improving mathematical problem solving skill

How to Improve Problem-Solving Skills: Mathematics and Critical Thinking

In today’s rapidly changing world, problem-solving has become a quintessential skill. When we discuss the topic, it’s natural to ask, “What is problem-solving?” and “How can we enhance this skill, particularly in children?” The discipline of mathematics offers a rich platform to explore these questions. Through math, not only do we delve into numbers and equations, but we also explore how to improve problem-solving skills and how to develop critical thinking skills in math. Let’s embark on this enlightening journey together.

What is Problem-Solving?

At its core, problem-solving involves identifying a challenge and finding a solution. But it’s not always as straightforward as it sounds. So, what is problem-solving? True problem-solving requires a combination of creative thinking and logical reasoning. Mathematics, in many ways, embodies this blend. When a student approaches a math problem, they must discern the issue at hand, consider various methods to tackle it, and then systematically execute their chosen strategy.

But what is problem-solving in a broader context? It’s a life skill. Whether we’re deciding the best route to a destination, determining how to save for a big purchase, or even figuring out how to fix a broken appliance, we’re using problem-solving.

How to Develop Critical Thinking Skills in Math

Critical thinking goes hand in hand with problem-solving. But exactly how to develop critical thinking skills in math might not be immediately obvious. Here are a few strategies:

• Contextual Learning: Teaching math within a story or real-life scenario makes it relevant. When students see math as a tool to navigate the world around them, they naturally begin to think critically about solutions.
• Open-ended Questions: Instead of merely seeking the “right” answer, encourage students to explain their thought processes. This nudges them to think deeply about their approach.
• Group Discussions: Collaborative learning can foster different perspectives, prompting students to consider multiple ways to solve a problem.
• Challenging Problems: Occasionally introducing problems that are a bit beyond a student’s current skill level can stimulate critical thinking. They will have to stretch their understanding and think outside the box.

What are the Six Basic Steps of the Problem-Solving Process?

Understanding how to improve problem-solving skills often comes down to familiarizing oneself with the systematic approach to challenges. So, what are the six basic steps of the problem-solving process?

• Identification: Recognize and define the problem.
• Analysis: Understand the problem’s intricacies and nuances.
• Generation of Alternatives: Think of different ways to approach the challenge.
• Decision Making: Choose the most suitable method to address the problem.
• Implementation: Put the chosen solution into action.
• Evaluation: Reflect on the solution’s effectiveness and learn from the outcome.

By embedding these steps into mathematical education, we provide students with a structured framework. When they wonder about how to improve problem-solving skills or how to develop critical thinking skills in math, they can revert to this process, refining their approach with each new challenge.

Making Math Fun and Relevant

At Wonder Math, we believe that the key to developing robust problem-solving skills lies in making math enjoyable and pertinent. When students see math not just as numbers on a page but as a captivating story or a real-world problem to be solved, their engagement skyrockets. And with heightened engagement comes enhanced understanding.

As educators and parents, it’s crucial to continuously ask ourselves: how can we demonstrate to our children what problem-solving is? How can we best teach them how to develop critical thinking skills in math? And how can we instill in them an understanding of the six basic steps of the problem-solving process?

The answer, we believe, lies in active learning, contextual teaching, and a genuine passion for the beauty of mathematics.

The Underlying Beauty of Mathematics

Often, people perceive mathematics as a rigid discipline confined to numbers and formulas. However, this is a limited view. Math, in essence, is a language that describes patterns, relationships, and structures. It’s a medium through which we can communicate complex ideas, describe our universe, and solve intricate problems. Understanding this deeper beauty of math can further emphasize how to develop critical thinking skills in math.

Why Mathematics is the Ideal Playground for Problem-Solving

Math provides endless opportunities for problem-solving. From basic arithmetic puzzles to advanced calculus challenges, every math problem offers a chance to hone our problem-solving skills. But why is mathematics so effective in this regard?

• Structured Challenges: Mathematics presents problems in a structured manner, allowing learners to systematically break them down. This format mimics real-world scenarios where understanding the structure of a challenge can be half the battle.
• Multiple Approaches: Most math problems can be approached in various ways . This teaches learners flexibility in thinking and the ability to view a single issue from multiple angles.
• Immediate Feedback: Unlike many real-world problems where solutions might take time to show results, in math, students often get immediate feedback. They can quickly gauge if their approach works or if they need to rethink their strategy.

Enhancing the Learning Environment

To genuinely harness the power of mathematics in developing problem-solving skills, the learning environment plays a crucial role. A student who is afraid of making mistakes will hesitate to try out different approaches, stunting their critical thinking growth.

However, in a nurturing, supportive environment where mistakes are seen as learning opportunities, students thrive. They become more willing to take risks, try unconventional solutions, and learn from missteps. This mindset, where failure is not feared but embraced as a part of the learning journey, is pivotal for developing robust problem-solving skills.

Incorporating Technology

In our digital age, technology offers innovative ways to explore math. Interactive apps and online platforms can provide dynamic problem-solving scenarios, making the process even more engaging. These tools can simulate real-world challenges, allowing students to apply their math skills in diverse contexts, further answering the question of how to improve problem-solving skills.

More than Numbers 

In summary, mathematics is more than just numbers and formulas—it’s a world filled with challenges, patterns, and beauty. By understanding its depth and leveraging its structured nature, we can provide learners with the perfect platform to develop critical thinking and problem-solving skills. The key lies in blending traditional techniques with modern tools, creating a holistic learning environment that fosters growth, curiosity, and a lifelong love for learning.

Join us on this transformative journey at Wonder Math. Let’s make math an adventure, teaching our children not just numbers and equations, but also how to improve problem-solving skills and navigate the world with confidence. Enroll your child today and witness the magic of mathematics unfold before your eyes!

FAQ: Mathematics and Critical Thinking

1. what is problem-solving in the context of mathematics.

Problem-solving in mathematics refers to the process of identifying a mathematical challenge and systematically working through methods and strategies to find a solution.

2. Why is math considered a good avenue for developing problem-solving skills?

Mathematics provides structured challenges and allows for multiple approaches to find solutions. This promotes flexibility in thinking and encourages learners to view problems from various angles.

3. How does contextual learning enhance problem-solving abilities?

By teaching math within a story or real-life scenario, it becomes more relevant for the learner. This helps them see math as a tool to navigate real-world challenges , thereby promoting critical thinking.

4. What are the six basic steps of the problem-solving process in math?

The six steps are: Identification, Analysis, Generation of Alternatives, Decision Making, Implementation, and Evaluation.

5. How can parents support their children in developing mathematical problem-solving skills?

Parents can provide real-life contexts for math problems , encourage open discussions about different methods, and ensure a supportive environment where mistakes are seen as learning opportunities.

6. Are there any tools or apps that can help in enhancing problem-solving skills in math?

Yes, there are various interactive apps and online platforms designed specifically for math learning. These tools provide dynamic problem-solving scenarios and simulate real-world challenges, making the learning process engaging.

7. How does group discussion foster critical thinking in math?

Group discussions allow students to hear different perspectives and approaches to a problem. This can challenge their own understanding and push them to think about alternative methods.

8. Is it necessary to always follow the six steps of the problem-solving process sequentially?

While the six steps provide a structured approach, real-life problem-solving can sometimes be more fluid. It’s beneficial to know the steps, but adaptability and responsiveness to the situation are also crucial.

9. How does Wonder Math incorporate active learning in teaching mathematics?

Wonder Math integrates mathematics within engaging stories and real-world scenarios, making it fun and relevant. This active learning approach ensures that students are not just passive recipients but active participants in the learning process.

10. What if my child finds a math problem too challenging and becomes demotivated?

It’s essential to create a supportive environment where challenges are seen as growth opportunities. Remind them that every problem is a chance to learn, and it’s okay to seek help or approach it differently.

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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.

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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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6 Tips for Teaching Math Problem-Solving Skills

Solving word problems is tougher than computing with numbers, but elementary teachers can guide students to do the deep thinking involved.

A growing concern with students is the ability to problem-solve, especially with complex, multistep problems. Data shows that students struggle more when solving word problems than they do with computation , and so problem-solving should be considered separately from computation. Why?

Consider this. When we’re on the way to a new destination and we plug in our location to a map on our phone, it tells us what lane to be in and takes us around any detours or collisions, sometimes even buzzing our watch to remind us to turn. When I experience this as a driver, I don’t have to do the thinking. I can think about what I’m going to cook for dinner, not paying much attention to my surroundings other than to follow those directions. If I were to be asked to go there again, I wouldn’t be able to remember, and I would again seek help.

If we can switch to giving students strategies that require them to think instead of giving them too much support throughout the journey to the answer, we may be able to give them the ability to learn the skills to read a map and have several ways to get there.

Here are six ways we can start letting students do this thinking so that they can go through rigorous problem-solving again and again, paving their own way to the solution. 

1. Link problem-solving to reading

When we can remind students that they already have many comprehension skills and strategies they can easily use in math problem-solving, it can ease the anxiety surrounding the math problem. For example, providing them with strategies to practice, such as visualizing, acting out the problem with math tools like counters or base 10 blocks, drawing a quick sketch of the problem, retelling the story in their own words, etc., can really help them to utilize the skills they already have to make the task less daunting.

We can break these skills into specific short lessons so students have a bank of strategies to try on their own. Here's an example of an anchor chart that they can use for visualizing . Breaking up comprehension into specific skills can increase student independence and help teachers to be much more targeted in their problem-solving instruction. This allows students to build confidence and break down the barriers between reading and math to see they already have so many strengths that are transferable to all problems.

2. Avoid boxing students into choosing a specific operation

It can be so tempting to tell students to look for certain words that might mean a certain operation. This might even be thoroughly successful in kindergarten and first grade, but just like when our map tells us where to go, that limits students from becoming deep thinkers. It also expires once they get into the upper grades, where those words could be in a problem multiple times, creating more confusion when students are trying to follow a rule that may not exist in every problem.

We can encourage a variety of ways to solve problems instead of choosing the operation first. In first grade, a problem might say, “Joceline has 13 stuffed animals and Jordan has 17. How many more does Jordan have?” Some students might choose to subtract, but a lot of students might just count to find the amount in between. If we tell them that “how many more” means to subtract, we’re taking the thinking out of the problem altogether, allowing them to go on autopilot without truly solving the problem or using their comprehension skills to visualize it. 

3. Revisit ‘representation’

The word “representation” can be misleading. It seems like something to do after the process of solving. When students think they have to go straight to solving, they may not realize that they need a step in between to be able to support their understanding of what’s actually happening in the problem first.

Using an anchor chart like one of these ( lower grade , upper grade ) can help students to choose a representation that most closely matches what they’re visualizing in their mind. Once they sketch it out, it can give them a clearer picture of different ways they could solve the problem.

Think about this problem: “Varush went on a trip with his family to his grandmother’s house. It was 710 miles away. On the way there, three people took turns driving. His mom drove 214 miles. His dad drove 358 miles. His older sister drove the rest. How many miles did his sister drive?”

If we were to show this student the anchor chart, they would probably choose a number line or a strip diagram to help them understand what’s happening.

If we tell students they must always draw base 10 blocks in a place value chart, that doesn’t necessarily match the concept of this problem. When we ask students to match our way of thinking, we rob them of critical thinking practice and sometimes confuse them in the process. 

4. Give time to process

Sometimes as educators, we can feel rushed to get to everyone and everything that’s required. When solving a complex problem, students need time to just sit with a problem and wrestle with it, maybe even leaving it and coming back to it after a period of time.

This might mean we need to give them fewer problems but go deeper with those problems we give them. We can also speed up processing time when we allow for collaboration and talk time with peers on problem-solving tasks. 

5. Ask questions that let Students do the thinking

Questions or prompts during problem-solving should be very open-ended to promote thinking. Telling a student to reread the problem or to think about what tools or resources would help them solve it is a way to get them to try something new but not take over their thinking.

These skills are also transferable across content, and students will be reminded, “Good readers and mathematicians reread.” 

6. Spiral concepts so students frequently use problem-solving skills

When students don’t have to switch gears in between concepts, they’re not truly using deep problem-solving skills. They already kind of know what operation it might be or that it’s something they have at the forefront of their mind from recent learning. Being intentional within their learning stations and assessments about having a variety of rigorous problem-solving skills will refine their critical thinking abilities while building more and more resilience throughout the school year as they retain content learning in the process. 

Problem-solving skills are so abstract, and it can be tough to pinpoint exactly what students need. Sometimes we have to go slow to go fast. Slowing down and helping students have tools when they get stuck and enabling them to be critical thinkers will prepare them for life and allow them multiple ways to get to their own destination.

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Maths Problem Solving: Engaging Your Students And Strengthening Their Mathematical Skills

Meriel Willatt

Maths problem solving can be challenging for pupils. There’s no ‘one size fits all’ approach or strategy and questions often combine different topic areas. Pupils often don’t know where to start. It’s no surprise that problem solving is a common topic teachers struggle to teach effectively to their pupils.

In this blog, we consider the importance of problem solving and share with you some ideas and resources for you to tackle problem solving in your maths classroom, from KS2 up to GCSE.

What is maths problem solving?

Why is maths problem solving so difficult, how to develop problem solving skills in maths, maths problem solving ks2, maths problem solving ks3, maths problem solving gcse.

Maths problem solving is when a mathematical task challenges pupils to apply their knowledge, logic and reasoning in unfamiliar contexts. Problem solving questions often combine several elements of maths.

We know from talking to the hundreds of school leaders and maths teachers that we work with as one to one online maths tutoring providers that this is one of their biggest challenges: equipping pupils with the skills and confidence necessary to approach problem solving questions.

The Ultimate Guide to Problem Solving Techniques

Download these 9 ready-to-go problem solving techniques that every pupil should know

The challenge with problem solving in maths is that there is no generic problem solving skill that can be taught in an isolated maths lesson. It’s a skill that teachers must explicitly teach to pupils, embed into their learning and revisit often.

When pupils are first introduced to a topic, they cannot start problem solving straight away using it. Problem solving relies on deep knowledge of concepts. Pupils need to become familiar with it and practice using it in different contexts before they can make connections, reason and problem solve with it. In fact, some researchers suggest that it could take up to two years to do this (Burkhardt, 2017). 

At Third Space Learning, we specialise in online one to one maths tutoring for schools, from KS1 all the way up to GCSE. Our lessons are designed by maths teachers and pedagogy experts to break down complex problems into their constituent parts. Our specialist tutors then carefully scaffold learning to build students’ confidence in key skills before combining them to tackle problem solving questions.

In order to develop problem solving skills in maths, pupils need lots of different contexts and word problems in which to practise them and the opportunity to engage in mathematical talk that draws on their metacognitive skills. 

The EEF suggests that to develop problem solving skills in maths, teachers need to teach pupils:

• To use different approaches to problem solving
• Use worked examples
• To use metacognition to plan, monitor and reflect on their approaches to problem solving

Below, we take a closer look at problem solving at each stage, from primary school all the way to GCSEs. We’ve also included links to maths resources and CPD to support you and your team’s classroom teaching.

At lower KS2, the National Curriculum states that pupils should develop their ability to solve a range of problems. However, these will involve simple calculations as pupils develop their numeracy skills. As pupils progress to upper KS2, the demand for problem solving skills increases. 

“At this stage, pupils should develop their ability to solve a wider range of problems, including increasingly complex properties of numbers and arithmetic, and problems demanding efficient written and mental methods of calculation. With this foundation in arithmetic, pupils are introduced to the language of algebra as a means for solving a variety of problems.” National curriculum in England: mathematics programmes of study (Upper key stage 2 – years 5 and 6)

KS2 problem solving can often fall into the trap of relying on acronyms, such as RICE, RIDE or even QUACK. The most popular is RUCSAC (Read, Underline, Calculate, Solve, Answer, Check). While these do aim to simplify the process for young minds, it encourages a superficial, formulaic approach to problem solving, rather than deep mathematical thinking. Also, consider how much is wrapped up within the word ‘solve’ – is this helpful?

We teach thousands of pupils KS2 maths problem solving skills every week through our one to one online tutoring programme for maths. In our interventions, we encourage deep mathematical thinking by using a simplified version of George Polya’s four stages of problem solving. Here are the four stages:

Understand the problem

• Devise a strategy for solving it
• Carry out the problem solving strategy
• Check the result

We use UCR as a simplified model: Understand, Communicate & Reflect. You may choose to adapt this depending on the age and ability of your class.

For example:

Maisy, Heidi and Freddie are children in the same family. The product of their ages is a score. How old might they be?

There are three people.

There are three numbers that multiply together to make twenty (a score is equal to 20). There will be lots of answers, but no ‘right’ answer.

Communicate

To solve the word problem we need to find the numbers that will go into 20 without a remainder (the factors).

The factors of 20 are 1, 2, 4, 5, 10 and 20.

Combinations of numbers that could work are: 1, 1, 20 1, 2, 10 1, 4, 5 2, 2, 5.

The question says children, which means ‘under 18 years’, so that would mean we could remove 1, 1, 20 from our list of possibilities. 

In our sessions, we create a nurturing learning environment where pupils feel safe to make mistakes. This is so important in the context of problem solving as the best problem solvers will be resilient and able to overcome challenges in the ‘Reflect’ stage. Read more: What is a growth mindset

Looking for more support teaching KS2 problem solving? We’ve developed a powerpoint on problem solving, reasoning and planning for depth that is designed to be used as CPD by primary school teachers, maths leads and SLT. 

The resource reflects on how metacognition can enhance reasoning and problem solving abilities, the ‘curse’ of real life maths (think ‘Carl buys 60 watermelons…) and how teachers can practically implement and teach strategies in the classroom.

You may also be interested in: 

• Developing Thinking Skills At KS2
• KS2 Maths Investigations
• Word problems for Year 6

At KS3, the importance of seeing mathematical concepts as interconnected with other skills, including problem solving, is foregrounded. The National Curriculum also stresses the importance of a strong foundation in maths before moving on to complex problem solving.

“Mathematics is an interconnected subject in which pupils need to be able to move fluently between representations of mathematical ideas. The programme of study for key stage 3 is organised into apparently distinct domains, but pupils should build on key stage 2 and connections across mathematical ideas to develop fluency, mathematical reasoning and competence in solving increasingly sophisticated problems” National curriculum in England: mathematics programmes of study (Key stage 3)

“Decisions about progression should be based on the security of pupils’ understanding and their readiness to progress to the next stage. Pupils who grasp concepts rapidly should be challenged through being offered rich and sophisticated problems before any acceleration through new content in preparation for key stage 4.” National curriculum in England: mathematics programmes of study (Key stage 3)

For many students, the transition from primary to secondary school can be a huge challenge.

Especially in the aftermath of the Covid-19 pandemic and the resultant school closures, students may arrive into Year 7 with various learning gaps and misconceptions that will hold them. Some students may need focused support to plug these gaps and grow in confidence.

You can give pupils a smoother transition from KS2 to KS3 with personalised one to one online tuition with specialist tutors with Third Space Learning. Our lessons cover content from Years 5-7 and build a solid foundation for pupils to develop their problem solving skills. Pupils are supported towards independent practice through worked examples, questioning and support slides.

The challenge for KS3 maths problem solving activities is that learners may struggle to get invested unless you start with a convincing hook. Engage your young mathematicians on topics you know well or you know they’ll be invested in and try your hand at designing your own mathematical problems. Alternatively, get some inspiration from our crossover ability and fun maths problems .

Since the new GCSE specification began in 2015, there has been an increased focus on non-routine problem solving questions. These questions demand students to make sense of lots of new information at once before they even move on to selecting the strategies they’ll use to find the correct answer. This is where many learners get stuck.

In recent years, teachers and researchers in pedagogy (including Ofsted) have recognised that open ended problem solving tasks do not in fact lead to improved student understanding. While they may be enjoyable and engage learners, they may not lead to improved results.

SSDD problems (Same Surface Different Depth) can offer a solution that develops students’ critical thinking skills, while ensuring they engage fully with the information they’re provided. The idea behind them is to provide a set of questions that look the same and use the same mathematical hook but each question requires a different mathematical process to be solved.

Read more about SSDD problems , tips on writing your own questions and download free printable examples. There are also plenty of more examples on the NRICH website.

Worked examples, careful questioning and constructing visual representations can help students to convert the information embedded in a maths challenge into mathematical notations. Read our blog on problem solving maths questions for Foundation, Crossover & Higher examples, worked solutions and strategies.

Remember that students can only move on to mathematics problem solving once they have secure knowledge in a topic. If you know there are areas your students need extra support, check our Secondary Maths Resources library for revision guides, teaching resources and worksheets for KS3 and GCSE topics.

DO YOU HAVE STUDENTS WHO NEED MORE SUPPORT IN MATHS?

Every week Third Space Learning’s specialist online maths tutors support thousands of students across hundreds of schools with weekly online 1 to 1 maths lessons designed to plug gaps and boost progress.

Since 2013 these personalised one to 1 lessons have helped over 150,000 primary and secondary students become more confident, able mathematicians.

Learn how the programmes are aligned to maths mastery teaching or request a personalised quote for your school to speak to us about your school’s needs and how we can help.

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9 Ways to Improve Math Skills Quickly & Effectively

Written by Ashley Crowe

• Parent Resources

• The importance of understanding basic math skills
• 9 Ways to improve math skills
• How to use technology to improve math skills

Math class can move pretty fast. There’s so much to cover in the course of a school year. And if your child doesn’t get a new math idea right away, they can quickly get left behind.

If your child is struggling with basic math problems every day, it doesn’t mean they’re destined to be bad at math. Some students need more time to develop the problem-solving skills that math requires. Others may need to revisit past concepts before moving on. Because of how math is structured, it’s best to take each year step-by-step, lesson by lesson.

This article has tips and tricks to improve your child’s math skills while minimizing frustrations and struggles. If your child is growing to hate math, read on for ways to improve their skills and confidence, and maybe even make math fun! 

But first, the basics.

Math is a subject that builds on itself. It takes a solid understanding of past concepts to prepare for the next lesson. 

That’s why math can become frustrating when you’re forced to move on before you’re ready. You’re either stuck trying to catch up or you end up falling further behind.

But with a strong understanding of basic math skills, your child can be set up for school success. If you’re unfamiliar with the idea of sets or whole numbers , this is a great place to start. 

What are considered basic math skills?

The basic math skills required to move on to higher levels of math learning are: 

• Addition — Adding to a set.
• Subtraction — Taking away from a set.
• Multiplication — Adding equal sets together in groups (2 sets of 3 is the same as 2x3, or 6).
• Division — How many equal sets can be found in a number (12 has how many sets of two in it? 6 sets of 2).
• Percentages — A specific amount in relation to 100.
• Fractions & Decimals — Fractions are equal parts of a whole set. Decimals represent a number of parts of a whole in relation to 10. These both contrast with whole numbers. 
• Spatial Reasoning — How numbers and shapes fit together.

How to improve math skills 

People aren’t bad at math — many just need more time and practice to gain a thorough understanding.

How can you help your child improve their math abilities? Use our top 9 tips for quickly and effectively improving math skills .

1. Wrap your head around the concepts

Repetition and practice are great, but if you don’t understand the concept , it will be difficult to move forward. 

Luckily, there are many great ways to break down math concepts . The trick is finding the one that works best for your child.

Math manipulatives can be a game-changer for children who are struggling with big math ideas. Taking math off the page and putting it into their hands can bring ideas to life. Numbers become less abstract and more concrete when you’re counting toy cars or playing with blocks. Creating these “sets” of objects can bring clarity to basic math learning.

2. Try game-based learning

During math practice, repetition is important — but it can get old in a hurry. No one enjoys copying their times tables over and over and over again. If learning math has become a chore, it’s time to bring back the fun! 

Game-based learning is a great way to practice new concepts and solidify past lessons. It can even make repetition fun and engaging.

Game-based learning can look like a family board game on Friday night or an educational app , like Prodigy Math .

Take math from frustrating to fun with the right game, then watch the learning happen easily!

3. Bring math into daily life

You use basic math every day. 

As you go about your day, help your child see the math that’s all around them:  

• Tell them how fast you’re driving on the way to school
• Calculate the discount you’ll receive on your next Target trip
• Count out the number of apples you need to buy at the grocery store
• While baking, explain how 6 quarter cups is the same amount of flour as a cup and a half — then enjoy some cookies!

Relate math back to what your child loves and show them how it’s used every day. Math doesn’t have to be mysterious or abstract. Instead, use math to race monster trucks or arrange tea parties. Break it down, take away the fear, and watch their interest in math grow.

4. Implement daily practice

Math practice is important. Once you understand the concept, you have to nail down the mechanics. And often, it’s the practice that finally helps the concept click. Either way, math requires more than just reading formulas on a page.

Daily practice can be tough to implement, especially with a math-averse child. This is a great time to bring out the game-based learning mentioned above. Or find an activity that lines up with their current lesson. Are they learning about squares? Break out the math link cubes and create them. Whenever possible, step away from the worksheets and flashcards and find practice elsewhere.

5. Sketch word problems

Nothing causes a panic quite like an unexpected word problem. Something about the combination of numbers and words can cause the brain of a struggling math learner to shut down. But it doesn’t have to be that way.

Many word problems just need to be broken down, step by step . One great way to do this is to sketch it out. If Doug has five apples and four oranges, then eats two of each, how many does he have left? Draw it, talk it out, cross them off, then count. 

If you’ve been talking your child through the various math challenges you encounter every day, many word problems will start to feel familiar. 

6. Set realistic goals

If your child has fallen behind in math, then more study time is the answer. But forcing them to cram an extra hour of math in their day is not likely to produce better results. To see a positive change, first identify their biggest struggles . Then set realistic goals addressing these issues . 

Two more hours of practicing a concept they don’t understand is only going to cause more frustration. Even if they can work through the mechanics of a problem, the next lesson will leave them feeling just as lost. 

Instead, try mini practice sessions and enlist some extra help. Approach the problem in a new way, reach out to their teacher or try an online math lesson . Make sure the extra time is troubleshooting the actual problem, not just reinforcing the idea that math is hard and no fun. 

Set Goals and Rewards in Prodigy Math

Did you know that parents can set learning goals for their child in Prodigy Math? And once they achieve them, they'll unlock in-game rewards of your choice!

7. Engage with a math tutor

If your child is struggling with big picture concepts, look into finding a math tutor . Everyone learns differently, and you and your child’s teacher may be missing that “aha” moment that a little extra time and the right tutor can provide.

It’s amazing when a piece of the math puzzle finally clicks for your child. If you’re ready to get that extra help, try a free 1:1 online session from Prodigy Math Tutoring. Prodigy’s tutors are real teachers who know how to connect kids to math. With the right approach, your child can become confident in math — and who knows, they may even begin to enjoy it. 

8. Focus on one concept at a time

Math builds on itself. If your child is struggling through their current lesson, they can’t skip it and come back to it later. This is the time to practice and repeat — re-examining and reinforcing the current concept until it makes sense.

Look for other ways to approach new math ideas. Use math manipulatives to bring numbers off the page. Or try a learning app with exciting rewards and positive reinforcement to encourage extra practice. 

Take a step back when frustrations get high — but resist the temptation to just let it go. Once the concept clicks, they’ll be excited to forge ahead.

9. Teach others math you already know

Even if your child is struggling in math, they’ve still learned so much since last year. Focus on the improvements they’ve made and let them showcase their knowledge. If they have younger siblings, your older child can demonstrate addition or show them how to use a number line. This is a great way to build their confidence and encourage them to keep going.

Or let them teach you how they solve new problems. Have your child talk you through the process while you solve a long division problem . You’re likely to find yourself a little rusty on the details. Play it up and get a little silly. They’ll love teaching you the ropes of this “new math.”

Embracing technology to improve math skills

Though much of your math learning was done with pencil to paper, there are many more ways to build number skills in today’s tech world. 

Your child can take live, online math courses to work through tough concepts. Or play a variety of online games, solving math puzzles and getting consistent practice while having fun.

These technical advances can help every child learn math, no matter their preferred learning or study style. If your child is a visual learner, there’s an app for that. Do they process best while working in groups? Jump online and find one. Don’t keep repeating the same lessons from their math class over and over. Branch out, try something new and watch the learning click. 

Look online for more math help

There are so many online resources, it can be hard to know where to start. 

At Prodigy, we’re happy to help you get the ball rolling on your child’s math learning, from kindergarten through 8th grade. It’s free to sign up, fun to play and exciting to watch as your child’s math understanding grows.

Sign up for a free parent account and get instant data on your child’s progress as they build more math skills with Prodigy Math Game . It’s time to take the math struggle out of your home and enjoy learning together!

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How to Improve Problem-Solving Skills in Math

Importance of Problem-Solving Skills in Math

Problem-solving skills are crucial in math education , enabling students to apply mathematical concepts and principles to real-world situations. Here’s why problem-solving skills are essential in math education:

1. Application of knowledge: Problem-solving in math requires encouraging students to apply the knowledge they acquire in the classroom to tackle real-life problems. It helps them understand the relevance of math in everyday life and enhances their critical thinking skills.

2. Developing critical thinking:  Problem-solving requires students to analyze, evaluate, and think critically about different approaches and strategies to solve a problem. It strengthens their mathematical abilities and improves their overall critical thinking skills.

3. Enhancing problem-solving skills:  Math problems often have multiple solutions, encouraging students to think creatively and explore different problem-solving strategies. It helps develop their problem-solving skills, which are valuable in various aspects of life beyond math.

4. Fostering perseverance:  Problem-solving in math often requires persistence and resilience. Students must be willing to try different approaches, learn from their mistakes, and keep trying until they find a solution. It fosters a growth mindset and teaches them the value of perseverance.

Benefits of strong problem-solving skills

Having strong problem-solving skills in math offers numerous benefits for students:

1. Improved academic performance:  Students with strong problem-solving skills are likelier to excel in math and other subjects that rely on logical reasoning and critical thinking.

2. Enhanced problem-solving abilities:  Strong problem-solving skills extend beyond math and can be applied to various real-life situations. It includes decision-making, analytical thinking, and solving complex problems creatively.

3. Increased confidence:  Successfully solving math problems boosts students’ self-confidence and encourages them to tackle more challenging tasks. This confidence spills over into other areas of their academic and personal lives.

4. Preparation for future careers:  Problem-solving skills are highly sought after by employers in various fields. Developing strong problem-solving skills in math sets students up for successful careers in engineering, technology, finance, and more.

Problem-solving skills are essential for math education and have numerous benefits for students. By fostering these skills, educators can empower students to become confident, critical thinkers who can apply their mathematical knowledge to solve real-world problems.

Understand the Problem

Breaking down the problem and identifying the key components.

To improve problem-solving math skills , it’s essential to first understand the problem at hand. Here are some tips to help break down the problem and identify its key components:

1. Read the problem carefully:  Take your time to read it attentively and ensure you understand what it asks. Pay attention to keywords or phrases that indicate what mathematical operation or concept to use.

2. Identify the known and unknown variables:  Determine what information is already given in the problem (known variables) and what you need to find (unknown variables). This step will help you analyze the problem more effectively.

3. Define the problem in your own words:  Restate the problem using your own words to ensure you clearly understand what needs to be solved. It can help you focus on the main objective and eliminate any distractions.

4. Break the problem into smaller parts:  Complex math problems can sometimes be overwhelming. Breaking them down into smaller, manageable parts can make them more approachable. Identify any sub-problems or intermediate steps that must be solved before reaching the final solution.

Reading and interpreting math word problems effectively

Many math problems are presented as word problems requiring reading and interpreting skills. Here are some strategies to help you effectively understand and solve math word problems:

1. Highlight key information:  As you read the word problem, underline or highlight any important details, such as numbers, units of measurement, or specific keywords related to mathematical operations.

2. Visualize the problem:  Create visual representations, such as diagrams or graphs, to help you understand the problem better. Visualizing the problem can make determining what steps to take and how to approach the solution easier.

3. Translate words into equations:  Convert the information in the word problem into mathematical equations or expressions. This translation step helps you transform the problem into a solvable math equation.

4. Solve step by step:  Break down the problem into smaller steps and solve each step individually. This approach helps you avoid confusion and progress toward the correct solution.

Improving problem-solving skills in math requires practice and patience. By understanding the problem thoroughly, breaking it into manageable parts, and effectively interpreting word problems, you can confidently enhance your ability to solve math problems.

Use Visual Representations

Using diagrams, charts, and graphs to visualize the problem.

One effective way to improve problem-solving skills in math is to utilize visual representations. Visual representations , such as diagrams, charts, and graphs, can help make complex problems more tangible and easily understood. Here are some ways to use visual representations in problem-solving:

1. Draw Diagrams:  When faced with a word problem or a complex mathematical concept, drawing a diagram can help break down the problem into more manageable parts. For example, suppose you are dealing with a geometry problem. In that case, sketching the shapes involved can provide valuable insights and help you visualize the problem better.

2. Create Charts or Tables:  For problems that involve data or quantitative information, creating charts or tables can help organize the data and identify patterns or trends. It can be particularly useful in analyzing data from surveys, experiments, or real-life scenarios.

3. Graphical Representations:  Graphs can be powerful tools in problem-solving, especially when dealing with functions, equations, or mathematical relationships. Graphically representing data or equations makes it easier to identify key features that may be hard to spot from a numerical representation alone, such as intercepts or trends.

Benefits of visual representation in problem-solving

Using visual representations in problem-solving offers several benefits:

1. Enhances Comprehension:  Visual representations provide a visual context for abstract mathematical concepts, making them easier to understand and grasp.

2. Encourages Critical Thinking:  Visual representations require active engagement and critical thinking skills. Students can enhance their problem-solving and critical thinking abilities by analyzing and interpreting visual data.

3. Promotes Pattern Recognition: Visual representations simplify identifying patterns, trends, and relationships within data or mathematical concepts. It can lead to more efficient problem-solving and a deeper understanding of mathematical principles.

4. Facilitates Communication:  Visual representations can be shared and discussed, helping students communicate their thoughts and ideas effectively. It can be particularly useful in collaborative problem-solving environments.

Incorporating visual representations into math problem-solving can significantly enhance understanding, critical thinking, pattern recognition, and communication skills. Students can approach math problems with a fresh perspective and improve their problem-solving abilities using visual tools.

Work Backwards

Understanding the concept of working backward in math problem-solving.

Working backward is a problem-solving strategy that starts with the solution and returns to the given problem. This approach can be particularly useful in math, as it helps students break down complex problems into smaller, more manageable steps. Here’s how to apply the concept of working backward in math problem-solving:

1. Identify the desired outcome : Start by clearly defining the goal or solution you are trying to reach. It could be finding the value of an unknown variable, determining a specific measurement, or solving for a particular quantity.

2. Visualize the result : Imagine the final step or solution. It will help you create a mental image of the steps needed to reach that outcome.

3. Trace the steps backward : Break down the problem into smaller steps, working backward from the desired outcome. Think about what needs to happen immediately before reaching the final solution and continue tracing the steps back to the beginning of the problem.

4. Check your work : Once you have worked backward to the beginning of the problem, double-check your calculations and steps to ensure accuracy.

Real-life examples and applications of working backward

Working backward is a valuable problem-solving technique in math and has real-life applications . Here are a few examples:

1. Financial planning : When creating a budget, you can work backward by determining your desired savings or spending amount and then calculating how much income or expenses are needed to reach that goal.

2. Project management : When planning a project, you can work backward by setting a fixed deadline and then determining the necessary steps and timelines to complete the project on time.

3. Game strategy : In games like chess or poker, working backward can help you anticipate your opponent’s moves and plan your strategy accordingly.

4. Recipe adjustments : When modifying a recipe, you can work backward by envisioning the final taste or texture you want to achieve and adjusting the ingredients or cooking methods accordingly.

By practicing working backward in math and applying it to real-life situations, you can enhance your problem-solving abilities and find creative solutions to various challenges.

Try Different Strategies

When solving math problems, it’s essential to have a repertoire of problem-solving strategies. You can improve your problem-solving skills and tackle various mathematical challenges by trying different approaches. Here are some strategies to consider:

Exploring Various Problem-Solving Strategies

1. Guess and Check:  This strategy involves making an educated guess and checking if it leads to the correct solution. It can be useful when dealing with trial-and-error problems.

2. Drawing a Diagram:  Visually representing the problem through diagrams or graphs can help you understand and solve it more effectively. This strategy is particularly useful in geometry and algebraic reasoning.

3. Using Logic:  Using logical reasoning is useful for breaking down complicated problems into smaller, more manageable components. This strategy is especially useful in mathematical proofs and logical puzzles.

4. Working Backwards:  Start with the desired outcome and return to the given information. When dealing with equations or word problems, this approach can assist.

5. Using Patterns:  Look for patterns and relationships within the problem to determine a solution. This approach can be used for different mathematical problems, such as sequences and numerical patterns.

When and How to Apply Different Strategies in Math Problem-Solving

Knowing when and how to apply different problem-solving strategies is crucial for success in math. Here are some tips:

• Understand the problem: Read the problem carefully and identify the key information and requirements.
• Select an appropriate strategy: Choose the most appropriate problem-solving strategy for the problem.
• Apply the chosen strategy: Implement the selected strategy, following the necessary steps.
• Check your solution: Verify your answer by double-checking the calculations or applying alternative methods.
• Reflect on the process: After solving the problem, take a moment to reflect and evaluate your problem-solving approach. Identify areas for improvement and consider alternative strategies that could have been used.

By exploring different problem-solving strategies and applying them to various math problems, you can enhance your problem-solving skills and develop a versatile toolkit for tackling mathematical challenges. Practice and persistence are key to honing your problem-solving abilities in math.

Key takeaways and tips for improving problem-solving skills in math

In conclusion, developing strong problem-solving skills in math is crucial for success in this subject. Here are some key takeaways and tips to help you improve your problem-solving abilities:

• Practice regularly:  The more you practice solving math problems, the better you will become at identifying patterns, applying strategies, and finding solutions.
• Break down the problem:  When faced with a complex math problem, break it into smaller, more manageable parts. It will make it easier to understand and solve.
• Understand the problem:  Before diving into a solution, fully understand the problem. Identify what information is given and what you are asked to find.
• Draw diagrams or visualize:  Use visual aids, such as diagrams or sketches, to help you better understand the problem and visualize the solution.
• Use logical reasoning:  Apply logical reasoning skills to analyze the problem and determine the most appropriate approach or strategy.
• Try different strategies:  If one approach doesn’t work, don’t be afraid to try different strategies or methods. There are often multiple ways to solve a math problem.
• Seek help and collaborate:  Don’t hesitate to seek help from your teacher, classmates, or online resources. Collaborating with others can provide different perspectives and insights.
• Learn from mistakes:  Mistakes are a valuable learning opportunity. Analyze your mistakes, understand where you went wrong, and learn from them to avoid making the same errors in the future.
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Problem Solving, Using and Applying and Functional Mathematics

Problem solving.

The problem-solving process can be described as a journey from meeting a problem for the first time to finding a solution, communicating it and evaluating the route. There are many models of the problem-solving process but they all have a similar structure. One model is given below. Although implying a linear process from comprehension through to evaluation, the model is more of a flow backward and forward, revisiting and revising on the problem-solving journey.

Comprehension

Representation.

• Can they represent the situation mathematically?
• What is it that they are trying to find?
• What do they think the answer might be (conjecturing and hypothesising)?
• What might they need to find out before they can get started?

Planning, analysis and synthesis

Having understood what the problem is about and established what needs finding, this stage is about planning a pathway to the solution. It is within this process that you might encourage pupils to think about whether they have seen something similar before and what strategies they adopted then. This will help them to identify appropriate methods and tools. Particular knowledge and skills gaps that need addressing may become evident at this stage.

Execution and communication

During the execution phase, pupils might identify further related problems they wish to investigate. They will need to consider how they will keep track of what they have done and how they will communicate their findings. This will lead on to interpreting results and drawing conclusions.

Pupils can learn as much from reflecting on and evaluating what they have done as they can from the process of solving the problem itself. During this phase pupils should be expected to reflect on the effectiveness of their approach as well as other people's approaches, justify their conclusions and assess their own learning. Evaluation may also lead to thinking about other questions that could now be investigated.

Using and Applying Mathematics

Aspects of using and applying reflect skills that can be developed through problem solving. For example:

In planning and executing a problem, problem solvers may need to:

• select and use appropriate and efficient techniques and strategies to solve problems
• identify what further information may be required in order to pursue a particular line of enquiry and give reasons for following or rejecting particular approaches
• break down a complex calculation problem into simpler steps before attempting a solution and justify their choice of methods
• make mental estimates of the answers to calculations
• present answers to sensible levels of accuracy; understand how errors are compounded in certain calculations.

During problem solving, solvers need to communicate their mathematics for example by:

• discussing their work and explaining their reasoning using a range of mathematical language and notation
• using a variety of strategies and diagrams for establishing algebraic or graphical representations of a problem and its solution
• moving from one form of representation to another to get different perspectives on the problem
• presenting and interpreting solutions in the context of the original problem
• using notation and symbols correctly and consistently within a given problem
• examining critically, improve, then justifying their choice of mathematical presentation
• presenting a concise, reasoned argument.

Problem solvers need to reason mathematically including through:

• exploring, identifying, and using pattern and symmetry in algebraic contexts, investigating whether a particular case may be generalised further and understanding the importance of a counter-example; identifying exceptional cases
• understanding the difference between a practical demonstration and a proof
• showing step-by-step deduction in solving a problem; deriving proofs using short chains of deductive reasoning
• recognising the significance of stating constraints and assumptions when deducing results
• recognising the limitations of any assumptions that are made and the effect that varying the assumptions may have on the solution to a problem.

Functional Mathematics

Functional maths requires learners to be able to use mathematics in ways that make them effective and involved as citizens, able to operate confidently in life and to work in a wide range of contexts. The key processes of Functional Skills reflect closely the problem solving model but within three phases:

• Making sense of situations and representing them
• Processing and using the mathematics
• Interpreting and communicating the results of the analysis

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Why It's So Important to Learn a Problem-Solving Approach to Mathematics

was invited to the Math Olympiad Summer Program (MOP) in the 10th grade. I went to MOP certain that I must really be good at math. But in my five weeks at MOP, I encountered over sixty problems on various tests and I didn’t solve a single one. That’s right—I was 0-for-60+. I came away no longer confident that I was good at math. I assumed that most of the other kids did better at MOP because they knew more tricks than I did. My formula sheets were pretty thorough, but perhaps they were missing something. By the end of MOP, I had learned a somewhat unsettling truth. The others knew fewer tricks than I did, not more. They didn’t even have formula sheets!

At another contest later that summer, a younger student, Alex, from another school asked me for my formula sheets. In my local and state circles, students’ formula sheets were the source of knowledge, the source of power that fueled the top students and the top schools. They were studied, memorized, revered. But most of all, they were not shared. But when Alex asked for my formula sheets I remembered my experience at MOP and I realized that formula sheets are not really math . Memorizing formulas is no more mathematics than memorizing dates is history or memorizing spelling words is literature. I gave him the formula sheets. (Alex must later have learned also that the formula sheets were fool’s gold—he became a Rhodes scholar.)

The difference between MOP and many of these state and local contests I participated in was the difference between problem solving and what many people call mathematics. For these people, math is a series of tricks to use on a series of specific problems. Trick A is for Problem A, Trick B for Problem B, and so on. In this vein, school can become a routine of learn tricks for a week, use tricks on a test, forget most tricks quickly. The tricks get forgotten quickly primarily because there are so many of them, and also because the students don’t see how these ‘tricks’ are just extensions of a few basic principles.

I had painfully learned at MOP that true mathematics is not a process of memorizing formulas and applying them to problems tailor-made for those formulas. Instead, the successful mathematician possesses fewer tools, but knows how to apply them to a much broader range of problems. We use the term problem solving to distinguish this approach to mathematics from the memorize, use, forget approach.

After MOP I relearned math throughout high school. I was unaware that I was learning much more. When I got to Princeton I enrolled in organic chemistry. There were over 200 students in the course, and we quickly separated into two groups. One group understood that all we would be taught could largely be derived from a very small number of basic principles. We loved the class—it was a year-long exploration of where these fundamental concepts could take us. The other, much larger, group saw each new destination not as the result of a path from the building blocks, but as yet another place whose coordinates had to be memorized if ever they were to visit again. Almost to a student, the difference between those in the happy group and those in the struggling group was how they learned mathematics. The class seemingly involved no math at all, but those who took a memorization approach to math were doomed to do it again in chemistry. The skills the problem solvers developed in math transferred, and these students flourished.

We use math to teach problem solving because it is the most fundamental logical discipline. Not only is it the foundation upon which sciences are built, it is the clearest way to learn and understand how to develop a rigorous logical argument. There are no loopholes, there are no half-truths. The language of mathematics is as precise as it is ‘right’ and ‘wrong’ (or ‘proven’ and ‘unproven’). Success and failure are immediate and indisputable; there isn’t room for subjectivity. This is not to say that those who cannot do math cannot solve problems. There are many paths to strong problem-solving skills. Mathematics is the shortest .

Problem solving is crucial in mathematics education because it transcends mathematics. By developing problem-solving skills, we learn not only how to tackle math problems, but also how to logically work our way through any problems we may face. The memorizer can only solve problems he has encountered already, but the problem solver can solve problems she’s never seen before. The problem solver is flexible; she can diversify. Above all, she can create .

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Study Skills

5 effective ways to improve math skills.

Mathematics is a subject that many children struggle with, and parents often find themselves at a loss for how to help their child improve their math skills. In fact, a large number of students experience math anxiety, which can cause them to avoid math-related tasks altogether. This anxiety can be caused by a variety of factors, such as a lack of understanding of math concepts or poor performance in previous math classes.

But don’t worry, there are effective ways to help your child overcome their math difficulties and develop a positive attitude towards math. In this blog post, we will explore some of the reasons why children find math hard and provide tips on how to improve math skills.

Firstly, math is a subject that builds on previous knowledge and skills, which means that if a student falls behind, they may find it difficult to catch up. This can lead to frustration and a lack of confidence in their ability to do math. Additionally, the abstract nature of math can make it difficult for some students to understand and apply concepts. Furthermore, math anxiety can be influenced by negative experiences with math, such as being ridiculed for making mistakes or feeling pressure to perform well on math tests.

Despite these challenges, there are several strategies parents can use to help their child improve their math skills and overcome math anxiety. These strategies include making math fun, finding real-life applications for math concepts, practicing with your child daily, getting a tutor, and encouraging your child to ask questions and seek help when needed.

By implementing these strategies and providing a supportive and positive environment, parents can help their children build confidence in their math abilities and develop a love for math.

5 Ways to Improve Math Skills

1. make math fun.

O ne effective approach is to add elements of fun and creativity to math classes. When students are engaged through creative and stimulating activities and puzzles, they are more likely to enjoy the learning experience and retain information better. This not only helps them to improve in math but also fosters a love for the subject.

To make math classes more engaging, educators can introduce games and activities that involve the use of props such as cards and dice. There are also several online portals that offer math-based games and puzzles that educators can use to supplement their teaching. These activities not only make math fun but also help students develop critical thinking and problem-solving skills, which are essential for success in math and in life.

By incorporating fun and creative activities in math classes, educators can help students overcome their fear of math and improve their overall math skills. This way, students can approach math with confidence and a positive attitude, leading to greater academic success and a lifelong love of learning (Meloney, 2022).

2. Practice Math Skills

T o improve your math skills, it’s important to make math a part of your daily life . Practice is key, and incorporating math into your daily routine can help you improve without even realizing it. You can take advantage of all the real-life opportunities you encounter to improve your math skills.

It can be as simple as calculating the total cost of your lunch based on what you’re buying or figuring out the discount percentage on a shirt that’s on sale. Even calculating the time you need to finish your errands can help you feel more comfortable with math itself. By using math in these practical situations, you’ll be able to improve your math skills and build confidence in your abilities ( 8 Strategies to Improve Your Mathematical Skills. , 2021). 

Additionally, if you’re struggling with a particular type of math problem, it’s helpful to work on solving additional problems of the same kind. By doing this, you’ll eventually come up with a strategy that works well for you, which will reduce the difficulty level. As you become more confident with the simplified problems, finding solutions will eventually become easier.

3. Apply Math to Real Life

As previously stated, math plays an essential role in our daily lives. For instance, the next time you go shopping, you can make your child compare prices, mentally calculate your total bill, and estimate the discount percentage to sharpen your mental math skills and overall math proficiency.

Another exciting and straightforward way to enhance your mathematical skills is to participate in cooking or baking activities. Measuring ingredients, doubling or reducing recipe quantities, converting cooking units, calculating temperatures and time-keeping all involve fundamental mathematical concepts that will help you gain confidence and familiarity with math. By incorporating these practices in your child’s daily life, your child can gradually enhance your math skills while performing routine tasks ( 8 Strategies to Improve Your Mathematical Skills. , 2021). 

4. Set Realistic Goals

I f your child is struggling in math, simply adding more study time may not be the solution. It’s essential to identify their specific areas of difficulty and set realistic goals to address them. Forcing them to spend an extra hour on math each day without addressing their underlying struggles is unlikely to produce positive results ( 9 Ways To Improve Math Skills Quickly , 2021).

Rather than simply increasing their study time, it’s important to focus on understanding their specific challenges. Spending two additional hours practicing a concept they don’t understand will only lead to more frustration. Even if they can work through the mechanics of a problem, they will likely feel lost when faced with the next lesson. By setting achievable goals and focusing on the areas where they need the most help, they will be more likely to see real progress and develop a stronger foundation in math ( 9 Ways To Improve Math Skills Quickly , 2021).

5. Get Math Help

If you’ve noticed that your child is struggling with big-picture concepts in math, it’s important to address the issue as soon as possible. One of the most effective ways to do this is to consider finding a math tutor or enrolling your child in a math program like MathProject.

It’s essential to understand that every child learns differently, and their struggles with math could be due to a variety of reasons. Perhaps they haven’t fully grasped a concept or are struggling to connect the dots between different topics. It could also be that they need someone to explain things in a way that makes more sense to them. Whatever the reason may be, a math program can help to provide the personalized attention and guidance that your child needs to succeed.

A math program can help your child develop good study habits and organizational skills. They can teach your child how to break down complex problems into more manageable parts, which can be especially helpful for children who struggle with big-picture concepts. Furthermore, a tutor can help your child learn how to think critically and logically, which will benefit them not only in math but also in other areas of their academic and personal life.

Check out what other parents have to say about us, here !

• Meloney, J. (2022, October 6). 10 Ways To Help Students Improve Math Skills . Bytelearn. Retrieved April 10, 2023, from https://www.bytelearn.com/articles/improve-math-skills/
• 8 strategies to improve your mathematical skills. (2021, December 21). Sherwood High. Retrieved April 10, 2023, from https://sherwoodhigh.com/blogs/8-strategies-to-improve-your-mathematical-skills/
• 9 Ways To Improve Math Skills Quickly . (2021, November 1). Prodigy. Retrieved April 10, 2023, from https://www.prodigygame.com/main-en/blog/improve-math-skills/

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4 Ways to Improve Your Problem-Solving Skills

The ability to problem-solve is indispensable. As you move forward with your education, it will prove to be an invaluable asset not only as you study for entrance exams and unit tests, but also in college and beyond, when you launch your career.

When you set out to solve a problem, you are essentially defining a question in need of a solution. Whether you solve that problem ultimately depends upon your ability to ask the right question and to take the necessary steps to find the most sensible solution.

Of course, problem-solving is not always straightforward. Often, it requires a great deal of insight and critical thinking to imagine possible solutions, while it takes creativity and decisiveness to implement the best solution. Learning when and how to employ these qualities involves discretion, but acquiring such skills can also be the key difference between acing and flunking an exam. Here are four ways to improve your problem-solving skills: 

1. Learn how to identify the problem

On tests, a significant amount of time is wasted when a student is unsure what the problem is about. Occasionally, the source of an incorrect answer is rooted not in misinformation, but rather in misunderstanding. When preparing to solve a problem, ensure you are certain of two things: its scope (what is the question truly asking?) and its limits (what is the question not asking?). You can then move on to defining the problem.

One way to define a problem is to rephrase the question. If you are dealing with a word problem whose sentences are long and convoluted, it may be helpful to break it into shorter, clearer portions. It may also be useful to mentally rearrange the word order so it makes sense to you. If you choose to do so, take care to avoid losing the original meaning. Correctly identifying a problem takes reading comprehension. To sharpen your reading comprehension skills, practice asking yourself questions when you read like, “Can I summarize this paragraph in two sentences?”

2. Draw connections

Once you have determined the correct question, you can then find the right answer. This can be a multi-step process. For instance, on a math test, you may encounter a complex problem that you have never encountered before. Instead of skipping it, assess whether there is any part of the question that resembles a math problem you have solved in the past. Break it into simpler steps, then think each through. Math is more than just memorizing formulas and functions and making calculations—much of math depends upon numerical reasoning and logic. As a result, to improve your problem-solving skills, sharpen your reasoning skills. You may be surprised by the results.

3. Develop good habits

To master any skill, you must first practice. Practice independently or with a mentor, like a tutor. Challenge yourself to practice problems in an area that is difficult for you. Here are a few of the best study habits for students. If solving antonym/synonym questions is simple, see if you can answer reading questions just as quickly. Or if algebra is your strong suit, perhaps you should devote more time to geometry. 

Solving such problems over and over again will help you strengthen pathways in your brain so you can do them again later—this time, more quickly.  Complete practice tests . Complete problem-solving exercises. Time yourself to see if you improve. Eventually, logical solutions will present themselves more readily to you, until solving problems feels like second nature.

4. Fuel your brain power

Your brain is like an engine. It is powerful, quick, and it can go far. However, without proper fuel, it will not function well. Do not underestimate the value of a good night’s sleep and eating well. Your brain and your body go hand-in-hand. To perform at your best mentally, you also need to take care of yourself physically.

You may be surprised to learn that problem-solving skills depend on logical reasoning. The rational thinking that is required to solve problems involves a mentality that can be applied to many different scenarios. Strengthen your ability to think clearly and to identify strategies. In the end, you will have a skill set that is not only vital, but will also point you to real solutions.

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How to Improve Problem Solving Skills

Last Updated: July 24, 2024 Fact Checked

This article was co-authored by Erin Conlon, PCC, JD . Erin Conlon is an Executive Life Coach, the Founder of Erin Conlon Coaching, and the host of the podcast "This is Not Advice." She specializes in aiding leaders and executives to thrive in their career and personal lives. In addition to her private coaching practice, she teaches and trains coaches and develops and revises training materials to be more diverse, equitable, and inclusive. She holds a BA in Communications and History and a JD from The University of Michigan. Erin is a Professional Certified Coach with The International Coaching Federation. There are 11 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 239,928 times.

The ability to solve problems applies to more than just mathematics homework. Analytical thinking and problem-solving skills are a part of many jobs, ranging from accounting and computer programming to detective work and even creative occupations like art, acting, and writing. While individual problems vary, there are certain general approaches to problem-solving like the one first proposed by mathematician George Polya in 1945. By following his principles of understanding the problem, devising a plan, carrying out the plan, and looking back, you can improve your problem-solving and tackle any issue systematically.

Define the problem clearly.

• Try to formulate questions. Say that as a student you have very little money and want to find an effective solution. What is at issue? Is it one of income – are you not making enough money? Is it one of over-spending? Or perhaps you have run into unexpected expenses or your financial situation has changed?

State your objective.

• Say that your problem is still money. What is your goal? Perhaps you never have enough to go out on the weekend and have fun at the movies or a club. You decide that your goal is to have more spending cash. Good! With a clear goal, you have better defined the problem.

Gather information systematically.

• To solve your money shortage, for example, you would want to get as detailed a picture of your financial situation as possible. Collect data through your latest bank statements and to talk to a bank teller. Track your earnings and spending habits in a notebook, and then create a spreadsheet or chart to show your income alongside your expenditures.

Analyze information.

• Say you have now collected all your bank statements. Look at them. When, how, and from where is your money coming? Where, when, and how are you spending it? What is the overall pattern of your finances? Do you have a net surplus or deficit? Are there any unexplained items?

Generate possible solutions.

• Your problem is a lack of money. Your goal is to have more spending cash. What are your options? Without evaluating them, come up with possible options. Perhaps you can acquire more money by getting a part-time job or by taking out a student loan. On the other hand, you might try to save by cutting your spending or by lowering other costs.
• Divide and conquer. Break the problem into smaller problems and brainstorm solutions for them separately, one by one.
• Use analogies and similarities. Try to find a resemblance with a previously solved or common problem. If you can find commonalities between your situation and one you've dealt with before, you may be able to adapt some of the solutions for use now.

Evaluate the solutions and choose.

• How can you raise money? Look at expenditures – you aren’t spending much outside of basic needs like tuition, food, and housing. Can you cut costs in other ways like finding a roommate to split rent? Can you afford to take a student loan just to have fun on the weekend? Can you spare time from your studies to work part-time?
• Each solution will produce its own set of circumstances that need evaluation. Run projections. Your money problem will require you to draw up budgets. But it will also take personal consideration. For example, can you cut back on basic things like food or housing? Are you willing to prioritize money over school or to take on debt?

Implement a solution.

• You decide to cut costs, because you were unwilling to take on debt, to divert time away from school, or to live with a roommate. You draw up a detailed budget, cutting a few dollars here and there, and commit to a month-long trial.

Review and evaluate the outcome.

• The results of your trial are mixed. On one hand, you have saved enough during the month for fun weekend activities. But there are new problems. You find that you must choose between spending cash and buying basics like food. You also need a new pair of shoes but can’t afford it, according to your budget. You may need to a different solution.

Adjust if necessary.

• After a month, you decide to abandon your first budget and to look for part-time work. You find a work-study job on campus. Making a new budget, you now have extra money without taking too much time away from your studies. You may have an effective solution.

Do regular mental exercises.

• Word games work great. In a game like “Split Words,” for example, you have to match word fragments to form words under a given theme like “philosophy.” In the game, “Tower of Babel,” you will need to memorize and then match words in a foreign language to the proper picture.
• Mathematical games will also put your problem solving to the test. Whether it be number or word problems, you will have to activate the parts of your brain that analyze information. For instance: “James is half as old now as he will be when he is 60 years older than he was six years before he was half as old as he is now. How old will James be when his age is twice what it was 10 years after he was half his current age?”

Play video games.

• Play something that will force you to think strategically or analytically. Try a puzzle game like Tetris. Or, perhaps you would rather prefer a role-playing or strategy game. In that case, something like “Civilization” or “Sim-City” might suit you better.

Take up a hobby.

• Web design, software programming, jigsaw puzzles, Sudoku, and chess are also hobbies that will force you to think strategically and systematically. Any of these will help you improve your overall problem solving.

Expert Q&A

You Might Also Like

• ↑ https://www.healthywa.wa.gov.au/Articles/N_R/Problem-solving
• ↑ https://asq.org/quality-resources/problem-solving
• ↑ https://ctb.ku.edu/en/table-of-contents/evaluate/evaluate-community-interventions/collect-analyze-data/main
• ↑ https://www.mindtools.com/pages/article/newCT_96.htm
• ↑ https://www.skillsyouneed.com/ips/problem-solving.html
• ↑ Erin Conlon, PCC, JD. Executive Life Coach. Expert Interview. 31 August 2021.
• ↑ https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5930973/
• ↑ https://www.theguardian.com/lifeandstyle/2018/oct/13/mental-exercises-to-keep-your-brain-sharp
• ↑ https://www.apa.org/monitor/2014/02/video-game
• ↑ https://www.nature.com/articles/d41586-018-05449-7

About This Article

To improve your problem-solving skills, start by clearly defining the problem and your objective or goal. Next, gather as much information as you can about the problem and organize the data by rewording, condensing, or summarizing it. Then, analyze the information you've gathered, looking for important links, patterns, and relationships in the data. Finally, brainstorm possible solutions, evaluate the solutions, and choose one to implement. For tips on implementing solutions successfully, read on! Did this summary help you? Yes No

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• General & Introductory Mathematics

Crossing the River with Dogs: Problem Solving for College Students, 3rd Edition

ISBN: 978-1-394-15314-5

Digital Evaluation Copy

Ken Johnson , Ted Herr , Judy Kysh

• Rewrote the language in several problems to make the problems easier to understand, or to make the problem situation more modern.
• Added new problems to Problem Set A in several chapters.
• Added problems to the More Practice section in several chapters.
• Replaced many Problem Set B problems with new problems.
• Examples are solved as dialogues between fictitious students so students can see common errors and learn to avoid pitfalls.
• Each chapter includes relevant examples of the strategy presented.
• Chapters conclude with problem sets made up of three levels: A (routine), B (lengthy), and Classic Problems whose level can be determined by instructors.
• Over a dozen distinct problem-solving strategies are developed in depth, including classic strategies as well as more contemporary strategies.
• Icons appear to indicate examples, important information, and difficult material.

Mathematical Problem-Solving: Techniques and Strategies

by Ali | Mar 8, 2023 | Blog Post , Blogs | 0 comments

Introduction to Mathematical Problem-Solving

Mathematical problem-solving is the process of using logical reasoning and critical thinking to find a solution to a mathematical problem. It is an essential skill that is required in a wide range of academic and professional fields, including science, technology, engineering, and mathematics (STEM).

Importance of Mathematical Problem-Solving Skills

Mathematical problem-solving skills are critical for success in many areas of life, including education, career, and daily life. It helps students to develop analytical and critical thinking skills, enhances their ability to reason logically, and encourages them to persevere when faced with challenges.

The Process of Mathematical Problem-Solving

The process of mathematical problem-solving involves several steps that include identifying the problem, understanding the problem, making a plan, carrying out the plan, and checking the answer.

Techniques and Strategies for Mathematical Problem-Solving

1.      identify the problem.

The first step in problem-solving is to identify the problem. It involves reading the problem carefully and determining what the problem is asking.

2.      Understand the problem

The next step is to understand the problem by breaking it down into smaller parts, identifying any relevant information, and determining what needs to be solved.

3.      Make a plan

After understanding the problem, the next step is to develop a plan to solve it. This may involve identifying a formula or method to use, drawing a diagram or chart, or making a list of steps to follow.

4.      Carry out the plan

Once a plan is developed, the next step is to carry out the plan by solving the problem using the chosen method. It is important to show all steps and work neatly to avoid making mistakes.

5.      Check the answer

Finally, it is essential to check the answer to ensure it is correct. This can be done by re-reading the problem, checking the solution for accuracy, and verifying that it makes sense.

Know About: HOW TO FIND PERFECT MATH TUTOR 

Importance of using online calculators while learning math.

Utilizing online calculators can prove to be a beneficial resource for learning mathematics. There are numerous reasons why incorporating them into your studies is a wise choice.

Firstly, online calculators offer the convenience of being easily accessible at any time and from anywhere. No longer do you need to carry a physical calculator with you; you can use them on any device that has internet connectivity.

In addition, online calculators excel in accuracy and can efficiently handle complex calculations that may be difficult to do manually. They can perform arithmetic at a faster speed, saving you time and increasing productivity.

Another advantage is that some online calculators include built-in visualizations such as graphs and charts, which can help students grasp mathematical concepts better.

Furthermore, feedback can be provided by certain online calculators, assisting students in identifying and rectifying errors in their calculations. This feature can be especially useful for students who are new to learning mathematics .

Online calculators have a versatile range of functions beyond basic arithmetic, including algebraic equations, trigonometry, and calculus . This makes them useful for students at all levels of math education.

Overall, online calculators are an invaluable tool for students learning math. They are convenient, accurate, efficient, and versatile, and aid in the understanding of mathematical concepts, making them an essential component of modern-day education.

Common Errors in Mathematical Problem-Solving

There are several common errors that can occur in mathematical problem-solving, including misunderstanding the problem, using incorrect formulas or methods, making computational errors, and not checking the answer. To avoid these errors, it is essential to read the problem carefully, use the correct formulas and methods, check all computations, and double-check the answer for accuracy.

Improving Mathematical Problem-Solving Skills

There are several ways to improve mathematical problem-solving skills, including practicing regularly, working with others, seeking help from a teacher or tutor, and reviewing past problems. It is also helpful to develop a positive attitude towards problem-solving, persevere through challenges, and learn from mistakes.

Must Know: WHICH IS THE BEST WAY OF LEARNING ONLINE TUTORING OR TRADITIONAL TUTORING

Mathematical problem-solving is a crucial skill that is required for success in many academic and professional fields. By following the process of problem-solving and using the techniques and strategies outlined in this article, individuals can improve their problem-solving skills and achieve success in their academic and professional endeavors.

Frequently Asked Questions

What is mathematical problem-solving.

Mathematical problem-solving is the process of using logical reasoning and critical thinking to find a solution to a mathematical problem.

Why are mathematical problem-solving skills important?

What are the steps involved in the process of mathematical problem-solving, how can online calculators aid in learning mathematics.

Online calculators can aid in learning mathematics by providing convenience, accuracy, and efficiency. They can also help students grasp mathematical concepts better through built-in visualizations and provide feedback to identify and rectify errors in their calculations.

What are common errors to avoid in mathematical problem-solving?

Common errors to avoid in mathematical problem-solving include misunderstanding the problem, using incorrect formulas or methods, making computational errors, and not checking the answer. To avoid these errors, it is essential to read the problem carefully, use the correct formulas and methods, check all computations, and double-check the answer for accuracy.

We are committed to help students by one on one online private tutoring to maximize their e-learning potential and achieve the best results they can.

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How to Improve Mathematical Thinking and General Problem Solving Skills?

I'm a sophomore in university and seriously feel that I'm bad at solving mathematical and algorithmic problems (be it discrete math, calculus or just puzzles). I noticed that I'm only good at solving questions that are similar to the ones that have been taught to us.

Here's how I generally approach it:

• What is the problem? What do I need to do here?
• Does it look like I've encountered this before?
• Can I think of a smaller problem to solve instead?

If the answer is no to all the above then I sort of blank out. I stare at it and force my brain to run through a wide variety of stuff, almost like a brute force attempt of solving it. Obviously that leads me to nowhere everytime. I simply can't think "outside the box."

What can I do to improve my situation?

• soft-question
• problem-solving
• $\begingroup$ What kinds of questions are you talking about? I think a lot of it does come down to recognizing certain tricks and patterns, and you build up this ability with experience. How often do people truly think outside of the box? $\endgroup$ –  wj32 Commented Nov 1, 2012 at 20:57
• 3 $\begingroup$ "What can I do to improve my situation?" (1) Do A LOT of problems. (2) Read George Polya's "How To Solve It" $\endgroup$ –  BobaFret Commented Nov 1, 2012 at 21:05
• $\begingroup$ Hmm not exactly sure how to answer this. Just questions in general on any topic say textbook practice problems or questions in: projecteuler.net/problems although these are more math puzzle types. $\endgroup$ –  Charles Khunt Commented Nov 1, 2012 at 21:06
• 1 $\begingroup$ I think that how we perceive ourselves, specifically how we perceive ourselves in terms of "what I'm good at" or "what I'm bad at" can be self-fulfilling. I think one's attitude when encountering novel situations, in general, like new problems, has a lot to do with how successful one is in handling the situation: if one develops confidence in one's competence, one is more likely to persevere . One can be fearful, intimidated (retreat); one can feel challenged and stimulated; etc... $\endgroup$ –  amWhy Commented Nov 1, 2012 at 21:07
• 1 $\begingroup$ I added a couple tags; hopefully, these tags will counter the "not constructive" close vote. $\endgroup$ –  Emily Commented Nov 1, 2012 at 21:08

6 Answers 6

You might want to read Thinking Mathematically . (I read it and it's excellent. It will teach you exactly what you're looking for.)

• 2 $\begingroup$ Hear hear! A terrific book: we've based part of a course on reading and doing maths for our first year undergrads on it. It helps them not just with problem solving, but also with understanding what it is we do when we do maths. $\endgroup$ –  user12477 Commented Nov 1, 2012 at 21:15
• 1 $\begingroup$ @amWhy That's a different book. See this instead. $\endgroup$ –  Michael Greinecker Commented Nov 1, 2012 at 21:24
• $\begingroup$ @Michael: thanks for checking that out and pointing it out!...oops, seems I've posted an incorrect link! I'll delete it at once! $\endgroup$ –  amWhy Commented Nov 1, 2012 at 21:38
• $\begingroup$ Dear @amWhy, I read your now deleted comment as an alternative recommendation. $\endgroup$ –  Rudy the Reindeer Commented Nov 2, 2012 at 7:34
• $\begingroup$ @MattN Thanks for your comment; I'll "repost" the link here as a different book, perhaps worth looking into. $\endgroup$ –  amWhy Commented Nov 2, 2012 at 13:19

I belonged to a school education system where we were made to do lots of different problems, but we were never told to try and understand the underlying theory behind the problems. This made me scared of math. What I basically had was a cookbook of a variety of wonderful recipes without realizing why I needed to add salt or sugar to a dish. May be you are facing the same problem? May be you are learning all these different techniques to solve problems without really understanding the theory behind why the problems can be solved using those techniques? Hence, because you don't understand the theory behind the techniques, once you get a problem that cannot be solved using the techniques you are familiar with, you get stuck.

While I agree with glebovg that trying to develop an intuition for how to write proofs is essential, I feel that you should make the effort to start reading proofs first. For instance, a book that really helped me understand Calculus was Spivak's Calculus. Try going through the proofs there, and learn the underlying theory. This is coming from someone who was in your position not too long ago.

I encourage you to read books that emphasize problem solving, but at some point you will just have muster the courage to open a book with proofs, and read through it.

Also, the issue of memorization is kind of a slippery slope. You will find that often even when you are trying to understand the theory, you will just have to memorize some computational techniques here and there. I think Terry Tao has a good post where he addresses the issue of memorization. I agree with him that certain basic things have to be memorized. For instance, you will have to memorize what the axioms of a group or a field are. I think memorization and understanding go hand in hand. Certainly your goal should not be to only memorize techniques to solve problems.

Here is more advice from a master:

http://terrytao.wordpress.com/career-advice/solving-mathematical-problems/

http://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/

http://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-grades-and-exams-and-methods/

http://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/

All the best!

• $\begingroup$ A very insightful answer and suggested readings! $\endgroup$ –  user1007190 Commented Dec 30, 2023 at 5:25

I think proving theorems really develops your thinking. Try to prove a few important theorems from calculus as well as discrete math, or try to understand someone's proof. Of course, the more you know the better, so that is why we say math is not a spectator sport. You need to do more than just the homework if you want to improve. Sometimes many results that you learn in, say discrete math, might seem confusing, but once you see why they are important in a different context, for example in number theory or algebra, you should remember them. To be honest, I think understanding and being able to prove theorems is actually relevant to math, whereas puzzles are just for fun. The best advice I can give is: Do not try to memorize math and simply remember everything for an exam because that way you might get a good grade, but you will forget everything a few days after the exam, instead try to understand why something is true. This way you will remember something practically forever, because you will be able to derive it when you forget.

I don't know about puzzles, so I write only about solving mathematics problems. In my experiences in this site, I find it far easier to solve problems in a field(like abstract algebra) I know well than in a field(like analysis) I know less. I think it's like walking in a town. If the town is where you live, you know every corner and you think you can almost walk with blindfold. On the other hand, if you are new in the town, you lose your way easily.

So the question is how we know a field well. Read textbooks, understand proofs, try to prove a theorem before reading the proof of a textbook. Reconstruct a proof without seeing a textbook. do exercises, try to find examples and counterexamples, try to find problems by yourself and solve them, etc.

I got a PhD and a postdoc in pure mathematics and I just can talk from my experience. I think that mathematical thinking can be improved with your experience solving problems and reading. For me, there are 2 options.

Option 1: improve your mathematical thinking by yourself. This means trying to approach the problem from all the possible points of view that you can imagine. Organize them, try to apply them one by one and draw a lot. Try to improvise and start solving similar problems in simpler versions. This is very hard to do, especially when you learn in a systematic way because this requires creativity but if you spend time doing this even if you don't solve it you can grow a lot and develop intuition. Warning! Do not spend more than 1 week with the same problem. Not all people solve problems quickly and that is just fine. If you cannot solve a problem after your hard work is a good idea to ask for help (books, mentors, the internet, etc) or just leave it in a special list and move forward. You will be able to solve that list in the future. Reading some comments reminds me that teaching to others the exercises that you can solve is a very powerful way to improve your mathematical thinking, it helps to organize your knowledge and discipline your mind. Please be patient with others. Some day someone will be patient with you and you will need it!!!

Option 2: improve your mathematical thinking using help. Some people may think that asking for help from others or books destroys your creativity and limit your mathematical thinking to the creativity of others. But only a few gifted can afford that. I think that there is nothing wrong with gathering some strategies of others to enrich your own bunch of tools. Consult a friend, professor, books or forums like this. When you ask for help the method to solve a problem is something that you haven't thought of, however, the new experience can help you to solve new problems in the future. When you are facing a new kind of problem and you do not have a clue where to start, look for examples and solved exercises. If you are in high school most likely there is a lot of reading material, examples and solved exercises for the topics that you are interested in.

In the end, your experience solving problems and your background will develop your mathematical thinking, and you can do it using options 1 and 2.

Old thread, but I came across this and wanted to pitch in my 2 cents.

I remember when I first got to college and was studying mechanical engineering. My high school education taught me the plug-n-chug method of thinking, so topics like differential equations, physics, let alone, linear algebra, dynamics, thermo, mechanics, etc. were really really difficult for me.

Somehow I struggled through it though, and graduated, but I always felt uneasy about having as solid of problem-solving skills in my educational foundation as I wanted on it. Especially since I was now working (tho my day-to-day work didn't require those specific skills). I ended up making a hack solution and practiced one math or physics problem a day on my own. I felt like I really came to understand those things since now I took the time to go through them myself, and see where all the formulas were derived from. Knowing that, I knew better when I could apply an equation, and in what manner.

I actually came across this site later: www.learnerds.com which pretty much was what I was looking for. An interesting (semi-realistic) math/engineering/science question a day with a good solution, and the authors are great at responding back to your comments, regardless of your level.

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• Title & authors

Caraan, Danielle Rencell C., et al. "Realistic Mathematics Education Approach on Improving Problem-Solving Skills of Students." The Research Probe , vol. 2, no. 2, pp. 88-96.

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The Philippines’ education system is still dominated by traditional mathematics teaching, which frequently overlooks the goal of mathematics education—to prepare students to deal successfully with real-life situations. This affects the declining performance of the students in their overall mathematical ability, especially in problem-solving. Hence, this study utilized a pre-experimental design to measure the effectiveness of the Realistic Mathematics Education (RME) approach in the problem-solving skills of the students in terms of understanding the problem, devising a plan, carrying out the plan, and looking back. Furthermore, the cluster sampling technique was used in choosing thirty-five (35) grade 9 students and evaluated their problem-solving ability using a pre-test and post-test assessment. Based on the result, there is a highly significant difference in the mean pre-test and post-test performance of the respondent before and after using the RME approach in all the four phases of problem-solving (p-value=0.000). This implies that the RME is an effective teaching approach that successfully improved the Mathematical proficiency of the students, especially in all aspects of problem-solving skills. According to the findings, the researchers may advise educators to use the RME approach to expose their students to more collaborative teaching-learning processes that incorporate real-world scenarios.

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