Catalogue Search | MBRL
Are you sure you want to remove the book from the shelf?
{{itemTitle}}
2,872
result(s) for
"Rational number"
Sort by:
Which Type of Rational Numbers Should Students Learn First?
Many children and adults have difficulty gaining a comprehensive understanding of rational numbers. Although fractions are taught before decimals and percentages in many countries, including the USA, a number of researchers have argued that decimals are easier to learn than fractions and therefore teaching them first might mitigate children's difficulty with rational numbers in general. We evaluate this proposal by discussing evidence regarding whether decimals are in fact easier to understand than fractions and whether teaching decimals before fractions leads to superior learning. Our review indicates that decimals are not generally easier to understand than fractions, though they are easier on some tasks. Learners have similar difficulty in understanding fraction and decimal magnitudes, arithmetic, and density, as well as with converting from either notation to the other. There was too little research on knowledge of percentages to include them in the comparisons or to establish the ideal order of instruction of the three types of rational numbers. Although existing research is insufficient to determine the best sequence for teaching the three rational number formats, we recommend several types of research that could help in addressing the issue in the future.
Journal Article
Is there a Gap or Congruency Effect? A Cross-Sectional Study in Students’ Fraction Comparison
Many studies have addressed the natural number bias in fraction comparison, focusing on the role of congruency. However, the congruency effect has been observed to operate in the opposite direction, suggesting that a deeper explanation must underlie students' different reasoning. We extend previous research by examining students' reasoning and by studying the effect of a gap condition in students' answers. A cross-sectional study was conducted on 438 students from 5th to 10th grade. Results showed that the gap effect could explain differences between congruent and incongruent items. Moreover, students' use of gap thinking decreased towards the end of Secondary Education.
Journal Article
THE RELATION BETWEEN LEARNERS’ SPONTANEOUS FOCUSING ON QUANTITATIVE RELATIONS AND THEIR RATIONAL NUMBER KNOWLEDGE
Many difficulties learners have with rational number tasks can be attributed to the \"natural number bias\", i.e. the tendency to inappropriately use natural number properties in rational numbers tasks (Van Hoof, 2015). McMullen and colleagues found a relevant source of individual differences in the learning of those aspects of rational numbers that are susceptible to the natural number bias, namely Spontaneous Focusing On quantitative Relations (SFOR) (McMullen, 2014). While McMullen and colleagues showed that SFOR relates to rational number knowledge as a whole, we studied its relation with several aspects of the natural number bias. Additionally, we 1) included test items addressing operations with rational numbers and 2) controlled for general mathematics achievement and age. Results showed that SFOR related strongly to rational number knowledge, even after taking into account several control variables. Results are discussed for each of the three aspects of the natural number bias separately.
Journal Article
Hard Lessons
Fraction and decimal arithmetic pose large difficulties for many children and adults. This is a serious problem because proficiency with these skills is crucial for learning more advanced mathematics and science and for success in many occupations. This review identifies two main classes of difficulties that underlie poor understanding of rational number arithmetic: inherent and culturally contingent. Inherent sources of difficulty are ones that are imposed by the task of learning rational number arithmetic, such as complex relations among fraction arithmetic operations. They are present for all learners. Culturally contingent sources of difficulty are ones that vary among cultures, such as teacher understanding of rational numbers. They lead to poorer learning among students in some places rather than others. We conclude by discussing interventions that can improve learning of rational number arithmetic.
Journal Article
Rational number understanding : the big picture, not the essence
Background: The learning of rational numbers is a complex and difficult process that begins in the early grades. This teaching often focuses on the mastery of essential knowledge, including particular skills (e.g. using fractions to describe part–whole diagrams) and interpretations (e.g. sharing), which often results in an incomplete and inflexible understanding of these numbers. Aim: This article proposes a holistic and relational perspective on rational number knowing and sense-making. Setting: This possibility emerged through research into the learning of rational number concepts by Foundation Phase and Grade 4 children. Methods: This research forms part of an ongoing, in-depth, exploratory research programme into the processes of learning rational numbers. Clinical interviews and classroom observations were the primary methods of data collection and an in-depth, constant comparative method of analysis was performed on the data. Results: Thinking relevant to rational numbers was identified within four different perspectives through which children make sense of their interactions with the world, namely, social, instrumental, personal and symbolic sense-making. Conclusion: The learning of rational numbers may be usefully seen as arising from the interrelation of multiple aspects of knowing and doing that develop as children balance these different ways of sense-making.
Journal Article
Multiply and Divide Rational Numbers 1 (CCSS 7.NS.A.2c)
Fill in the gaps of your Common Core curriculum! Each ePacket has reproducible worksheets with questions, problems, or activities that correspond to the packet's Common Core standard. Download and print the worksheets for your students to complete. Then, use the answer key at the end of the document to evaluate their progress. Look at the product code on each worksheet to discover which of our many books it came from and build your teaching library! This ePacket has 9 activities that you can use to reinforce the standard CCSS 7.NS.A.2c: Multiply and Divide Rational Numbers. To view the ePacket, you must have Adobe Reader installed. You can install it by going to http://get.adobe.com/reader/.
eBook
Existence of minimal models for varieties of log general type
We prove that the canonical ring of a smooth projective variety is finitely generated.
Journal Article
Student Teachers’ and Experienced Teachers’ Professional Vision of Students’ Understanding of the Rational Number Concept
The aim of this study was to investigate differences in student teachers’ and experienced teachers’ professional vision in natural settings and to elicit clues of the relation of in-the-moment noticing and instruction quality of students’ understanding of rational number concept. Rational number concept challenges both students and teachers because of natural number bias that learning of rational numbers is vulnerable to. Accurate professional vision and adequate instructions are needed to enhance students’ understanding of rational number concept. Mobile eye-tracking technique enables video recording of natural teaching situations from a teacher’s perspective with more specific information of teacher’s in-the-moment noticing. Combined with cued retrospective reporting, this approach can gather more explicit evidence of teachers’ professional vision and instructions. Results indicated that both student teachers and experienced teachers attended to mathematical and fraction-related aspect similarly but differed in interpreting and instructing students’fraction understanding. Student teachers made more advanced interpretations but their instructions were less adequate, whereas among experienced teachers, it was just the opposite. Furthermore, student teachers made more attempts to shared attention when using fraction understanding non-supporting instructions, whereas experienced teachers’ attempts to shared attention were related to fraction understanding supporting instructions. Results indicate student teachers’ difficulty to transfer pedagogical content knowledge from noticing to actions and experienced teachers to have more enhanced in-the-moment professional vision and its application to teaching. Practical implications for teacher training as well as methodological decisions of in-the-moment professional vision studies in natural settings are discussed.
Journal Article
Analysis Concerning the Problem of the Optimal Delivery in Three Towns
The aim intimately pertinent to this research program is to use constructive mathematical notions to analyze how constructive real numbers and rational numbers are likely to have an influence on the outcomes involving economic problems. Researchers endeavor to find a computer program to determine the optimal location of a production plant and to find out that provided that the distances from the production plant to the sales location are rational numbers, then the problem is algorithmically solvable, though, if the distances are constructive real numbers, then the existence of the computer program that solves all such problems will not be possible.
Journal Article