Adults' Health-Related Problem Solving Is Facilitated by Number Lines, But Not Risk Ladders and Icon Arrays

Visual displays, such as icon arrays and risk ladders, are often used to communicate numerical health information. Number lines improve reasoning with rational numbers but are seldom used in health contexts. College students compared rates for information related to COVID-19 (e.g., number of deaths and number of cases) in one of four randomly-assigned conditions: icon arrays, risk ladders, number lines, or no accompanying visual display. As predicted, number lines facilitated performance on these problems – the number line condition outperformed the other visual display conditions, which did not perform any better than the no visual display condition. In addition, higher performance on the health-related math problems was associated with higher COVID-19 worry for oneself and others, higher perceptions of COVID-19 severity, and higher endorsement of intentions to engage in preventive health behaviors, even when controlling for baseline math skills. These findings have important implications for effectively presenting health statistics.

Promoting Discourse: Fractions on Number Lines

Examine fourth graders’ thinking about the unit, partitioning, order, and equivalence on the number line and consider ways to orchestrate mathematical discussions through the Five Practices.

Number line tasks and their relation to arithmetics in second to fourth graders

Considering the importance of mathematical knowledge for STEM careers, we aimed to better understand the cognitive mechanisms underlying the commonly observed relation between number line estimations (NLEs) and arithmetics. We used a within-subject design to model NLEs in an unbounded and bounded task and to assess their relations to arithmetics in second to fourth grades. Our results mostly agree with previous findings, indicating that unbounded and bounded NLEs likely index different cognitive constructs at this age. Bounded NLEs were best described by cyclic power models including the subtraction bias model, likely indicating proportional reasoning. Conversely, mixed log-linear and single scalloped power models provided better fits for unbounded NLEs, suggesting direct estimation. Moreover, only bounded but not unbounded NLEs related to addition and subtraction skills. This thus suggests that proportional reasoning probably accounts for the relation between NLEs and arithmetics, at least in second to fourth graders. This was further confirmed by moderation analysis, showing that relations between bounded NLEs and subtraction skills were only observed in children whose estimates were best described by the cyclic power models. Depending on the aim of future studies, our results suggest measuring estimations on unbounded number lines if one is interested in directly assessing numerical magnitude representations. Conversely, if one aims to predict arithmetic skills, one should assess bounded NLEs, probably indexing proportional reasoning, at least in second to fourth graders. The present outcomes also further highlight the potential usefulness of training the positioning of target numbers on bounded number lines for arithmetic development.

Plato’s Intellectual Development and Visual Thinking

Plato has devised texts which call the readers to collaborate cognitively with them. An important epistemic stimulation is the schematization, the line segment, which summarizes Plato’s idea of intellectual development. In this research, visual thinking will help us to make the most of the Platonic invitation to investigate further cognitive growth. It will be analyzed how visual discoveries are rendered possible by mental number lines, realizing the epistemological importance of visualization. Thanks to visualization, structuralism will be grasped. It will reveal a connection with Plato’s philosophy which suggests a novel elaboration of the Platonic concept of intellectual growth.

What Diagrams Are Considered Useful for Solving Mathematical Word Problems in Japan?

Previous studies have shown that diagram use is effective in mathematical word problem solving. However, they have also revealed that students manifest many problems in using diagrams for such purposes. A possible reason is an inadequacy in students’ understanding of variations in types of problems and the corresponding kinds of diagrams appropriate to use. In the present study, a preliminary investigation was undertaken of how such correspondences between problem types and kinds of diagrams are represented in textbooks. One government-approved textbook series for elementary school level in Japan was examined for the types of mathematical word problems, and the kinds of diagrams presented with those problems. The analyses revealed significant differences in association between kinds of diagrams and types of problems. More concrete diagrams were included with problems involving change, combination, variation, and visualization of quantities; while number lines were more often used with comparison and variation problems. Tables and graphs corresponded to problems requiring organization of quantities; and more concrete diagrams and graphs to problems involving quantity visualization. These findings are considered in relation to the crucial role of textbooks and other teaching materials in facilitating strategy knowledge acquisition in students.

Fixated in more familiar territory: Providing an explicit midpoint for typical and atypical number lines

Does providing an explicit midpoint affect adults’ performance differently for typical and atypical number line tasks? Participants ( N = 29) estimated the location of target numbers on typical (i.e., 0–10,000) and atypical (i.e., 0–7,000) number lines with either an explicitly labelled midpoint or no midpoint. For the typical number line, estimation accuracy did not differ for the explicit- and implicit-midpoint conditions. For the atypical number line, participants in the explicit-midpoint condition were more accurate than those in the implicit-midpoint condition and their pattern of error was similar to that seen for typical number lines (i.e., M-shaped). In contrast, for participants in the implicit-midpoint condition, the pattern of error on the atypical line was tent-shaped, with less accurate estimates around the midpoint and quartiles than the endpoints. Eye-tracking data showed that, for all number lines, participants used the middle of the line to guide their estimates, but participants in the explicit-midpoint condition were more likely to make their first fixation around the true midpoint than those in the implicit–midpoint condition. We conclude that adults have difficulty in estimating on atypical number lines because they incorrectly calculate the numerical value of the midpoint.

Fraction magnitude: Mapping between symbolic and spatial representations of proportion

Fraction notation conveys both part-whole (3/4 is 3 out of 4) and magnitude (3/4 = 0.75) information, yet evidence suggests that both children and adults find accessing magnitude information from fractions particularly difficult. Recent research suggests that using number lines to teach children about fractions can help emphasize fraction magnitude. In three experiments with adults and 9-12-year-old children, we compare the benefits of number lines and pie charts for thinking about rational numbers. In Experiment 1, we first investigate how adults spontaneously visualize symbolic fractions. Then, in two further experiments, we explore whether priming children to use pie charts vs. number lines impacts performance on a subsequent symbolic magnitude task and whether children differentially rely on a partitioning strategy to map rational numbers to number lines vs. pie charts. Our data reveal that adults very infrequently spontaneously visualize fractions along a number line and, contrary to other findings, that practice mapping rational numbers to number lines did not improve performance on a subsequent symbolic magnitude comparison task relative to practice mapping the same magnitudes to pie charts. However, children were more likely to use overt partitioning strategies when working with pie charts compared to number lines, suggesting these representations did lend themselves to different working strategies. We discuss the interpretations and implications of these findings for future research and education. All materials and data are provided as Supplementary Materials.