From Non-symbolic to Symbolic Proportions and Back: A Cuisenaire Rod Proportional Reasoning Intervention Enhances Continuous Proportional Reasoning Skills

Abstract

The persistent educational challenges that fractions pose call for developing novel instructional methods to better prepare students for fraction learning. Here, we examined the effects of a 24-session, Cuisenaire rod intervention on a building block for symbolic fraction knowledge, continuous and discrete non-symbolic proportional reasoning, in children who have yet to receive fraction instruction. Participants were 34 second-graders who attended the intervention (intervention group) and 15 children who did not participate in any sessions (control group). As attendance at the intervention sessions was irregular (median = 15.6 sessions, range = 1–24), we specifically examined the effect of the number of sessions completed on their non-symbolic proportional reasoning. Our results showed that children who attended a larger number of sessions increased their ability to compare non-symbolic continuous proportions. However, contrary to our expectations, they also decreased their ability to compare misleading discretized proportions. In contrast, children in the Control group did not show any change in their performance. These results provide further evidence on the malleability of non-symbolic continuous proportional reasoning and highlight the rigidity of counting knowledge interference on discrete proportional reasoning.

Middle-schoolers' misconceptions in discretized nonsymbolic proportional reasoning explain fraction biases better than their continuous reasoning: Evidence from correlation and cluster analyses

“…The mismatch between students' areas of difficulty and instruction may explain null results in intervention studies. For example, a recent intervention study targeting nonsymbolic continuous proportional skills in second graders using physical manipulatives (Abreu-Mendoza et al, 2021) showed that students improved their continuous proportional skills; however, their discretized skills declined after the intervention. As second graders are more likely to show a strong whole-number bias (Abreu-Mendoza et al, 2020;Jeong et al, 2007), a possibility is that an intervention targeting misconceptions about proportions (e.g., overgeneralization of whole-number rules) might have been more effective.…”

“…For instance, before formal instruction, children can compare and match objects based on their proportional magnitudes (Boyer et al, 2008;Hurst & Cordes, 2018;Jeong et al, 2007). Combined with correlational evidence for the positive relationship between nonsymbolic and symbolic proportional reasoning (Begolli et al, 2020;Matthews et al, 2016;Möhring et al, 2016), these findings have led researchers to develop fraction instructions that scaffold on nonsymbolic representations (Abreu-Mendoza et al, 2021;Braithwaite & Siegler, 2021;Gouet et al, 2020;Hamdan & Gunderson, 2017); for a review, see Abreu-Mendoza and Rosenberg-Lee, (2022). However, the mechanisms underlying the relationship between nonsymbolic and symbolic proportional reasoning remain unknown.…”

Non-symbolic Ratio Reasoning in Kindergarteners: Underlying Unidimensional Heuristics and Relations With Math Abilities

“…That link between ratio reasoning and fraction understanding is also supported by studies that have specifically investigated whether ratio reasoning can be harnessed and whether improvements in ratio reasoning translate to improvements in math and fraction understanding. For instance, Szkudlarek and Brannon (2021) found that non-symbolic ratio reasoning may function as a scaffold for symbolic ratio reasoning in children who lack an understanding of fractions (see also Gouet et al, 2020;Abreu-Mendoza et al, 2021). Other training studies involving mapping non-symbolic continuous representations of proportions with fractions have resulted in improvements in fraction knowledge in older children (Fazio et al, 2016;Soni and Okamoto, 2020).…”

Measuring Fences and Sharing Pizzas: Current Advances in Nonsymbolic Fraction Interventions

“…First, we focused on the six papers that employed numberlines as their primary instructional tool (Numberline Interventions, Figure 2), in which participants were students who had already received some fraction instruction (Barbieri et al, 2020;Bush, 2021;Dyson et al, 2020;Fuchs et al, 2016;Jayanthi et al, 2021;Malone et al, 2019). Then, we described one paper that centered on developing an understanding of proportional magnitudes using nonsymbolic representations of children who have not received fraction instruction, (Nonsymbolicto-Symbolic Intervention, Figure 2) (Abreu-Mendoza et al, 2021). Fractions are formally introduced in the third grade in the U.S., although part-whole relationships are presented as early as first grade (Common Core State Standards Initiative, 2020).…”