Where do Fractions Encounter their Equivalents? – Can this Encounter Take Place in Elementary-School?

The concept of equivalence class plays a significant role in the structure of Rational Numbers. Piaget taught that in order to help elementary school children develop mathematical concepts, concrete objects and concrete reflection-enhancing-activities are needed. The “Shemesh” software was specially designed for learning equivalence-classes of fractions. The software offers concrete representations of such classes, as well as activities which cannot be constructed without a computer. In a discrete Cartesian system students construct points on the grid and learn to identify each such point as a fraction-numeral (a denominator-numerator pair). The children then learn to construct sets of such points, all of which are located on a line through the origin point. They learn to identify the line with the set of its constituent equivalent fractions. Subsequently, they investigate other phenomena and constructions in such systems, developing these constructions into additional fraction concepts. These concrete constructions can be used in solving traditional fraction problems as well as in broadening the scope of fraction meaning. Fifth-graders who used “Shemesh” in their learning activities were clinically interviewed several months after the learning sessions ended. These interviews revealed evidence indicating initial actual development of the desired mathematical concepts. This revised version was published online in July 2006 with corrections to the Cover Date.     

Teacher and students’ joint production of a reversible fraction conception

Within a constructivist perspective, I conducted a teaching experiment with two fourth graders to study how a teacher and students can jointly produce the reversible fraction conception. Ongoing and retrospective analysis of the data revealed the non-trivial process by which students can abstract multiplicative reasoning about fractions. The study articulates a conception in a developmental sequence of iteration-based fraction conceptions and the teacher’s role in fostering such a conception in students.     

Students’ whole number multiplicative concepts: A critical constructive resource for fraction composition schemes

This article reports on the activity of two pairs of sixth grade students who participated in an 8-month teaching experiment that investigated the students’ construction of fraction composition schemes. A fraction composition scheme consists of the operations and concepts used to determine, for example, the size of 1/3 of 1/5 of a whole in relation to the whole. Students’ whole number multiplicative concepts were found to be critical constructive resources for students’ fraction composition schemes. Specifically, the interiorization of two levels of units, a particular multiplicative concept, was found to be necessary for the construction of a unit fraction composition scheme, while the interiorization of three levels of units was necessary for the construction of a general fraction composition scheme. These findings contribute to previous research on students’ construction of fraction multiplication that has emphasized partitioning and conceptualizing quantitative units. Implications of the findings for teaching are considered.     

A Comparison of Textbooks' Presentation of Fractions

In the United States, fractions are an important part of the middle school curriculum, yet many middle school students struggle with fraction concepts. Teachers also have difficulty with the conceptual understanding needed to teach fractions and rely on textbooks when making instructional decisions. This reliance on textbooks, the idea that teaching and learning of fractions is a complex process, and that fraction understanding is the foundation for later topics such as proportionality, algebra, and probability, makes it important to examine the variation in presentation of fraction concepts in U.S. textbooks, especially the difference between traditional and standards‐based curricula. The purpose of this study is to determine if differences exist in the presentation of fractions in conventional and standards‐based textbooks and how these differences align with the recommendations of National Council of Teachers of Mathematics, Common Core State Standards, and the research on the teaching and learning of fractions.     

Fifth graders’ understanding of fractions on the number line

The goal of this research was to examine fifth graders’ understanding of fractions on the number line. This case‐study design focused on the various ways that students represented fractions on number lines. Students responded to task‐based interview questions by identifying fractions as a number on the number line as well as equivalency and problem solving. The tasks were administered individually to 26 fifth‐grade students over a 15‐minute time frame in their respective schools. The two groups of 10‐year‐old students answered most questions in written form with pencil and paper and were often asked to explain how they arrived at an answer. Student performance was highest when instructed to plot ½ on a number line of 0 to 1 as well as naming a fraction less than ½. The students performed lowest when they attempted to plot ½, ¼, and 1 on a number line with a predetermined unit 0 to 1/3. Other low performing concepts consisted of plotting ¼ on a number line from 0‐3, identifying ¼ on a non‐routine number line, and plotting a unit fraction with an equivalent fraction as well as an improper fraction on a common number line.     

Analyzing connections between teacher and student topic-specific knowledge of lower secondary mathematics

The interpretive cross-case study focused on the examination of connections between teacher and student topic-specific knowledge of lower secondary mathematics. Two teachers were selected for the study using non-probability purposive sampling technique. Teachers completed the Teacher Content Knowledge Survey before teaching a topic on the division of fractions. The survey consisted of multiple-choice items measuring teachers’ knowledge of facts and procedures, knowledge of concepts and connections, and knowledge of models and generalizations. Teachers were also interviewed on the topic of fraction division using questions addressing their content and pedagogical content knowledge. After teaching the topic on the division of fractions, two groups of 6th-grade students of the participating teachers were tested using similar items measuring students’ topic-specific knowledge at the level of procedures, concepts, and generalizations. The cross-case examination using meaning coding and linguistic analysis revealed topic-specific connections between teacher and student knowledge of fraction division. Results of the study suggest that students’ knowledge could be associated with the teacher knowledge in the context of topic-specific teaching and learning of mathematics at the lower secondary school.     

An empirically-based trajectory for fostering abstraction of equivalent-fraction concepts: A study of the Learning Through Activity research program

Promoting deep understanding of equivalent-fractions has proved problematic. Using a one-on-one teaching experiment, we investigated the development of an increasingly sophisticated, sequentially organized set of abstractions for equivalent fractions. The article describes the initial hypothetical learning trajectory (HLT) which built on the concept of recursive partitioning (anticipation of the results of taking a unit fraction of a unit fraction), analysis of the empirical study, conclusions, and the resulting revised HLT (based on the conclusions). Whereas recursive partitioning proved to provide a strong conceptual foundation, the analysis revealed a need for more effective ways of promoting reversibility of concepts. The revised HLT reflects an approach to promoting reversibility derived from the empirical and theoretical work of the researchers.     

Students as toolmakers: refining the results in the accuracy and precision of a trigonometric activity

Smartphones used as tools provide opportunities for the teaching of the concepts of accuracy and precision and the mathematical concept of arctan. The accuracy and precision of a trigonometric experiment using entirely mechanical tools is compared to one using electronic tools, such as a smartphone clinometer application and a laser pointer. This research has demonstrated how two classroom activities based on tool-making can enhance student measurement and application of accuracy and precision considerations through a trigonometric activity investigating arctan.     

Children’s sense-making of division of fractions

The purpose of this paper is to share some results from a year-long teaching experiment in which fourth grade students were given the opportunity to understand fraction concepts prior to the introduction of algorithmic instruction. In particular, this paper focuses on the means by which children solved problems involving division of fractions. Children were given a task-based activity specifically designed to promote solutions that would be grounded in conceptual understanding. Three distinct solution methods, all related to counting, emerged. When the activity was replicated as part of regular classroom practice seven and a half years later, the same solution methods were observed.     

Effect of self-constructed concept map with peer feedback on chemistry learning outcome

Concept mapping remains widely used in education. However, little is known about how self-constructed concept maps and peer feedback can improve student learning outcomes in chemistry. We investigated the effects of peer feedback on concept mapping and how it improves students' learning performance in a large second-semester, introductory chemistry course. Three hundred and twenty students were randomly assigned to one of two concept mapping conditions: self-constructed concept map with peer feedback and self-constructed concept map without peer feedback. Each group constructed concept maps that depicted the relationship between concepts on the topic of intermolecular force. The results showed that students in the self-constructed concept map with peer feedback condition outperformed students in the no peer feedback condition in chemistry learning outcome. Overall, this study demonstrates that peer feedback enhances the effectiveness of learning with generative concept maps. The implications and future directions are discussed.     

Towards a framework for developing students' fraction proficiency

The importance of the knowledge of fractions in mathematical learning, coupled with the difficulties students have with them, has prompted researchers to focus on this particular area of mathematics. The term ‘fraction proficiency' used in this article refers to a person's conceptual comprehension, procedural skills and the ability to approach daily situations involving fractions. In the area of fractions, there has been a call for more research to study how, and where, efforts should be focused in order to integrate the various aspects of fraction knowledge for students, and even for teachers, to help them develop proficiency in fractions. Thus, the article presents a theoretical synthesis of the specialized literature in the learning and teaching of fractions, with the aim of proposing a framework for developing students' fraction proficiency. The frameworks presented in the article may shed light upon the implications for the design of fraction instruction, which should focus on developing a multi-faceted knowledge of fractions, rather than simply isolating one facet from the others.     

The fractional knowledge and algebraic reasoning of students with the first multiplicative concept

To understand relationships between students’ quantitative reasoning with fractions and their algebraic reasoning, a clinical interview study was conducted with 18 middle and high school students. Six students with each of three different multiplicative concepts participated. This paper reports on the fractional knowledge and algebraic reasoning of six students with the most basic multiplicative concept. The fractional knowledge of these students was found to be consistent with prior research, in that the students had constructed partitioning and iteration operations but not disembedding operations, and that the students conceived of fractions as parts within wholes. The students’ iterating operations facilitated their work on algebra problems, but the lack of disembedding operations was a significant constraint in writing algebraic equations and expressions, as well as in generalizing relationships. Implications for teaching these students are discussed.     

Making sense of instruction on fractions when a student lacks necessary fractional schemes: The case of Tim

This paper critically examines the discrepancies among the pre-requisite fractional concepts assumed by a curricular unit on operations with fractions, the teacher's assumptions about those concepts and a particular student's understanding of fractions. The paper focuses on the case of one student (Tim) in the teacher's 6th grade class who was interviewed by one of the authors once a week during the teaching of the unit. The teaching materials and the teacher's instruction were based on the assumption that students understood the concept of a unit fraction as being one of several equal parts of a given whole. The teacher neither emphasized the need for equal parts nor the part-to-whole relation. The teacher's reasonable assumptions about her students’ understanding of fractions were severely challenged by the cognitive constructs that Tim exhibited during his first two interviews. When she viewed tapes of the class instruction and the interviews with Tim she realized Tim lacked essential constructs to make sense of her instruction. She subsequently made adjustments in her instruction, making effective use of more appropriate representations based on tasks from the unit that we modified and used with Tim in our interviews. These adjustments helped Tim to construct partitioning operations and an appropriate unit fractional scheme. This study illustrates the importance of coming to understand a student's mathematical activity in terms of possible conceptual schemes and modifying instructional strategies to build on those schemes. The coordinated design of the research study facilitated these instructional modifications.     

Empirically-based hypothetical learning trajectories for fraction concepts: Products of the Learning Through Activity research program

The Measurement Approach to Rational Number (MARN) Project, a project of the ongoing Learning Through Activity (LTA) research program, produced eleven hypothetical learning trajectories (HLTs) for promoting fraction concepts. Four of these HLTs are the subject of research reports. In this article, we present the other seven HLTs We judged that the data and analyses of these seven would not separately make sufficient contributions to merit individual research reports. However, presenting these seven HLTs together was intended to meet the following goals:1. To give a broad set of examples of HLTs developed based on the LTA theoretical framework.2. To complete a set of HLTs that provide a comprehensive example of HLTs built on prior HLTs.3. To make available for future research and development the full set of HLTs generated by the MARN Project.LTA researchers have focused on how learners abstract a concept through their mathematical activity and how the abstractions can be promoted. The MARN Project continued this inquiry with rigorous single-subject teaching experiments.     

The impact of using multiple modalities on students’ acquisition of fractional knowledge: An international study in embodied mathematics across semiotic cultures

Principled by the Embodied, Situated, and Distributed Cognition paradigm, the study investigated the impact of using a research-based curriculum that employs multiple modalities on the performance of grade 5 students on 3 subscales: concept of unit, fraction equivalence, and fraction comparison. The sample included five schools randomly selected from a population of 14 schools in Lebanon. Eighteen 5th grade classrooms were randomly assigned to experimental (using multimodal curriculum) and control (using a monomodal curriculum) groups. Three data sources were used to collect quantitative and qualitative data: tests, interviews, and classroom observations. Quantitative data were analyzed using two methods: reliability and MANOVA. Results of the quantitative data show that students taught using the multimodal curriculum outperformed their counterparts who were instructed using a monomodal curriculum on the three aforementioned subscales (at an alpha level = .001). Additionally, fine-grained analysis using the semiotic bundle model revealed different semiotic systems across experimental and control groups. The study findings support the multimodal approach to teaching fractions as it facilitates students’ conceptual understanding.     

Impact of computer animations in cognitive learning: differentiation

In mathematic courses, construction of some concepts by the students in a meaningful way may be complicated. In such circumstances, to embody the concepts application of the required technologies may reinforce learning process. Onset of learning process over daily life events of the student's environment may lure their attention and may enable them to gain from the preliminary knowledge. Therefore, a good initiation may be realized in the course of meaningful learning. The underlying meaning of the abstract concepts by computer animations may be accomplished in class environments. That study is conducted searching out to discover the effects of animations over the learning process in mathematic courses. The study was performed over the 58 university freshman students selected randomly. Thirty-two students constituted the experiment group and 26 students constituted the control group. Computer animations-aided instruction model in constructive form were applied on the experiment group and non-computer-aided instruction model in constructive form were implemented on the control group. Student academic success via a test method developed by explored group with confidence rate .819 (Cronbach's alpha) revealed that data were evaluated by two-way variance analyses. The findings provided from the final test shows that the experiment group students were significantly higher according to the control group students in terms of academic success average scores. Computer animations were observed to be significant to assimilate the derivative concept in a discrete way over the students, to appeal their attention, animations of real life events observed to transform the abstract meanings in the events to a concrete manner. Students of whom the concrete stage is constructed meaningfully found to be tactful in reaching to semi-abstract and abstract stages.     

STUDENT TEACHERS’ UNDERSTANDING OF RATIONAL NUMBER: PART-WHOLE AND NUMERICAL CONSTRUCTS

Previous research has shown that secondary school students’ understanding of fractions is dominated by the part-whole concept to the possible detriment of their understanding of a fraction as a number in its own right. The present paper reports on an investigation into the understanding of intending primary teachers in this area. Four representatives of a cohort of sixty students on a PGCE course specialising in the lower primary age range were asked detailed questions probing their knowledge of fractions. The conclusion was that the part-whole concept dominates. All of the students were familiar with the numerical concept from their work on the PGCE course, but they reverted to the more familiar part-whole ideas in attempting to solve problems.     

Fostering construction of a meaning for multiplication that subsumes whole-number and fraction multiplication: A study of the Learning Through Activity research program

We report on a teaching experiment intended to foster a concept of multiplication that would both subsume students’ multiple-groups concept of whole number multiplication and provide a conceptual basis for understanding multiplication of fractions. The teaching experiment, which used a rigorous single-subject methodology, began with an attempt to build on students’ multiple-groups concept by promoting generalizing assimilation. This was not totally successful and led to a redesign aimed at promoting reflective abstraction. Analysis of this latter phase led to several significant conclusions, which in turn led to a revised hypothetical learning trajectory. The revised trajectory aims to foster a concept of multiplication as a change in units.     

The Effect of Area Measurement Tools on Student Strategies: The Role of a Computer Microworld

This study focuses on the role of tools, provided by a computer microworld (C.AR.ME), on the strategies developed by 14-year-old students for the area measurement of a non-convex polygon. Students' strategies on a transformation and a comparison task were interpreted and classified into categories in terms of the tools used for their development. The analysis of the data shows that an environment providing the students with the opportunity to select various tools and asking them to produce solutions `in any possible way' can stimulate them to construct a plurality of solution strategies. The students selected tools appropriate for their cognitive development and expressed their own individual approaches regarding the concept of area measurement. The nature of tools used affected the nature of solution strategies that the students constructed. Moreover, all students were involved in the tasks and succeeded in completing them with more than one correct solution strategy thereby developing a broader view of the concept, although not all of them realized the same strategies. Three different approaches to area measurement emerged from the strategies which were constructed by the students in this microworld: automatic area measurement, provided by the environment, the operation of area measurement using spatial units and the use of area formulae. Almost all the students experienced qualitative aspects of area measurement through being involved in the process of covering areas using spatial units. Students also managed to use the area formulae meaningfully by studying it in relation to automatic area measurement and to area measurement using spatial units. Through these strategies, the concepts of conservation of area and its measurement as well as area formulae were viewed by the students as interrelated. Finally, some basic difficulties regarding area measurement were overcome in this computer environment.This revised version was published online in September 2005 with corrections to the Cover Date.     

The effect of activity-based instruction on conceptual development of seventh grade students in probability

The purpose of this study is to investigate and compare the effects of activity-based and traditional instructions on students’ conceptual development of certain probability concepts. The study was conducted using a pretest–posttest control group design with 80 seventh graders. A developed ‘Conceptual Development Test’ comprising 12 open-ended questions was administered on both groups of students before and after the intervention. The data were analysed using analysis of covariance, with the pretest as covariate. The results revealed that activity-based instruction (ABI) outperformed the traditional counterpart in the development of probability concepts. Furthermore, ABI was found to contribute students’ conceptual development of the concept of ‘Probability of an Event’ the most, whereas to the concept of ‘Sample Space’ the least. As a consequence, it can be deduced that the designed instructional process was effective in the instruction of probability concepts.