## Key Ideas > [!abstract] Core Concepts > > - Examining a diamond from different angles reveals its full beauty; mathematics has similar multifaceted beauty revealed through different approaches > - Comparing different methods helps students recognise the most elegant or appropriate approaches > - Can be built into practice sequences or used when showcasing student work ## Definition Solution comparison involves examining and evaluating different mathematical approaches to the same problem to develop understanding of efficiency, elegance, and appropriateness of various methods (Rittle-Johnson & Star, 2007). ## Connected to [[Practice]] | [[Mini-Whiteboards]] | [[Concrete Pictorial Abstract]] | [[Activity-Based Curriculum]] --- ## Purpose and value Eddie Woo uses a diamond analogy to describe mathematical understanding: looking at a diamond from one angle is insufficient, so we twist and turn it to appreciate its beauty. Mathematics has similar multifaceted beauty revealed through different approaches (Marton & Booth, 1997). However, looking at a diamond from many angles does not necessarily make one a better jeweller. See [[Activity-Based Curriculum|18×5]] - multiple methods do not automatically improve mathematical ability (Silver et al., 2005). Appreciation differs from expertise. ## Personal teaching example One teacher devised a question where students would use Pythagoras' theorem to find the area of a square given its diagonal. An insightful student used the area of a kite to solve the problem more efficiently. The student's alternative method was more elegant than the intended approach, demonstrating the value of comparing solutions. ## Implementation strategies Solution comparison can be planned or opportunistic. Teachers can build it into [[Practice]] sequences by including problems designed to elicit different solution methods, anticipating various approaches students might use, and ordering problems strategically to highlight efficiency differences between methods. Opportunistic use occurs when using [[Mini-Whiteboards]] to showcase different student approaches, capitalising on unexpected student methods as learning opportunities, and helping students evaluate which methods work best in different contexts. ## Educational benefits Students learn to assess the efficiency of different approaches (Rittle-Johnson & Star, 2007) and recognise when certain methods are more appropriate than others (Hiebert & Grouws, 2007). This develops mathematical judgement and strategic thinking. Seeing multiple approaches reinforces conceptual understanding (Rittle-Johnson & Star, 2009). Students develop mathematical flexibility and creativity (Star & Rittle-Johnson, 2009), and connections between different mathematical representations become clearer (Ainsworth, 2006; Cobb et al., 1992). ## Key considerations Quality matters more than quantity. Teachers should focus on genuinely different approaches rather than minor variations that provide little insight. The focus should be on helping students recognise the most elegant or efficient methods for specific contexts. Some methods are better suited to particular problem types or mathematical contexts, so strategic selection matters more than knowing many methods. ## References Ainsworth, S. (2006). DeFT: A conceptual framework for considering learning with multiple representations. *Learning and Instruction*, 16(3), 183-198. https://doi.org/10.1016/j.learninstruc.2006.03.001 Marton, F., & Booth, S. (1997). *Learning and awareness*. Lawrence Erlbaum Associates. Rittle-Johnson, B., & Star, J. R. (2007). Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. *Journal of Educational Psychology*, 99(3), 561-574. https://doi.org/10.1037/0022-0663.99.3.561 Rittle-Johnson, B., & Star, J. R. (2009). Compared with what? The effects of different comparisons on conceptual knowledge and procedural flexibility for equation solving. *Journal of Educational Psychology*, 101(3), 529-544. https://doi.org/10.1037/a0014224 Star, J. R., & Rittle-Johnson, B. (2009). It pays to compare: An experimental study on computational estimation. *Journal of Experimental Child Psychology*, 102(4), 408-426. https://doi.org/10.1016/j.jecp.2008.11.004 Silver, E. A., Ghousseini, H., Gosen, D., Charalambous, C., & Strawhun, B. T. F. (2005). Moving from rhetoric to praxis: Issues faced by teachers in having students consider multiple solutions for problems in the mathematics classroom. *Journal of Mathematical Behavior*, 24(3-4), 287-301. https://doi.org/10.1016/j.jmathb.2005.09.009 Hiebert, J., & Grouws, D. A. (2007). The effects of classroom mathematics teaching on students' learning. In F. K. Lester (Ed.), *Second handbook of research on mathematics teaching and learning* (pp. 371-404). Information Age Publishing. Cobb, P., Yackel, E., & Wood, T. (1992). A constructivist alternative to the representational view of mind in mathematics education. *Journal for Research in Mathematics Education*, 23(1), 2-33. https://doi.org/10.2307/749161