scholarly journals The relational nature of rational numbers

Pythagoras ◽  
2015 ◽  
Vol 36 (1) ◽  
Author(s):  
Bruce Brown
Keyword(s):  
Conceptual Change ◽  
Number Field ◽  
Rational Number ◽  
Teaching And Learning ◽  
Field Research ◽  
Mathematical Structure ◽  
Rational Numbers ◽  
Number Line ◽  
Complex Nature ◽  
Research Projects

It is commonly accepted that the knowledge and learning of rational numbers is more complex than that of the whole number field. This complexity includes the broader range of application of rational numbers, the increased level of technical complexity in the mathematical structure and symbol systems of this field and the more complex nature of many conceptual properties of the rational number field. Research on rational number learning is divided as to whether children’s difficulties in learning rational numbers arise only from the increased complexity or also include elements of conceptual change. This article argues for a fundamental conceptual difference between whole and rational numbers. It develops the position that rational numbers are fundamentally relational in nature and that the move from absolute counts to relative comparisons leads to a further level of abstraction in our understanding of number and quantity. The argument is based on a number of qualitative, in-depth research projects with children and adults. These research projects indicated the importance of such a relational understanding in both the learning and teaching of rational numbers, as well as in adult representations of rational numbers on the number line. Acknowledgement of such a conceptual change could have important consequences for the teaching and learning of rational numbers.

scholarly journals Gaussian Integers

2013 ◽  
Vol 21 (2) ◽  
pp. 115-125
Author(s):  
Yuichi Futa ◽  
Hiroyuki Okazaki ◽  
Daichi Mizushima ◽  
Yasunari Shidama
Keyword(s):  
Number Field ◽  
Rational Number ◽  
Rational Numbers ◽  
Gaussian Integer ◽  
Algebraic Integers ◽  
Quotient Field ◽  
Gaussian Integers ◽  
Integer Ring

Summary Gaussian integer is one of basic algebraic integers. In this article we formalize some definitions about Gaussian integers [27]. We also formalize ring (called Gaussian integer ring), Z-module and Z-algebra generated by Gaussian integer mentioned above. Moreover, we formalize some definitions about Gaussian rational numbers and Gaussian rational number field. Then we prove that the Gaussian rational number field and a quotient field of the Gaussian integer ring are isomorphic.

Special Characteristics of Rational Numbers

1984 ◽  
Vol 31 (6) ◽  
pp. 10-12
Author(s):  
Dora Helen B. Skypek
Keyword(s):  
Rational Number ◽  
Teaching And Learning ◽  
Rational Numbers ◽  
Unlimited Number ◽  
Coding Schemes ◽  
Whole Number

The characteristics of rational numbers must be considered in a variety of interpretations and coding schemes. It is this variety that, if not sorted and carefully developed in appropriate contexts, results in confusion in teaching and learning about rational numbers. Although a discussion of interpretations necessarily involves the use of coding conventions, the two will be treated separately as special characteristics of the rational numbers. Another important charac-teristic. which is difficult to separate from interpretations and coding conventions. is this: unlike a whole number (or an integer), a rational number has an unlimited number of “behavioral clones.” These clones have different names, but they behave in exactly the same way. Still other Still other characteristics to be considered are the density and order of the rational number.

IMPLEMENTASI PENGEMBANGAN KECERDASAN SPIRITUAL ANAK MELALUI METODE PEMBIASAAN DI SD ISLAM PLUS MASYITHOH KROYA KABUPATEN CILACAP

2018 ◽  
Vol 7 (2) ◽  
pp. 157-176
Author(s):  
Fita Tri Wijayanti
Keyword(s):  
Extracurricular Activities ◽  
Teaching And Learning ◽  
Field Research ◽  
Routine Activities ◽  
Spiritual Intelligence ◽  
Learning Activities ◽  
Outdoor Learning ◽  
Special Events ◽  
Grade 5 ◽  
Analysis Technique

This study aims to describe and analyze it critical about the implementation of the development of children's spiritual intelligence through habituation methods at SD Islam Plus Masyithoh Kroya, Cilacap district. This type of research is field research or field research. This research is presented in descriptive form with the aim to describe a process that occurs in the field. While the approach taken is a qualitative approach. Data collection techniques used: observation, interviews, and documentation. While the data analysis technique uses the Miles and Huberman Model, which consists of: Data Reduction, Data Display and Verification (Conclusion Drawing). The results of this study found that the forms of development of children's spiritual intelligence through habituation methods at SD Islam Plus Masyithoh Kroya were divided into two activities, namely: first, programmed habituation activities, including extracurricular activities scheduled every Saturday namely extracurricular tilawah, tambourine and calligraphy . In addition, outdoor learning, activities that have been scheduled each year for grade 5 (five) students, are religious tourism. Second, habituation activities are not programmed in the development of children's spiritual intelligence through habituation methods. a) routine activities, carried out continuously and scheduled. The routine activities include: morning munajat activities (asmaul husna, daily prayers, tartil juz 30, and memorizing selected hadith) which are carried out before teaching and learning activities, dhuha prayer, dzuhur prayer in congregation, and social service activities, b) spontaneous namely activities that occur when experiencing special events. In this case the spontaneous activities carried out included: greeting, apologizing before asking for help, always dhikr, and thanking, c) exemplary is a habituation activity shown by the teacher in daily actions. Exemplary here is shown by the performance of the teacher both in the classroom providing subject matter as well as outside the classroom.

scholarly journals On the integer part of the reciprocal of the Riemann zeta function tail at certain rational numbers in the critical strip

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
WonTae Hwang ◽  
Kyunghwan Song
Keyword(s):  
Zeta Function ◽  
Rational Number ◽  
Riemann Zeta Function ◽  
Rational Numbers ◽  
Critical Strip ◽  
Integer Part ◽  
Integral Points ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

Abstract We prove that the integer part of the reciprocal of the tail of $\zeta (s)$ ζ ( s ) at a rational number $s=\frac{1}{p}$ s = 1 p for any integer with $p \geq 5$ p ≥ 5 or $s=\frac{2}{p}$ s = 2 p for any odd integer with $p \geq 5$ p ≥ 5 can be described essentially as the integer part of an explicit quantity corresponding to it. To deal with the case when $s=\frac{2}{p}$ s = 2 p , we use a result on the finiteness of integral points of certain curves over $\mathbb{Q}$ Q .

Call for Manuscripts for the 2014 Focus Issue: Rational Number Sense – October 2012

2012 ◽  
Vol 18 (3) ◽  
pp. 189
Keyword(s):  
Rational Number ◽  
Number Sense ◽  
Rational Numbers

This call for manuscripts is requesting articles that address how to make sense of rational numbers in their myriad forms, including as fractions, ratios, rates, percentages, and decimals.

Teacher Professional Dispositions: Much Assemblage Required

2019 ◽  
Vol 121 (11) ◽  
pp. 1-28
Author(s):  
Kathryn Strom ◽  
Jason Margolis ◽  
Nihat Polat
Keyword(s):  
Teacher Preparation ◽  
Teaching And Learning ◽  
Assessment Tools ◽  
Teacher Preparation Programs ◽  
Teacher Dispositions ◽  
Complex Nature ◽  
Preparation Programs ◽  
Professional Dispositions ◽  
State Education ◽  
Nature Of Teaching

Background/Context Despite noted difficulties with defining and assessing teacher dispositions, U.S. state education departments and national accreditation agencies have included dispositions in mandates and standards both for determining teacher quality and for assessing the quality of the teacher preparation programs that certify them. Thus, there remains a significant impetus to specify dispositions to assess, identify what “good” dispositions look like in practice, and determine the best way to measure them. Purpose The purpose of this paper is twofold. First, we aim to problematize the construct of “teacher dispositions” through a critical synthesis of literature and a discussion of a rhizomatic perspective to generate a (re)conceptualization that is more closely aligned with the immensely complex nature of teaching and learning. Second, we draw on samples of university-generated teacher disposition assessment tools to provide concrete examples that “put to work” this complex perspective on dispositions. Research Design To apply ideas introduced in our rhizomatic framework focused on multiple, dynamic assemblages, we conducted a qualitative textual analysis of a sample of 16 widely available assessment tools used by university-based teacher preparation programs to measure teachers’ professional dispositions. Findings and Conclusions Overall, the vast majority of disposition criteria included in the tools reviewed were temporal and relational, seeking to assess the interactions of the teacher candidate amidst a variety of potential circumstances as well as material and discursive factors. This reveals a paradox, however, since, despite their more contextual phrasing, these criteria ultimately seek to assess an individual and are high-stakes only for that teacher. Yet, we suggest that the results of this review may be an indication that the field is moving toward a more multifaceted vision of teaching that can better take into account the dynamic, situated, and relational nature of teaching activity. We also suggest the language accounting for some of the complexity of teaching in the disposition assessment tools we reviewed may be an entry point into a more dynamic, vital materialist vision of the profession.

scholarly journals Automorphy factors for a Hilbert modular group

1988 ◽  
Vol 30 (2) ◽  
pp. 231-236
Author(s):  
Shigeaki Tsuyumine
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Rational Number ◽  
Modular Group ◽  
Algebraic Number Field ◽  
Maximal Order ◽  
Hilbert Modular Group ◽  
Hilbert Modular ◽  
Totally Real

Let K be a totally real algebraic number field of degree n > 1, and let OK be the maximal order. We denote by гk, the Hilbert modular group SL2(OK) associated with K. On the extent of the weight of an automorphy factor for гK, some restrictions are imposed, not as in the elliptic modular case. Maass [5] showed that the weight is integral for K = ℚ(√5). It was shown by Christian [1] that for any Hilbert modular group it is a rational number with the bounded denominator depending on the group.

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