Humans are able to make sense of extraordinarily abstract concepts in mathematics (e.g., negative numbers). What is the underlying representation of these concepts? Integers extend natural numbers by including zero and negative numbers. To study the mental representation of integers, we employed a number comparison task in an online context. We replicated the previously-reported distance effect, in that far comparisons were faster than near comparisons. Namely, we observed reliable distance effects for positive and negative comparisons, and critically, an inverse distance effect for mixed comparisons. We conclude that the mental representation of integers may align with a hypothesis proposing the mental number line for negative numbers mirrors the natural number line. Moreover, we conclude that web-based data collection is a promising tool for future numerical cognition research.

Math talk has implications for the development of numerical concepts. Research suggests that when caregivers talk about natural numbers (1, 2, 3…), it may enhance children’s later math knowledge. Natural numbers have physical quantities that children can observe, yet abstract numerical concepts do not have such observable quantities. In this analysis, we examined how zero occurs in math talk. Using the CHILDES American English corpora (MacWhinney, 2000), we examined the frequency and nature of math talk about zero in naturalistic interactions between 2- to 6-year-olds and other speakers. Input from other speakers increased in frequency and complexity across development. Input with zero in symbolic sentential contexts (e.g., “one and zero make ten”) and cardinal sentential contexts (e.g., “zero means nothing”) increased with development. Children’s production of zero did not change in frequency or context. These results have implications for the concepts about zero children may bring to formal education.

One pedagogical technique that promotes conceptual understanding in mathematics learners is self-explanation integrated with worked examples (e.g., Rittle-Johnson et al., 2017). In this work, we implemented self-explanations with worked examples (correct and erroneous) in a software-based Intelligent Tutoring System (ITS) for learning algebra. We developed an approach to eliciting self-explanations in which the ITS guided students to select explanations that were conceptually rich in nature. Students who used the ITS with self-explanations scored higher on a posttest that included items tapping both conceptual and procedural knowledge than did students who used a version of the ITS that included only traditional problem-solving practice. This study replicates previous findings that self-explanation and worked examples in an ITS can foster algebra learning (Booth et al., 2013). Further, this study extends prior work to show that guiding students towards conceptual explanations is beneficial.

Diagrams are a potentially valuable tool for helping students understand mathematical concepts and procedures. Onetype of diagram that is sometimes used to depict mathematical relationships is tape diagrams, which depict quantitiesin continuous strips. This study investigated whether tape diagrams with different visual features differentially supportreasoning about equations, and explored whether people have preferences for tape diagrams with different visual features.Undergraduates (N = 50) were asked (1) to generate equations to correspond with tape diagrams with varying visualfeatures, and (2) to select the diagram they preferred from pairs that differed in visual features. Variations in visualfeatures (color, presence of outer lines, and position of the constant) did not affect participants success at generatingequations to correspond to the tape diagrams. However, participants displayed systematic preferences for most visualfeatures considered. Future research should examine the effects of these visual features on performance while solvingequations.

Although visual representations are generally beneficial for learners, past research also suggests that often only a subset of learners benefits from visual representations. In this work, we designed and evaluated anticipatory diagrammatic self-explanation, a novel form of instructional scaffolding in which visual representations are used to guide learners’ inference generation as they solve algebra problems in an Intelligent Tutoring System. We conducted a classroom experiment with 84 students in grades 5-8 in the US to investigate the effectiveness of anticipatory diagrammatic self-explanation on algebra performance and learning. The results show that anticipatory diagrammatic self-explanation benefits learners on problem-solving performance and the acquisition of formal problem-solving strategies. These effects mostly did not depend on students’ prior knowledge. We analyze and discuss how performance with the visual representation may have influenced the enhanced problem-solving performance.

Integrating visual representations in an interactive learning activity effectively scaffolds performance and learning. However, it is unclear whether and how sustaining or interleaving visual scaffolding helps learners solve problems efficiently and learn from problem solving. We conducted a classroom study with 63 middle-school students in which we tested whether sustaining or interleaving a particular form of visual scaffolding, called anticipatory diagrammatic self-explanation in an Intelligent Tutoring System, helps students’ learning and performance in the domain of early algebra. Sustaining visual scaffolding during problem solving helped students solve problems efficiently with no negative effects on learning. However, in-depth log data analyses suggest that interleaving visual scaffolding allowed students to practice important skills that may help them in later phases of algebra learning. This paper extends scientific understanding that sustaining visual scaffold does not over-scaffold student learning in the early phase of skill acquisition in algebra.