Division in a binary representation for complex numbers

Computer operations involving complex numbers, essential in such applications as Fourier transforms or image processing, are normally performed in a ‘divide-and-conquer’ approach dealing separately with real and imaginary parts. A number of proposals have treated complex numbers as a single unit but all have foundered on the problem of the division process without which it is impossible to carry out all but the most basic arithmetic. This paper resurrects an early proposal to express complex numbers in a single ‘binary’ representation, reviews basic complex arithmetic and is able to provide a fail-safe procedure for obtaining the quotient of two complex numbers expressed in the representation. Thus, while an outstanding problem is solved, recourse is made only to readily accessible methods. A variety of extensions to the work requiring similar basic techniques are also identified. An interesting side-line is the occurrence of fractal structures, and the power of the ‘binary’ representation in analysing the structure is briefly discussed.     

Dynamic mathematics and the blending of knowledge structures in the calculus

This paper considers the role of dynamic aspects of mathematics specifically focusing on the calculus, including computer software that responds to physical action to produce dynamic visual effects. The development builds from dynamic human embodiment, uses arithmetic calculations in computer software to calculate ‘good enough’ values of required quantities and algebraic manipulation to develop precise symbolic values. The approach is based on a developmental framework blending human embodiment, with the symbolism of arithmetic and algebra leading to the formalism of real numbers and limits. It builds from dynamic actions on embodied objects to see the effect of those actions as a new embodiment that needs to be calculated accurately and symbolised precisely. The framework relates the growth of meaning in history to the mental conceptions of today’s students, focusing on the relationship between potentially infinite processes and their consequent embodiment as mental concepts. It broadens the strategy of process-object encapsulation by blending embodiment and symbolism.     

A Longitudinal Study Revisiting the Notion of Early Number Sense: Algebraic Arithmetic AS a Catalyst for Number Sense Development

The aim of this study was to propose a new conceptualization of early number sense. Six-year-old students’ (n = 204) number sense was tracked from the beginning of Grade 1 through the beginning of Grade 2. Data analysis suggested that elementary arithmetic, conventional arithmetic, and algebraic arithmetic contributed to the latent construct early number sense, and the invariance of the model over time was validated empirically. Algebraic arithmetic represents the dimension of early number sense that moves beyond conventional arithmetic and encompasses an abstract understanding of the relations between numbers. A parallel process growth model showed that the three components of number sense adopt a linear growth rate. A structural model showed that the growth rate of the algebraic arithmetic component has a direct effect on the growth rate of conventional arithmetic, and subsequently the growth rate of conventional arithmetic predicts the growth rate of elementary arithmetic.     

Unique factorization and the fundamental theorem of arithmetic

The fundamental theorem of arithmetic is one of those topics in mathematics that somehow ‘falls through the cracks’ in a student's education. When asked to state this theorem, those few students who are willing to give it a try (most have no idea of its content) will say something like ‘every natural number can be broken down into a product of primes’. The fact that this breakdown always results in the same primes is viewed as ‘obvious’. The purpose of this paper is to illustrate with a number of examples that the ‘Unique Factorization Property’ is a rare property and the fact that the natural numbers possess this property is ‘fundamental’ to our understanding of this number system.     

A tale of two algorithms

Real numbers are often a missing link in mathematical education. The standard working assumption in calculus courses is that there exists a system of ‘numbers’, extending the rational number system, adequate for measuring continuous quantities. Moreover, that such ‘numbers’ are in one-to-one correspondence with points on a ‘number line’. But typically real ‘numbers’ are not systematically presented via any constructive method. While taken for granted, they are one of the most commonly used mathematical objects. This paper proposes a geometric algorithm, extending the long division algorithm, which leads to a constructive definition of real numbers. It proceeds to describe a direct algorithm for adding ‘real numbers’. Combined use of the two algorithms enables a smooth and meaningful presentation, offering a double image (geometric and numerical) of real numbers in decimal notation. An early such presentation is of both conceptual and practical importance.     

Using Metaphors to Understand and Solve Arithmetic Problems: Novices and Experts Working With Negative Numbers

In this study, novices and experts used the same metaphors to understand and solve problems with negative numbers. However, they used them differently. Twenty-four participants (12 middle school children and 12 postsecondary adults) computed arithmetic expressions during the problem-solving task. During this task, children used metaphors more often than adults did to compute, detect and correct errors, and justify their answers. Metaphorical computations were more accurate but slower than other methods. The participants explained 6 arithmetic expressions during the understanding task. During this task, the adults used more metaphors (with fewer details) and used them more often than the children did. Compared to the median child, the median adult showed a more integrated understanding of arithmetic through multiple metaphors, mathematical rules, and transformations. These results suggest that the metaphors used by both the children and the adults are central to understanding arithmetic. Thus, these metaphors are likely candidates for theory-constitutive metaphors.     

Using Metaphors to Understand and Solve Arithmetic Problems: Novices and Experts Working With Negative Numbers

In this study, novices and experts used the same metaphors to understand and solve problems with negative numbers. However, they used them differently. Twenty-four participants (12 middle school children and 12 postsecondary adults) computed arithmetic expressions during the problem-solving task. During this task, children used metaphors more often than adults did to compute, detect and correct errors, and justify their answers. Metaphorical computations were more accurate but slower than other methods. The participants explained 6 arithmetic expressions during the understanding task. During this task, the adults used more metaphors (with fewer details) and used them more often than the children did. Compared to the median child, the median adult showed a more integrated understanding of arithmetic through multiple metaphors, mathematical rules, and transformations. These results suggest that the metaphors used by both the children and the adults are central to understanding arithmetic. Thus, these metaphors are likely candidates for theory-constitutive metaphors.     

Writing practitioner case studies to help behavioural OR researchers ground their theories: application of the mangle perspective

Behavioural research into the practice of OR needs to be grounded. Case studies written by practitioners can potentially help address this need but currently most do not. The paper explores a way of describing OR projects that place the emphasis on the ‘actors’ who provide the motivating force and the consequences of their actions. The ‘mangle’ perspective focuses on the dynamic intertwining of people, technology and concepts; this can provide the basis for an insightful narrative describing the reality of the project in terms of the planned approach, the problems met and the outcomes. Two examples are given, one of a conventional model building exercise, the second of a ‘soft OR’ intervention: both describe projects conducted by practitioners for commercial purposes. It is concluded that, by using the mangle perspective, the OR case writer can winnow the wheat from the chaff in order to write a succinct informative narrative, a narrative that could be utilized by behavioural OR (BOR) researchers. It is further concluded that BOR researchers should engage with ‘practice theory’ to deepen their understanding of what actually happens in projects.     

Relative arithmetic

In nonstandard mathematics, the predicate ‘x is standard’ is fundamental. Recently, ‘relative’ or ‘stratified’ nonstandard theories have been developed in which this predicate is replaced with ‘x is y ‐standard’. Thus, objects are not (non)standard in an absolute sense, but (non)standard relative to other objects and there is a whole stratified universe of ‘levels’ or ‘degrees’ of standardness. Here, we study stratified nonstandard arithmetic and the related transfer principle. Using the latter, we obtain the ‘reduction theorem’ which states that arithmetical formulas can be reduced to equivalent bounded formulas. Surprisingly, the reduction theorem is also equivalent to the transfer principle. As applications, we obtain a truth definition for arithmetical sentences and we formalize Nelson's notion of impredicativity (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Addition and subtraction of three-digit numbers: adaptive strategy use and the influence of instruction in German third grade

Empirical findings show that many students do not achieve the level of a flexible and adaptive use of arithmetic computation strategies during the primary school years. Accordingly, educators suggest a reform-based instruction to improve students’ learning opportunities. In a study with 245 German third graders learning by textbooks with different instructional approaches, we investigate accuracy and adaptivity of students’ strategy use when adding and subtracting three-digit numbers. The findings indicate that students often choose efficient strategies provided they know any appropriate strategies for a given problem. The proportion of appropriate and efficient strategies students use differs with respect to the instructional approach of their textbooks. Learning with an investigative approach, more students use appropriate strategies, whereas children following a problem-solving approach show a higher competence in adaptive strategy choice. Based on these results, we hypothesize that different instructional approaches have different advantages and disadvantages regarding the teaching and learning of adaptive strategy use.     

OR in strategic land-use planning

An important historic strategic application of OR has been in the field of land-use and development plan production. Changes in Government policy and legislation have led to varying levels of interest in plan production. Three post-war cycles of ‘enthusiasm for plans’ can be identified. Whilst the first was rooted very much in the Architectural Design tradition, the second led to significant developments in OR, with far wider application. Subsequent reduced Governmental enthusiasm for ‘Development Plan production’ led to considerable atrophy of relevant skills in the planning community, including those derived from OR. However, the current ‘third period of post-war enthusiasm for planning’, reinforced by environmental concerns, has revived the need for relevant skills. It is suggested that, whilst the deficit in skills and their application remains high, there are some encouraging signs. Moreover, substantial progress in the field of ‘soft OR’ offers opportunities to both the OR and planning communities.     

Telling and illustrating stories of parity: a classroom-based design experiment on young children’s use of narrative in mathematics

This paper examines ways to engage young children in constructing and interpreting narratives to develop their understanding of parity. It reports on a teaching intervention that was developed over three research cycles of a classroom-based design experiment, and focuses on the last of these cycles. The teaching intervention set out to investigate how young children (5–6-year-olds) can be supported to draw on narrative in their explanations of whether a whole number less than 20 is odd or even. Evidence of the effectiveness of the intervention is provided through comparison of children’s performance on pre- and post-tests in the form of semi-structured individual interviews. Also, authentic examples are provided of how children utilised their power of ‘imagining and expressing’ to tell stories of whether a whole number is odd or even, using either a counting, partitive, or quotitive model for division. Implications for research and practice are discussed in light of these findings.     

Multivariate analysis using ‘and’ and ‘or’

The connectives ‘and’ and ‘or’ are potentially useful in multivariate analysis and theory construction. They are simple, logical ways to connect two or more variables together. However, until recently there has been no framework for operationalizing these connectives for continuous variables, and this lack has severely limited their use. Using fuzzy set theory as a basis for such a framework, this paper lays out the necessary tools and models to permit the use of ‘and’ and ‘or’ in multivariate analysis.Section 1 introduces conventional operators for ‘and’ and ‘or’, and Section 2 provides suitable extensions and generalizations of them. Section 3 sets out the required least-squares techniques for fitting these generalized operators to data, first in the context of ANOVA problems and then in regression contexts, for single-connective (three-variable) models. The theoretical developments and examples from real data-sets demonstrate the utility of ‘and’ and ‘or’ as a means to cell-specific interpretations of interaction effects which can also readily be translated into English. Section 4 extends these developments to multivariate, multiple-connective models and discusses issues of generalizability. The paper concludes (Section 5) with a brief discussion of remaining unsolved problems, future prospects for more sophisticated models, and computer programs.     

Frameworks of Inquiry: OR Practice Across the Hard—Soft Divide

The growing interest in understanding the practice of OR has, not unnaturally, tended to concentrate upon experience with those ‘soft’ methodologies which address both process and content management issues. This paper uses a detailed account of one practitioner's work in a ‘traditional’ area of OR (linear programming) to demonstrate how process-related issues are handled there, and argues that more extensive reporting of such conventional practice is essential for the health of the discipline. In particular, it suggests that an emphasis on discussing the development of working relationships between OR practitioners and their clients might usefully supplant the contemporary emphasis on the ‘project’.     

Finding Sum of Powers on Arithmetic Progressions with Application of Cauchy’s Equation

In this paper we introduce a method to find the sum of powers on arithmetic progressions by using Cauchy’s equation and obtain a general formula. Then we apply our results to show how to determine some other sums of powers and sums of products. Our results are more general than those in [9]. Finally we discuss the sum of powers on arithmetic progressions in commmutative rings with characteristic 2 and find ‘full polynomials’.     

From Fractions to Rational Numbers of Arithmetic: A Reorganization Hypothesis

Nathan and Arthur, 2 children in a 3-year teaching experiment on children's construction of the rational numbers of arithmetic (RNA), developed their operations for multiplying, dividing, and simplifying fractions over the last 2 years (Grades 4 and 5) of the experiment. The 2 children worked in the context of specially developed computer microworlds with a teacher/researcher for approximately 45 min a week for 50 weeks over the 2-year period. The children's construction of multiplicative fractional schemes was investigated in a retrospective analysis of each of the 50 videotaped teaching episodes. Four distinct modifications of the children's fractional schemes were discerned that contributed to their construction of the RNA. The investigation suggested that the operations and unit types associated with the children's whole-number sequences did not interfere with the reorganization of their fractional schemes but rather contributed to those schemes. The reorganization involved an integration of their whole-number knowledge with their fractional schemes whereby whole-number division was regarded as the same as multiplication by the reciprocal fraction.     

Designing Effective Simulation-based Decision Support Systems: An Empirical Assessment of Three Types of Decision Support Systems

This paper addresses the design of effective simulation-based decision support systems (DSS). An experiment was conducted using three different DSS tools developed around three types of simulation model—traditional, conventional visual interactive simulation (VIS), and ‘paired-systems’ VIS. Subjects were asked to perform a decision making task and their performance was evaluated. Subjects who used the DSS based on a ‘paired systems’ VIS model were found to be both the most effective and the most efficient at the problem-solving task. Subjects provided with the DSS based upon a conventional VIS model were found to be more effective at the task than the group provided with the traditional simulation-based DSS.     

From Fractions to Rational Numbers of Arithmetic: A Reorganization Hypothesis

Nathan and Arthur, 2 children in a 3-year teaching experiment on children's construction of the rational numbers of arithmetic (RNA), developed their operations for multiplying, dividing, and simplifying fractions over the last 2 years (Grades 4 and 5) of the experiment. The 2 children worked in the context of specially developed computer microworlds with a teacher/researcher for approximately 45 min a week for 50 weeks over the 2-year period. The children's construction of multiplicative fractional schemes was investigated in a retrospective analysis of each of the 50 videotaped teaching episodes. Four distinct modifications of the children's fractional schemes were discerned that contributed to their construction of the RNA. The investigation suggested that the operations and unit types associated with the children's whole-number sequences did not interfere with the reorganization of their fractional schemes but rather contributed to those schemes. The reorganization involved an integration of their whole-number knowledge with their fractional schemes whereby whole-number division was regarded as the same as multiplication by the reciprocal fraction.     

CHILDREN'S PROPORTIONAL REASONING AND TENDENCY FOR AN ADDITIVE STRATEGY: THE ROLE OF MODELS

In this paper we report on 10 –14 year old children's strategies while solving two versions of ratio and proportion tasks: one ‘with models’ thought to facilitate proportional reasoning and one ‘without’. Rasch methodology was used to develop ‘with’ and ‘without models’ test versions which were given to a linked sample involving 673 children. We examine the pupils’ additive errors, their effect on ratio reasoning and how contingent on ‘model’ presentation this is. First, we provide a single scale on which pupils, item-difficulty and additive errors can be located. We then provide a new scale constructed from the error prone items, which we name the ‘tendency for additive strategy’. The measurement data is supported by qualitative data showing that the presence of ‘models’ can sometimes affect children's strategies, both positively and negatively but rarely makes a significant measurement difference on this, untutored, sample.     

Additive relations and function tables

We present work with a second grade classroom where we carried out a teaching experiment that attempted to bring out the algebraic character of arithmetic. In this paper, we specifically illustrate our work with the second graders on additive relations, through the children’s work with function tables. We explore the different ways in which the children represented the information of a problem in the form of a self-designed function table. We argue that the choices children make about the kind of information to represent or not, as well as the way in which they constructed their tables, highlight some of the issues that children may find relevant in their construction of function tables. This open-ended format pointed to how they were understanding and appropriating tables into their thinking about additive relations.