Undergraduate Students’ Combinatorial Proof of Binomial Identities

2021 ◽  
Vol 52 (5) ◽  
pp. 539-580
Author(s):  
Elise Lockwood ◽  
Zackery Reed ◽  
Sarah Erickson
Keyword(s):  
Undergraduate Students ◽  
Teaching Experiment ◽  
Combinatorial Proof ◽  
Representation System ◽  
Representation Systems ◽  
Symbolic Reasoning ◽  
Binomial Identities

Combinatorial proof serves both as an important topic in combinatorics and as a type of proof with certain properties and constraints. We report on a teaching experiment in which undergraduate students (who were novice provers) engaged in combinatorial reasoning as they proved binomial identities. We highlight ways of understanding that were important for their success with establishing combinatorial arguments; in particular, the students demonstrated referential symbolic reasoning within an enumerative representation system, and as the students engaged in successful combinatorial proof, they had to coordinate reasoning within algebraic and enumerative representation systems. We illuminate features of the students’ work that potentially contributed to their successes and highlight potential issues that students may face when working with binomial identities.

scholarly journals Novel Binary Signed-Digit Addition Algorithm for FPGA Implementation

2019 ◽  
Vol 29 (09) ◽  
pp. 2050136
Author(s):  
Yuuki Tanaka ◽  
Yuuki Suzuki ◽  
Shugang Wei
Keyword(s):  
Field Programmable Gate Array ◽  
High Speed ◽  
Number System ◽  
Logic Circuit ◽  
Fpga Implementation ◽  
Number Representation ◽  
Representation System ◽  
Representation Systems ◽  
Field Programmable ◽  
Gate Array

Signed-digit (SD) number representation systems have been studied for high-speed arithmetic. One important property of the SD number system is the possibility of performing addition without long carry chain. However, many numbers of logic elements are required when the number representation system and such an adder are realized on a logic circuit. In this study, we propose a new adder on the binary SD number system. The proposed adder uses more circuit area than the conventional SD adders when those adders are realized on ASIC. However, the proposed adder uses 20% less number of logic elements than the conventional SD adder when those adders are realized on a field-programmable gate array (FPGA) which is made up of 4-input 1-output LUT such as Intel Cyclone IV FPGA.

Situation-Theoretic Account of Valid Reasoning with Venn Diagrams

1996 ◽  
Author(s):  
Sun-Joo Shin
Keyword(s):  
Semantic Analysis ◽  
Formal System ◽  
Venn Diagram ◽  
Formal Proofs ◽  
Representation System ◽  
Venn Diagrams ◽  
Linguistic Representation ◽  
Deductive Systems ◽  
Representation Systems ◽  
Mathematical Tradition

Venn diagrams are widely used to solve problems in set theory and to test the validity of syllogisms in logic. Since elementary school we have been taught how to draw Venn diagrams for a problem, how to manipulate them, how to interpret the resulting diagrams, and so on. However, it is a fact that Venn diagrams are not considered valid proofs, but heuristic tools for finding valid formal proofs. This is just a reflection of a general prejudice against visualization which resides in the mathematical tradition. With this bias for linguistic representation systems, little attempt has been made to analyze any nonlinguistic representation system despite the fact that many forms of visualization are used to help our reasoning. The purpose of this chapter is to give a semantic analysis for a visual representation system—the Venn diagram representation system. We were mainly motivated to undertake this project by the discussion of multiple forms of representation presented in Chapter I More specifically, we will clarify the following passage in that chapter, by presenting Venn diagrams as a formal system of representations equipped with its own syntax and semantics:. . . As the preceding demonstration illustrated, Venn diagrams provide us with a formalism that consists of a standardized system of representations, together with rules of manipulating them. . . . We think it should be possible to give an informationtheoretic analysis of this system, . . . . In the following, the formal system of Venn diagrams is named VENN. The analysis of VENN will lead to interesting issues which have their ana logues in other deductive systems. An interesting point is that VENN, whose primitive objects are diagrammatic, not linguistic, casts these issues in a different light from linguistic representation systems. Accordingly, this VENN system helps us to realize what we take for granted in other more familiar deductive systems. Through comparison with symbolic logic, we hope the presentation of VENN contributes some support to the idea that valid reasoning should be thought of in terms of manipulation of information, not just in terms of manipulation of linguistic symbols.

scholarly journals Integrating Data Rescue into the Classroom

2018 ◽  
Vol 99 (9) ◽  
pp. 1757-1764 ◽  
Author(s):  
Ciara Ryan ◽  
Catriona Duffy ◽  
Ciaran Broderick ◽  
Peter W. Thorne ◽  
Mary Curley ◽  
...  
Keyword(s):  
Learning Outcomes ◽  
Undergraduate Students ◽  
Scientific Community ◽  
Daily Precipitation ◽  
Learning Experience ◽  
Teaching Experiment ◽  
Precipitation Data ◽  
Academic Institutions ◽  
Daily Precipitation Data ◽  
Temporal Extent

AbstractOver much of the globe, the temporal extent of meteorological records is limited, yet a wealth of data remains in paper or image form in numerous archives. To date, little attention has been given to the role that students might play in efforts to rescue these data. Here we summarize an ambitious research-led, accredited teaching experiment in which undergraduate students successfully transcribed more than 1,300 station years of daily precipitation data and associated metadata across Ireland over the period 1860–1939. We explore i) the potential for integrating data rescue activities into the classroom, ii) the ability of students to produce reliable transcriptions and, iii) the learning outcomes for students. Data previously transcribed by Met Éireann (Ireland’s National Meteorological Service) were used as a benchmark against which it was ascertained that students were as accurate as the professionals. Details on the assignment, its planning and execution, and student-aids used are provided. The experience highlights the benefits that can accrue for data rescue through innovative collaboration between national meteorological services and academic institutions. At the same time, students have gained valuable learning outcomes and firsthand understanding of the processes that underpin data rescue and analysis. The success of the project demonstrates the potential to extend data rescue in the classroom to other universities, thus providing both an enriched learning experience for the students and a lasting legacy to the scientific community.

scholarly journals A Study of Piano Timbre Teaching in the Context of Artificial Intelligence Interaction

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Cui Wei
Keyword(s):  
Artificial Intelligence ◽  
Undergraduate Students ◽  
Teaching Practice ◽  
Development Trend ◽  
Teaching Experiment ◽  
Experimental Results ◽  
Piano Teaching ◽  
Teaching Objectives ◽  
Depth Analysis ◽  
Software Modules

This paper provides an in-depth analysis and research on piano timbre teaching in the context of artificial intelligence interaction, a bold vision of piano teaching, proposes a feasible solution in terms of teaching modules in intelligent piano teaching for senior teachers, and proposes an implementation path for the integration of intelligent piano and piano teaching from the four main blocks of piano teaching. Based on the multiplicative harmonic model of monophonic signal, combined with the variability of timbre characteristics, an audio synthesis model with editable timbre is proposed, and the experimental results show that editing the timbre parameters in the model can realize timbre modification, and the synthesized timbre conforms to the piano timbre characteristics. Based on the timbre analysis and the timbre synthesis model, a piano timbre library generation system is designed. The detailed design of the software modules such as audio file reading and writing, audio information analysis, timbre parameter acquisition, timbre synthesis, and simulated performance is given. The system can generate piano timbre libraries of different qualities flexibly and meet the requirements of timbre realism. The teaching experiment designed for teaching practice from solo teaching, and the practice target is first-year undergraduate students in the university, and the practice period is six weeks, and finally, the feasibility of intelligent piano teaching application is analysed by combining the experimental results. Through the teaching objectives, teaching content, and teaching methods, teaching environment reflects intelligent piano teaching to make up for the limitations of traditional piano teaching. Analyse the development trend of intelligent piano teaching in the context of artificial intelligence interaction, and explore the value of intelligent piano teaching.

scholarly journals Una Experiencia de Inclusión Educativa en el Aula de Expresión Corporal con Alumnado Universitario (An Educational Inclusion Experience in a Body Expression course with undergraduate students)

Retos ◽  
2019 ◽  
pp. 702-705
Author(s):  
Alejandro Muñoz Moreno ◽  
Eulisis Smith ◽  
Michelle Matos Duarte
Keyword(s):  
Students With Disabilities ◽  
Undergraduate Students ◽  
Teaching And Learning ◽  
Positive Impact ◽  
Teaching Experiment ◽  
Nonverbal Expression ◽  
Inclusive Learning ◽  
Discapacidad Intelectual ◽  
Educational Inclusion ◽  
Sports Sciences

Las últimas investigaciones en el ámbito de la Expresión Corporal (EC) como parte del currículum del área de Educación Física constatan la mejora de la calidad de enseñanza y aprendizaje en el aula (Archilla & Pérez, 2017; Coterón & Sánchez, 2013). Especialmente interesante son aquellas que utilizan el contexto de esta materia para crear entornos de aprendizajes inclusivos, pues al favorecerse las relaciones sociales, la expresión corporal y no verbal es posible contribuir a la eliminación de las barreras que impiden el aprendizaje y participación de todos los estudiantes (Booth & Ainscow, 2002). La posibilidad de vivenciar una clase universitaria plenamente inclusiva puede llegar a tener una incidencia positiva sobre el autoconcepto (Navajas, 2016) y sociabilización (Pérez & Urdampilleta, 2012). En este contexto, el objetivo planteado ha sido el de ofrecer una experiencia didáctica inclusiva en Expresión Corporal para el segundo curso del Grado de CAFyD. Participaron treinta y cinco alumnos de la asignatura de Actividades Físicas Artístico-Expresivas y siete alumnos con discapacidad intelectual durante cinco sesiones en las diferentes temáticas de la asignatura. Esta primera experiencia ha sido beneficiosa para atestiguar que es posible incluir al alumnado con discapacidad intelectual en las aulas universitarias y aprovechar los beneficios que ello conlleva para el resto del alumnado.Abstract: The latest research in the field of Corporal Expression (EC) as a part of the curriculum in the area of Physical Education area confirms an improvement in the quality of teaching and learning in the classroom (Archilla & Pérez, 2017; Coterón & Sánchez, 2013). Those studies using the context of this subject to create inclusive learning environments are of particular interest, for they improve social relationships, and corporal and nonverbal expression, making a contribution to the elimination of learning and participation barriers among students (Booth & Ainscow, 2002). The chance to experience a fully inclusive university class can have a positive impact on self-concept (Navajas, 2016) and socialization (Pérez & Urdampilleta, 2012). In this context, we carried out a teaching experiment during the second year of the Physical Activity and Sports Sciences (CAFyD) bachelor degree. Thirty-five students taking the course of Artistic-Expressive Physical Activities and seven students with disabilities participated in five sessions focused on different subject topics. This initial experience has been positive, showing that it is possible to include students with intellectual disability in university classrooms and take advantage of the benefits this brings to the rest of the students.

scholarly journals Caracterización del significado de múltiplo por maestros en formación

2016 ◽  
Vol 10 (2) ◽  
pp. 111-134
Author(s):  
Ángel López ◽  
Encarnación Castro ◽  
María C. Cañadas
Keyword(s):  
Elementary Teachers ◽  
Elementary School Teachers ◽  
Prospective Teachers ◽  
Teaching Experiment ◽  
Practice Session ◽  
School Teachers ◽  
Prospective Elementary Teachers ◽  
Structure Representation ◽  
Representation Systems ◽  
Multiple Product

Este trabajo forma parte de una investigación centrada en la divisibilidad en Z+. Los sujetos participantes son maestros en formación. Uno de los objetivos de la investigación consiste en caracterizar los significados que muestran los maestros en formación sobre el concepto de múltiplo. Este artículo recoge los resultados obtenidos en relación con dicho objetivo. Analizamos las producciones escritas de 37 maestros en formación obtenidas en una sesión práctica de aula, diseñada y desarrollada en el contexto de un experimento de enseñanza. Realizamos la caracterización de los significados a través de los elementos del análisis didáctico: estructura conceptual, sistemas de representación y fenomenología. Los maestros en formación mostraron mayoritariamente tres significados de múltiplo: producto, relación y dividendo en una división exacta. Characterizing the meaning of multiple by pre-service elementary school teachers This paper is part of a wider study focused on divisibility. Participants were prospective elementary teachers. One of the aims of the research is to characterize the meanings of multiple shown by prospective teachers. In this paper, we present the results concerning this aim. We analyse the productions of 37 prospective elementary teachers collected in a practice session, designed and developed in the context of a teaching experiment. We characterize the meanings through the following elements of the didactic analysis: conceptual structure, representation systems and phenomenology. Prospective teachers showed mostly three meanings of multiple: product, relationship and dividend in an exact division.Handle: http://hdl.handle.net/10481/39495WOS-ESCI

Two Undergraduate Students' Reinvention of the Multiplication Principle

2019 ◽  
Vol 50 (3) ◽  
pp. 225-267 ◽  
Author(s):  
Elise Lockwood ◽  
Branwen Purdy
Keyword(s):  
Undergraduate Students ◽  
Teaching Experiment ◽  
Fundamental Aspect ◽  
Future Research ◽  
Combinatorial Enumeration ◽  
Counting Problems ◽  
Guided Reinvention ◽  
Combinatorial Formulas ◽  
Research Questions ◽  
Composite Outcomes

The multiplication principle (MP) is a fundamental aspect of combinatorial enumeration, serving as an effective tool for solving counting problems and underlying many key combinatorial formulas. In this study, the authors used guided reinvention to investigate 2 undergraduate students' reasoning about the MP, and they sought to answer the following research questions: How do students come to understand and make sense of the MP? Specifically, while a pair of students reinvented a statement of the MP, how did they attend to and reason about key mathematical features of the MP? The students participated in a paired 8-session teaching experiment during which they progressed from a nascent to a sophisticated statement of the MP. Two key mathematical features emerged for the students through this process, including independence and distinct composite outcomes, and we discuss ways in which these ideas informed the students' reinvention of the statement. In addition, we present potential implications and directions for future research.

A framework for design verification and evaluation systems

1989 ◽  
Vol 3 (2) ◽  
pp. 71-84 ◽  
Author(s):  
Steven M. Medoff ◽  
Michael S. Register ◽  
Mark W. Swartwout
Keyword(s):  
Knowledge Representation ◽  
Evaluation System ◽  
Design Verification ◽  
Representation System ◽  
Knowledge Engineer ◽  
Evaluation Systems ◽  
Representation Systems ◽  
Evaluation Problem ◽  
Knowledge Representation System ◽  
Knowledge Representation Systems

Although current efforts have indicated the need for design verification and evaluation systems, little has been done to develop knowledge representation systems for implementing this class of expert systems. In this paper, we describe CANDLE, a knowledge representation system built to solve design verification and evaluation problems. We define the class of design verification and evaluation systems and the functionality required of knowledge representation systems used to develop these systems. We present an example design verification and evaluation problem, illustrating how this problem imposes these functionality requirements on the knowledge representation system used. In the remainder of the paper we discuss CANDLE, a knowledge representation system created to develop design verification and evaluation systems. We show that CANDLE meets the knowledge representation requirements for this class of systems. We also present some of CANDLE'S functionality that exceeds the defined knowledge representation requirements and provides a knowledge engineer with the capability to expand the functionality of a design verification and evaluation system.

scholarly journals Learning sparse representations on the sphere

2019 ◽  
Vol 621 ◽  
pp. A73
Author(s):  
F. Sureau ◽  
F. Voigtlaender ◽  
M. Wust ◽  
J.-L. Starck ◽  
G. Kutyniok
Keyword(s):  
Learning Approach ◽  
Data Sets ◽  
Complex Structures ◽  
Representation System ◽  
Scientific Application ◽  
The Past ◽  
Representation Systems ◽  
Data Representations ◽  
Adaptive Approaches ◽  
Parametrized Family

Many representation systems on the sphere have been proposed in the past, such as spherical harmonics, wavelets, or curvelets. Each of these data representations is designed to extract a specific set of features, and choosing the best fixed representation system for a given scientific application is challenging. In this paper, we show that one can directly learn a representation system from given data on the sphere. We propose two new adaptive approaches: the first is a (potentially multiscale) patch-based dictionary learning approach, and the second consists in selecting a representation from among a parametrized family of representations, the α-shearlets. We investigate their relative performance to represent and denoise complex structures on different astrophysical data sets on the sphere.

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