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Ricardo Martinez and Ji Yeong I

Ratios and proportions are important concepts that occur in real life, but they are difficult to learn and complicated to teach (Lamon 2007). In general, proportional reasoning in the middle grades is an area in need of attention because of its connection to later concepts (Ojose 2015). In this article, we explain the meanings of ratios, proportions, and equivalent ratios and then provide useful methods to determine equivalent ratios using students' examples. Because multiple definitions exist, it is necessary to determine definitions to avoid possible confusion. Johnson (2010) defines a ratio as a pair of positive nonzero real numbers such that there are a units for every b units, written a:b; read a to b; and represented as a fraction, a/b. We follow definitions suggested by the writers of the Common Core (CCSSI 2010): Fraction representation is known as the associated unit rate or the value of a ratio A:B and is found by dividing B into A, where “equivalent ratios have the same unit rate” (McCallum, Zimba, and Daro 2011). Using both contextual and noncontextual proportion tasks, we saw various ways that students found the equivalency of ratios as well as misconceptions they have.

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Hyunyi Jung and I Ji-Yeong

You are asked to participate in a contest to design a parking lot in the downtown area of your city. The space is limited, and the civic design department wants to maximize the number of cars that can be safely parked in the space. The required conditions are specified below.

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Ji-Yeong I and Jasmine Stanford

Using visuals is a well-known strategy to teach emergent bilinguals (EBs). This study examined how preservice teachers (PSTs) implemented visuals to help EBs understand mathematical problems and how an innovative intervention cultivated PSTs' capability of using visuals for EBs. Four middle school mathematics PSTs were engaged in a _ eld experience with EBs to work on mathematical problems; during the _ eld experience, the PSTs received interventions. In one intervention session, the PSTs were asked to make sense of a word problem written in an unknown language with different visuals. After this intervention, they changed their use of visuals when modifying tasks for EBs. The results suggest that immersive experiences where PSTs can experience learning from the perspective of EBs helps PSTs implement mathematically meaningful visuals in a way that makes mathematical problems accessible to EBs.

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Ji Yeong I, Barbara J. Dougherty and Zaur Berkaliev

The meat and potatoes of multiplication is the change of units.