Using the concrete-representational-abstract approach to support students with intellectual disability to solve change-making problems
Introduction
Functional (or life skills) mathematics – the mathematics related to living, working, participating, and accessing services in everyday life – is important for all individuals, but particularly students with disabilities (Burton, Anderson, Prater, & Dyches, 2013). Functional mathematics, like all functional domains (e.g., vocational skills and daily living skills), teaches students to maximize their own independence (Bouck & Joshi, 2012). Additional research, such as this study, is needed to find research-based approaches to teaching functional mathematical content to students with intellectual disability.
One important domain of life skills mathematics is purchasing skills (Alwell & Cobb, 2009; Xin, Grasso, Dipipi-Hoy, & Jitendra, 2005). Purchasing skills refer to the skills related to engaging in financial transactions to obtain services or goods (Browder, Spooner, & Trela, 2011); purchasing skills include such skills as navigating a store, comparing prices, and using money (e.g., making change; Mechling & Gast, 2003). Previous researchers suggested the value in helping students to become more independent with purchasing skills, including making change (c.f., Cihak & Grim, 2008). Cihak and Grim (2008) successfully taught purchasing skills to secondary students with autism and intellectual disability via the counting-on and next-dollar strategies. The students acquired as well as maintained and generalized the skills. In another study, Burton et al. (2013) found video self-modeling as an effective instructional strategy to teach students with autism and intellectual disability to estimate the amount they would need to pay for an item as well as the change they would receive.
As found by Burton et al. (2013), video modeling was an effective strategy for helping students with autism and intellectual disability determine the change they would receive from making a purchase. Video modeling is considered an evidence-based practice (Wong et al., 2014) and is used extensively to provide instruction to students with disabilities, particularly students with autism and intellectual disability (Ayres, Mechling, & Sansosti, 2013; Odom et al., 2015), including in mathematics. For example, Yakubova, Hughes, and Hornberger (2015) successfully used video modeling to teach three high school students with autism to solve mixed fraction word problems. However, video modeling possesses limitations as well as benefits (e.g., effective strategy). For some schools the implementation of video modeling can be cost prohibitive, as they would need to purchase tablets (Weng & Bouck, 2014). Teachers have also expressed concerns that creating video models is both time consuming and challenging (Carnahan, Basham, Christman, & Hollinghead, 2012; Weng, Savage, & Bouck, 2014). This study examined an alternative approach to the effective and evidence-based approach of video modeling with regards to supporting students with intellectual disability in learning functional mathematics content, to provide teachers options to consider when teaching such mathematical content.
Researchers sought to find other effective mathematical strategies for students with disabilities. For example, some researchers explored the Concrete-Representational-Abstract (CRA) approach to teaching mathematics to students with autism and/or intellectual disability (Flores, Hinton, Stroizer, & Terry, 2014; Stroizer, Hinton, Flores, & Terry, 2015; Yakubova, Hughes, & Shinaberry, 2016). The previous studies of the CRA approach involving students with intellectual disability and/or autism suggest the benefit of such approach, with all demonstrating a functional relation between the CRA approach and students’ solving mathematical problems. However, the three previous studies examining the CRA approach for students with intellectual disability and/or autism involve basic operations, such as addition, subtraction, and multiplication (Flores et al., 2014, Stroizer et al., 2015; Yakubova et al., 2016). To date, no study examines the CRA approach to support functional mathematical content, such as change-making problems with students with intellectual disability or other disabilities. Hence, this study fills an important niche in the research base for both the CRA instructional strategy and for teaching mathematical content to students with intellectual disability.
The CRA approach is a graduated sequence of instruction, which moves students from solving mathematical problems (e.g., subtraction with regrouping) with concrete manipulatives (e.g., base 10 blocks) to solving the problems with drawings or representations of the objects (e.g., lines and dots). Finally, students learn to solve the problems abstractly, without any support or aids (Agrawal & Morin, 2016). The CRA approach is built upon the instructional strategy in mathematics known as explicit instruction; explicit instruction is considered an effective or recommended instructional approach for students with disabilities (Gersten et al., 2009). When a teacher uses explicit instruction in mathematics s/he models how to solve mathematical problems, such as subtraction with regrouping, through demonstrations and think-alouds. Next, the teacher moves onto providing prompts and cues as needed when students solve the problems themselves (i.e., guided instruction). Last, the students engage in independent practice in solving the mathematical problems (Doabler & Fien, 2013).
The CRA begins with teachers teaching students to solve a type of mathematics problem (e.g., multiplication with regrouping) with concrete manipulatives. To do so, teachers use the principles of explicit instruction to first model (i.e., demonstrate how to solve with the concrete manipulatives and use think-alouds to explain) with a few problems. Next, teachers guide students as they solve a few problems with the concrete manipulatives, meaning they provide prompts and cues as needed. Finally, students solve some mathematical problems independently with the concrete manipulatives. The same procedures are repeated for the representational phase in which students draw pictures or images to represent the concrete manipulatives, and then finally the abstract phase where students solve the problems without additional supports (Agrawal & Morin, 2016; Mancl, Miller, & Kennedy, 2012). Generally, students need to achieve 80% correct or better across three sets of independently solved math problems to move from concrete to representational and then from representational to abstract (Mancl et al., 2012).
The CRA approach has an extensive research base, spawning multiple decades and across different mathematical skill areas (e.g., place value, fractions, subtraction with regrouping, algebra) (Bouck & Park, 2016). Across the multiple studies – both single case design studies and group design studies, researchers found students were able to make gains in their ability to solve the mathematics problems when using the CRA approach (c.f., Butler, Miller, Crehan, Babbitt, & Pierce, 2003; Flores, 2010; Maccini & Hughes, 2000; Mancl et al., 2012; Witzel, Mercer, & Miller, 2003). However, the majority of the research on CRA was conducted with students with learning disabilities. The majority of the research on CRA also focuses on largely academic mathematical concepts, such as place value, basic operations (e.g., subtraction, multiplication), and algebra (Bouck & Park, 2016; Bouck, Satsangi, & Park, 2016). Little-to-no research exists that explores the CRA approach to support students with disabilities in learning more functional-based mathematics, such as purchasing skills, and limited research (n = 3) examines the CRA instructional sequence for students with intellectual disability. Yet, the CRA approach is an evidence-based practice for students with learning disabilities and works to build conceptual understanding – a goal of all mathematics education, including functional mathematics (Bouck et al., 2016; Mancl et al., 2012). This study sought to explore the potential of an evidence-based mathematical approach for students with learning disabilities to educate students with intellectual disability in functional mathematics content.
As a result of the potential CRA instruction to support students in understanding mathematic concepts and solving real-world problems, this study investigated how students with intellectual disability solved word problems involving making change with coins with the CRA approach. The research questions for this study include: (a) To what extent do secondary students with disabilities improve their performance on solving change-making with coins word problems following instruction via the CRA sequence?; (b) To what extent do secondary students with disabilities maintain performance on solving change-making with coins word problems two weeks after instruction ends?; and (c) To what extent do secondary students with disabilities and their teacher find the CRA strategy beneficial?
Section snippets
Participants
Four middle school students participated in the study. The same teacher taught all students in the same pullout special education classroom. The middle school the students attended provided three pullout special education options; the teacher reported her class was the middle option, meaning the students’ functioning was deemed between the higher level functioning room in which more academics were taught and the lower functional pullout classroom in which the total focus was on life skills
Results
CRA instruction results showed an increased percentage of accurately solved problems for all four students as compared to their baseline levels (see Fig. 1). In other words, a functional relation existed between students being taught via the CRA instructional strategy and students’ accurately solving change-making problems. There was a zero overlap in all students’ baseline and intervention data (Tau-U = 100% for all students). All four students maintained solving making change with coins for up
Discussion
This study explored the effectiveness of CRA instruction to teach four middle school students with disabilities to solve change-making with coins problems. Given the repeated demonstrations of the effectiveness of the CRA approach to support students with disabilities in learning different mathematical concepts (c.f., Flores, 2010; Flores et al., 2014, Witzel et al., 2003, Yakubova et al., 2016), the researchers explored the use for more functional-based mathematical content. In this study, the
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