Fraction Arithmetic Research Articles

Individual differences in fraction arithmetic learning

Understanding fractions is critical to mathematical development, yet many children struggle with fractions even after years of instruction. Fraction arithmetic is particularly challenging. The present study employed a computational model of fraction arithmetic learning, FARRA (Fraction Arithmetic Reflects Rules and Associations; Braithwaite, Pyke, and Siegler, 2017), to investigate individual differences in children’s fraction arithmetic. FARRA predicted four qualitatively distinct patterns of performance, as well as differences in math achievement among the four patterns. These predictions were confirmed in analyses of two datasets using two methods to classify children’s performance—a theory-based method and a data-driven method, Latent Profile Analysis. The findings highlight three dimensions of individual differences that may affect learning in fraction arithmetic, and perhaps other domains as well: effective learning after committing errors, behavioral consistency versus variability, and presence or absence of initial bias. Methodological and educational implications of the findings are discussed.

Surveying Middle-Grades Teachers' Reasoning About Fraction Arithmetic in Terms of Measured Quantities

We report a novel survey that narrows the gap between information about teachers' knowledge of fraction arithmetic provided, on the one hand, by measures practical to administer at scale and, on the other, by close analysis of moment-to-moment cognition. In particular, the survey measured components that would support reasoning directly with measured quantities, not by executing computational algorithms, to solve problems. These components—each of which was grounded in past research—were attention to referent units, partitioning and iterating, appropriateness, and reversibility. A second part of the survey asked about teachers' professional preparation and history. We administered the survey to a national sample of in-service middle-grades mathematics teachers in the United States and received responses from 990 of those teachers. We analyzed responses to items in the first part of the survey using the log-linear diagnostic classification model to estimate each teacher's profile of strengths and weaknesses with respect to the four components of reasoning. We report on the diversity of profiles that we found and on relationships between those profiles and various aspects of teachers' professional preparation and history. Our results provide insight into teachers' knowledge resources for enacting standards-based instruction in fraction arithmetic and an example of new possibilities for mathematics education research afforded by recent advances in psychometric modeling.

The problems of teaching fractional arithmetic operations for disabled student using Realistic Mathematics Education

This study aims to describe the problems of teaching a blind student using Realistic Mathematics Education (RME) in the arithmetic’s operations ability of many fractions. The method of this study was a descriptive analysis of implementation RME for a blind student. The subject of the implementation of the RME was a 15-year-old student in grade VI at the Muhammadiyah Purworejo Special School (SLB) who were randomly selected. The imple-mentation of RME for the blind students has characteristics that are in accordance with the principles of teaching blind students. However, sometimes there are many factors that make RME could not optimize the ability of blind students. These factors are the student itself and external factors. As a teacher, knowing these factors is one way to plan the next learning. Therefore, this study focuses on explaining the problems when the RME has been implemented in the learning of arithmetic operations of many fractions for a blind student.

A Fraction Sense Intervention for Sixth Graders With or At Risk for Mathematics Difficulties

The efficacy of a research-based fraction sense intervention for sixth graders with or at risk for mathematics difficulties ( N = 52) was examined. The intervention aimed to build understanding of fraction magnitudes on the number line. Key concepts were taught with a narrow range of denominators to develop deep understanding. The intervention was centered on a visual number line in the meaningful context of a color run race. Students were randomly assigned to the fraction sense intervention ( n = 25) or a business-as-usual control group ( n = 27). Students in the intervention condition received 21 lessons in small groups (45 min each) during their regular mathematics intervention period. Students in the intervention group performed significantly better than those in the control group on a measure of fraction number line estimation and a more general measure of fraction concepts, both at immediate posttest and delayed posttest, with large effect sizes; lesser effects were shown for fraction arithmetic.

Children learn spurious associations in their math textbooks: Examples from fraction arithmetic.

Fraction arithmetic is among the most important and difficult topics children encounter in elementary and middle school mathematics. Braithwaite, Pyke, and Siegler (2017) hypothesized that difficulties learning fraction arithmetic often reflect reliance on associative knowledge-rather than understanding of mathematical concepts and procedures-to guide choices of solution strategies. They further proposed that this associative knowledge reflects distributional characteristics of the fraction arithmetic problems children encounter. To test these hypotheses, we examined textbooks and middle school children in the United States (Experiments 1 and 2) and China (Experiment 3). We asked the children to predict which arithmetic operation would accompany a specified pair of operands, to generate operands to accompany a specified arithmetic operation, and to match operands and operations. In both countries, children's responses indicated that they associated operand pairs having equal denominators with addition and subtraction, and operand pairs having a whole number and a fraction with multiplication and division. The children's associations paralleled the textbook input in both countries, which was consistent with the hypothesis that children learned the associations from the practice problems. Differences in the effects of such associative knowledge on U.S. and Chinese children's fraction arithmetic performance are discussed, as are implications of these differences for educational practice. (PsycINFO Database Record (c) 2018 APA, all rights reserved).

Teacher knowledge experiment: Testing mechanisms underlying the formation of preservice elementary school teachers’ pedagogical content knowledge concerning fractions and fractional arithmetic.

Pedagogical content knowledge forms the core of teachers’ professional knowledge; it refers to knowledge about making subject matter accessible to students. Thus, the formation of pedagogical content knowledge constitutes a crucial issue for educational research and practice. We investigated the contributions of content knowledge and pedagogical knowledge to the formation of pedagogical content knowledge about fractions and fractional arithmetic in 6th grade mathematics in a between-participants study with 100 German preservice teachers. The three experimental and two control groups received 7 hr of intervention spread out over two days. We assessed participants’ pedagogical content knowledge before intervention, between the two days, after intervention, as well as at 6-week follow-up. The control groups exclusively received instruction on either pedagogical knowledge or pedagogical content knowledge; each of the experimental groups embodied a specific hypothesis about the formation of pedagogical content knowledge. Providing support for a mechanism of amalgamation, a combination of instruction on content knowledge and pedagogical knowledge produced small but statistically significant growth in pedagogical content knowledge. Similarly, instruction on content knowledge exclusively was sufficient to cause small but statistically significant growth in pedagogical content knowledge. Prior instruction on content knowledge did not facilitate learning from instruction on pedagogical content knowledge. Nevertheless, direct instruction on pedagogical content knowledge caused medium and statistically significant growth in pedagogical content knowledge. (PsycINFO Database Record (c) 2018 APA, all rights reserved)

Five-Level Inverter-Fed Space Vector Based Direct Torque Control of Open-End Winding Induction Motor Drive

This work proposes a space vector based direct torque control scheme implemented on an open-end winding type induction motor fed by a five-level inverter. The modulation algorithm uses a 60-degree coordinate system to avoid fractional arithmetic as present in the standard Cartesian coordinate system, which reduces the arithmetic calculation complexity. Making use of the natural fractal structure of a hexagonal space vector diagram of the multilevel inverter, this scheme uses a fractal-based algorithm for sector identification and switching vector generation. In this scheme, three-phase signals to the space vector modulator are generated based on a torque-flux error by means of transformation logic. Two three-level inverters are fed from both open ends of an induction motor for realizing a five-level inverter. The scheme includes implementation of different levels of inverter-fed direct torque control drive. The control scheme is implemented on a laboratory prototype 2hp induction motor through a TMS320F240 DSP processor based dSPACE DS-1104 controller. The inverter gating signals are generated by using a Virtex-II FPGA board. The hardware results of different direct torque control configurations are presented for validating the algorithm and all the measurements are done under closed-loop operation.

Supplemental Material for Children Learn Spurious Associations in Their Math Textbooks: Examples From Fraction Arithmetic

Supplemental Material for Children Learn Spurious Associations in Their Math Textbooks: Examples From Fraction Arithmetic

A computational model of fraction arithmetic.

Many children fail to master fraction arithmetic even after years of instruction, a failure that hinders their learning of more advanced mathematics as well as their occupational success. To test hypotheses about why children have so many difficulties in this area, we created a computational model of fraction arithmetic learning and presented it with the problems from a widely used textbook series. The simulation generated many phenomena of children's fraction arithmetic performance through a small number of common learning mechanisms operating on a biased input set. The biases were not unique to this textbook series-they were present in 2 other textbook series as well-nor were the phenomena unique to a particular sample of children-they were present in another sample as well. Among other phenomena, the model predicted the high difficulty of fraction division, variable strategy use by individual children and on individual problems, relative frequencies of different types of strategy errors on different types of problems, and variable effects of denominator equality on the four arithmetic operations. The model also generated nonintuitive predictions regarding the relative difficulties of several types of problems and the potential effectiveness of a novel instructional approach. Perhaps the most general lesson of the findings is that the statistical distribution of problems that learners encounter can influence mathematics learning in powerful and nonintuitive ways. (PsycINFO Database Record

Do children understand fraction addition?

Many children fail to master fraction arithmetic even after years of instruction. A recent theory of fraction arithmetic (Braithwaite, Pyke, & Siegler, 2017) hypothesized that this poor learning of fraction arithmetic procedures reflects poor conceptual understanding of them. To test this hypothesis, we performed three experiments examining fourth to eighth graders' estimates of fraction sums. We found that roughly half of estimates of sums were smaller than the same child's estimate of one of the two addends in the problem. Moreover, children's estimates of fraction sums were no more accurate than if they had estimated each sum as the average of the smallest and largest possible response. This weak performance could not be attributed to poor mastery of arithmetic procedures, poor knowledge of individual fraction magnitudes, or general inability to estimate sums. These results suggest that a major source of difficulty in this domain is that many children's learning of fraction arithmetic procedures develops unconstrained by conceptual understanding of the procedures. Implications for education are discussed.

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.

On the periodicity of continued fractions in hyperelliptic fields

Let L be a function field of a hyperelliptic curve defined over an arbitrary field characteristic different from 2. We construct an arithmetic of continued fractions of an arbitrary quadratic irrationality in field of formal power series with respect to linear finite valuation. The set of infinite valuation and finite linear valuation of L is denoted by S. As an application, we have found a relationship between the issue of the existence of nontrivial S-units in L and periodicity of continued fractions of some key elements of L.

Supplemental Material for A Computational Model of Fraction Arithmetic

Supplemental Material for A Computational Model of Fraction Arithmetic

Rational-number comparison across notation: Fractions, decimals, and whole numbers.

Although fractions, decimals, and whole numbers can be used to represent the same rational-number values, it is unclear whether adults conceive of these rational-number magnitudes as lying along the same ordered mental continuum. In the current study, we investigated whether adults' processing of rational-number magnitudes in fraction, decimal, and whole-number notation show systematic ratio-dependent responding characteristic of an integrated mental continuum. Both reaction time (RT) and eye-tracking data from a number-magnitude comparison task revealed ratio-dependent performance when adults compared the relative magnitudes of rational numbers, both within the same notation (e.g., fractions vs. fractions) and across different notations (e.g., fractions vs. decimals), pointing to an integrated mental continuum for rational numbers across notation types. In addition, eye-tracking analyses provided evidence of an implicit whole-number bias when we compared values in fraction notation, and individual differences in this whole-number bias were related to the individual's performance on a fraction arithmetic task. Implications of our results for both cognitive development research and math education are discussed. (PsycINFO Database Record

Representing Numbers as Continued Fractions and an N-spire .tns Document To Do Some Basic Continued Fraction Arithmetic

Representing Numbers as Continued Fractions and an N-spire .tns Document To Do Some Basic Continued Fraction Arithmetic

Knowledge and motivation as mediators in mathematics teaching practice: the case of drawn models for fraction arithmetic

Past studies have suggested that in light of recent curriculum standards, many US teachers make limited use of drawn models in their mathematics instruction. To gain insight into this phenomenon, we investigated relationships between US teachers’ opportunities to learn about, knowledge of, motivation for, and instructional use of drawn models for representing multiplication and division of fractions. A national sample of 990 practicing middle-grade teachers was administered a three-part survey that contained a knowledge assessment; a professional history and teaching practices questionnaire that included questions about opportunities to learn to use drawn models; and a motivation questionnaire that measured teachers’ value, anxiety, and self-concept of ability for using such models in instruction. In regression models without motivation, opportunity to learn significantly predicted the teachers’ knowledge, frequency of use, and purpose for use of drawn models. In structural equation models that included motivation, knowledge and motivation substantially accounted for relationships between the teachers’ opportunity to learn and their self-reported use of drawn models in instruction. These findings are consistent with the general hypothesis that teacher’ opportunities to learn teaching practices indirectly affect their instructional practices. Teachers’ knowledge and motivation also play a central role.

Conceptual knowledge of fraction arithmetic.

Understanding an arithmetic operation implies, at minimum, knowing the direction of effects that the operation produces. However, many children and adults, even those who execute arithmetic procedures correctly, may lack this knowledge on some operations and types of numbers. To test this hypothesis, we presented preservice teachers (Study 1), middle school students (Study 2), and math and science majors at a selective university (Study 3) with a novel direction of effects task with fractions. On this task, participants were asked to predict without calculating whether the answer to an inequality would be larger or smaller than the larger fraction in the problem (e.g., “True or false: 31/56 17/42 31/56”). Both preservice teachers and middle school students correctly answered less often than chance on problems involving multiplication and division of fractions below 1, though they were consistently correct on all other types of problems. In contrast, the math and science students from the selective university were consistently correct on all items. Interestingly, the weak understanding of multiplication and division of fractions below 1 was present even among middle school students and preservice teachers who correctly executed the fraction arithmetic procedures and had highly accurate knowledge of the magnitudes of individual fractions, which ruled out several otherwise plausible interpretations of the findings. Theoretical and educational implications of the findings are discussed.

What to Do About Canada's Declining Math Scores?

The declining performance of Canadian students on international math assessments should worry Canadians and their provincial governments. Strong mathematics knowledge is required for success in the workforce, and early achievement in math is one of the best predictors of later academic success and future career options. Between 2003 and 2012, all but two Canadian provinces showed statistically significant declines in math scores on international exams administered by the Organization for Economic Cooperation and Development. In several provinces, the percentage of students performing at the lowest levels in math significantly increased while the percentage of students performing at the highest levels significantly decreased, suggesting that more students are struggling and fewer students are excelling in math. It should be a policy priority to halt these trends and to improve math achievement for Canadian children. In this Commentary, I examine domestic and international evidence regarding three areas of provincial education programs that could play an important role in halting the downward trend in math scores. I make three main recommendations regarding best teaching practices in math, the math curriculum, and the math knowledge of future teachers. Best teaching practices in math have been at the forefront of discussions regarding declining math scores in Canada. Discovery-based instruction – also called problem-based, inquiry, experiential, and constructivist learning – has become popular in North America in recent years, pushing aside direct instruction techniques, like times table memorization, explicit teacher instruction, pencil-and-paper practice, and mastery of standard mathematical procedures. Based on international and domestic evidence, this Commentary finds that studies consistently show direct instruction is much more effective than discovery-based instruction, which leads to straightforward recommendations on how to tilt the balance toward best instructional techniques. Student fluency with particular math concepts, such as fraction arithmetic, in early and middle years has been shown to predict future math success. This Commentary recommends that provincial math curricula be rewritten to remove ineffective pedagogical directives and to stress specific topics, at appropriate grade levels, that are known to lead to later success in math. Evidence shows that teachers who are most comfortable and knowledgeable with the content they are required to teach tend to transmit that knowledge best to students. This Commentary suggests that future early and middle-years teachers be required to pass a math-content licensure exam prior to receiving certification to teach mathematics.

High Efficiency Video Coding

In recent years,with the development of video coding technology and the increasing popularity of high definition(HD)video content,the ISO/IEC Moving Picture Experts Group(MPEG)and the ITU-T Video Coding Experts Group(VCEG)formed the joint collaborative team on video coding(JCV-VC)in 2010which aims to develop the next-generation video coding standard for HD video application,called High Efficiency Video Coding(HEVC).So far the test model HM has been used for performance evaluation and algorithm test.The final international standard of HEVC has been completed and published in the end of 2012based on the working plan of JCV-VC.In this paper,the key technologies about HEVC will be surveyed,specifically including the quad-tree structure of coding unit,prediction unit and transform unit,the advanced motion vector prediction and merging technology,angular intra prediction,DCT-based fractional pixel interpolation filter and context adaptive arithmetic coding etc.Finally the coding performance of some coding tools and complexity of HEVC are analyzed in detail.

Early predictors of middle school fraction knowledge.

Recent findings that earlier fraction knowledge predicts later mathematics achievement raise the question of what predicts later fraction knowledge. Analyses of longitudinal data indicated that whole number magnitude knowledge in first grade predicted knowledge of fraction magnitudes in middle school, controlling for whole number arithmetic proficiency, domain general cognitive abilities, parental income and education, race, and gender. Similarly, knowledge of whole number arithmetic in first grade predicted knowledge of fraction arithmetic in middle school, controlling for whole number magnitude knowledge in first grade and the other control variables. In contrast, neither type of early whole number knowledge uniquely predicted middle school reading achievement. We discuss the implications of these findings for theories of numerical development and for improving mathematics learning.