Representing Fractions with Standard Notation: A Developmental Analysis
In order to deeply understand a given concept or phenomenon, one must carry out an intensive research. This study was aimed at investigating the developmental relationship between how to use fractions notations and their understandings of part-whole relations. It also aimed to produce an analysis of the role played by fractions teaching in students’ use of notation to represent parts of an area. This paper seeks to analyze the basic ideas in the content of the report, the fundamental ideas about learning as brought out by the author as well as the clarity of the research questions used. It will also highlight the main findings in light of the research questions.
This particular report concentrates on finding out how the students develop in the use of standard notation to represent fractional parts of given areas from before instruction to after instruction (Saxe, Taylor, McIntosh, & Gearhart, 2005). The inquiry into the representation of fractional parts of areas is simplified by the introduction of two distinct parts-notation and reference.
Under notation which in general consists of marks or symbols, the study focuses on the notations for fractions which are known to vary across the various cultural groups and historical time periods (Saxe et al., 2005). Reference, on the other hand, means the conceptual work of using notational form to point to index physical objects or mathematical ideas. It is also brought forward that when individuals use Hindu-Arabic notations to refer to whole numbers, then they connect them to a system of values in which additive relations are of great importance, as manifested in the addition and subtraction of natural numbers. Also when individuals use the Hindu-Arabic notations to refer to fractions, they connect these numerals to a system in which multiplicative relations are of primary importance.
In this article, the author clearly states that learning fractions is a major issue for learners especially those in the middle elementary grades and above. He points out that one of the major causes may be the students’ difficulty in acquiring flexible use and comprehending of the written notations of fractions (Saxe et al., 2005). The author brings the concept of studying the developmental patterns in the students’ use of standard notation to represent fractional parts of areas and the role of instruction in their development of notation.
In order to achieve these, the article clearly states three research questions. The first one is to find out whether students treat parts of area as discreet or continuous quantities when they produce part-whole reference on Equal and Unequal Area problems. The second is to investigate the relation between conventional notation and part-whole reference. Thirdly, the research seeks to find out the effect of students beginning fractions instruction with similar notational approaches on their trajectories as compared to the students’ participation in classrooms that stress inquiry as contrasted with classrooms that stress skills (Saxe et al., 2005). To investigate these questions, the research followed up a number of students from Grades 4, 5, and 6 where the students were administered with Equal and Unequal Area problems prior to and at the conclusion of their fractions units and the appropriate data collected using reliable methods.
The findings of the research reveal that many students cannot differentiate between discreet and continuous quantities (Saxe et al., 2005). Some significant independence between notation and reference is also revealed. The student performance approached a ceiling on the Equal Area problems at the posttest where most of them used conventional notation to represent part-whole relations. Concerning the instructional conditions, the findings indicate that students’ pretest categories and classroom practice are predictive of their trajectories of change though they are not indicative of the effect of the change over the various instructional practices. Further analysis reveal that there is greater shift to higher inquiry instruction in that a greater proportion in the inquiry classrooms tend to shift to Uncertain in order to adequately represent the Unequal Area problems.
Instructional Practices and Prospective Teachers’ Motivation for Fractions
Any prospective worker in a given field of specialization ought to be adequately motivated if maximum output is to be achieved. Prospective elementary school teachers, therefore, need to be motivated especially in areas perceived to be having a higher level of difficulty. This paper seeks to analyze the study in general as far as motivating the prospective elementary school teachers is concerned.
The basic objectives of this investigative study is to find out the motivations for prospective elementary school teachers for working with fractions before and after taking a course aimed at enhancing their grasp of mathematics, fractions in particular. It also seeks to establish the instructional practices that might be associated with any changes detected in their motivations.
The fundamental learning ideas brought into this study by the author include the belief that elementary education majors tend to be among the most anxious college students with regard to mathematics (Newton, 2009). It is further thought that the students feel less confident about mathematics and view it as a set of arbitrary facts instead of interconnected concepts. Most students blame their inadequate knowledge in mathematics on the perceived arbitrariness of the subject. This, however, may not be the case since it depends mainly upon the individual attitudes towards any given subject and hence affecting delivery of content in a classroom setting. It is also of concern, the author observes, whether the negative attitude towards mathematics can be topic-specific. Most anxiety studies have used the Mathematics Anxiety Rating Scale (MARS), which includes some academic situations but is primarily interested in the practical aspects of mathematics outside of classroom. It is therefore crucial to ensure that both cognitive and motivational factors are taken into consideration if changes are to be made in the education sector.
This study developed three research questions which were aimed at building on other existing literature. First, it sought to investigate the level of prospective elementary teachers’ motivations for fractions at the start of a course designed to promote understanding in mathematics. Secondly, it studied the possible effect of participating in the course on the prospective elementary teachers’ motivations for fractions. Thirdly, whether there are discernible profiles for instructors teaching different sections of the course, and how the profiles contribute to prospective teachers’ fraction motivation at the end of the course (Newton, 2009). These questions were developed from the findings of prior studies into the same. Alsup’s study had been established that prospective teachers had lowered anxiety after taking a course in mathematics which focused on understanding. This, however, was not based on topic-specific anxiety. Furthermore, Alsup had mixed findings on the effect of instruction on anxiety hence the need for deeper research using newer methods to collect data both at pretest and posttest.
This study reveals that value, anxiety, and self-concept of ability are all average at the beginning of the semester (Newton, 2009). Another study on younger students revealed that prospective teachers’ anxiety levels concerning fractions are raised and their self-concept of ability and valuing of fractions are significantly depressed. It is established that motivation for fractions improved generally as a result of students participating in the course from pretest to posttest. It is, however, still very low compared to former studies with younger students. The study also shows that the motivation variables; self-concept, anxiety, and value are related to one another (Newton, 2009). Profiles are created and used to analyze students’ differences with the instructors. The results also indicate that different instructors influenced prospective teachers’ motivation even when they deliver the same content over the same period of time depending on their instructional approaches and personality.
An Integrated Study of Children’s Construction of Improper Fractions and the Teacher’s Role
There can be a very significant role that a teacher can play as far as learning some mathematical concepts is concerned. There is a relationship between teaching and the learner’s construction of a specific conception that supports the development of improper fractions. The ability of the children to pose and solve tasks in a computer micro-world promoted a modification in their fraction schemes. The children improved from thinking about a unit fraction as a part of a whole to thinking about it as standing in a multiplicative relationship with a reference whole. This work seeks to discuss the article in light of its content, fundamental ideas, research questions, and the main findings.
The article examines how teaching and children’s construction of fraction knowledge co-emerge (Tzur, 1999). It notes that a number of scholars have always stressed that studies of children’s learning should be done concurrently with their teachers’ involvement in the process as well as the effects on the children’s learning of other people interacting with children.
The author believes that unlike children’s construction of early number knowledge that in most instances occurs without a teacher, a great deal of children’s fraction learning takes place through instruction. He proposes that it is more important to study the problem if one is to offer any suggestions on understanding and improving the current perceptions of fractions. Many have been convinced that the teaching and learning of mathematics especially fractions is not only very complex, but also a depressing failure especially in the case of young learners.
This research carried out by a researcher-teacher and some other team members were responsible for the stating of research objectives. It was an integrative research where there was the teaching aspect and the children’s continual construction of a specific mathematical scheme. The experimental set up was that of a typical classroom. The author discusses the nature of the tasks and the interventions as well as how the learners did or did not manage possibilities for that reorganization. The research questions were to be addressed for a period of three consecutive years (Tzur, 1999). This was with an aim of understanding children’s conceptions of fractions, in particular, the children’s conceptions that can lead to the generation of improper fractions. The sequence of iteration-based tasks and how they helped in the reorganization of the children’s conceptions were also discussed. A scheme was developed for the purposes of explaining the three types of tasks, the initial, reflective, and anticipatory which were used to engender the children’s learning. While seeking to understand the relationships, the researchers responded to prior studies by developing models of teaching which are consistent with a constructivist theory of learning.
Significant findings were obtained from the research. The study brought about twofold learning process- the children’s learning as well as the learning of researcher-teacher (Tzur, 1999). The children’s activities during the research period made a great transformation in their way of handling fractions. The researcher’s inquiry into the conceptual nature of the children’s difficulties brought forth a key distinction between two fraction schemes. The children’s limited conceptions of improper fractions occurred through their iterating a nonunit fraction. The same efforts ensured that the children could conceive of the result of iterating a unit fraction beyond the whole although it was not an obvious extension of the previous episode. This illustrates clearly that if one is to understand the significance of the transformation in the children’s conceptions, one must reexamine the limited conception of fractions that was changed- the partitive scheme. Under this scheme, the learner establishes a substantial but limited understanding of fractions as parts of a specific partitioned whole (Tzur, 1999). Furthermore, this understanding is crucial because the child is able to use his or her number knowledge to work on unit and nonunit fractions in a meaningful way.
Initial Fraction Learning: A comparison
There can be a significant difference in the achievement of students if different curricula are used in instruction. There are two main curricula that can be used in the initial fraction learning; the Commercial Curricula (CC) and Rational number Project (RNP). This paper will discuss the impact of the two curricula on the achievement of students, the fundamental ideas about learning brought out in the article, the research questions that were formulated for the research as well as the main findings based on the research questions.
In this study, the accomplishments of students either using the Commercial Curricula for the original encounter with fractions and those using the Rational Number Project fraction curriculum are contrasted (Cramer, Post, & delMas, 2002). The RNP curriculum stresses the use of various physical models and conversions within and between many modes of representation. The period that the teaching program had to last was about a month and 1600 fourth and fifth graders were engaged from 66 classrooms. They were randomly categorized into treatment clusters during the study.
In this particular article, the author appreciates the fact that the teaching and learning of fractions have traditionally been difficult (Cramer et al., 2002). He quotes the results from large-scale assessments such as the National Assessment of Educational Progress (NAEP) which reveal that fourth-grade students have limited understandings of fractions. The article further points out that students’ understanding of fundamental concepts of equivalent fractions should include more than just a knowledge of generating equal fractions but should also be rich in connections among symbols, models, pictures, and context. The NAEP sample, however, revealed that only 42% of the fourth-graders were able to choose a picture that represented a fraction equivalent to a given fraction, while only about 18% could shade a rectangular region to produce a representation of a given fraction.
The author also brings in another learning concept from the NAEP which showed that the fourth-graders had notable difficulty with a problem that assessed their understanding of the importance of the unit in determining the quantity associated with a fraction. Investigations into understanding instructional issues in the learning of fractions by mathematics teachers, according to the author, reveal that the difficulties are related in part to teaching practices that stress on syntactic knowledge or rules over semantic knowledge or meaning and the notable discouragement of children from spontaneous attempts to make sense of rational numbers. These practices have been mainly rampant at fourth-grade where CC is used and hence the students are not allowed to actively participate in the classroom and thus cannot communicate their solutions to problems.
In order to further understand these learning difficulties, the researcher identified major research questions under the RNP and the CC curricula (Cramer et al., 2002). The RNP investigated, firstly, whether fraction-related differences in student achievement exist when fourth- and fifth-grade students using the conceptually oriented RNP curriculum are compared with students using district-adopted commercial mathematics curricula and secondly, if differences in student achievement do exist, what the nature of the differences in students’ thinking and understanding could be. The research identified the best methods to obtain valid and reliable results. Both qualitative and quantitative analyses were conducted.
The research findings reveal that there are significant differences in student achievement between students using the RNP curriculum and those using a commercial curriculum (Cramer et al., 2002). It is noted from analysis that students using the RNP outperformed students using the commercial curriculum on both the overall posttest and retention test results. The RNP students have a stronger conceptual understanding of fractions, are better able to judge the relative sizes of two fractions, use the knowledge to estimate sums or differences, and are better able to transfer their understanding of fractions to tasks not directly taught to them. Despite the fact that the CC group spent most of the instructional time on addition and subtraction of fractions, there is no significant difference between them and those of the RNP group who used only 5 out of the total 23 lessons dealing with the same topic. Concerning the groups’ number sense, the findings reveal that the RNP students relied mostly on conceptual or mental images in ordering of fractions compared to their CC counterparts who relied on a rote algorithm that did not consider the numbers in the problem as fractions, that is they used procedural methods.
Conceptual Units Analysis: Teachers and a Rational-Number-as-Operator Task
The understanding of any given concept requires some degree of strategies purposely designed for a given context. Mathematical concepts are not an exception to this requirement. This paper seeks to analyze the conceptual units of preservice elementary school teachers’ strategies on a rational-number-as-operator task presented in the article.
The study generally investigates preservice teachers’ perception of the operator construct of rational number (Behr, Khoury, Harel, Post & Lesh, 1997). It involves the giving of three related problems in 1-on-1 clinical interviews. The article describes two major rational number operator subconstruct; the duplicator/partition-reducer (DPR) subconstruct and the stretcher/shrinker (SS) subconstruct. This study goes ahead to offer a confirmation that students use the two rational number operator sub-constructs. The stretcher/shrinker strategies are recognized when the rational number is spread over a uniting operation in the form of an operator. Rational number is conceptualized as a rate when the SS strategies are used. During the study, the stretcher/shrinker strategies were not used as much as the duplicator/partition-reducer strategies. The investigation also produced comprehensive mental models of these strategies in terms of the underlying abstract units, their configuration, and their adjustments.
The author in the article notes that children’s and teachers’ understanding of multiplicative concepts; multiplication, division, ratio, rational number, and others are crucial to their ability to gain mathematical understanding. Although extensive research has been carried out on the knowing, learning, and teaching of these concepts among these groups during the previous decade, much work remains to be done. The author notes that although the question of what a rational number is can be easily answered from the mathematical perspective, the same question considered from a psychological or developmental perspective is less clear. Psychologists observe that the concept of rational number consists of a number of possible subconstructs- part-whole, quotient, ratio number, operator, and measure. Little research has been done to investigate learners’ understanding of these separate rational subconstructs. Other researchers have argued that explicit information is lacking among researchers and instructors concerning the concepts that underlie the understanding of the separate subconstructs.
The research questions in this article are not stated clearly. However, the study sought to find out the separate rational number operator subconstructs of preservice elementary school teachers (Behr et al., 1997). The article refers the understanding of the operator rational number as the “operator construct.” It explores, analyzes, and describes the rational number operator subconstructs of a group of preservice elementary school teachers, within a problem-solving context involving the finding three quarters of a quantity presented as eight bundles of four sticks. The study mainly intended to; make out the preservice teachers’ strategies with the two projected theoretical rational number operator subconstructs, illustrate and analyze the underlying abstract unit structures, and their adjustment, and to implement the “mathematics of quantity” notational system, to represent the entrenched abstractions of the constructed conceptual unit formation and to model the dynamics of thinking.
The research findings reveal that two major types of strategies were used; the number-exchange strategies and size-exchange strategies. Variations in the use of DPR and SS are reported from the study. The results from the study also support the idea that the several constructs interact in the course of resolving a task (Behr et al., 1997). It is therefore important for teachers to ensure that students form cognitive entities since it is then that they can be taught to form and extend the necessary cognitive structures.
Low-Performing Students and Teaching Fractions for Understanding: An Analysis
Individual differences play a great role in capturing most concepts as far as learning experiences is concerned. It is important for any instructor to take note of the differences among the children in a classroom setting especially when teaching mathematics for understanding since the rates of understanding do vary greatly. This paper will discuss the article with emphasis on the low-performing students and the teaching of fractions for the purposes of understanding.
The article provides an analysis of two low-performing students’ experiences in a first-grade classroom oriented toward teaching mathematics for understanding (Empson, 2003). It investigates the students’ participation in classroom discussions, specifically about fractions. The author proposes three main factors that account for the two students’ success: the use of tasks that elicited the students’ prior understanding, creation of a variety of participant frameworks in which students were treated as mathematically competent, and finally, the frequency of opportunities for identity-enhancing interactions.
In this article, the author notes the historically unique current standards movement in the United States which claims that all students can learn mathematics with understanding (Empson, 2003). However, almost every study of reform-oriented classrooms includes students who, based on criterion-referenced measures, do struggle. Although the presence of low-achieving students in a classroom setting is of concern to the teachers, they are rarely the focus of mainstream mathematics education research. The author cites several studies that have been done in the field of teaching mathematics for understanding.
It is important to understand how students’ success and failure basically depend on understanding the dynamics of instructional interactions and their consequences for the students. This study sought to investigate two low-performing students’ participation in a first-grade classroom oriented toward teaching mathematics for understanding in relation to their interaction with the teacher. The author argue that although success may not be assured, understanding fractions in mathematics greatly depend fundamentally on the teachers’ role in making space and meaning for students’ contributions to classroom discussions. This requires a fine-grained analysis of teachers’ and students’ interactive talk.
The findings reveal that the low-performing students can benefit from positive interaction with the instructors and being allowed to participate fully in the classroom in solving and discussing problems despite the cognitive or social skills that they may have lacked (Empson, 2003). The findings also indicate that there is growing evidence that positive interaction between instructors and the students leads to greater mathematical understanding and problem-solving achievement, especially for middle- and high-achieving students. Therefore, the author thinks that such classrooms offer some hope for understanding the nature of success for students who are mathematically low achieving. The analysis of the findings of pre- and post-interview results revealed that there was significant gain in understanding mathematics by the two low-performing students due to participation in instructional interactions. This was made possible by positioning them such that they could contribute to the discussion and were animated as problem solvers, claim makers, and solution reporters. These roles were viewed as central to doing mathematics in a group. Additionally, it was found that the frequency with which low-performing students made their contributions was central to developing the children’s understanding. The findings conclude that students’ understanding and identity are to a non-negligible extent a function of classroom interactions.
Understanding Rational Numbers: A new Model and an Experimental Curriculum
Understanding some concepts in mathematics has proved difficult and hence new methods and approaches need to be devised if meaningful understanding is to be achieved. This paper seeks to analyze the article on developing children’s understanding of rational numbers in light of its content, fundamental ideas about learning that the author brings into the research, its research questions and most importantly, the research findings.
The article focuses on the new curriculum that was devised to introduce rational numbers using developmental theory as a guide (Moss &Case, 1997). The content of the article consists of the use of the new curriculum where the 1st topic is percent in linear-measurement context. The second is the study of problems dealing with two decimal places, and then 3- and 1-places of decimals are introduced. Finally, the study focuses on fractional notations and is an alternative way of dealing with decimals.
The author in this article notes that the domain of rational numbers has traditionally been a difficult one for middle school students to master. Moreover, although most of them eventually learn the specific algorithms that they are taught, their conceptual knowledge, in general, often remains remarkably deficient (Case, 1985). The author describes how significant the difficulty that students face is when it comes to estimations of values of fractions, percentages, and decimals. It is noted that although the foregoing errors are quite diverse, they all show a profound lack of conceptual understanding that extends across all the three rational number symbolic representations and calls the existing teaching methods of these representations into critical question. The article does well in providing some explanations for the difficulties that students face in understanding the rational numbers with current methods of teaching. These will provide guidelines for future researchers. The explanations provide some hint towards the solution to the central problem of improving the teaching of rational numbers.
The article states the objectives in a clear manner. The first is the description of the general theory that the entire project was based, the new curriculum is described, and the subsequent results are discussed. These objectives were formulated with regard to the existing literature which has concentrated on the analyses domain of rational numbers. The two general stances include; epistemological and psychological. The researchers in the psychological thrust have been devoted to identifying the schemas that children bring to the domain of rational numbers and the way in which these schemas develop when the children are introduced to the domain in a more formal way. Those with an epistemological thrust have been devoted to clarifying; the nature of rational numbers as mathematical constructs and the subconstructs of which they are comprised, for instance ratio, measure, quotient, and operator. The study focused on the psychological tradition.
The results indicate that there is significant improvement in the treatment group for both pretest and posttest items in the Rational Number Test (Moss & Case, 1997). The author points out that, Piaget, a developmental psychologist emphasized the importance of including misleading features in any test if true understanding is to be assessed.
The author offers an acceptable recommendation that, in any successful attempt to teach a deeper understanding of rational numbers, instructors must emphasize the semantics of the rational numbers and on the proportional nature, in a context that is child centered and using alternative forms of representation instead of the standard pie chart. It is also established that children can be helped to understand and construct a rapid and serviceable overview of the rational numbers from the time of their first introduction to them. This involves the starting with percents and decimals rather than fractions.
References
Behr, M., Khoury, H. A., Harel, G., Post, T., & Lesh, R. (1997). Conceptual Units Analysis of Preservice Elementary School Teachers’ Strategies on a Rational-Number-as-Operator Task. Journal for Research in Mathematics Education, 28, 48-69.
Case, R. (1985). Intellectual development: Birth to adulthood. Orlando, FL: Academic Press.
Cramer, K. A., Post, T. R., & delMas, R. C. (2002). Initial Fraction Learning: A comparison. Journal for Research in Mathematics Education, 33 (2), 111-144.
Empson, S. B. (2003). Science and Mathematics Education. Journal for Research in Mathematics Education, 34 (4) 305-343.
Moss, J. & Case, R. (1997). Developing children’s rational number sense: A new approach and an experimental program. Unpublished master’s thesis, University of Toronto, Toronto, ON, Canada.
Newton, K. J. (2009). Instructional practices related to prospective elementary school teachers’ motivation for fractions. Journal for Mathematics Teacher Education, 12, 89-109.
Saxe, G. B., Taylor, E. V., McIntosh, C., & Gearhart, M. (2005). Representing Fractions with Standard Notion: A Developmental Analysis. Journal for Research in Mathematics Education, 36 (2), 137-157.
Tzur, R. (1999). An Integrated Study of Children’s Construction of Improper Fractions and the Teacher’s Role in Promoting That Learning. Journal for Research in Mathematics Education, 30 (4), 390-416.