Symbolic Notation Research Articles

Comparing Symbolic and Nonsymbolic Number Lines: Consistent Effects of Notation Across Output Measures.

The mental number line (MNL) is a popular metaphor for magnitude representation in numerical cognition. Its shape has frequently been reported as being nonlinear, based on nonlinear response functions in magnitude estimation. We investigated whether this shape reflects a phenomenon of the mapping from stimulus to internal magnitude representation or of the mapping from internal representation to response. In five experiments, participants (total N = 66) viewed stimuli that represented numerical magnitude either in a symbolic notation (i.e., Arabic digits) or in a nonsymbolic notation (i.e., clouds of dots). Participants estimated these magnitudes by either adjusting the position of a mark on a ruler-like response bar (nonsymbolic response) or by typing the corresponding number on a keyboard (symbolic response). Responses to symbolic stimuli were markedly different from responses to nonsymbolic stimuli, in that they were mostly powershaped. We investigated whether the nonlinearity could be explained by effects of previous trials, but such effects were (a) not strong enough to explain the nonlinear responses and (b) existed only between trials of the same input notation, suggesting that the nonlinearity is due to input mappings. Introducing veridical feedback improved the accuracy of responses, thereby showing a calibration based on the feedback. However, this calibration persisted only temporarily, and responses to nonsymbolic stimuli remained nonlinear. Overall, we conclude that the nonlinearity is a phenomenon of the mapping from nonsymbolic input format to internal magnitude representation and that the phenomenon is surprisingly robust to calibration.

Deepening students understanding of triangle topic through ‘application’ component of ELPSA (Experience, Language, Pictorial, Symbol and Application) framework

ELPSA (Experience, Language, Pictorial, Symbol and Application) is a learning framework that was introduced as a sequence of a learning process which presents mathematical ideas through lived experience, mathematical conversation, visual stimuli, symbolic notation and the application of applied knowledge. Employing the “application” component of ELPSA framework in the mathematics classroom emphasizes how the understanding of certain material can be applied to a new situation. This study investigates the learning process during the class activity about the triangle. The study was conducted in one of the junior high school in Central Lombok, West Nusa Tenggara. The data were collected from teaching and learning videos and students’ work during the learning processes in the classroom. We transcribed and analysed the video as well as looked at the students’ works. The result shows that the activity of composing a 2D shape from various types of triangle support students in deepening their understanding about the types of triangles. Also, the activity of composing 2D shape can be a promising foreground for students in their following learning about the area of the composite figure.

Synthesis of regular expression problems and solutions

ABSTRACT Formal languages and automata (FLA) theory is perceived by many as one of the hardest topics to teach or learn at the undergraduate level, due to the abstract nature of its contents; often containing proofs, lots of symbolic notations, and specifications of abstract computing machines that are difficult for many students to relate with. One way of approaching this problem is to administer numerous practice problems to students to aid learning. In this paper, we present: definitions of formal rules (using context-free grammar notations) for the synthesis of practice problems and solutions in a sub-domain of formal language theory – regular expressions, new algorithms for implementing these rules with iterations of generated problem instances and associated solutions. Generated problems are then embedded in predefined, short text response and multiple-choice question templates. The implementation of this technique produced numerous problems and solutions that can be used as practice, test or examination questions to support the teaching of regular expressions as a topic in FLA at the undergraduate level.

A multimodal approach to visual thinking: the scientific sketchnote

There is a growing interest in the use of visual thinking techniques for promoting conceptual thinking in problem solving tasks as well as for reducing the complexity of ideas expressed in scientific and technical formats. The products of visual thinking, such as sketchnotes, graphics and diagrams, consist of ‘multimodal complexes’ that combine language, images, mathematical symbolism and various other semiotic resources. This article adopts a social semiotic perspective, more specifically a Systemic Functional Multimodal Discourse Analysis approach, to study the underlying semiotic mechanisms through which visual thinking makes complex scientific content accessible. To illustrate the approach, the authors analyse the roles of language, images, and mathematical graphs and symbolism in four sketchnotes based on scientific literature in physics. The analysis reveals that through the process of resemiotization, where meanings are transformed from one semiotic system to another, the abstractness of specialized discourses such as physics and mathematics is reduced by multimodal strategies which include reformulating the content in terms of entities which participate in observable (i.e. tangible) processes and enhancing the reader/viewer’s engagement with the text. Moreover, the compositional arrangement creates clear stages in the development of the ideas and arguments that are presented. In this regard, visual thinking is a form of cultural communication through which abstract ideas are translated and explained using a multimodal outline or summary of essential parts by adapting resources (e.g. linguistic resources and mathematical graphs), using new resources (e.g. stick figures and other simple schematic drawings) and maintaining others from the original text (e.g. mathematical symbolic notation), resulting in a congruent (or concrete) depiction of abstract concepts and ideas for a non-specialist audience.

An insider’s view of a mathematics learning disability: Compensating to gain access to fractions

ABSTRACTResearch on mathematics learning disability (i.e., dyscalculia) tends to adopt a deficit model of the learner and quantify the ways in which these students are deficient. In this study we adopt a radically different approach. We understand disabilities to result from issues of access, and we explore mathematics learning disability (MLD) by leveraging the expertise of an individual with first-hand experience with this disability. Dylan (second author) has a unique perspective because, in addition to having an MLD, she also majored in statistics in college. Through videotaped interviews we documented the issues of access Dylan encountered and the ways in which she compensated to gain access. In this article we focus specifically on the topic of fractions, and identify Dylan’s issues of access with standard representations of fractions, including symbolic notation, area models, and fraction manipulatives. These issues of access mirrored the difficulties documented in other students with MLD, but were not similarly problematic for Dylan. This research provides novel insight into the difficulties experienced by students with MLD, reframes deficits in terms of issues of access, and demonstrates how an individual with an MLD may learn to compensate.

A Comparative Study of Frequent and Maximal Periodic Pattern Mining Algorithms in Spatiotemporal Databases

Detecting regular and efficient cyclic models is the demanding activity for data analysts due to unstructured, vigorous and enormous raw information produced from web. Many existing approaches generate large candidate patterns in the occurrence of huge and complex databases. In this work, two novel algorithms are proposed and a comparative examination is performed by considering scalability and performance parameters. The first algorithm is, EFPMA (Extended Regular Model Detection Algorithm) used to find frequent sequential patterns from the spatiotemporal dataset and the second one is, ETMA (Enhanced Tree-based Mining Algorithm) for detecting effective cyclic models with symbolic database representation. EFPMA is an algorithm grows models from both ends (prefixes and suffixes) of detected patterns, which results in faster pattern growth because of less levels of database projection compared to existing approaches such as Prefixspan and SPADE. ETMA uses distinct notions to store and manage transactions data horizontally such as segment, sequence and individual symbols. ETMA exploits a partition-and-conquer method to find maximal patterns by using symbolic notations. Using this algorithm, we can mine cyclic models in full-series sequential patterns including subsection series also. ETMA reduces the memory consumption and makes use of the efficient symbolic operation. Furthermore, ETMA only records time-series instances dynamically, in terms of character, series and section approaches respectively. The extent of the pattern and proving efficiency of the reducing and retrieval techniques from synthetic and actual datasets is a really open & challenging mining problem. These techniques are useful in data streams, traffic risk analysis, medical diagnosis, DNA sequence Mining, Earthquake prediction applications. Extensive investigational outcomes illustrates that the algorithms outperforms well towards efficiency and scalability than ECLAT, STNR and MAFIA approaches.

A linguistic analysis of the sample numeracy skills test items for pre-service teachers issued by the Australian Council for Educational Research (ACER)

In 2015, the Australian Council for Educational Research (ACER) was tasked with developing literacy and numeracy skills testing for pre-service teachers. All undergraduate and postgraduate trainee teachers are now required to pass these literacy and numeracy tests at some stage on their journey to becoming a teacher; for commencing students from 2017 in all states (except New South Wales), successfully passing these tests is a requirement for graduation. Following this announcement, ACER released a number of sample test items for both literacy and numeracy. This study used multimodal analysis to study the nature of the language, symbolic notation and visual representations in the sample test items for numeracy, with a focus on the linguistic complexity of the questions. The findings suggest that the types of linguistic constructions found in the assessment questions create significant challenges in terms of reading and comprehension which could impact on how the students engage with the numeracy assessment items.

Do we have a sense for irrational numbers?

Number sense requires, at least, an ability to assess magnitude information represented by number symbols. Most educated adults are able to assess magnitude information of rational numbers fairly quickly, including whole numbers and fractions. It is to date unclear whether educated adults without training are able to assess magnitudes of irrational numbers, such as the cube root of 41. In a computerized experiment, we asked mathematically skilled adults to repeatedly choose the larger of two irrational numbers as quickly as possible. Participants were highly accurate on problems in which reasoning about the exact or approximate value of the irrational numbers’ whole number components (e.g., 3 and 41 in the cube root of 41) yielded the correct response. However, they performed at random chance level when these strategies were invalid and the problem required reasoning about the irrational number magnitudes as a whole. Response times suggested that participants hardly even tried to assess magnitudes of the irrational numbers as a whole, and if they did, were largely unsuccessful. We conclude that even mathematically skilled adults struggle with quickly assessing magnitudes of irrational numbers in their symbolic notation. Without practice, number sense seems to be restricted to rational numbers.

Formal Verification of Automotive Design in Compliance With ISO 26262 Design Verification Guidelines

ISO 26262, an automotive functional safety standard, ensures the functional safety of automotive systems by providing requirements and processes to govern the software lifecycle. Each functional system must be classified in terms of safety goals, risks, and automotive safety integrity level (ASIL: A, B, C, and D), with ASIL D, denoting the most stringent safety level. As the risk of the system increases, the ASIL level increases, and the standard highly recommends more stringent methods to ensure safety. ISO 26262 highly recommends that ASIL C and D-classified systems utilize semiformal and formal verification among other techniques to verify software unit design and implementation. In this paper, we compare industrial design verification steps of WatchDog Manager in an effort to be ASIL B-compliant with a proposed nondisruptive methodology to semiformally verify WatchDog Manager UML design via an automated formal framework backbone. This semiformal verification framework will allow automotive software to comply with ASILs C and D formal and semiformal unit design and implementation verification recommended guidelines in ISO 26262. Semiformal UML finite-state machines are automatically compiled into formal notations based on the Symbolic Analysis Laboratory formal notation. We capture requirements in the UML design and compile them automatically into theorems. Model checkers are run against the compiled formal model and theorems to detect counterexamples that violate the requirements in the UML model. We show that semi-formal verification of the design allows us to uncover issues that were detected in testing and production stages of ASIL B-compliant Watchdog Manager existing implementation.

Análisis De Representaciones Gráficas Cartesianas En Libros De Texto De Biología

Graphical representations Cartesian (RGC) are important communication tools in different areas of knowledge and in particular in the Population Biology (BP). The teaching of this area can be strongly affected by the difficulties that students encounter in the interpretation and use of the RGC. In this paper the analysis of RGC in ten textbooks and eight Secondary Education Teacher Training Biology, most widespread in the Province of Mendoza (Argentina), intended to compare and contrast their results is made. The analysis of character ontosemiotic involves both the text and the images, and uses a set of indicators associated with the semiotic concepts of relevance, appropriateness and didactic suitability, based on previous research. We conclude that the books of Teacher Education use a greater amount of RGC and generate as many positive interactions in their school books readers. They are therefore most flattering of inter-linguistic semiotic interactions and relationships between mathematical and biological language. However, they are a source of potential conflict, like school, largely because of lack of clarification of meanings of symbolic notations.

The role of visual representations in college students' understanding of mathematical notation.

Developing understanding of fractions involves connections between nonsymbolic visual representations and symbolic representations. Initially, teachers introduce fraction concepts with visual representations before moving to symbolic representations. Once the focus is shifted to symbolic representations, the connections between visual representations and symbolic notation are considered to be less useful, and students are rarely asked to connect symbolic notation back to visual representations. In 2 experiments, we ask whether visual representations affect understanding of symbolic notation for adults who understand symbolic notation. In a conceptual fraction comparison task (e.g., Which is larger, 5 / a or 8 / a?), participants were given comparisons paired with accurate, helpful visual representations, misleading visual representations, or no visual representations. The results show that even college students perform significantly better when accurate visuals are provided over misleading or no visuals. Further, eye-tracking data suggest that these visual representations may affect performance even when only briefly looked at. Implications for theories of fraction understanding and education are discussed.

Method of equipment graphic representation in the process of preproduction engineering

We consider the process of order determining of elements connection as a stage of preproduction engineering for the use in the design phase of production department organization structure. The initial data are: given design object (production department), operating conditions and options of equipment. Conventional approaches are considered; a new method is proposed notable for better visualization. The essence of the approach lies in replacement of the simplified contour shapes and symbolic notations on unified three-dimensional models (UTM). UTM is a three-dimensional model created from the real object - prototype by means of generalizing abstraction. The best known representative of the group is chosen as a prototype. Stepwise method of UTM creating is presented. Testing results of the method for two specific groups of technological equipment are given: screw-cutting equipment and industrial manipulators with an anthropomorphic arm. An original program module is developed that works in conjunction with additional means for information content enhancement according to the algorithm of creation of multi-component 3D-models. The proposed solution is intended for developers of automated preproduction engineering systems.

Higher-order Convolution Identities for Cauchy Numbers

Euler's famous formula written in symbolic notation as $(B_0+B_0)^n=-n B_{n-1}-(n-1)B_n$ was extended to $(B_{l_1}+\cdots+B_{l_m})^n$ for $m\ge 2$ and arbitrary fixed integers $l_1,\dots,l_m\ge 0$. In this paper, we consider the higher-order recurrences for Cauchy numbers $(c_{l_1}+\cdots+c_{l_m})^n$, where the $n$-th Cauchy number $c_n$ ($n\ge 0$) is defined by the generating function $x/\ln(1+x)=\sum_{n=0}^\infty c_n x^n/n!$. In special, we give an explicit expression in the case $l_1=\cdots=l_m=0$ for any integers $n\ge 1$ and $m\ge 2$. We also discuss the case for Cauchy numbers of the second kind $\widehat c_n$ in similar ways.

Subalternation and existence presuppositions in an unconventionally formalized canonical square of opposition

An unconventional formalization of the canonical (Aristotelian-Boethian) square of opposition in the notation of classical symbolic logic secures all but one of the canonical square’s grid of logical interrelations between four A-E-I-O categorical sentence types. The canonical square is first formalized in the functional calculus in Frege’s Begriffsschrift, from which it can be directly transcribed into the syntax of contemporary symbolic logic. Difficulties in received formalizations of the canonical square motivate translating I categoricals, ‘Some S is P’, into symbolic logical notation, not conjunctively as \({\exists x[Sx\wedge Px]}\), but unconventionally instead in an ontically neutral conditional logical symbolization, as \({\exists x[Sx\rightarrow Px]}\). The virtues and drawbacks of the proposal are compared at length on twelve grounds with the explicit existence expansion of A and E categoricals as the default strategy for symbolizing the canonical square preserving all original logical interrelations.

Non-symbolic and symbolic notations in simple arithmetic differentially involve intraparietal sulcus and angular gyrus activity

Addition problems can be solved by mentally manipulating quantities for which the bilateral intraparietal sulcus (IPS) is likely recruited, or by retrieving the answer directly from fact memory in which the left angular gyrus (AG) and perisylvian areas may play a role. Mental addition is usually studied with problems presented in the Arabic notation (4+2), and less so with number words (four+two) or dots (:: +·.). In the present study, we investigated how the notation of numbers influences processing during simple mental arithmetic. Twenty-five highly educated participants performed simple arithmetic while their brain activity was recorded with functional magnetic resonance imaging. To reveal the effect of number notation, arithmetic problems were presented in a non-symbolic (Dots) or symbolic (Arabic; Words) notation. Furthermore, we asked whether IPS processing during mental arithmetic is magnitude specific or of a more general, visuospatial nature. To this end, we included perception and manipulation of non-magnitude formats (Colors; unfamiliar Japanese Characters). Increased IPS activity was observed, suggesting magnitude calculations during addition of non-symbolic numbers. In contrast, there was greater activity in the AG and perisylvian areas for symbolic compared to non-symbolic addition, suggesting increased verbal fact retrieval. Furthermore, IPS activity was not specific to processing of numerical magnitude but also present for non-magnitude stimuli that required mental visuospatial processing (Color-mixing; Character-memory measured by a delayed match-to-sample task). Together, our data suggest that simple non-symbolic sums are calculated using visual imagery, whereas answers for simple symbolic sums are retrieved from verbal memory.

Arbitrary mechanical system description by a symbolic line

A single-line symbolic notation is proposed for description of an arbitrary multibody system. The kinematics is represented by a sequence of elementary transformations, each of those being marked by a reserved alphabetic character. Force and constraint links between the bodies are also defined by reserved characters. The parameters of the system, such as identifiers of degrees of freedom, inertia parameters and others, are assigned default names if not specified. However, user-defined names, parameters and functions can be placed instead if needed. The proposed description in its shortest form is suitable for academic purpose to identify only the essential properties of a multibody system. In an extended form, by explicit mentioning names of variables and parameters and other data like initial conditions, this description can serve as input data for a multibody analysis software. Lots of examples from the academic area and technical applications are given to show the applicability of the description.

Symbolic, Nonsymbolic and Conceptual: An Across-Notation Study on the Space Mapping of Numerals.

Previous studies suggested that there are interconnections between two numeral modalities of symbolic notation and nonsymbolic notation (array of dots), differences and similarities of the processing, and representation of the two modalities have both been found in previous research. However, whether there are differences between the spatial representation and numeral-space mapping of the two numeral modalities of symbolic notation and nonsymbolic notation is still uninvestigated. The present study aims to examine whether there are differences between the spatial representation and numeral-space mapping of the two numeral modalities of symbolic notation and nonsymbolic notation; especially how zero, as both a symbolic magnitude numeral and a nonsymbolic conceptual numeral, mapping onto space; and if the mapping happens automatically at an early stage of the numeral information processing. Results of the two experiments demonstrate that the low-level processing of symbolic numerals including zero and nonsymbolic numerals except zero can mapping onto space, whereas the low-level processing of nonsymbolic zero as a semantic conceptual numeral cannot mapping onto space, which indicating the specialty of zero in the numeral domain. The present study indicates that the processing of non-semantic numerals can mapping onto space, whereas semantic conceptual numerals cannot mapping onto space.

Compromise: An Alternative Solution Strategy for Contradiction Problems in the Butterfly Model

The introduction of symbols in mathematical problems eliminates the ambiguity of natural language by representing the core of the problems simply. Mathematical proofs use a chain of deduction. Despite the unique contribution of TRIZ professionals to innovative problem solving, many researchers’ effort about the development of TRIZ was based on induction rather than deduction. The Butterfly model for contradiction problem solving uses symbolic notations, and the model carries out the logical proofs of the validities of the conditional propositions by using truth tables. The Butterfly model provides the right solution strategy to solve the problems and reduces the solution space of the problems systematically. Many TRIZ researchers have argued that for a given contradiction problem, the contradiction resolution was much more innovative than any other compromise solutions which accommodated both conflicting requirements. In effect, they treated compromise-based solution strategies as naive methods. Unlike the viewpoint of the traditional TRIZ, this study suggests that the compromise is an effective alternative to solve contradictions. To prove this assertion, we firstly analyze a few contradiction problems with the Butterfly diagram. And then, we classify the types of contradiction relations and identify the conditions how the compromise approach resolves the contradiction systematically. In order to achieve the research objectives, this study explains the Butterfly model and its relevant cases.

Методика оценки доступного уровня символизации в общении у детей с нарушениями слуха и зрения и множественными нарушениями развития

In this article, we describe a possible way of assessing the level of the means of communication in children with impaired sight and hearing in the structure of multiple developmental disorders. The method is designed for examination of children with complex and multiple disabilities. It is advisable to use a methodology for assessing the level of language development in children who do not enjoy the expanded verbal or sign language, especially in cases where the surrounding is difficult to assess what treatment and the extent to which the child understands. The proposed character level evaluation procedure, the child can understand, can be used for initial diagnosis, as well as for the dynamic tracking of language development in the learning process. This version of the method may be used with children who have a substantive vision. Addressing age: school-aged children. Stimulus material technique selected on the basis of the model of symbolic stairs (M. Skelly), according to which the first level of symbolization is the real object (the demonstration of an object becomes a permanent sign of the beginning of the corresponding event, such as a cap – a symbol of a walk), and the highest level of symbolization becomes concept language word. It includes 15 real objects, 5 items of clothing, 5 pieces of dishes and 5 educational supplies, pictures of these objects, realistic color images, as well as the outline of the image. In addition, the plates are attached to the printed names of the same things and pictures on their sign symbols. The procedure of conducting the survey and recording results are discribed. Evaluation protocol allows you to determine what level of symbolic notation of items most successful child acts and what is the maximum level of symbolization it is available. Examples of description and analysis of the results are presented.

An efficient service composition using frequent service sequence patterns over extended web service architecture

In the current business scenario business processes are realised using a stack of simple services selected from service repository to satisfy the business requirement of an enterprise or organisation. The service composition methodologies in common build the composite service using composition tree-based functional and non-functional requirements of a business process. These methodologies do not focus in storing the composition tree and reusability. In addition the discovery and composition are repeated for all instances of similar requirement. This paper proposes an efficient service composition methodology over extended web service architecture to ensure reusability, quick response, efficient space utilisation and availability of services. The composition tree is transformed to symbolic notations and stored in service sequence registry. The frequent service sequence patterns are then extracted from service sequence registry and stored for future reference. These patterns are used for service allocation during composition. The contents of the service sequence registry and frequent service sequence pattern registry are replicated in different geographical location to ensure availability. The contents of replicated instances are consistently updated using push methodology. Experiments showed significant improvement of infrastructure of web service architecture in terms of availability, space requirement and time to allocate service for composition.