Kristen Pilner Blair

Beyond natural numbers: negative number representation in parietal cortex

Unlike natural numbers, negative numbers do not have natural physical referents. How does the brain represent such abstract mathematical concepts? Two competing hypotheses regarding representational systems for negative numbers are a rule-based model, in which symbolic rules are applied to negative numbers to translate them into positive numbers when assessing magnitudes, and an expanded magnitude model, in which negative numbers have a distinct magnitude representation. Using an event-related functional magnetic resonance imaging design, we examined brain responses in 22 adults while they performed magnitude comparisons of negative and positive numbers that were quantitatively near (difference <4) or far apart (difference >6). Reaction times (RTs) for negative numbers were slower than positive numbers, and both showed a distance effect whereby near pairs took longer to compare. A network of parietal, frontal, and occipital regions were differentially engaged by negative numbers. Specifically, compared to positive numbers, negative number processing resulted in greater activation bilaterally in intraparietal sulcus (IPS), middle frontal gyrus, and inferior lateral occipital cortex. Representational similarity analysis revealed that neural responses in the IPS were more differentiated among positive numbers than among negative numbers, and greater differentiation among negative numbers was associated with faster RTs. Our findings indicate that despite negative numbers engaging the IPS more strongly, the underlying neural representation are less distinct than that of positive numbers. We discuss our findings in the context of the two theoretical models of negative number processing and demonstrate how multivariate approaches can provide novel insights into abstract number representation.

Go Math! How Research Anchors New Mobile Learning Environments

A reciprocal research and design process (RR&D) became central to the development of mobile learning environments for families. Go Math! applications were developed to support families in the situations they face in their daily activities where problem solving involves mathematics. The RR&D process is effective for synthesizing design and development choices with consideration of the results of basic research on mathematics in everyday life, the voices of users, the social context of use, and mobile affordances. The RR&D process is described, and two mobile mathematics applications illustrate how the process maintains fidelity among research, the development of design criteria, and user voice and practice. We consider the process important in the development of mobile learning environments.

Learning in critter corral: evaluating three kinds of feedback in a preschool math app

We describe a freely available iPad app for preschoolers aimed at helping children develop robust early number concepts for the numbers 1-10. We will present results of an in-progress study, involving pre-post learning measures, in-game learning measures, and video data, aimed at evaluating the effectiveness of the app and of comparing three models of feedback. We created three versions of the app, each of which employs a different kind of feedback when the learner solves a problem incorrectly. In the Implication Feedback condition (IF), the learner sees that their incorrect answer resulted in too many or too few, and they add more or take some away to fix it. In the Corrective Feedback condition (CF) the learner is shown the correct answer after a mistake and why it is correct, and they imitate the correct answer. In the Answer until Correct condition (AUC), learners are given an indication that they have made a mistake, and they continue trying until they answer correctly.

Learning to "See" Less than Nothing: Putting Perceptual Skills to Work for Learning Numerical Structure.

How can childrens natural perceptuo-motor skills be harnessed for teaching and learning mathematical structure? We address this question in the case of the integers. Existing research suggests that adult mental representations of integers recruit perceptuo-motor functionalities involving symmetry. Building on these findings, we designed a hands-on curriculum that emphasizes symmetry to teach integer concepts to fourth graders. Compared to two control conditions, children who went through the experimental curriculum showed evidence of incorporating symmetry into their mental representations of integers and performed higher on problems beyond the scope of instruction, including negative fractions and algebra-readiness problems. Gains did not come at the expense of basic integer computation skill. This study has direct practical implications, as current integers curricula generally omit symmetry. The research demonstrates an approach to designing instruction that involves identifying perceptuo-motor functionalities underlying numerical cognition and creating learning activities to recruit them.

Assessing Whether Students Seek Constructive Criticism: The Design of an Automated Feedback System for a Graphic Design Task

We introduce a choice-based assessment strategy that measures students’ choices to seek constructive feedback and to revise their work. We present the feedback system of a game we designed to assess whether students choose positive or negative feedback and choose to revise their posters in the context of a poster design task, where they learn graphic design principles from feedback. We then describe an empirical study that sampled one hundred and six students from a US middle school to evaluate the feedback system. We make the following contributions: (1) describe the design and implementation of a novel feedback system embedded in an assessment game, Posterlet, (2) outline an approach to analyze graphic design principles automatically to provide contextual feedback in a novel poster design domain, (3) show that choices to seek negative feedback and to revise correlate with in-game performance, and most importantly, (4) show that choices correlate with in-school achievement: the choice to revise correlated with both in-school performance measures (Science and Mathematics grades), while the choice to seek negative feedback correlated with students’ prior standardized scores in Mathematics.

Testing the Effectiveness of iPad Math Game: Lessons Learned from Running a Multi-Classroom Study

Many educational products designed for young children go through extensive user testing, but rarely through a rigorous examination of whether they improve learning. We describe our experiences and lessons learned from conducting a multi-classroom study to examine learning from an iPad math app we developed for preschool and kindergarten children. Focusing on the research experience itself, we describe six common challenges to conducting learning research with technology and young children, as well as six principles to help mitigate the challenges. This paper is intended to help others who wish to assess learning from educational games for children.

Applying cognitive developmental psychology to middle school physics learning: The rule assessment method

Cognitive developmental psychology often describes children’s growing qualitative understanding of the physical world. Physics educators may be able to use the relevant methods to advantage for characterizing changes in students’ qualitative reasoning. Siegler developed the “rule assessment” method for characterizing levels of qualitative understanding for two factor situations (e.g., volume and mass for density). The method assigns children to rule levels that correspond to the degree they notice and coordinate the two factors. Here, we provide a brief tutorial plus a demonstration of how we have used this method to evaluate instructional outcomes with middle-school students who learned about torque, projectile motion, and collisions using different instructional methods with simulations.

A value of concrete learning materials in adolescence.

There are disagreements about the appropriate role of concrete materials for learning. We clarify the assumptions underlying different uses of concrete materials and abstract symbols in instruction for learning new mathematical ideas. We argue that despite the potential structural isomorphism of concrete materials and abstract representations, they engage different psychological processes. We describe several studies highlighting the unique psychological properties of concrete materials and abstract symbols for learning. We propose the co-evolution hypothesis, where concrete materials and symbols work together to help students discover the structure both in the world and in the abstract representations. In a classroom study with sixth grade students learning about proportion and ratio, students who received instruction consistent with the co-evolution hypothesis showed improved initial learning and greater transfer than students who first learned abstract symbols and mathematical principles, and then applied them to concrete materials as practice.