A comparison of behavioral and constructivist interventions for increasing math‐fact fluency in a second‐grade classroom

Although basic math skill deficits are commonly encountered across elementary and secondary school students, few empirically validated, group‐administered interventions are available for educators attempting to prevent or remedy such problems. This study compared the effectiveness of two theoretically distinct interventions for improving the subtraction‐fact fluency of a general education class of second‐grade students. An alternating treatments design was used to compare a behavioral intervention, Cover, Copy, and Compare (CCC), to an intervention from a constructivist‐oriented resource, Facts That Last (FTL). Results demonstrated that CCC led to increases in math‐fact fluency, whereas the classwide response to FTL activities did not differ from the control condition. Two months postintervention, maintenance data revealed that the fluency increases associated with CCC were sustained. Discussion focuses on practical implications of the study as well as possible theoretical explanations for the documented results. © 2010 Wiley Periodicals, Inc.     

The differential effects of two self‐managed math instruction procedures: Cover,Copy, and Compare versus Copy,Cover, and Compare

This study compared the fluency and error rates produced when using the Cover, Copy, and Compare (CCC) and a modified CCC procedure (MCCC) called Copy, Cover, and Compare to complete subtraction math problems. Two second‐grade classrooms consisting of 47 total students participated in the study. The following items were administered to participants: (a) a timed pretest, (b) a timed CCC worksheet, (c) a timed MCCC worksheet, and (d) a timed posttest. Then the participants were asked which procedure they liked best. Results revealed significantly higher digits correct per minute (i.e., fluency scores) on the posttest when compared with the pretest scores. Likewise, students' fluency scores were significantly higher under the CCC condition when compared to the MCCC condition. There were no significant differences in accuracy from pretest to posttest nor between the CCC and MCCC conditions. Discussion focuses on implications for modifying instructional strategies, measuring student progress, implications for practice, and directions for future research. © 2009 Wiley Periodicals, Inc.     

An investigation of Detect,Practice, and Repair to remedy math‐fact deficits in a group of third‐grade students

A multiple‐probe‐across‐problem‐sets (tasks) design was used to evaluate the effects of the Detect, Practice, and Repair (DPR) on multiplication‐fact fluency development in seven third‐grade students nominated by their teacher as needing remediation. DPR is a multicomponent intervention and begins with a group‐administered, metronome‐paced assessment used to identify specific facts in need of repair. Next, Cover, Copy, and Compare (CCC) procedures are used to enhance automaticity with those specific facts. Lastly, students complete a 1‐min speed drill and self‐graph their fluency performance. Results showed large level and trend increases in fact fluency after DPR was applied across all three sets of multiplication problems. Discussion focuses on the importance of developing effective and efficient basic‐skill‐remediation procedures and directions for future research. © 2010 Wiley Periodicals, Inc.     

A meta‐analytic review of the cover‐copy‐compare and variations of this self‐management procedure

Studies that examined copy‐cover‐compare (CCC) and variations of this procedure were reviewed and analyzed. This review revealed a substantial number of studies that validated the use of CCC across spelling and math skills and across students with and without disabilities. A meta‐analysis of findings indicated that CCC and variations of this procedure were effective for helping students acquire and become fluent in academic skills. The strongest effects were evident when CCC and variations of this procedure were combined with other evidence‐based instructional components. Limitations, future directions for research, and recommendations for practice are offered. © 2011 Wiley Periodicals, Inc.     

The Effect of Goal-Line Presentation and Goal Selection on First-Grader Subtraction Fluency

Math proficiency is related to math calculation fluency. Explicit timing provides repeated practice for math fluency. It is enhanced through goal setting, graphic feedback, and rewards. Self-selected goals have potential to increase performance for math fluency. This study compared the effect of goal lines, and researcher goals versus self-selected goals. The authors compared three groups: (a) researcher-selected goals only, (b) researcher-selected goals with goal lines, and (c) self-selected goals with goal lines. First-grade students completed subtraction probes and colored in a bar graph with their performance. When participants met or exceeded their goals, participants received rewards twice per week. Results indicated that the researcher-selected goals with goal lines procedures with explicit timing had the greatest effect on subtraction fluency for first-grade students. The researcher-selected goals with goal lines and self-selected goals with goal lines groups outperformed the researcher-selected goals-only group.     

ANALYZING MATH‐TO‐MASTERY THROUGH BRIEF EXPERIMENTAL ANALYSIS

The current study evaluated the effectiveness of individualized math‐to‐mastery (MTM) interventions, selected though brief experimental analysis (BEA), at increasing math fluency skills for three elementary‐aged females. As MTM has only been investigated as a multicomponent intervention, the present study utilized BEA to identify those specific components that led to math skills gains in the most efficient manner possible. BEA results indicated that for two of three participants only a partial MTM intervention was necessary to prompt fluency gains, while the entire intervention was the most effective for the third. During extended analysis all three participants displayed math skills gains above those seen during repeated baseline assessments. Results are discussed in terms of further refining MTM through BEA procedures so as to individually target math skill deficits by considering both intervention effectiveness and efficiency.     

Special education students’ use of indirect addition in solving subtraction problems up to 100—A proof of the didactical potential of an ignored procedure

In this study, we examined special education students’ use of indirect addition (subtraction by adding on) for solving two-digit subtraction problems. Fifty-six students (8- to 12-year-olds), with a mathematical level of end grade 2, participated in the study. They were given a computer-based test on subtraction with different types of problems. Although most students had not been taught indirect addition for solving subtraction problems, they frequently applied this procedure spontaneously. The item characteristics were the main prompt for using indirect addition. Context problems that reflect an adding-on situation and problems that have a small difference between the minuend and subtrahend most strongly elicited the use of the indirect addition procedure. Moreover, indirect addition was identified as a highly successful procedure for special education students, and the best predictor of a correct answer was found in combination with a stringing strategy.     

Mathematical power of special‐needs pupils: An ICT‐based dynamic assessment format to reveal weak pupils' learning potential1

This paper reports a study aimed at revealing special‐educational‐needs pupils' learning potential by means of an ICT‐based assessment including a dynamic visual tool that might help pupils when solving mathematics problems. The study focused on subtraction problems up to 100, which require ‘borrowing’. These problems, in which the value of the ones‐digit of the subtrahend is larger than the ones‐digit of the minuend, are known as a serious difficulty for weak pupils in mathematics. Seven of such problems from a standardised test were placed in the ICT environment. Data were collected from two test conditions: the standardised written test format and the ICT version of the test items including the tool that provided pupils with a set of virtual manipulatives. The 37 pupils involved in the study were 8–12 years old and from two special‐education schools in the Netherlands. Comparison of the performance scores in the two formats showed that an ICT‐based assessment format, including a dynamic visual tool, can reveal weak pupils' learning potential and strategy use. The study also pointed out that ‘partial‐tool use’, ie, not carrying out the complete subtraction operation with the tool, can provide sufficient support to find the correct answer.     

Does speed matter? A comparison of the effectiveness of fluency and discrete trial training for teaching noun labels to children with autism

This study used an alternating‐treatment design to compare the efficacy of discrete trial training (DTT) with fluency training (FT) for the acquisition, stimulus generalization, and retention of noun labels in children with autism. Four elementary‐age students diagnosed with autism were taught to expressively label nouns using a DTT format and a FT format. A between‐treatments comparison of the total number of nouns retained at 6 weeks post intervention was also conducted to compare retention. The results of this study showed that FT was superior to DTT in all of these areas. © 2010 Wiley Periodicals, Inc.     

Coordinating Procedural and Conceptual Knowledge to Make Sense of Word Equations: Understanding the complexity of a ‘simple’ completion task at the learner’s resolution

The present paper discusses the conceptual demands of an apparently straightforward task set to secondary‐level students—completing chemical word equations with a single omitted term. Chemical equations are of considerable importance in chemistry, and school students are expected to learn to be able to write and interpret them. However, it is recognised that many students find them challenging. The present paper explores students’ accounts of their attempts to identify the missing terms, to illuminate why working with chemical word equations is so challenging from the learner’s perspective. Three hundred secondary‐age students responded to a five‐item exercise based on chemicals and types of reactions commonly met at school level. For each item they were asked to identify the missing term in a word equation, and to explain their answers. This provided a database containing more than 1,000 student accounts of their rationales. Analysis of the data led to the identification of seven main classes of strategy used to answer the questions. Most approaches required the coordination of chemical knowledge at several different levels for a successful outcome; and there was much evidence both for correct answers based on flawed chemical thinking, and appropriate chemical thinking being insufficient to lead to the correct answer. It is suggested that the model reported here should be tested by more in‐depth methods, but could help chemistry teachers appreciate learners’ difficulties and offer them explicit support in selection and application of strategies when working with chemical equations.     

An examination of group‐based treatment packages for increasing elementary‐aged students' reading fluency

Reading fluency has been described as one of the essential ingredients for ensuring that students become successful readers. Unfortunately, a large number of elementary‐aged students in this country do not fluently read age‐appropriate material. Because of this, small‐group interventions are practical and more time efficient than individualized intervention programs, but very few small‐group interventions have been developed to target students' reading fluency of connected text. The primary purpose of this study was to examine four group‐based treatment packages containing two or more of the following reading interventions: repeated reading, listening passage preview, and practicing difficult words in isolation. Effects of each treatment package were evaluated with 4 third graders using an alternating‐treatments design, and effects were evaluated for both immediate and retained reading‐fluency gains. Findings indicated that the combination of all three intervention components was most effective. Results also suggested some inconsistent relationships between immediate and retained reading gains for 3 of the 4 students. Implications of these findings, limitations of the study, and directions for future research are discussed. © 2006 Wiley Periodicals, Inc. Psychol Schs 43: 183–195, 2006.     

A group math intervention: The modification of cover,copy, and compare for group application

A drill and practice intervention was applied in a small-group setting to improve students' fluency (accuracy and speed) with basic math facts. The Cover, Copy, and Compare intervention (Skinner, Turco, Beatty, & Rasavage, 1989) was modified and implemented with a small, middle school Chapter I math class. Data are presented showing digits correct per minute (DCPM) using a multiple baseline design across sets of division facts. Component parts of DCPM, speed and accuracy, are further discussed. Results of this study provide the scientist-practitioner with ideas about implementing group interventions using drill for math facts. Ways to refine the intervention based on baseline data are discussed.     

Math fluency is etiologically distinct from untimed math performance, decoding fluency, and untimed reading performance: evidence from a twin study

The authors examined whether math fluency was independent from untimed math and from reading using 314 pairs of school-aged twins drawn from the Western Reserve Reading and Math Projects. Twins were assessed through a 90-min home visit at approximately age 10 and were reassessed in their homes approximately 1 year later. Results suggested that the shared environment and genetics influenced the covariance among math fluency, untimed math measures, and reading measures. However, roughly two thirds of the variance in math fluency was independent from untimed math measures and reading, including reading fluency. The majority of this independent variance was the result of genetic factors that were longitudinally stable across two measurement occasions. These results suggest that math fluency, although related to other math measures, may also be a genetically distinct dimension of mathematics performance.     

Teaching Subtraction and Multiplication with Regrouping Using the Concrete‐Representational‐Abstract Sequence and Strategic Instruction Model

Based on Common Core Standards (2010), mathematics interventions should emphasize conceptual understanding of numbers and operations as well as fluency. For students at risk for failure, the concrete‐representational‐abstract (CRA) sequence and the Strategic Instruction Model (SIM) have been shown effective in teaching computation with an emphasis on conceptual understanding. No studies have investigated the effects of CRA and SIM for teaching multiplication with regrouping. Therefore, the purpose of this study was to replicate and extend the literature, teaching subtraction and multiplication with regrouping using CRA and SIM. Three students receiving tier three mathematics interventions participated. A multiple‐probe across behaviors design was used to show a functional relation. All of the students demonstrated increases in fluency across all regrouping tasks.     

EXAMINATION OF A ONE‐TRIAL BRIEF EXPERIMENTAL ANALYSIS TO IDENTIFY READING FLUENCY INTERVENTIONS

This study sought to evaluate whether a one‐trial brief experimental analysis (OTBEA) would reliably and validly identify effective treatments to improve oral reading fluency for 6 elementary school students referred for reading problems. An OTBEA was conducted with each participant to assess the effects of skill‐ and performance‐based treatment components. Following the OTBEA, an extended analysis was conducted using an alternating treatments design to experimentally assess whether predictions generated by the OTBEA would be accurate across sessions, time, and stimulus materials. OTBEA results suggested that 3 participants required a treatment package containing both performance‐based and skill‐based instructional components and that the other 3 participants might only need performance‐based interventions. Results indicate that predictions were accurate for those participants requiring both types of intervention components, but that predictions for the remaining participants met with mixed results in the extended analysis. Results are discussed in terms of further refining brief experimental analysis methods to make them more efficient for educators while ensuring valid results.     

Listening previewing in reading to read: Relative effects on oral reading fluency

An intervention designed to increase oral reading fluency (Reading to Read; RTR) as well as the additional component of listening previewing (PRV) was evaluated in an alternating treatments design with 3 African-American male elementary students (9, 10, and 12 years of age; 2 fourth grade and 1 third grade) who were reading 2 to 3 years below their current grade placement. Considering the relatively brief amount of time spent during intervention (4 to 9 weeks; 2 to 3 30-minute sessions per week), results evidenced substantial increases for all 3 students on correct words per minute and percent comprehension accuracy, and substantial decreases in mean number of errors. The PRV condition yielded higher correct words per minute rates and lower error rates for 2 of the students, whereas the third student's performance was superior with the regular RTR intervention without PRV. The efficacy of PRV within RTR is discussed with regard to factors related to students' attentional problems.     

Assessment of subtraction scene understanding using a story‐generation task

The present study used a new assessment technique, the story‐generation task, to examine students’ understanding of subtraction scenes. The students from four grade levels (110 first‐, 107 third‐, 110 fourth‐ and 119 sixth‐graders) generated stories under the constraints provided by a picture (representing Change, Combine or Compare scene) and a number sentence. The results showed that the Change scene was the most intuitive, and the students tended to reinterpret Combine and Compare scenes as a form of Change scene, even when they understood that the identity of the objects in the Combine and the Compare scenes was different. The number of students generating correct stories for the Combine scene increased through the elementary school grades while for the Compare scene the number of students generating correct stories remained fairly constant from Grade 3 to Grade 6. This study also examined students’ exposure to subtraction scenes in mathematics textbooks and discussed educational implications for supporting students’ understanding of subtraction scenes.     

Effects of Fact Retrieval Tutoring on Third‐Grade Students with Math Difficulties with and without Reading Difficulties

The purpose of this study was to assess the efficacy of fact retrieval tutoring as a function of math difficulty (MD) subtype, that is, whether students have MD alone (MD‐only) or have concurrent difficulty with math and reading (MDRD). Third graders (n= 139) at two sites were randomly assigned, blocking by site and MD subtype, to four tutoring conditions: fact retrieval practice, conceptual fact retrieval instruction with practice, procedural computation/estimation instruction, and control (no tutoring). Tutoring occurred for 45 sessions over 15 weeks for 15–25 minutes per session. Results provided evidence of an interaction between tutoring condition and MD subtype status for assessment of fact retrieval. For MD‐only students, students in both fact retrieval conditions achieved comparably and outperformed MD‐only students in the control group as well as those in the procedural computation/estimation instruction group. By contrast, for MDRD students, there were no significant differences among intervention conditions.     

Elicitation of personal probabilities and their assessment

A student's choice of an answer to a test question is a coarse measure of his knowledge about the subject matter of the question. Much finer measurement might be achieved if the student were asked to estimate, for each possible answer, the probability that it is the correct one. Such a procedure could yield two classes of benefits: (a) students could learn the language of numerical probability and use it to communicate uncertainty, and (b) the learning of other subjects could be facilitated. This report describes the rationale underlying a procedure for eliciting personal estimates of probabilities utilizing a proper scoring rule, and illustrates some new techniques for calibrating those probabilities and providing better feedback to students learning to assess uncertainty. In addition, new results are presented compring the incentive for study, rehearsal, and practice provided by the proper scoring rule with that provided by the simple choice procedure, and concerning the potential effect of cutoff scores and prizes upon student behavior.     

Enhancing science performance in students with learning disabilities using cover,copy, and compare: A student shows the way

A student who participated in a previous study where he was trained to use a self‐managed academic intervention known as cover, copy, and compare (CCC) to enhance his mathematics performance developed the specific intervention used in the current study. Without prompting, this student adapted and employed this procedure to learn bone in the human body. In the current study a multiple baseline across tasks design was used to determine if CCC could be used to enhance accuracy in identifying parts of the human heart in three students with learning disabilities. Three students learned how to perform the self‐management procedures during one session that lasted less than 15 minutes. Results showed that immediately after implementing the CCC intervention students' accuracy improved. Following 11 brief sessions, students were able to identify 15 parts of a human heart with an overall accuracy of 80% over 2 consecutive days. Discussion focuses on limitations and implications of this study and school psychologists' and students' contributions to the development and empirical validation of academic interventions. © 2002 Wiley Periodicals, Inc.