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The Importance of Making Links Visible

- Begolli, K. N., Richland, L. E. (2015). Teaching Mathematics by Comparison: Analog Visibility as a Double-Edged Sword.
*Journal of Educational Psychology*, 107(3), No Pagination Specified. Retrieved September 3, 2015. [DOI]

The spatial organization of visual information in mathematics plays an important and often unrecognized role in classroom learning. Visual information in mathematics instruction includes, for example, the form of problems or formulas written on the board, presentations of multiple students’ solutions to a single problem, manipulatives used to show connections between mathematics and physical instantiations, or gestures used to make instructional aims more visible (see Figure 1A-C for examples). While using visual information may not always be a conscious part of teachers’ practices, the spatial literature would suggest that it is important to student learning, and may impact whether students gain usable knowledge from instruction and are able to use that content in new settings (Alfieri, Nokes-Malach, & Schunn, 2013).

**Figure 1.** A, B, C.

In one study, we investigated whether spatial organization used by teachers would make a difference to student learning in the context of a 5th grade lesson on rate and ratio. The lesson focused on a highly recommended instructional practice: leading mathematical discussions to compare student solution strategies to a single mathematics problem (Carpenter et al., 1999; Groβe & Renkl, 2006; Siegler, 2002; Smith & Stein, 2011). Specifically, comparing and contrasting solution arguments is posited to foster expert-like, flexible, and conceptual mathematical knowledge (NRC, 2001; National Mathematics Panel, 2008; Schwartz & Bransford, & Sears, 2015), partly by supporting explanation, developing more integrative mathematical content knowledge, or reducing use of misconceptions. This recommendation is evident in the Common Core Mathematics Standards within several of the practice standards, as well as in the more integrative content standards (Common Core State Standards Initiative, 2010).

At the same time, leading these classroom discussions is highly challenging to all but very experienced teachers (Stein et al., 2008), and many U.S. teachers do not capitalize on opportunities for relational reasoning in their classrooms (Hiebert et al, 2003; Richland, Zur, & Holyoak, 2007). More broadly, despite decades of agreement that teaching students mathematics that is flexible, transferrable, and connected across topics is crucial to high quality instruction, U.S. education is not yet successfully teaching all students such mathematical thinking (National Mathematics Panel, 2008). Capitalizing on the best ways to implement visual information in the classroom may be a key contribution to bridging this gap.

Using a novel video-based paradigm, we videotaped a teacher leading a class in comparing three different solutions to a single ratio problem and edited the video to create three versions of the lesson. These versions varied the visibility of the solutions during instruction, while keeping the verbal information constant.

**Figure 2.**

Data from this study supported our hypothesis that manipulating the visual information impacted student learning. As shown in Figure 2, gain scores revealed large differences between the conditions, despite all including the same audio stream. Most importantly, enabling spatial comparison by making all solutions visible simultaneously led to the greatest gains in conceptual knowledge and transfer, and the lowest rates of misconception use.

However, this advantage does not seem to stem merely from the visibility of the solutions, but rather their simultaneous spatial alignment. When the same solutions were presented sequentially, so that only one was visible at a time, students actually had fewer gains and higher rates of misconceptions than seeing no solutions visible at all. Thus presence of visual representations during a discussion comparing multiple solutions to a problem can serve as a double-edged sword; visual information that is not aligned spatially could actually create more misconceptions and reduce learning.

**Figure 3.**

Specifically, the ability to see all compared representations simultaneously throughout the discussion increased the likelihood of schema formation and optimized learning when compared with seeing representations only sequentially. This was evidenced by greater ability to: 1) use taught procedures, 2) understand multiple accurate solution strategies and select the most efficient strategy, 3) explain and use the concepts underlying taught mathematics, and 4) minimize use of a misconception.

Strikingly, presenting mathematical solutions sequentially led to even lower rates of learning than having no visual representations present during any of the comparison episodes, though these differences were not present on all measures. Having the solution strategies presented only verbally (Not Visible condition) did lead to some retention of taught procedures and some schema formation, but not as universally as in the All Visible condition. At the same time, participants in the Not Visible condition were less likely to produce the misconception than in the Sequential condition, suggesting that they did not retain the instructed representations as well or uncritically as in the Sequential condition. It may be that having no visible solutions was most effortful for students, and thus only those students who were able to perform the necessary effort showed strong learning.

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