In their paper titled Levels of students’ ‘conception’ of fractions, Marilena Pantziara & George Philippou examine sixth grade students’ degree of conceptualization of fractions. In their study, they developed test items aimed to measure students’ understanding of fractions along the three stages proposed by Sfard (1991). It 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. Their analysis revealed six levels of understanding that teachers and researchers can use to analyze and assess understanding of the fraction concept.

###### The six levels of understanding fractions

Level 1 – is characterized by procedural understanding. Students placed in this level are able to perform a simple step-by-step procedure for computing the sum of fractions with like denominators.

Level 2 – students at this level could apply a step-by-step procedure to fill the missing numerator in two equivalent fractions and to find the largest fraction among two fractions with same denominators.

Level 3 – students placed in this level were able to find the fraction of a set of discrete objects (A3) and to select the correct representation of a fraction as a part of the equally divided whole, a first step towards the conceptual understanding of fractions. Students were also able to locate a fraction (3/5) on a number line from 0 to 1 divided into five equal parts.

Level 4 -students in this level could combine various processes and could reconstruct the whole from a given quantity, 2/3 equals four objects, finding the value of 1/3 and then the 3/3. Moreover students were able to alternate between different representations.

Level 5 -students placed in this level exhibited a perception of the fraction as an abstract construct to a certain extent and a possession of flexible thought. They could compare two fractions in more than one way. In addition, students placed in this level could find the fraction represented as a continuous quantity and place it on a number line divided into different parts than the ones presented by the fraction’s denominator.

Level 6 - students in this level could find a fraction between two consecutive fractions (C6), an activity that requires the perception of the fraction as an object and the acquisition and coordination of various sub procedures and representation. For example, a student used the comparison of fractions with the same numerator to find a fraction between two consecutive fractions, 1/9=2/18 and 1/8=2/16 and wrote 2/17 as an answer. Another student used fraction equivalence and turned fractions to fractions with the same denominators using 72 as the LCM in the first attempt and 144 in the second one, 1/9=8/72=16/144 and 1/8=9/72=18/144 and gave 17/144 as an answer.

Their paper was published online in Educational Studies of Mathematics (2012) 79:61–83.

This study on levels of students conception of fraction contributes to the on-going research about students learning trajectory for specific concepts of mathematics. A similar study which I conducted as part of my dissertation deals with students understanding of function (not fraction). You can find the article in Mathematics Education Research Journal Vol 21 published in 2009. I described them briefly in How to assess understanding of function in equation form in my blog Mathematics for Teaching.