The Development of Proportional Reasoning: Effect of Continuous Versus Discrete Quantities

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Neurocognitive Architectures and the Nonsymbolic Foundations of Fractions Understanding (2016, unknown): “Because these types of ratios are inherently uncountable, their use should help prevent the inappropriate application of whole-number knowledge (see also, Boyer et al., 2008; Jeong et al., 2007).”
The development of students’ use of additive and proportional methods along primary and secondary school (2012, background): “instead of a comparison format, for which stronger effects have been reported in previous research (Boyer et al., 2008; Jeong et al., 2007).”
How to make ‘more’ better? Principles for effective use of multiple representations to enhance students’ learning about fractions (2017, background): “This finding suggests that young children possess the necessary perceptual abilities for rudimentary proportional reasoning but that, partitioned representations are less effective than continuous representations for supporting this reasoning (see also Boyer and Levine 2012; Jeong et al. 2007).”