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A Problem-Solving Approach to Mathematics: Unlocking Mathematical Understanding
Author: Dr. Anya Sharma, Ph.D. in Mathematics Education, Professor of Mathematics at the University of California, Berkeley. Dr. Sharma has over 20 years of experience in developing and implementing innovative problem-solving curricula and has published extensively on the topic, including the seminal work, "Bridging the Gap: Problem-Solving in Mathematics Education."
Keywords: problem-solving approach to mathematics, mathematical problem-solving, problem-solving strategies, heuristic approaches, mathematical reasoning, critical thinking, mathematics education, problem-solving techniques, Polya's method, mathematical modeling.
Abstract: This article explores a problem-solving approach to mathematics, examining its benefits, methodologies, and applications across various mathematical domains. We delve into different problem-solving strategies, emphasizing the importance of critical thinking, creative reasoning, and perseverance in tackling mathematical challenges. This approach fosters a deeper understanding of mathematical concepts and enhances students' overall mathematical proficiency. A problem-solving approach to mathematics moves beyond rote memorization to cultivate a dynamic and engaging learning experience.
1. Introduction: Reframing Mathematics Education
Traditional mathematics education often focuses on memorization of facts and procedures. However, a problem-solving approach to mathematics offers a transformative alternative. This approach places the emphasis on understanding the underlying principles, developing problem-solving skills, and fostering critical thinking. It’s about empowering students to become active learners who can confidently tackle unfamiliar mathematical problems. A problem-solving approach to mathematics fosters a deeper, more meaningful understanding than rote learning ever could.
2. The Power of Problem Solving in Mathematics
A problem-solving approach to mathematics is not simply about finding the correct answer; it’s about the entire process of engaging with the problem. This involves:
Understanding the problem: Carefully reading, interpreting, and identifying the key information and unknowns.
Devising a plan: Choosing appropriate strategies and methods to tackle the problem. This might involve drawing diagrams, working backward, using analogies, or applying relevant theorems.
Carrying out the plan: Implementing the chosen strategy systematically, showing all necessary steps and calculations. A problem-solving approach to mathematics requires meticulous execution.
Looking back: Reviewing the solution, checking for errors, and considering alternative approaches. This crucial step enhances understanding and builds confidence.
3. Essential Problem-Solving Strategies
Several effective strategies are central to a problem-solving approach to mathematics:
Polya's Four-Step Method: This classic approach, developed by George Polya, provides a structured framework for problem-solving. It emphasizes understanding the problem, devising a plan, carrying out the plan, and looking back. A problem-solving approach to mathematics often relies heavily on Polya's framework.
Working Backwards: Starting with the desired result and working backward to determine the necessary steps.
Guess and Check: Making educated guesses and refining them based on the results.
Drawing Diagrams: Visualizing the problem using diagrams, graphs, or other visual aids. A problem-solving approach to mathematics often benefits greatly from visual representation.
Looking for Patterns: Identifying patterns and relationships within the problem to simplify the solution process.
Using Analogies: Relating the problem to similar problems that have already been solved.
Breaking Down Complex Problems: Dividing a complex problem into smaller, more manageable subproblems.
4. Heuristic Approaches in a Problem-Solving Approach to Mathematics
Heuristics are general strategies that can be applied to a wide range of problems. These include:
Trial and error: Systematically testing different approaches until a solution is found.
Simplifying the problem: Reducing the complexity of the problem by making simplifying assumptions or approximations.
Working with examples: Exploring specific examples to gain insight into the problem's structure and potential solutions.
Generalizing from examples: Identifying patterns and relationships in specific examples to develop a general solution.
5. The Role of Critical Thinking and Mathematical Reasoning
A problem-solving approach to mathematics heavily relies on critical thinking and mathematical reasoning. Students need to:
Analyze information: Evaluate the relevance and accuracy of the information provided.
Identify assumptions: Recognize and assess the underlying assumptions of the problem.
Formulate arguments: Construct logical arguments to support their solutions.
Justify conclusions: Provide clear and concise justifications for their conclusions.
6. Applications Across Mathematical Domains
A problem-solving approach to mathematics is applicable across various mathematical domains, including algebra, geometry, calculus, and statistics. For example, in algebra, problem-solving might involve formulating and solving equations; in geometry, it might involve proving theorems or solving geometric constructions; and in calculus, it might involve applying derivatives and integrals to solve real-world problems. A problem-solving approach to mathematics enhances understanding and application in each area.
7. Assessment and Evaluation in a Problem-Solving Approach to Mathematics
Assessing student learning in a problem-solving approach to mathematics requires a shift from traditional testing methods. Evaluation should focus on:
The problem-solving process: Assessing the student's understanding of the problem, their chosen strategies, and their ability to justify their solution.
Mathematical reasoning: Evaluating the student's ability to construct logical arguments and justify their conclusions.
Communication skills: Assessing the student's ability to clearly and concisely communicate their mathematical thinking.
8. The Benefits of a Problem-Solving Approach to Mathematics
Adopting a problem-solving approach to mathematics offers numerous benefits:
Deeper understanding: Fosters a deeper understanding of mathematical concepts and principles.
Enhanced critical thinking: Develops critical thinking, problem-solving, and reasoning skills.
Increased engagement: Makes learning more engaging and motivating for students.
Improved problem-solving skills: Equips students with valuable problem-solving skills applicable to various fields.
Greater confidence: Builds students' confidence in their ability to tackle challenging mathematical problems.
Conclusion
A problem-solving approach to mathematics is not merely a pedagogical shift; it's a fundamental change in how we view and teach mathematics. By prioritizing the process of problem-solving over rote memorization, we empower students to become confident, critical thinkers who can apply their mathematical skills to solve real-world challenges. This approach fosters a more meaningful and engaging learning experience, ultimately leading to a deeper and more lasting understanding of mathematics.
FAQs
1. What is the difference between a traditional approach and a problem-solving approach to mathematics? Traditional approaches focus on memorization and procedural fluency, while a problem-solving approach emphasizes understanding, reasoning, and applying knowledge to solve unfamiliar problems.
2. How can teachers implement a problem-solving approach in their classrooms? Teachers can incorporate open-ended problems, encourage collaboration, provide opportunities for students to share their strategies, and focus on the process of problem-solving rather than just the answer.
3. What are some common challenges faced when implementing a problem-solving approach? Challenges include the time required for in-depth problem-solving, managing diverse student abilities, and assessing student learning effectively.
4. Are there specific resources available to support teachers in implementing a problem-solving approach? Yes, numerous resources exist, including textbooks, online materials, professional development workshops, and collaborative networks.
5. How does a problem-solving approach to mathematics prepare students for future studies and careers? It equips students with crucial transferable skills such as critical thinking, problem-solving, and communication, valuable in any field.
6. Can a problem-solving approach be used with all levels of mathematics? Yes, from elementary school to advanced university courses, a problem-solving approach can be adapted to different levels of mathematical understanding.
7. What role does technology play in a problem-solving approach to mathematics? Technology can provide tools for exploration, visualization, and simulation, enhancing the problem-solving experience.
8. How can we assess a student's understanding when using a problem-solving approach? Assessment should focus on the process, reasoning, and justification, not just the final answer. This might include observing students' work, analyzing their solution strategies, and having them explain their reasoning.
9. What are some examples of real-world problems that can be solved using mathematics? Real-world applications include modeling population growth, designing structures, analyzing data, and optimizing resource allocation.
Related Articles:
1. "The Art of Problem Solving: A Comprehensive Guide": This article provides a detailed overview of various problem-solving techniques and strategies.
2. "Problem-Solving in Algebra: A Practical Approach": This article focuses on applying problem-solving techniques to algebraic problems.
3. "Geometric Problem Solving: Visualizing and Reasoning": This article explores the use of visual reasoning in solving geometric problems.
4. "Calculus Problem Solving: Applications and Techniques": This article demonstrates the use of calculus in solving real-world problems.
5. "Statistical Problem Solving: Data Analysis and Interpretation": This article explores problem-solving within the context of statistical analysis.
6. "Polya's Method: A Timeless Approach to Problem Solving": This article delves into the four-step method developed by George Polya.
7. "Developing Problem-Solving Skills in Mathematics Education": This article discusses pedagogical approaches for fostering problem-solving skills in students.
8. "The Role of Heuristics in Mathematical Problem Solving": This article explores the use of heuristic methods in solving complex mathematical problems.
9. "Assessment and Evaluation in a Problem-Solving Based Mathematics Curriculum": This article provides guidance on assessing student learning in a problem-solving environment.
Publisher: Springer Nature - a leading global publisher of scientific and academic resources, known for its high-quality publications in mathematics and mathematics education.
Editor: Dr. David Chen, Ph.D. in Applied Mathematics, experienced editor with Springer Nature and specialist in mathematics education research.
| a problem solving approach to mathematics: A Problem Solving Approach to Mathematics for Elementary School Teachers Rick Billstein, Shlomo Libeskind, Johnny W. Lott, 1984-01-01 |
| a problem solving approach to mathematics: Maple and Mathematica Inna K. Shingareva, Carlos Lizárraga-Celaya, 2007-12-27 By presenting side-by-side comparisons, this handbook enables Mathematica users to quickly learn Maple, and vice versa. The parallel presentation enables students, mathematicians, scientists, and engineers to easily find equivalent functions on each of these algebra programs. The handbook provides core material for incorporating Maple and Mathematica as working tools into many different undergraduate mathematics courses. |
| a problem solving approach to mathematics: A Problem Solving Approach to Mathematics Rick Billstein, Shlomo Libeskind, Johnny W. Lott, 1993 |
| a problem solving approach to mathematics: A Problem Solving Approach to Mathematics for Elementary School Teachers Rick Billstein, Shlomo Libeskind, Johnny Lott, 2015-02-25 NOTE: You are purchasing a standalone product; MyMathLab does not come packaged with this content. If you would like to purchase both the physical text and MyMathLab search for ISBN-10: 0321990595/ISBN-13: 9780321990594 . That package includes ISBN-10: 0321431308/ISBN-13: 9780321431301, ISBN-10: 0321654064/ISBN-13: 9780321654069 and ISBN-10: 0321987292//ISBN-13: 9780321987297 . For courses in mathematics for elementary teachers. The Gold Standard for the New Standards A Problem Solving Approach to Mathematics for Elementary School Teachers has always reflected the content and processes set forth in today’s new state mathematics standards and the Common Core State Standards (CCSS). In the Twelfth Edition, the authors have further tightened the connections to the CCSS and made them more explicit. This text not only helps students learn the math by promoting active learning and developing skills and concepts—it also provides an invaluable reference to future teachers by including professional development features and discussions of today’s standards. Also available with MyMathLab MyMathLab is an online homework, tutorial, and assessment program designed to work with this text to engage students and improve results. MyMathLab includes assignable algorithmic exercises, the complete eBook, tutorial and classroom videos, eManipulatives, tools to personalize learning, and more. |
| a problem solving approach to mathematics: A Problem Solving Approach to Mathematics Rick Billstein, 2004 |
| a problem solving approach to mathematics: A Problem-Solving Approach to Mathematics for Elementary School Teachers Rick Billstein, 2025 Perhaps the largest change for the 14th edition is the new feature Illustrative Mathematics K-8 Student Tasks, which replaces the previous School Book Pages feature. Illustrative Mathematics is a series of open education resources (OER) available online and free to access by anyone. It covers all grades K-12, but we will focus on K-8 content for this book. The content from Illustrative Mathematics is immersive and engaging when implemented in the classroom. In this text, we have extracted elements from the series that focus on activities and student tasks. Students of this text have the opportunity to see what content that elementary and middle school students can actually see in their classrooms. Each section that contains this new feature will also have a block of exercises within the Mathematical Connections portions of the exercise sets that ask students questions around this content. The Activity Manual is all new for this edition. Written by the authors themselves, new classroom-tested activities demonstrate ways to engage students through active learning. New! Illustrative Mathematics (IM) K-8 Student Tasks replace the School Book Pages from the previous edition. This new content from open educational resources (OER) material is included to show how various topics are introduced to the K-8 pupil. Icons within the text link the narrative to the appropriate IM K-8 Student Task. Students are asked to complete many of the activities on the student pages so they can see what is expected in elementary school-- |
| a problem solving approach to mathematics: Discovering Mathematics Jiří Gregor, Jaroslav Tišer, 2010-12-21 The book contains chapters of structured approach to problem solving in mathematical analysis on an intermediate level. It follows the ideas of G.Polya and others, distinguishing between exercises and problem solving in mathematics. Interrelated concepts are connected by hyperlinks, pointing toward easier or more difficult problems so as to show paths of mathematical reasoning. Basic definitions and theorems can also be found by hyperlinks from relevant places. Problems are open to alternative formulations, generalizations, simplifications, and verification of hypotheses by the reader; this is shown to be helpful in solving problems. The book presents how advanced mathematical software can aid all stages of mathematical reasoning while the mathematical content remains in foreground. The authors show how software can contribute to deeper understanding and to enlarging the scope of teaching for students and teachers of mathematics. |
| a problem solving approach to mathematics: Problem-Solving Strategies Arthur Engel, 2008-01-19 A unique collection of competition problems from over twenty major national and international mathematical competitions for high school students. Written for trainers and participants of contests of all levels up to the highest level, this will appeal to high school teachers conducting a mathematics club who need a range of simple to complex problems and to those instructors wishing to pose a problem of the week, thus bringing a creative atmosphere into the classrooms. Equally, this is a must-have for individuals interested in solving difficult and challenging problems. Each chapter starts with typical examples illustrating the central concepts and is followed by a number of carefully selected problems and their solutions. Most of the solutions are complete, but some merely point to the road leading to the final solution. In addition to being a valuable resource of mathematical problems and solution strategies, this is the most complete training book on the market. |
| a problem solving approach to mathematics: Teaching Mathematics Through Problem-Solving Akihiko Takahashi, 2021-03-31 This engaging book offers an in-depth introduction to teaching mathematics through problem-solving, providing lessons and techniques that can be used in classrooms for both primary and lower secondary grades. Based on the innovative and successful Japanese approaches of Teaching Through Problem-solving (TTP) and Collaborative Lesson Research (CLR), renowned mathematics education scholar Akihiko Takahashi demonstrates how these teaching methods can be successfully adapted in schools outside of Japan. TTP encourages students to try and solve a problem independently, rather than relying on the format of lectures and walkthroughs provided in classrooms across the world. Teaching Mathematics Through Problem-Solving gives educators the tools to restructure their lesson and curriculum design to make creative and adaptive problem-solving the main way students learn new procedures. Takahashi showcases TTP lessons for elementary and secondary classrooms, showing how teachers can create their own TTP lessons and units using techniques adapted from Japanese educators through CLR. Examples are discussed in relation to the Common Core State Standards, though the methods and lessons offered can be used in any country. Teaching Mathematics Through Problem-Solving offers an innovative new approach to teaching mathematics written by a leading expert in Japanese mathematics education, suitable for pre-service and in-service primary and secondary math educators. |
| a problem solving approach to mathematics: Primary Problem-Solving in Mathematics George Booker, Denise Bond, 2010-01-29 A photocopiable series to develop problem solving skills and mathematical thinking in primary pupils. It provides activities that develop spatial visualisation, logical reasoning, establishing criteria, interpreting, analysing, organising and using information, strategic thinking and using patterns. |
| a problem solving approach to mathematics: Maple and Mathematica Inna K. Shingareva, Carlos Lizárraga-Celaya, 2009-08-14 In the history of mathematics there are many situations in which cal- lations were performed incorrectly for important practical applications. Let us look at some examples, the history of computing the number ? began in Egypt and Babylon about 2000 years BC, since then many mathematicians have calculated ? (e. g. , Archimedes, Ptolemy, Vi` ete, etc. ). The ?rst formula for computing decimal digits of ? was disc- ered by J. Machin (in 1706), who was the ?rst to correctly compute 100 digits of ?. Then many people used his method, e. g. , W. Shanks calculated ? with 707 digits (within 15 years), although due to mistakes only the ?rst 527 were correct. For the next examples, we can mention the history of computing the ?ne-structure constant ? (that was ?rst discovered by A. Sommerfeld), and the mathematical tables, exact - lutions, and formulas, published in many mathematical textbooks, were not veri?ed rigorously [25]. These errors could have a large e?ect on results obtained by engineers. But sometimes, the solution of such problems required such techn- ogy that was not available at that time. In modern mathematics there exist computers that can perform various mathematical operations for which humans are incapable. Therefore the computers can be used to verify the results obtained by humans, to discovery new results, to - provetheresultsthatahumancanobtainwithoutanytechnology. With respectto our example of computing?, we can mention that recently (in 2002) Y. Kanada, Y. Ushiro, H. Kuroda, and M. |
| a problem solving approach to mathematics: Problem Solving Approach to Mathematics, A (Recover) Plus MyMathLab Rick Billstein, Shlomo Libeskind, Johnny W. Lott, 2009-12-16 The new edition of this best-selling text includes a new focus on active and collaborative learning, while maintaining its emphasis on developing skills and concepts. With a wealth of pedagogical tools, as well as relevant discussions of standard curricula and assessments, this book will be a valuable textbook and reference for future teachers. With this revision, two new chapters are included to address the needs of future middle school teachers, in accordance to the NCTM Focal Points document. |
| a problem solving approach to mathematics: A Problem Solving Approach to Mathematics for Elementary School Teachers Rick Billstein, Shlomo Libeskind, Johnny W. Lott, 1997 |
| a problem solving approach to mathematics: Algebraic Geometry Thomas A. Garrity, 2013-02-01 Algebraic Geometry has been at the center of much of mathematics for hundreds of years. It is not an easy field to break into, despite its humble beginnings in the study of circles, ellipses, hyperbolas, and parabolas. This text consists of a series of ex |
| a problem solving approach to mathematics: A Problem Solving Approach to Mathematics for Elementary School Teachers Shiomo Liebeskind, Johnny W. Lott, 1984 |
| a problem solving approach to mathematics: Problem-Solving Through Problems Loren C. Larson, 2012-12-06 This is a practical anthology of some of the best elementary problems in different branches of mathematics. Arranged by subject, the problems highlight the most common problem-solving techniques encountered in undergraduate mathematics. This book teaches the important principles and broad strategies for coping with the experience of solving problems. It has been found very helpful for students preparing for the Putnam exam. |
| a problem solving approach to mathematics: Activities Manual for a Problem Solving Approach to Mathematics for Elementary School Teachers Dan Dolan, Jim Williamson, Mari Muri, 2019-01-12 |
| a problem solving approach to mathematics: Mathematical Problem Solving Peter Liljedahl, Manuel Santos-Trigo, 2019-02-12 This book contributes to the field of mathematical problem solving by exploring current themes, trends and research perspectives. It does so by addressing five broad and related dimensions: problem solving heuristics, problem solving and technology, inquiry and problem posing in mathematics education, assessment of and through problem solving, and the problem solving environment. Mathematical problem solving has long been recognized as an important aspect of mathematics, teaching mathematics, and learning mathematics. It has influenced mathematics curricula around the world, with calls for the teaching of problem solving as well as the teaching of mathematics through problem solving. And as such, it has been of interest to mathematics education researchers for as long as the field has existed. Research in this area has generally aimed at understanding and relating the processes involved in solving problems to students’ development of mathematical knowledge and problem solving skills. The accumulated knowledge and field developments have included conceptual frameworks for characterizing learners’ success in problem solving activities, cognitive, metacognitive, social and affective analysis, curriculum proposals, and ways to promote problem solving approaches. |
| a problem solving approach to mathematics: The Art and Craft of Problem Solving Paul Zeitz, 2017 This text on mathematical problem solving provides a comprehensive outline of problemsolving-ology, concentrating on strategy and tactics. It discusses a number of standard mathematical subjects such as combinatorics and calculus from a problem solver's perspective. |
| a problem solving approach to mathematics: Conceptual Model-Based Problem Solving Yan Ping Xin, 2013-02-11 Are you having trouble in finding Tier II intervention materials for elementary students who are struggling in math? Are you hungry for effective instructional strategies that will address students’ conceptual gap in additive and multiplicative math problem solving? Are you searching for a powerful and generalizable problem solving approach that will help those who are left behind in meeting the Common Core State Standards for Mathematics (CCSSM)? If so, this book is the answer for you. • The conceptual model-based problem solving (COMPS) program emphasizes mathematical modeling and algebraic representation of mathematical relations in equations, which are in line with the new Common Core. • “Through building most fundamental concepts pertinent to additive and multiplicative reasoning and making the connection between concrete and abstract modeling, students were prepared to go above and beyond concrete level of operation and be able to use mathematical models to solve more complex real-world problems. As the connection is made between the concrete model (or students’ existing knowledge scheme) and the symbolic mathematical algorithm, the abstract mathematical models are no longer “alien” to the students.” As Ms. Karen Combs, Director of Elementary Education of Lafayette School Corporation in Indiana, testified: “It really worked with our kids!” • “One hallmark of mathematical understanding is the ability to justify,... why a particular mathematical statement is true or where a mathematical rule comes from” (http://illustrativemathematics.org/standards). Through making connections between mathematical ideas, the COMPS program makes explicit the reasoning behind math, which has the potential to promote a powerful transfer of knowledge by applying the learned conception to solve other problems in new contexts. • Dr. Yan Ping Xin’s book contains essential tools for teachers to help students with learning disabilities or difficulties close the gap in mathematics word problem solving. I have witnessed many struggling students use these strategies to solve word problems and gain confidence as learners of mathematics. This book is a valuable resource for general and special education teachers of mathematics. - Casey Hord, PhD, University of Cincinnati |
| a problem solving approach to mathematics: A Problem-Solving Approach to Supporting Mathematics Instruction in Elementary School Sheldon N. Rothman, 2019 The book takes a problem-solving approach to learning elementary school mathematics, and develops many concepts using patterns established by examples. In many cases, the METHOD of problem-solving used is indicated. |
| a problem solving approach to mathematics: Lesson Study: Challenges In Mathematics Education Maitree Inprasitha, Masami Isoda, Patsy Wang-iverson, Ban Har Yeap, 2015-03-25 Classroom Innovations through Lesson Study is an APEC EDNET (Asia-Pacific Economic Cooperation Education Network) project that aims to improve the quality of education in the area of mathematics. This book includes challenges of lesson study implementation from members of the APEC economies.Lesson study is one of the best ways to improve the quality of teaching. It is a model approach for improvement of teacher education across the globe. This book focuses on mathematics education, teacher education, and curriculum implementation and reforms. |
| a problem solving approach to mathematics: The Stanford Mathematics Problem Book George Polya, Jeremy Kilpatrick, 2013-04-09 Based on Stanford University's well-known competitive exam, this excellent mathematics workbook offers students at both high school and college levels a complete set of problems, hints, and solutions. 1974 edition. |
| a problem solving approach to mathematics: A Problem Solving Approach to Mathematics for Elementary School Teachers Dan Dolan, Jim Williamson, Mari Muri, 2015-02-04 This manual provides hands-on, manipulative-based activities keyed to the text. These activities involve future elementary school teachers discovering concepts, solving problems, and exploring mathematical ideas. Colorful perforated, paper manipulatives are bound in a convenient storage pouch. Activities can also be adapted for use with elementary students at a later time. References to these activities are located in the margin of the Annotated Instructor s Edition. |
| a problem solving approach to mathematics: Street-Fighting Mathematics Sanjoy Mahajan, 2010-03-05 An antidote to mathematical rigor mortis, teaching how to guess answers without needing a proof or an exact calculation. In problem solving, as in street fighting, rules are for fools: do whatever works—don't just stand there! Yet we often fear an unjustified leap even though it may land us on a correct result. Traditional mathematics teaching is largely about solving exactly stated problems exactly, yet life often hands us partly defined problems needing only moderately accurate solutions. This engaging book is an antidote to the rigor mortis brought on by too much mathematical rigor, teaching us how to guess answers without needing a proof or an exact calculation. In Street-Fighting Mathematics, Sanjoy Mahajan builds, sharpens, and demonstrates tools for educated guessing and down-and-dirty, opportunistic problem solving across diverse fields of knowledge—from mathematics to management. Mahajan describes six tools: dimensional analysis, easy cases, lumping, picture proofs, successive approximation, and reasoning by analogy. Illustrating each tool with numerous examples, he carefully separates the tool—the general principle—from the particular application so that the reader can most easily grasp the tool itself to use on problems of particular interest. Street-Fighting Mathematics grew out of a short course taught by the author at MIT for students ranging from first-year undergraduates to graduate students ready for careers in physics, mathematics, management, electrical engineering, computer science, and biology. They benefited from an approach that avoided rigor and taught them how to use mathematics to solve real problems. Street-Fighting Mathematics will appear in print and online under a Creative Commons Noncommercial Share Alike license. |
| a problem solving approach to mathematics: Mathematical Thinking and Problem Solving Alan H. Schoenfeld, Alan H. Sloane, 2016-05-06 In the early 1980s there was virtually no serious communication among the various groups that contribute to mathematics education -- mathematicians, mathematics educators, classroom teachers, and cognitive scientists. Members of these groups came from different traditions, had different perspectives, and rarely gathered in the same place to discuss issues of common interest. Part of the problem was that there was no common ground for the discussions -- given the disparate traditions and perspectives. As one way of addressing this problem, the Sloan Foundation funded two conferences in the mid-1980s, bringing together members of the different communities in a ground clearing effort, designed to establish a base for communication. In those conferences, interdisciplinary teams reviewed major topic areas and put together distillations of what was known about them.* A more recent conference -- upon which this volume is based -- offered a forum in which various people involved in education reform would present their work, and members of the broad communities gathered would comment on it. The focus was primarily on college mathematics, informed by developments in K-12 mathematics. The main issues of the conference were mathematical thinking and problem solving. |
| a problem solving approach to mathematics: 21st Century Skills Bernie Trilling, Charles Fadel, 2012-02-07 This important resource introduces a framework for 21st Century learning that maps out the skills needed to survive and thrive in a complex and connected world. 21st Century content includes the basic core subjects of reading, writing, and arithmetic-but also emphasizes global awareness, financial/economic literacy, and health issues. The skills fall into three categories: learning and innovations skills; digital literacy skills; and life and career skills. This book is filled with vignettes, international examples, and classroom samples that help illustrate the framework and provide an exciting view of twenty-first century teaching and learning. Explores the three main categories of 21st Century Skills: learning and innovations skills; digital literacy skills; and life and career skills Addresses timely issues such as the rapid advance of technology and increased economic competition Based on a framework developed by the Partnership for 21st Century Skills (P21) The book contains a video with clips of classroom teaching. For more information on the book visit www.21stcenturyskillsbook.com. |
| a problem solving approach to mathematics: Understanding and Enriching Problem Solving in Primary Mathematics Patrick Barmby, David Bolden, Lynn Thompson, 2014-05-19 This up to date book is essential reading for all those teaching or training to teach primary mathematics. Problem solving is a key aspect of teaching and learning mathematics, but also an area where teachers and pupils often struggle. Set within the context of the new primary curriculum and drawing on research and practice, the book identifies the key knowledge and skills required in teaching and learning problem solving in mathematics, and examines how these and can be applied in the classroom. It explores the issues in depth while remaining straightforward and relevant, emphasises the enrichment of maths through problem-solving, and provides opportunities for teachers to reflect on and further develop their classroom practice. |
| a problem solving approach to mathematics: Approaches to Algebra N. Bednarz, C. Kieran, L. Lee, 2012-12-06 In Greek geometry, there is an arithmetic of magnitudes in which, in terms of numbers, only integers are involved. This theory of measure is limited to exact measure. Operations on magnitudes cannot be actually numerically calculated, except if those magnitudes are exactly measured by a certain unit. The theory of proportions does not have access to such operations. It cannot be seen as an arithmetic of ratios. Even if Euclidean geometry is done in a highly theoretical context, its axioms are essentially semantic. This is contrary to Mahoney's second characteristic. This cannot be said of the theory of proportions, which is less semantic. Only synthetic proofs are considered rigorous in Greek geometry. Arithmetic reasoning is also synthetic, going from the known to the unknown. Finally, analysis is an approach to geometrical problems that has some algebraic characteristics and involves a method for solving problems that is different from the arithmetical approach. 3. GEOMETRIC PROOFS OF ALGEBRAIC RULES Until the second half of the 19th century, Euclid's Elements was considered a model of a mathematical theory. This may be one reason why geometry was used by algebraists as a tool to demonstrate the accuracy of rules otherwise given as numerical algorithms. It may also be that geometry was one way to represent general reasoning without involving specific magnitudes. To go a bit deeper into this, here are three geometric proofs of algebraic rules, the frrst by Al-Khwarizmi, the other two by Cardano. |
| a problem solving approach to mathematics: Problem Solving in Mathematics Education Peter Liljedahl, Manuel Santos-Trigo, Uldarico Malaspina, Regina Bruder, 2016-06-27 This survey book reviews four interrelated areas: (i) the relevance of heuristics in problem-solving approaches – why they are important and what research tells us about their use; (ii) the need to characterize and foster creative problem-solving approaches – what type of heuristics helps learners devise and practice creative solutions; (iii) the importance that learners formulate and pursue their own problems; and iv) the role played by the use of both multiple-purpose and ad hoc mathematical action types of technologies in problem-solving contexts – what ways of reasoning learners construct when they rely on the use of digital technologies, and how technology and technology approaches can be reconciled. |
| a problem solving approach to mathematics: Differential Equations P. Mohana Shankar, 2018-04-17 The book takes a problem solving approach in presenting the topic of differential equations. It provides a complete narrative of differential equations showing the theoretical aspects of the problem (the how's and why's), various steps in arriving at solutions, multiple ways of obtaining solutions and comparison of solutions. A large number of comprehensive examples are provided to show depth and breadth and these are presented in a manner very similar to the instructor's class room work. The examples contain solutions from Laplace transform based approaches alongside the solutions based on eigenvalues and eigenvectors and characteristic equations. The verification of the results in examples is additionally provided using Runge-Kutta offering a holistic means to interpret and understand the solutions. Wherever necessary, phase plots are provided to support the analytical results. All the examples are worked out using MATLAB® taking advantage of the Symbolic Toolbox and LaTex for displaying equations. With the subject matter being presented through these descriptive examples, students will find it easy to grasp the concepts. A large number of exercises have been provided in each chapter to allow instructors and students to explore various aspects of differential equations. |
| a problem solving approach to mathematics: A Problem Solving Approach to Mathematics Fof Elementary School Teachers Louis L. Levy, Richard Billstein, 1997 |
| a problem solving approach to mathematics: Precalculus Mathematics Walter Fleming, Dale E. Varberg, 1989 |
| a problem solving approach to mathematics: Solving Mathematical Problems Terence Tao, 2006-07-28 Authored by a leading name in mathematics, this engaging and clearly presented text leads the reader through the tactics involved in solving mathematical problems at the Mathematical Olympiad level. With numerous exercises and assuming only basic mathematics, this text is ideal for students of 14 years and above in pure mathematics. |
| a problem solving approach to mathematics: Unlocking Creativity in Solving Novel Mathematics Problems Carol R. Aldous, 2019-06-25 Unlocking Creativity in Solving Novel Mathematics Problems delivers a fascinating insight into thinking and feeling approaches used in creative problem solving and explores whether attending to ‘feeling’ makes any difference to solving novel problems successfully. With a focus on research throughout, this book reveals ways of identifying, describing and measuring ‘feeling’ (or ‘intuition’) in problem-solving processes. It details construction of a new creative problem-solving conceptual framework using cognitive and non-cognitive elements, including the brain’s visuo-spatial and linguistic circuits, conscious and non-conscious mental activity, and the generation of feeling in listening to the self, identified from verbal data. This framework becomes the process model for developing a comprehensive quantitative model of creative problem solving incorporating the Person, Product, Process and Environment dimensions of creativity. In a world constantly seeking new ideas and new approaches to solving complex problems, the application of this book’s findings will revolutionize the way students, teachers, businesses and industries approach novel problem solving, and mathematics learning and teaching. |
| a problem solving approach to mathematics: Doing Physics with Scientific Notebook Joseph Gallant, 2012-03-05 The goal of this book is to teach undergraduate students how to use Scientific Notebook (SNB) to solve physics problems. SNB software combines word processing and mathematics in standard notation with the power of symbolic computation. As its name implies, SNB can be used as a notebook in which students set up a math or science problem, write and solve equations, and analyze and discuss their results. Written by a physics teacher with over 20 years experience, this text includes topics that have educational value, fit within the typical physics curriculum, and show the benefits of using SNB. This easy-to-read text: Provides step-by-step instructions for using Scientific Notebook (SNB) to solve physics problems Features examples in almost every section to enhance the reader's understanding of the relevant physics and to provide detailed instructions on using SNB Follows the traditional physics curriculum, so it can be used to supplement teaching at all levels of undergraduate physics Includes many problems taken from the author’s class notes and research Aimed at undergraduate physics and engineering students, this text teaches readers how to use SNB to solve some everyday physics problems. |
| a problem solving approach to mathematics: Learning MATLAB Walter Gander, 2015-11-21 This comprehensive and stimulating introduction to Matlab, a computer language now widely used for technical computing, is based on an introductory course held at Qian Weichang College, Shanghai University, in the fall of 2014. Teaching and learning a substantial programming language aren’t always straightforward tasks. Accordingly, this textbook is not meant to cover the whole range of this high-performance technical programming environment, but to motivate first- and second-year undergraduate students in mathematics and computer science to learn Matlab by studying representative problems, developing algorithms and programming them in Matlab. While several topics are taken from the field of scientific computing, the main emphasis is on programming. A wealth of examples are completely discussed and solved, allowing students to learn Matlab by doing: by solving problems, comparing approaches and assessing the proposed solutions. |
| a problem solving approach to mathematics: Digital Media Rimon Elias, 2014-03-27 Focusing on the computer graphics required to create digital media this book discusses the concepts and provides hundreds of solved examples and unsolved problems for practice. Pseudo codes are included where appropriate but these coding examples do not rely on specific languages. The aim is to get readers to understand the ideas and how concepts and algorithms work, through practicing numeric examples. Topics covered include: 2D Graphics 3D Solid Modelling Mapping Techniques Transformations in 2D and 3D Space Illuminations, Lighting and Shading Ideal as an upper level undergraduate text, Digital Media – A Problem-solving Approach for Computer Graphic, approaches the field at a conceptual level thus no programming experience is required, just a basic knowledge of mathematics and linear algebra. |
| a problem solving approach to mathematics: Advanced Problems in Mathematics Stephen Siklos, 2019-10-16 This new and expanded edition is intended to help candidates prepare for entrance examinations in mathematics and scientific subjects, including STEP (Sixth Term Examination Paper). STEP is an examination used by Cambridge Colleges for conditional offers in mathematics. They are also used by some other UK universities and many mathematics departments recommend that their applicants practice on the past papers even if they do not take the examination. Advanced Problems in Mathematics bridges the gap between school and university mathematics, and prepares students for an undergraduate mathematics course. The questions analysed in this book are all based on past STEP questions and each question is followed by a comment and a full solution. The comments direct the reader's attention to key points and put the question in its true mathematical context. The solutions point students to the methodology required to address advanced mathematical problems critically and independently. This book is a must read for any student wishing to apply to scientific subjects at university level and for anyone interested in advanced mathematics. |
| a problem solving approach to mathematics: Maths Problems Galore Prim-Ed Publishing Staff, 1995 Features 50 photocopiable problem solving activities. This title covers number, handling data, shape, space and measurement. It provides interesting activities for early finishers. |
PROBLEM Definition & Meaning - Merriam-Webster
The meaning of PROBLEM is a question raised for inquiry, consideration, or solution. How to use problem in a sentence. Synonym Discussion of Problem.
PROBLEM | English meaning - Cambridge Dictionary
PROBLEM definition: 1. a situation, person, or thing that needs attention and needs to be dealt with or solved: 2. a…. Learn more.
Problem - definition of problem by The Free Dictionary
1. Difficult to deal with or control: a problem child. 2. Dealing with a moral or social problem: a problem play.
problem, n. meanings, etymology and more | Oxford English …
What does the noun problem mean? There are nine meanings listed in OED's entry for the noun problem, three of which are labelled obsolete. See ‘Meaning & use’ for definitions, usage, and …
672 Synonyms & Antonyms for PROBLEM | Thesaurus.com
Find 672 different ways to say PROBLEM, along with antonyms, related words, and example sentences at Thesaurus.com.
problem - Wiktionary, the free dictionary
May 17, 2025 · problem (plural problems) A difficulty that has to be resolved or dealt with. Hypernyms: challenge, issue, obstacle She's leaving because she faced numerous problems …
What does Problem mean? - Definitions.net
A problem can be defined as a situation or an issue that needs to be resolved or dealt with. It typically involves a discrepancy between the current state or desired situation and the actual …
problem - WordReference.com Dictionary of English
any question or matter involving doubt or difficulty: has financial and emotional problems. a statement requiring a solution, usually by means of mathematical operations: simple problems …
PROBLEM Definition & Meaning | Dictionary.com
What is a basic definition of problem? A problem is a situation, question, or thing that causes difficulty, stress, or doubt. A problem is also a question raised to inspire thought. In …
Problem Definition & Meaning | YourDictionary
Problem definition: A question to be considered, solved, or answered.
PROBLEM Definition & Meaning - Merriam-Webster
The meaning of PROBLEM is a question raised for inquiry, consideration, or solution. How to use problem in a sentence. Synonym Discussion of Problem.
PROBLEM | English meaning - Cambridge Dictionary
PROBLEM definition: 1. a situation, person, or thing that needs attention and needs to be dealt with or solved: 2. a…. Learn more.
Problem - definition of problem by The Free Dictionary
1. Difficult to deal with or control: a problem child. 2. Dealing with a moral or social problem: a problem play.
problem, n. meanings, etymology and more | Oxford English …
What does the noun problem mean? There are nine meanings listed in OED's entry for the noun problem, three of which are labelled obsolete. See ‘Meaning & use’ for definitions, usage, and …
672 Synonyms & Antonyms for PROBLEM | Thesaurus.com
Find 672 different ways to say PROBLEM, along with antonyms, related words, and example sentences at Thesaurus.com.
problem - Wiktionary, the free dictionary
May 17, 2025 · problem (plural problems) A difficulty that has to be resolved or dealt with. Hypernyms: challenge, issue, obstacle She's leaving because she faced numerous problems …
What does Problem mean? - Definitions.net
A problem can be defined as a situation or an issue that needs to be resolved or dealt with. It typically involves a discrepancy between the current state or desired situation and the actual …
problem - WordReference.com Dictionary of English
any question or matter involving doubt or difficulty: has financial and emotional problems. a statement requiring a solution, usually by means of mathematical operations: simple problems …
PROBLEM Definition & Meaning | Dictionary.com
What is a basic definition of problem? A problem is a situation, question, or thing that causes difficulty, stress, or doubt. A problem is also a question raised to inspire thought. In …
Problem Definition & Meaning | YourDictionary
Problem definition: A question to be considered, solved, or answered.
