Abstract
Number sense refers to one’s general comprehension of numbers and operations and the ability to use the understanding in making mathematical judgments in handling mathematical operations. It encompasses a broad range of mathematically relevant concepts. This essay describes different authors’ perspectives on equivalent fractions in number sense. Different approaches, hypotheses, and theories have been used to describe teacher and student comprehension of fractions and number sense. The general purpose of the authors’ contexts is to make additions to the literature on mathematical information relating to fractions and number sense. Students’ comprehension of number sense and equivalent fractions is reinforced through practical applications of real-world examples given by teachers. A summation of the authors’ perspectives reveals that knowledge on equivalent fractions in number sense is still an ongoing struggle both for prospective elementary teachers and students. The papers answers questions such as; What is so difficult about teaching fractions. The teachers need to familiarize themselves with better techniques for aiding student comprehension. The important perspectives and theories used include research-based instructional materials, instructional theory, and tasks designs involving comparing fractions.
Keywords : Equivalent fractions, mathematical knowledge for teaching, number sense.
Summaries
Bobos, G., & Sierpinska, A. (2017). Measurement approach to teaching fractions: A design experiment in a pre-service course for elementary teachers. International Journal for Mathematics Teaching and Learning , 18 (2).
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The authors use a measurement approach that focuses on the conceptualization of mathematical content. The authors deduce that developing a concept of fractions with sources, in reality, can be used to guide children’s attention to multiplicative relationships. The authors use such relationships to define systemic operations between them. The context is aimed at projecting equivalent fractions as abstract numbers that present the measure of a relationship between two quantities. The approach is proposed to teachers and is expected to foster the development of quantitative reasoning and theoretical thinking. The conceptualization concept rests on the assumption that future prospective elementary teachers understanding of fractions comprises of two disconnected parts; material conception or visual based ideas of a fraction of something, the other part is the formal conception which showcases the calculus expressions of fractions using whole numbers. The fraction of a quantity in number sense gives an intuitive idea of the magnitude of the fraction in material conception.
Chinnappan, M., & Forrester, T. (2014). Generating procedural and conceptual knowledge of fractions by pre-service teachers. Mathematics Education Research Journal , 26 (4), 871-896.
The authors focus on teacher knowledge as it is a significant contributor to levels of numeracy attained by learners. The authors affirm that teachers need robust content and pedagogical knowledge in mathematics to improve learning in students. Studies show that teacher knowledge in mathematics significantly contributes to student knowledge. The authors’ focused more on the detailed knowledge of teachers in the context of fractions. Fractions pose a challenge to students as they are forced to transfer the concept of whole numbers into a new class of numbers. Teacher knowledge of fractions can, therefore, affect student comprehension positively or negatively. Equivalent fractions enable students to make number sense of other fractions due to their similarity properties. The authors document that comprehension of equivalent fractions in number sense entails both procedural pieces of knowledge that describe mathematical rules and conceptual knowledge which describes mathematical relationships. Teachers use fractional knowledge to enable students to understand concepts such as equivalence. The authors’ analyze teacher knowledge and experience through a study that explicates tutorial concepts.
Da Ponte, J. P., & Quaresma, M. (2016). Teachers’ professional practice conducting mathematical discussions. Educational Studies in mathematics , 93 (1), 51-66.
The authors document the actions of teachers that can be regarded as building elements of classroom practice and how they can be combined to provide fruitful learning opportunities for students. The authors use examples of teacher-assigned tasks in which students are required to solve equations. The purpose of the tasks is to foster the development of connections and reasoning. The teachers through the tasks found that the students had intuitive knowledge of computing fractions from whole numbers. The tasks encouraged students to reason using representations and contextual elements. The use of equivalent fractions in assigning tasks to students encouraged the use of informal procedures and practical knowledge to compute and discern relationships in fractions and support the development of student’s reasoning. The teacher activities discussed by the authors were aimed at creating conditions for student learning about number sense, representations through fractions, and procedures. Through the tasks, the students were able to learn about representing and comparing rational numbers, the equivalence of fractions, and generalizations and justifying.
Lewis, C., & Perry, R. (2017). Lesson study to scale up research-based knowledge: A randomized, controlled trial of fractions learning. Journal for research in mathematics education , 48 (3), 261-299.
The authors’ document that resources provided for instructors and students in self-managed learning over months improve teacher and student fractions knowledge. Resources are derived from research-based instructional improvements and materials aimed at scaling up fractional learning. An analysis of the authors’ take on number sense and equivalent fractions reveals that integration of research-based resources offers new approaches to scaling up equivalent fractions. The context is useful for instructors as a new method for expanding equivalent fractions knowledge for learners. The significant approach used by the authors includes the use of research-based knowledge to scale up fraction learning. The observation made is that implementing research-based knowledge resources into lesson study for controlled environments yield better results compared to uncontrolled premises. The perspectives used include linear models. Linear models describe continuous response variables as predictors of one or more variable. Linear models are used as instructional materials by instructors to up-scale learning of equivalent fractions. More research is suggested to identify the resources needed to improve students’ comprehension of fractions.
Olanoff, D., Lo, J. J., & Tobias, J. (2014). Mathematical content knowledge for teaching elementary mathematics: A focus on fractions. The Mathematics Enthusiast , 11 (2), 267-310.
The authors present a summary of the prospective elementary teacher’s knowledge of fractions. The authors’ use literature from 43 articles written across different points in time and that focus on prospective teachers knowledge of fractions. The findings show a strong performance procedure in prospective teachers but show shortcomings in performing procedures using fraction number sense. The authors deduce that fractions along with ratios and proportions are the most cognitively challenging areas of mathematical knowledge for prospective elementary teachers to teach. The authors identify lacking in the literature procedures for teachers to expand their knowledge of operations and fraction concepts. Content and methods are suggested for teachers as well as more research into how prospective teachers’ fraction content knowledge develops — the authors’ use Shulman’s perspective of teacher knowledge to assess prospective teachers’ knowledge of fractions. The perspective encompasses subject matter knowledge which comprises basic concepts, pedagogical content knowledge which comprises useful representations, and curricular knowledge which is the full range of programs designed for teaching.
Rosli, R., Goldsby, D., & Capraro, M. M. (2015). Using manipulatives in solving and posing mathematical problems. Creative Education , 6 (16), 1718
The authors’ focus on how teachers of mathematics use various representations to help students’ develops abstract concepts. Students’ manipulate objects in search of solutions. Concrete manipulative exercises in class are used to teaching and learning of equivalent fractions. Students construct knowledge on unit wholes and recognize relationships when manipulating objects. Activities such as paper folding into equal halves, aid the students in comprehending the relational aspects of fractions. The use of real-world examples in teaching fractions enables students to understand equivalent fractions in number sense. According to the authors, teachers can probe students thinking by having them explain the relationships between fractions. This allows students to create mental images of the relationships that exist between fractions. The authors’ suggest that the use of manipulations in teaching facilitates belter conceptual understandings of equivalent fractions and the relationships between fractions. The idea of fair sharing is a virtual manipulative concept that guides students to make discernments concerning equivalent fractions.
Thanheiser, E., Olanoff, D., Hillen, A., Feldman, Z., Tobias, J. M., & Welder, R. M. (2016). Reflective analysis as a tool for task redesign: The case of prospective elementary teachers solving and posing fraction comparison problems. Journal of Mathematics Teacher Education , 19 (2-3), 123-148.
The authors explored the critical aspects of mathematical design such as planning, implementing, and reflecting in the comparison of fractions using reasoning and sense-making. The authors make contributions in literature with reasoning strategies for comparing fractions and an exploration of prospective elementary teacher’s mathematical content knowledge. The authors use literature on research in tasks and design, and the research on teacher’s knowledge of fraction concepts to assess the development of mathematical ideas. Comparison of fractions is felt necessary for achieving an intuitive feel of fractions. One comprehends a fractional number sense when estimating the magnitude of fractions. Other studies indicate that teacher and student knowledge of fraction concepts is limited. Teachers and students rely more on finding common denominators in fractions when other number sense strategies are more efficient. The reliance on whole number reasoning leads to a gap in thinking and may skew the reasoning of equivalent fractions. It is a common assumption in young children that all fractions are less than one.
Whitacre, I., & Nickerson, S. D. (2016). Investigating the improvement of prospective elementary teachers’ number sense in reasoning about fraction magnitude. Journal of Mathematics Teacher Education , 19 (1), 57-77.
The authors’ were concerned with improving the number of sense in the population under study. The study shows the application of instruction theory as a framework for developing number sense and reasoning for fraction magnitude and content course. The authors discuss the local instruction theory as a significant influencer of fractions understanding. An interview of students before and after instruction shows significant improvement. The theory is proposed to help the mathematical preparation of prospective elementary teachers. Prospective elementary teachers developed an improved number sense as a result of the whole number to rational number computation. The highlights of the authors take on number sense, and fractions include highlighting how prospective teachers become more flexible in reasoning about fractions, and a shift to non-standard strategies. The authors contribute to literature through documenting successful improvement of fraction sense and describing a design of instruction for incorporating number sense and comprehension of fractions. The authors’ also showcase the productive changes in the magnitude of fraction reasoning.
Synthesis and Discussions
The various researches conducted on equivalent fractions in number sense shows that mathematical knowledge in prospective elementary teachers determines the level of comprehension in students. Mathematical thinking in students can be reinforced through the use of practical examples and activities. Research suggests that children have a hard time understanding fractions and transference of whole numbers into fractions. The teaching of fractions has been described as one of the most challenging topics in mathematics and requires a great deal of mathematical knowledge and thinking for both teachers and students. Intuitive teachers use real-world examples to improve comprehension of fractions in students.
A critical evaluation of the authors’ perspectives is concurrent with underlying perspectives on mathematical knowledge of fractions. Fractions deviate from the norm of whole numbers and therefore pose a challenge for students to comprehend. The teaching of fractions requires instruction to go beyond class limits and incorporate other methods for enabling students to understand fractions. The use of resources in teaching fractions increases reasoning about fraction magnitude. Some of the studies have been conducted before and after instructional materials were used on students. The results suggest that reasoning about fractions increases after the use of instructional materials. Participants of such studies showed better mental computations, reasoned flexibly, and favored sophisticated strategies used in teaching fractions.
The authors’ perspectives provide information on how to answer focus questions regarding equivalent fractions in number sense. The contexts give an overview of why learning fractions are considered burdensome. Some of the reasons given are that fractions give a lot to understand. Students have to be taught to ask a fraction of what? For them to comprehend fractions in number sense, teachers have to go out of their way to use unconventional methods to project abstract knowledge of fractions.
The relevant questions that instructors should be asking include how to find the fraction of a quantity? What fractions are quantities of others? Moreover, how are fractions added? Talking fractions using the language of fractions can help make students understand the vocabulary associated with fractions. Teachers can draw attention to how words are used in dealing with fractions, and they get focused on getting students to have discussions on the language of fractions. The next step would be to develop an understanding of fractions or embodiment. This is the step that includes practical classroom applications of fractions to help students understand fractions better.
The main themes observed in the authors’ contexts of equivalent fractions in number sense include the use of research-based resources, instructional perspectives, task designs, and manipulative resources to aid students in learning fractions. Research-based resources mentioned include linear models and controlled environments as ways through which instructors can better the understanding of students about fractions. The instructional perspective focuses on using instructional materials to scale up the understanding of fractions. The other recurring themes include the mathematical knowledge of teachers as the influencer of students’ comprehension of fractions.
The assumption made by the researchers is that the majority of prospective elementary teachers lack sufficient mathematical knowledge of fractions. Other assumptions include the fact that all students were able to comprehend equivalent fractions after the various methods of reinforcement were used. Some of the theoretical ideas featured in the contexts include instructional theory. The instructional theory is a tool used to offer explicit guidance to students on how to learn and develop. Instructors focus on the best structures they can use to facilitate learning. What has probably not been well addressed in fractions about number sense? The majority of authors fail to give a rationale for equivalent fractions in number sense.
References
Bobos, G., & Sierpinska, A. (2017). Measurement approach to teaching fractions: A design experiment in a pre-service course for elementary teachers. International Journal for Mathematics Teaching and Learning , 18 (2). http://www.cimt.org.uk/ijmtl/index.php/IJMTL/article/view/65/39
Chinnappan, M., & Forrester, T. (2014). Generating procedural and conceptual knowledge of fractions by pre-service teachers. Mathematics Education Research Journal , 26 (4), 871-896. http://ro.uow.edu.au/cgi/viewcontent.cgi?article=2341&context=sspapers
Da Ponte, J. P., & Quaresma, M. (2016). Teachers’ professional practice conducting mathematical discussions. Educational Studies in mathematics , 93 (1), 51-66. http://repositorio.ul.pt/bitstream/10451/24730/1/Ponte%20Quaresma%20ESM%202016.pdf
Lewis, C., & Perry, R. (2017). Lesson study to scale up research-based knowledge: A randomized, controlled trial of fractions learning. Journal for research in mathematics education , 48 (3), 261-299. https://www.jstor.org/stable/10.5951/jresematheduc.48.3.0261?seq=1#page_scan_tab_contents
Olanoff, D., Lo, J. J., & Tobias, J. (2014). Mathematical content knowledge for teaching elementary mathematics: A focus on fractions. The Mathematics Enthusiast , 11 (2), 267-310. https://scholarworks.umt.edu/cgi/viewcontent.cgi?article=1304&context=tme
Rosli, R., Goldsby, D., & Capraro, M. M. (2015). Using manipulatives in solving and posing mathematical problems. Creative Education , 6 (16), 1718. http://file.scirp.org/pdf/CE_2015091611514383.pdf
Thanheiser, E., Olanoff, D., Hillen, A., Feldman, Z., Tobias, J. M., & Welder, R. M. (2016). Reflective analysis as a tool for task redesign: The case of prospective elementary teachers solving and posing fraction comparison problems. Journal of Mathematics Teacher Education , 19 (2-3), 123-148.
Whitacre, I., & Nickerson, S. D. (2016). Investigating the improvement of prospective elementary teachers’ number sense in reasoning about fraction magnitude. Journal of Mathematics Teacher Education , 19 (1), 57-77. http://www.sci.sdsu.edu/crmse/msed/papers/Whiteacre_Nickerson.pdf
