Quantitative Algebraic Reasoning

We develop a quantitative analogue of equational reasoning which we call quantitative algebra. We define an equality relation indexed by rationals: a = ϵ b which we think of as saying that "a is approximately equal to b up to an error of ϵ ". We have 4 interesting examples where we have a quantitative equational theory whose free algebras correspond to well known structures. In each case we have finitary and continuous versions. The four cases are: Hausdorff metrics from quantitive semilattices; p-Wasserstein metrics (hence also the Kantorovich metric) from barycentric algebras and also from pointed barycentric algebras and the total variation metric from a variant of barycentric algebras

Hidden-Markov Program Algebra with iteration

We use Hidden Markov Models to motivate a quantitative compositional semantics for noninterference-based security with iteration, including a refinement- or "implements" relation that compares two programs with respect to their information leakage; and we propose a program algebra for source-level reasoning about such programs, in particular as a means of establishing that an "implementation" program leaks no more than its "specification" program. This joins two themes: we extend our earlier work, having iteration but only qualitative, by making it quantitative; and we extend our earlier quantitative work by including iteration. We advocate stepwise refinement and source-level program algebra, both as conceptual reasoning tools and as targets for automated assistance. A selection of algebraic laws is given to support this view in the case of quantitative noninterference; and it is demonstrated on a simple iterated password-guessing attack

One Teacher\u27s Transformation of Practice Through the Development of Covariational Thinking and Reasoning in Algebra : A Self-Study

CCSSM (2010) describes quantitative reasoning as expertise that mathematics educators should seek to develop in their students. Researchers must then understand how to develop covariational reasoning. The problem is that researchers draw from students’ dialogue as the data for understanding quantitative relationships. As a result, the researcher can only conceive the students’ reasoning. The objective of using the self-study research methodology is to examine and improve existing teaching practices. To improve my practice, I reflected upon the implementation of my algebra curriculum through a hermeneutics cycle of my personal history and living educational theory. The critical friend provoked through dialogues and narratives the reconceptualization of my smooth covariational reasoning from a “transformational perspective” to a “solving algebraic equations” perspective. This study showed that by creating images in motion, graphs, or algebraic representation, I recognized the importance of students’ cognitive development in the conceptual embodied and proceptual symbolic worlds. The results presented the transformation of my teaching practices by building new algebraic connections. By using these findings, researchers can gain additional understanding as to how they can transform their teaching practices

Exploring the episodic structure of algebra story problem solving

This paper analyzes the quantitative and situational structure of algebra story problems, uses these materials to propose an interpretive framework for written problem-solving protocols, and then presents an exploratory study of the episodic structure of algebra story problem solving in a sizable group of mathematically competent subjects. Analyses of written protocols compare the strategic, tactical, and conceptual content of solution attampts, looking within these attempts at the interplay between problem comprehension and solution. Comprehension and solution of algebra story problems are complimentary activities, giving rise to a succession of problem solving episodes. While direct algebraic problem solving is sometimes effective, results suggest that the algebraic formalism may be of little help in comprehending the quantitative constraints posed in a problem text. Instead, competent problem solvers often reason within the situational context presented by a story problem, using various forms of "model-based reasoning" to identify, pursue, and verify quantitative constraints required for solution. The paper concludes by discussing the implications of these findings for acquiring mathematical concepts (e.g., related linear functions) and for supporting their acquisition through instruction

Exploring the episodic structure of algebra story problem solving

This paper analyzes the quantitative and situational structure of algebra story problems, uses these materials to propose an interpretive framework for written problem-solving protocols, and then presents an exploratory study of the episodic structure of algebra story problem solving in a sizable group of mathematically competent subjects. Analyses of written protocols compare the strategic, tactical, and conceptual content of solution attempts, looking within these attempts at the interplay between problem comprehension and solution. Comprehension and solution of algebra story problems are complimentary activities, giving rise to a succession of problem solving episodes. While direct algebraic problem solving is sometimes effective, results suggest that the algebraic formalism may be of little help in comprehending the quantitative constraints posed in a problem text. Instead, competent problem solvers often reason within the situational context presented by a story problem, using various forms of "model-based reasoning" to identify, pursue, and verify quantitative constraints required for solution. The paper concludes by discussing the implications of these findings for acquiring mathematical concepts (e.g., related linear functions) and for supporting their acquisition through instruction

Design and Implementation of Conchoid and Offset Processing Maple Packages

Annals of Mathematics and Artificial Intelligence presents a range of topics of concern to scholars applying quantitative, combinatorial, logical, algebraic and algorithmic methods to diverse areas of Artificial Intelligence, from decision support, automated deduction, and reasoning, to knowledge-based systems, machine learning, computer vision, robotics and planning

Karakteristik Penalaran Kuantitatif siswa dalam menyelesaikan Masalah Matematika ditinjau dari Jenjang sekolah

There are various types of thinking skills that can be developed by studying mathematics, one of which is the algebraic thinking skills. One of the abilities in algebraic thinking is quantitative reasoning ability. Quantitative reasoning ability is a fundamental ability for students to be successful in learning mathematics. However, the facts on the field show that there are still many students who have low mathematical reasoning abilities. The purpose of this study was to obtain an overview and characteristics of students' mathematical reasoning abilities, especially students' quantitative reasoning. This research uses a qualitative approach, while this type of research is a phenomenological study. Phenomenology is the study that describes what a person receives, feels and knows in his consciousness about the experiences he has experienced. The subjects of this study were students of class VI elementary school and class VII junior high school. Based on the results of the research and discussion, it can be concluded that there are similarities and differences in the characteristics of quantitative reasoning between students at the final level of elementary school and students at the initial level of junior high school.Terdapat berbagai jenis kemampuan berpikir yang dapat dikembangkan dengan mempelajari matematika, salah satunya adalah kemampuan berpikir aljabar. Salah satu kemampuan dalam berpikir aljabar adalah kemampuan penalaran kuantitatif. Kemampuan penalaran kuantitatif adalah kemampuan yang fundamental agar siswa sukses dalam pembelajaran matematika. Namun demikian, fakta di lapangan menunjukan bahwa masih banyak siswa yang memiliki kemampuan penalaran matematis yang rendah. Tujuan penelitian ini adalah untuk memperoleh gambaran dan karakteristik tentang kemampuan penalaran matematis siswa khususnya penalaran kuantitatif peserta didik. Penelitian ini menggunakan pendekatan kualitatif, sementara jenis penelitian ini adalah studi fenomenologis. Fenomenologi adalah studi yang menggambarkan apa yang seseorang terima, rasakan dan ketahui di dalam kesadarannya tentang pengalaman yang dialaminya. Subjek penelitian ini adalah siswa kelas VI sekolah dasar dan kelas VII sekolah menengah pertama. Berdasarkan hasil penelitian dan pembahasan, diperoleh kesimpulan terdapat persamaan dan perbedaan karakteristik penalaran kuantitatif antara siswa pada level akhir jenjang sekolah dasar dengan siswa pada level awal jenjang sekolah menengah pertama

Sources of Mature Students’ Difficulties in Solving Different Types of Word Problems in Mathematics

There are many different types of research done on algebra learning. In particular, word problems have been used to analyze students’ thought process and to identify difficulties in algebraic thinking. In this thesis, we show the importance of quantitative reasoning in problem solving. We gave 14 mature students, who were re-taking an introductory course on algebra, four word problems of different types to solve: a connected problem, a disconnected problem, a problem with contradictory data and a problem where students were asked to assess the correctness of a fictional solution. In selecting these types of problems we have drawn on the research of Sylvine Schmidt and Nadine Bednarz on the difficulties of passing from arithmetic to algebra in mathematical problem solving. We present the students’ solutions and a detailed analysis of these solutions, seeking to identify the sources of the difficulty these students had in producing correct solutions. We sought these sources in the defects of quantitative reasoning, arithmetic mistakes, and algebraic mistakes. The attention to quantitative reasoning was inspired by the research of Pat Thompson and Stacey Brown. Defects of quantitative reasoning appeared to be an important reason why the students massively failed to solve the problems correctly, more important than their lack of technical algebraic skills. Therefore, teaching procedures and algebraic technical skills is not enough for students to develop problem solving skills. There should be a focus on developing students’ quantitative reasoning. Students need to have a good understanding of relations between quantities. Defects of quantitative reasoning create obstacles that prevent mature students from successfully solving any type of word problem

Effects of Geometer’s Sketchpad on Algebraic Reasoning Competency amongst Students in Malaysia

This study investigates the impact of incorporating Geometer\u27s Sketchpad into GSP-based instruction on the algebraic reasoning proficiency of students. The research outlines the application of GSP-based instruction, utilizing Geometer\u27s Sketchpad in Mathematics education, with a specific focus on solving quadratic problems. A quasi-experimental design, involving non-equivalent pre-test and post-test assessments, was conducted on 60 second-grade students at a private secondary school in Kuala Lumpur. The experimental group, consisting of 30 students, utilized Geometer\u27s Sketchpad in their learning, while the control group, comprising 30 students, received traditional instruction. Data analysis was performed using ANATES 4 and SPSS 25.0 software, incorporating inferential statistics such as paired t-tests and one-way ANCOVA for quantitative data analysis. Both groups underwent an initial pre-test assessment. The research findings reveal a significant disparity in algebraic reasoning proficiency between the two groups, indicating substantial improvement after the intervention. Consequently, the integration of Geometer\u27s Sketchpad in GSP-based instruction contributed to the enhancement of students\u27 algebraic reasoning skills, particularly in quadratic problem-solving. In conclusion, this study underscores the potential of Geometer\u27s Sketchpad as an instructional intervention, urging mathematics educators to contemplate its incorporation as an alternative teaching tool