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Unlocking the Power of Math Learning: Strategies and Tools for Success

posted on September 20, 2023

Mathematics, the foundation of all sciences and technology, plays a fundamental role in our everyday lives. Yet many students find the subject challenging, causing them to shy away from it altogether. This reluctance is often due to a lack of confidence, a misunderstanding of unclear concepts, a move ahead to more advanced skills before they are ready, and ineffective learning methods. However, with the right approach, math learning can be both rewarding and empowering. This post will explore different approaches to learning math, strategies for success, and cutting-edge tools to help you achieve your goals.

Math Learning

Math learning can take many forms, including traditional classroom instruction, online courses, and self-directed learning. A multifaceted approach to math learning can improve understanding, engage students, and promote subject mastery. A 2014 study by the National Council of Teachers of Mathematics found that the use of multiple representations, such as visual aids, graphs, and real-world examples, supports the development of mathematical connections, reasoning, and problem-solving skills.

Moreover, the importance of math learning goes beyond solving equations and formulas. Advanced math skills are essential for success in many fields, including science, engineering, finance, health care, and technology. In fact, a report by Burning Glass Technologies found that 71% of high-salary, entry-level positions require advanced math skills.

Benefits of Math Learning

In today’s 21st-century world, having a broad knowledge base and strong reading and math skills is essential. Mathematical literacy plays a crucial role in this success. It empowers individuals to comprehend the world around them and make well-informed decisions based on data-driven understanding. More than just earning good grades in math, mathematical literacy is a vital life skill that can open doors to economic opportunities, improve financial management, and foster critical thinking. We’re not the only ones who say so:

• Math learning enhances problem-solving skills, critical thinking, and logical reasoning abilities. (Source: National Council of Teachers of Mathematics )
• It improves analytical skills that can be applied in various real-life situations, such as budgeting or analyzing data. (Source: Southern New Hampshire University )
• Math learning promotes creativity and innovation by fostering a deep understanding of patterns and relationships. (Source: Purdue University )
• It provides a strong foundation for careers in fields such as engineering, finance, computer science, and more. These careers generally correlate to high wages. (Source: U.S. Bureau of Labor Statistics )
• Math skills are transferable and can be applied across different academic disciplines. (Source: Sydney School of Education and Social Work )

How to Know What Math You Need to Learn

Often students will find gaps in their math knowledge; this can occur at any age or skill level. As math learning is generally iterative, a solid foundation and understanding of the math skills that preceded current learning are key to success. The solution to these gaps is called mastery learning, the philosophy that underpins Khan Academy’s approach to education .

Mastery learning is an educational philosophy that emphasizes the importance of a student fully understanding a concept before moving on to the next one. Rather than rushing students through a curriculum, mastery learning asks educators to ensure that learners have “mastered” a topic or skill, showing a high level of proficiency and understanding, before progressing. This approach is rooted in the belief that all students can learn given the appropriate learning conditions and enough time, making it a markedly student-centered method. It promotes thoroughness over speed and encourages individualized learning paths, thus catering to the unique learning needs of each student.

Students will encounter mastery learning passively as they go through Khan Academy coursework, as our platform identifies gaps and systematically adjusts to support student learning outcomes. More details can be found in our Educators Hub . 

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How to learn math.

Learning at School

One of the most common methods of math instruction is classroom learning. In-class instruction provides students with real-time feedback, practical application, and a peer-learning environment. Teachers can personalize instruction by assessing students’ strengths and weaknesses, providing remediation when necessary, and offering advanced instruction to students who need it.

Learning at Home

Supplemental learning at home can complement traditional classroom instruction. For example, using online resources that provide additional practice opportunities, interactive games, and demonstrations, can help students consolidate learning outside of class. E-learning has become increasingly popular, with a wealth of online resources available to learners of all ages. The benefits of online learning include flexibility, customization, and the ability to work at one’s own pace. One excellent online learning platform is Khan Academy, which offers free video tutorials, interactive practice exercises, and a wealth of resources across a range of mathematical topics.

Moreover, parents can encourage and monitor progress, answer questions, and demonstrate practical applications of math in everyday life. For example, when at the grocery store, parents can ask their children to help calculate the price per ounce of two items to discover which one is the better deal. Cooking and baking with your children also provides a lot of opportunities to use math skills, like dividing a recipe in half or doubling the ingredients. 

Learning Math with the Help of Artificial Intelligence (AI) 

AI-powered tools are changing the way students learn math. Personalized feedback and adaptive practice help target individual needs. Virtual tutors offer real-time help with math concepts while AI algorithms identify areas for improvement. Custom math problems provide tailored practice, and natural language processing allows for instant question-and-answer sessions. 

Using Khan Academy’s AI Tutor, Khanmigo

Transform your child’s grasp of mathematics with Khanmigo , the 24/7 AI-powered tutor that specializes in tailored, one-on-one math instruction. Available at any time, Khanmigo provides personalized support that goes beyond mere answers to nurture genuine mathematical understanding and critical thinking. Khanmigo can track progress, identify strengths and weaknesses, and offer real-time feedback to help students stay on the right track. Within a secure and ethical AI framework, your child can tackle everything from basic arithmetic to complex calculus, all while you maintain oversight using robust parental controls.

Get Math Help with Khanmigo Right Now

You can learn anything .

Math learning is essential for success in the modern world, and with the right approach, it can also be enjoyable and rewarding. Learning math requires curiosity, diligence, and the ability to connect abstract concepts with real-world applications. Strategies for effective math learning include a multifaceted approach, including classroom instruction, online courses, homework, tutoring, and personalized AI support. 

So, don’t let math anxiety hold you back; take advantage of available resources and technology to enhance your knowledge base and enjoy the benefits of math learning.

National Council of Teachers of Mathematics, “Principles to Actions: Ensuring Mathematical Success for All” , April 2014

Project Lead The Way Research Report, “The Power of Transportable Skills: Assessing the Demand and Value of the Skills of the Future” , 2020

Page. M, “Why Develop Quantitative and Qualitative Data Analysis Skills?” , 2016

Mann. EL, Creativity: The Essence of Mathematics, Journal for the Education of the Gifted. Vol. 30, No. 2, 2006, pp. 236–260, http://www.prufrock.com ’

Nakakoji Y, Wilson R.” Interdisciplinary Learning in Mathematics and Science: Transfer of Learning for 21st Century Problem Solving at University ”. J Intell. 2020 Sep 1;8(3):32. doi: 10.3390/jintelligence8030032. PMID: 32882908; PMCID: PMC7555771.

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Mathematical Problem Solving

Current Themes, Trends, and Research

• © 2019
• Peter Liljedahl 0 ,
• Manuel Santos-Trigo 1

Faculty of Education, Simon Fraser University, Burnaby, Canada

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Department of Mathematics Education, Cinvestav-Instituto Politecnico Nacional, Mexico City, Mexico

• Brings together some of the latest research on problem solving
• Offers international perspectives on current themes, trends, and research on problem solving
• Presents multiple frameworks and views on problem solving

Part of the book series: ICME-13 Monographs (ICME13Mo)

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About this book

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. 

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Can Mathematical Problem Solving Be Taught? Preliminary Answers from 30 Years of Research

Reflections on Problem-Solving

Problem solving in mathematics education: tracing its foundations and current research-practice trends

• Mathematical problem solving
• Problem solving heuristics
• Problem posing in mathematics education
• Problem solving environment
• Future-oriented Thinking in Mathematical Problem Solving
• Mathematical problem solving with technology
• Spreadsheet affordances in solving complex word problems
• Didactic analysis by means of problem posing
• Mathematical problem solving competencies
• Activating Problem Solving in the Classroom
• Creating and sustaining online problem-solving forums
• Methods in Evaluating Metacognitive Strategies

Table of contents (15 chapters)

Front matter, problem solving heuristics, “looking back” to solve differently: familiarity, fluency, and flexibility.

• Hartono Tjoe

Future-Oriented Thinking and Activity in Mathematical Problem Solving

• Wes Maciejewski

Problem Solving and Technology

A model of mathematical problem solving with technology: the case of marco solving-and-expressing two geometry problems.

• Susana Carreira, Hélia Jacinto

Mathematical Problem Solving and the Use of Digital Technologies

Manuel Santos-Trigo

The Spreadsheet Affordances in Solving Complex Word Problems

• Nélia Amado, Susana Carreira, Sandra Nobre

Inquiry and Problem Posing in Mathematics Education

Is an inquiry-based approach possible at the elementary school.

• Magali Hersant, Christine Choquet

How to Stimulate In-Service Teachers’ Didactic Analysis Competence by Means of Problem Posing

• Uldarico Malaspina, Carlos Torres, Norma Rubio

Assessment of and Through Problem Solving

The impact of various methods in evaluating metacognitive strategies in mathematical problem solving.

• Mei Yoke Loh, Ngan Hoe Lee

Assessing Inquiry-Based Mathematics Education with Both a Summative and Formative Purpose

• Maud Chanudet

Beyond the Standardized Assessment of Mathematical Problem Solving Competencies: From Products to Processes

• Pietro Di Martino, Giulia Signorini

Toward Designing and Developing Likert Items to Assess Mathematical Problem Solving

• James A. Mendoza Álvarez, Kathryn Rhoads, R. Cavender Campbell

The Problem Solving Environment

Creating and sustaining online problem solving forums: two perspectives.

• Boris Koichu, Nelly Keller

Conditions for Supporting Problem Solving: Vertical Non-permanent Surfaces

Peter Liljedahl

The ARPA Experience in Chile: Problem Solving for Teachers’ Professional Development

• Patricio Felmer, Josefa Perdomo-Díaz, Cristián Reyes

“The volume will be a useful addition to the libraries of education researchers and to mathematics teacher educators who are charged with supporting preservice or inservice teachers’ development of capacities to incorporate problem solving into their instruction. … this volume offer interesting insights into the phenomenon of problem solving and the technological tools and learning environments that support students to engage in problem solving.” (Duane Gaysay, MAA Reviews, August 16, 2020)

Editors and Affiliations

Bibliographic information.

Book Title : Mathematical Problem Solving

Book Subtitle : Current Themes, Trends, and Research

Editors : Peter Liljedahl, Manuel Santos-Trigo

Series Title : ICME-13 Monographs

DOI : https://doi.org/10.1007/978-3-030-10472-6

Publisher : Springer Cham

eBook Packages : Education , Education (R0)

Copyright Information : Springer Nature Switzerland AG 2019

Hardcover ISBN : 978-3-030-10471-9 Published: 21 February 2019

eBook ISBN : 978-3-030-10472-6 Published: 12 February 2019

Series ISSN : 2520-8322

Series E-ISSN : 2520-8330

Edition Number : 1

Number of Pages : XI, 362

Number of Illustrations : 30 b/w illustrations, 82 illustrations in colour

Topics : Mathematics Education , Teaching and Teacher Education , Study and Learning Skills , International and Comparative Education

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5 Teaching Mathematics Through Problem Solving

Janet Stramel

In his book “How to Solve It,” George Pólya (1945) said, “One of the most important tasks of the teacher is to help his students. This task is not quite easy; it demands time, practice, devotion, and sound principles. The student should acquire as much experience of independent work as possible. But if he is left alone with his problem without any help, he may make no progress at all. If the teacher helps too much, nothing is left to the student. The teacher should help, but not too much and not too little, so that the student shall have a reasonable share of the work.” (page 1)

What is a problem  in mathematics? A problem is “any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method” (Hiebert, et. al., 1997). Problem solving in mathematics is one of the most important topics to teach; learning to problem solve helps students develop a sense of solving real-life problems and apply mathematics to real world situations. It is also used for a deeper understanding of mathematical concepts. Learning “math facts” is not enough; students must also learn how to use these facts to develop their thinking skills.

According to NCTM (2010), the term “problem solving” refers to mathematical tasks that have the potential to provide intellectual challenges for enhancing students’ mathematical understanding and development. When you first hear “problem solving,” what do you think about? Story problems or word problems? Story problems may be limited to and not “problematic” enough. For example, you may ask students to find the area of a rectangle, given the length and width. This type of problem is an exercise in computation and can be completed mindlessly without understanding the concept of area. Worthwhile problems  includes problems that are truly problematic and have the potential to provide contexts for students’ mathematical development.

There are three ways to solve problems: teaching for problem solving, teaching about problem solving, and teaching through problem solving.

Teaching for problem solving begins with learning a skill. For example, students are learning how to multiply a two-digit number by a one-digit number, and the story problems you select are multiplication problems. Be sure when you are teaching for problem solving, you select or develop tasks that can promote the development of mathematical understanding.

Teaching about problem solving begins with suggested strategies to solve a problem. For example, “draw a picture,” “make a table,” etc. You may see posters in teachers’ classrooms of the “Problem Solving Method” such as: 1) Read the problem, 2) Devise a plan, 3) Solve the problem, and 4) Check your work. There is little or no evidence that students’ problem-solving abilities are improved when teaching about problem solving. Students will see a word problem as a separate endeavor and focus on the steps to follow rather than the mathematics. In addition, students will tend to use trial and error instead of focusing on sense making.

Teaching through problem solving  focuses students’ attention on ideas and sense making and develops mathematical practices. Teaching through problem solving also develops a student’s confidence and builds on their strengths. It allows for collaboration among students and engages students in their own learning.

Consider the following worthwhile-problem criteria developed by Lappan and Phillips (1998):

• The problem has important, useful mathematics embedded in it.
• The problem requires high-level thinking and problem solving.
• The problem contributes to the conceptual development of students.
• The problem creates an opportunity for the teacher to assess what his or her students are learning and where they are experiencing difficulty.
• The problem can be approached by students in multiple ways using different solution strategies.
• The problem has various solutions or allows different decisions or positions to be taken and defended.
• The problem encourages student engagement and discourse.
• The problem connects to other important mathematical ideas.
• The problem promotes the skillful use of mathematics.
• The problem provides an opportunity to practice important skills.

Of course, not every problem will include all of the above. Sometimes, you will choose a problem because your students need an opportunity to practice a certain skill.

Key features of a good mathematics problem includes:

• It must begin where the students are mathematically.
• The feature of the problem must be the mathematics that students are to learn.
• It must require justifications and explanations for both answers and methods of solving.

Problem solving is not a  neat and orderly process. Think about needlework. On the front side, it is neat and perfect and pretty.

But look at the b ack.

It is messy and full of knots and loops. Problem solving in mathematics is also like this and we need to help our students be “messy” with problem solving; they need to go through those knots and loops and learn how to solve problems with the teacher’s guidance.

When you teach through problem solving , your students are focused on ideas and sense-making and they develop confidence in mathematics!

Mathematics Tasks and Activities that Promote Teaching through Problem Solving

Choosing the Right Task

Selecting activities and/or tasks is the most significant decision teachers make that will affect students’ learning. Consider the following questions:

• Teachers must do the activity first. What is problematic about the activity? What will you need to do BEFORE the activity and AFTER the activity? Additionally, think how your students would do the activity.
• What mathematical ideas will the activity develop? Are there connections to other related mathematics topics, or other content areas?
• Can the activity accomplish your learning objective/goals?

Low Floor High Ceiling Tasks

By definition, a “ low floor/high ceiling task ” is a mathematical activity where everyone in the group can begin and then work on at their own level of engagement. Low Floor High Ceiling Tasks are activities that everyone can begin and work on based on their own level, and have many possibilities for students to do more challenging mathematics. One gauge of knowing whether an activity is a Low Floor High Ceiling Task is when the work on the problems becomes more important than the answer itself, and leads to rich mathematical discourse [Hover: ways of representing, thinking, talking, agreeing, and disagreeing; the way ideas are exchanged and what the ideas entail; and as being shaped by the tasks in which students engage as well as by the nature of the learning environment].

The strengths of using Low Floor High Ceiling Tasks:

• Allows students to show what they can do, not what they can’t.
• Provides differentiation to all students.
• Promotes a positive classroom environment.
• Advances a growth mindset in students
• Aligns with the Standards for Mathematical Practice

Examples of some Low Floor High Ceiling Tasks can be found at the following sites:

• YouCubed – under grades choose Low Floor High Ceiling
• NRICH Creating a Low Threshold High Ceiling Classroom
• Inside Mathematics Problems of the Month

Math in 3-Acts

Math in 3-Acts was developed by Dan Meyer to spark an interest in and engage students in thought-provoking mathematical inquiry. Math in 3-Acts is a whole-group mathematics task consisting of three distinct parts:

Act One is about noticing and wondering. The teacher shares with students an image, video, or other situation that is engaging and perplexing. Students then generate questions about the situation.

In Act Two , the teacher offers some information for the students to use as they find the solutions to the problem.

Act Three is the “reveal.” Students share their thinking as well as their solutions.

“Math in 3 Acts” is a fun way to engage your students, there is a low entry point that gives students confidence, there are multiple paths to a solution, and it encourages students to work in groups to solve the problem. Some examples of Math in 3-Acts can be found at the following websites:

• Dan Meyer’s Three-Act Math Tasks
• Graham Fletcher3-Act Tasks ]
• Math in 3-Acts: Real World Math Problems to Make Math Contextual, Visual and Concrete

Number Talks

Number talks are brief, 5-15 minute discussions that focus on student solutions for a mental math computation problem. Students share their different mental math processes aloud while the teacher records their thinking visually on a chart or board. In addition, students learn from each other’s strategies as they question, critique, or build on the strategies that are shared.. To use a “number talk,” you would include the following steps:

• The teacher presents a problem for students to solve mentally.
• Provide adequate “ wait time .”
• The teacher calls on a students and asks, “What were you thinking?” and “Explain your thinking.”
• For each student who volunteers to share their strategy, write their thinking on the board. Make sure to accurately record their thinking; do not correct their responses.
• Invite students to question each other about their strategies, compare and contrast the strategies, and ask for clarification about strategies that are confusing.

“Number Talks” can be used as an introduction, a warm up to a lesson, or an extension. Some examples of Number Talks can be found at the following websites:

• Inside Mathematics Number Talks
• Number Talks Build Numerical Reasoning

Saying “This is Easy”

“This is easy.” Three little words that can have a big impact on students. What may be “easy” for one person, may be more “difficult” for someone else. And saying “this is easy” defeats the purpose of a growth mindset classroom, where students are comfortable making mistakes.

When the teacher says, “this is easy,” students may think,

• “Everyone else understands and I don’t. I can’t do this!”
• Students may just give up and surrender the mathematics to their classmates.
• Students may shut down.

Instead, you and your students could say the following:

• “I think I can do this.”
• “I have an idea I want to try.”
• “I’ve seen this kind of problem before.”

Tracy Zager wrote a short article, “This is easy”: The Little Phrase That Causes Big Problems” that can give you more information. Read Tracy Zager’s article here.

Using “Worksheets”

Do you want your students to memorize concepts, or do you want them to understand and apply the mathematics for different situations?

What is a “worksheet” in mathematics? It is a paper and pencil assignment when no other materials are used. A worksheet does not allow your students to use hands-on materials/manipulatives [Hover: physical objects that are used as teaching tools to engage students in the hands-on learning of mathematics]; and worksheets are many times “naked number” with no context. And a worksheet should not be used to enhance a hands-on activity.

Students need time to explore and manipulate materials in order to learn the mathematics concept. Worksheets are just a test of rote memory. Students need to develop those higher-order thinking skills, and worksheets will not allow them to do that.

One productive belief from the NCTM publication, Principles to Action (2014), states, “Students at all grade levels can benefit from the use of physical and virtual manipulative materials to provide visual models of a range of mathematical ideas.”

You may need an “activity sheet,” a “graphic organizer,” etc. as you plan your mathematics activities/lessons, but be sure to include hands-on manipulatives. Using manipulatives can

• Provide your students a bridge between the concrete and abstract
• Serve as models that support students’ thinking
• Provide another representation
• Support student engagement
• Give students ownership of their own learning.

Adapted from “ The Top 5 Reasons for Using Manipulatives in the Classroom ”.

any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method

should be intriguing and contain a level of challenge that invites speculation and hard work, and directs students to investigate important mathematical ideas and ways of thinking toward the learning

involves teaching a skill so that a student can later solve a story problem

when we teach students how to problem solve

teaching mathematics content through real contexts, problems, situations, and models

a mathematical activity where everyone in the group can begin and then work on at their own level of engagement

20 seconds to 2 minutes for students to make sense of questions

Mathematics Methods for Early Childhood Copyright © 2021 by Janet Stramel is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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Mathematics Through Problem Solving

What Is A ‘Problem-Solving Approach’?

• interactions between students/students and teacher/students (Van Zoest et al., 1994)
• mathematical dialogue and consensus between students (Van Zoest et al., 1994)
• teachers providing just enough information to establish background/intent of the problem, and students clarifing, interpreting, and attempting to construct one or more solution processes (Cobb et al., 1991)
• teachers accepting right/wrong answers in a non-evaluative way (Cobb et al., 1991)
• teachers guiding, coaching, asking insightful questions and sharing in the process of solving problems (Lester et al., 1994)
• teachers knowing when it is appropriate to intervene, and when to step back and let the pupils make their own way (Lester et al., 1994)
• A further characteristic is that a problem-solving approach can be used to encourage students to make generalisations about rules and concepts, a process which is central to mathematics (Evan and Lappin, 1994).

Schoenfeld (in Olkin and Schoenfeld, 1994, p.43) described the way in which the use of problem solving in his teaching has changed since the 1970s:

My early problem-solving courses focused on problems amenable to solutions by Polya-type heuristics: draw a diagram, examine special cases or analogies, specialize, generalize, and so on. Over the years the courses evolved to the point where they focused less on heuristics per se and more on introducing students to fundamental ideas: the importance of mathematical reasoning and proof…, for example, and of sustained mathematical investigations (where my problems served as starting points for serious explorations, rather than tasks to be completed).

Schoenfeld also suggested that a good problem should be one which can be extended to lead to mathematical explorations and generalisations. He described three characteristics of mathematical thinking:

• valuing the processes of mathematization and abstraction and having the predilection to apply them
• developing competence with the tools of the trade and using those tools in the service of the goal of understanding structure – mathematical sense-making (Schoenfeld, 1994, p.60).
• As Cobb et al. (1991) suggested, the purpose for engaging in problem solving is not just to solve specific problems, but to ‘encourage the interiorization and reorganization of the involved schemes as a result of the activity’ (p.187). Not only does this approach develop students’ confidence in their own ability to think mathematically (Schifter and Fosnot, 1993), it is a vehicle for students to construct, evaluate and refine their own theories about mathematics and the theories of others (NCTM, 1989). Because it has become so predominant a requirement of teaching, it is important to consider the processes themselves in more detail.

The Role of Problem Solving in Teaching Mathematics as a Process

Problem solving is an important component of mathematics education because it is the single vehicle which seems to be able to achieve at school level all three of the values of mathematics listed at the outset of this article: functional, logical and aesthetic. Let us consider how problem solving is a useful medium for each of these.

It has already been pointed out that mathematics is an essential discipline because of its practical role to the individual and society. Through a problem-solving approach, this aspect of mathematics can be developed. Presenting a problem and developing the skills needed to solve that problem is more motivational than teaching the skills without a context. Such motivation gives problem solving special value as a vehicle for learning new concepts and skills or the reinforcement of skills already acquired (Stanic and Kilpatrick, 1989, NCTM, 1989). Approaching mathematics through problem solving can create a context which simulates real life and therefore justifies the mathematics rather than treating it as an end in itself. The National Council of Teachers of Mathematics (NCTM, 1980) recommended that problem solving be the focus of mathematics teaching because, they say, it encompasses skills and functions which are an important part of everyday life. Furthermore it can help people to adapt to changes and unexpected problems in their careers and other aspects of their lives. More recently the Council endorsed this recommendation (NCTM, 1989) with the statement that problem solving should underly all aspects of mathematics teaching in order to give students experience of the power of mathematics in the world around them. They see problem solving as a vehicle for students to construct, evaluate and refine their own theories about mathematics and the theories of others.

According to Resnick (1987) a problem-solving approach contributes to the practical use of mathematics by helping people to develop the facility to be adaptable when, for instance, technology breaks down. It can thus also help people to transfer into new work environments at this time when most are likely to be faced with several career changes during a working lifetime (NCTM, 1989). Resnick expressed the belief that ‘school should focus its efforts on preparing people to be good adaptive learners, so that they can perform effectively when situations are unpredictable and task demands change’ (p.18). Cockcroft (1982) also advocated problem solving as a means of developing mathematical thinking as a tool for daily living, saying that problem-solving ability lies ‘at the heart of mathematics’ (p.73) because it is the means by which mathematics can be applied to a variety of unfamiliar situations.

Problem solving is, however, more than a vehicle for teaching and reinforcing mathematical knowledge and helping to meet everyday challenges. It is also a skill which can enhance logical reasoning. Individuals can no longer function optimally in society by just knowing the rules to follow to obtain a correct answer. They also need to be able to decide through a process of logical deduction what algorithm, if any, a situation requires, and sometimes need to be able to develop their own rules in a situation where an algorithm cannot be directly applied. For these reasons problem solving can be developed as a valuable skill in itself, a way of thinking (NCTM, 1989), rather than just as the means to an end of finding the correct answer.

Many writers have emphasised the importance of problem solving as a means of developing the logical thinking aspect of mathematics. ‘If education fails to contribute to the development of the intelligence, it is obviously incomplete. Yet intelligence is essentially the ability to solve problems: everyday problems, personal problems … ‘(Polya, 1980, p.1). Modern definitions of intelligence (Gardner, 1985) talk about practical intelligence which enables ‘the individual to resolve genuine problems or difficulties that he or she encounters’ (p.60) and also encourages the individual to find or create problems ‘thereby laying the groundwork for the acquisition of new knowledge’ (p.85). As was pointed out earlier, standard mathematics, with the emphasis on the acquisition of knowledge, does not necessarily cater for these needs. Resnick (1987) described the discrepancies which exist between the algorithmic approaches taught in schools and the ‘invented’ strategies which most people use in the workforce in order to solve practical problems which do not always fit neatly into a taught algorithm. As she says, most people have developed ‘rules of thumb’ for calculating, for example, quantities, discounts or the amount of change they should give, and these rarely involve standard algorithms. Training in problem-solving techniques equips people more readily with the ability to adapt to such situations.

A further reason why a problem-solving approach is valuable is as an aesthetic form. Problem solving allows the student to experience a range of emotions associated with various stages in the solution process. Mathematicians who successfully solve problems say that the experience of having done so contributes to an appreciation for the ‘power and beauty of mathematics’ (NCTM, 1989, p.77), the “joy of banging your head against a mathematical wall, and then discovering that there might be ways of either going around or over that wall” (Olkin and Schoenfeld, 1994, p.43). They also speak of the willingness or even desire to engage with a task for a length of time which causes the task to cease being a ‘puzzle’ and allows it to become a problem. However, although it is this engagement which initially motivates the solver to pursue a problem, it is still necessary for certain techniques to be available for the involvement to continue successfully. Hence more needs to be understood about what these techniques are and how they can best be made available.

In the past decade it has been suggested that problem-solving techniques can be made available most effectively through making problem solving the focus of the mathematics curriculum. Although mathematical problems have traditionally been a part of the mathematics curriculum, it has been only comparatively recently that problem solving has come to be regarded as an important medium for teaching and learning mathematics (Stanic and Kilpatrick, 1989). In the past problem solving had a place in the mathematics classroom, but it was usually used in a token way as a starting point to obtain a single correct answer, usually by following a single ‘correct’ procedure. More recently, however, professional organisations such as the National Council of Teachers of Mathematics (NCTM, 1980 and 1989) have recommended that the mathematics curriculum should be organized around problem solving, focusing on:

• developing skills and the ability to apply these skills to unfamiliar situations
• gathering, organising, interpreting and communicating information
• formulating key questions, analyzing and conceptualizing problems, defining problems and goals, discovering patterns and similarities, seeking out appropriate data, experimenting, transferring skills and strategies to new situations
• developing curiosity, confidence and open-mindedness (NCTM, 1980, pp.2-3).

One of the aims of teaching through problem solving is to encourage students to refine and build onto their own processes over a period of time as their experiences allow them to discard some ideas and become aware of further possibilities (Carpenter, 1989). As well as developing knowledge, the students are also developing an understanding of when it is appropriate to use particular strategies. Through using this approach the emphasis is on making the students more responsible for their own learning rather than letting them feel that the algorithms they use are the inventions of some external and unknown ‘expert’. There is considerable importance placed on exploratory activities, observation and discovery, and trial and error. Students need to develop their own theories, test them, test the theories of others, discard them if they are not consistent, and try something else (NCTM, 1989). Students can become even more involved in problem solving by formulating and solving their own problems, or by rewriting problems in their own words in order to facilitate understanding. It is of particular importance to note that they are encouraged to discuss the processes which they are undertaking, in order to improve understanding, gain new insights into the problem and communicate their ideas (Thompson, 1985, Stacey and Groves, 1985).

It has been suggested in this chapter that there are many reasons why a problem-solving approach can contribute significantly to the outcomes of a mathematics education. Not only is it a vehicle for developing logical thinking, it can provide students with a context for learning mathematical knowledge, it can enhance transfer of skills to unfamiliar situations and it is an aesthetic form in itself. A problem-solving approach can provide a vehicle for students to construct their own ideas about mathematics and to take responsibility for their own learning. There is little doubt that the mathematics program can be enhanced by the establishment of an environment in which students are exposed to teaching via problem solving, as opposed to more traditional models of teaching about problem solving. The challenge for teachers, at all levels, is to develop the process of mathematical thinking alongside the knowledge and to seek opportunities to present even routine mathematics tasks in problem-solving contexts.

Example #1 – Mathematical Treasure Hunt

Objective – The objective of this activity is to encourage students to apply their problem-solving skills while having fun exploring mathematical concepts in a real-world context.

Materials Needed

Paper and pencils for each student Treasure map (could be a printed map or drawn by hand) Clues (math-related questions or puzzles) Optional: Small prizes or rewards for completing the treasure hunt Instructions:

Introduction (5 minutes)

Begin by introducing the activity to the students. Explain that they will be going on a mathematical treasure hunt where they will solve math problems to uncover hidden clues leading them to the treasure. Emphasize that this activity will require their problem-solving skills and teamwork.

Setting Up the Treasure Hunt (10 minutes)

Prepare a treasure map with different locations marked on it. These locations could be scattered around the classroom, school, or any other designated area. Hide clues at each location that will lead the students to the next destination.

Creating Clues (15 minutes)

Create math-related clues or puzzles that the students will need to solve to uncover the next location on the treasure map. The clues should be age-appropriate and aligned with the students’ math skills. For example:

Solve the following addition problem to reveal the next clue: 15 + 27 – 9 = ?

Count the number of chairs in the classroom and multiply by 3 to find the next location.

Find the area of the square-shaped rug in the library to unlock the next clue.

Starting the Treasure Hunt (5 minutes)

Divide the students into small groups or pairs, depending on the class size. Provide each group with a treasure map and the first clue. Explain the rules of the treasure hunt and encourage students to work together to solve the clues.

Exploring and Solving Clues (30 minutes)

Allow the students to begin the treasure hunt. As they solve each clue, they will uncover the location of the next clue on the treasure map. Encourage them to discuss and collaborate on the solutions to the math problems. Circulate around the room to provide assistance and guidance as needed.

Finding the Treasure (10 minutes)

Once the students have solved all the clues and reached the final location on the treasure map, they will discover the hidden treasure.

Congratulate them on their problem-solving skills and teamwork. You can optionally reward the students with small prizes or certificates for completing the treasure hunt successfully.

Reflection and Discussion (10 minutes)

After the treasure hunt, gather the students together for a brief reflection and discussion. Ask them about their favorite part of the activity, the challenges they faced, and what they learned from solving the math problems. Encourage them to share their strategies and insights with the class.

Extension Ideas

Create themed treasure hunts based on specific mathematical concepts such as geometry, fractions, or measurement.

Invite students to design their own treasure hunts for their classmates, incorporating math problems and creative clues.

Integrate technology by using QR codes or digital maps to lead students to each clue location.

By engaging students in a fun and interactive math problem-solving activity like the “Mathematical Treasure Hunt,” educators can foster a positive attitude towards mathematics while strengthening students’ critical thinking and collaboration skills.

Example #2 – Math Maze Adventure

Objective – The objective of this activity is to challenge students’ problem-solving abilities while navigating through a maze filled with math-related obstacles and puzzles.

Large maze layout (could be drawn on a poster board or printed) Dice Game tokens or markers for each student Math problem cards (with varying difficulty levels) Stopwatch or timer Optional: Prizes or rewards for completing the maze within a certain time limit

Instructions

Begin by introducing the “Math Maze Adventure” to the students. Explain that they will embark on a thrilling journey through a maze filled with mathematical challenges that they must overcome using their problem-solving skills.

Setting Up the Maze (10 minutes)

Create a large maze layout on a poster board or print one from a maze generator website. Designate a starting point and an endpoint within the maze. Place obstacles and challenges throughout the maze, such as math problems, riddles, or puzzles.

Preparing Math Problem Cards (15 minutes)

Create a set of math problem cards with varying difficulty levels. These problems could involve arithmetic operations, geometry concepts, fractions, or any other relevant math topics. Write each problem on a separate card and mix them up.

Starting the Adventure (5 minutes)

Divide the students into small groups or pairs, depending on the class size. Provide each group with a game token or marker to represent their position in the maze. Explain the rules of the game and how to navigate through the maze.

Navigating the Maze (30 minutes)

Start the timer and allow the students to begin their “Math Maze Adventure.” They will roll the dice to determine how many spaces they can move in the maze. When they land on a space with a math problem, they must draw a problem card and solve it correctly to proceed.

Solving Math Problems (30 minutes)

As students encounter math problems in the maze, they will work together to solve them. Encourage them to discuss strategies, share ideas, and check each other’s work. If they solve the problem correctly, they can continue moving through the maze. If not, they must stay in place until they solve it.

Reaching the Endpoint (10 minutes)

The goal of the “Math Maze Adventure” is to reach the endpoint of the maze within a certain time limit. Students must use their problem-solving skills and teamwork to overcome obstacles and challenges along the way. If they reach the endpoint before time runs out, they win the game!

After completing the maze, gather the students together for a reflection and discussion. Ask them about their experience navigating through the maze, the math problems they encountered, and the strategies they used to solve them. Encourage them to share their insights and lessons learned.

Create multiple versions of the maze with different layouts and levels of difficulty to provide ongoing challenges for students.

Integrate storytelling elements into the maze adventure, with each space representing a different part of the story that unfolds as students progress.

Incorporate technology by using a digital maze app or online platform to create and navigate through virtual mazes with math challenges.

The “Math Maze Adventure” offers an exciting and interactive way for students to practice their problem-solving skills while embarking on a thrilling journey through a maze filled with mathematical challenges. Through teamwork, critical thinking, and perseverance, students will navigate their way to success!

Carpenter, T. P. (1989). ‘Teaching as problem solving’. In R.I.Charles and E.A. Silver (Eds), The Teaching and Assessing of Mathematical Problem Solving, (pp.187-202). USA: National Council of Teachers of Mathematics.

Clarke, D. and McDonough, A. (1989). ‘The problems of the problem solving classroom’, The Australian Mathematics Teacher, 45, 3, 20-24.

Cobb, P., Wood, T. and Yackel, E. (1991). ‘A constructivist approach to second grade mathematics’. In von Glaserfield, E. (Ed.), Radical Constructivism in Mathematics Education, pp. 157-176. Dordrecht, The Netherlands: Kluwer Academic Publishers.

Cockcroft, W.H. (Ed.) (1982). Mathematics Counts. Report of the Committee of Inquiry into the Teaching of Mathematics in Schools, London: Her Majesty’s Stationery Office.

Evan, R. and Lappin, G. (1994). ‘Constructing meaningful understanding of mathematics content’, in Aichele, D. and Coxford, A. (Eds.) Professional Development for Teachers of Mathematics , pp. 128-143. Reston, Virginia: NCTM.

Gardner, Howard (1985). Frames of Mind. N.Y: Basic Books.

Lester, F.K.Jr., Masingila, J.O., Mau, S.T., Lambdin, D.V., dos Santon, V.M. and Raymond, A.M. (1994). ‘Learning how to teach via problem solving’. in Aichele, D. and Coxford, A. (Eds.) Professional Development for Teachers of Mathematics , pp. 152-166. Reston, Virginia: NCTM.

National Council of Teachers of Mathematics (NCTM) (1980). An Agenda for Action: Recommendations for School Mathematics of the 1980s, Reston, Virginia: NCTM.

National Council of Teachers of Mathematics (NCTM) (1989). Curriculum and Evaluation Standards for School Mathematics, Reston, Virginia: NCTM.

Olkin, I. & Schoenfeld, A. (1994). A discussion of Bruce Reznick’s chapter. In A. Schoenfeld (Ed.). Mathematical Thinking and Problem Solving. (pp. 39-51). Hillsdale, NJ: Lawrence Erlbaum Associates.

Polya, G. (1980). ‘On solving mathematical problems in high school’. In S. Krulik (Ed). Problem Solving in School Mathematics, (pp.1-2). Reston, Virginia: NCTM.

Resnick, L. B. (1987). ‘Learning in school and out’, Educational Researcher, 16, 13-20..

Romberg, T. (1994). Classroom instruction that fosters mathematical thinking and problem solving: connections between theory and practice. In A. Schoenfeld (Ed.). Mathematical Thinking and Problem Solving. (pp. 287-304). Hillsdale, NJ: Lawrence Erlbaum Associates.

Schifter, D. and Fosnot, C. (1993). Reconstructing Mathematics Education. NY: Teachers College Press.

Schoenfeld, A. (1994). Reflections on doing and teaching mathematics. In A. Schoenfeld (Ed.). Mathematical Thinking and Problem Solving. (pp. 53-69). Hillsdale, NJ: Lawrence Erlbaum Associates.

Stacey, K. and Groves, S. (1985). Strategies for Problem Solving, Melbourne, Victoria: VICTRACC.

Stanic, G. and Kilpatrick, J. (1989). ‘Historical perspectives on problem solving in the mathematics curriculum’. In R.I. Charles and E.A. Silver (Eds), The Teaching and Assessing of Mathematical Problem Solving, (pp.1-22). USA: National Council of Teachers of Mathematics.

Swafford, J.O. (1995). ‘Teacher preparation’. in Carl, I.M. (Ed.) Prospects for School Mathematics , pp. 157-174. Reston, Virginia: NCTM.

Thompson, P. W. (1985). ‘Experience, problem solving, and learning mathematics: considerations in developing mathematics curricula’. In E.A. Silver (Ed.), Teaching and Learning Mathematical Problem Solving: Multiple Research Perspectives, (pp.189-236). Hillsdale, N.J: Lawrence Erlbaum.

Van Zoest, L., Jones, G. and Thornton, C. (1994). ‘Beliefs about mathematics teaching held by pre-service teachers involved in a first grade mentorship program’. Mathematics Education Research Journal. 6(1): 37-55.

Related Article on Teaching Values  |  Other Articles

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Problem-Solving Strategies

October 16, 2019

There are many different ways to solve a math problem, and equipping students with problem-solving strategies is just as important as teaching computation and algorithms. Problem-solving strategies help students visualize the problem or present the given information in a way that can lead them to the solution. Solving word problems using strategies works great as a number talks activity and helps to revise many skills.

Problem-solving strategies

1. create a diagram/picture, 2. guess and check., 3. make a table or a list., 4. logical reasoning., 5. find a pattern, 6. work backward, 1. create a diagram/draw a picture.

Creating a diagram helps students visualize the problem and reach the solution. A diagram can be a picture with labels, or a representation of the problem with objects that can be manipulated. Role-playing and acting out the problem like a story can help get to the solution.

Alice spent 3/4 of her babysitting money on comic books. She is left with $6. How much money did she make from babysitting?

2. Guess and check

Teach students the same strategy research mathematicians use.

With this strategy, students solve problems by making a reasonable guess depending on the information given. Then they check to see if the answer is correct and they improve it accordingly.  By repeating this process, a student can arrive at a correct answer that has been checked. It is recommended that the students keep a record of their guesses by making a chart, a table or a list. This is a flexible strategy that works for many types of problems. When students are stuck, guessing and checking helps them start and explore the problem. However, there is a trap. Exactly because it is such a simple strategy to use, some students find it difficult to consider other strategies. As problems get more complicated, other strategies become more important and more effective.

Find two numbers that have sum 11 and product 24.

Try/guess  5 and 6  the product is 30 too high

  adjust  to 4 and 7 with product 28 still high

  adjust  again 3 and 8 product 24

3. Make a table or a list

Carefully organize the information on a table or list according to the problem information. It might be a table of numbers, a table with ticks and crosses to solve a logic problem or a list of possible answers. Seeing the given information sorted out on a table or a list will help find patterns and lead to the correct solution.

To make sure you are listing all the information correctly read the problem carefully.

Find the common factors of 24, 30 and 18

Logical reasoning is the process of using logical, systemic steps to arrive at a conclusion based on given facts and mathematic principles. Read and understand the problem. Then find the information that helps you start solving the problem. Continue with each piece of information and write possible answers.

Thomas, Helen, Bill, and Mary have cats that are black, brown, white, or gray. The cats’ names are Buddy, Lucky, Fifi, and Moo. Buddy is brown. Thoma’s cat, Lucky, is not gray. Helen’s cat is white but is not named Moo. The gray cat belongs to Bill. Which cat belongs to each student, and what is its color?

A table or list is useful in solving logic problems.

Since Lucky is not gray it can be black or brown. However, Buddy is brown so Lucky has to be black.

Buddy is brown so it cannot be Helen’s cat. Helen’s cat cannot be Moo, Buddy or Lucky, so it is Fifi.

Therefore, Moo is Bill’s cat and Buddy is Mary’s cat.

5. Find a pattern.

Finding a pattern is a strategy in which students look for patterns in the given information in order to solve the problem. When the problem consists of data like numbers or events that are repeated then it can be solved using the “find a pattern” problem-solving strategy. Data can be organized in a table or a list to reveal the pattern and help discover the “rule” of the pattern.

 The “rule” can then be used to find the answer to the question and complete the table/list.

Shannon’s Pizzeria made 5 pizzas on Sunday, 10 pizzas on Monday, 20 pizzas on Tuesday, and 40 pizzas on Wednesday. If this pattern continues, how many pizzas will the pizzeria make on Saturday?

6. Working backward

Problems that can be solved with this strategy are the ones that  list a series of events or a sequence of steps .

In this strategy, the students must start with the solution and work back to the beginning. Each operation must be reversed to get back to the beginning. So if working forwards requires addition, when students work backward they will need to subtract. And if they multiply working forwards, they must divide when working backward.

Mom bought a box of candy. Mary took 5 of them, Nick took 4 of them and 31 were given out on Halloween night. The next morning they found 8 pieces of candy in the box. How many candy pieces were in the box when mom bought it.

For this problem, we know that the final number of candy was 8, so if we work backward to “put back” the candy that was taken from the box we can reach the number of candy pieces that were in the box, to begin with.

The candy was taken away so we will normally subtract them. However, to get back to the original number of candy we need to work backward and do the opposite, which is to add them.

8 candy pieces were left + the 31 given out + plus the ones Mary took + the ones Nick took

8+31+5+4= 48   Answer: The box came with 48 pieces of candy.

Selecting the best strategy for a problem comes with practice and often problems will require the use of more than one strategies.

Print and digital activities

I have created a collection of print and digital activity cards and worksheets with word problems (print and google slides) to solve using the strategies above. The collection includes 70 problems (5 challenge ones) and their solution s and explanations.

sample below

How to use the activity cards

Allow the students to use manipulatives to solve the problems. (counters, shapes, lego blocks, Cuisenaire blocks, base 10 blocks, clocks) They can use manipulatives to create a picture and visualize the problem. They can use counters for the guess and check strategy. Discuss which strategy/strategies are better for solving each problem. Discuss the different ways. Use the activities as warm-ups, number talks, initiate discussions, group work, challenge, escape rooms, and more.

Ask your students to write their own problems using the problems in this resource, and more, as examples. Start with a simple type. Students learn a lot when trying to compose a problem. They can share the problem with their partner or the whole class. Make a collection of problems to share with another class.

For the google slides the students can use text boxes to explain their thinking with words, add shapes and lines to create diagrams, and add (insert) tables and diagrams.

Many of the problems can be solved faster by using algebraic expressions. However, since I created this resource for grades 4 and up I chose to show simple conceptual ways of solving the problems using the strategies above. You can suggest different ways of solving the problems based on the grade level.

Find the free and premium versions of the resource below. The premium version includes 70 problems (challenge problems included) and their solutions

There are 2 versions of the resource

70 google slides with explanations + 70 printable task cards

70 google slides with explanations + 11 worksheets

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Mathematicians Are Edging Close to Solving One of the World's 7 Hardest Math Problems

And there’s $1 million at stake.

• In new research, mathematicians have narrowed down one of the biggest outstanding problems in math.
• Huge breakthroughs in math and science are usually the work of many people over many years.
• Seven math problems were given a $1 million bounty each in 2000, and just one has been solved so far.

The “Millennium Problems” are seven infamously intractable math problems laid out in the year 2000 by the prestigious Clay Institute, each with $1 million attached as payment for a solution. They span all areas of math , as the Clay Institute was founded in 1998 to push the entire field forward with financial support for researchers and important breakthroughs.

But the only solved Millennium Problem so far, the Poincare conjecture, illustrates one of the funny pitfalls inherent to offering a large cash prize for math. The winner, Grigori Perelman, refused the Clay prize as well as the prestigious Fields Medal. He withdrew from mathematics and public life in 2006, and even in 2010, he still insisted his contribution was the same as the mathematician whose work laid the foundation on which he built his proof, Richard Hamilton.

Math, all sciences, and arguably all human inquiries are filled with pairs or groups that circle the same finding at the same time until one officially makes the breakthrough. Think about Sir Isaac Newton and Gottfried Leibniz, whose back-and-forth about calculus led to the combined version of the field we still study today. Rosalind Franklin is now mentioned in the same breath as her fellow discoverers of DNA, James Watson and Francis Crick. Even the Bechdel Test for women in media is sometimes called the Bechdel-Wallace Test, because humans are almost always in collaboration.

That’s what makes this new paper so important. Two mathematicians—Larry Guth of the Massachusetts Institute of Technology (MIT) and James Maynard of the University of Oxford—collaborated on the new finding about how certain polynomials are formed and how they reach out into the number line. Maynard is just 37, and won the Fields Medal himself in 2022. Guth, a decade older, has won a number of important prizes with a little less name recognition.

The Riemann hypothesis is not directly related to prime numbers , but it has implications that ripple through number theory in different ways (including with prime numbers). Basically, it deals with where and how the graph of a certain function of complex numbers crosses back and forth across axes. The points where the function crosses an axis is called a “zero,” and the frequency with which those zeroes appear is called the zero density.

In the far reaches of the number line, prime numbers become less and less predictable (in the proverbial sense). They are not, so far, predictable in the literal sense—a fact that is an underpinning of modern encryption , where data is protected by enormous strings of integers made by multiplying enormous prime numbers together. The idea of a periodic table of primes, of any kind of template that could help mathematicians better understand where and how large primes cluster together or not, is a holy grail.

In the new paper, Maynard and Guth focus on a new limitation of Dirichlet polynomials. These are special series of complex numbers that many believe are of the same type as the function involved in the Riemann hypothesis involves. In the paper, they claim they’ve proven that these polynomials have a certain number of large values, or solutions , within a tighter range than before.

In other words, if we knew there might be an estimated three Dirichlet values between 50 and 100 before, now we may know that range to be between 60 and 90 instead. The eye exam just switched a blurry plate for a slightly less blurry one, but we still haven’t found the perfect prescription. “If one knows some more structure about the set of large values of a Dirichlet polynomial, then one can hope to have improved bound,” Maynard and Guth conclude.

No, this is not a final proof of the Riemann hypothesis. But no one is suggesting it is. In advanced math, narrowing things down is also vital. Indeed, even finding out that a promising idea turns out to be wrong can have a lot of value—as it has a number of times in the related Twin Primes Conjecture that still eludes mathematicians.

In a collaboration that has lasted 160 years and counting, mathematicians continue to take each step together and then, hopefully, compare notes.

Caroline Delbert is a writer, avid reader, and contributing editor at Pop Mech. She's also an enthusiast of just about everything. Her favorite topics include nuclear energy, cosmology, math of everyday things, and the philosophy of it all. 

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7 of the hardest problems in mathematics that have been solved

Mathematics is full of problems, some of which have been solved and others that haven’t. here, we focus on 7 of the hardest problems that have been solved..

Tejasri Gururaj

What are the hardest problems in math that have already been solved?

intararit  

• Some problems in mathematics have taken centuries to be solved, due to their complexity.
• Although there are some complex math problems that still elude solutions, others have now been solved.
• Here are 7 of the hardest math problems ever solved.

Some mathematical problems are challenging even for the most accomplished mathematicians.

From the Poincaré conjecture to Fermat’s last theorem , here we take a look at some of the most challenging math problems ever solved.

1. Poincaré conjecture

Salix alba  

The Poincaré conjecture is a famous problem in topology, initially proposed by French mathematician and theoretical physicist Henri Poincaré in 1904. It asserts that every simply connected, closed 3-manifold is topologically homeomorphic (a function that is a one-to-one mapping between sets such that both the function and its inverse are continuous) to a 3-dimensional sphere. 

In simpler terms, the conjecture asserts that a particular group of three-dimensional shapes can be continuously transformed into a sphere without any gaps or holes.

The problem was solved by the reclusive Russian mathematician Grigori Perelman in 2003. He built upon the work of American mathematician Richard S. Hamilton’s program involving the Ricci flow.

What makes this achievement even more remarkable is that Perelman declined the prestigious Fields Medal and the Clay Millennium Prize reward that came with it. He chose to stay away from the spotlight and mathematical acclaim, but his proof withstood rigorous scrutiny from the mathematical community.

The resolution of the Poincaré conjecture confirmed the fundamental role of topology in understanding the shape and structure of spaces, impacting fields like geometry and manifold theory.

2. The prime number theorem 

Britannica  

The prime number theorem long stood as one of the fundamental questions in number theory. At its core, this problem is concerned with unraveling the distribution of prime numbers. 

The question at hand revolves around the distribution pattern of these primes within the realm of natural numbers. Are there any discernible patterns governing the distribution of prime numbers, or do they appear to be entirely random? The theorem states that for large values of x, π(x) is approximately equal to x/ln(x).

The breakthrough in solving this theorem came in the late 19th century, thanks to the independent work of two mathematicians, Jacques Hadamard and Charles de la Vallée-Poussin. In 1896, both mathematicians presented their proofs of the theorem. 

Their work demonstrated that prime numbers exhibit a remarkable, asymptotic distribution pattern . Their solution produced the insight that, as one considers larger and larger numbers, the density of prime numbers diminishes. 

The theorem precisely characterized the rate of this decrease, showing that prime numbers become less frequent as we move along the number line. It’s as if they gradually thin out, although they never entirely vanish.

The prime number theorem was a turning point in the study of number theory . It provided profound insights into the distribution of prime numbers and put to rest the notion that there might exist a formula predicting each prime number. 

Instead, it proposed a probabilistic approach to understanding the distribution of prime numbers. The theorem’s significance extends into various mathematical fields, especially in cryptography, where the properties of prime numbers play a pivotal role in securing communications.

3. Fermat’s last theorem

Charles Rex Arbogast/AP via NPR  

Fermat’s last theorem is one of the problems on this list many people are most likely to have heard of. The conjecture, proposed by French mathematician Pierre de Fermat in the 17th century, states that it’s impossible to find three positive integers, a, b, and c, that can satisfy the equation a n + b n = c n for any integer value of n greater than 2.

For instance, there are no whole number values of a, b, and c that can make 3 3 + 4 3 = 5 3 true.

This problem remained unsolved for centuries and became one of the most challenging problems in mathematics. It was made more enticing because Fermat apparently wrote a note in his copy of the  Arithmetica  by Diophantus of Alexandria, saying, “I have discovered a truly remarkable proof [of this theorem], but this margin is too small to contain it.”

Numerous mathematicians attempted to prove or disprove Fermat’s conjecture, but it wasn’t until 1994 that the breakthrough came.

The solution was achieved by British mathematician Andrew Wiles, who built upon the work of many other mathematicians who had contributed to the field of number theory. Wiles’ proof was extraordinarily complex and required intricate mathematical concepts and theorems, particularly those related to elliptic curves and modular forms.

Wiles’ remarkable proof of Fermat’s last theorem confirmed that the conjecture was indeed true. Having taken more than three centuries to be solved, it had a profound impact on the world of mathematics, demonstrating the power of advanced mathematical techniques in solving long-standing problems.

Z. Ziegler, M. Ondrachek  

Before Wiles presented its proof, it was in the Guinness Book of World Records as the “most difficult mathematical problem,” in part because the theorem has seen the greatest number of unsuccessful proofs.

4. Classification of finite simple groups

Jakob.scholbach/Pbroks13  

This one is a bit different from the others on the list. The classification of finite simple groups , also known as the “enormous theorem,” set out to classify all finite simple groups, which are the fundamental building blocks of group theory.

Finite simple groups are those groups that cannot be divided into smaller non-trivial normal subgroups. The goal was to understand and categorize all the different types of finite simple groups that exist.

The solution to this problem is not straightforward. The proof is a collaborative effort by hundreds of mathematicians covering tens of thousands of pages in hundreds of journal articles published between 1955 and 2004.

It is one of the most extensive mathematical proofs ever produced and marks a monumental achievement in group theory.

The proof outlines the structure of finite simple groups and demonstrates that they can be classified into several specific categories. This achievement paved the way for a deeper understanding of group theory and its applications in various mathematical fields. 

5. The four color theorem

Inductiveload  

The four color theorem tackles an intriguing question related to topology and stands as one of the first significant theorems proved by a computer. 

It states that any map in a plane can be colored using four colors so that no two adjacent regions share the same color while using the fewest possible colors. Adjacent, in this context, means that two regions share a common boundary curve segment, not merely a corner where three or more regions meet.

The theorem doesn’t focus on the artistic aspect of map coloring but rather on the fundamental mathematical principles that underlie it.

The solution to this theorem arrived in 1976, thanks to the combined efforts of mathematicians Kenneth Appel and Wolfgang Haken. However, the proof was not widely accepted due to the infeasibility of checking it by hand.

Appel and Haken’s achievement confirmed that any map, regardless of its complexity, can be colored with just four colors such that no two neighboring regions share the same color. While the idea seems simple, proving it rigorously was complex and time-consuming.

To address any lingering skepticism about the Appel–Haken proof, a more accessible proof using similar principles and still utilizing computer assistance was presented in 1997 by Robertson, Sanders, Seymour, and Thomas. 

Additionally, in 2005, Georges Gonthier achieved a proof of the theorem using general-purpose theorem-proving software, reinforcing the credibility of the four color theorem.

This theorem is not actually used in map-making but has far-reaching implications in various fields, from graph theory to computer science, where it finds applications in scheduling, circuit design, and optimization problems.

6. Gödel’s incompleteness theorems

Andrew Das Arulsamy/Research Gate  

Gödel’s incompleteness theorems , formulated by Austrian mathematician Kurt Gödel in the 20th century, delve into the mysteries of formal systems and their inherent limitations. 

In mathematics, a formal system has a structured and well-defined framework or language that comprises a set of symbols, rules, and axioms employed for representing and manipulating mathematical or logical expressions.

Gödel’s first incompleteness theorem explores a fundamental question: In any consistent formal system, are there true mathematical statements that are undecidable within that system? In other words, do statements exist that cannot be proven as either true or false using the rules and axioms of that system?

The second incompleteness theorem takes this further: Can any consistent formal system prove its own consistency?

Gödel not only posed these questions but also provided the answers. He established, through rigorous mathematical proofs , that there exist true statements within formal systems that cannot be proven within those very systems.

In essence, the first theorem asserts that there are statements that cannot be proven as either true or false using the rules and axioms of a system. The second theorem demonstrates that no consistent formal system can prove its own consistency.

Gödel’s theorems introduced a profound paradox within the realm of mathematical logic: There are truths that exist beyond the reach of formal proofs, and there are limits to what can be achieved through mathematical systems alone.

Gödel’s contributions to mathematical logic influenced the philosophy of mathematics and our understanding of the inherent limits of formal systems.

7. The goat problem

Mnchnstnr  

The goat problem is a much more recently solved mathematical problem. It involves calculating the grazing area for a tethered goat. Despite its initial simplicity, mathematicians have pondered this problem for over a century.

In its basic form, a goat on a rope can graze in a semicircle with an area of A = 1/2πr 2 , where r is the rope’s length. However, the problem becomes more complex when you change the shape of the area the goat can access.

For instance, when tethered to a square barn, the goat can access more than just a semicircle. The goat can also go around the corners of the barn, creating additional quarter circles.

Mathematician Ingo Ullisch recently unraveled the goat grazing problem, introducing complex analysis into the equation. However, the solution is far from elementary.

It involves intricate calculations, relying on the ratio of contour integral expressions and involves numerous trigonometric terms . Although the solution may not offer a practical guide for goat owners, it represents a significant achievement in the world of mathematics.

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What makes the goat problem truly fascinating is its capacity to act as a mathematical Rosetta stone , transcending boundaries between various fields and serving as a versatile challenge for experts from diverse disciplines.

From age-old conundrums that took centuries to crack to enigmas that continue to elude solutions, mathematical mysteries remind us that the pursuit of knowledge is an ever-evolving journey.

So, the next time you find yourself pondering a difficult math problem, remember that you are in good company, following in the footsteps of the greatest mathematical explorers!

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ABOUT THE EDITOR

Tejasri Gururaj Tejasri is a versatile Science Writer & Communicator, leveraging her expertise from an MS in Physics to make science accessible to all. In her spare time, she enjoys spending quality time with her cats, indulging in TV shows, and rejuvenating through naps.

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The most rigorous math program you've never heard of.

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Math-M-Addicts students eagerly dive into complex math problems during class.

In the building of the Speyer Legacy School in New York City, a revolutionary math program is quietly producing some of the city's most gifted young problem solvers and logical thinkers. Founded in 2005 by two former math prodigies, Math-M-Addicts has grown into an elite academy developing the skills and mindset that traditional schooling often lacks.

"We wanted to establish the most advanced math program in New York," explains Ruvim Breydo, co-founder of Math-M-Addicts. "The curriculum focuses not just on mathematical knowledge, but on developing a mastery of problem-solving through a proof-based approach aligned with prestigious competitions like the International Mathematical Olympiad."

From its inception, Math-M-Addicts took an unconventional path. What began as an attempt to attract only the highest caliber high school students soon expanded to offer multiple curriculum levels. "We realized we couldn't find enough kids at the most advanced levels," says Breydo. "So we decided to develop that talent from an earlier age."

The program's approach centers on rigor. At each of the 7 levels, the coursework comprises just a handful of fiendishly difficult proof-based math problems every week. "On average, we expect them to get about 50% of the solutions right," explains instructor Natalia Lukina. "The problems take hours and require grappling with sophisticated mathematical concepts."

But it's about more than just the content. Class sizes are small, with two instructors for every 15-20 students. One instructor leads the session, while the other teacher coordinates the presentation of the homework solutions by students. The teachers also provide customized feedback by meticulously reviewing each student's solutions. "I spend as much time analyzing their thought processes as I do teaching new material," admits instructor Bobby Lee.

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Lee and the Math-M-Addicts faculty embrace an unconventional pedagogy focused on developing logic, creativity, and a tenacious problem-solving mindset over procedures. "We don't dumb it down for them," says Breydo. "We use technical math language and allow students to struggle through the challenges because that's where real learning happens."

Impressive results of Math-M-addicts students in selective math competitions highlight their ... [+] preparation and dedication.

For the Math-M-Addicts team, finding the right teachers is as essential as shaping brilliant students. Prospective instructors go through a rigorous multi-stage vetting process. "We seek passionate mathematical problem solvers first," says program director Sonali Jasuja. "Teaching experience is great, but first and foremost, we need people who deeply understand and enjoy the reasoning behind mathematics."

Even exceptional instructors undergo extensive training by co-teaching for at least a year alongside veteran Math-M-Addicts faculty before taking the lead role. "Our approach is different from how most US teachers learned mathematics," explains instructor Tanya Gross, the director of Girls Adventures in Math (GAIM) competition. "We immerse them in our unique math culture, which focuses on the 'why' instead of the 'how,' empowering a paradigm shift."

That culture extends to the students as well. In addition to the tools and strategies imparted in class, Math-M-Addicts alumni speak of an unshakable confidence and camaraderie that comes from up to several thousands of hours grappling with mathematics at the highest levels alongside peers facing the same challenges.

As Math-M-Addicts ramps up efforts to expand access through online classes and global partnerships, the founders remain devoted to their core mission. "Math education should not obsess with speed and memorization of math concepts," argues Breydo. "This is not what mathematics is about. To unlock human potential, we must refocus on cognitive reasoning and problem-solving skills. We are seeking to raise young people unafraid to tackle any complex challenge they face"

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Carnegie Mellon Informatics and Mathematics Competition

The Carnegie Mellon Informatics and Mathematics Competition (CMIMC) is an annual high school math competition organized by students at Carnegie Mellon University. The math competition has run annually in the spring since 2016 and was last held on April 6, 2024 .

The tournament consists of a TCS round, a team round, and individual rounds.

TCS (Theoretical Computer Science) Round

This is a 60 minute test with 2-3 problems to be completed as a team. Each question is worth the same number of points. Teams will submit at most one bound per problem and should provide an accompanying proof.

This is a 40 minute test with 10 problems to be completed as a team. Each question is worth the same number of points.

Individual Rounds

There are three different rounds: Algebra and Number Theory, Combinatorics and Computer Science, and Geometry. Each student will take all 3 exams.

Each round is a 50 minute test with 10 problems (and a tiebreaker). Harder/less-solved questions are worth more points.

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July 1, 2024

The Biggest Problem in Mathematics Is Finally a Step Closer to Being Solved

Number theorists have been trying to prove a conjecture about the distribution of prime numbers for more than 160 years

By Manon Bischoff

Weiquan Lin/Getty Images

The Riemann hypothesis is the most important open question in number theory—if not all of mathematics. It has occupied experts for more than 160 years. And the problem appeared both in mathematician David Hilbert’s groundbreaking speech from 1900 and among the “Millennium Problems” formulated a century later. The person who solves it will win a million-dollar prize.

But the Riemann hypothesis is a tough nut to crack. Despite decades of effort, the interest of many experts and the cash reward, there has been little progress. Now mathematicians Larry Guth of the Massachusetts Institute of Technology and James Maynard of the University of Oxford have posted a sensational new finding on the preprint server arXiv.org. In the paper, “the authors improve a result that seemed insurmountable for more than 50 years,” says number theorist Valentin Blomer of the University of Bonn in Germany.

Other experts agree. The work is “a remarkable breakthrough,” mathematician and Fields Medalist Terence Tao wrote on Mastodon , “though still very far from fully resolving this conjecture.”

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The Riemann hypothesis concerns the basic building blocks of natural numbers: prime numbers, values greater than 1 that are only divisible by 1 and themselves. Examples include 2, 3, 5, 7, 11, 13, and so on.*

Every other number, such as 15, can be clearly broken down into a product of prime numbers: 15 = 3 x 5. The problem is that the prime numbers do not seem to follow a simple pattern and instead appear randomly among the natural numbers. Nineteenth-century German mathematician Bernhard Riemann proposed a way to deal with this peculiarity that explains how prime numbers are distributed on the number line—at least from a statistical point of view.

A Periodic Table for Numbers

Proving this conjecture would provide mathematicians with nothing less than a kind of “periodic table of numbers.” Just as the basic building blocks of matter (such as quarks, electrons and photons) help us to understand the universe and our world, prime numbers also play an important role, not just in number theory but in almost all areas of mathematics.

There are now numerous theorems based on the Riemann conjecture. Proof of this conjecture would prove many other theorems as well—yet another incentive to tackle this stubborn problem.

Interest in prime numbers goes back thousands of years. Euclid proved as early as 300 B.C.E. that there are an infinite number of prime numbers. And although interest in prime numbers persisted, it was not until the 18th century that any further significant findings were made about these basic building blocks.

As a 15-year-old, physicist Carl Friedrich Gauss realized that the number of prime numbers decreases along the number line. His so-called prime number theorem (not proven until 100 years later) states that approximately n / ln( n ) prime numbers appear in the interval from 0 to n . In other words, the prime number theorem offers mathematicians a way of estimating the typical distribution of primes along a chunk of the number line.

The exact number of prime numbers may differ from the estimate given by the theorem, however. For example: According to the prime number theorem, there are approximately 100 / ln(100) ≈ 22 prime numbers in the interval between 1 and 100. But in reality there are 25. There is therefore a deviation of 3. This is where the Riemann hypothesis comes in. This hypothesis gives mathematicians a way to estimate the deviation. More specifically, it states that this deviation cannot become arbitrarily large but instead must scale at most with the square root of n , the length of the interval under consideration.

The Riemann hypothesis therefore does not predict exactly where prime numbers are located but posits that their appearance on the number line follows certain rules. According to the Riemann hypothesis, the density of primes decreases according to the prime number theorem, and the primes are evenly distributed according to this density. This means that there are no large areas in which there are no prime numbers at all, while others are full of them.

You can also imagine this idea by thinking about the distribution of molecules in the air of a room: the overall density on the floor is somewhat higher than on the ceiling, but the particles—following this density distribution—are nonetheless evenly scattered, and there is no vacuum anywhere.

A Strange Connection

Riemann formulated the conjecture named after him in 1859, in a slim, six-page publication (his only contribution to the field of number theory). At first glance, however, his work has little to do with prime numbers.

He dealt with a specific function, the so-called zeta function ζ( s ), an infinitely long sum that adds the reciprocal values of natural numbers that are raised to the power of s :

Even before Riemann’s work, experts knew that such zeta functions are related to prime numbers. Thus, the zeta function can also be expressed as a function of all prime numbers p as follows:

Riemann recognized the full significance of this connection with prime numbers when he used not only real values for s but also complex numbers. These numbers contain both a real part and roots from negative numbers, the so-called imaginary part.

You can imagine complex numbers as a two-dimensional construct. Rather than mark a point on the number line, they instead lie on the plane. The x coordinate corresponds to the real part and the y coordinate to the imaginary part:

Никита Воробьев/Wikimedia

The complex zeta function that Riemann investigated can be visualized as a landscape above the plane. As it turns out, there are certain points amid the mountains and valleys that play an important role in relation to prime numbers. These are the points at which the zeta function becomes zero (so-called zeros), where the landscape sinks to sea level, so to speak.

The colors represent the values of the complex zeta function, with the white dots indicating its zeros.

Jan Homann/Wikimedia

Riemann quickly found that the zeta function has no zeros if the real part is greater than 1. This means that the area of the landscape to the right of the straight line x = 1 never sinks to sea level. The zeros of the zeta function are also known for negative values of the real part. They lie on the real axis at x = –2, –4, –6, and so on. But what really interested Riemann—and all mathematicians since—were the zeros of the zeta function in the “critical strip” between 0 ≤ x ≤ 1.

In the critical strip (dark blue), the Riemann zeta function can have “nontrivial” zeros. The Riemann conjecture states that these are located exclusively on the line x = 1/2 (dashed line).

LoStrangolatore/Wikimedia ( CC BY-SA 3.0 )

Riemann knew that the zeta function has an infinite number of zeros within the critical strip. But interestingly, all appear to lie on the straight line x = 1 / 2 . Thus Riemann hypothesized that all zeros of the zeta function within the critical strip have a real part of x = 1 / 2 . That statement is actually at the crux of understanding the distribution of prime numbers. If correct, then the placement of prime numbers along the number line never deviates too much from the prime number set.

On the Hunt for Zeros

To date, billions and billions of zeta function zeros have now been examined— more than 10 13 of them —and all lie on the straight line x = 1 / 2 .

But that alone is not a valid proof. You would only have to find a single zero that deviates from this scheme to disprove the Riemann hypothesis. Therefore we are looking for a proof that clearly demonstrates that there are no zeros outside x = 1 / 2 in the critical strip.

Thus far, such a proof has been out of reach, so researchers took a different approach. They tried to show that there is, at most, a certain number N of zeros outside this straight line x = 1 / 2 . The hope is to reduce N until N = 0 at some point, thereby proving the Riemann conjecture. Unfortunately, this path also turns out to be extremely difficult. In 1940 mathematician Albert Ingham was able to show that between 0.75 ≤ x ≤ 1 there are at most y 3/5+ c zeros with an imaginary part of at most y , where c is a constant between 0 and 9.

In the following 80 years, this estimation barely improved. The last notable progress came from mathematician Martin Huxley in 1972 . “This has limited us from doing many things in analytic number theory,” Tao wrote in his social media post . For example, if you wanted to apply the prime number theorem to short intervals of the type [ x , x + x θ ], you were limited by Ingham’s estimate to θ > 1 / 6 .

Yet if Riemann’s conjecture is true, then the prime number theorem applies to any interval (or θ = 0), no matter how small (because [ x , x + x θ ] = [ x , x + 1] applies to θ = 0).

Now Maynard, who was awarded the prestigious Fields Medal in 2022 , and Guth have succeeded in significantly improving Ingham’s estimate for the first time. According to their work, the zeta function in the range 0.75 ≤ x ≤ 1 has at most y (13/25)+ c zeros with an imaginary part of at most y . What does that mean exactly? Blomer explains: “The authors show in a quantitative sense that zeros of the Riemann zeta function become rarer the further away they are from the critical straight line. In other words, the worse the possible violations of the Riemann conjecture are, the more rarely they would occur.”

“This propagates to many corresponding improvements in analytic number theory,” Tao wrote . It makes it possible to reduce the size of the intervals for which the prime number theorem applies. The theorem is valid for [ x , x + x 2/15 ], so θ > 1 / 6 = 0.166... becomes θ > 2 ⁄ 15 = 0.133...

For this advance, Maynard and Guth initially used well-known methods from Fourier analysis for their result. These are similar techniques to what is used to break down a sound into its overtones. “The first few steps are standard, and many analytic number theorists, including myself, who have attempted to break the Ingham bound, will recognize them,” Tao explained . From there, however, Maynard and Guth “do a number of clever and unexpected maneuvers,” Tao wrote.

Blomer agrees. “The work provides a whole new set of ideas that—as the authors rightly say—can probably be applied to other problems. From a research point of view, that’s the most decisive contribution of the work,” he says.

So even if Maynard and Guth have not solved Riemann’s conjecture, they have at least provided new food for thought to tackle the 160-year-old puzzle. And who knows—perhaps their efforts hold the key to finally cracking the conjecture.

This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.

*Editor’s Note (7/9/24): This sentence was edited after posting to better clarify that prime numbers exclude 1.