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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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All you need to know to successfully implement a mastery approach to mathematics in your primary school, at whatever stage of your journey.

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5. Teaching Mathematical Reasoning: Critical Math Thinking Through Problem-Solving and Modeling

• Mathematical problem-solving : This approach makes students think conceptually about problems before applying tools they’ve learned.
• Mathematical modeling : Modeling projects give students experience in weighing several factors against one another and using mathematical knowledge to make decisions.

I. Mathematical Problem-Solving

An emphasis on open-ended mathematical problem-solving can help develop mathematical reasoning skills and address a problem teachers have long been concerned about: too much “rote” learning in math. 

Too often students spend time in math class memorizing procedures and applying them mindlessly to problems. This leads to errors when students are confronted with unfamiliar problems. It also contributes to a widespread misperception of math as boring and lacking relevance to everyday life. 

On the other hand, attempting to remedy this problem by giving students open-ended problems has its own drawbacks. Without the conceptual and methodological tools to solve these problems students become frustrated and disengaged. It can end up being an inefficient way to spend class time.  

Although learning fundamental math skills like algorithms for adding, subtracting, multiplying, and dividing is absolutely critical for students in the early grades, the deeper mathematical problem-solving skills are the ones we really want students to graduate with. How can we ensure they do?

The deeper mathematical problem-solving skills are the ones we really want students to graduate with.

Evidence suggests that skills in mathematical problem-solving lead to more general improvements in outcomes related to math. They help students acquire a deeper understanding of mathematical reasoning and concepts. 

For instance, the commutative property, which most students learn applies to addition and multiplication problems (changing the order of the operations doesn’t affect the outcome), also applies to other logical and practical situations. A familiarity with some of these situations fosters deeper conceptual understanding, and deeper conceptual understanding leads to better critical thinking.

And learning these skills helps students improve outcomes related to critical thinking more generally. For example, students who become skilled in mathematical problem-solving tend to also:

• Create beneficial habits of mind — persistence, thoroughness, creativity in solution-finding, and improved self-monitoring.
• Break down hard problems into easier parts or reframing problems so that they can think about them more clearly. 
• Some problem solving tactics are applicable to situations well beyond math: making a visualization of a situation to understand it more clearly; creating a simplified version of the problem to more easily address the essence of the problem; creating branches of possibilities to solve the problem; creating “what if” example cases to test key assumptions, etc.
• Elevate the value of discussion and argumentation over simple appeals to authority.

Small-group mathematical problem solving targets skills that traditional mathematics instruction doesn’t. Instead of just finding a match between an algorithm and a question, students must: adapt or create an algorithm; evaluate and debate the merits of different solution paths; and verify their solution through additional evidence.

Small-group mathematical problem solving targets skills that traditional mathematics instruction doesn’t.

This process continues until the class has thoroughly explored the problem space, revealing multiple solution paths and exploring variations on the problem or contrasting problem-types.

Of course, the usefulness of a question like this depends on what students already know. If students don’t already know that chickens have two legs and pigs have four, they’re just going to be confused by the problem (and the explanation of the solution). It also requires some other basic skills—for instance, that if one chicken has two legs, four chickens would have eight.

As a way of evaluating student growth, teachers could also include some of these open-ended problems in homework assignments or as extra credit assignments.

Lesson Plan Outline

An example that might be appropriate for fifth grade is something like the following: A farmer has some pigs and some chickens. He finds that together they have 70 heads and 200 legs. How many pigs and how many chickens does he have? Divide the class into student groups of three to four. Have students spend a few minutes reading over the problem individually. Then let student groups discuss possible solution paths. The teacher walks around the classroom, monitoring the groups. Then the teacher leads a whole-class discussion about the problem.

• So how did you go about thinking about the problem?
• Show us how you got your answer and why you think it’s right. This might mean that a student goes up to the board to illustrate something if a verbal explanation is inadequate.
• And what was the answer you got?
• Does anyone else have a different way of thinking about the problem? If there are other ways of solving the problem that students didn’t come up with, teachers can introduce these other ways themselves.

Developing Math Problem-Solving Skills

Teachers should keep in mind the following as they bring mathematical problem-solving activities into their classrooms:

• Problem selection . Teachers have to select grade-appropriate problems. A question like “John is taller than Mary. Mary is taller than Peter. Who is the shortest of the three children?” may be considered an exercise to older students — that is, a question where the solutions steps are already known — but a genuine problem to younger students. It’s also helpful when problems can be extended in various ways. Adding variation and complexity to a problem lets students explore a class of related problems in greater depth.
• Managing student expectations . Introducing open-ended math problems to students who haven’t experienced them before can also be confusing for the students. Students who are used to applying algorithms to problems can be confused about what teachers expect them to do with open-ended problems, because no algorithm is available.
• Asking why . Asking students to explain the rationale behind their answer is critical to improving their thinking. Teachers need to make clear that these rationales or justifications are even more important than the answer itself. These justifications give us confidence that an answer is right. That is, if the student can’t justify her answer, it almost doesn’t matter if it’s correct, because there’s no way of verifying it.

II. Mathematical Modeling

Another approach is mathematical modeling. Usually used for students in middle or high school, mathematical modeling brings math tools to bear on real-world problems, keeping students engaged and helping them to develop deeper mathematical reasoning and critical thinking skills.

Math modeling is an extremely common practice in the professional world. Investors model returns and the effects of various events on the market; business owners model revenue and expenses, buying behavior, and more; ecologists model population growth, rainfall, water levels, and soil composition, among many other things. 

But, despite these many applications and the contributions it can make to general mathematical reasoning and critical thinking skills, mathematical modeling is rarely a main component of the math curriculum. Although textbook examples occasionally refer to real-world phenomena, the modeling process is not commonly practiced in the classroom.

Modeling involves engaging students in a big, messy real-world problem. The goals are for students to:

• refine their understanding of the situation by asking questions and making assumptions,
• leverage mathematical tools to solve the problem,
• make their own decisions about how to go about solving the problem,
• explain whether and how their methods and solutions make sense,
• and test or revise their solutions if necessary.

Mathematical modeling typically takes place over the course of several class sessions and involves working collaboratively with other students in small groups.

Modeling is not just about getting to a “right” answer — it’s about considering factors beyond mathematics as well.

Modeling also offers the opportunity to integrate other material across the curriculum and to “think mathematically” in several different contexts. Modeling is not just about getting to a “right” answer — it’s about considering factors beyond mathematics as well. For example, students deal with questions like:

• What is a “fair” split? 
• What level of risk should someone tolerate?
• What tradeoffs should a society make?

In others words, students come to see mathematics as the socially indispensable tool that it is, rather than an abstract (and sometimes frustrating) school subject.

Mathematical Modeling and Critical Thinking

Research suggests that the ability to solve abstractly framed academic math problems is not necessarily related to mathematical reasoning more broadly: that is, the ability to use math well in everyday life or to integrate mathematical thinking into one’s decision-making. Students may be able to follow procedures when given certain cues, but unable to reason about underlying concepts. 

It’s also very common to hear complaints from students about math — that either they aren’t “ math people ,” that math is irrelevant, or that math is simply boring.

Mathematical modeling is one approach to resolving both these problems. It asks students to move between the concreteness of real — or at least relatively realistic — situations and the abstraction of mathematical models. Well-chosen problems can engage student interest. And the practice emphasizes revision, step-by-step improvement, and tradeoffs over single solution paths and single right-or-wrong answers.

Mathematical modeling often begins with a general question, one that may initially seem only loosely related to mathematics:

• how to design an efficient elevator system, given certain constraints;
• what the best gas station is to visit in our local area;
• how to distinguish between two kinds of flies, given some data about their physical attributes.

Then, over the course of the modeling process, students develop more specific questions or cases, adding constraints or assumptions to simplify the problem. Along the way, students identify the important variables — what’s changing, and what’s not changing? Which variables are playing the biggest role in the desired outcomes?

Students with little experience in modeling can leap too quickly into looking for a generalized solution, before they have developed a feel for the problem. They may also need assistance in developing those specific cases. During this part of the process, it can be easiest to use well-defined values for some variables. These values may then become variables later on.

After students explore some simplifying cases, then they work on extensions of these cases to reach ever more general solutions.

A key part of this activity is letting students be creative — students will often come up with unusual or especially innovative solutions.

Throughout the modeling process, the teacher may need to point out missing assumptions or constraints, or offer other ways of reframing the problem. For any given modeling problem, some solutions are usually more obvious than others, which leads to common stages students may reach as they solve the problem. But a key part of this activity is letting students be creative — students will often come up with unusual or especially innovative solutions.

A sample problem, from the Guidelines for Assessment and Instruction in Mathematical Modeling Education is below:

This problem involves variables that aren’t necessarily immediately apparent to students. For instance, the size of the gas tank, and how much gas you fill up on per trip. As students manage this specific case, they can take other hypothetical scenarios to generalize their solution: if it’s 10 miles away, how cheap would the gas have to be to make it worth it? What about the time spent in the car — is there a value to put on that?

Many modeling problems can be arbitrarily extended in various directions. Instead of just considering the best gas station to go to for a single car, for instance, students can explore the behavior of a fleet of trucks on set routes or seasonal changes to gas prices.

It’s also possible to include shorter modeling activities, where students work together in pairs or small groups to extend a problem or interpret the meaning of a solution.

These kinds of modeling activities are not reserved solely for older students. One example of a modeling problem for students in elementary school might be something like: what should go in a lunchbox? Students can talk about what kinds of things are important to them for lunch, “mathematize” the problem by counting student preferences or coming up with an equation (e.g., lunch = sandwich + vegetable + dessert + drink); and even explore geometrically how to fit such items into a lunchbox of a certain size.

Teaching Mathematical Modeling: Further Key Factors

Mathematical modeling activities can be challenging for both teachers and students. 

Often, mathematical modeling activities stretch over several class periods. Fitting modeling activities in, especially if standardized tests are focused on mathematical content, can be challenging. One approach is to design modeling activities that support the overall content goals.

The teacher’s role during mathematical modeling is more like a facilitator than a lecturer. Mathematical modeling activities are considerably more open-ended than typical math activities, and require active organization, monitoring, and regrouping by the teacher. Deciding when to let students persevere on a problem for a bit longer and when to stop the class to provide additional guidance is a key skill that only comes with practice.

The teacher’s role during math modeling is more like a facilitator than a lecturer.

Students — especially students who have traditionally been successful in previous math classes — may also experience frustration when encountering modeling activities for the first time. Traditional math problems involve applying the right procedure to a well-defined problem. But expertise at this kind of mathematical reasoning differs markedly from tackling yet-to-be-defined problems with many possible solutions, each of which has tradeoffs and assumptions. Students might feel unprepared or even that they’re being treated unfairly.

Students also have to have some knowledge about the situation to reason mathematically about it. If the question is about elevators, for example, they need to know that elevators in tall buildings might go to different sets of floors; that elevators have a maximum capacity; that elevators occasionally break and need to be repaired. 

Finally, the mathematical question needs to be tailored to students’ experience and interests. Asking a group of students who don’t drive about how to efficiently purchase gas won’t garner student interest. Teachers should use their familiarity with their students to find and design compelling modeling projects. This is chance for both students and teachers to be creative. 

To download the PDF of the Teachers’ Guide

(please click here)

Sources and Resources

O’Connell, S. (2000). Introduction to Problem Solving: Strategies for The Elementary Classroom . Heinemann. A recent handbook for teachers with tips on how to implement small-group problem solving.

Youcubed.org , managed by Jo Boaler.  A community with lots of resources for small-group problem solving instruction.

Yackel, E., Cobb, P., & Wood, T. (1991). Small group interactions as a source of learning opportunities in second-grade mathematics . Journal for research in mathematics education , 390-408. Education research that illustrates how small-group problem solving leads to different kinds of learning opportunities than traditional instruction.

Guidelines for Assessment and Instruction in Mathematical Modeling Education , 2nd ed. (2019). Consortium for Mathematics and its Applications & Society for Industrial and Applied Mathematics.  An extensive guide for teaching mathematical modeling at all grade levels.

Hernández, M. L., Levy, R., Felton-Koestler, M. D., & Zbiek, R. M. (March/April 2017). Mathematical modeling in the high school curriculum . The variable , 2(2). A discussion of the advantages of mathematical modeling at the high school level.

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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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Khan Academy Blog

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

posted on September 20, 2023

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

Math Learning

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

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

Benefits of Math Learning

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

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

How to Know What Math You Need to Learn

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

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

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

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How to learn math.

Learning at School

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

Learning at Home

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

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

Learning Math with the Help of Artificial Intelligence (AI) 

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

Using Khan Academy’s AI Tutor, Khanmigo

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

Get Math Help with Khanmigo Right Now

You can learn anything .

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

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

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

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

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

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

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

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Mathematical Reasoning & Problem Solving

In this lesson, we’ll discuss mathematical reasoning and methods of problem solving with an eye toward helping your students make the best use of their reasoning skills when it comes to tackling complex problems.

Previously Covered:

• Over the course of the previous lesson, we reviewed some basics about chance and probability, as well as some basics about sampling, surveys, etc. We also covered some ideas about data sets, how they’re represented, and how to interpret the results.

Approaches to Problem Solving

When solving a mathematical problem, it is very common for a student to feel overwhelmed by the information or lack a clear idea about how to get started.

To help the students with their problem-solving “problem,” let’s look at some examples of mathematical problems and some general methods for solving problems:

Identify the following four-digit number when presented with the following information:

• One of the four digits is a 1.
• The digit in the hundreds place is three times the digit in the thousands place.
• The digit in the ones place is four times the digit in the ten’s place.
• The sum of all four digits is 13.
• The digit 2 is in the thousands place.

Help your students identify and prioritize the information presented.

In this particular example, we want to look for concrete information. Clue #1 tells us that one digit is a 1, but we’re not sure of its location, so we see if we can find a clue with more concrete information.

We can see that clue #5 gives us that kind of information and is the only clue that does, so we start from there.

Because this clue tells us that the thousands place digit is 2, we search for clues relevant to this clue. Clue #2 tells us that the digit in the hundreds place is three times that of the thousands place digit, so it is 6.

So now we need to find the tens and ones place digits, and see that clue #3 tells us that the digit in the ones place is four times the digit in the tens place. But we remember that clue #1 tells us that there’s a one somewhere, and since one is not four times any digit, we see that the one must be in the tens place, which leads us to the conclusion that the digit in the ones place is four. So then we conclude that our number is:

If you were following closely, you would notice that clue #4 was never used. It is a nice way to check our answer, since the digits of 2614 do indeed add up to be thirteen, but we did not need this clue to solve the problem.

Recall that the clues’ relevance were identified and prioritized as follows:

• clue #3 and clue #1

By identifying and prioritizing information, we were able to make the information given in the problem seem less overwhelming. We ordered the clues by relevance, with the most relevant clue providing us with a starting point to solve the problem. This method also utilized the more general method of breaking a problem into smaller and simpler parts to make it easier to solve.

Now let’s look at another mathematical problem and another general problem-solving method to help us solve it:

Two trees with heights of 20 m and 30 m respectively have ropes running from the top of each tree to the bottom of the other tree. The trees are 40 meters apart. We’ll assume that the ropes are pulled tight enough that we can ignore any bending or drooping. How high above the ground do the ropes intersect?

Let’s solve this problem by representing it in a visual way , in this case, a diagram:

You can see that we have a much simpler problem on our hands after drawing the diagram. A, B, C, D, E, and F are vertices of the triangles in the diagram. Now also notice that:

b = the base of triangle EFA

h = the height of triangle EFA and the height above the ground at which the ropes intersect

If we had not drawn this diagram, it would have been very hard to solve this problem, since we need the triangles and their properties to solve for h. Also, this diagram allows us to see that triangle BCA is similar to triangle EFC, and triangle DCA is similar to triangle EFA. Solving for h shows that the ropes intersect twelve meters above the ground.

Students frequently complain that mathematics is too difficult for them, because it is too abstract and unapproachable. Explaining mathematical reasoning and problem solving by using a variety of methods , such as words, numbers, symbols, charts, graphs, tables, diagrams, and concrete models can help students understand the problem better by making it more concrete and approachable.

Let’s try another one.

Given a pickle jar filled with marbles, about how many marbles does the jar contain?

Problems like this one require the student to make and use estimations . In this case, an estimation is all that is required, although, in more complex problems, estimates may help the student arrive at the final answer.

How would a student do this? A good estimation can be found by counting how many marbles are on the base of the jar and multiplying that by the number of marbles that make up the height of the marbles in the jar.

Now to make sure that we understand when and how to use these methods, let’s solve a problem on our own:

How many more faces does a cube have than a square pyramid?

Reveal Answer

The answer is B. To see how many more faces a cube has than a square pyramid, it is best to draw a diagram of a square pyramid and a cube:

From the diagrams above, we can see that the square pyramid has five faces and the cube has six. Therefore, the cube has one more face, so the answer is B.

Before we start having the same problem our model student in the beginning did—that is, being overwhelmed with too much information—let’s have a quick review of all the problem-solving methods we’ve discussed so far:

• Sort and prioritize relevant and irrelevant information.
• Represent a problem in different ways, such as words, symbols, concrete models, and diagrams.
• Generate and use estimations to find solutions to mathematical problems.

Mathematical Mistakes

Along with learning methods and tools for solving mathematical problems, it is important to recognize and avoid ways to make mathematical errors. This section will review some common errors.

Circular Arguments

These involve drawing a conclusion from a premise that is itself dependent on the conclusion. In other words, you are not actually proving anything. Circular reasoning often looks like deductive reasoning, but a quick examination will reveal that it’s far from it. Consider the following argument:

• Premise: Only an untrustworthy man would become an insurance salesman; the fact that insurance salesmen cannot be trusted is proof of this.
• Conclusion: Therefore, insurance salesmen cannot be trusted.

While this may be a simplistic example, you can see that there’s no logical procession in a circular argument.

Assuming the Truth of the Converse

Simply put: The fact that A implies B doesn’t not necessarily mean that B implies A. For example, “All dogs are mammals; therefore, all mammals are dogs.”

Assuming the Truth of the Inverse

Watch out for this one. You cannot automatically assume the inverse of a given statement is true. Consider the following true statement:

If you grew up in Minnesota , you’ve seen snow.

Now, notice that the inverse of this statement is not necessarily true:

If you didn’t grow up in Minnesota , you’ve never seen snow.

Faulty Generalizations

This mistake (also known as inductive fallacy) can take many forms, the most common being assuming a general rule based on a specific instance: (“Bridge is a hard game; therefore, all card games are difficult.”) Be aware of more subtle forms of faulty generalizations.

Faulty Analogies

It’s a mistake to assume that because two things are alike in one respect that they are necessarily alike in other ways too. Consider the faulty analogy below:

People who absolutely have to have a cup of coffee in the morning to get going are as bad as alcoholics who can’t cope without drinking.

False (or tenuous) analogies are often used in persuasive arguments.

Now that we’ve gone over some common mathematical mistakes, let’s look at some correct and effective ways to use mathematical reasoning.

Let’s look at basic logic, its operations, some fundamental laws, and the rules of logic that help us prove statements and deduce the truth. First off, there are two different styles of proofs: direct and indirect .

Whether it’s a direct or indirect proof, the engine that drives the proof is the if-then structure of a logical statement. In formal logic, you’ll see the format using the letters p and q, representing statements, as in:

If p, then q

An arrow is used to indicate that q is derived from p, like this:

This would be the general form of many types of logical statements that would be similar to: “if Joe has 5 cents, then Joe has a nickel or Joe has 5 pennies “. Basically, a proof is a flow of implications starting with the statement p and ending with the statement q. The stepping stones we use to link these statements in a logical proof on the way are called axioms or postulates , which are accepted logical tools.

A direct proof will attempt to lay out the shortest number of steps between p and q.

The goal of an indirect proof is exactly the same—it wants to show that q follows from p; however, it goes about it in a different manner. An indirect proof also goes by the names “proof by contradiction” or reductio ad absurdum . This type of proof assumes that the opposite of what you want to prove is true, and then shows that this is untenable or absurd, so, in fact, your original statement must be true.

Let’s see how this works using the isosceles triangle below. The indirect proof assumption is in bold.

Given: Triangle ABC is isosceles with B marking the vertex

Prove: Angles A and C are congruent.

Now, let’s work through this, matching our statements with our reasons.

• Triangle ABC is isosceles . . . . . . . . . . . . Given
• Angle A is the vertex . . . . . . . . . . . . . . . . Given
• Angles A and C are not congruent . . Indirect proof assumption
• Line AB is equal to line BC . . . . . . . . . . . Legs of an isosceles triangle are congruent
• Angles A and C are congruent . . . . . . . . The angles opposite congruent sides of a triangle are congruent
• Contradiction . . . . . . . . . . . . . . . . . . . . . . Angles can’t be congruent and incongruent
• Angles A and C are indeed congruent . . . The indirect proof assumption (step 3) is wrong
• Therefore, if angles A and C are not incongruent, they are congruent.

“Always, Sometimes, and Never”

Some math problems work on the mechanics that statements are “always”, “sometimes” and “never” true.

Example: x < x 2 for all real numbers x

We may be tempted to say that this statement is “always” true, because by choosing different values of x, like -2 and 3, we see that:

Example: For all primes x ≥ 3, x is odd.

This statement is “always” true. The only prime that is not odd is two. If we had a prime x ≥ 3 that is not odd, it would be divisible by two, which would make x not prime.

• Know and be able to identify common mathematical errors, such as circular arguments, assuming the truth of the converse, assuming the truth of the inverse, making faulty generalizations, and faulty use of analogical reasoning.
• Be familiar with direct proofs and indirect proofs (proof by contradiction).
• Be able to work with problems to identify “always,” “sometimes,” and “never” statements.

Student Workbooks

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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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Unit 1: Problem solving with arithmetic

Comparing with multiplication.

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Multiplication, division word problems

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Multi-step word problems

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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 Skills: Meaning, Examples & Techniques

Table of Contents

26 January 2021

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Do your children have trouble solving their Maths homework?

Or, do they struggle to maintain friendships at school?

If your answer is ‘Yes,’ the issue might be related to your child’s problem-solving abilities. Whether your child often forgets his/her lunch at school or is lagging in his/her class, good problem-solving skills can be a major tool to help them manage their lives better.

Children need to learn to solve problems on their own. Whether it is about dealing with academic difficulties or peer issues when children are equipped with necessary problem-solving skills they gain confidence and learn to make healthy decisions for themselves. So let us look at what is problem-solving, its benefits, and how to encourage your child to inculcate problem-solving abilities

Problem-solving skills can be defined as the ability to identify a problem, determine its cause, and figure out all possible solutions to solve the problem.

• Trigonometric Problems

What is problem-solving, then? Problem-solving is the ability to use appropriate methods to tackle unexpected challenges in an organized manner. The ability to solve problems is considered a soft skill, meaning that it’s more of a personality trait than a skill you’ve learned at school, on-the-job, or through technical training. While your natural ability to tackle problems and solve them is something you were born with or began to hone early on, it doesn’t mean that you can’t work on it. This is a skill that can be cultivated and nurtured so you can become better at dealing with problems over time.

Problem Solving Skills: Meaning, Examples & Techniques are mentioned below in the Downloadable PDF. 

Benefits of learning problem-solving skills  

Promotes creative thinking and thinking outside the box.

Improves decision-making abilities.

Builds solid communication skills.

Develop the ability to learn from mistakes and avoid the repetition of mistakes.

Problem Solving as an ability is a life skill desired by everyone, as it is essential to manage our day-to-day lives. Whether you are at home, school, or work, life throws us curve balls at every single step of the way. And how do we resolve those? You guessed it right – Problem Solving.

Strengthening and nurturing problem-solving skills helps children cope with challenges and obstacles as they come. They can face and resolve a wide variety of problems efficiently and effectively without having a breakdown. Nurturing good problem-solving skills develop your child’s independence, allowing them to grow into confident, responsible adults. 

Children enjoy experimenting with a wide variety of situations as they develop their problem-solving skills through trial and error. A child’s action of sprinkling and pouring sand on their hands while playing in the ground, then finally mixing it all to eliminate the stickiness shows how fast their little minds work.

Sometimes children become frustrated when an idea doesn't work according to their expectations, they may even walk away from their project. They often become focused on one particular solution, which may or may not work.

However, they can be encouraged to try other methods of problem-solving when given support by an adult. The adult may give hints or ask questions in ways that help the kids to formulate their solutions. 

Encouraging Problem-Solving Skills in Kids

Practice problem solving through games.

Exposing kids to various riddles, mysteries, and treasure hunts, puzzles, and games not only enhances their critical thinking but is also an excellent bonding experience to create a lifetime of memories.

Create a safe environment for brainstorming

Welcome, all the ideas your child brings up to you. Children learn how to work together to solve a problem collectively when given the freedom and flexibility to come up with their solutions. This bout of encouragement instills in them the confidence to face obstacles bravely.

Invite children to expand their Learning capabilities

 Whenever children experiment with an idea or problem, they test out their solutions in different settings. They apply their teachings to new situations and effectively receive and communicate ideas. They learn the ability to think abstractly and can learn to tackle any obstacle whether it is finding solutions to a math problem or navigating social interactions.

Problem-solving is the act of finding answers and solutions to complicated problems. 

Developing problem-solving skills from an early age helps kids to navigate their life problems, whether academic or social more effectively and avoid mental and emotional turmoil.

Children learn to develop a future-oriented approach and view problems as challenges that can be easily overcome by exploring solutions. 

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Frequently Asked Questions (FAQs)

How do you teach problem-solving skills.

Model a useful problem-solving method. Problem solving can be difficult and sometimes tedious. ... 1. Teach within a specific context. ... 2. Help students understand the problem. ... 3. Take enough time. ... 4. Ask questions and make suggestions. ... 5. Link errors to misconceptions.

What makes a good problem solver?

Excellent problem solvers build networks and know how to collaborate with other people and teams. They are skilled in bringing people together and sharing knowledge and information. A key skill for great problem solvers is that they are trusted by others.

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Mathematical Skills: What They Are And Examples

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Find a Job You Really Want In

Mathematical skills are important to improve if you want to increase your chances for professional success, no matter what career path you pursue. Many jobs use mathematical skills regularly, and even for the rare jobs that never directly deal with numbers and figures, you’ll often need the same problem-solving and critical thinking abilities used in math to succeed.

So, if you’re a job seeker who wants to know more about how to make your mathematical skills shine, stay tuned. In this article, we’ll cover the most important mathematical skills to master for the workplace and discuss how to improve and highlight your math skills during the job-search process .

Key Takeaways:

10 mathematical skills that are useful in the workplace are time management, mental arithmetic, constructing logical arguments, abstract thinking, data analysis, research, visualization, creativity, forecasting, and attention to detail.

Improve your mathematical skills by acquiring conceptual understandings of the skills and solving practice problems.

A mathematical skill should be listed on a resume when the job listing states the skill as a requirement.

Most mathematical skills are transferable and help you stand out in a crowd of applicants.

What are mathematical skills?

How to improve your mathematical skills, how to highlight mathematical skills on a resume, mathematical skills resume example, mathematical skills faq.

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The term “mathematical skills” doesn’t just refer to nebulous topics taught in school, such as calculus. They’re the practical abilities that are useful no matter the industry or size of business you work in.

This includes skills such as:

Time management. Being able to manage your time efficiently is critical for your day-to-day activities, in addition to long-term planning success.

The average person wastes three entire hours each day due to inefficient time management.

Not only does that immediately translate to wasted money, but to wasted time that could be devoted to your non-work passions and activities.

Mental arithmetic. Being able to do mental math quickly will serve you well in a variety of professions.

Retail workers may need to quickly and accurately figure out a customer’s change when given a large sum of money.

Constructing logical arguments. Many careers demand the same precise, logical reasoning that’s used to solve math problems.

An attorney needs to ensure that their legal argument logically follows from the facts and evidence provided.

Abstract thinking. Abstract thinking is the ability to understand and compare non-physical concepts, such as freedom or honesty.

Improving your abstract thinking skills is useful for any career that involves creativity or navigating through complex rules.

Data analysis. A large variety of professions will require you to interpret and analyze data at some point.

Any scientific career will involve heavy interpretation of complex sets of data.

Research. Knowing how to effectively research information is crucial for developing solutions for the many problems you’ll face in your career.

In the age of the internet, there is a nearly infinite wealth of information about any topic you could wish to learn about.

Visualization. The same ability to visualize problems and outcomes is critical for finding solutions in the workplace.

Any problem you face during your career will present a variety of possible solutions with which to tackle it.

Creativity. Improving your creativity skills allows you to come up with new ideas and innovations.

Presenting fresh ideas and solutions will also help you stand out among the competition in the workplace.

Forecasting. Forecasting is the ability to extrapolate events into the future based on available data and knowledge.

This skill is critical for any job that involves planning for the future.

Attention to detail. Some jobs require more attention to small details than others.

Mathematical skills can be improved in the same way that you would improve any other skill – through consistent practice.

More specifically, there are two key actions you should follow:

Acquire conceptual understanding. You can’t improve a mathematical skill if you don’t even know what that math skill entails.

For example, suppose that you wish to improve your data analysis skills.

A quick Google search reveals that the main elements of data analysis include understanding statistics, visuals such as charts and graphs, and how to apply the data to practical problems.

Solve practice problems. It’s not enough to understand a concept to master it; you must practice practically applying it.

This piece of advice applies to certain math skills more than others. You can find plenty of online games to help you improve your hard skills , such as mental arithmetic, but maybe not your creativity.

Mathematical skills positively effect your work performance, especially when you improve them.

However, we still want to find a way to highlight them to recruiters, so they know that we’ve mastered them.

There are a few important guidelines to follow:

List or prove on your resume. The skills section of your resume can be an okay place to mention your math skills.

The job listing states the skill as a requirement. If your resume doesn’t contain the specific term, some companies’ applicant tracking systems (ATS) may automatically filter you out.

Not significant enough to waste additional resume space. Despite being required, some skills may not be essential enough to waste more than a single bullet point talking about.

For example, a job may require basic clerical skills such as multiplying and dividing small figures.

“Critical-thinking skills” and “ problem-solving skills ” are generically added to so many resumes that the terms often become meaningless.

A better way to highlight your math skills on a resume is to prove it through the results you’ve achieved. Use numbers to emphasize the positive value you created for a past employer.

Prove them in your cover letter. You want to give examples of when you used mathematical skills to create value for a past employer.

This differs from the resume strategy in that cover letters are narrower in scope.

Your resume needs to fit many examples on a single page , while your cover letter can target a few key skills to demonstrate with greater detail.

To figure out which math skills to focus on, pay attention to the essential requirements and duties listed in the job listing. Make your best judgment on the most important skills to highlight.

Explain in-depth during your job interview. Job interviews allow you the time to dive much deeper into examples of how you’ve utilized mathematical skills.

Consider the previous resume example about developing a new marketing strategy using data-analysis skills.

During the interview, you could expand on the specific technical skills and tools you used. Explain the initial problem and the thought process you employed to tackle it.

Mathematical skills can be more tangible when you can see them on a resume. Luckily, we’ve provided an mathematical skills resume for you:

Finnegan Bennett 117 Melrose Ave., Austin, TX , 73301 (662)-280-0092 [email protected] Detail-orientated and organized mathematics teacher with over 10 years of experience working in high schools. Possesses a Masters in Education from Austin University. Strong skills in problem-solving and time management. Professional Experience Austin Independent School District , Austin, TX Geometry Teacher , September 2016 — Present Presented HSPA specific lessons to various classes within the mathematics department. Assist with teaching students as an entire class or in small groups as the lessons are planned. Designed, developed, and implemented courses in coordination with science curriculum Designed and implement classroom management strategies at a school wide level. Educate , Austin, TX Math Teacher , September 2013 — August 2016 Delivered and graded assessments in multiple subject areas meeting educational standards. Provide tutoring services by facilitating small groups or individual students for children in grades kindergarten through high school. Skills Pre-Calculus Classroom Management Student Learning Mathematics Special Education State Standards Clear Objectives Test Scores In-Service Training Small Groups Education University of Austin , Austin, TX Masters Degree Education , May 2016 Baker University , Austin, TX Bachelor’s Degree Business , May 2012 Graduated with honors

What are the most important math skills?

The most important math skills in the workplace depend on your needs. The four fundamental arithmetic operations of adding, subtracting, multiplying and dividing are very important for all adults to have a basic understanding because they appear in many of our lives daily.

Time management, logic, and abstract thinking are also very important for most adults to know, regardless of profession, because they help provide structure to your life and prepare you for critical thinking.

Why are mathematical skills important?

Mathematical skills are important because they provide structure to solving problems rationally. Mathematical skills can be used everyday to make sense of a chaotic world. Recognizing patterns, using logic, building on abstract concepts all are what help keep society moving.

How do you list math skills on a resume?

List math skills under the skills section of a resume. In order to be efficient with space, make sure to only list relevant skills that are found in the job description and avoid general terms.

MAA – 10 Skills and Abilities Every Math Major Should Include on Their Resume

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Chris Kolmar is a co-founder of Zippia and the editor-in-chief of the Zippia career advice blog. He has hired over 50 people in his career, been hired five times, and wants to help you land your next job. His research has been featured on the New York Times, Thrillist, VOX, The Atlantic, and a host of local news. More recently, he's been quoted on USA Today, BusinessInsider, and CNBC.

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18: Solutions to Selected Exercises

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• 18.1: Problem Solving
• 18.2: Voting Theory
• 18.3: Weighted Voting
• 18.4: Fair Division
• 18.5: Apportionment
• 18.6: Graph Theory
• 18.10: Statistics
• 18.11: Describing Data
• 18.12: Probability
• 18.13: Sets
• 18.14: Historical Counting
• 18.15: Fractals
• 18.16: Cryptography
• 18.17: Logic
• 18.7: Scheduling
• 18.8: Growth Models
• 18.9: Finance

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International Journal for Lesson and Learning Studies

ISSN : 2046-8253

Article publication date: 3 January 2017

Problem solving is a skill in mathematics which although always relevant has heightened priority due to the changes in the new mathematics GCSE (Department for Education, 2013). It has previously been a skill which is deemed underdeveloped within mathematics and therefore is a theme which teachers are seeking to improve and nurture in order to align with the new changes. The GCSE is the formal qualification that students take at the end of Key Stage 4 (KS4) in the UK. The paper aims to discuss these issues.

Design/methodology/approach

The focus of the enquiry was to explore, using lesson studies, the differences in students’ approaches to problem solving. Consequently, key themes relating to the mediation of gender, ability, and academic motivation surfaced. Considering these themes, the paper subsequently reflects upon pedagogical practices which might effectively develop students’ ability to problem solve. The study took part in a mixed gender comprehensive secondary school with students taking part in the observation lesson ranging in age from 11 to 12 years old. The authors are the teachers who took part in the lesson study. The teachers implemented observation techniques in the form of video and peer observation with the accompanying teacher. In addition, students provided feedback on how they approached the problem-solving tasks through a form of semi-structured interviews, conducted via the use of video diaries where no teachers were present to prevent power bias. Following this, a thematic analysis of both the observations and student video diaries generated conclusions regarding how said key themes shaped the students’ approaches to problem solving.

Students’ frustration and competitive need to find a specific answer inhibited their ability to thoroughly explore the problem posed thus overseeing vital aspects needed to solve the problem set. Many students expressed a passion for problem solving due to its freedom and un-rigid nature, which is something teachers should nurture.

Originality/value

Generally, teachers are led by a culture in which attainment is the key. However, an atmosphere should be developed where the answer is not the key and students can explore the vibrant diversity mathematics and problem solving can offer.

• Lesson study
• Mathematics
• Problem-solving skills
• Video diaries

Bradshaw, Z. and Hazell, A. (2017), "Developing problem-solving skills in mathematics: a lesson study", International Journal for Lesson and Learning Studies , Vol. 6 No. 1, pp. 32-44. https://doi.org/10.1108/IJLLS-09-2016-0032

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Copyright © 2017, Emerald Publishing Limited

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