The impact of a conceptual model-based mathematics computer tutor on multiplicative reasoning and problem-solving of students with learning disabilities☆
Introduction
According to the 2017 National Report Card (NAEP, 2017), the mathematics performance of fourth-grade students who are at the 10th and 25th percentile showed a decreasing trend from 2005 to 2017. The gap between these students and their high performing peers is getting wider rather than closing. There is a serious shortage of intervention programs that focus on building conceptual understanding of fundamental mathematical ideas--concepts that are essential to enabling struggling students to understand and solve mathematics problems. In particular, little research has been done on identifying and nurturing multiplicative reasoning and problem solving for students with learning disabilities (LD). Multiplicative reasoning (MR), a foundational skill from which students proceed to more advanced mathematical thinking, is developed through students’ conceptual understanding of multiplication (Carrier, 2010).
Based on existing literature, two indicators can be used to evaluate students’ developmental level of multiplicative reasoning: (a) students’ counting schemes, and (b) their cognitive levels of operations (Carrier, 2010; Clark & Kamii, 1996; Steffe, 1988; Steffe & Cobb, 1988). A counting scheme in multiplication refers to students’ counting acts while solving multiplicative word problems. According to the constructivist point of view, students undergo numerous adjustments to re-establish their counting acts through their personal experience of counting (Steffe, 1994). According to Steffe and Cobb’s study (1988), elementary students undergo several stages of number sequence development as they encounter various numerical situations. Particularly, students continually refine the notion of units from singletons: “1, 2… 3, 4… 5, 6” to iterable units: “2, 4, 6” (one iterated two times leads to one two's, that can also be broken down to two ones) to forming composite units (CU): “2 × 3 is two units of three” (Steffe, 1992). Students in this stage can mentally understand that three ones are taken as one three (Steffe, 1992). Composite unit (CU) can be defined as a set of equal quantity of singletons (1 s) (Carrier, 2010; Clark & Kamii, 1996; Steffe, 1988; Steffe & Cobb, 1988). Students who have acquired multiplicative reasoning will be able to identify units of one (singletons) and units of units (composite units) and understand the relation between the two quantities (Steffe & Cobb, 1988). In particular, to understand multiplication and division, a child would need to view a collection or a group of individual items as one thing (the composite unit) and be able to coordinate a number of equal sized groups (Sullivan, Clarke, Cheeseman & Mulligan, 2001). Acquisition of an equal–grouping structure is at the core of multiplicative reasoning (Downton, 2008).
On the other hand, cognitive levels of operation indicate students’ different levels of representation when engaging in intellectual work. In the case of mathematics learning, prominent educators (Bruner, 1996; Inhelder and Piaget, 1958) believed that mathematics is learned by going through a sequence of concrete, semi-concrete, and abstract stages (Miller & Mercer, 1993). According to Olive (2001), students also undergo concrete, semi-concrete, and abstract stages (CSA) stages as they continue to internalize their counting acts. Students would represent their counting acts by using the concrete manipulative (e.g., unifix cubes) and gradually progress to semi-concrete (e.g., drawing, pictures) and to abstract level (e.g., math sentence).
To meet current challenging high academic standards such as mathematics Common Core, it is important for students to establish the concept of CU, coordinate the units between the CU and 1 s, and move and beyond the concrete level of operation to symbolic representation of mathematical relations for generalized problem-solving skills. To date, little is known about how the counting scheme-focused multiplicative reasoning instruction affects students with LD on their mathematics word problem solving.
Through the support from the National Science Foundation (NSF)1, PGBM-COMPS, a computer tutor by Xin, Tzur, and Si (2008) along with the research team, was developed to nurture multiplicative reasoning (MR) of students with learning disabilities (LD). The PGBM-COMPS computer tutor draws on three research-based frameworks: a constructivist view of learning from mathematics education (Steffe, 1994), data (or statistical) learning from computer sciences, and Conceptual Model-based Problem Solving (COMPS, Xin, 2012) that generalizes word-problem underlying structures from special education. Rooted in guided constructivist perspective (Mayer, 2004) on learning, we focused on how a student-adaptive teaching approach (Steffe, 1990), which tailors goals and activities for students’ learning to their available conceptions, can foster advances in multiplicative reasoning.
The PGBM-COMPS computer tutor approaches the fundamental MR concepts in an explicit manner. The Please Go Bring Me (PGBM) component engaged students with mathematics activities where they manipulated the unifix cubes to form same-sized towers. These activities provided students the chance to build their understanding of CU. On the other hand, the Conceptual Model-based Problem Solving (COMPS) component (Xin, Wiles, & Lin, 2008; Xin, 2012) emphasized modeling the multiplicative concept through translating the mathematical relation within a word problem into an algebraic model equation.
This study attempted to investigate the effect of the PGBM-COMPS computer tutor on multiplicative reasoning and problem solving of students with LD. Specific research questions were: (a) was there a functional relationship between the PGBM-COMPS computer tutor and students’ multiplicative word problem-solving performance? and (b) how did the computer tutor affect participating students’ multiplicative reasoning, measured by their counting schemes and cognitive levels of operation?
Section snippets
Participants & settings
Participants were three, third and fourth grade students who had school-identified learning disabilities (LD) from a Midwest urban public elementary school in the United States. Participant selection was based on: (a) school identification of students experiencing substantial problems in mathematics word-problem solving; and (b) scores below the 30th percentile on the Mathematics Problem Solving subtest of the Stanford Achievement Test (SAT-10, Harcourt Assessment Inc., 2004; Bryant et al., 2011
MR criterion tests and module specific probes
The second author scored all the test sheets, including the MR criterion test and the module-specific probes. Then, a graduate student in special education who was naïve to this experiment and the assessments, conducted inter-rater reliability check on both MR-criterion test and module specific probes. The graduate student independently scored and coded 30 % of all the test sheets and sessions using the keys developed by the research team. An agreement on problem solving accuracy was made when
Effects on word problem solving performance
The present study aimed to evaluate the impact of the PGBM-COMPS computer tutor on multiplicative problem solving and reasoning of students with LD. Overall, the results showed a gradual increase in participating students’ word problem-solving performance from baseline to intervention and post-intervention phases. There was no overlapping of data points between the baseline and post-intervention across all three participants. This indicates a strong treatment effect (percentage of
Limitations
This study contained a few limitations. First, due to limited school days available for the study, it was difficult to allow each student to reach mastery during each phase of instruction before moving on to the next phase. As shown in Fig. 1, most participants did not reach 100 % during the Modules B and C & D phases. Most participants received only about three to four sessions of each problem types (e.g., UDS, MUC, QD, PD) before proceeding to the next module during the experiment. Second,
Implications for future research and practice
It is important for future research to extend this preliminary study by further improving the curriculum design in the PGBM-COMPS program. In particular, further effort is needed to (1) refine Module B instruction by breaking down the two-step problems into smaller tasks, or adjust the sequences of the modules (e.g., perhaps Module B should be scheduled after students’ learning of quotative division problems so that they are prepared to solve for the CU before combining the CUs as required in
Conclusion
The integration of heuristic instruction (the PGBM) and explicit model-based problem solving (COMPS) allowed students with LD to engage in constructivist-oriented learning. Given students’ fundamental concept of composite units, it is necessary to help students understand underlying problem structures and move toward mathematical model-based problem representation and solving for generalized problem solving skills. The findings from this study provide preliminary evidence to support the
CRediT authorship contribution statement
Yan Ping Xin: Funding acquisition, Conceptualization, Methodology. Joo Young Park: . Ron Tzur: Funding acquisition. Luo Si: Funding acquisition.
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This research was supported by the National Science Foundation, under grant DRL 0822296. The opinions expressed do not necessarily reflect the views of the Foundation. The authors would like to thank the administrators, teachers, and staff at Lafayette School Corporation, as well as many graduate students at Purdue University who facilitated this study.