Effects of mathematical modelling based project production and management program on gifted students’ mathematical modelling and reflective thinking for real-life problem solving

Abstract

This study aimed to examine the relationship between mathematical modelling and reflective thinking and to examine the effects of employing mathematical modelling processes on perception of mathematical modelling competencies and reflective thinking skills for real-life problem solving of gifted students participating in Project Production and Management Program at Turkish Science and Art Centers. Correlational path analysis of 300 students data revealed that Mathematical modelling is a significant predictor of reflective thinking skills, and the constructed model was found to be of a good fit with excellent path coefficients. Experimental design with pre-post-maintenance-tests with comparison group was employed with 60 students purposefully selected chosen from the above regression study. Reflective thinking and mathematical modelling levels of experimental group students who employed mathematical modelling processes for Project Production and Management increased significantly more from pre- to post-tests, and the significant difference was found to be maintained.

Keywords

Gifted and talented students mathematically gifted mathematical modelling reflective thinking math project production project production program

Introduction

Even though gifted students might have potential to produce creative solutions for real-life problems, it cannot be assumed that all gifted students will do (Reis & Renzulli, 1991). Many gifted students may end-up synthesizing existing literature on the problem or produce typical solutions which were already presented by others. Not all their talents, skills, and knowledge may be transformed into original products and projects (Bishop, 2000; Özbek & Dağyar, 2022).

Unprecedented challenges in the 21st century call for innovative solutions. UNESCO, OECD, and EU have published documents outlining educational goals for the 21st century emphasizing creativity as an important goal. Gifted students in science can create new ideas and products that can be used for solving the problems and challenges of our society. Gifted students should be provided with opportunities to learn and encouraged to produce original solutions and products. For some gifted students whose talents could not be transformed into original products and solutions, more innovative approaches for facilitating production of original projects might be needed. Creative productivity in STEM domain requires motivation, general knowledge and skills and knowledge and skills in STEM domain (Cho, 2003, 2007; Sternberg, 1988; Urban, 2003) These components have been found to be critical predictor of creative productivity of scientifically talented students (Ahn & Cho, 2021; Kim et al., 2021) at the secondary schools. Reflective thinking (Lia, 2020; Özbek & Köse, 2019; Sternberg, 1988; Urban, 2003) is one of the main components of general knowledge and skills needed for creative productivity, meanwhile mathematical modelling is another main component of knowledge and skills in STEM domain (Blum & Niss, 1991; Cho, 2003, 2007; Özbek & Köse, 2022; Sternberg, 1988; Urban, 2003). There have been studies on the relationship between creative productivity and academic achievements in general. However few studies examined effects of more specific strategies such as mathematical modelling and reflective thinking on creative productivity. Mathematical modelling and reflective thinking can be great problem-solving strategies for gifted students can and should apply to produce original project products. However, previous studies focused mostly on the level of mathematical modelling and reflective thinking, but not enough on the effects of educational practices for improving gifted students’ mathematical modelling and reflective thinking.

In Türkiye, gifted students are admitted to Science and Art Centers (SACs) through identification process of recommendation, and group abilities testing. SACs provide educational programs in five stages and its final stage provides project production and management program (PPMP). PPMP is project-based, interdisciplinary, enriched and differentiated program, and is aimed for gifted students to produce original projects (MONE, 2019). In PPMP, students are expected to produce original projects in the field of their talent fields and develop new ideas, produce utility models or patents as a result of projects. Students carry out projects individually or in groups. Most gifted students at SACs experienced difficulties in applying their talents for project production. Some students had ideas for solving real-life problems, but could not transform the ideas into original products. Producing original projects, products and solutions for real-life problems might be challenging for most gifted youths at the SACs (Özbek & Dağyar, 2022).

Review of Literature

Mathematical Modelling is a process of creating models and explaining and solving real-life problems mathematically (Blum, 1993, 1996; Blum & Niss, 1991; Erbaş et al., 2014; Maaß, 2006). It simplifies and expresses real-life problems mathematically to solve, interpret, and validate the real-life problems through developing interdisciplinary models (Borromeo Ferri, 2006; Maaß, 2006; Tekin Dede, 2017). This study is grounded on the theoretical framework (Borromeo Ferri, 2006) of mathematical modelling cycle (see Figure 1) and competencies based on multi-steps of the modelling process (Özbek & Köse, 2022). It starts project with the ‘identifying the real-life problem’, continues with the ‘understanding and simplifying the problem’, ‘mathematizing’, ‘working mathematically’ and culminates project with ‘interpretation and validation’. “Identifying real-life problems” is about how real-life problems are identified to produce solutions using mathematical knowledge. “Understanding and simplifying” is about how appropriate strategies are identified. “Mathematizing” is about how graphs and tables are represented to develop models to real-life situations created. “Working mathematically” is about how problems encountered in real life are associated with mathematics. Finally, “interpretation and validation” is about how applicability of the solution in real life is demonstrated.

Figure 1.

Mathematical modelling cycle.

Mathematical Modeling Cycle

Reflective thinking is a kind of metacognitive processes and an important skill for problem solving. Reflective thinking is aimed at removing or improving the sense of discomfort from experiences, actions, or practices. Cyclically repeated questioning is taken as a basis of problem solving (Bolton & Delderfield, 2018; Boyd & Fales, 1983; Brookfield, 2017; Dewey, 1933; Johns, 2017; Kolb, 2014; Lia, 2020; Mezirow, 2006). Reflective thinking skills include questioning, reasoning and evaluation. It questions all aspects of problem solving such as what is done and what will be done. It rationalizes reasons for doing them, and evaluates both sufficiency and insufficiency of solutions for improving the problem situation (Dewey, 1933; Kızılkaya & Aşkar, 2009; Rodgers, 2002; Schön, 1983).

Reviewing literature on the theoretical framework both reflective thinking in the problem-solving context (Kızılkaya & Aşkar, 2009) and mathematical modelling (Borromeo Ferri, 2006; Özbek & Köse, 2022) revealed they both share a similar cyclical structured thinking which allows to progress practically along their stages and also questioning what is done and why. They seem to share a related structure in terms of being a solution- generating producing process which starts with real-life problem. However, relationship between them has not been studied much and also effects of educational practices for improving both of them have not been studied much.

In previous studies, reflection skills (Gözeten, 2017; Güneş & Aybek, 2018; Hendriana et al., 2019; Saygılı & Atahan, 2014) and modelling competencies of gifted students have been examined (Blomhøj & Højgaard Jensen, 2003; Haines & Crouch, 2007; Hidayat et al., 2018; Kaiser & Grünewald, 2015). However, few study was conducted on mathematical modelling and reflective thinking for real-life problem solving in the context of gifted students’ project production and management. Findings of this study may contribute to the field since this study investigates the effects of mathematical modelling on gifted students' reflective thinking for real-life problem solving.

Reflective thinking skills of gifted students towards problem solving has been studied and found that gifted students have good reflective thinking skills (Gözeten, 2017; Güneş & Aybek, 2018; Saygılı & Atahan 2014). These skills can be developed through project-based learning programs (Hendriana et al., 2019). Studies on mathematical modelling competencies in each step or combined steps of mathematical modelling process (Blomhøj & Højgaard Jensen, 2003; Haines & Crouch, 2007; Kaiser & Grünewald, 2015) have revealed that students face more struggles working with three stages of MM (interpretation, validation and mathematizing) than other mathematical modelling stages, such as understanding the problem, simplifying the problem, working mathematically, and interpreting (Biccard & Wessels, 2011; Bloom, 1976; Blum, 2011; Maaß, 2006). It is observed that identifying current skills and competency levels of mathematical modelling of gifted students and the relationship between them has provided insights for the educational needs and decisions to facilitate their development (Kaiser, 2020; Zedan & Bitar, 2017). It has been found that a project producing process develops mathematical skills, abilities, and competencies of gifted students when the process was developed based on mathematical modelling (Yu & Yun, 2017). However, there is not enough experimental studies on how to facilitate the development of competencies and skills of gifted students in mathematical modelling and reflective thinking (Pierce et al., 2011).

Study I

Research design

To answer for the first research question “Is mathematical modelling a significant predictor of reflective thinking in problem solving for gifted students?“, a linear regression analysis was conducted. It is because linear regression analysis provides amount of variances in the criterion variable explained by a predictor variable.

Participants

300 participants were selected from the Science and Art Centres (SACs) of 39 cities located in seven geographical regions throughout Türkiye. From each Center, students who were participating in the Project Product Management Program (PPMP) in the field of mathematics were invited voluntarily to participate in the study. Numbers of students from each city and region vary since participation was voluntary and some Centers have not implemented PPMP, as of yet (See Table 1).

Table 1.

Gender and region of participants.

Characteristics	Category	Participants
		f	%
Gender	Female	115	38.3
	Male	185	61.7
Region	Eastern anatolia	25	8.3
	Central anatolia	47	15.6
	Black sea	34	11.3
	Mediterranean	130	43.3
	Aegean	39	13.0
	Marmara	18	6.0
	South-eastern anatolia	7	2.3
	Total	300	100.0

Instruments

Mathematical modelling competences scale (MMCS)

The scale was developed by Özbek & Köse, 2022 with 486 gifted students and has a structure of 5-factors with 31 items. Five point Likert scale survey ranged from ‘Strongly Agree (5)’, ‘Agree (4)’, ‘Moderately Agree (3)’, ‘Disagree (2)’, and ‘Strongly Disagree (1)’, and has no items which required reverse coding. Examples of each of five sub-factors of the scale are as follows: identifying the real-life problem (IP) (1, 2, 3, e.g., “I can identify real-life problems that can be solved using mathematical knowledge”), understanding and simplifying the problem (US) (8, 10–14, 16, e.g., “I can explain the relationships between variables in real-life problems.“, mathematizing (M) (17-19, e.g., “I can create graphics to develop models that represent real-life situations.“), working mathematically (WM) (4-7, 9, 15, e.g., “I can apply mathematical models of real life problems to find mathematical solutions” “I can get mathematical results by solving models that represent real life problems.“), and interpretation and validation (IV) (20-31, e.g., “I can show how applicable the solution of the mathematical model is in real-life.“). The internal consistency coefficients, Cronbach’s alpha, were α = 0.958 for the whole scale and 0.811, 0.900, 0.883, 0.820, and 0.927 respectively for each sub-factor of IP, US, M, WM, and IV respectively. Fit indices ( $χ^{2 / d f}$ = 2.00, GFI = 0.90, RMSEA = 0.075, SRMR = 0.063, IFI = 0.97, NNFI = 0.97, CFI = 0.97, NFI = 0.94, PNFI = 0.86) demonstrated that the instrument has a valid structure (Özbek & Köse, 2022). The Content Validity Index, which represents the mean value of the items' Content Validity Ratio, was found to be very high with 0.95 (Özbek & Köse, 2022).

Previous studies examined interpretation and validation of mathematical modelling competencies as two separate sub-factors (Blomhøj & Højgaard Jensen, 2003; Haines & Crouch, 2007; Kaiser & Grünewald, 2015). However, this study combined ‘interpretation and validation’ as one sub-factor based on the confirmatory factor analysis results.

Reflective thinking towards problem solving skill scale (RTTPSSS)

This scale was developed by Kızılkaya & Aşkar (2009) with 339 students has a 5-factor structure with 14 items. The scale is five point Likert scale ranging from ‘Never (1)’, ‘Rarely (2)’, ‘Sometimes (3)’, ‘Often (4)’, and ‘Always (5)’, and has no item which requires reverse coding. The three sub-factors of the scale include Questioning (1, 3, 7, 9, 13, e.g., “I question regarding my solutions to the problem.“), Reasoning (5, 8, 11, 12, e.g., “I find out relationship among results by thinking about the logics of my operations.“), and Evaluation (2, 4, 6, 10, 14, e.g., “I check my operations after solving the problems.“). The internal consistency coefficients, Cronbach’s alpha, were α = 0.83 for the overall scale, and 0.73 for questioning, 0.71 for reasoning, and 0.69 for evaluation. The fit indices ( $χ^{2 / d f}$ = 2.69, GFI = 0.92, AGFI = 0.89, NNFI = 0.93, CFI = 0.95, RMSR = 0.08, RMSEA = 0.071) demonstrated that the instrument has a valid structure (Kızılkaya & Aşkar, 2009).

The validity, reliability and confirmatory factor analysis of the RTTPSSS was conducted with participants of this study. The calculated t values were significant at the level of p < 0.01 for all items, and the standardized factor loading values ranged from 0.46 to 0.72. The fit indices [ $χ^{2}$ = 217.30, df = 74, p<.001, $χ^{2 / d f}$ = 2.93], GFI = 0.91, IFI = 0.96, NFI = 0.94, NNFI = 0.95 CFI = 0.96, AGFI = 0.87, SRMR = 0.053, RMSR = 0.054 and RMSEA = 0.08] demonstrated that the scale had a valid structure for this sample (Kline, 2005; Sümer, 2000). The internal consistency coefficients, Cronbach’s alpha, were 0.881 for the overall scale, 0.726 for questioning, 0.721 for reasoning, and 0.760 for evaluation. It was concluded that the scale was reliable for this sample as well (Özdamar, 2004).

Statistical analyses

Descriptive statistics and Pearson’s correlational path analyses were conducted to examine whether mathematical modelling is a significant predictor of reflective thinking in problem solving of gifted students. Data set had no missing values. As a result of outlier analysis, 10 individuals were excluded from the data set. Parametric techniques were used as a result of examining the normality of the total scores and the assumptions. Path analysis as a part of structural equation analysis was used to evaluate explanatory theoretical models in a global way, since regression analysis does not allow generating causal or global hypotheses. Fit indices (residual-based as SRMR, GFI, AGFI; independence-model-based as NFI, NNFI, CFI; root mean square error of approximation as RMSEA) were examined to evaluate the model fit for the Path analysis which is chosen to model the causal relationships of data (Çokluk et al., 2010).

Findings

Descriptive statistics of MM and RT scores of participants

Participants’ scores in the sub-dimensions of MM were ranked from the highest to the lowest in the order of IP, US, WM, IV and M respectively (see Table 2). In addition, their scores in the sub-dimensions of RT were ranked from the highest to the lowest in the order of ‘reasoning’, ‘questioning’, and ‘evaluation’ respectively.

Table 2.

Mean and SD of Sub-Factors of MM and RT of Participants.

	MM							RT
	Total	IP	US	M	WM	IV	Total	Questioning	Reasoning	Evaluation
Mean	3.96	4.28	4.10	3.71	3.98	3.85	4.08	4.16	4.17	3.93
SD	0.63	0.60	0.70	0.94	0.70	0.84	0.62	0.66	0.69	0.77

Significant predictor between MM and RT scores of participants

A positive relationship was found to be significant at the level of p < .01 between MM and RT scores (see Table 3).

Table 3.

Pearson’s correlational coefficients among sub-factors of MM and RT.

R
Variables	1	2	3	4	5	6	7	8	9	10
1.MM total		.735^**	.807^**	.657^**	.790^**	.901^**	.754^**	.548^**	.556^**	.503^**
2.MM_IP			.631^**	.346^**	.677^**	.460^**	.529^**	.485^**	.447^**	.464^**
3.MM_US				.459^**	.657^**	.554^**	.562^**	.561^**	.482^**	.450^**
4.MM_M					.387^**	.540^**	.394^**	.328^**	.378^**	.342^**
5.MM_WM						.558^**	.519^**	.491^**	.443^**	.440^**
6.MM_IV							.470^**	.401^**	.468^**	.390^**
7.RT total								.892^**	.858^**	.892^**
8.RT_ questioning									.700^**	.666^**
9.RT_ reasoning										.633^**
10.RT_ evaluation
**p < .01

Participants' total RT scores were positively and moderately correlated with the scores of three sub-factors WM, US, and IP of MM. Five sub-factors of MM (IP, US, M, WM, and IV) and three sub-dimensions of RT (questioning, reasoning and evaluation) were included in the model for path analyses. The t values are shown (see Figure 2) with the path diagram and the standardized values of the model are illustrated (see Figure 3). It was determined that the t values were significant for the structural model (see Figure 2). It was revealed that the p value in relation to the $χ_{}^{2}$ value ( $χ_{(18)}^{2}$ ₌ 52.74) was significant (p < .01) (see Figure 3).

Figure 2.

The t Values of the Model.

Figure 3.

The path diagram.

The fit was very good as indicated in a result of evaluating the model according to the ratio of

χ^{2}

to the degrees of freedom (

χ^{2 / d f}

) (see Table 4). GFI and AGFI values are very sensitive to sample size and give better values as the sample size increases (Tabachnick & Fidell, 2001). When the path diagram in Figure 3 is examined based on the structural model, it can be asserted that MM achieved good path coefficients in explaining RT, and the constructed model was validated. In the model, the path coefficient/total effect of the MM variable on RT was calculated as d=.74, which is moderate (0.20-0.80) according to Cohen (1988). In addition, the MM explained a total of 49% of the change in reflective thinking towards problem-solving (R² = .49). In other words, 49% of students’ reflective thinking needed for problem solving can be explained by their mathematical modelling. So, if students are good at mathematical modelling, then it is possible that they will also be good at reflective thinking which is needed for problem solving.

Table 4.

Indices of the model fit for the path analysis.

Fit Indices	Value	Fit
χ²/Sd	2.93	Perfect	≤3 perfect (Kline, 2005)
RMSEA	0.08	Good	≤0.08 good (Sümer, 2000)
GFI	0.96	Perfect	≥0.95 perfect (Hooper et al., 2008; Sümer, 2000)
AGFI	0.92	Good	≥0.90 good (Hooper et al., 2008; Sümer, 2000)
SRMR	0.035	Perfect	≤.05 perfect (Brown, 2015; Byrne, 1994)
IFI	0.98	Perfect	≥0.90 good (Tabachnick & Fidell, 2001)
NNFI	0.98	Perfect	≥0.95 perfect (Hu and Bentler, 1999)
CFI	0.98	Perfect

Study II

Research design

To answer for the second research question, “How does employment of mathematical modelling processes effect gifted students' perception of their mathematical modelling competencies and reflective thinking skills for real-life problem solving?“, quasi-experimental study was conducted by rigorously applying MM for experimental group students in Project Producing and Management Program (PPMP) and comparing experimental and comparison groups’ pre-post-maintenance (maint) test scores on the same instruments for Study I.

Participants

Participants were selected purposefully from the participants of the Study I. Only those who were ready to participate in the PPMP in the field of mathematics at the same centre were selected. PPMP is the last stage and most challenging program for gifted students at the SACs (MONE, 2019). Students of various grades participated in PPMP because of individual differences in learning progress from beginning (Orientation) stage to final (PPMP) stages at the SACs. Stratified random sampling was used to assign participants to experimental groups (EG) and comparison groups (CG) according to their demographic characteristics (see Table 5).

Table 5.

Participants’ gender, school type, grade level, age, and years of experience at the SAC.

Characteristics	Category	Experiment		Comparison
		f	%	f	%
Gender	Male	22	73.3	20	66.6
Gender	Female	8	26.6	10	33.3
Grade level	9	16	53.3	16	53.3
	10	10	33.3	8	26.6
	11	2	6.6	5	16.6
	12	2	6.6	1	3.3
Age	14	9	30.0	12	40.0
	15	14	46.6	11	36.6
	16	3	10.0	4	13.3
	17	2	6.6	1	3.3
	18	2	6.6	2	6.6
Years of experience at the SACs	4	2	6.6	0	0.0
Years of experience at the SACs	5	2	6.6	3	10.0
	6	4	13.3	6	20.0
	7	11	36.6	11	36.6
	8	6	20.0	3	10.0
	9	3	10.0	2	6.6
	10	2	6.6	3	10.0
	11	0	0.0	2	6.6
	Total	30	100.0	30	100.0

Data Collection Procedure

All students participating in PPMP were required to create and conduct original mathematics projects (MONE, 2019). Intervention group students started their PPMP by identifying a real-life problem by themselves. The teaching process was structured cyclically taking into consideration of the theoretical structure of reflective thinking as well as modelling. In every weekly modelling session, students of EG were required to apply mathematical modelling rigorously. Each weekly mathematical modelling session included activities such as peer presentations; keeping logs; and filling out modelling form following the cycle of trigger, frame decisions, reframe decisions, and plan for future action within itself. They were assisted in using textbooks on mathematics and online resources, met and received feedbacks from field experts they needed, collaboratively worked and shared feedbacks with their peers, and received feedbacks from teachers (see Table 6). In every weekly mathematics modelling session, participants are asked to present to peers what they accomplished and planned to do next for 5 minutes, in order to increase motivation and to ensure planned actions. In comparison group, students participated in PPMP of the SACs, in which students’ goal was to produce projects. Compared with EG students, CG students were not required to apply MM rigorously to produce their projects. Other than that, CG students were also provided with resources, experts, peers, and teachers whenever needed to produce projects as a requirement of the PPMP’s regular project producing process.

Table 6.

Project production activities and data collection procedure.

Activities	Weeks	Components Included in the Process of Project Production
		Experiment Group	Comparison Group
Orientation and planning	−2	-Introduction of data collection tools by reading aloud, and explanation of contents; securing participants’ consent forms
Pre-test administration	−1	Administration of pre-test
Weekly project production sessions	1–4	Identifying a real-life problem	Introductory review activities of PPMP
	5–7	Understanding the problem	Determining each student’s project topic
	8–10	Simplifying (converting the assumptions to be identified into a mathematical problem by using concepts)	Specifying the project problem situation
	11–15	Mathematizing (using appropriate methods to come up with a solution)	Working on the projects (using methods of their choice)
	16–22	Working mathematically (finding solutions by using and applying mathematical methods)	Continue working on the project (achieving original results, converting the results into product)
	23–26	Interpretation (interpreting the mathematical model, the solution of the model and the results in the real-life situation)	Finalizing (confirming the results, reporting project)
	27–30	Validation (Validating the assumptions, the mathematical models built, solution of the model, and the whole process)	Disseminating (sharing the results, through project-based competitions and with patent)
Post-test	31	Post-test administration
Maint-test	38	Maint-test administration

Data analysis

To examine the impact of running different approaches for PPMP on MM and reflective thinking, two-way ANOVA for mixed measures and Bonferroni multiple comparison tests were conducted.

Findings

No significant difference in the pre-test scores between EG and CG students was found (p > 0.05) in both MM and RT levels (see Table 7). These scores were used as baseline to determine the impact of the experiment on their post-test and maintenance test.

Table 7.

Mean and SD of MM and RT in pre-test, post-test and maintenance-test.

		Group				t	df	p
		Exp (n = 30)		Com (n = 30)		t	df	p
		M	(SD)	M	(SD)
MM	Pre	109.73	(18.059)	112.37	(25.587)	.461	58	.647^*
	Post	149.16	(3.343)	121.60	(22.245)
	Maint	148.73	(3.289)	121.60	(21.906)
RT	Pre	51.37	(7.915)	50.57	(8.649)	.374	58	.710^*
	Post	65.10	(2.202)	52.03	(7.712)
	Maint	63.80	(2.171)	51.70	(6.649)

Note. *p < .05.

There was an increase in the MM and RT mean scores in both groups from the pre-test to post-test and to maintenance -test. This increase of EG students were found to be significantly higher than that of CG students. It was found that the MM mean scores in the EG decreased from the post-test to the maintenance-test while the scores remained constant in the CG. In addition, RT mean scores in both groups decreased from the post-test to the maintenance-test (see Table 7).

As the results of the two-way mixed ANOVA, the MM and RT levels of the participants in both groups differed significantly from pre-test to post-test (see Table 8). The effect sizes (η²) value of the difference between EG and CG calculated for MM (η² = .426) and for RT (η² = .379) were considered a large effect size (Cohen, 1988).

Table 8.

Two-way ANOVA results between experimental and comparison groups and among pre-, post-, and maintenance tests.

	MM					RT
Source of Variance	SS	sd	MS	F	η ²	SS	sd	MS	F	η ²
Between subjects	39,282.13	59				7221.39	59
Group (EG/CG)	13,554.68	1	13,554.68	30.5^***	.35	3371.33	1	3371.34	50.8^***	.47
Error	25,729.44	58	443.61			3850.05	58	66.38
Within subjects	64,088.66	120				6905.33	120
Measurement (pre/Post/Maintenance)	23,475.43	2	11,737.71	43.05^***	.42	2088.81	2	1044.40	35.4^***	.38
Group*Measurement	8991.41	2	4495.70	16.49^***	.22	1395.47	2	697.73	23.6^***	.29
Error	31,621.82	116	272.60			3421.04	116	29.49
Total	103,370.79	179				14,126.73	179

Note.

^***p < .001.

Bonferroni multiple comparison test was conducted and it was found that, for EG students, the MM mean scores differed significantly by increasing 39 points and RT mean scores differed significantly by increasing 11 points from the pre-implementation to the post-implementation and that the improvement was maintained despite the decrease over time. For CG students, change in MM levels from pre- to post-tests was found to be insignificant, even though there was an increase of nine points, and it was maintained from post- to maintenance tests. It was also found that the RT levels of the participants in the CG differed significantly by increasing one point and that this increase was maintained. In other words, there was a smaller increase in the mean score of RT of the CG compared to the EG, but this increase was not significant. Both MM and RT levels that increased in both groups after the experiment, decreased slightly over time. Despite the decrease in both groups, most of the improvement experienced in the EG was maintained compared to the one in the CG.

Discussion and Conclusion

It has been concluded that mathematical modelling is a significant predictor of reflective thinking in problem solving for gifted students and that gifted students’ perception of their abilities to perform the stages of the mathematical modelling process can be ordered from the highest to the lowest as IP, US, WM, IV and M respectively. Their perception of competencies to perform the stages of RT can be ordered from the highest to the lowest as reasoning, questioning and evaluation. Similar to the results in previous studies (Gözeten, 2017; Güneş & Aybek, 2018; Hendriana et al., 2019; Saygılı & Atahan 2014), this study also found that the reflective thinking skills of gifted students are at a good level and these skills can be improved by applying mathematical modelling processes to project production and management. These results demonstrate different levels of skills and competencies between mathematical modelling and reflective thinking skills in relation with real-life problem solving. This information can be useful for determining the needs of education for developing these skills and competencies of each gifted individual student (Kaiser, 2020; Zedan & Bitar, 2017). The research results have shown that gifted students perceive their competencies to perform mathematical modelling process in the similar order as general education students do (Biccard & Wessels, 2011; Blum, 2011; Maaß, 2006)

It was revealed that EG students’ perception of their reflective thinking and mathematical modelling increased significantly more from pre- to post-tests due to an impact of the rigorous application of mathematical modelling processes to the project producing process. In comparison, CG students did not show significant increase or decrease throughout from pre-, post-, to maintenance-tests. Even though EG students’ reflective thinking and mathematical modelling levels from post- to maintenance-test decreased at p < .05 level, the decrease in actual score was as minimal as .433. Considering the much longer temporal distance of 8 weeks between post-test and maintenance-test compared to typical two to 3 weeks of duration in other studies, it can be said the effects of applying mathematical modelling were significant The results of this research support that rigorous application of mathematical modelling processes to project producing process improved gifted students’ perception of their reflective thinking skills (Hendriana et al., 2019; Thahir et al., 2019) and mathematical modelling (Yu & Yun, 2017). This study found that students’ mathematical modelling and reflective thinking skills can be developed. This development can be facilitated by applying appropriate educational approaches (Altıntaş & Özdemir, 2015; Callahan et al., 2015; Cho et al., 2015; Deringöl & Davaslıgil, 2020; Gavin et al., 2007; Kim et al., 2016; Little et al., 2018; Manuel & Freiman, 2017; Özçelik, 2018; Özdemir & Gürlen, 2019; Özyaprak & Davaslıgil, 2015; Sengil-Akar & Yetkin-Ozdemir, 2022).

Since the research was conducted on a sample of gifted students who are at the secondary and high school levels in the SACs in Türkiye, this limitation should be taken into account for any future research. In the future, studies to examine the effects of gifted students' demographic factors on their performance in each stage of the mathematical modelling and reflecting thinking might be conducted. Effects of applying mathematical modelling processes to project production and management program on gifted students' mathematical modelling competencies and reflective thinking skills for real-life problem solving need to be examined further. More studies on how gifted students' project production and management process can be developed to be more productive and effective.

Footnotes

Acknowledgement

Special thanks are given to The Scientific and Technological Research Council of Türkiye (TUBİTAK) for their support and Republic of Türkiye Ministry of National Education, General Directorate of Special Education and Guidance Services, Department for Research, Development, and Projects for their permission and support for the research.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Gülnur Özbek

Seokhee Cho

Author Biographies

Seokhee Cho, Ph.D. Professor and Director, Center for Creativity and Gifted Education at St. John’s University. She has studied nurturing STEM creativity of specialized science high school students and developing math talent of young gifted English learners with Javits Grants sponsored by the US DOE.

Gülnur Özbek, Visiting Scholar, Center for Creativity and Gifted Education at St. John’s University. She has been supported with a research scholarship sponsored by The Scientific and Technological Research Council of Türkiye on gifted education, math talent development and curriculum differentiation for gifted. She has been serving at the Science and Art Center in Türkiye and also a member of Project BRIDGE sponsored by the Javits Grant at St. John’s University.

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