**2. Background**

#### **2.1 Meaning and components of number sense**

Number sense refers to an individual understanding of numbers, operations, the relations between numbers and operations, and the ability to solve real-world problems that involve numbers [1, 17].

Based on previous studies [1, 7, 9, 16, 17], the current study's definition of number sense comprises four components.

#### **C1. Understanding the basic meaning of numbers and operations**

This implies an ability to fully understand the meaning of the base-10 number system (e.g., integer, fraction, and decimal), place value, patterns of numbers, multiple methods of representation, and the four basic operations [9]. For example, students should realize that infinite decimals and fractions are found between 0.41 and 0.42 for middle-grade students [18].

#### **C2. Composing and decomposing numbers**

Decomposing numbers means to decompose numbers to facilitate the computation, such as 18 = 2 + 16, 18 = 15 + 3; composing numbers means to add numbers to become a number, such 19 + 1 = 20, 37 + 3 = 40 [9]. For example, when encountering a question such as 96 + 76 =? Students can first decompose 76 into 72 and 4, then add 4 to 96, which equals to 100, and finally, add 72. The result is 172.

#### **C3. Ability to judge the reasonableness of a computational result**

After obtaining an answer, students can use the information given by a question to determine the reasonableness of a computational result [9]. For example, when students were asked to answer: "How many digits is the sum of 2 three-digit numbers?" They should know that a small three-digit number plus another small three-digit number could be a three-digit number and a large three-digit number plus another large three-digit number could become a four-digit number.

#### **C4. Recognizing relative number size**

Students can determine relative number size or determine which number is closer to the target number. For example, "Arrange the following rational numbers in order from the lowest to the highest: <sup>1</sup> 2 , 0.65, <sup>5</sup> 4." Students should know that 0.65 is greater than <sup>1</sup> 2 , and <sup>5</sup> <sup>4</sup> is greater than 1. Therefore, the order is <sup>5</sup> <sup>4</sup> > 0.65 > <sup>1</sup> <sup>2</sup> [18].

#### **2.2 Gifted students' characteristics and related studies**

In this study, gifted students were defined as students who passed two levels test instituted by the Ministry of Education in Taiwan [MEiT] [19]. The first level is the Intelligence Quotient Test, with a passing score of ranking above 93%, and the second level is the Wechsler Intelligence Scale (fourth edition) for Children, with a passing score of ranking above 97%. The IQ test was designed by educators in Taiwan and includes three subtests (e.g., language, mathematics, and graphics) [19]. The IQ test

was used to assess students' intelligence. In this test, student's IQ score ranked over 93% is considered to have a high IQ. The Wechsler Intelligence Scale (fourth edition) was designed by Wechsler [20] to measure a child's intellectual ability. The test includes five primary index scores: the Verbal Comprehension Index (**VCI**), Visual Spatial Index (**VSI**), Fluid Reasoning Index (**FRI**), Working Memory Index (**WMI**), and Processing Speed Index (**PSI**). In this study, a student whose score ranked above 93% on the IQ test and ranked above 97% on the Wechsler Intelligence Scale (fourth edition) was defined as a gifted student.

Clark [21] discussed the characteristics of a gifted student from four perspectives, including cognitive, affective, physical, and intuitive perspectives. From a cognitive perspective, a gifted student has excellent memory and comprehension abilities, being capable of fast and flexible thinking, producing different ideas and problem-solving strategies, and so on (Clark). From an affective perspective, a gifted student has a strong motivation to explore new knowledge (Clark). From a physical perspective, a gifted student can absorb a lot of new information at a same time (Clark). From an intuitive perspective, a gifted student has a higher degree of creativity than peers (Clark). Davis and Rimm [22] indicated that gifted students possess many different characteristics, such as excellent analysis, reasoning, and problem-solving ability, ability to use abstract, complex, and high-level logical thinking abilities, and producing effective strategies to solve questions, having good meta-cognitive abilities, and so on. Earlier studies in Taiwan also showed that Taiwanese gifted students are better in abstract thinking, logical reasoning, fast and flexible thinking, using multiple and effective strategies, having good meta-cognitive abilities, and so on [23, 24].

Based on the above studies, the characteristics of gifted students include excellent abilities on concentration, comprehension, and creativity; flexible thinking, good abstract and logical reasoning ability, and insights; strong learning motivation; having the ability to self-reflection and meta-cognition, and so on. In fact, the gifted students in mathematics also have some characteristics that gifted students have [14, 25–29]. Number sense is a foundational content area in mathematics education [5, 9, 16, 30]. However, there is no research that focuses on the examination of the relationship between gifted students and number sense. This motivated the conduct of this study.

#### **2.3 Number sense and gifted students-related studies**

Earlier studies showed that there are several common methods used by genera gifted students when solving questions [13, 14, 28, 31]. These methods included (1) composing and decomposing, (2) finding patterns, (3) connecting to prior experience, (4) graphic representation, (5) eliminating possibilities, (6) making and testing conjectures, (7) intuition, and (8) logical reasoning. Some of the methods are similar to the number sense strategies.

Some methods are not specific to number sense. For example, flexibly using pictorial representations, which consists of drawing figures, is typically considered a problem-solving strategy [32]. "Making and testing conjectures" means that students make guesses by observing patterns, test these guesses, and then evaluate the result [33]. Regarding logical reasoning, Greeno [34] asserted that number sense is a set of capabilities for constructing and reasoning with a mental model. Students can estimate the area of a given region by using benchmarks and reasoning, which is an example of logical reasoning. "Intuition" is a type of number sense [28]. According to the previous discussion, the variations in performance and strategies between the

gifted and general students in solving number sense questions are not definitive. Therefore, this study is relevant.

Study related to the gifted students on variations in number sense and use of strategies for solving number sense-related questions is lacking. Examining the difference and recognizing the characteristics between the gifted and general students would contribute to the future studies. Hence, this lack of research encouraged us to conduct this study.

## **3. Methods**

A mixed-method approach was used in the current study. For quantitative analysis, the number sense data on the students in this study were collected using a number sense web-based two-tier test system. A statistical analysis was used to evaluate the performance of the gifted students and general students. For qualitative analysis, data were collected through semi-structured interviews.

#### **3.1 Sample**

Fourth graders from two public elementary schools (A and B) in Southern Taiwan were selected. Student numbers in each school are over 1,000. School A had 28 gifted students and 320 general students in fourth grade; School B had 20 gifted students and 200 general students in fourth grade. All 48 gifted students from both schools and three classes with 95 general students from School A and B (two classes from school A and one class from school B) were randomly selected to join this study. The families from schools have a wide range of socioeconomic backgrounds. All of the participants were voluntary to join the test and under the agreement of parents and the school administration.

According to the results of the number sense web-based two-tier test, the students in each group were classified into three categories: high-level (top 20%), middle-level (middle 50–60%), and low-level (bottom 20%). Three students at each level of both the gifted and general student groups were randomly selected and interviewed to examine their methods of thinking about number sense problems. Therefore, the sample for the interviews consisted of nine gifted students, coded as low (GL1–3), middle (GM1–3), and high (GH1–3) level students, and nine general students, coded as low (NGL1–3), middle (NGM1–3), and high (NGH1–3).

#### **3.2 Instrument**

A Number Sense Web-Based Two-Tier Test System for fourth graders designed by Lin [35, 36] was adopted in this study. The online test system consisted of a two-phase evaluation. The first-tier test (answer-tier) in the two-tier test assesses children's responses to number sense-related questions, and the second-tier test (reason-tier) examines children's reasons for their related choice made in the first-tier test [16, 37]. One example is shown in **Figure 1**.

This test included four components, with eight questions for each component, resulting in 32 total questions. The test was divided into two subtests. Each subtest included 16 items. The items in the web-based two-tier test were written in Chinese and translated into English for writing this manuscript. Each question in the test was
