Unlike Denominators Research Articles

There’s a Thin Line Between Numerators and Denominators: Addressing Academic and Behavioral Needs of Students With EBD Using SRSD Fractions

Fractions are an imperative skill for students to master to achieve success in future mathematics concepts and classes. Yet many students, especially those experiencing math difficulties and/or characteristics of emotional and behavioral disorders, continue to exit elementary school without a concrete foundation of fractions skills required to successfully transition into higher level math courses. In this article, we present an overview of SRSD Fractions, a strategy instructional framework to teach adding and subtracting fractions with unlike denominators. We highlight the benefits of using the framework to promote student engagement and skill mastery and identify free-access resources for educators to successfully implement the math intervention in their own classrooms.

Teaching Rational Number Concepts to Fifth-Grade Students Who Struggle With Mathematics

This study’s purpose was to examine the effects of the concrete-representational-abstract integrated (CRA-I) sequence on the performance of students who struggled with rational number concepts. Three students in southeastern U.S. fifth grade class participated in the study. The CRA-I intervention was grounded in the principles of explicit instruction and showed rational number concepts related to fractions and decimals using fraction blocks, number lines, base 10 blocks, and coins. Students learned about unit fractions, fraction magnitude, fraction equivalence, addition of fractions with unlike denominators, equivalent decimals, and notation of fractions as decimals. The researchers used a multiple probe across behaviors design and demonstrated a functional relation between CRA-I and three behaviors: decreased fraction estimation error, accuracy in adding fractions with unlike denominators, and accuracy writing fractions as decimals.

Using CRA-I to Teach Fraction and Decimal Concepts to Students With Learning Disabilities

The purpose of this study was to investigate the effects of the concrete-representational-abstract–integrated (CRA-I) sequence on students with learning disabilities’ performance when learning fraction and decimal concepts. Three elementary-level students in Grades 4 and 6 participated in a single-case multiple probe across behaviors study. The intervention involved explicit use of fraction blocks, coins, base ten blocks, number lines, pictures, and abstract symbols to teach unit fractions, fraction and decimal equivalence, addition of fractions with unlike denominators, and writing fractions as decimals. The researchers demonstrated a functional relation between CRA-I and three different behaviors related to fraction concepts: decreased error in estimating fraction magnitude, addition of unlike fractions, and writing fractions as decimals. The findings show promise in the use of CRA-I for teaching fraction concepts to students with learning disabilities.

Strategies and Procedural and Conceptual Knowledge of Addition of Unlike Denominator Fractions: The Case of Grade 8 Children in Two Districts of the Central Region of Ghana

This study investigated children’s knowledge of the addition of fractions with unlike denominators. It focussed on the strategies used by children to add fractions with unlike denominators. A cross-sectional survey of Grade 8 children in schools in two districts in Ghana was used. A test on the addition of fractions with unlike denominators was administered to 543 high-, average- and low-achieving children, and their responses were categorised based on the characterisation of correct strategies by Braithwaite et al. and Siegler and Pykes. Cluster bar charts and analysis of variance were used to analyse the data. The results showed that the children’s use of correct strategies for adding proper fractions with unlike denominators was higher than their use of correct strategies for adding improper fractions with unlike denominators. Thus, the children’s procedural knowledge of the addition of proper fractions with unlike denominators was higher than that of improper fractions with unlike denominators. Conceptual knowledge of the addition of fractions with unlike denominators was much lower than procedural knowledge of the same. The less advantaged (females, rural and low achievers) used fewer correct strategies and demonstrated less procedural and conceptual knowledge. Among other recommendations, curricula and classroom teaching should emphasise conceptual knowledge and its support for procedural knowledge of the addition of fractions with unlike denominators. Research interventions focussed on the less advantaged are advised.

The impact of the MathMagic learning method on students’ mathematics cognitive learning outcomes

Low cognitive learning outcomes of students can hinder student learning processes. This obstacle occurs because the learning method has yet to facilitate student learning trajectories resulting in low cognitive mathematics learning outcomes for students. The purpose of this research was to determine the impact of the MathMagic learning method on the mathematics cognitive learning outcomes of the students. The method used in this research was a quasi-experimental research design with a nonequivalent control group design. Data collection was done by test technique of students' cognitive learning outcomes. Data analysis techniques in this study used instrument tests in the form of validity and reliability tests, prerequisite tests in the form of normality tests and homogeneity tests, and hypothesis tests in the form of independent sample t-tests. The results show that the MathMagic learning method influenced students' cognitive learning outcomes in mathematics, especially in adding fractions with different denominators. This MathMagic learning method effectively improves understanding of the basic addition of fractions with unlike denominators.

Using SRSD to Improve the Fraction Computations of Students With and At-Risk for EBD

This article provides results from a study examining the impact of a self-regulated strategy development (SRSD) intervention on the fraction calculations of three Grade 5 male students with or at-risk for emotional or behavioral disorders using a multiple-baseline across students, single-case design. The teacher-led SRSD Fractions intervention addressed adding and subtracting fractions with unlike denominators using the FILMS (Find the denominator, Identify the multiples, Locate the least common multiple, Multiply to make new fractions, Solve the problem) strategy. Results of fraction probes are reported along with a discussion of the study with respect to the research question posed, limitations, and future directions for research.

SRSD Fractions: Using Self-Regulated Strategy Development to Support Students’ Conceptual and Procedural Fraction Knowledge

Fractions are a cornerstone skill for student success in higher-level mathematics and many practical skills. However, most students do not have proficient fraction conceptual or computational knowledge. Effective and efficient interventions are needed to remediate these skills, especially for students with or at-risk for disabilities. We present a curricular overview of a self-regulated strategy development (SRSD) framework, SRSD Fractions. SRSD Fractions addresses adding and subtracting fractions with unlike denominators, simplifying fractions, and converting fractions to mixed numbers. We summarize research on SRSD Fractions and lessons learned, including implications for future research and practice, especially for students with comorbid behavior and academic challenges.

SRSD Fractions: Helping Students at Risk for Disabilities Add/Subtract Fractions With Unlike Denominators

Many students fall below benchmarks in the area of fractions computation, particularly students with disabilities. Self-regulated strategy development (SRSD) is one strategy with proven effectiveness for improving outcomes for students with disabilities, although very few studies have applied SRSD to the area of mathematics. In this study, we used SRSD Fractions to teach adding and subtracting fractions with unlike denominators, simplifying fractions, and converting fractions to mixed numbers using the mnemonics FILMS, CUT, and EDIT. A researcher provided instruction in small groups to fifth-grade students at risk for disabilities. The results from use of a multiple-baseline-across-groups design suggest a functional relationship between SRSD Fractions and 8 fifth-grade students’ digits correct on timed fraction probes. Results from treatment fidelity, social validity, and academic engagement during the intervention are also reported. We also discuss limitations and directions for future researchers.

Learning Fraction Concepts Through the Virtual-Abstract Instructional Sequence

Fractions are an important mathematical concept for both more advanced mathematical concepts and daily living activities (e.g., cooking) and thus are important for all students to learn. In this study, researchers explored the use of the virtual-abstract (VA) instructional sequence to teach three middle school students with disabilities three fraction concepts—equivalent fractions, addition of fractions with unlike denominators, and multiplication of fractions by a whole number. For each student for each fraction concept, a functional relation existed between the intervention (i.e., the VA instructional sequence) and the dependent variable (i.e., accuracy on problems of the fraction concept). The students acquired the fraction concepts, but there were struggles to maintain their performance when no instruction was provided.

An Investigation of SRSD to Teach Fractions to Students With Disabilities

In the current study we examined the effect of a self-regulated strategy development intervention on the fraction calculations of students with or at risk for disabilities using a multiple-baseline-across-sites, single-case design. Specifically, the intervention package addressed the following skills: adding and subtracting fractions with unlike denominators, simplifying fractions, and converting improper fractions to mixed numbers. The intervention was implemented with high levels of treatment fidelity and social validity across three separate intervention agents. Results of the study showed the intervention to be effective, with 15 of the 16 participants making marked gains on fraction probes. We discuss the results of the study with respect to the research questions, provide limitations to the study, and propose areas for future research.

Adding It Up

Manipulatives are a common tool in mathematics teaching and learning, including for students with disabilities. The most common manipulatives are concrete manipulatives, yet app-based manipulatives are a viable age-appropriate option for secondary students with disabilities. Through an adapted alternating treatment design with three middle school students—two with mild intellectual disability and one with a learning disability, researchers explored the impact of virtual and concrete manipulatives on students’ accuracy, independence, and task completion time for solving addition of fractions with unlike denominators. Students were equally successful in terms of accuracy and differences with independence were minimal. When comparing the two manipulative types, the results were idiosyncratic; two students were more independent with the concrete manipulative and one with the app-based manipulative. Implications for research regarding mathematics instruction and use of concrete and app-based manipulatives are discussed.

Using the virtual-abstract instructional sequence to teach addition of fractions

Background/aims/methodsLimited literature examines mathematics education for students with mild intellectual disability. This study investigated the effects of using the Virtual-Abstract instructional sequenceto teach middle school students, predominantly with mild intellectual disability, to add fractions of unlike denominators. Researchers used a multiple probe across participants design to determine if a functional relation existed between the Virtual-Abstract instructional sequence strategy and students’ ability to add fractions with unlike denominators. Procedures/outcomesThe study of consisted of three-to-nine baseline sessions, 6–11 intervention sessions, and two maintenance sessions for each student. Data were collected on accuracy across five addition of fractions with unlike denominators problems. Results/conclusionsThe VA instructional strategy was effective in thestudents to add fractions with unlike denominators; a functional relation existed between the VA instructional sequence and adding fractions with unlike denominators for three of the four students. ImplicationsThe Virtual-Abstract instructional sequencemay be appropriate to support students with mild intellectual disability in learning mathematics, especially when drawing or representing the mathematical concepts may prove challenging.

Error Patterns with Fraction Calculations at Fourth Grade as a Function of Students' Mathematics Achievement Status.

The goal of the present study was to describe fraction-calculation errors among 4th-grade students and determine whether error patterns differed as a function of problem type (addition vs. subtraction; like vs. unlike denominators), orientation (horizontal vs. vertical), or mathematics-achievement status (low- vs. average- vs. high-achieving). We specifically addressed whether mathematics-achievement status was related to students' tendency to operate with whole number bias. We extended this focus by comparing low-performing students' errors in two instructional settings that focused on two different types of fraction understandings: core instruction that focused on part-whole understanding vs. small-group tutoring that focused on magnitude understanding. Results showed students across the sample were more likely to operate with whole number bias on problems with unlike denominators. Students with low or average achievement (who only participated in core instruction) were more likely to operate with whole number bias than students with low achievement who participated in small-group tutoring. We suggest instruction should emphasize magnitude understanding to sufficiently increase fraction understanding for all students in the upper elementary grades.

The codevelopment of children’s fraction arithmetic skill and fraction magnitude understanding.

The importance of fraction knowledge to later mathematics achievement, along with U.S. students’ poor knowledge of fraction concepts and procedures, has prompted research on the development of fraction learning. In the present study, participants’ (N = 536) development of fraction magnitude understanding and fraction arithmetic skills was assessed over 4 time points between 4th and 6th grades. Latent state-trait modeling was used to examine codevelopment of these 2 areas of fraction knowledge. Fraction arithmetic skill predicted later fraction magnitude understanding, and conversely, fraction magnitude understanding predicted later fraction arithmetic skill. The results are consistent with a bidirectional model of the development of fraction concepts and procedures, in which knowledge of one type facilitates learning of the other type. However, transfer in both directions between fraction arithmetic skill and fraction magnitude understanding was more likely to occur later in the development of fraction knowledge, after fraction arithmetic with unlike denominators had been taught in school (during 5th grade in the current sample). Furthermore, the effects of previous knowledge of the other type were small and not nearly as substantial as the effects of previous knowledge on later knowledge of the same type. Findings suggest a need for instruction to link fraction magnitude understanding to fraction arithmetic skill and vice versa.

The Power of Problem Choice

Fourth- and fifth-grade learners can use differentiated number sets within CGI problem structures to add and subtract fractions with unlike denominators.

Discourse + Technology/Collaborative Learning = Fraction Success

Recent national public policy addresses the need for strengthening mathematics communication and reasoning skills for elementary students. This study analyzed the effects of using podcasts to increase mathematical discourse on the fraction abilities of a sixth grade mathematics class. Students collaborated within small heterogeneous groups that completed a flow map to show steps in the process representing the sequence of a fraction problem‟s solution. Students grappled with their understanding of mathematics by turning the flow map into a radio script which was ultimately posted as a podcast on the classroom‟s website. Scores from district and school based assessments measured the growth of mathematical ability to add, subtract, and regroup mixed numbers with unlike denominators over a nine week period during the first half of the sixth grade. A paired sample t-test of the sixth grade pretest and posttest scores on a criterion-referenced assessment, revealed a statically significant difference. The use of precise mathematical vocabulary, displays of proper sequence to solve for the correct answer, and accuracy of answers increased. District created measures of progress remained consistent as the level of difficulty in the curriculum increased.

Toward a Good or Better Understanding of Best Practices

Recent national public policy addresses the need for strengthening mathematics communication and reasoning skills for elementary students. This study analyzed the effects of using podcasts to increase mathematical discourse on the fraction abilities of a sixth grade mathematics class. Students collaborated within small heterogeneous groups that completed a flow map to show steps in the process representing the sequence of a fraction problem‟s solution. Students grappled with their understanding of mathematics by turning the flow map into a radio script which was ultimately posted as a podcast on the classroom‟s website. Scores from district and school based assessments measured the growth of mathematical ability to add, subtract, and regroup mixed numbers with unlike denominators over a nine week period during the first half of the sixth grade. A paired sample t-test of the sixth grade pretest and posttest scores on a criterion-referenced assessment, revealed a statically significant difference. The use of precise mathematical vocabulary, displays of proper sequence to solve for the correct answer, and accuracy of answers increased. District created measures of progress remained consistent as the level of difficulty in the curriculum increased.

Extending a rich web-based cognitive tool to a mobile computer-supported collaborative learning environment

A web-based cognitive tool was previously built for learners to help them comprehend the procedural knowledge of fraction addition and subtraction with unlike denominators. A series of experiments was conducted to validate the effectiveness of the web-based cognitive tool in assisting learners to understand the concept of fractions. In this study, we attempt to extend this rich web-based cognitive tool to a collaborative environment that allows learners to interact with each other and co-construct a procedural knowledge of fraction addition with unlike denominators by using the cognitive tool on a mobile platform. We introduce the web-based cognitive tool, describe its features on the mobile platform, illustrate how it can support collaborative interaction in a designed learning activity, and describe a scenario of using the mobile computer-supported cognitive tool to attain new procedural knowledge of fraction addition.

A cognitive tool for teaching the addition/subtraction of common fractions: a model of affordances

The aim of this research is to devise a cognitive tool for meeting the diverse needs of learners for comprehending new procedural knowledge. A model of affordances on teaching fraction equivalence for developing procedural knowledge for adding/subtracting fractions with unlike denominators was derived from the results of a case study of an initial prototype of a graphical partitioning model. Offering affordances in our model makes available profitable spaces for learners to interact in ways that meet their needs. This model of affordances was evaluated by a pre-test–post-test control group design to study the performance of the experimental group in learning with the model. Results of the study indicated that the model afforded learners, with various abilities for learning, knowledge of fraction equivalence. The key for mediating the generation of procedural knowledge for adding/subtracting fractions with unlike denominators in working with our cognitive tool was the concept of fraction equivalence and the capability of computing this.

Modeling a cognitive tool for teaching the addition/subtraction of common fractions

The case study reported here models a Cognitive Tool (CT) for primary school children to learn common fraction addition/subtraction with unlike denominators. In the study, we developed a CT based on a cognitive task analysis of the domain. We then observed how 12 learners used this CT to understand fraction operations based on a knowledge of fraction equivalence. Results of the study indicate that the support offered by the CT aids learners with higher ability in mathematics to produce cognitive residues. The graphical partitioning model helps to link the concrete operations of partitioning with the abstract idea of a common denominator required for adding/subtracting fractions with unlike denominators. However, learners who have failed to develop the knowledge of fraction equivalence, which includes the concept of equivalence and the ways of finding it, cannot gain much from working with the CT. A model of affordances for improving the design of the CT to meet the diverse needs of learners is discussed.