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Results from a 2-year longitudinal study of 181 children from 4th through 5th grade are reported. Levels of growth in children's computation, word problem, and estimation skills by means of common fractions were predicted by working memory, attentive classroom behavior, conceptual knowledge about fractions, and simple arithmetic fluency. Comparisons of 55 participants identified as having mathematical difficulties to those without mathematical difficulties revealed that group differences in emerging fraction skills were consistently mediated by attentive classroom behavior and conceptual knowledge about fractions. Neither working memory nor arithmetic fluency mediated group differences in growth in fraction skills. It was also found that the development of basic fraction skills and conceptual knowledge are bidirectional in that conceptual knowledge exerted strong influences on all 3 types of basic fraction skills, and basic fraction skills exerted a more modest influence on subsequent conceptual knowledge. Results are discussed with reference to how the identification of potentially malleable student characteristics that contribute to the difficulties that some students have with fractions informs interventions. Also, results will contribute to a future theoretical account concerning how domain-general and domain-specific factors influence the development of basic fraction skills. Keywords: fraction skills, mathematical learning disabilities, mathematical learning difficulties, mathematical development DOI: 10.1037/a0019824

We review the growing literature on health numeracy, the ability to understand and use numerical information, and its relation to cognition, health behaviors, and medical outcomes. Despite the surfeit of health information from commercial and noncommercial sources, national and international surveys show that many people lack basic numerical skills that are essential to maintain their health and make informed medical decisions. Low numeracy distorts perceptions of risks and benefits of screening, reduces medication compliance, impedes access to treatments, impairs risk communication (limiting prevention efforts among the most vulnerable), and, based on the scant research conducted on outcomes, appears to adversely affect medical outcomes. Low numeracy is also associated with greater susceptibility to extraneous factors (i.e., factors that do not change the objective numerical information). That is, low numeracy increases susceptibility to effects of mood or bow information is presented (e.g., as frequencies vs. percentages) and to biases in judgment and decision making (e.g., framing and ratio bias effects). Much of this research is not grounded in empirically supported theories of numeracy or mathematical cognition, which are crucial for designing evidence-based policies and interventions that are effective in reducing risk and improving medical decision making. To address this gap, we outline four theoretical approaches (psychophysical, computational, standard dual-process, and fuzzy trace theory), review their implications for numeracy, and point to avenues for future research. Keywords: risk perception, risk communication, mathematical cognition, intuition, dual processes DOI: 10.1037/a0017327

The goal of the current research was to explore whether monkeys possess conceptual precursors necessary for understanding zero. We trained rhesus monkeys on a nonsymbolic numerical matching-to-sample task, and on a numerical ordering task. We then introduced nondifferentially reinforced trials that contained empty sets to determine whether monkeys would treat empty sets as numerical values. All monkeys successfully matched and ordered the empty sets without any training. Accuracy showed distance effects, indicating that they treated empty sets as values on a numerical continuum. Keywords: zero, empty sets, numerical cognition, distance effects, numerical continuum

The mathematical performance of 182 third and fourth graders in 8 different areas of mathematics was examined. The children belonged to 4 achievement groups: children with mathematic difficulties (MD only), children with both mathematic and reading difficulties (MD-RD), children with reading difficulties (RD only), and normally achieving children (control group). Both MD groups performed worse than the normally achieving children in all but 1 area, place value knowledge. The MD-only and the MD-RD children performed equally in all areas of mathematics. The RD-only group performed at the same level as the control group on all areas of mathematics. The study provides further evidence that fact retrieval deficits are a cardinal characteristic of children with MD. The MD children's substantial difficulties with mathematic word problem solving can be attributed to several processes involved in problem solving. Besides poor skills in multidigit calculation, arithmetic fact retrieval, and poor understanding of calculation principles, children with MD might have deficits related to specific problem-solving processes such as establishing a problem representation and developing a solution plan. Keywords: mathematical difficulties, reading difficulties, calculation, mathematic problem solving, telling time

The purpose of this study was to explore patterns of difficulty in 2 domains of mathematical cognition: computation and problem solving. Third graders (n = 924; 47.3% male) were representatively sampled from 89 classrooms; assessed on computation and problem solving; classified as having difficulty with computation, problem solving, both domains, or neither domain; and measured on 9 cognitive dimensions. Difficulty occurred across domains with the same prevalence as difficulty with a single domain; specific difficulty was distributed similarly across domains. Multivariate profile analysis on cognitive dimensions and chi-square tests on demographics showed that specific computational difficulty was associated with strength in language and weaknesses in attentive behavior and processing speed; problem-solving difficulty was associated with deficient language as well as race and poverty. Implications for understanding mathematics competence and for the identification and treatment of mathematics difficulties are discussed. Keywords: calculations, word problems, cognitive predictors, mathematics

Automatic processing of 2-digit numbers was demonstrated using the size congruency effect (SiCE). The SiCE indicates the processing of the irrelevant (numerical) dimension when 2 digits differing both numerically and physically are compared on the relevant (physical) dimension. The SiCE was affected by the compatibility between unit and decade digits but was unaffected by the global magnitude of the numbers. Together these results suggest automatic processing of the magnitudes of the components of the 2-digit numbers but not of whole numbers. Finally, the SiCE was affected more by the magnitude of the decade digits compared with the unit digits, indicating that the syntactic roles of the digits were represented. The implications of these results for understanding the numerical representations are discussed. Keywords: automaticity, numerical processing, two-digit numbers, size congruency effect

L. L. Bonatti, M. Pena, M. Nespor, and J. Mehler (2006) argued that P. Perruchet, M. D. Tyler, N. Galland, and R. Peereman (2004) confused the notions of segmentation and generalization by ignoring the evidence for generalization in M. Pena, L. L. Bonatti, M. Nespor, and J. Mehler (2002). In this reply, the authors reformulate and complement their initial arguments, showing that their way of dealing with segmentation and generalization is not due to confusion or ignorance but rather to the fact that the tests used in Pena et al. make it likely that neither segmentation nor generalization were captured in their experiments. Finally, the authors address the challenge posed by Pena et al. of accounting for the whole pattern of their results without invoking rule-based, algebraic-like computations. Keywords: statistical computation, rule-based computation, associative learning, language

This article considers 3 claims that cognitive sex differences account for the differential representation of men and women in high-level careers in mathematics and science: (a) males are more focused on objects from the beginning of life and therefore are predisposed to better learning about mechanical systems; (b) males have a profile of spatial and numerical abilities producing greater aptitude for mathematics; and (c) males are more variable in their cognitive abilities and therefore predominate at the upper reaches of mathematical talent. Research on cognitive development in human infants, preschool children, and students at all levels fails to support these claims. Instead, it provides evidence that mathematical and scientific reasoning develop from a set of biologically based cognitive capacities that males and females share. These capacities lead men and women to develop equal talent for mathematics and science. Keywords: sex differences, mathematics, science, cognitive development, spatial reasoning

The number skills of groups of 7- to 9-year-old children with specific language impairment (SLI) attending mainstream or special schools were compared with an age and nonverbal reasoning matched group (age control [AC]) and with a younger group matched on oral language comprehension. The SLI groups performed below the AC group on every skill. They also showed lower working memory functioning and had received lower levels of instruction. Nonverbal reasoning, working memory functioning, language comprehension, and instruction accounted for individual variation in number skills to differing extents depending on the skill. These factors did not explain the differences between SLI and AC groups on most skills. Keywords: number development, working memory, specific language impairment, instruction

This study examined the efficacy of preventive 1st-grade tutoring in mathematics, estimated the prevalence and severity of mathematics disability, and explored pretreatment cognitive characteristics associated with mathematics development. Participants were 564 first graders, 127 of whom were designated at risk (AR) for mathematics difficulty and randomly assigned to tutoring or control conditions. Before treatment, all participants were assessed on cognitive and academic measures. Tutoring occurred 3 times weekly for 16 weeks; treatment fidelity was documented; and math outcomes were assessed. Tutoring efficacy was supported on computation and concepts/applications, but not on fact fluency. Tutoring decreased the prevalence of math disability, with prevalence and severity varying as a function of identification method and math domain. Attention accounted for unique variance in predicting each aspect of end-of-year math performance. Other predictors, depending on the aspect of math performance, were nonverbal problem solving, working memory, and phonological processing. Keywords: mathematics, learning disability, prevention, first grade, intervention

Recent research suggests that humans and other animals have sophisticated abilities to extract both statistical dependencies and rule-based regularities from sequences. Most of this research stresses the flexibility and generality of such processes. Here the authors take up an equally important project, namely, to explore the limits of such processes. As a case study for rule-based generalizations, the authors demonstrate that only repetition-based structures with repetitions at the edges of sequences (e.g., ABCDEFF but not ABCDDEF) can be reliably generalized, although token repetitions can easily be discriminated at both sequence edges and middles. This finding suggests limits on rule-based sequence learning and new interpretations of earlier work alleging rule learning in infants. Rather than implementing a computerlike, formal process that operates over all patterns equally well, rule-based learning may be a highly constrained and piecemeal process driven by perceptual primitives--specialized type operations that are highly sensitive to perceptual factors. Keywords: grammar acquisition, perceptual primitives, symbol manipulation, statistical learning, modularity

This longitudinal study tracked 1,110 adolescents identified as mathematically precocious at Age 13 (top 1%) with plans for a math-science undergraduate major. Participants' high school educational experiences, abilities, and interests predicted whether their attained undergraduate degrees were within math-science or nonmath-nonscience areas. More women than men eventually completed undergraduate degrees outside math-science, but many individuals who completed nonmath-nonscience degrees ultimately chose math-science occupations (and vice versa). At Age 33, the 2 degree groups reported commensurate and uniformly high levels of career satisfaction, success, and life satisfaction. Assessing individual differences is critical for modeling talent development and life satisfaction; it reveals that equal male-female representation across disciplines may not be as simple to accomplish as many policy discussions imply.

The task of rats (Rattus norvegicus) was to enter 1 box of a defined ordinal number among an array of boxes. In Experiments 1 and 2, the rats correctly chose the 4th box from arrays of 6 and 12 boxes, respectively. In Experiments 3 and 4, in which the ordinal number of the correct box was increased in a graduated fashion, they were able to select the correct box even when its position was higher than 10th among 12 and 18 boxes, respectively. In Experiment 5, the possibility that the rats had used cues to the openability of the box doors was ruled out. In Experiments 6 and 7, the rats succeeded in the task even when the total number of boxes was varied from trial to trial. In Experiment 8, both small- and large-sized boxes were used to control for the possibility of using the cumulative length of the boxes as a cue. Overall, the results suggest that performance was based on numerical cues. Intentional acts were occasionally observed but appeared not to be essential for solving the task.

When do students detect algebra word problems with missing information, and how do they do it? To determine whether such detection operates automatically or requires conscious effort and attention, this study manipulated whether a hint was provided that problems might be unsolvable. A signal detection analysis revealed that the hint resulted in a large improvement in the detection of unsolvable problems, implying that many individuals possess the means to detect missing information but that conscious effort is required for those means to be deployed. Participants with moderate mathematical ability had good success at detecting missing information when given the hint, but only when the problems had familiar cover stories. High math ability individuals were successful regardless of the cover story's familiarity. In addition, participants were more likely to conclude that a solvable problem was unsolvable when the hint was provided, when the problems' cover story was unfamiliar, and with the passage of time. As a result, problem solving of solvable problems was sometimes abandoned prematurely.

Three studies were conducted to examine the effects of individual differences and language differences on children's understanding of teen quantities (11 [less than or equal to] n [less than or equal to] 19) as counted cardinal tens and ones (embedded-ten cardinal understanding). At age 4, most Chinese children, using named-ten number words (e.g., 12 is said as 'ten two'), did not show such understanding on a task in which y quantifies were added to 10 quantities. At age 5, half the children of average or above intelligence who had high rote-counting sequences (M = 90) did show such understanding; those with lower rote-counting sequences did not. English-speaking 5-year-old children in England and in the United States, whose teen words obfuscate the tens and ones, showed no evidence of understanding teen quantities as cardinal tens and ones.

The unique associations between 3 families of mathematical knowledge and individual differences in 7th and 8th graders' fraction skills were examined. Procedural knowledge regarding the steps to solve fraction computation problems and conceptual knowledge of part-whole and measurement interpretations of fractions were measured. Math fact knowledge was indexed as the speed of solving single-digit addition and multiplication problems. Conceptual knowledge and procedural knowledge uniquely explained variability in fraction computation solving and fraction word problem set-up accuracy. Only conceptual knowledge uniquely explained individual differences in fraction estimation skills. The findings are discussed within a framework that describes the 3 necessary steps toward an information-processing account of individual differences in fraction skills. Four important limitations to this study were described.

An important question in learning is the nature of the information required to support skilled, or automatic, performance. In 2 experiments, participants counted patterns of 8-10 objects displayed on a computer screen for 4 sessions of 240 trials each before being transferred to a different set of patterns for a final session of 240 trials. The patterns in the final session differed from those seen in the 4 training sessions in either overall configuration (Experiment 1) or identity of constituents (Experiment 2). Results indicated that both types of information are important in learning and automatization of a counting task and support the idea that what is attended to during training will be necessary to support subsequent automatic performance.

In Experiment 1, 2 squirrel monkeys (Saimiri sciureus) were given choices between all possible pairs of the arabic numbers 0, 1, 3, 5, 7, and 9, with choice of any number yielding that number of pieces of peanut as a reward. Both monkeys learned to choose the larger number in all pairings and learned to choose the largest number within a set of 4 numbers. In Experiments 2-4, the monkeys were tested on problems in which they chose between pairs of stimuli containing 2 numbers versus 2 numbers, 1 number versus 2 numbers, and 3 numbers versus 3 numbers. Both monkeys showed a significant tendency to choose the stimulus that contained the largest sum. Various tests indicated that this effect could not be explained by choice of the stimulus with the largest single number, by avoidance of the stimulus with the smallest single number, or by experimenter cuing.

Researchers have assumed that adults solve simple arithmetic problems by retrieving answers from a network of stored facts. In 2 studies, undergraduates described their solutions of single-digit multiplication problems. They reported direct retrieval on approximately 80% of trials, but also reported rules (e.g., anything times 0 is 0), repeated addition (e.g., 2 x 4 = 4 + 4), number series (e.g., 3 x 5 = 5, 10, 15), and derived facts (e.g., 6 x 7 = [6 x 6] + 6). Participants were slower to retrieve problems that were most likely to be solved by nonretrieval procedures and faster to retrieve problems that were usually solved by retrieval. These results indicate that direct-retrieval models are incomplete accounts of adults' performance and support a continuing influence of learning and experience on the mental representation of simple multiplication problems.

For this study of the organization of cognitive abilities and gender differences in young children advanced in mathematical reasoning, parents identified 778 preschoolers and kindergartners. After screening with 2 arithmetic subtests of standard intelligence tests, 310 high scorers (55% boys) were given 15 additional measures. Mean performance of these high scorers on all standardized measures was 1 to 2 standard deviations above the mean of the norming samples. Boys scored higher on 8 of 11 quantitative measures, 0 of 3 verbal measures, and 1 of 3 spatial measures, including quantitative and spatial working memory span. Three factors (quantitative, verbal, and spatial) were modeled using confirmatory factor analysis; patterns of relationships were similar for older and younger groups and for girls and boys. Spatial and quantitative factors were highly correlated; the verbal factor correlated weakly with the others but showed a stronger relationship with the spatial factor for boys than girls.

The relationship between mental rotation ability and gender differences in Scholastic Aptitude Test - Math (SAT-M) across diverse samples was investigated. Talented preadolescents, college students, and high- and low-ability college-bound youths, totaling 760, were administered the Vandenberg Mental Rotation Test. Gender comparisons showed male outperforming female students in both mental rotation and SAT-M for all 3 high-ability groups but not for the low-ability group. For all female samples, mental rotation predicted math aptitude even when SAT - Verbal was entered first into the regression. For male samples, the relationship varied as a function of sample. When mental rotation ability was statistically adjusted for, the significant gender difference in SAT-M was eliminated for the college sample and the high-ability college-bound students. This suggests that spatial ability may be responsible in part for mediating gender differences in math aptitude among these groups.

In this study, 391 students were asked to provide 3 types of mathematics self-efficacy judgments: confidence to solve mathematics problems, confidence to succeed in math-related courses, and confidence to perform math-related tasks. Criterial tasks were solution of math problems and choice of math-related majors. As hypothesized, students' reported confidence to solve the problems they were later asked to solve was a more powerful predictor of that performance than was either their confidence to perform math-related tasks or to succeed in math-related courses. Similarly, confidence to succeed in math-related courses was a stronger predictor of choice of math-related majors than was either confidence to solve problems or to perform math-related tasks. Results support A. Bandura's (1986) contention that, because judgments of self-efficacy are task specific, measures of self-efficacy should be tailored to the criterial task being assessed and the domain of functioning being analyzed to increase prediction.

The influence of school- and age-related variables was examined separately on 2 tasks involving elementary quantitative skills: conservation of number and mental addition. Performance on these tasks was compared by using a cutoff design with 3 groups of kindergarten and Grade I children who differed in age but not amount of schooling (grade), in schooling but not age, or in both age and schooling. The effects of age and schooling were distinct. On conservation of number, performance improved as a function of age but not schooling. On mental arithmetic, accuracy improved with schooling rather than age, but children's use of various solution procedures (e.g., retrieval, counting) was not influenced by schooling. Thus, in the domain of elementary mathematical skills, the influence of schooling can be very specific, and age-related variables other than schooling play an important role in the development of elementary mathematical skills. Results illustrate the utility of the cutoff design for investigating instructional and developmental influences on cognitive development.

Two experiments examined preschool children's ability to draw inferences about numerosity from correspondences between sets. In Experiment 1, 3- and 4-year-old children made numerical inferences about a hidden set from their own counts of a corresponding visible set and also from numerical information about that set stated by the experimenter. Experiment 2 contrasted a count condition with a move condition, in which children's attention was not explicitly drawn to the numerosity of the visible set. Again, children were able to make numerical inferences as early as 3 years of age. However, differences between the 2 conditions implicate production deficiencies in young children's use of counting as a problem-solving strategy when they are not explicitly told to count.

In 2 experiments, college Ss practiced extensively on single-digit multiplication and division problems (e.g., ____ = 6 X 9; 42 = ____ X 6) and were tested on both practice problems and several altered versions of those problems, which were constructed by changing the required operation, operand order, or arithmetic symbol. There was strong positive transfer to test problem that had exactly the same elements (the numbers and the required operation) as practice problem, regardless of whether other factors such as operand order or symbol were changed, but little if any positive transfer to test problems that did not have the same elements as a practice problem. An identical elements framework is used to interpret these results and implications for existing computational models of arithmetic fact retrieval and for the development of arithmetic skill are discussed.

Pigeons were trained on a psychophysical choice task to discriminate between 2 fixed ratios and tested with probe ratios intermediate to the training requirements. Absolute and relative ratio size were varied over conditions. In a final condition, subjects were transferred from the number-based discrimination to a time-based discrimination. The times taken to emit the ratios were recorded, and ratio value and ratio time were used to predict choice in a multiple regression analysis. Psychophysical analyses of the response number dimension suggested a scalar counting principle of numerosity discrimination, similar to that found with temporal dimensions. The multiple regression analysis indicated that both number and time contributed to the discrimination of ratio value. The results of the transfer test indicated individual differences in the degree to which responding was controlled by temporal versus numerical factors.

Ss evaluated either numerical size or physical size of stimuli varying along both dimensions. Size congruity, distance, and semantic congruency effects were obtained for numerical comparisons of digit pairs and for comparisons of digits with an internal standard (5). Only the size congruity effect was obtained for physical judgments. It was smaller for pairs in which both stimuli were either both smaller or both larger than 5 than for pairs that contained the digit 5. The results are consistent with the notion that intentional processing is mainaly algorithm based, whereas autonomous processing is mainly memory based. Implications of the results for models for numerical processing are discussed.

In Experiment 1, with the number of sides or angles of irregular polygons as cues, programmed training, and a 90% correct criterion (36 of 40), 2 squirrel monkeys' (Saimiri sciureus sciureus and S. boliviensus boliviensus) best performances were to discriminate heptagons from octagons, a 3rd's best was hexagons from heptagons, and a 4th's best was pentagons from heptagons. In Experiment 2, on most trials 2 polygons on one or both discriminanda had to be summed to determine which discriminandum had the total fewer sides. Only 1 monkey met criterion (27 of 30) on the 2 tasks, 6 vs. 8 and 7 vs. 8 sides, but the other 3 performed better than chance on the 6 vs. 8 task. We conclude that previous studies of animals' discrimination of polygons in terms of complexity were minimally relevant to this work, and counting and subitizing were rejected in favor of a prototype-matching process to explain our monkeys' performances.

Thirty younger and 29 older adults were tested on reaction times for set of display of 1 to 9 digits. The results suggested that, with advancing age, the size of the focus of attention may shrink, but speed of access to elements in the focus of attention may remain constant.

Three experiments were performed to test if tactile stimuli could serve as the basis for a numerical discrimination in rats (Rattus norvegicus). In Experiment 1, touch delivered symmetrically to both sides of the animal's body yielded no evidence of numerical discrimination. In Experiment 2, the restriction of tactile cues to one side of the animal's body resulted in marginally better results, although performance remained below conventional levels of significance. In Experiment 3, tactile contact with the animal's vibrissae yielded statistically evidence of numerical processing. Subjects learned to enter one arm of a Y-maze when three vibrissal deflections were presented, and the other arm when either two or four stimuli occurred. The demonstration of a two-three-four discrimination extends the use of this relatively complex intermediate number procedure from a previous demonstration in rats involving auditory stimuli (Davis & Albert, 1986) and indicates for the first time in any species that touch may be used as the basis for numberical behavior.

A chimpanzee (Pan toglodytes), trained to count foods and objects by using Arabic numbers demonstrated the ability to sum arrays of 0-4 food items placed in 2 of 3 possible sites. To address representational use of numbers, we next baited sites with Arabic numbers as stimuli. In both cases performance was significantly above chance from the first sessions, which suggests that without explicit training in combining arrays, the animal was able to select the correct arithmetic sum for arrays of foods or Arabic numbers under novel test conditions. These findings demonstrate that counting strategies and the representational use of numbers lie within the cognitive domain of the chimpanzee and compare favorably with the spontaneous use of addition algorithms of demonstrated in preschool children.

Studied the role of self-efficacy in mediating gender differences on Investigative and Realistic General Occupational Themes of the Strong-Campbell Interest Inventory, Ss were 77 male and 71 female entering university freshmen. Canonical correlations identified a science/technical factor relating high Investigative and Realistic theme standard scores to high Investigative and Realistic theme efficacy rating. Path analyses suggested a correlational structure wherein mathematics self-efficacy and high school mathematics preparation mediated gender differences on these themes. Implications for counselor interpretation of Investigative and Realistic General Occupational Themes were presented. The importance of integrating self-efficacy career development research and traditional self-concept vocational theory was emphasized.

This study examined relations between children's conceptual understanding of mathematical equivalence and their procedures for solving equivalence problems (e.g., 3 + 4 + 5 = 3 + ___). Students in 4th and 5th grades completed assessments of their conceptual and procedural knowledge of equivalence, both before and after a brief lesson. The instruction focused either on the concept of equivalence or on a correct procedure for solving equivalence problems. Conceptual instruction led to increased conceptual understanding and to generation and transfer of a correct procedure. Procedural instruction led to increased conceptual understanding and to adoption, but only limited transfer, of the instructed procedure. These findings highlight the causal relations between conceptual and procedural knowledge and suggest that conceptual knowledge may have a greater influence on procedural knowledge than the reverse.

Three- to 7-year-olds' ability to calculate with whole-number, fraction, and mixed-number amounts was tested using a nonverbal task in which an amount was displayed and then hidden (J. Huttenlocher, N. C. Jordan, & S.C. Levine, 1994). Next, an amount was added to or subtracted from the hidden amount. The child's task was to determine the hidden amount that resulted from the transformation. Although fraction problems were more difficult than whole-number problems, competence on all problem types emerged in the early childhood period. Furthermore, there were striking parallels between the development of whole-number and fraction calculation. This is inconsistent with the hypothesis that early representations of quantity promote learning about whole numbers but interfere with learning about fractions (e.g., R. Gelman, 1991; K. Wynn, 1995, 1997).

This study investigated patterns of change in children's strategies for solving mathematical equivalence problems. The strategies children expressed in speech and in gesture were assessed both before and after an instructional intervention. In the intervention, children received either no input, accuracy feedback, or feedback plus instruction about a principle, an analogy, or a procedure. From pretest to posttest, many children changed both the variability of their strategy use and the content of their strategy repertoires. Patterns of change depended on type of instruction and on children's initial level of variability. Children who received instruction were especially likely to generate new strategies, and children with high variability were especially likely to abandon prior strategies. Gradual change was most common; however, many children modified their repertoires abruptly. Abrupt strategy change was especially prevalent among children who received procedure-based instruction and among children with low initial variability.