*Number Sense Performance of Gifted and General Fourth Graders in Taiwan DOI: http://dx.doi.org/10.5772/intechopen.111752*


#### **Figure 1.**

*One example for the two-tier number sense test.*

reviewed by mathematics educators and experienced elementary school teachers. They all agreed that the tests are appropriate for the sample students.

To deeply explore students' thinking, a semi-structured interview was used to collect data [38]. Three questions were derived from each of the four components, and a total of 12 questions from the test were selected and used in the interviews to examine the gifted and general students' methods when solving number sense-related problems.

#### **3.3 Procedure**

The web-based two-tier test was conducted via an online setting in which students were asked to complete 16 items on computers individually for each subtest. The test included two subtests with 32 items in total. Due to the test included answer-tier and reason-tier, the answer-tier limited 40 seconds and reason-tier limited 60 seconds for students to answer the questions [36, 39]. Hence, participants required about 35 minutes each subtest to complete the test. During the test, students were required to follow the testing rules and procedures: (1) the papers and pencils were not allowed for students to use; (2) log on to the web-based system; (3) key in the individual data; (4) review the rules for the on-line test; (5) practice one item presented on the computer; and (6) begin the formal test.

Each participant was given a booklet during the interview. Each page of the booklet included one item and ample space for allowing students to record their thinking and methods. Each interview took about 40 minutes. Before the interview, the following directions were read aloud to each interviewee: 1. You are encouraged to estimate or mentally compute and do not necessarily to use written computation to find an exact answer on each item; 2. You can write an answer to the question and then briefly explain how you arrived at the answer; and 3. You are welcome to use different approaches to solve questions; the time on each item for you to answer was 3 minutes, so you should not turn to the next page without permission. The interviewer controlled the time to ensure that all interviewees would have an opportunity to answer each question.

#### **3.4 Data collection and analysis**

Data were collected through online tests and interviews. Based on the results of the online test, computer software was used to assemble statistical data; in the interview segment, video- and audio-recorded information of the interviews was transcribed into written records.

#### *3.4.1 Quantitative data analysis*

The scoring criteria of the two-tier test was calculated based on the students' answer and reason choices. In the first tier, if the students chose the correct answer, 4 points were given. In the second tier, if the students selected the number sense-based method, 4 points were given because the purpose of this study was to examine students' performance on the use of number sense-based method. If the students selected the rule-based method, 2 points were given. Therefore, the highest score was 8 points, and the lowest score was 4 points. If the students chose the wrong answer in the first tier, then 0 points were given in the first and second tiers.

Two independent groups (the gifted and general students) were used in this study; therefore, SPSS statistical software was used to perform the *t* test to determine the variation in number sense performance between the gifted students and the general students. In addition, an ANOVA was used to detect any variation between the gifted students and general students in the use of four components of number sense.

#### *3.4.2 Qualitative data analysis*

The students' responses were examined and sorted carefully. In an effort to identify the different methods used by the interviewees, each response (whether correct or incorrect) was sorted according to one of the following categories [17, 38]:

1.Number sense-based method: The students who used meaningful approaches to solve questions were coded as number sense-based method. For example:

Question 4: "Which answer is equal to 2 42 + 2 58? (1) 2 100 (2) 4 100 (3) 2 44 58 (4) 86 58".

GH1: Because 42 and 58 are all multiplied by 2; therefore, 2 42 + 2 58 equals 2 (42 + 58). The answer is 2 100.

GH1 knew that "2 42 + 2 58 equals 2 (42 + 58)." This was coded as "being able to decompose and compose numbers."


Two researchers independently reviewed the transcripts and categorized the students' responses for each correct and incorrect answer. These initial reviews produced agreement in over 92% of the categorization of student responses. The remaining responses were reexamined and discussed by the coders until agreement was reached.

#### **3.5 Reliability and validity**

The Cronbach's *α* coefficient of reliability was 0.828, and the construct reliability indices derived from structural equation modeling analysis for the two-tier test was 0.875. In addition, the difficulty indices of the test items were .26–.67, and the discrimination power was .48–.80.

Regarding the content validity, the options (both answer options and response options) in the NS came from earlier number sense studies (e.g., [36]). Especially, the options, including number sense-based method, misconceptions, and so on, used in the reason selections of the test were collected from interviewing over 100 fourth graders from earlier studies (e.g., [39]). Therefore, these options represented students' most frequent responses. In addition, the web-based test was reviewed by several experienced teachers, researchers, and mathematics educators who are experts in number sense to check whether those questions in the test were appropriate and relevant to the fourth graders. They all agreed that all the 32 questions in the test including wording, content, and the reasons for were appropriate for fourth graders.

### **4. Results**

### **4.1 Variation in number sense performance between the gifted and general students**

In **Table 1**, the number sense performance of the gifted and general students is reported. The *t* test results show statistically significant differences in the number sense performance of the gifted and general students for each number sense component (F1: *t* = 9.5, *p* < .000; F2: *t* = 9.51, *p* < .000; F3: *t* = 8.3, *p* < .000; and F4: *t* = 8.96, *p* < .000) and total score (*t* = 11.65, *p* < .000). This indicates that the gifted students significantly outperformed the general students in each number sense component and overall number sense performance. Moreover, the results also reached a high effect size (*η* <sup>2</sup> = .44). It indicates that the gifted students significantly outperformed the general students in number sense.

#### **4.2 Variations in number sense performance for each component for both groups**

To further examine the variations in number sense performance for each component for both groups, a one-factor repeated measures analysis of variance was used. Before proceeding with the statistical analysis, we ensured that these data did not violate the sphericity assumption. The Mauchly values were W = .847 (*χ* <sup>2</sup> = 7.613, *p* > .05) for the gifted students and W = .969 (*χ* <sup>2</sup> = 2.90, *p* > .05) for the general students. The results show that the data did not violate the sphericity assumption. Therefore, the one-factor repeated measures analysis of variance could be performed.

In **Tables 2** and **3**, the results of the one-factor repeated measures analysis of variance for both groups are shared. In **Table 2**, the results of ANOVA show that a value of the 48 gifted students did not reach the significance level [*F*(3, 141) = 2.444, *p* > .05]. Therefore, no significant variation was found among the four number sense components for the gifted students. In **Table 3**, the ANOVA value of the general students reached the significance level [*F*(3, 282) = 2.962, *p* < .05], indicating a significant variation among the four number sense components for the general students. The results of post hoc tests showed a significant variation between F4 (*M* = 27.95; recognizing the relative number size) and F2 (*M* = 24.08; ability to decompose and compose numbers). This implies that the general students performed higher on F4 than on F2.

#### **4.3 Similarities and differences in methods used by students of both groups**

In **Table 4**, the interview results regarding the methods used by the students of both groups are shown. To explain the three types of methods used by the students, their responses are reported as follows. The interview Question A10 (F4, recognizing the relative number size) asked: "A box had 24 moon cakes. John bought 0.4 of a box, and Mary bought <sup>1</sup> <sup>2</sup> of a box. Who bought more moon cakes?: (a) John; (b) Mary; (c) John bought as many moon cakes as Mary; (d) Cannot be compared."


Note*. The total score was 256; each dimension score was 64. \*\**<sup>p</sup> *<sup>&</sup>lt; .01.*

#### **Table 1.**

*The statistical analysis of number sense between the gifted students and general students.*

#### *Number Sense Performance of Gifted and General Fourth Graders in Taiwan DOI: http://dx.doi.org/10.5772/intechopen.111752*


Note*. Post hoc: F3 (Recognizing the relative number size) > F2 (Being able to decompose and compose numbers). \*< .05.*

#### **Table 2.**

*ANOVA analysis of components of number sense for the gifted students.*


Note*. Post hoc: F3 (Recognizing the relative number size) > F2 (Being able to decompose and compose numbers). \*< .05.*

#### **Table 3.**

*ANOVA analysis of components of number sense for the general students.*


#### **Table 4.**

*The frequencies of number sense methods used by students of both groups.*

#### *4.3.1 Number sense-based method*

The following student response is an example of using a number sense (NS)-based method.

GH3: I think Mary bought more moon cakes than John, because Mary bought half of the box. But what John bought, 0.4 of a box, is less than half. Therefore, Mary bought more moon cakes.

GH3 responded that "0.4 is less than half." This indicated that GH3 could apply <sup>1</sup> <sup>2</sup> as a benchmark and knew that 0.4 is less than <sup>1</sup> 2 . Therefore, the response of GH3 was coded as a NS-based method.

#### *4.3.2 Rule-based method*

The following student response is an example of using a rule-based method.

GM1: Mary bought more moon cakes, because <sup>1</sup> <sup>2</sup> ¼ 0*:*5, and compared with 0.4, 0.5 is greater. That is why Mary bought more moon cakes.

R: Can you explain it another way?

GM1: It can be solved by comparing 24 � 0.4 = 9.6 and 24 � 0.5 = 12. Therefore, 12 > 9.6.

As seen in the previous exchange, GM1 had two ways to solve the problem. One was converting <sup>1</sup> <sup>2</sup> to 0.5 and then comparing it with 0.4. The other was converting <sup>1</sup> <sup>2</sup> to 0.5 and then multiplying it by 24. GM1 also multiplied 0.4 by 24. Both solutions were based on written computation. Therefore, GM1's responses were coded as a rule-based method.

#### *4.3.3 Misconception*

The following student response is an example of a misconception.

NGL2: John bought more moon cakes because 24 � 0.4 = 96, and Mary bought <sup>1</sup> <sup>2</sup> of them, which is less.

R: Can you do it another way?

NGL2: 0.4 is greater, and <sup>1</sup> <sup>2</sup> is less.

R: How do you determine that 0.4 is greater, and <sup>1</sup> <sup>2</sup> is less?

NGL2: By multiplying the two numbers (meaning 24 � 0.4). I do not know how to explain it.

The previous explanation shows that NGL2 had a misconception when solving this problem. This was coded as a misconception.

The data showed the gifted students used number sense methods more frequently (84%) than did the general students (37%). By contrast, the general students had more misconceptions (46%) than did the gifted students (6%). The λ<sup>2</sup> test (λ2 cri = 5.991, *df* = 2, *p* = .000) showed significant variation between the methods used by the gifted and general students. The results of the Marascuilo post hoc test showed significant variation in the use of the NS-based method between the gifted and general students. Moreover, a significant variation was found in the number of misconceptions between the gifted and the general students. However, no significant variation was found in the use of written methods between the gifted and the general students.
