Levels of students’ “conception” of fractions

Abstract

In this paper, we examine sixth grade students’ degree of conceptualization of fractions. A specially developed test aimed to measure students’ understanding of fractions along the three stages proposed by Sfard (1991) was administered to 321 sixth grade students. The Rasch model was applied to specify the reliability of the test across the sample and cluster analysis to locate groups by facility level. The analysis revealed six such levels. The characteristics of each level were specified according to Sfard’s framework and the results of the fraction test. Based on our findings, we draw implications for the learning and teaching of fractions and provide suggestions for future research.

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Acknowledgments

We wish to thank Dr. Leonidas Kyriakides and Dr. Demetra Pitta-Pantazi of the University of Cyprus for their constructive comments during the research of this study.

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Correspondence to Marilena Pantziara.

Appendices

Appendix A

Fraction test items

Appendix B

The method applied for finding clusters

Suppose that I 1, I 2, I 3, … I n represent the items to be clustered into groups. First we find the range of the observed measurements that is (I maxI 2min). Next, we change the item values to a standardized 0–1 scale, using the formula \( {S_i} = \left( {{I_i} - {I_{{\min }}}} \right)/({I_{{\max }}} - {I_{{\min }}}) \), a transformation that conserves the relative item standing. We next sort the values S i in ascending order and calculate the gaps between two consecutive items, using the formula \( i\Delta = {S_{{i + 1}}} - {S_i} \) (where i = 1, 2, 3, … n). Finally, we sort the values of Δi in descending order (Δ1, Δ2, Δ3 …); the largest term Δ1 divides the items into two groups according to the largest gap identified among these items. The second largest term Δ2 further splits one of the two resulting groups into two subgroups based on the second largest gap, and so on. Hence, when the first k largest Δs are considered, the items are split into k + 1groups. The number of clusters that can be formed is determined by exploring the contribution of each iΔ to the cumulative Δ, which is expressed as a percentage and represents the explained variance.

Table 4 Grouping of the 21 items into clusters based on the procedure of pattern clustering

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Pantziara, M., Philippou, G. Levels of students’ “conception” of fractions. Educ Stud Math 79, 61–83 (2012). https://doi.org/10.1007/s10649-011-9338-x

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Keywords

  • Procedural and conceptual understanding
  • Fraction
  • Part–whole subconstruct
  • Measurement
  • Equivalence
  • Comparison
  • Rasch model