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Abstract
Purpose of Study: Comparison plays a critical role in the cognitive process (Goldstone, Day, & Son, 2010; Kotovsky & Genterner, 1996), and supports effective learning of mathematics (Ritter-Johnson & Star, 2009; Star & Ritter-Johnson, 2009). Although comparisons are widely used in US classrooms, they are often used with minimal efficiency (Richland et al., 2007, Richland et al., 2012). This study takes a cross-cultural perspective systematically examining comparisons in US and Chinese classrooms. We focus on inverse relations (CCSSI, 2010) allows for an in-depth analysis.
Theoretical Framework: Star and Ritter-Johnson (2009) suggest that effectiveness of comparison lies on “what” to compare. Among three types of comparisons—different solutions/same problem, different problem/same structure, and different problem/different structure—they found comparing solutions most effective for algebraic computations. We seek to determine if this finding holds true for other topics.
The goal of comparisons (“why”) might include similarities or differences involving a literal or structural level. However, deep learning only occurs when comparisons target structural information (Richland et al., 2012).
Comparisons can be made by teachers, students, or jointly as a class. Since effective learning calls for students’ thinking, it matters “who” makes comparisons.
To enhance effectiveness, teachers should consider the “how” of comparisons (Richland et al., 2012). Visual representations (e.g., diagrams) are frequently used in high-achieving classrooms (Richland et al., 2007), and expert teachers often ask deep questions to facilitate comparisons (Chen & Ding, 2016).
The effectiveness of comparisons depends on prior knowledge (Ritter-Johnson & Star, 2009), and thus “when” comparisons occur (e.g., review, new lesson, or practice) is critical.
Methods: Our study analyzes 4 lessons on inverse relations from 8 US and 8 Chinese expert elementary teachers. All 64 videotaped lessons were transcribed and coded as follows:
1. Identified segments of discourse of comparison (episodes);
2. Analyzed five aspects for each episode: what, why, who, how, and when;
3. Quantified the qualitative data to identify aspect percentages,
Partial Results and Conclusions: Table 1 presents the results of one Chinese and one US teacher. We note the following similarities and differences.
• “What” All three comparisons were used in both classrooms, but the Chinese classroom contained more comparisons on different problems than solutions. This is different from the literature (Star & Ritter-Johnson, 2009), possibly because our targeted topic was different.
• “Why” Although both expert teachers focused on structural information, the Chinese teacher focused more on structural similarities; US more on differences.
• “Who” Comparison in the Chinese classroom always involved students, and were often suggested by students themselves. In contrast, the US teacher often conducted comparison for students.
• “How” The Chinese teacher asked questions on concrete representations (e.g., tape diagrams) to facilitate comparisons. The US teacher sometimes provided explanations without the use of aids.
• “When” Comparisons tended to occur during the Chinese teacher’s worked examples, whereas more comparisons occurred during US practice problems.
Scientific Significance: Our findings suggest that instead of making comparisons for students, it is important for teachers to ask deep questions about concrete representations in order to encourage structural comparisons by students.
References:
Chen, W., & Ding, M. (2016, April). Transitioning textbooks into classroom teaching: An action research on Chinese elementary mathematics lessons. Proposals presented at 2016 AERA conference. Washington, DC.
Goldstone, R. L., Day, S., & Son, J. Y. (2010). Comparison. In B. Glatzeder, V. Goel, & A. von Müller (Eds.) On thinking: Volume II, towards a theory of thinking (pp. 103-122).
Heidelberg, Germany: Springer Verlag GmbH. Kotovsky, L., & Gentner, D. (1996). Comparison and categorization in the development of relational similarity. Child Development, 67, 2797–2822.
Richland, L. E., Zur, O., & Holyoak, K. J. (2007). Cognitive supports for analogies in the mathematics classroom. Science, 316, 1128–1129.
Richland, L. E., Stigler, J. W., & Holyoak, K. J. (2012). Teaching the conceptual structure of mathematics. Educational Psychologist, 47(3), 189-203.
Rittle-Johnson, B., & Star, J. R. (2009). Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology, 99, 561–574.
Star, J. R., & Rittle-Johnson, B. (2009). It pays to compare: An experimental study on computational estimation. Journal of Experimental Child Psychology 102, 408–426.