The purpose of this paper is to share with the mathematics community what I discovered from analyzingone of myGrade 11 students’ approach to solving exponential equations of the form a a k x p x q , where a and k are positive integers greater than 1, and p , q . The student got a correct answer using a procedure which does not conform with the known exponential laws, thereby making it difficult to evaluate the student’s work. I gave the student’s script to fellow mathematics educators and they marked the student wrong, arguing that laws of exponents cannot be extended to a sum or difference of exponentials with the same base. Ithen decided to present the student’s solution method to other mathematics experts (through the Math Forum)for further evaluation. The responses and comments I received were far from being conclusive. It was suggested that there was need to use mathematical proof to verify whether the student’s approach was valid or not. I then set out to investigate why what looked like invalid reasoning on the surface gave the right answer in the end.After a careful analysis of the student’s approach, coupled with some kind of empirical investigations and mathematical proof, I eventually discovered that there was some logic in the student’s approach, only that it was not supported by the known theory of exponents. I therefore recommend that mathematics educators should not take students’ solution methods for granted. We might be marking some of the students’ solution methods wrong when they are valid, and robbing them of their precious marks. It is possible for students to come up with new and valid methods of solving mathematics problems which are not known to the educators.Good mathematics educators should therefore be on the lookout for new unanticipated approaches to solving mathematical problems that students of exceptional abilities may use in class. The famous German mathematician and astronomer, Carl Friedrich Gauss (1777-1855)amazed his teachers when he discovered a quick way of summing the integers from 1 to 100, at the age of seven. Such exceptional intellectual abilities still existeven in school children of today.
This document discusses innovative practices in teaching discrete structures/mathematics and data structures to undergraduate computer science students. It describes course structures at various universities and suggests focusing discrete mathematics on fundamental concepts like logic, proofs, and counting before more advanced topics. For data structures, it recommends teaching implementation to build understanding but also focusing on usage. Projects, multimedia, and games are presented as motivating teaching techniques. Historical sources are proposed to provide context for abstract concepts.
Mathematics is defined as a science of patterns and relationships that reveals hidden patterns in the world and relies on logic and creativity. The two main goals of teaching mathematics are developing critical thinking and problem solving skills. Learning mathematics is most effective when done through active learning and problem solving. The document then outlines the key stage standards and grade level standards for mathematics in terms of the concepts, skills, and applications students are expected to understand at each level.
Mathematical skills such as arithmetic, geometry, and graphing are important foundations for students. Key skills include number sense, measurement, patterns, problem-solving, and computational fluency. Higher-order thinking skills (HOTS) like problem-solving, reasoning, and conceptualizing are valued as they better prepare students for challenges. HOTS involve skills like critical thinking, creativity, and systems thinking. Teachers should focus on developing students' HOTS through open-ended learning activities.
Nature and principles of teaching and learning mathJunarie Ramirez
This document discusses effective teaching of mathematics. It outlines three phases of mathematical inquiry: (1) abstraction and symbolic representation, (2) manipulating mathematical statements, and (3) application. It also discusses the nature and principles of teaching mathematics, including that mathematics relies on both logic and creativity. Effective teaching requires understanding what students know and challenging them, as well as using worthwhile tasks to engage them intellectually. Teachers must have mathematical knowledge and commit to students' understanding.
A description of the analytical tools developed in Physics Education Research for understanding students use of and difficulties with mathematics as used in science.
The document discusses the distinction between theoretical and practical thinking in mathematics. It presents a model that defines theoretical thinking as focusing on conceptual connections, logical reasoning, and developing consistent theories, while practical thinking focuses on solving particular problems and understanding factual relationships. The author argues that successful mathematics students demonstrate a "practical understanding of theory" by applying theoretical knowledge flexibly to efficiently solve problems. Examples of student work show the importance of both theoretical and practical thinking skills for high achievement in university mathematics.
The document discusses mathematical skills and higher order thinking skills (HOTS) in mathematics. It begins by defining key arithmetic skills like number sense, measurement, and patterns that are important for students to learn. It then discusses different types of graphs and charts used to visualize data. The document outlines the five fundamental HOTS: problem solving, inquiring, reasoning, communicating, and conceptualizing skills. It notes that developing HOTS better prepares students for challenges in work and life. The document provides examples of how to incorporate HOTS into mathematics teaching and assessments in order to improve student performance.
Nature of Mathematics and Pedagogical practicesLaxman Luitel
I presented this paper in mathematics education and society conference 2019 (Jan 28 - Feb 2) at University of Hyderabad, Hyderabad, India. Paper is available in the website of conference and the link given below.
https://www.researchgate.net/publication/331113612
This document discusses innovative practices in teaching discrete structures/mathematics and data structures to undergraduate computer science students. It describes course structures at various universities and suggests focusing discrete mathematics on fundamental concepts like logic, proofs, and counting before more advanced topics. For data structures, it recommends teaching implementation to build understanding but also focusing on usage. Projects, multimedia, and games are presented as motivating teaching techniques. Historical sources are proposed to provide context for abstract concepts.
Mathematics is defined as a science of patterns and relationships that reveals hidden patterns in the world and relies on logic and creativity. The two main goals of teaching mathematics are developing critical thinking and problem solving skills. Learning mathematics is most effective when done through active learning and problem solving. The document then outlines the key stage standards and grade level standards for mathematics in terms of the concepts, skills, and applications students are expected to understand at each level.
Mathematical skills such as arithmetic, geometry, and graphing are important foundations for students. Key skills include number sense, measurement, patterns, problem-solving, and computational fluency. Higher-order thinking skills (HOTS) like problem-solving, reasoning, and conceptualizing are valued as they better prepare students for challenges. HOTS involve skills like critical thinking, creativity, and systems thinking. Teachers should focus on developing students' HOTS through open-ended learning activities.
Nature and principles of teaching and learning mathJunarie Ramirez
This document discusses effective teaching of mathematics. It outlines three phases of mathematical inquiry: (1) abstraction and symbolic representation, (2) manipulating mathematical statements, and (3) application. It also discusses the nature and principles of teaching mathematics, including that mathematics relies on both logic and creativity. Effective teaching requires understanding what students know and challenging them, as well as using worthwhile tasks to engage them intellectually. Teachers must have mathematical knowledge and commit to students' understanding.
A description of the analytical tools developed in Physics Education Research for understanding students use of and difficulties with mathematics as used in science.
The document discusses the distinction between theoretical and practical thinking in mathematics. It presents a model that defines theoretical thinking as focusing on conceptual connections, logical reasoning, and developing consistent theories, while practical thinking focuses on solving particular problems and understanding factual relationships. The author argues that successful mathematics students demonstrate a "practical understanding of theory" by applying theoretical knowledge flexibly to efficiently solve problems. Examples of student work show the importance of both theoretical and practical thinking skills for high achievement in university mathematics.
The document discusses mathematical skills and higher order thinking skills (HOTS) in mathematics. It begins by defining key arithmetic skills like number sense, measurement, and patterns that are important for students to learn. It then discusses different types of graphs and charts used to visualize data. The document outlines the five fundamental HOTS: problem solving, inquiring, reasoning, communicating, and conceptualizing skills. It notes that developing HOTS better prepares students for challenges in work and life. The document provides examples of how to incorporate HOTS into mathematics teaching and assessments in order to improve student performance.
Nature of Mathematics and Pedagogical practicesLaxman Luitel
I presented this paper in mathematics education and society conference 2019 (Jan 28 - Feb 2) at University of Hyderabad, Hyderabad, India. Paper is available in the website of conference and the link given below.
https://www.researchgate.net/publication/331113612
Nature, characteristics and definition of mathsAngel Rathnabai
This document discusses various views of mathematics, including student, parent, and teacher views. It also covers the nature, characteristics, development, and applications of mathematics. Some key points include:
- Mathematics involves finding and studying patterns, and can be seen as a language, way of thinking, and problem solving approach.
- It has developed over time from ancient subjects like geometry to a more modern field incorporating diverse areas.
- Major subfields include algebra, analysis, applied math, with connections to many other domains. Real-world applications span fields like imaging, cryptography, simulation, and bioinformatics.
The document is a 12-page extended essay on Vedic mathematics and its methods for multiplication. It explores several Vedic multiplication techniques including vertically and crosswise, multiplying numbers above 10, multiplying by one more than the number before for squaring numbers ending in 5. The essay finds that Vedic multiplication can shorten and ease both basic and complex multiplication problems by using numerical patterns, though some problems require adapting the methods.
1) The document discusses studies conducted in Indonesia from 2012-2013 that found most high school students and teachers neglected mathematical proofs in their instruction and assessments.
2) A key finding was that the national standardized exam, which is high-stakes, does not assess skills in mathematical reasoning and proof, so teachers do not focus on teaching these skills.
3) Diagnostic assessments with students found they lacked the ability to prove math statements and provide rigorous reasoning due to proofs not being adequately addressed in textbooks or classroom instruction. The document proposes ways to better teach and assess proof abilities.
This document provides an overview of the 5th grade mathematics standards for North Carolina related to the Common Core. It is intended to help educators understand what students are expected to know and be able to do under the new standards. The document explains that the standards describe the essential knowledge and skills students should master in order to be prepared for 6th grade. It also provides examples for how the standards can be unpacked to clarify their meaning and intent. Educators are encouraged to provide feedback to help improve the usefulness of the document.
This document provides an overview of the 6th-8th grade Virginia SOLs and Common Core State Standards for mathematics. It summarizes the key focuses and differences between the two sets of standards, including their approaches to problem solving, use of technology, and emphasis on different mathematical concepts across grades 6-8 such as foundations of algebra, rational numbers, and geometric properties. The document also includes perspectives on the benefits and drawbacks of each set of standards.
This document discusses mental representations in learning, including logic, rules, concepts, analogies, and images. It analyzes how two of David Perkins' seven principles of teaching - working on hard parts and learning from teams - can be applied when teaching systems of equations and the distributive property in mathematics. For systems of equations, focusing on moving from algebraic solutions to graphical representations addresses a hard part. Group work and feedback helps learning. For the distributive property, playing the whole game ensures students understand when and how to apply the rule in different situations.
This document discusses integrating technology and the arts into reading and math instruction to address challenges in schools. It reviews literature on using literacy strategies in math, engaging career and technical education in the Common Core, and interactive art websites. The document also outlines the eight mathematical practices in the Common Core, and provides examples of how literacy standards are addressed in math at various grade levels.
The document summarizes a student paper examining teaching mathematics for conceptual understanding versus procedural efficiency. It defines the two approaches, provides an example lesson for each, and analyzes the differences. Teaching for procedural efficiency emphasizes memorizing and applying formulas, while teaching for conceptual understanding focuses on reasoning, making connections to prior knowledge, and understanding why procedures work.
The document discusses principles and methods for teaching mathematics. It covers:
1) The spiral progression approach which revisits math basics each grade level with increasing depth and breadth.
2) Principles like balancing standard-based and integrated approaches, using problem-solving, and assessment-driven instruction.
3) Bruner's three-tiered learning theory of enactive, iconic, and symbolic representation.
4) Teaching methods like problem-solving, concept attainment strategies, concept formation strategies, direct instruction, and experiential/constructivist approaches.
The document provides the conceptual framework for mathematics education in the K to 12 Basic Education Curriculum in the Philippines. It outlines that the twin goals of mathematics education from K-10 are critical thinking and problem solving. It also describes the five content areas, specific skills and processes, values and attitudes, appropriate tools, and contexts that make up the framework. The framework is supported by theories of experiential learning, situated learning, reflective learning, constructivism, cooperative learning, and discovery and inquiry-based learning.
Correlations of Students’ Academic Achievement in Mathematics with Their Erro...paperpublications3
Abstract: While there have been many researches done with regards to error analysis in mathematics, little is known about analysis of error detection abilities. It is necessary that mathematics educators understand the error detection abilities of the students so that appropriate instructions and materials can be given to the learners. In particular, this study investigates the error detection abilities of private school students in mathematics and the correlations with their written test results. A total of 41 private candidates who are reading mathematics sat for three written tests. The first test is an arithmetic test, which assesses the students’ understanding in number operations. The second test is on algebra, and the third test is an error detection test. The results from these three tests are consolidated and analyzed using Excel spreadsheets and a graphic calculator TI-84Plus. Statistical tools such as the regression line, the Pearson product-moment correlation coefficient and the reliability coefficient are used to analyze the data and evaluate the error detection instrument. At the end of the study, it is found that the academic performance of the students in mathematics is highly correlated to their error detection abilities. At the same time, it is also found that the designed error detection test is a reliable instrument which is suitable to predict the future success of a student in mathematics with some limitations. This study is part of a growing body of research on error analysis in mathematics. In using the largely untapped source of students studying in a local private education institution, this project will definitely contribute to future research on similar topics.
This document discusses frameworks for children's mathematical thinking and the importance of experiences in developing conceptual understanding of mathematics. It argues that experiences should illustrate concepts in multiple contexts to build representations and connections. Concrete materials can help students understand concepts but must be closely linked to the intended concept. The teacher's role is to provide engaging experiences that emphasize conceptual understanding and a positive attitude towards math.
This is part of the professional development for the team that translate My Pals Are Here into Dutch and also people who are going to provide professionald evelopment for teachers using Singapore textbooks in the future.
This document outlines the K to 12 mathematics curriculum guide for the Philippines from Kindergarten to Grade 10. It discusses the conceptual framework, course description, learning area and grade level standards, time allotment, and sample content for Grade 1. The goals are critical thinking and problem solving. Key concepts covered include numbers, measurement, geometry, patterns and algebra, and statistics. The curriculum is supported by theories of experiential learning and constructivism.
Dr. M.THIRUNAVUKKARASU
Research Associate
Department of Education
Bharathidasan University,
Tiruchirappalli - 620 024, Tamil Nadu, India
E-mail: edutechthiru@gmail.com
Dr. S. SENTHILNATHAN
Director (FAC),
UGC - Human Resource Development Centre
(HRDC)
Bharathidasan University
Khajamalai Campus
Tiruchirappalli - 620 023
E-mail: edutechsenthil@gmail.com
The document discusses key aspects of the mathematical process. It defines mathematical process as thinking, reasoning, calculation, and problem solving using mathematical methods. The main components discussed are reasoning, logical thinking, problem solving, and making connections. Reasoning involves making conjectures, investigating findings, and justifying conclusions. Problem solving requires applying previously learned skills to new situations. Problem posing encourages students to write and solve their own problems to improve problem solving abilities.
The document discusses changes to Louisiana's standardized tests to better align with the Common Core State Standards. Key points include:
- Math tests will assess only content common to current standards and the CCSS, narrowing the focus areas.
- Some tests will no longer include the Iowa Test of Basic Skills. Grades 4 and 8 tests will be grade-specific rather than grade-span.
- Test difficulty and cut scores will remain the same during the transition period to the CCSS. New CCSS content will not be added until 2014-2015.
The document provides an overview of the K to 12 Mathematics curriculum guide for grade 1 in the Philippines. It discusses the conceptual framework, which identifies critical thinking and problem solving as the twin goals of mathematics education. It also outlines the five content areas - numbers and number sense, measurement, geometry, patterns and algebra, and probability and statistics. The guide is grounded in learning principles like experiential learning, reflective learning, constructivism, cooperative learning, and discovery-based learning.
The document discusses the nature and certainty of mathematics. It describes how mathematics was traditionally viewed through Euclid's formal model of starting from axioms and reaching theorems through deductive reasoning. However, philosophers have debated whether mathematics is discovered or invented, and whether it is empirical, analytic, or synthetic a priori. While mathematics allows for precise calculations and patterns, the exact status of mathematical concepts remains an open and fundamental question.
Improving Communication about Limit Concept in Mathematics through Inquisitio...IOSR Journals
In this action research study, where the subjects are our undergraduate grade mathematics students,
w e try to investigate the impact of direct ‘inquisition’ instruction on their communication and achievement.
We will strategically implement the addition of ‘replication’ study into each concept of limit over a four-month
time period and thus conclusion can be making for the rest of the Mathemat ics . The students practiced using
inquiry in verbal discussions, review activities, and in mathematical problem explanations. We discovered
that a majority of students improved their overall understanding of mathematical concepts based on an analysis
of the data we collected. We also found that in general, students felt that knowing the definition of
mathematical words are important and that it increased their achievement when they understood the concept as a
whole. In addition, students will be more exact in their communication after receiving inquiry instructions. As
a result of this research, we plan to continue to implement inquisition into daily lessons and keep replication
communication as a focus of the mathematics class
IOSR Journal of Mathematics(IOSR-JM) is an open access international journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Nature, characteristics and definition of mathsAngel Rathnabai
This document discusses various views of mathematics, including student, parent, and teacher views. It also covers the nature, characteristics, development, and applications of mathematics. Some key points include:
- Mathematics involves finding and studying patterns, and can be seen as a language, way of thinking, and problem solving approach.
- It has developed over time from ancient subjects like geometry to a more modern field incorporating diverse areas.
- Major subfields include algebra, analysis, applied math, with connections to many other domains. Real-world applications span fields like imaging, cryptography, simulation, and bioinformatics.
The document is a 12-page extended essay on Vedic mathematics and its methods for multiplication. It explores several Vedic multiplication techniques including vertically and crosswise, multiplying numbers above 10, multiplying by one more than the number before for squaring numbers ending in 5. The essay finds that Vedic multiplication can shorten and ease both basic and complex multiplication problems by using numerical patterns, though some problems require adapting the methods.
1) The document discusses studies conducted in Indonesia from 2012-2013 that found most high school students and teachers neglected mathematical proofs in their instruction and assessments.
2) A key finding was that the national standardized exam, which is high-stakes, does not assess skills in mathematical reasoning and proof, so teachers do not focus on teaching these skills.
3) Diagnostic assessments with students found they lacked the ability to prove math statements and provide rigorous reasoning due to proofs not being adequately addressed in textbooks or classroom instruction. The document proposes ways to better teach and assess proof abilities.
This document provides an overview of the 5th grade mathematics standards for North Carolina related to the Common Core. It is intended to help educators understand what students are expected to know and be able to do under the new standards. The document explains that the standards describe the essential knowledge and skills students should master in order to be prepared for 6th grade. It also provides examples for how the standards can be unpacked to clarify their meaning and intent. Educators are encouraged to provide feedback to help improve the usefulness of the document.
This document provides an overview of the 6th-8th grade Virginia SOLs and Common Core State Standards for mathematics. It summarizes the key focuses and differences between the two sets of standards, including their approaches to problem solving, use of technology, and emphasis on different mathematical concepts across grades 6-8 such as foundations of algebra, rational numbers, and geometric properties. The document also includes perspectives on the benefits and drawbacks of each set of standards.
This document discusses mental representations in learning, including logic, rules, concepts, analogies, and images. It analyzes how two of David Perkins' seven principles of teaching - working on hard parts and learning from teams - can be applied when teaching systems of equations and the distributive property in mathematics. For systems of equations, focusing on moving from algebraic solutions to graphical representations addresses a hard part. Group work and feedback helps learning. For the distributive property, playing the whole game ensures students understand when and how to apply the rule in different situations.
This document discusses integrating technology and the arts into reading and math instruction to address challenges in schools. It reviews literature on using literacy strategies in math, engaging career and technical education in the Common Core, and interactive art websites. The document also outlines the eight mathematical practices in the Common Core, and provides examples of how literacy standards are addressed in math at various grade levels.
The document summarizes a student paper examining teaching mathematics for conceptual understanding versus procedural efficiency. It defines the two approaches, provides an example lesson for each, and analyzes the differences. Teaching for procedural efficiency emphasizes memorizing and applying formulas, while teaching for conceptual understanding focuses on reasoning, making connections to prior knowledge, and understanding why procedures work.
The document discusses principles and methods for teaching mathematics. It covers:
1) The spiral progression approach which revisits math basics each grade level with increasing depth and breadth.
2) Principles like balancing standard-based and integrated approaches, using problem-solving, and assessment-driven instruction.
3) Bruner's three-tiered learning theory of enactive, iconic, and symbolic representation.
4) Teaching methods like problem-solving, concept attainment strategies, concept formation strategies, direct instruction, and experiential/constructivist approaches.
The document provides the conceptual framework for mathematics education in the K to 12 Basic Education Curriculum in the Philippines. It outlines that the twin goals of mathematics education from K-10 are critical thinking and problem solving. It also describes the five content areas, specific skills and processes, values and attitudes, appropriate tools, and contexts that make up the framework. The framework is supported by theories of experiential learning, situated learning, reflective learning, constructivism, cooperative learning, and discovery and inquiry-based learning.
Correlations of Students’ Academic Achievement in Mathematics with Their Erro...paperpublications3
Abstract: While there have been many researches done with regards to error analysis in mathematics, little is known about analysis of error detection abilities. It is necessary that mathematics educators understand the error detection abilities of the students so that appropriate instructions and materials can be given to the learners. In particular, this study investigates the error detection abilities of private school students in mathematics and the correlations with their written test results. A total of 41 private candidates who are reading mathematics sat for three written tests. The first test is an arithmetic test, which assesses the students’ understanding in number operations. The second test is on algebra, and the third test is an error detection test. The results from these three tests are consolidated and analyzed using Excel spreadsheets and a graphic calculator TI-84Plus. Statistical tools such as the regression line, the Pearson product-moment correlation coefficient and the reliability coefficient are used to analyze the data and evaluate the error detection instrument. At the end of the study, it is found that the academic performance of the students in mathematics is highly correlated to their error detection abilities. At the same time, it is also found that the designed error detection test is a reliable instrument which is suitable to predict the future success of a student in mathematics with some limitations. This study is part of a growing body of research on error analysis in mathematics. In using the largely untapped source of students studying in a local private education institution, this project will definitely contribute to future research on similar topics.
This document discusses frameworks for children's mathematical thinking and the importance of experiences in developing conceptual understanding of mathematics. It argues that experiences should illustrate concepts in multiple contexts to build representations and connections. Concrete materials can help students understand concepts but must be closely linked to the intended concept. The teacher's role is to provide engaging experiences that emphasize conceptual understanding and a positive attitude towards math.
This is part of the professional development for the team that translate My Pals Are Here into Dutch and also people who are going to provide professionald evelopment for teachers using Singapore textbooks in the future.
This document outlines the K to 12 mathematics curriculum guide for the Philippines from Kindergarten to Grade 10. It discusses the conceptual framework, course description, learning area and grade level standards, time allotment, and sample content for Grade 1. The goals are critical thinking and problem solving. Key concepts covered include numbers, measurement, geometry, patterns and algebra, and statistics. The curriculum is supported by theories of experiential learning and constructivism.
Dr. M.THIRUNAVUKKARASU
Research Associate
Department of Education
Bharathidasan University,
Tiruchirappalli - 620 024, Tamil Nadu, India
E-mail: edutechthiru@gmail.com
Dr. S. SENTHILNATHAN
Director (FAC),
UGC - Human Resource Development Centre
(HRDC)
Bharathidasan University
Khajamalai Campus
Tiruchirappalli - 620 023
E-mail: edutechsenthil@gmail.com
The document discusses key aspects of the mathematical process. It defines mathematical process as thinking, reasoning, calculation, and problem solving using mathematical methods. The main components discussed are reasoning, logical thinking, problem solving, and making connections. Reasoning involves making conjectures, investigating findings, and justifying conclusions. Problem solving requires applying previously learned skills to new situations. Problem posing encourages students to write and solve their own problems to improve problem solving abilities.
The document discusses changes to Louisiana's standardized tests to better align with the Common Core State Standards. Key points include:
- Math tests will assess only content common to current standards and the CCSS, narrowing the focus areas.
- Some tests will no longer include the Iowa Test of Basic Skills. Grades 4 and 8 tests will be grade-specific rather than grade-span.
- Test difficulty and cut scores will remain the same during the transition period to the CCSS. New CCSS content will not be added until 2014-2015.
The document provides an overview of the K to 12 Mathematics curriculum guide for grade 1 in the Philippines. It discusses the conceptual framework, which identifies critical thinking and problem solving as the twin goals of mathematics education. It also outlines the five content areas - numbers and number sense, measurement, geometry, patterns and algebra, and probability and statistics. The guide is grounded in learning principles like experiential learning, reflective learning, constructivism, cooperative learning, and discovery-based learning.
The document discusses the nature and certainty of mathematics. It describes how mathematics was traditionally viewed through Euclid's formal model of starting from axioms and reaching theorems through deductive reasoning. However, philosophers have debated whether mathematics is discovered or invented, and whether it is empirical, analytic, or synthetic a priori. While mathematics allows for precise calculations and patterns, the exact status of mathematical concepts remains an open and fundamental question.
Improving Communication about Limit Concept in Mathematics through Inquisitio...IOSR Journals
In this action research study, where the subjects are our undergraduate grade mathematics students,
w e try to investigate the impact of direct ‘inquisition’ instruction on their communication and achievement.
We will strategically implement the addition of ‘replication’ study into each concept of limit over a four-month
time period and thus conclusion can be making for the rest of the Mathemat ics . The students practiced using
inquiry in verbal discussions, review activities, and in mathematical problem explanations. We discovered
that a majority of students improved their overall understanding of mathematical concepts based on an analysis
of the data we collected. We also found that in general, students felt that knowing the definition of
mathematical words are important and that it increased their achievement when they understood the concept as a
whole. In addition, students will be more exact in their communication after receiving inquiry instructions. As
a result of this research, we plan to continue to implement inquisition into daily lessons and keep replication
communication as a focus of the mathematics class
IOSR Journal of Mathematics(IOSR-JM) is an open access international journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Man-made resources like calculators, computers, and technology tools can help reshape mathematics education. They allow students to more deeply learn mathematics by making and testing conjectures and working at higher levels of abstraction. Teachers facilitate active learning by challenging students to make connections and think more deeply about problems. Man-made resources assist mathematics teachers in teaching the subject more easily.
Assessment Of Mathematical Modelling And ApplicationsMartha Brown
The document discusses assessment of mathematical modeling and applications in Dutch education. It describes two projects - HEWET and HAWEX - that developed new curricula focusing on realistic mathematics involving modeling, applications, and problem-solving. The projects found appropriate ways to assess these skills in timed written exams. Students learning through modeling activities become good problem solvers who create their own solution strategies, which sometimes surprise teachers. Assessments must test these modeling skills and not just technical algorithms.
Problem-Solving Capacity of Students: A Study of Solving Problems in Differen...theijes
Training towards the development of the capacity of learners has become an inevitable trend of world education. Vietnamese education also emphasizes the comprehensive development of the capacity and the quality of students. In mathematics teaching, there are some notable capacities such as problem-solving capacity, cooperation capacity, capacity for using mathematical language, computing capacity and so on. In particular, the problem-solving capacity is very important to students because it helps them to solve problems not only in mathematics but also in practice. In this paper, we want to investigate the problem-solving capacity of students in primary schools through a problem required to solve in different ways. The results of the study showed that students had enough the problem-solving capacity to find out various solutions to the given problem
Intuition – Based Teaching Mathematics for EngineersIDES Editor
It is suggested to teach Mathematics for engineers
based on development of mathematical intuition, thus, combining
conceptual and operational approaches. It is proposed to teach
main mathematical concepts based on discussion of carefully
selected case studies following solving of algorithmically generated
problems to help mastering appropriate mathematical tools.
The former component helps development of mathematical intuition;
the latter applies means of adaptive instructional technology
to improvement of operational skills. Proposed approach is applied
to teaching uniform convergence and to knowledge generation
using Computer Science object-oriented methodology.
This study investigated the effects of using a graphical calculator (Microsoft Math Tool) on students' performance in linear functions. Ninety-eight students from two schools participated, with one group receiving instruction using the graphical calculator (experimental group) and the other receiving traditional lecture-based instruction (control group). Pre-tests showed no significant difference between the groups. Post-test results revealed significantly higher scores for the experimental group compared to the control group, suggesting that using graphical calculators can improve students' understanding of linear functions and performance on related tasks. The study recommends integrating graphical calculators into mathematics instruction.
Authentic Tasks And Mathematical Problem SolvingJim Webb
This document discusses authentic tasks in mathematical problem solving and their role in developing mathematical literacy. It describes four key dimensions of authentic tasks: thinking and reasoning, discourse, mathematical tools, and attitudes and dispositions. Each of these dimensions supports meaningful learning and prepares students to solve everyday problems. The document provides examples of lessons and programs that incorporate these dimensions through real-world, problem-based activities.
This document provides an overview of Realistic Mathematics Education (RME), including its key characteristics and principles for designing lessons based on this approach. RME stresses starting with real-world contexts that are meaningful to students, and having students explore problems and develop mathematical concepts through guided reinvention that incorporates both horizontal and vertical mathematization. Lessons based on RME should include contextual problems for student exploration, opportunities for students to develop and use their own models and strategies, and an interactive teaching process that weaves together different mathematical strands.
An ICT Environment To Assess And Support Students Mathematical Problem-Solvi...Vicki Cristol
1) The study used an ICT environment to assess and support 24 fourth-grade students' problem-solving performance on non-routine word problems.
2) The ICT environment allowed students to work in pairs, producing and experimenting with solutions while their dialogues and actions were analyzed.
3) The study aimed to better understand students' problem-solving processes and how the ICT environment could improve their problem-solving skills, though pre- and post-tests did not conclusively show its impact on performance.
MATD611 Mathematics Education In Perspective.docxstirlingvwriters
The document discusses four views of mathematics:
1) Logicism view - Mathematics is a continuation of logic and can be reduced to logic.
2) Intuitionist view - Mathematics is a product of human imagination and must be mentally constructed.
3) Instrumentalist view - Mathematics is a collection of unrelated rules and facts that serve practical purposes.
4) The document also discusses teachers' views on the nature of mathematics, the benefits of teaching history of mathematics, and ways to implement student-centered learning in mathematics classrooms.
Algebraic Thinking A Problem Solving ApproachCheryl Brown
This document discusses developing algebraic thinking through a problem solving approach in mathematics education. It argues that solving mathematical problems can help students develop a deeper understanding of algebraic concepts by encouraging them to consider the generalities and relationships within problems. The document outlines several key aspects of algebraic thinking, such as reasoning about patterns, generalizing, and thinking about mathematical relations. It advocates facilitating student discourse and communication of mathematical ideas to promote algebraic reasoning. Finally, it suggests students progress through stages of developing algebraic thinking, starting with describing generalities verbally and moving to using diagrams, symbols, and formal algebraic notation.
Assessing Algebraic Solving Ability A Theoretical FrameworkAmy Cernava
This document presents a theoretical framework for assessing algebraic solving ability. It discusses three phases of algebraic processes: 1) investigating patterns in numerical data, 2) representing patterns in tables and equations, and 3) interpreting and applying equations to new situations. It also reviews different approaches for developing algebraic solving ability in students, such as game-based learning, building conceptual understanding of variables/functions, and representing problems in multiple ways. The theoretical framework is based on these algebraic processes and how they align with levels of the SOLO model for assessing student understanding.
Constructivist Approach Vs Expository Teaching: Exponential Functionsinventionjournals
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Solving Exponential Equations: Learning from the Students We Teach
1. International Journal of Engineering Science Invention
ISSN (Online): 2319 – 6734, ISSN (Print): 2319 – 6726
www.ijesi.org ||Volume 6 Issue 5|| May 2017 || PP. 01-07
www.ijesi.org 1 | Page
Solving Exponential Equations: Learning from the Students We Teach
Eric Machisi
Institute for Science & Technology EducationUniversity of South Africa
Abstract: The purpose of this paper is to share with the mathematics community what I discovered from
analyzingone of myGrade 11 students’ approach to solving exponential equations of the form
kaa
qxpx
, where a and k are positive integers greater than 1, and qp , . The student got a
correct answer using a procedure which does not conform with the known exponential laws, thereby making it
difficult to evaluate the student’s work. I gave the student’s script to fellow mathematics educators and they
marked the student wrong, arguing that laws of exponents cannot be extended to a sum or difference of
exponentials with the same base. Ithen decided to present the student’s solution method to other mathematics
experts (through the Math Forum)for further evaluation. The responses and comments I received were far from
being conclusive. It was suggested that there was need to use mathematical proof to verify whether the student’s
approach was valid or not. I then set out to investigate why what looked like invalid reasoning on the surface
gave the right answer in the end.After a careful analysis of the student’s approach, coupled with some kind of
empirical investigations and mathematical proof, I eventually discovered that there was some logic in the
student’s approach, only that it was not supported by the known theory of exponents. I therefore recommend
that mathematics educators should not take students’ solution methods for granted. We might be marking some
of the students’ solution methods wrong when they are valid, and robbing them of their precious marks. It is
possible for students to come up with new and valid methods of solving mathematics problems which are not
known to the educators.Good mathematics educators should therefore be on the lookout for new unanticipated
approaches to solving mathematical problems that students of exceptional abilities may use in class. The famous
German mathematician and astronomer, Carl Friedrich Gauss (1777-1855)amazed his teachers when he
discovered a quick way of summing the integers from 1 to 100, at the age of seven. Such exceptional intellectual
abilities still existeven in school children of today.
Key Terms: Exponents, exponential equations, anticipated approach, unanticipated approach
I. Introduction
Exponents, which indicate how many times a number multiplies itself, are fundamental in the modern
technological world.Exponents are used by Computer Programmers, Bankers, Investors, Accountants,
Economists, Insurance Risk Assessors, Chemists, Geologists, Biologists, Physicists, artisans, Sound Engineers
and Mathematicians, to solve real life problems [1]. For instance, investors simply plug in numbers into
exponential equations and they are able to figure out how much they are earning on their savings. Banks use
exponential equations to calculate how much borrowers have to repay each month to settle their loans.
Geoscientists use exponents to calculate the intensity of an earthquake. Builders and carpenters use exponents
daily to calculate the quantity of material needed to construct buildings. Aeronautical engineers apply
knowledge of exponents to predict how rockets and jets will perform during a flight.Clearly, exponents are a
necessary component of the school mathematics curriculum to prepare students for these critical careers.
In South Africa, Grade 10 and 11 students are expected to use laws of exponents to solve exponential
equations[2], including those of the form kaa
qxpx
. Examples of such equations are:
1033
2
xx
11333
312
xxx
322
2
xx
655
1
xx
In solving these equations, we rely mainly on the exponential theorem which states that:
If
cx
aa , then cx .
It is not known whether this could be extended toa sum or difference of exponentials with the same base.For
instance, we do not have theorems suggesting that:
srqxpxaaaa
srqxpx
2. Solving Exponential Equations: Learning from the Students We Teach
www.ijesi.org 2 | Page
However, one of my Grade 11 students used a technique that seemed to suggest the existence of such
theorems. The student got a correct answer using a procedure that seemed to be invalid because it did not
conform with the known theorems of exponents. I therefore set out to investigate the reason why an approach
that seemed invalid on the surface, gave a correct answer in the end. In this paper, I intend to share with the
mathematics community what I discovered from analysing the student method of solution.
II. Theoretical Framework
Mathematics is “a human activity that involves observing, representing and investigating patterns and
qualitative relationships in the physical and social phenomena and between mathematical objects themselves”
[2, p. 8]. Mathematical problem solvingteaches us to think critically, logically and creatively [2].Contemporary
views of mathematics education encourage students to use their own methods to solve problems rather than
imitate their teachers. This is in line with the constructivist view of mathematics teaching and learning which
states that students have the ability to construct their own knowledge through discovery and problem solving.
Constructivism “gives preeminent value to the development of students’ personal mathematical ideas” [3,p.
9].In a traditional mathematics classroom, it is predetermined that students will solve mathematics problems
using the method(s) shown to them by their teacher. This is in sharp contrast to a constructivist mathematics
classroom where students employ different and multiple solution methods. “Students may use unanticipated
solution-methods and unforeseen difficulties may arise”[4, p. 3].The role of the teacher in this context becomes
more demanding and unless the teacherlooks more closely at what the students have written, some solutionsrisk
being marked wrong when they are valid.A good mathematics teacher understands that there are multiple ways
of solving mathematics problems, and hence seriously considersevery attempt that students make towards
solving a mathematics problem.
III. Purpose of The Study
The purpose of this study was to explore the logic behind one of my Grade 11 students’ unanticipated
approach to solving exponential equations of the form kaa
qxpx
,which resulted in a correct answer,
using what seemed to be an invalid procedure. Findings of this study were intended to assist the researcher in
deciding whether or not the student’s method could be accepted as valid.
IV. Methodology
This investigation utilized the single-case study design with only one Grade 11 student as the unit of
analysis. According to Yin [5], a single-case study is appropriate under the following circumstances: (a) where
it represents a critical case that can be used to confirm, challenge or extend a well-formulated theory, (b) where
it represents a unique case, and (c) where it makes a significant contribution to knowledge or theory building.
All the three circumstances were found to match the present investigation perfectly.Data were collected from the
student’s classwork book. The focus of the study was on the student’s approach to solving exponential equations
of the form kaa
qxpx
. The student’s solution method which was a unique case, was presented to the
Math Forum for evaluation. Two mathematics experts, Doctor X and Doctor Y (not their real names), responded
through email. After analyzing their comments, I decided to conduct my own empirical investigations by
applying the student’s unanticipated approach to several other similar exponential equations and compared the
results with those obtained using the anticipated (usual) approach.The findings of the study are presented in the
next section.
V. Findings
3. Solving Exponential Equations: Learning from the Students We Teach
www.ijesi.org 3 | Page
SOLVING EXPONENTIAL EQUATIONS OF THE FORM kaa
qxpx
The following is what the student wrote in her classwork:
Figure 1. The Student’s Unanticipated Approach
There is one thing in the student’s approach that puzzled me, that is, concluding that:
If then
There is no known theorem which supports this proposition. I was expecting the following approach:
Figure 2. The Anticipated Approach
Clearly, it can be seen that the student’s unanticipated approach gave the same answer as the anticipated
approach. This prompted me to do further investigations. I presented both approaches to mathematics experts
(through the Math Forum) for evaluation, and received the following comments:
COMMENT 1:
Doctor X:The second approach is correct. The first is…um…not quite. In the second approach, each equation is
logically equivalent to the one before it. Therefore, it is a valid mathematical argument (or proof) that the
solution set contains exactly one solution, namely
In the first, the statement is logically equivalent to the one before it. But there is no
reasonable way to get from that statement to except perhaps by first proving that the only
solution to the former is If someone had instead gotten to and then said, ‘And then
if and , then we have a solution!’ then I would say, ‘Yes! That is correct. And that proves
that there are no other solutions’. But when you put the + in between those two pieces then there is no sense to
this.
COMMENT 2:
Doctor Y:We’re talking about this step
2
42
55
6255
3750)6(5
3750)15(5
375055.5
375055
42
2
2
2
212
212
x
x
x
x
x
x
xx
xx
4. Solving Exponential Equations: Learning from the Students We Teach
www.ijesi.org 4 | Page
:
45212
5555
45212
xx
xx
In the absence of any explicit reason given for this step, I can only guess that whoever did this work was relying
on a supposed theorem that:
dcbaxxxx
dcba
A similar theorem is well known to be true, and a critical part of solving many exponential equations:
baxx
ba
But then this cannot be extended to a SUM of exponentials with the same base, unless we can prove it. I doubt
that it is true, but I’ll need to look into it to be sure.
I could see that the experts’ comments were far from being conclusive. The question of why the
student’s method which appeared to be invalid gave a correct answer in the end was not answered. Was it by
coincidence or was there some hidden logic behind it? I decided to spent more time analyzing the student’s
approach to figure out why it gave a correct answer. After several hours of critical thinking, I discovered that
there was a pattern in the manner in which the student wrote the exponents. I noticed that the difference between
the exponents on the left-hand side was the same as the difference between the exponents on the right-hand side.
I conjectured that this was the magic behind the student’s correct answer. I then set out to conduct my own
empirical investigations by applying the student’s approach to several other similar exponential equations. Here
are my findings:
Equation 1: Solve
655
1
xx
Equation 2: Solve
9622
1
xx
Anticipated Approach Student’s Unanticipated Approach
0
55
15
6)6(5
6)15(5
655.5
0
x
x
x
x
x
xx
0
02
112
011
5555
011
x
x
x
xx
xx
Anticipated Approach Student’s Unanticipated Approach
5
22
322
96)3(2
96)12(2
9622.2
5
1
x
x
x
x
x
xx
5
102
1112
561
2222
561
x
x
x
xx
xx
5. Solving Exponential Equations: Learning from the Students We Teach
www.ijesi.org 5 | Page
Equation 3: Solve
322
2
xx
Equation 4: Solve
1033
2
xx
Equation 5: Solve
9
10
33
11
xx
Anticipated Approach Student’s Unanticipated Approach
2
242
3
4
32
3
4
3
2
3
4
1
12
3)21(2
32.22
2
2
2
x
x
x
x
x
x
xx
2
42
222
022
2222
022
x
x
x
xx
xx
Anticipated Approach Student’s Unanticipated Approach
0
33
13
10)10(3
10)13(3
1033.3
0
2
2
x
x
x
x
x
xx
0
02
222
022
3333
022
x
x
x
xx
xx
Anticipated Approach Student’s Unanticipated Approach
1
3
3
1
10
3
9
10
3
9
10
3
10
3
9
10
3
1
33
9
10
)33(3
9
10
3.33.3
1
11
11
x
x
x
x
x
xx
1
22
)2(011
3333
33333
2011
20211
x
x
xx
xx
xx
6. Solving Exponential Equations: Learning from the Students We Teach
www.ijesi.org 6 | Page
VI. Discussionof Results
After trying out several cases, I concluded that it was not by coincidence that the student’s
unanticipated approach to solving exponential equations gave a correct answer. Indeed, the student’s technique
works under certain conditions. The only problem with the student’s presentation was that it was not backed up
by the known exponential laws and lacked the necessary details to assist the educator in marking thestudent’s
work. The student therefore risked being marked wrong because the logic behind the student’s approach was not
easily discernible. There is no known theorem which suggests that:
If , then .
Based on my findings from using the student’s technique in several similar cases, I would like to propose the
possibility of developing such a theorem, under certain restrictions:
Proof:
Suppose where ,
Then we have: (*)
(1)
If then from equation (*), , which implies that: (2)
If results (1) and (2) are true, then it is also holds true that (3)
Result (3) explains why the student’s unanticipated approach gave a correct answer in the end. I therefore argue
here that my student’s solution method is acceptable and I am inviting the mathematics community to criticize
these findings.
VII. Recommendations
Based on the findings of this investigation, I strongly urge fellow mathematics educators to closely
examine students’ solutions to avoid marking them wrong when they are right. Some of the methods of
solutions that students may use might be completely new and unfamiliar to the educator. Like the famous
German mathematician Carl Friedrich Gauss (1777-1855) who discovered a quick way of adding natural
numbers from 1 to 100 at the age of seven, these young minds should not be looked down upon. Some of the
students we teach possess exceptional intellectual abilities that need to be nurtured and not suppressed. Good
mathematics teachers should therefore be ready to learn even from their own students. Finally, I recommend
further studies on the possibility of extending exponential laws to a sum or difference of exponentials.
References
[1] Passy, “Exponents in the Real World,” 17 May 2013. [Online]. Available:
http://passyworldofmathematics.com/exponents-in-the-real-world/. [Accessed 1 April 2017].
[2] Department of Basic Education, Curriculum and Assessment Policy Statement, Further Education and
Training Phase, Grades 10-12 Mathematics, Pretoria: Department of Basic Education, 2011.
[3] B. Monoranjan, “Constructivism approach in mathematics teaching and assessment of mathematical
understanding,” Basic Research Journal of Education Research and Review, vol. 4, no. 1, pp. 8-12, January
7. Solving Exponential Equations: Learning from the Students We Teach
www.ijesi.org 7 | Page
2015.
[4] S. Evans and M. Swan, “Developing Students Strategies for Problem Solving in Mathematics: the role of
pre-designed "Sample Student Work",” Journal of International Society for Design and Development in
Education, vol. 2, no. 7, pp. 1-31, 2014.
[5] R. K. Yin, Case Study Research: Design and Methods, 5th ed., Los Angeles: SAGE Publications, 2013.