Darcy Hallett

Disentangling the course of epistemic development: parsing knowledge by epistemic content

Abstract Over the past three decades, research into the developmental course by means of which persons come to an increasingly mature conception of the knowing process has yielded an highly defracted picture. Despite some concert of opinion about the general bill of particulars, what remains deeply problematic is the increasingly radical disagreement that has arisen regarding the ages at which major milestones in the course of epistemic development are said to be reached. As a way of making some sense of these competing claims, it is argued that the emerging insight that knowledge is ineluctably shaped by those doing the knowing (i.e., that there is an unavoidable “world-to-mind direction of fit” (J.R. Searle, Intentionality: An Essay in the Philosophy of Mind, Cambridge University Press, Cambridge, 1983) between things in the world and the manner of their understanding) does not arrive in a single piece. Instead, as the data presented here help to illustrate, an appreciation of the constructed character of knowledge more commonly arrives piecemeal and at different ontogenetic moments, the times of which are governed by the place that different objects of knowledge occupy along an envisioned continuum of diverse epistemic contents. On this account, not all “facts of the matter” are ordinarily seen to occupy the same epistemic footing. Rather, some so-called facts are commonly understood to be of an “institutional” sort, where “representational” diversity is early expected and widely tolerated. By contrast, other objects of knowledge are imagined to be more like “brute” facts that, on some less mature readings, fully escape the clutches of subjective opinion. Viewed against the backcloth of this proposed continuum, a developmental sequence hypothesized according to which growing persons first come to view “institutional” facts as humanly constructed before subsequently coming to a similar view about presumptively “brute” facts. To test this hypothesis, 242 young persons were administered a paper and pencil measure of epistemic reasoning (the EDQ). Results strongly support the hypothesis that respondents understood the interpretive nature of beliefs about “institutional” facts at an earlier age than so-called “brute” facts.

Deaf children's informal knowledge of multiplicative reasoning.

Multiplicative reasoning is required in different contexts in mathematics: it is necessary to understand the concept of multipart units, involved in learning place value and measurement, and also to solve multiplication and division problems. Measures of hearing childrens multiplicative reasoning at school entry are reliable and specific predictors of their mathematics achievement in school. An analysis of deaf childrens informal multiplicative reasoning showed that deaf children under-perform in comparison to the hearing cohorts in their first two years of school. However, a brief training study, which significantly improved their success on these problems, suggested that this may be a performance, rather than a competence difference. Thus, it is possible and desirable to promote deaf childrens multiplicative reasoning when they start school so that they are provided with a more solid basis for learning mathematics.

Teaching Children About the Inverse Relation Between Addition and Subtraction

Two intervention studies are described. Both were designed to study the effects of teaching children about the inverse relation between addition and subtraction. The interventions were successful with 8-year-old children in Study 1 and to a limited extent with 5-year-old children in Study 2. In Study 1 teaching children about inversion increased their success not just in Inverse problems (a + b − b = ?) but also in Transfer complement problems (a + b = c; c − b = ?).

Individual Differences in Conceptual and Procedural Fraction Understanding: The Role of Abilities and School Experience.

Recent research on childrens conceptual and procedural knowledge has suggested that there are individual differences in the ways that children combine these two types of knowledge across a number of mathematical topics. Cluster analyses have demonstrated that some children have more conceptual knowledge, some children have more procedural knowledge, and some children have an equal level of both. The current study investigated whether similar individual differences exist in childrens understanding of fractions and searches for explanations for these differences. Grade 6 students (n=119) and Grade 8 students (n=114) were given measures of conceptual and procedural knowledge of fractions as well as measures of general fraction knowledge, general conceptual ability, and general procedural ability. Grade 6 children demonstrated a four-cluster solution reflecting those who do poorly on procedural and conceptual fraction knowledge, those who do well on both, those whose strength is procedural knowledge, and those whose strength is conceptual knowledge. Grade 8 children demonstrated a two-cluster solution reflecting those whose strength is procedural knowledge and those whose strength is conceptual knowledge. Cluster in either grade, however, did not vary in distribution across schools and was not related to general conceptual ability or general procedural ability. Overall, these results provide a more detailed picture of individual differences in conceptual and procedural knowledge in mathematical cognition.

Cross-Cultural and Intra-Cultural Differences in Finger-Counting Habits and Number Magnitude Processing: Embodied Numerosity in Canadian and Chinese University Students

Recent work in numerical cognition has shown-that number magnitude is not entirely abstract, and at least partly rooted in embodied and situated experiences, including finger-counting. The current study extends previous cross-cultural research to address within-culture individual differences in finger counting habits. Results indicated that Canadian participants demonstrated an additional cognitive load when comparing numbers that require more than one hand to represent, and this pattern of performance is further modulated by whether they typically start counting on their left hand or their right hand. Chinese students typically count on only one hand and so show no such effect, except for an increase in errors, similar to that seen in Canadians, for those whom self-identify as predominantly two-hand counters. Results suggest that the impact of finger counting habits extend beyond cultural experience and concord in predictable ways with differences in number magnitude processing for specific number-digits. We conclude that symbolic number magnitude processing is partially rooted in learned finger-counting habits, consistent with a motor simulation account of embodied numeracy and that argument is supported by both cross-cultural and within-culture differences in finger-counting habits.

The role of spatial and temporal information in learning interval time-place tasks

In time-place learning (TPL) paradigms animals are thought to form tripartite memory codes consisting of the spatiotemporal characteristics of biologically significant events. In Phase I, rats were trained on a modified TPL task in which either the spatial or temporal component was constant, while the other component varied randomly. If the memory codes are tripartite then when one aspect of the code is random the rats should have difficulty learning the constant aspect of the code. However, rats that were trained with a fixed spatial sequence of food availability and a random duration did in fact learn the task. Rats that were trained with a fixed duration and a random sequence did not learn the task. In Phase II all rats were placed on a TPL task in which food availability was contingent upon both spatial and temporal information. According to the tripartite theory, prior knowledge of either aspect of the code should have little effect on the acquisition of the task. The rats that received fixed spatial training learned the task relatively more quickly. The use of bipartite, rather than tripartite codes, is better able to explain the results of the current study.

Interval time-place learning in young children

While previous research has investigated the ability of animals to learn the spatial and temporal contingencies of biologically significant events (known as time-place learning), this ability has not been studied in humans. Children ranging from 5 to 10 years old were tested on a modified interval time-place learning task using a touchscreen computer. Results demonstrate the children were able to quickly learn both the timing and the sequence of this task. Despite a lack of anticipation on baseline trials, the children continued to follow the spatio-temporal contingencies in probe sessions where these contingencies were removed. Performance on the probe sessions provide strong evidence that the children had learned the spatio-temporal contingencies. Future research is needed to determine what age-related changes in iTPL occur. Furthermore, it is argued that this procedure can be used to extend interval timing in research in children, including, but not limited to, investigation of scalar timing with longer durations than have previously been investigated.

Finger-counting habits, not finger movements, predict simple arithmetic problem solving

Previous research in embodied mathematical cognition has found differences between those who start counting on their left hand and those who start counting on the right hand. However, if starting hand is a finger-embodied effect, then finger-specific interference may affect these differences between left and right starters. Furthermore, cultures that demonstrate different finger-counting habits may also be differently affected by this interference. In the current study, a total of 66 Canadians and 60 Chinese participants completed a single/dual-task paradigm and were also assessed on their starting hand for counting. The primary task was to verbally answer simple arithmetic problems, while the dual task was to either sequentially tap their fingers or their foot. Contrary to predictions, a specific finger-movement interference pattern that had previously been reported was not evident in this study, despite a much larger sample. Nevertheless, Canadians left starters outperformed right starters for every operation type, which may be further evidence of individual differences in the lateralization of arithmetic processes. Derived from a combination of a replication, a conceptual replication, and a cross-cultural comparison, this investigation suggests that embodied effects in the published literature are in need of both independent replication as well as investigation of individual differences. This study also further validates the differences between left and right starters, and suggests that more research is needed to understand the influence of embodied cognition on mathematical understanding.

Math Anxiety Is Related to Some, but Not All, Experiences with Math

Math anxiety has been defined as unpleasant feelings of tension and anxiety that hinder the ability to deal with numbers and math in a variety of situations. Although many studies have looked at situational and demographic factors associated with math anxiety, little research has looked at the self-reported experiences with math that are associated with math anxiety. The present study used a mixed-methods design and surveyed 131 undergraduate students about their experiences with math through elementary school, junior high, and high school, while also assessing math anxiety, general anxiety, and test anxiety. Some reported experiences (e.g., support in high school, giving students plenty of examples) were significantly related to the level of math anxiety, even after controlling for general and test anxiety, but many other factors originally thought to be related to math anxiety did not demonstrate a relation in this study. Overall, this study addresses a gap in the literature and provides some suggestive specifics of the kinds of past experiences that are related to math anxiety and those that are not.

Normative pediatric data for three tests of functional vision

OBJECTIVE To provide the normative pediatric data for three tests of functional vision. DESIGN Prospective, cross-sectional study. PARTICIPANTS The participants were 281 children between 3 and 5 years of age (mean 4.2 ± 0.8 years) participating in a preschool vision screening study. METHODS Participants completed monocular testing with the Precision Vision Visual Acuity Testing (PVVAT) system, a computerized test of visual acuity. Stereoacuity was assessed using the Pass Test 3 Stereo Test. Noncycloplegic refractive error was measured using the Plusoptix S09 Vision Screener (PS09). The mean and 95% tolerance limits were determined for each test. RESULTS Visual acuity improved significantly from 0.31 logMAR at 3 years to 0.18 logMAR at 5 years (p < 2.4 × 10-15). Stereoacuity improved from 104 arcsec at 3 years to 81 arcsec at 5 years (p = 0.0058). Spherical refractive error remained relatively stable at 0.29 D at 3 years and 0.24 D at 5 years (p = 0.543). Cylindrical refractive error was also relatively stable, measuring 0.39 D at 3 years and 0.31 D at 5 years (p = 0.109). CONCLUSIONS Both visual acuity and stereoacuity improved significantly from 3 to 5 years of age. Note, however, that the norms obtained using the PVVAT system were somewhat lower than those reported in previous studies using other tests. The norms for the PS09 were stable and showed relatively good agreement with the manufacturer-suggested referral criteria.