Examining Application Relationships: Differences in Mathematical Elements and Compound Performance between American, Japanese, and Taiwanese Students

Application refers to the process of behavioral elements, sometimes referred to as component behaviors, combining to form a behavioral compound, also referred to as a composite behavior (Binder, 1996; Haughton, 1980). Figure 1 shows how element behaviors can combine to form a compound behavior. In the example, two discrete elements behaviors, (A) using finger on right hand to press a piano key and (B) reading the C note, may exist in a person's repertoire. When they combine the new behavior (AB) is a behavioral compound, seeing a C note on a piece of sheet music and pressing the correct key on the piano. Other behavioral compounds may have two, three or more elements. Application, like any synthesis reaction in chemistry when the combination of two or more substances results in a compound, requires activation energy for the two behaviors to combine. The activation energy necessary in application appears in the form a frequency measure (Johnson & Pennypacker, 1993). This initial discovery came from Haughton (1972) who observed one student performing multiplication facts (i.e., multiplying 0--5 facts) at a frequency of 30 correct per minute and another answering facts below 20 correct per minute. The first student went on to successfully answer more complex math facts (i.e., mixed multiplication facts) while the second student failed to learn the more complex math facts. In other words the higher frequency permitted the application of the skill element multiplication of basic facts, to a skill compound, multiplication of mixed facts. [FIGURE 1 OMITTED] Haughton's seminal observation of application coincides with the collision theory. With chemical reactions, reactants must collide to break their existing bonds and form new bonds resulting in a product. If too little energy exists, molecules will bounce off one another and not combine (Brown, LeMay, & Bursten, 1994). Haughton (1972) indicated that even though both students could perform the element behavior with some degree of accuracy, the math fact performance in the second student would not successfully apply to the compound behavior because it did not possess the necessary frequency. Based on the application concept and a number of previous application studies (Berens, Boyce, Berens, Doney, & Kenzer, 2003; Bucklin, Dickinson, & Brethower, 2000; Chiesa & Robertson, 2000; Evans & Evans, 1985; Evans, Mercer, & Evans, 1983; Kubina, Young, & Kilwein, 2004; McDade, Rubenstein, & Olander, 1983; McDowell & Keenan, 2001; McDowell, & Keenan, 2002; McDowell, Keenan, & Kerr, 2002; McDowell, McIntyre, Bones, & Keenan, 2002; Smyth, & Keenan, 2002), Lin and Kubina (2005) hypothesized that a relationship should exist between elements and compound behaviors in mathematical behavior. In their study Lin and Kubina examined the frequency of one behavioral element, basic facts in multiplication (i.e., x 0--9 facts), and the compound behavior of complex multiplication facts (i.e., multiplication problems involving single, two-digit, and three-digit factors). After measuring the 1minute frequencies of the 156 fifth grade students' single and multi-digit multiplication facts, a positive .745 correlation between the element and compound skill was found, which supports the essential role of basic skill frequencies in contributing to the learning in compound behaviors. Additionally, when comparing the 5th grade sample against a performance standard for basic fact fluency (i.e., 80120 correct digits per minute, Mercer, Mercer, & Evans, 1982), 14% of the sample met or exceeded the criterion. When contrasting the 5th grade students with the performance standard for multi-digit fact fluency (i.e., 40-60 correct digit per minute, Kubina & Lin, 2003) only 3% met or exceeded the criterion. Such dataindicate that like chemical reactions, with the minimum energy necessary to initiate a reaction varying from reaction to reaction (Brown, et al. …