The optimal achievement model and underachievement in Hong Kong: an application of the Rasch model

Abstract Termed the optimal achievement model, a new method for the estimation of underachievement relies on measuring student potential (P) and achievement (A) using Rasch models and calculating an achievement index, I^sub A^, for each individual. This study extends a previous report (Phillipson & Tse, 2007) that estimated the proportion of Hong Kong students in Primary 5 to now include a sample of students from Primary 3 (n = 1406), Secondary 1 (n = 756) and Secondary 3 (n = 578), across six districts of Hong Kong. The students were administered a standardized test of mathematical achievement and the Ravens Progressive Matrices Test. Using the optimal achievement model, estimates of underachievement at six percentile bands showed that the proportion of students who were underachieving ranged from 10 % at the 50-59th percentile band up to 30 % at the >95th percentile band for Primary 3, Primary 5 and Secondary 1 students, and 50 % of Secondary 3 students. The estimation of I^sub A^ at the level of the individual allows the researcher the possibility to directly study the interaction of the environment on student potential. Key words: Mathematics, optimal achievement model, underachievement Introduction The study of student underachievement continues to diversify, including the characterization of gifted underachieves (Dixon, Craven, & Martin, 2006), the EEG differences between students of average and above average intelligence (Staudt & Neubauer, 2006), motivational orientations amongst high- and under-achievers (Ziegler & Stoeger, 2004), selfregulation in gifted mathematics underachievers (Stoeger & Ziegler, 2005), students with emotional and behaviour disorders (Lane, Gresham, & O'Shaughnessy, 2002), students with learning disabilities (Fletcher, Denton, & Francis, 2005), and the relationship between parenting styles and achievement (Jacobs & Harvey, 2005). Despite general agreement that underachievement is the discrepancy between what can be expected and what is actually achieved, there is a growing realization that the current methods used to identify underachievement are themselves problematic (Fletcher et al, 2005; Lau & Chan, 2001; Phillipson & Tse, 2007; Stoeger & Ziegler, 2005; Ziegler & Stoeger, 2003) with some studies beginning to propose alternate methods (Fletcher et al., 2005; Lau & Chan, 2001; Phillipson & Tse, 2007). At a fundamental level, this article argues that our understanding of the conceptual basis of underachievement is not well defined, and that there is a need to rethink the nexus between potential and achievement. Furthermore, it is significant that over-achievement has not received the same degree of attention as underachievement, except that they are seen as opposite ends of a continuum, with the achievement of intellectual potential being arbitrarily defined as the mid-point between the two. However, any conceptual basis for the detection of underachievement must be a unified model, including explanations for both the phenomenon of under- and over-achievement. Furthermore, there appears to be little adherence to the requirements of fundamental measurement when using instruments that address issues of underachievement, despite progress in related domains such as intelligence (Styles, 1998; van der Ven & Ellis, 2000; Vigneau & Bors, 2005), science education (Liu & Boone, 2006) and school achievement (Lissitz, 2005). These measurement requirements include the need for unidimensionality of the measurement instrument, invariance in the order of the items used to construct the instrument, and the units of measurement should correspond to an interval level scale (Bond & Fox, 2001, 2007; Styles, 1999). In the development of measurement models that meet these requirements, Georg Rasch (1980) and others ( see Bond & Fox, 2007; Smith & Smith, 2004) have developed the methodology to assess the specific objectivity of the test instruments that produce dichotomous (True/False) data and polytomous data, for example, and to transform raw counts of participant responses from these instruments into interval level measures. …