基礎數感測驗編製之研究

Fundamental Number Sense Test for First- and Second-Grade Primary School Students

本研究以數學概念性知識搭配基礎數感，並輔以數學認知能力，編製適用於國小低年級學生的基礎數感測驗（Fundamental Number Sense Test for G1 & G2，簡稱FNST G1 & G2）。本測驗以數值概念與計算為理論架構，正式測驗採叢集取樣，在8校16班取得376名有效樣本，測驗結果經探索性因素分析結果顯示理論架構與測驗資料結構相符，另以結構方程模式進行驗證性因素分析，結果顯示測驗結果與理論模式具良好的模式適配性，跨年級分析結果亦具有良好的模式適配性，故證實本測驗適用國小低年級學生。本測驗有8項分測驗，各分測驗均有6題，全測驗共計48題，施測時間為40分鐘。研究結果發現：1.本測驗由「數值概念」與「計算」等構念組成，因素結構穩定且模式適配性良好。2.「數值概念」包含數量計數、量序對應、心理數線、相對距離及數列排序；「計算」包含數字分合、數字矩陣、數學事實。3.本測驗具有良好的信效度。4.各分測驗難度偏易及鑑別度佳。5.全量表與分測驗結果分布為常態至負偏態，有利於偵測基礎數感較弱的群體。6.本測驗以百分等級25為篩選基準，低於篩選基準者，可評估補救教學的必要性。

Number sense is an innate understanding of quantities and numbers that is an essential component of cognition. This ability does not exclusively occur in humans; some animals can also comprehend basic quantities. Dehaene (2011) asserted that this ability is inherent but evolves with mathematical learning, developing with the understanding of primary nonsymbolic representations. Chan et al. (2022) and Siemann and Petermann (2018) have indicated that this development comprises three forms of intuition: innate intuition, basic numerical intuition, and full number sense. Environmental factors influence the development of these three forms of intuition; however, variations in innate intuition may occur among students in the same environment and can result in differences in mathematical performance. According to Butterworth (2011) and Jordan et al. (2010), the development of number sense is catalyzed by primary mathematical education, which facilitates the shift from innate intuition to full numeracy. Concepts such as approximate number systems are based on the human capacity for intuitive magnitude comparisons, and understanding of these concepts are the basis for assessing mathematical performance . The present study developed a fundamental number sense test (FNST) for lower elementary school students that combines core mathematical concepts and research insights to evaluate number sense proficiency accurately and reliably.
The present study highlights the importance of inherent number sense and its role in achieving numerical fluency when combined with acquired mathematical knowledge. The FNST covers the core mathematical competencies young students are expected to acquire. It focuses on cardinality, ordinality, arithmetic, and representational fluency and was developed with reference to Clements et al. (2019), Dehaene (2011), and other scholars. The FNST integrates global mathematical ability evaluations centered on representational transformation, and it was developed with a focus on clearly presenting items, reducing cognitive load, and ensuring understanding. The test comprises the following eight subtests, each of which contains six items:
1. Numerical counting: This subtest evaluates student fluency in finger counting, strategic counting, and numerical representation. It encompasses foundational concepts such as cardinality and sequencing.
2. Magnitude-order matching: This subtest evaluates innate numerical intuition and students’fluency in transforming this intuition into symbolic representations, evaluating whether students understand basic concepts of cardinality by asking them to arrange numbers in ascending order.
3. Mental number line: This subtest evaluates students’abilities to spatially place numbers.
4. Sequential numbering: This subtest evaluates the sensitivity of students to number magnitude. Students with higher numerical sensitivity sequence numbers more efficiently, and their performance is influenced by the numerical distance effect.
5. Relative distance: This subtest measures the intuitive ability of students to categorize numbers in relation to a reference point.
6. Numerical matrix: This subtest evaluates students’intuitive comprehension of specific numerical combinations. Students are presented with a 3×3 matrix constructed from single-digit numbers that is used to evaluate their understanding of the combination and decomposition of numbers and arithmetic problems.
7. Number composition and decomposition: This subtest evaluates students’ability to composition and decomposition specific numbers. Students with stronger numerical intuition can more swiftly construct or deconstruct numbers.
8. Mathematical facts: This subtest evaluates students’single-digit flexibility and numeric sensitivity through decomposition, arithmetic, and intuitive combination tasks.
The test content was reviewed by three scholars specializing in special education and educational psychology, a special education teacher, and a regular primary school teacher after preliminary research was conducted. The review was used to evaluate whether the test questions matched the intended assessment dimensions, determine the validity of the questions, assess the appropriateness of the presentation of the items, and determine the correlations between the assessment dimensions and test content.
A total of 189 first-grade and 187 second-grade students were asked to complete the formal test. Exploratory factor analysis was used for both groups of students and revealed two factors—grasp of basic numerical concepts and calculation ability— that explained more than 50% of the variance. The factor loading value for the two grades indicated similarity and alignment with the theoretical framework of the test. Multigroup analysis with structural equation modeling also revealed excellent model fit indices, indicating construct validity. Pearson’s correlation analysis and subsequent scores on the New Elementary School Mathematics Achievement Test (Chou, 2021) revealed a moderate correlation, supporting the criterion-related validity of the FNST. Correlation analysis between the subtests and total scores revealed low to moderate correlations, indicating that each subtest measured distinct but related concepts. Overall, the FNST exhibited high reliability, with standardized internal consistency reliability values of .84 and .83 for the first- and second-grade students, respectively, and a Guttman’s split-half reliability value of .80.
For the first- and second-grade students, the distribution of scores indicated negative skewness and varying degrees of kurtosis. The second-grade students had a more concentrated distribution of high scores than the first-grade students did. Difficulty and discrimination indices were calculated on the basis of the scores of the students in the top and bottom 27%. The discrimination was excellent for several subtests for the first-grade students, whereas a ceiling effect was observed for some subtests for the second-grade students. In terms of difficulty, numerous second-grade students reported that the test was generally easy.
In summary, the FNST exhibited high validity and reliability with respect to evaluating the fundamental mathematical skills of first- and second-grade students. Its content validity was confirmed through expert review, and its construct validity was confirmed through factor analysis. The FNST’s criterion-related validity was established through correlation analysis with an established mathematical achievement test, and its reliability was consistently high for both the first- and second-grade students. The test results indicated that the FNST is a robust and reliable tool for evaluating the basic number sense of lower elementary school students.
With consideration of the limited attention span of younger students, the FNST was developed in several steps. First, it was administered to six first-grade students to determine whether it required modifications. Second, it was administered to a group of 15 students and further refined. Third, a pilot test was conducted with first- and second-grade students to calibrate the FNST before official testing. Students were recruited from schools in several districts of Kaohsiung City to ensure the sample was diverse. A cross-sectional research design was adopted to ensure that the sample adequately represented the characteristics of the target population. A linear mixed model was used to examine the stagnation or growth in the skills of the FNST subtests between grades. The intercept was treated as a fixed effect to ensure covariance between grades. The results indicated significant differences in basic numerical ability between the first- and second-grade students. Further comparisons revealed substantial growth in ordinal correspondence, number decomposition, and mathematical facts, as indicated by the scores on the subtests, indicating that major progress could be achieved through learning and mastery. However, the scores on the numerical counting, mental number line, relative distance, sequential numbering, and numerical matrix subtest scores remained relatively unchanged across grades, indicating that the students exhibited no major improvements after training. Additionally, students demonstrated better performance in the subtests for ordinal correspondence, number decomposition, and mathematical facts, indicating they possess greater numerical flexibility.
In terms of performance on the FNST, the second-grade students consistently outperformed the first-grade students on the eight subtests, indicating they had progressed in their numerical comprehension. Generally, first-grade students who struggle with numerical intuition require extra support. According to the literature, foundational difficulties with understanding numbers often originate from difficulty in making quantity associations and number comparisons. The FNST combines assessments of cognitive abilities with evaluations of the grasp of fundamental mathematical principles, rendering it an excellent tool for evaluating the performance of young primary school students.
In conclusion, the present study developed a reliable FNST for young primary school students focusing on the numerical concepts and calculation domains of grasp of numerical concepts and calculation ability. Although it has limitations, as well as being during a period of mandated infectious disease control, purposive sampling was adopted for sample collection. In the future, we plan to continue expanding the number of samples to establish a national sample and its replicas. This test has established normative data for Kaohsiung City but has not yet collected large-scale normative data from other regions or nationwide. Therefore, when applying this test in other counties and cities in Taiwan, the results should be interpreted with caution.