Computational Fluency Research Articles

Students’ understanding of the associative property and its applications: noticing, doubling and halving, and place value

Understanding the properties of multiplication is a critical precursor to students’ thinking algebraically. However, these properties have not been the focus of extensive rigorous research, particularly the associative property. In this study, we report on follow-up interviews with 25 year 5–6 students who had completed five items taken from an assessment of mental computational fluency. This assessment required students to reason from the perspective of a fictional student (Emma), who had applied the associative property in various ways to solve multiplication problems. In the interviews, students had to explain their understanding of Emma’s thinking. Coding of this interview data revealed distinct continuums of understanding of each of the three applications of the associative property (noticing, doubling and halving, and place value), which teachers could use to inform their planning. The findings reveal that students best understood the noticing application, closely followed by doubling and halving. By contrast, the place value application was not well understood, which we attribute to students relying on truncation procedures applied with little or no conceptual understanding.

Integrating polar research into undergraduate curricula using computational guided inquiry

Polar research plays a vital role in developing our understanding of Earth’s climate system. It is intrinsically interdisciplinary, lending itself to integration into existing undergraduate courses. Here we explore introducing undergraduates to polar research through computational guided inquiry (CGI) modules taught in a variety of courses and disciplines. Students apply course disciplinary techniques to analysis of polar data or research, in the context of climate change, by working through educational modules that include spreadsheets (ExcelTM) or interactive computer programing (Python in a Jupyter Notebook), over a few class or lab periods. The goals of this exploratory curriculum project are to determine instructor perceptions of effectiveness of the educational modules for teaching preexisting disciplinary course objectives, as well as student perceptions of enjoyment and learning. Evaluation consisted of a student questionnaire and interviews with instructors by an external evaluator. Students and instructors overall reported positive experiences with the modules, highlighted the importance of polar data and climate literacy, and noted increases in student understanding of course learning goals and comfort with the computational tools. Professors further reported that students found the modules motivating, fun and engaging. Taken together, this suggests that the modules are an effective means of bringing polar research into undergraduate classrooms while satisfying instructor goals for course learning objectives. Lessons learned include the importance of providing material such as videos to help transition to the topics of polar research and climate change and of supporting widely varying computational fluency.

Parents’ characteristics, the home environment, and children’s numeracy skills: How are they related in low- to middle-income families in the Philippines?

The home numeracy environment is often assumed to play a role in early numeracy development. This study aimed to provide a comprehensive picture of the home numeracy environment of children living in low- to middle-income communities in the Philippines. Moreover, it examined the relations among various parents’ characteristics, the home numeracy environment parents created, and their young children’s numeracy skills. Data were collected from 290 families through parental questionnaires, numeracy tests for children, and numeracy tests for parents. Results of structural equation modeling showed that parents’ education level, calculation fluency, own reported math activities, home numeracy resources and home numeracy activities all had direct associations with children’s numeracy skills. Parents’ calculation fluency and own math activities also had indirect links with children’s numeracy skills through home numeracy resources and home numeracy activities. These findings provide insights into the question of which children are at risk for poor numeracy development and how to support children’s early numeracy learning.

Brief Experimental Analysis of Math Interventions: A Synthesis of Evidence

Brief experimental analysis (BEA) is a quick method used to identify the function of student learning difficulties and match effective interventions to students’ needs. Extensive work has been done to explore the use of this methodology to determine effective reading interventions; however, a smaller number of published studies have examined the use of BEAs in math. The purpose of the current review was to identify all studies that have used BEA methodology in math. Fifteen studies that included 63 participants and used BEA methodology to identify the most effective math intervention for students were located. Results of the synthesis indicate that the majority of BEAs compared skill and performance interventions on computational fluency; however, the methodology across the included studies varied. Strengths and limitations of the research, in addition to implications for research and practice, are discussed.

Do the processes engaged during mathematical word-problem solving differ along the distribution of word-problem competence?

Conventional research methods for understanding sources of individual differences in word-problem solving (WPS) only permit estimation of average relations between component processes and outcomes. The purpose of the present study was instead to examine whether and if so how the component processes engaged in WPS differ along the spectrum of WPS performance. Second graders (N = 1130) from 126 classrooms in 17 schools were assessed on component processes (reasoning, in-class attentive behavior, working memory, language comprehension, calculation fluency, word reading) and WPS. Multilevel, unconditional quantile multiple regression indicated that 3 component processes, calculation fluency, language comprehension, and working memory, are engaged in WPS differentially depending on students’ overall word-problem skill. The role of calculation fluency and language comprehension was stronger with more competent word-problem solving ability. By contrast, the role of working memory was stronger with intermediate-level than for strong problem solving. Results deepen insight into the role of these processes in WPS and provide the basis for hypothesizing how instructional strategies may be differentiated depending on students’ overall level of WPS competence.

Verbal counting skill predicts later math performance and difficulties in middle school

This study examined the role of verbal counting skill as an early predictor of math performance and difficulties (at or below −1.5 standard deviation in basic math skills) in middle school. The role of fourth-grade level arithmetical skills (i.e., calculation fluency, multi-digit arithmetic i.e. procedural calculation, and word problem solving) as mediators was also investigated. The participants included 207 children in central Finland who were studied from kindergarten to the seventh grade. Path modeling showed that verbal counting in kindergarten is a strong predictor for basic math performance in seventh grade, explaining even 52% of the variance in these skills after controlling for the mothers’ education levels. This association between early verbal counting skill and basic math performance was partly mediated through fourth-grade procedural calculation and word problem solving skills. Furthermore, verbal counting had an unique predictive relation to middle school math performance above and beyond the basic arithmetical and problem solving skills in fourth grade. Poor kindergarten verbal counting skill was a significant indicator for later difficulties in mathematics.

Effects of Interventions with Manipulatives on Immediate Learning, Maintenance, and Transfer in Children with Mathematics Learning Disabilities: A Systematic Review

Manipulatives are concrete or virtual objects (e.g., blocks and chips) often used in elementary grades to illustrate abstract mathematical concepts. We conducted a systematic review to examine the effects of interventions delivered with manipulatives on the learning of children with mathematics learning disabilities (MLD). The outcomes observed in the sample (N = 38) were learning, maintenance, and transfer in a variety of mathematical domains. Interventions using manipulatives were reported to be effective for a range of learning objectives (e.g., conceptual understanding and computational fluency), but several methodological weaknesses were observed. Analyses also highlighted considerable heterogeneity in the studies reviewed in terms of participant characteristics, intervention approaches, and methodology. We discuss overall effects of interventions with manipulatives in the MLD population, the methodological quality across the sample, and implications for practice.

Misaligned core body temperature rhythms impact cognitive performance of hospital shift work nurses.

Circadian rhythms greatly influence 24-h variation in cognition in nearly all organisms, including humans. Circadian clock impairment and sleep disruption are detrimental to hippocampus-dependent memory and negatively influence the acquisition and recall of learned behaviors. The circadian clock can become out of sync with the environment during circadian misalignment. Shift work represents a real-world model of circadian misalignment that can be studied for its physiological implications. The present study aimed to test the hypothesis that circadian misalignment disrupts vigilance and cognitive performance on occupationally relevant tasks using shift work as a model. As such, we sought to (1) explore the general effects of night- and day-shift worker schedules on sleep-wake parameters and core body temperature (CBT) phase, and (2) determine whether shift-type and CBT phase impact cognitive performance and vigilance at the end of a 12-h shift. We observed a sample of day-shift and night-shift hospital nurses over a 10-day period. At the end of three, consecutive, 12-h shifts (7 pm-7am or 7am-7 pm), participants completed a cognitive battery assessing vigilance, cognitive throughput, and medication calculation fluency (via an investigator developed and tested metric). Night-shift nurses exhibited significantly greater sleep fragmentation as well as a greater disparity between their wake-time and time of CBT minimum compared to day-shift nurses. Night-shift nurses exhibited significantly slower cognitive proficiency at the end of their shifts, even after adjustment for CBT phase. These results suggest that circadian disruption and reduced sleep quality both contribute to cognitive functioning and performance.

The effects of teaching mental calculation in the development of mathematical abilities

Many countries include mental calculation within their curriculum, as the literature shows benefits related to it. However, evidence mainly focuses on the effects of teaching mental calculation on computational fluency. Therefore, the authors aimed to assess the effects of teaching mental calculation on double-digit computation, number line estimation, and computational fluency. Fifty second-grade students from a Uruguayan private school were randomly assigned to a control or treatment condition (n = 25 each). The treatment group participated in 15 sessions of mental calculation intervention, while the control group received “teaching as usual.” Assessment took place before and after the intervention. The authors found no significant impact of the treatment on the mathematical outcome variables. The results are explained in the context of the available literature and the methodological decisions taken. From a perspective of evidence-based teaching practices, the present study contributes to a better understanding of effective instructional principles.

The Role of Approximate Number System in Different Mathematics Skills Across Grades

Although approximate number system (ANS) has been found to predict mathematics ability, it remains unclear if both aspects of ANS (symbolic and non-symbolic estimation) contribute equally well to mathematics performance and if their contribution varies as a function of the mathematics outcome and grade level. Thus, in this study, we examined the effects of both aspects of ANS on different mathematics skills across three grade levels. Three hundred eleven children (100 children from kindergarten, 107 children from Grade 2, and 104 children from Grade 4) from two kindergartens and three elementary schools in Shanghai, China, were assessed on measures of ANS (dot estimation and number line estimation), general cognitive ability (nonverbal intelligence, inhibition, and working memory), and mathematics abilities (numerical operations and mathematical problem solving in all grades, early mathematical skills in kindergarten, and calculation fluency in Grades 2 and 4). Results of hierarchical regression analyses showed that, in kindergarten, non-symbolic estimation predicted all mathematics skills even after controlling for age, gender, and general cognitive ability. In Grades 2 and 4, symbolic estimation accounted for unique variance in mathematical problem solving, but not in calculation fluency. Symbolic estimation also predicted numerical operations in Grade 4. Taken together, these findings suggest that in the early phases of mathematics development different aspects of ANS contribute to different mathematics skills.

Understanding Students' Computational Fluency: Synechistically Using Test Scores and Interviews for a Richer Picture

Mathematics education researchers are tasked with solving practical research problems involving complex constructs in complex settings. The effective integration of quantitative and qualitative data allows researchers to draw more nuanced conclusions about these complex phenomena. This article describes the use of a convergent parallel mixed methods design to integrate two seemingly conflicting data sources that measured six second-grade students' development of computational fluency. The mixed methods analysis of students' computational fluency assessments and interviews showed that there was variation in students' assessment scores, strategy use, and engagement of number sense. Within these variations, the quantitative and qualitative data converged or diverged at various measurement points, and the results highlight the importance of merging the two data sets to capture a richer picture of students' computational fluency. Implications for using mixed methods in understanding how mathematics learning occurs in classrooms are discussed.

Does Multi-Component Strategy Training Improve Calculation Fluency Among Poor Performing Elementary School Children?

The aim of the present study was to extend the previous intervention research in math by examining whether elementary school children with poor calculation fluency benefit from strategy training focusing on derived fact strategies and following an integrative framework, i.e., integrating factual, conceptual, and procedural arithmetic knowledge. It was also examined what kind of changes can be found in frequency of using different strategies. A quasi-experimental design was applied, and the study was carried out within the context of the school and its schedules and resources. Twenty schools in Finland volunteered to participate, and 1376 children were screened in for calculation fluency problems. Children from second to fourth grades were recruited for the math intervention study. Children with low performance (below the 20th percentile) were selected for individual assessment, and indications of using counting-based strategies were the inclusion criteria. Altogether, 69 children participated in calculation training for 12 weeks. Children participated in a group based strategy training twice a week for 45 min. In addition, they had two short weekly sessions for practicing basic addition skills. Along with pre- and post-intervention assessments, a 5-month follow-up assessment was conducted to exam the long-term effects of the intervention. The results showed that children with dysfluent calculation skills participating in the intervention improved significantly in their addition fluency during the intervention period, showing greater positive change than business-as-usual or reading intervention controls. They also maintained the reached fluency level during the 5-month follow-up but did not continue to develop in addition fluency after the end of the intensive training program. There was an increase in fact retrieval and derived fact/decomposition as the preferred strategies in math intervention children and a decrease of the use of counting-based strategies, which were the most common strategies for them before the intervention. No transfer effect was found for subtraction fluency.

Protocol for a systematic review: Interventions to improve mathematics achievement in primary school-aged children: a systematic review.

Protocol for a systematic review: Interventions to improve mathematics achievement in primary school-aged children: a systematic review.

Direct and indirect effects of self-concept of ability on math skills

The purpose of this study was to examine the direct and indirect effects of self-concept of ability on two math skills (calculation fluency and problem solving) in three grade levels (2, 4, and 7) in China. Two hundred forty-one children (80 Grade 2, 76 Grade 4, and 85 Grade 7) from Shanghai (China) were assessed on calculation fluency and problem solving. They also responded to a questionnaire measuring their math self-concept and interest. Teachers rated the children's task-focused behavior. The results of path analysis showed that self-concept of ability had a direct effect on calculation fluency in Grade 7 and on problem solving in Grades 4 and 7. The results of multiple mediation analyses further showed that part of self-concept of ability's effects on problem solving in Grades 4 and 7 were mediated by task-focused behavior. Taken together, our findings suggest that, even in the context of Chinese culture, self-concept of ability is important for math skills, particularly when predicting problem solving.

Selecting effective intervention strategies for escape-maintained academic-performance problems: Consider giving 'em a break!

This study compared consequence-and antecedent-based strategies to determine which treatments or combination of treatments produced the strongest improvements in math computation fluency with four elementary-aged students whose math computation was under the control of an escape contingency. Functional analyses were conducted to identify elementary-school students whose academic responding was under a negative-reinforcement contingency. A multielement design was then used to examine the impact of four treatments (DNRA, DRA, task choice, and task choice plus DRA) on each student's rate of correct digits per min. All four treatments increased rate of responding. Differentiated results were obtained for all participants, indicating a reliable effect. Yet, participants responded differently to the treatments, illustrating the need to investigate and adapt interventions for escape-motivated behavior on a case-by-case basis. Results are also discussed in terms of the effectiveness of choice relative to reinforcement procedures, whether there were additional benefits to combining treatments, and which type of reinforcement procedures (DRA or DNRA) appears to be more effective for students whose behavior is under the control of an escape contingency.

Number Line Estimation Predicts Mathematical Skills: Difference in Grades 2 and 4.

Studies have shown that number line estimation is important for learning. However, it is yet unclear if number line estimation predicts different mathematical skills in different grades after controlling for age, non-verbal cognitive ability, attention, and working memory. The purpose of this study was to examine the role of number line estimation on two mathematical skills (calculation fluency and math problem-solving) in grade 2 and grade 4. One hundred and forty-eight children from Shanghai, China were assessed on measures of number line estimation, non-verbal cognitive ability (non-verbal matrices), working memory (N-back), attention (expressive attention), and mathematical skills (calculation fluency and math problem-solving). The results showed that in grade 2, number line estimation correlated significantly with calculation fluency (r = -0.27, p < 0.05) and math problem-solving (r = -0.52, p < 0.01). In grade 4, number line estimation correlated significantly with math problem-solving (r = -0.38, p < 0.01), but not with calculation fluency. Regression analyses indicated that in grade 2, number line estimation accounted for unique variance in math problem-solving (12.0%) and calculation fluency (4.0%) after controlling for the effects of age, non-verbal cognitive ability, attention, and working memory. In grade 4, number line estimation accounted for unique variance in math problem-solving (9.0%) but not in calculation fluency. These findings suggested that number line estimation had an important role in math problem-solving for both grades 2 and 4 children and in calculation fluency for grade 2 children. We concluded that number line estimation could be a useful indicator for teachers to identify and improve children’s mathematical skills.

Number Strings: Daily Computational Fluency

Second graders use this beneficial instructional routine alongside the Standards for Mathematical Practice.

The role of cognitive processes, foundational math skill, and calculation accuracy and fluency in word-problem solving versus prealgebraic knowledge.

The purpose of this study was to examine child-level pathways in development of prealgebraic knowledge versus word-problem solving, while evaluating the contribution of calculation accuracy and fluency as mediators of foundational skills/processes. Children (n = 962; mean 7.60 years) were assessed on general cognitive processes and early calculation, word-problem, and number knowledge at start of Grade 2; calculation accuracy and calculation fluency at end of Grade 2; and prealgebraic knowledge and word-problem solving at end of Grade 4. Important similarities in pathways were identified, but path analysis also indicated that language comprehension is more critical for later word-problem solving than prealgebraic knowledge. We conclude that pathways in development of these forms of 4th-grade mathematics performance are more alike than different, but demonstrate the need to fine-tune instruction for strands of the mathematics curriculum in ways that address individual students' foundational mathematics skills or cognitive processes. (PsycINFO Database Record

The Effects of Training on Computational Fluency and Working Memory on Students’ Achievement and Retention in Algebra

Many students, despite a good understanding of mathematical concepts, are hindered by simple calculations. Recent cognitive researches claimed that working memory which is the limited capacity system of the brain that allows simultaneous storage and processing of temporary information may be improved through trainings and, consequently, improved mathematics performance. The purpose of this study is to investigate the effect of the dual task training on computational fluency with a load of working memory task compared to the traditional basic mathematics drill on the students’ achievement and retention in high school algebra. This research made use of two randomly chosen intact groups of fourth year high school students who were given both pretest and posttest using three validated instruments. A 3-week follow-up test was administered to assess algebra retention. This study found that the students who received the dual task training have significantly higher gains in their computational fluency and working memory ability. However, both groups were comparable in terms of their algebra achievement and retention after the 6-week training. The computational fluency was also found to be a predictor of the students’ achievement and retention in algebra. Thus, the working memory training has no direct effect on the students’ algebra achievement but mediated the effect of computational fluency. A longer and uninterrupted experimental period is recommended for further studies with a focus on binary number tasks as this was found to predict computational fluency.

Behavioral and Neural Effects of Game-Based Learning on Improving Computational Fluency With Numbers

Abstract. This study examines the short-term behavioral and neural effects of an educational mobile game called MathDash on computational fluency. A portable functional near-infrared spectroscopy (fNIR) device was used to monitor changes in the prefrontal cortices of middle school students due to game-based training. Our aim is to explore changes in neural activity at regions in the prefrontal cortex associated with the management of working memory and attentional resources due to arithmetic training with MathDash. Overall, our results indicated that playing MathDash for a short duration of time elicited behavioral improvements as well as functional changes in bilateral dorsolateral prefrontal cortex.