Mathematical Fluency Research Articles

Collaborative learning with multi-touch technology: Developing adaptive expertise

Developing fluency and flexibility in mathematics is a key goal of upper primary schooling, however, while fluency can be developed with practice, designing activities that support the development of flexibility is more difficult. Drawing on concepts of adaptive expertise, we developed a task for a multi-touch classroom, NumberNet, that aimed to support both fluency and flexibility. Results from a quasi-experimental study of 86 students (44 using NumberNet, 42 using a paper-based comparison activity) indicated that all students increased in fluency after completing these activities, while students who used NumberNet also increased in flexibility. Video analysis of the NumberNet groups indicate that the opportunity to collaborate, and learn from other groups' expressions, may have supported this increase in flexibility. The final phase of the task suggests future possibilities for engaging students in mathematical discourse to further support the development of mathematical adaptive expertise.

A comparison of class‐wide taped problems and cover, copy, and compare for enhancing mathematics fluency

AbstractAn adapted alternating treatments design was used to compare the effects of class‐wide applications of Taped Problems (TP) and Cover, Copy, and Compare (CCC) procedures designed to enhance subtraction fact fluency in an intact third‐grade classroom. During the TP procedure, a tape provided an auditory prompt (i.e., the problem), followed by a 2‐second delay and then the correct answer. Students attempted to supply the correct answer before the answer was provided by the tape. During the CCC procedure, students would cover a math‐fact family triangle, attempt to write one of two possible problems and answer, write the reciprocal fact and answer, and evaluate their responses. Results show that TP caused greater increases in subtraction fact fluency than CCC or a control condition. Discussion focuses on preventing student math deficits, issues regarding alternating treatments designs, and future directions for research. © 2012 Wiley Periodicals, Inc.

Meta–Analysis of Mathematic Basic–Fact Fluency Interventions: A Component Analysis

Mathematics fluency is a critical component of mathematics learning yet few attempts have been made to synthesize this research base. Seventeen single–case design studies with 55 participants were reviewed using meta–analytic procedures. A component analysis of practice elements was conducted and treatment intensity and feasibility were examined. Findings suggest that drill and practice with modeling produced the largest effect sizes. Treatments with more than 3 components yielded higher effect sizes than those with fewer than 3, and a combination of treatment agents lead to better outcomes than a single agent. Other findings pertaining to prebaseline assessment, treatment time, experimental design employed, and treatment setting are also discussed.

Efficacy of Two Mathematics Interventions for Enhancing Fluency with Elementary Students

An alternating treatments design was used to evaluate two curriculum-based mathematics interventions designed to enhance fluency with three elementary school students. Results indicate that both the Math to Mastery (MTM) intervention and the Cover, Copy, Compare (CCC) intervention were effective at increasing mathematics fluency, as measured by digits correct per min, for all students. However, MTM was more effective than CCC for two of the three students. Follow-up data taken 6 and 18 days following termination of the intervention phase indicated that all participants achieving mastery performance maintained mastery levels across both interventions.

The role of fluency in a mathematics item with an embedded graphic: interpreting a pie chart

The purpose of this study was to identify the pedagogical knowledge relevant to the successful completion of a pie chart item. This purpose was achieved through the identification of the essential fluencies that 12–13-year-olds required for the successful solution of a pie chart item. Fluency relates to ease of solution and is particularly important in mathematics because it impacts on performance. Although the majority of students were successful on this multiple choice item, there was considerable divergence in the strategies they employed. Approximately two-thirds of the students employed efficient multiplicative strategies, which recognised and capitalised on the pie chart as a proportional representation. In contrast, the remaining one-third of students used a less efficient additive strategy that failed to capitalise on the representation of the pie chart. The results of our investigation of students’ performance on the pie chart item during individual interviews revealed that five distinct fluencies were involved in the solution process: conceptual (understanding the question), linguistic (keywords), retrieval (strategy selection), perceptual (orientation of a segment of the pie chart) and graphical (recognising the pie chart as a proportional representation). In addition, some students exhibited mild disfluencies corresponding to the five fluencies identified above. Three major outcomes emerged from the study. First, a model of knowledge of content and students for pie charts was developed. This model can be used to inform instruction about the pie chart and guide strategic support for students. Second, perceptual and graphical fluency were identified as two aspects of the curriculum, which should receive a greater emphasis in the primary years, due to their importance in interpreting pie charts. Finally, a working definition of fluency in mathematics was derived from students’ responses to the pie chart item.

Examining a classwide application of cover-copy-compare with and without goal setting to enhance mathematics fluency.

Examining a classwide application of cover-copy-compare with and without goal setting to enhance mathematics fluency.

Degree of Mathematics Fluency and Success in Second-Semester Introductory Chemistry

We report on the contents and results for 360 students of a mathematics assessment administered at the start of the second-semester introductory chemistry course required for science and engineering majors at the University of Minnesota. This calculator-free, 20-question, 30-minute, multiple-choice, diagnostic quiz includes questions selected specifically for their relevance to this course, concerning logarithms, scientific notation, graphs, and algebra. For the 325 students in degree-granting programs, significant correlations are reported between their mathematics assessment scores and success in this course, as measured by performance on exams (for which scientific calculators were permitted) and course grades. These observations suggest that responses to the simple questions on this unannounced quiz have some predictive utility as signatures of underlying thinking and learning patterns that are associated with success in this course. In addition, we argue that these results indicate an inadequate degree of mathematics fluency for the majority of the students tested, which can seriously impede their abilities to develop a firm conceptual understanding of quantitative introductory chemistry. Several mathematical and pedagogical misconceptions highlighted by these results are discussed, and possible methods are suggested for enhancing students' mental and pencil-and-paper mathematics skills during, and prior to, their participation in this course.

Comparing mathematics interventions: Does initial level of fluency predict intervention effectiveness?

A paucity of research has directly compared empirically supported interventions to examine their effectiveness among students with different mathematics fluency skills. The purpose of this study was to compare the effects of two empirically supported interventions and a control condition on the mathematics fluency of 98 second and third grade students. Specifically, effects of explicit timing and cover-copy-compare on subtraction fluency were examined each session, which occurred twice weekly for a total of six weeks. HLM was used to examine progress during the intervention sessions and determine the effect of initial fluency level on intervention effectiveness. Results suggested students' initial level of fluency impacted intervention effectiveness. That is, for students whose fluency levels fell within the frustrational range CCC and control conditions resulted in the best performance over time but for children whose fluency was in the instructional range explicit timing was the most effective treatment.

Comparing Mathematics Interventions: The Effects of Cover-Copy-Compare Alone and Combined with Performance Feedback on Digits Correct and Incorrect

The purpose of this study was to compare the isolated effects of cover-copy-compare (CCC) with the combined effects of CCC and two types of performance feedback (i.e., digits correct and incorrect per minute) on sixth grade students’ mathematics fluency and accuracy. An alternating treatments design was employed to compare treatments across 16 weeks. Participants included three general education sixth grade students whose teachers recommended them for additional help with mathematics calculation. No differentiation between treatments was demonstrated for any of the participants and individual differences pertaining to the number of sessions needed to reach and consistently perform at mastery levels were found. Follow-up data at 4 and 12 days following termination of the intervention phase indicated that all participants continued to perform at mastery levels. Generalization to a slightly more difficult mathematics skill yielded pre-intervention to post-intervention increases, albeit small. Students rated all three treatment conditions favorably.

Inchworm and a Half: Developing Fraction and Measurement Concepts Using Mathematical Representations

Children's conceptual understanding, strategic competence, and procedural fluency in mathematics are highly influenced by experiences in the early grades. During these years, many important mathematics concepts, including measurement and rational-number sense, should be a part of children's informal investigations. By informally exploring the same concepts in a variety of ways—with concrete manipulatives, visual models, and abstract representations—first graders can connect their initial youthful notions to sophisticated mathematical thinking.

Effectiveness of a multi-component treatment for improving mathematics fluency.

Effectiveness of a multi-component treatment for improving mathematics fluency.

The mathematical knowledge of physics graduates: Primary data and conclusions

Systematic observations were made of the mathematical knowledge of physics students from the U.S. and other countries during their first years of graduate study at Ohio University. It was found that all were deficient in general and in ‘‘modern’’ mathematical concepts, and in problem-solving skills. Sizable fractions of them did not even possess adequate concepts of ‘‘derivative,’’ ‘‘integration,’’ and ‘‘truth.’’ Nearly all were limited to some familiarity with rather elementary calculus, and with equally elementary differential and linear equations, but they showed some ability and a pronounced willingness to perform manipulations. Roughly, they regarded mathematics as mechanical method, not as constructive thinking. In view of the significantly higher levels of mathematical fluency demanded by contemporary advances in physics and in computer usage, none of these students was adequately prepared for future-oriented study, or for research and employment in physics and related areas at the close of the 20th century. It is intended to discuss the likely causes of this state of affairs elsewhere with a view toward remedial actions.

Performance on creativity measures with examination‐taking instructions intended to induce high or low levels of test anxiety

The major purpose of this investigation was to ascertain whether differential levels of performance on three tests of creativity portraying divergent production of verbal (semantic), figural (visual), and mathematical (symbolic) responses would be associated with two sets of test‐taking instructions: (a) one intended to induce a relatively high level of anxiety or stress and (b) the other designed to minimize the degree of anxiety or stress in the testing experience. Participants were 132 private secondary school students (82 females and 50 males) in a college preparatory program who were divided into experimental and control groups. On the basis of three univariate four‐way factorial designs at each of two levels per variable (treatment x sex x test anxiety x mathematics anxiety) with the pretest serving as a covariate, it was observed that only in the instance of a mathematics fluency test did the experimental group exposed to anxiety‐inducing test‐taking instructions achieve a significantly lower mean score on the posttest than did the control group that had been administered instructions intended to minimize anxiety (p < .001). A marginally significant interaction occurred between gender and mathematics anxiety on the mathematics fluency test with highly‐anxious males showing a greater decrement in performance than highly‐anxious females when exposed to anxiety‐inducing instructions. The results were interpreted primarily within a conceptual framework developed by Tobias (1985), who proposed an integration of cognitive capacity and interference as explanatory constructs for decrements in the performance of tasks involving anxiety.

Pengaruh Model Pembelajaran Peer Lesson terhadap Kemampuan Procedural Fluency Matematika

This study aims to determine (1) the mathematical fluency procedural ability of students taught using the peer lesson learning model and those not taught with the peer lesson learning model in class VIII, and (2) the effect of the peer lesson learning model on the mathematics procedural fluency ability of grade VIII students . The population in this study was grade VIII students of SMP Negeri 11 Sungai Penuh, for the study sample consisted of 2 classes from 3 classes in the population, the samples in this study were taken by random sampling technique in which class VIII B was an experimental class and class VIII C was a control class. . Based on the results of research using the t-test for hypothesis testing, it was found that the peer lesson learning model affected the mathematical procedural fluency ability of students in class VIII.