The role of pattern recognition in children's exact enumeration of small numbers

Enumeration can be accomplished by subitizing, counting, estimation, and combinations of these processes. We investigated whether the dissociation between subitizing and counting can be observed in 4‐ to 6‐year‐olds and studied whether the maximum number of elements that can be subitized changes with age. To detect a dissociation between subitizing and counting, it is tested whether task manipulations have different effects in the subitizing than in the counting range. Task manipulations concerned duration of presentation of elements (limited, unlimited) and configuration of elements (random, line, dice). In Study 1, forty‐nine 4‐ and 5‐year‐olds were tested with a computerized enumeration task. Study 2 concerned data from 4‐, 5‐, and 6‐year‐olds, collected with Math Garden, a computer‐adaptive application to practice math. Both task manipulations affected performance in the counting, but not the subitizing range, supporting the conclusion that children use two distinct enumeration processes in the two ranges. In all age groups, the maximum number of elements that could be subitized was three. The strong effect of configuration of elements suggests that subitizing might be based on a general ability of pattern recognition.     

Assessing Mathematics Misunderstandings via Bayesian Inverse Planning

Online educational technologies offer opportunities for providing individualized feedback and detailed profiles of students' skills. Yet many technologies for mathematics education assess students based only on the correctness of either their final answers or responses to individual steps. In contrast, examining the choices students make for how to solve the equation and the ways in which they might answer incorrectly offers the opportunity to obtain a more nuanced perspective of their algebra skills. To automatically make sense of step-by-step solutions, we propose a Bayesian inverse planning model for equation solving that computes an assessment of a learner's skills based on her pattern of errors in individual steps and her choices about what sequence of problem-solving steps to take. Bayesian inverse planning builds on existing machine learning tools to create a generative model relating (mis)-understandings to equation solving choices. Two behavioral experiments demonstrate that the model can interpret people's equation solving and that its assessments are consistent with those of experienced teachers. A third experiment uses this model to tailor guidance for learners based on individual differences in misunderstandings, closing the loop between assessing understanding, and using that assessment within an educational technology. Finally, because the bottleneck in applying inverse planning to a new domain is in creating the model of possible student misunderstandings, we show how to combine inverse planning with an existing production rule model to make inferences about student misunderstandings of fraction arithmetic.     

Evidence for the Phase Transition from Rule I to Rule II on the Balance Scale Task

A central and recurrent theme in developmental psychology is the question whether development proceeds continuously or discontinuously. This question is difficult to answer because the two types of development may be hard to distinguish. To investigate whether change is discontinuous, one requires a formal model for discontinuous development. Such a model should provide operational, empirical criteria to investigate the nature of development. The cusp model, which is derived from catastrophe theory, is a formal model of discontinuity that provides such criteria. Eight criteria, so-called catastrophe flags, can be derived from the model. Some of these flags are predicted in other models of discontinuity, but others, notably hysteresis, are unique to catastrophe theory. In the present research, it is tested whether the transition from Rule I to Rule II on the balance scale task proceeds discontinuously. The present research centers around five catastrophe flags: bimodality, inaccessible region, sudden jump, divergence, and hysteresis. Two experiments are reported. In Experiment 1, a paper-and-pencil version of the balance scale task was administered to 314 children who were 6 to 10 years old. In Experiment 2, an adapted version of the test was administered to 302 children who were 6 to 10 years old. Bimodality, inaccessible region, hysteresis, and sudden jump were clearly observed. Divergence was not observed. The presence of four of the five flags strongly supports the hypothesis that the transition from Rule I to Rule II is discontinuous.     

Comprehension of algebraic expressions by experienced users of mathematics

Little is known about how people comprehend mathematical expressions. In the present study we investigate the internal representations used by experienced users of mathematics to encode algebraic expressions. Initially, a memory recognition task was conducted that examined the role of mathematical syntax in the encoding of algebraic expressions. The results indicate that participants could more readily identify those parts of a previously seen expression that were syntactically well formed than those that were not well formed, suggesting that syntax plays an important role. To determine the level of syntactic structure involved, a second recognition task was conducted. The results indicate that algebraic expressions are encoded into components that represent the phrasal constituents of the expression. However, the results of these experiments did not rule out the possibility that the visual processes of perceptual organization were used to encode the expressions, or that the data were a consequence of a mathematical “lexicon” of common mathematical “phrases”. Three further experiments were conducted to examine this, the results of which indicate that the encoding of algebraic expressions is based primarily on processes that occur beyond the level of visual or “lexical” processing. This is consistent with our finding that syntactic structure plays a key role.     

The influence of juggling on mental rotation performance in children with spina bifida

This study examined the influence of juggling training on mental rotation ability in children with spina bifida. Children between the ages of 8 and 12 solved a chronometric mental rotation test. Half of the children received juggling training (EG) over an 8week time period; the other half did not receive training (CG). Afterwards, all participants completed the mental rotation test again. Children of the EG showed a significant decrease in reaction time and an increase in mental rotation speed compared to the control group. This indicates that juggling improves the rotation in the mental rotation process in children with spina bifida.