5746,050 | 48,892 | ,000 |

Greenhouse-Geisser | 5746,050 | 1,000 | 5746,050 | 48,892 | ,000 | |

Huynh-Feldt | 5746,050 | 1,000 | 5746,050 | 48,892 | ,000 | |

Lower-bound | 5746,050 | 1,000 | 5746,050 | 48,892 | ,000 | |

Measurement * Group | Sphericity Assumed | 451,250 | 1 | 451,250 | 3,840 | ,086 |

Greenhouse-Geisser | 451,250 | 1,000 | 451,250 | 3,840 | ,086 | |

Huynh-Feldt | 451,250 | 1,000 | 451,250 | 3,840 | ,086 | |

Lower-bound | 451,250 | 1,000 | 451,250 | 3,840 | ,086 | |

Error(Measurement) | Sphericity Assumed | 940,200 | 8 | 117,525 | ||

Greenhouse-Geisser | 940,200 | 8,000 | 117,525 | |||

Huynh-Feldt | 940,200 | 8,000 | 117,525 | |||

Lower-bound | 940,200 | 8,000 | 117,525 |

**Measure: MEASURE_1**

In Table 2, measurement section shows whether there is a meaningful difference between the pretest and posttest results of students. In this study, there is a meaningful differentiation between all the ten students’ pre test average and post test average. (F=48,892, p<0.01).

Again, in Table 2, the bold line shows whether there is a meaningful difference between the five students’ achievement studying with their parents and the students’ achievement studying individually. In short, it shows whether the group has meaningful influence on the measurement. In this application, the influence of the group on the measurement is not meaningful. (F=3,840, p=0.086>0.01).

**Table 3.**

**Tests of Within-Subjects Contrasts**

**Measure: MEASURE_1**

Source | Measurement | Type III Sum of Squares | df | Mean Square | F | Sig. |

Measurement | Linear | 5746,050 | 1 | 5746,050 | 48,892 | ,000 |

Measurement * Group | Linear | 451,250 | 1 | 451,250 | 3,840 | ,086 |

Error(Measurement) | Linear | 940,200 | 8 | 117,525 |

Table 4, shows the graph of the achievement process of the two groups of students SP (Students with Parents) and Student) pointing out the difference of the pretest and posttest results. As seen in the graph, in the first measurement students who studied individually perform better than the students who studied with their parents. In the second measurement, the situation has göne the students who studied with parents favour.

**Sample of the Activities and their analysis**

- Activity: Sharing activity

When the 11 cake are shared with 8 children equally how much cake does each child have?

The aim of this activity is to relate the concept of division with the concept of fraction with different representation by using creative thinking.

In this activity, the students tried to demonstrate various solutions. These are;

- Division
- Awareness of fraction
- Visualization-using model or models

For the students who studied with their parents; two of these students both did visualization and fraction; one of these students shows the share by using just fraction statement and make relation with division by emphasizing the sharing. One of these students just did a division operation and did not state any expression. One of these students used models, division and fraction.

**The third and the fifth students mentioned above almost achieved this goal**

Two of the students who studied individually could not do the activity. One of the students just did a division without any expression. One of the students tried to make visualization and relate to it with a division representation and state it as a fraction. The last student of this group made shares by visualization and division but not represent it with fraction.

The last two students of the second group who studied individually, tried to make connections between the two concepts division and fraction by using different strategies.

**Activity: Placing the figs**

Aybüke has 100 figs. She wants to place her figs with packets which have place for 8 figs.

In order to place all the figs, how many packet at least should Aybüke have?

The aim of this activity is to think flexible and fluent in order to make relations with daily life and reach the solution.

In other words, reasoning and creative thinking are the basis for this activity.

For the students who studied with their parents;

Two of the students drawed 12 packets consisting of 8 figs and draw 1 packet consisting 4 figs and she reach the solution that Aybüke needs 13 packets to place her figs. One of the students similar to the first one drawn 12 packets consisting 8 figs in each of them. However, she did not place the rest of the 4 figs in any packet. She reached the solution that Aybüke needs 12 packets to place her figs. One of the students just did the division to find the solution. She states that to place the rest of the figs she needs 4 figs. This statement shows a different thinking way for the activity. The last student made division and found 12 packets that Aybüke needs and she also thought about the rest and place the rest in a packet. Totally, she reached the solution by reasoning and thinking flexibly. However, this student did not use drawing.

In this activity, these students performance shows that they were able to reason and think flexible and fluently.

**For the students who studied individually,**

Two of these students made division to find the right solution. One of them states that Aybüke needs 12 packets without not mentioning the rest of the 4 figs. One of them wrote the answer as 12.5 by making division. This situation shows that she could not make reasoning and does not have flexible thinking. One of these students did this activity by drawing models, but he also could not think of the last packet involving 4 figs. Just one of these students was able to reason and think flexibly. He drawed packets for all the figs and place the rest of the 4 figs in a packet.

This activity was designed based on the TIMSS study. The activity is the most selective activity during the study. The reason is that the students who studied with their parents performed remarkably well compared to the other students who studied individually. In this result, the nature of the activity play an important role. The activity is a good example to see the mathematics problems as a daily life problem. Also, different thinking strategies can be represented. Based on the discussions with the parents, the interaction of the student and the parent is the most crucial factor in this result. The students were able to relate the activity with the real life when they study with their parents.

On the other hand, the other students who studied individually tended to solve the problem by making just operation mostly without reasoning and thinking flexibly and fluently.

**Analysis of the Diaries for Parents**

Five parents were given consisting of five questions for each of the activities. The questions written on diaries are;

- Do you feel that your child enjoy studying with you?
- Did you observe that your child produce original ideas while studying on this activity? How did your interaction with your child affect this situation?
- Did you observe that your child produce ideas by thinking flexibly? How did your interaction with your child affect this situation?
- Did you observe that your child produce ideas by thinking fluently? How did your interaction with your child affect this situation?
- After this study, do you think that your child’s awareness and curiosity changed? How did this change happen?

*Activity:*

- All of the parents answered this question positively.
- One of the parents states that his daughter tried to solve the problem with different strategies, such as, drawing models. An other parent just stated she talked to her daughter about the activity. One of the parents also mentioned her daughter’s imagination while doing the activity. One of the parents stated that her daughter produce original ideas based on real life. One of the parents stated that her daughter produced some original ideas during the activity.
- Four of the parents stated that they observed their children thinking flexibly while doing the activity.
- All of the parents stated that they observed their children while thinking fluently.
- Four of the parents stated that they observed awareness and curiosity towards mathematics on their children. One of them stated that his daughter began to correct her mistakes. One of them stated that her daughter enjoyed studying with her mother. She stated that her daughter began to ask interesting question to her mother.

*2. Activity*

- All the parents answered this question positively.
- Two of the parents stated that their daughters produce original ideas during the activity. One of them stated that her daughter can produce different ideas after the interaction with her mother. The rest of the parents stated that they did not observe originality in their children’s thinking.
- All the parents stated that their children were able think flexibly in this activity.
- Four of the parents stated that their children thought fluently.
- One of the parents stated that her daughter began tried to solve mathematics problems with different strategies. One of the parents stated that her daughter began to be self confident in mathematics. Also studying with her mother made her more comfortable in producing ideas. One of the parents stated that her daughter began to make operations fluently and drawing for the problems.

**Analysis of Focus Group Discussion with Students Studied with Their Parents**

In the discussion, the students were asked about the study whether they enjoy or not by studying with their parents.

Four of the students have positive attitude towards this type of study. One of them stated that she get easily angry with someone while studying. Because of this reason she was not sure about the positive impact of the study for her.

The four students stated that they enjoyed studying with their parents. This was the first time for them taking part in such a study. They stated that they began to feel comfortable to talk about mathematics. They also stated that studying with parents provided them to think differently; such as, making relationships with real life and mathematics.

**Discussion and Conclusion**

In this study mathematical creativity of the two groups of students one of them involves students studying with parents and the other involves students studying individually. The importance of the mathematical creativity is reported in many of the researchers studies (Ervynck, 1991; Sriraman, 2005; Türkan, 2010).

In this study, five sixth grade students were required to study with their parents on five activities which were designed based on the TIMSS and the Turkish Mathematics Curriculum. Five sixth grade students were required to study individually. Each of the activities were required to study in a week day. At the beginning of the study, pre test was applied to all the ten students. The average of the students studying with their parents was below the average while the average of the students studying individually was above the average. However, in the post test the average of the first group (studying with their parents) was found above the average while the average of the second group (studying individually) below the average.

Two of the activities were given in the study as samples. Each of them were explained in terms of the performance of each student. Additionally, the answers of the parents to the diary questions were also presented as samples. The results of the two activities and the parents’ observations about their students’ performance shows the importance of studying with parents from different views.

This study also shows the emphasis of procedural knowledge for students rather than conceptual knowledge as the approach for problem solving. Thus, students have found weak in terms of creating ideas, thinking deeply and flexibly in terms of approaching the problems. Another issue was the relationship between real life and the mathematics as a subject. Because the students see the mathematics as a separate subject from the everyday life, they cannot relate some of the activities which requires attention as the 4. Activity in this study. In this sense, studying with parents provides them to be aware of the place of mathematics in everyday experiences. Therefore, the students prefer to study with their parents. As Sriraman (2004) states the importance of the social interaction for the creativity, in this study studying with parents provided students to began to think creatively. On the other hand, in this study, Torrance (1974)’s idea of the components of creativity as originality, flexibility, fluency have been used as the focus of the study.

As a result, studying with parents provided students to be more aware of the importance of mathematics. On the other hand, the parents have also began to be aware of the importance of the studying with their children. They have learned the significance of the mathematical creativity by observing their children’s thought processes during the activities. However, all the parents state that they were not enough to help their child while studying mathematics in general. They stated that they need this kind of studies to help their children and provide their students full potential.

**Limitations of the Study**

First of all peers of the students and the teachers were also wanted to be involved in the study in addition to the parents. However, because of the time limitation the planned process could not be carried out. Thus, only parents were able to involve in the study. On the other hand, the study wanted to be video recorded for each of the activities for students studying with their parents. However, time constraints and the some inconvenience the parents could not do video recording for the study. The other limitation was the time constraint in terms of selecting the students and reaching deeper results.

**REFERENCES**

Aiken, J. R. and Lewis, R. (1977). Mathematics as a creative language. The Arithmetic Teacher, Vol. 24, No. 3 (MARCH 1977), pp. 251-255. Accessed through http://www.jstor.org/stable/41189258

Andzans, A. and Koichu, B. (undated). Mathematical creativity and giftedness in out-of-school activities. Civil, M. (2008). Retrieved from http://math.arizona.edu/~civil/ on October 17th 2015

Civil, M. (2001). Mathematics for parents: Issues and pedagogy and content. In Proceedings of the Eighth International Conference of Adults Learning Maths – A Research Forum.Roskilde University, Denmark, June 2001.

Civil, M. (1998). Parents as resources for mathematical instruction. Proceedings of the Adult Learning Mathematics-A research Forum’, Utrecht, The Netherlands, July 1998, van Groenestijn, M. & Coben, D. (Eds.)

Civil, M. (1998). Bridging in school mathematics and out of school mathematics: A reflection. Presentation at the Annual Meeting of AERA in San Diego, April 1998.

Cohen, L., Manion, L. and Morrison, K. (2011). Research Methods in Education. Routledge: London and New York

Ervynck, G. (1991). Mathematical creativity advanced mathematical thinking (pp. 42-53), Springer Mann, E. (2009). The search for mathematical creativity: Identifying creative

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Münz, M. (2014). Non-canonical solutions in children–Adult Interactions – A case study of the emergence of mathematical creativity. U. Kortenkamp et al. (eds.), Early Mathematics Learning, Springer Science+Business Media New York 2014.

Singh, B. (1987). The development of tests to measure mathematical creativity. International Journal of Mathematical Education in Science and Technology Volume 18, Issue 2, pp. 181-186, 1987 Sriraman, B. (2004). The characteristics of mathematical creativity. Mathematics Educator, 14(1), 19-34 Torrance, E. P. (1974). Torrance tests of creative thinking. Bensenville, IL: Scholastic Testing Service.