An Investigation of the Measurement Estimation Strategies Used by Gifted Students

This study aimed to investigate the measurement estimation strategies used by gifted students. Case study was used. 17 seventh grade students who were studying in the Science and Art Center located in a province in Eastern Anatolia Region of Turkey and who were identified as gifted participated in this study. The data was obtained through “The Measurement Estimation Skill Form” which includes nine open-ended tasks. In addition, clinical interviews were conducted with five students. The data were analyzed using descriptive analysis. The findings of this study indicates that students use eight different strategies, which rough guess ,breaking down, using prior knowledge, reference point, unit iteration, comparison with referents, subdivision clues and squeezing, in cases requiring measurement estimation. It was also found that while gifted students mostly used strategy “rough guess”, strategies “subdivision clues” and “squeezing” are used very little. Furthermore, the findings of this study show that gifted students did not use different strategies at the expected level in the measurement estimation situations.


Introduction
In recent years, giftedness has been one of the most controversial concepts (Smedsrud, 2018). Different definitions are made about giftedness and gifted students (e.g.,

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practices, is emphasized in the education of gifted students (Renzulli, 2012). With the development of mathematical creativity, which is one of the basic elements of these approaches, mathematical thinking potentials of gifted students can be supported (Singer et al., 2016).
Mathematical thinking consists of processes such as reasoning, deduction, induction, assumption, generalization, proof, and estimation (Liu & Niess, 2006). Estimation is an important skill for mathematically gifted students because the concept of creativity, which is closely related to the concept of giftedness, includes estimation skill (Davis & Rimm, 2004;Torrence, 1974).
In the literature, it has been reported that the most remarkable abilities about creativity are "fluency, flexibility, originality, and elaboration" (Pitta-Pantazi et al., 2018).
However, apart from these four abilities, estimation is also stated to be in creative abilities (Davis & Rimms, 2004). Torrance (1974) stated that the creativity process includes skills such as being sensitive to problems and seeking solutions, making estimations or developing hypotheses for deficiencies. In this regard, Starko (2005) emphasized that both mathematical thinking and creative thinking ability include estimating, generating ideas and developing a multi-faceted perspective on events.
Another relationship between estimation skill and creativity emerges in the process of constructing mathematical concept structures. Estimation is required in the development of rich and flexible mathematical concept structures. Because, estimation plays a role in the development of intuitive, internally born concept structures (Meissner, 2000). Consequently, it is accepted that mathematically gifted students have the ability to estimate within the scope of creativity, which is among the remarkable characteristics.
Although the estimation is deemed important in terms of mathematical development of gifted students, it is noteworthy that the research on this subject is limited (Akar, 2017; Baroody & Gatzke, 1991;Dai, Moon, & Feldhusen, 1998;Montague & van Garderen;2003;Wang, Halberda, & Feigenson, 2017). In some of these studies, measurement skills of gifted students were examined (Baroody & Gatzke, 1991;Montague & van Garderen;2003). In some other studies, the number sense in the context of estimation of gifted students has been considered (Wang et al., 2017). In the studies in the third group, the use of estimation skills in problem solving situations of gifted students was analyzed. In these studies, it has been determined that gifted students make more consistent estimations in the context of mathematical problem solving (Dai et al., 1998). Akar (2017), in her study, stated that gifted students, in the process of solving modeling problems, use estimations as a problem solving strategy.

Purpose and Importance of the Present Study
Although measurement estimation skill has been proved to have positive contributions to mathematical development, there is a lack of studies focusing on this topic (Hartono, 2015;Jones et al., 2009;2012). A similar situation is observed in studies conducted in Turkey. A limited number of studies investigating estimation skill have been found since 2000 (Boz-Yaman & Bulut, 2017). In these studies, predominantly on computational estimation skill, therefore it was determined that the number sense was addressed (Aytekin & Toluk-Uçar, 2014). Studies on estimation skills have been found to be limited (e.g., Kılıç & Olkun, 2013).
The literature on gifted students and mathematics education also shows that there are limited number of international studies on the estimation skills of gifted students (e.g., Baroody & Gatzke, 1991;Dai et al., 1998;Montague & van Garderen;2003;Wang et al., 2017).
In addition, no study that has paid attention to estimation skills of gifted students can be found in Turkey. In the studies conducted in the field of giftedness, it is stated by the researchers that the learning and mathematical thinking processes of the gifted students are not examined sufficiently (Sheffield, 2018;Wang et al., 2017). This requires investigation of estimation skills of gifted students. The present study focused on the strategies used by gifted students in measurement estimation tasks. Therefore, this study is important since it bridges an important gap in the literature on giftedness and mathematics education.

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Estimation skill is within the abilities that express mathematical giftedness (Davis & Rimm, 2004). Also, measurement estimation is an important real-life skill (Gooya et al., 2011;van de Walle et al., 2016). Accordingly, determining the measurement estimation performances of the gifted students is considered to have a crucial importance. The results of this study are expected to contribute to the awareness of the gifted students' measurement estimation skills. It is also thought that the results of this study will be a starting point for the researchers who will carry out studies on this topic.
Teachers are required to have knowledge about the estimation ability and to design a teaching environment based on this information (Boz-Yaman & Bulut, 2017). In addition, it is clear that gifted students have different needs in mathematics education than their peers (Gutierrez et al., 2018;Hu, 2019;Smedsrud, 2018). Teacher competencies are an important factor in meeting these learning needs (Leikin et al., 2017). Based on this information, the current study is expected to provide teachers with information about the estimation strategies used by gifted students. In addition, these study findings are thought to present new ideas to teachers in designing measurement estimation tasks for gifted students.
Developing differentiated mathematics programs to meet the needs of gifted students is one of the topics highlighted in recent years (Hu, 2019;Sheffield, 2018). However, in Turkey, the development of special programs for gifted students appears to be insufficient (Özçelik, 2017). The current findings of this study may provide program development experts with an understanding into the process of designing tasks for estimation skill.
Motivated by the aforementioned concerns, this study aimed to investigate the measurement estimation strategies used by gifted students. Within the scope of the study, the answer to the question "What are the measurement estimation strategies used by gifted students in real life situations?" was sought.

Model
Case study was used in the study. Case study is an empirical inquiry that examines a current phenomenon in its real-life context (Yin, 2017, p. 16). While gifted students are the analysis unit of the study, the measurement estimation strategies used by gifted students constitute the analyzed situation. 208 Participants 17 seventh grade students who were studying in the Science and Art Center located in a province in Eastern Anatolia Region of Turkey and who were identified as gifted participated in this study. Participants of the study were determined by appropriate sampling method since they are easily accessible. Conducting the study has become easier with the appropriate sampling method (Mcmillan & Schumacher, 2010). The reason why the participants were selected among seventh grade students is that they have seen the estimation gains in the mathematics curriculum at previous grade levels. Totally 21 seventh grade students are enrolled in the Science and Art Center. However, two students with a talent field "painting" and two students who were absent could not be taken as participants of the study. All of the participants were educated in the Science and Art Center in their city, within the scope of the program to realize their individual talents. In addition, six of the students were female (35%) and 11 were male (65%). Eight of the students were in public secondary school, and nine were in a private secondary school.
In the second sampling process, five gifted students were determined by purposeful sampling method. Among the purposeful sampling methods, according to the criterion sampling, the participants were included in the group according to the predefined criteria (Patton, 2002). Accordingly, the number of using different strategies was taken as a criterion.
The aim is to obtain detailed information on different strategies. Accordingly, clinical interviews were conducted with five gifted students who use the most different strategies.
Throughout the whole study, gifted students were provided on a voluntary basis. In the findings part of the study, the term "students" was used in order to ensure fluency instead of gifted students.

Data Collection Tools
The data was obtained through "Measurement Estimation Skill Form ( and 7), and weights (task 9). The length-related tasks are often due to the fact that the studies in the literature are mostly done on length estimation (Desli & Giakoumi, 2017;Jones et al., 2009;Hartono, 2015). In this study, by preparing tasks in different contexts related to length estimation, diversity in students' estimation strategies was determined.
MESF was presented to the opinion of four specialist faculty members (who had studies in mathematics education and special education) and three mathematics teachers (one of the teachers working in secondary school and two of the teachers working in Science and Art Center). According to the specialist and teacher feedback, it was determined that the MESF was appropriate for the student level and the purpose of the study. In addition, MESF was applied as a pilot study to two seventh grade students who were not study participants.
At the end of the pilot study, it has been determined that the duties are understandable and the MESF is applicable. Besides, the purpose of considering the mathematics curriculum and studies in the literature is to raise the validity of the tasks.
Clinical interviews were also conducted with students in order to reach more detailed data and support the study, which are used by gifted students in measurement estimation tasks. Clinical interview is a mutual interview to examine thoughts in depth and to investigate the structure of information (Clement, 2000). During clinical interviews, gifted students were shown their responses to the tasks in the MESF, and they were asked to explain their responses to each task. In addition, additional questions were asked (e.g., how?, why?). The interview with each gifted student took about 20 minutes, the interview was recorded with a voice recorder.
Student responses were coded according to this framework, and the findings became clear.
The strategies and explanations that emerged in student responses are given in Table 1. This strategy is for students to estimate the measurements of other objects by taking advantage of the measurements of the objects they knew well before. In this strategy, the student makes a measurement estimation using a mental image of a non-standard unit.
If there is a pencil that the student knows the length of, the student can estimate the length of any box by comparing it with the length of the pencil.

Using prior knowledge
In this strategy, the student needs to know in advance about the length, area, or volume measure of the object to be estimated. The student makes measurement estimations about the object by using his prior knowledge.
In the task of estimating the height of a multi-storey building, the student makes the estimation, using this information, if student knows in advance that the height of a floor is three meters.

Unit iteration
With the unit iteration strategy, the student mentally or physically performs the measurement estimation of the object by repeating a standard unit.
The student guesses by repeating the movements of one hand while estimating the length of the blackboard.

Comparison with referents
It is the comparison of physically existing or abstract objects with the measure of the object to be estimated. Although the student's prior knowledge is important, expressions such as "equal, greater or smaller" can be used.
It is a student who guesses the height of a tree, saying that the height of the tree is equal to the height of the school.

Squeezing
The compression strategy is placing the size of the object whose size is to be estimated, between the two measures.
The student expresses that the length of the object related to a measurement with a length of 70 cm is between half a meter and a meter.

Breaking down
The breaking down strategy is to estimate the size of the object using smaller, equal, or unequal parts.
In the case of estimating the length of a wall, the length of the wall is to estimate the length of the shorter sections, such as windows, boards, located along the wall.

Subdivision clues
This strategy is similar to the breaking down strategy. However, in the subdivision clues strategy, there are no useful sections to estimate the size of the object, and the student mentally divides the object into pieces.
In the task of estimating the length of a wall, the wall is divided into two, then four, and eight, and the estimate is made.

Rough guess
In the rough guess strategy, the student does not express what he/she has guessed.
The student explains his/her estimation on grounds such as "I looked at the eye (eyeballed it)" or "I made a guess".
incompatible codes until they reached a common view. In addition, direct transfers were made from participant responses. While presenting the findings, instead of the real names of the students, coding was done in the form of "S1 (first-line student)".

Findings
In this part of the study, the analysis results of the data obtained from MESF are presented. First of all, in situations that require students' measurement estimation, the measurement estimation strategies they use are examined on a task basis, and their strategy distributions are given in Table 2.  Table 2, a total of 144 answers were obtained in the measurement estimation tasks, whereas 9 tasks were not answered. It was found that students use eight different strategies, namely rough guess, breaking down, using prior knowledge, reference point, unit iteration, comparison with referents, subdivision clues and squeezing, in cases requiring measurement estimation. In addition, it has been determined that some students use two strategies together. The strategies used together are "using prior knowledge + breaking down" and "using prior knowledge + subdivision clues".
"Rough guess" strategy appeared to be the most used strategy of the students (f=61).
The rough strategy is followed by "breaking down" (f=21) and "using prior knowledge"

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(f=15) strategies, respectively. "Squeezing" strategy was found to be the least used strategy (f=2). In addition, the "subdivision clues" strategy was shown to be one of the less preferred strategies (f=6).
After the measurement estimation strategies are analyzed on a task basis, the number of different strategies used by each student is analyzed and the results are shown in Table 3.
At the end of this analysis, clinical interviews were made with S1, S5, S10, S14, and S17 which uses the most strategy. S4, S9, S12 4 S6, S7, S13, S15, S16 5 S8, S11, S14 6 S5 7 S1, S10, S17 The distribution of the students' measurement estimation strategies according to the tasks reveals that that the strategies varied according to the tasks. In the first task, "subdivision clues" (35%) and "using prior knowledge + subdivision clues" (24%) strategies came to the fore. In strategy "subdivision clues", the students made the estimate by dividing the rectangle into pieces to estimate the area of the rectangle. The statements of S1 using this strategy and the drawing for the task are given below.  In the second and eighth tasks, more than half of the students (59%) were observed to use the "breaking down" strategy. "Beraking down" strategy was similar to strategy "subdivision clues". However, the main difference between the two strategies was that in 213 "breaking down" strategy, there were useful parts to estimate the size of the object.
Accordingly, it has been observed that students make their estimaitons in the second and eighth tasks, using unit squares and dotted paper as useful sections. In the second task, the statements of S10 using the breaking down strategy and the drawing for the task are presented below.  Approximately, it will be like this.
In the eighth task, the statements of S5 using the "breaking down" strategy and the drawing for the task are presented below.
(Content of student response: I smash squares. I count how many squares are in them. I guess a square is 1 cm 2 . That is, about…)  In the third, fourth, and fifth tasks, it was determined that mostly the "rough guess" strategy was used (41% for third task, 71% for fourth task, and 65% for fifth task). Students using the "rough guess" strategy do not provide a justification for their estimations. In the third task, the clinical interview and sample estimations of S14 using the rough guess strategy are presented.
(Content of student response: I guess, the eye decision can be 1.30 cm wide and 2.15 cm tall.) . Student's (S17) estimation using the strategy of rough guess for fourth task (Content of student response: When I look at this, I guess that the big glass will take 300 ml of water.) Figure 6. Student's (S14) estimation using the strategy of rough guess for fifth task Task 3: Estimate the width and length of the door of the class you are in.
Task 4: How tall is the Science and Art Center building?
Task 5: The small glass takes 90 ml of water. Accordingly, how many milliliters of water does the other glass on the table take?
I: The small glass provided takes 90 ml of water. Accordingly, in the question of how many ml of water the glass will take, can you explain its estimation? In the sixth task, the "unit iteration" strategy is the most used strategy (35%). In the "unit iteration" strategy, students counted repetitive units, physically or mentally, to estimate the size of an object. The statements of S17 using the unit iteration strategy and the drawing Figure 7 for the task are presented below.
(Content of student response: If the length of the eraser is three fingers, if we put a line after three fingers, it will be 3 times.) Figure 7. Student's (S17) estimation using the strategy of unit iteration Task 6: How many erasers does the length of the pencil given above correspond to?
Please explain. In the seventh task, more than half of the students used the "using prior knowledge" strategy (59%). In this strategy, students have made an estimate of the object to be measured using the knowledge they have. The statements for clinical interview of the student (S10) applying "using prior knowledge" strategy and Figure 8 for the task are presented below. "using prior knowledge + breaking down" and "using prior knowledge + subdivision clues".
It was found that very few students left the tasks unanswered.
In accordance with findings, it was also found that while gifted students mostly used strategy "rough guess", strategies "subdivision clues" and "squeezing" are used very little.
In addition, it was concluded that gifted students used different strategies in tasks requiring length, area, volume and weight estimation. It is thought that this finding is related to the context of the tasks demanded from the gifted students and their experiences. Previous studies support this notion (Gooya et al., 2011;Jones et al., 2012). Gooya et al. (2011) stated that students use different strategies depending on the context in situations that require measurement estimation. Jones et al. (2012), on the other hand, stated that their students' estimation performance is related to their past lives.
A remarkable result reached in the study is that gifted students use the "rough guess" strategy intensively. In the "rough guess" strategy, there is no justification for how the estimate is made (Gooya et al., 2011). Estimation skill is related to creativity, which is one of the important characteristic features of gifted students (Davis & Rimm, 2004;Torrance, 1974).
Therefore, it is an expected situation that gifted students use more different estimation strategies. However, this result of the study may be due to the fact that gifted students did not have enough experience and developed different strategies with their measurement estimation activities. Based on this idea, it is suggested to design learning environments where gifted students can discuss different measurement estimation strategies. In addition, it is expected that studies examining the effects of various learning methods and techniques on measurement estimation strategies used by gifted students are expected.
The findings of this study show that gifted students did not use different strategies at the expected level in the measurement estimation situations. This result of the study is consistent with the findings of Montague and van Garderen (2003). Montague and van Garderen (2003) stated that gifted students had low "numerosity estimation" performances.
However, it was determined that this result obtained in the study differed from the results of Baroody and Gaztzke (1991). The researchers had revealed that gifted students showed success in terms of measurement estimation. This difference in study results may have resulted from the sample. Because, one of these studies included preschool students (Baroody & Gaztzke, 1991), while the other study included secondary school students In the literature, it is emphasized that the studies on mathematics education with gifted students are inadequate (Sheffield, 2018;Singer et al., 2016;Wang et al., 2017). In this regard, Leikin (2011) states that mathematics education is not adequately represented in the giftedness literature. However, studies examining the estimation ability of gifted students are quite limited (e.g., Baroody & Gatzke, 1991;Dai et al., 1998;Montague & van Garderen;2003). Therefore, it can be said that there is not enough knowledge accumulating the results of the current study, which would allow for a more detailed discussion. However, comparisons can be made with studies conducted with non-gifted students. Accordingly, differences are observed in the measurement estimation strategies frequently used by gifted students and non-gifted students. Gooya et al. (2011) stated that students mostly use the "comparison with referents", "using prior knowledge", and "reference point" strategies. Joram et al. (2005), on the other hand, talked about the "unit iteration" strategy being used frequently. Both Joram et al. (2005) and Kılıç and Olkun (2013) found that the "refence point" strategy was not used frequently.
As participants of this study, only the handling of seventh grade students can be considered as the limitation of the study. However, gender differences and age (grade level) variables are frequently examined in the literature on mathematics education. In studies examining students' estimation skills, the age and gender variable is considered important (Aytekin & Toluk-Uçar, 2014;Jones et al., 2012;Yun-hing, 2007). Accordingly, the future studies may address the issue whether the gifted students' measurement estimation skills differ according to age and gender. Also, the future studies can examine the accuracy of the gifted students' measurement estimation results.
In studies conducted with non-gifted students, it has been determined that the measurement estimation skill is related to motivation, conceptual knowledge and spatial reasoning skills (e.g., Jones et al., 2012;Hogan & Brezinski, 2003). From this point forth, the future studies should also focus on variables such as motivation, spatial reasoning skill, which are thought to be related to students' computational or measurement estimation skills.
In addition, it is recommended to conduct studies to examine the computational estimation skills and number sense of gifted students.

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Teacher competencies are an important factor affecting the math learning processes of gifted students (Gutierrez et al., 2018;Leikin et al., 2017). In order to respond to the needs of gifted students, which differ from other students, there is a need for teachers who have indepth knowledge and talent in their field (Subotnik, Olszewski-Kubilius, & Worrell, 2011). In addition, one of the reasons for the low performance of students' estimation is the teacher competencies (Desli & Giakoumi, 2017). Based on this information, studies examining the measurement estimation skills of mathematics teachers who deliver instruction to gifted students can also be carried out. Thus, necessary precautions can be planned by seeing the deficiencies of the teachers. Finally, it is suggested that mathematics teachers working with gifted students should be provided with seminars, courses or in-service trainings on topics such as mathematical creativity and estimation skills.

Acknowledgement
The earlier version of this paper was presented at International Congress on Gifted and Talented Education at Inönü University, Malatya-Turkey (November 1-3, 2019).
The data used in this study was confirmed by the researchers that it belongs to the years before 2020.