mathematical and problem solving skills examples

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Mathematical Skills: What They Are And Examples

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

posted on September 20, 2023

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

Math Learning

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

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

Benefits of Math Learning

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

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

How to Know What Math You Need to Learn

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

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

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

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How to learn math.

Learning at School

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

Learning at Home

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

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

Learning Math with the Help of Artificial Intelligence (AI) 

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

Using Khan Academy’s AI Tutor, Khanmigo

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

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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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1.1: Introduction to Problem Solving

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• Michelle Manes
• University of Hawaii

The Common Core State Standards for Mathematics ( http://www.corestandards.org/Math/Practice ) identify eight “Mathematical Practices” — the kinds of expertise that all teachers should try to foster in their students, but they go far beyond any particular piece of mathematics content. They describe what mathematics is really about, and why it is so valuable for students to master. The very first Mathematical Practice is:

Make sense of problems and persevere in solving them.

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary.

This chapter will help you develop these very important mathematical skills, so that you will be better prepared to help your future students develop them. Let’s start with solving a problem!

Draw curves connecting A to A, B to B, and C to C. Your curves cannot cross or even touch each other,they cannot cross through any of the lettered boxes, and they cannot go outside the large box or even touch it’s sides.

Think / Pair / Share

After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it).

• What did you try?
• What makes this problem difficult?
• Can you change the problem slightly so that it would be easier to solve?

Problem Solving Strategy 1 (Wishful Thinking).

Do you wish something in the problem was different? Would it then be easier to solve the problem?

For example, what if ABC problem had a picture like this:

Can you solve this case and use it to help you solve the original case? Think about moving the boxes around once the lines are already drawn.

Here is one possible solution.

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Common Core Math Explained: 8 Common Core Math Examples To Use In The Classroom 

Samantha dock.

Common Core math examples can be a tricky world to navigate for teachers trying to meet the individual needs of their students. Having a bank of Common Core math examples to hand can be helpful when planning your lessons. 

Embraced by the majority of states in the U.S., the Common Core math standards help to develop students’ conceptual understanding, problem-solving skills, and real-world applications.  In this article, we explore what Common Core math is, 8 Common Core math examples and top tips for educators teaching Common Core math.

What is Common Core math?

How is common core math different from traditional math, common core standards and 8 common core math examples, 1. make sense of problems and persevere in solving them, 2. reason abstractly and quantitatively, 3. construct viable arguments and critique the reasoning of others, 4. model mathematics, 5. use appropriate tools strategically, 6. attend to precision, 7. look for and make use of structure, 8. look for and express regularity in repeated reasoning, tips for teaching common core math, common core math and the wider world.

Common Core math standards are a set of educational standards for mathematics adopted by forty states in the United States. Each standard outlines the math knowledge students should know and be able to do at each grade level, from kindergarten through to high school. 

These standards aim to provide a more focused and coherent set of learning goals for students, emphasizing conceptual understanding, problem-solving, and critical thinking skills.  

Often, Common Core math involves multiple strategies and approaches to solving problems. In turn, this encourages students to understand the underlying concepts rather than simply memorizing algorithms.  

One aim of the Common Core State Standards is to move away from traditional memorization of procedures and algorithms towards a deeper understanding of connections between mathematical concepts.  

Common Core math standards are organized by grade level and cover a wide range of mathematical topics, including:  

• Arithmetic 
• Probability

Each standard is divided into domains, which represent broad categories of mathematical content such as: 

• Counting and cardinality
• Operations and algebraic thinking
• Numbers and operations in base ten
• Measurement and data

3rd to 6th grade Common Core math test

Help your students prepare for their state math test with these Common Core practice math tests for 3rd - 6th grade.

Common Core math standards have been controversial in some areas due to concerns about curriculum changes, standardized testing, and complexity. But some argue that these standards provide a more coherent approach to mathematics education and better prepare students for higher education than traditional math. 

Read more: Why is Math Important?

Common Core math and traditional math represent two different approaches to teaching mathematics.

Traditional math typically refers to methods of teaching mathematics that were used before the adoption of the Common Core standards. These methods often focused on rote memorization of formulas and procedures, with less emphasis on understanding the concepts or on real-life application of mathematical skills.

Here are some key differences between the two:

Focus on Conceptual Understanding vs. Memorization

A strong emphasis is placed on developing students’ conceptual understanding of math concepts under the common core. It aims to help learners understand the “why” behind mathematical procedures rather than just memorizing algorithms. Traditional math often focuses more on rote memorization of formulas and procedures without necessarily understanding the underlying concepts needed to approach math questions.

Problem-Solving and Critical Thinking vs. Rote Practice

Common Core math problems encourage critical thinking skills. They promote multiple approaches to solving new math problems and require students to justify their reasoning. Often, traditional math involves repetitive practice of standard procedures with less emphasis on problem-solving and critical thinking.

Real-World Applications vs. Abstract Exercises

Connections between mathematical concepts to real-world situations is valued under the common core. This helps students see the relevance of the math skills they are learning. Tasks and problems require the application of mathematical skills in practical contexts. Traditional math lessons focus more on abstract exercises and textbook problems that may not always have clear real-world connections.

Depth of Understanding vs. Breadth of Coverage

Rather than covering a wide range of topics, Common Core math aims for depth of understanding and maths mastery. Fewer topics at each grade level allow for deeper exploration and mastery of key concepts. In contrast, traditional math tends to cover a broader range of topics in less depth.

Flexibility and Multiple Strategies vs. One Correct Method

Students are encouraged to use multiple strategies and approaches to solve problems through the Common Core math standards. Flexibility and creativity are valued when approaching problem-solving. Emphasis on a single “correct” method or algorithm for solving problems is the general approach in traditional math. Overall, Common Core State Standards aim to develop students’ mathematical proficiency in alignment with the demands of the modern world. This includes the need for critical thinking, problem-solving, and application of mathematical concepts to real-world situations

Overall, Common Core State Standards aim to develop students’ mathematical proficiency in alignment with the demands of the modern world. This includes the need for critical thinking, problem-solving, and application of mathematical concepts to real-world situations. 

Third Space Learning provides one-on-one math instruction for students who need it most. Personalized one-on-one math lessons are designed by math experts and aligned to your state’s math standards — including the Common Core State Standards. 

Students should not only be able to understand problems and make sense of them, but persevere in finding solutions. 

Finding solutions may involve math skills such as: 

• Analyzing problems
• Making conjectures
• Planning approaches to solving math problems 

Common Core math example 1

A student is faced with a word problem about finding the area of a garden. They must take the time to carefully read and understand the problem before attempting to solve it. 

This problem may require several approaches to answer the math question. Small group work and discussion can encourage students to persevere through the challenge and try different strategies until they find a solution.

In order to reason abstractly, students need to be able to make sense of quantities and their relationships in mathematical situations. 

This will be easier for students if they can take abstract information from context and quantify information. Being able to decontextualize and contextualize mathematical ideas will benefit students.

Common Core math example 2

A graph shows the relationship between the number of hours worked and the amount earned. Students can analyze the graph to determine patterns and make predictions about future earnings based on proportional relationships between hours worked and money earned. 

For example, if John worked for 13 hours, how much money would he earn?

Introducing math vocabulary in the classroom helps students construct viable arguments and critique the mathematical reasoning of others. Exposure to mathematics language and sentence stems will help students to reason mathematically, construct arguments, and justify their thinking, without creating cognitive overload. 

Common Core math example 3

During a class discussion about strategies for solving a particular math problem, you might ask students to present their solutions — justifying and explaining their reasoning. 

They can also be encouraged to critique each other’s approaches, identify strengths and weaknesses in their arguments and offer alternative methods. 

Math lessons should prepare students to use math to solve real-world problems. It may help students to do this if you represent mathematical concepts with visual models and math manipulatives . 

Common Core math example 4

Subtraction of fractions is a skill that many students struggle with. Using a visual model to describe and analyze the word problem can release cognitive load for students. 

For example, Paul had 11 ⅔ yards of twine. He used 6 ½ yards to make macrame wall hangings, how many yards of twine does Paul have left?

To solve math problems effectively and efficiently, students must be able to select and use appropriate tools. This includes recognizing when and how to use tools, as well as evaluating effectiveness and efficiency.

Common Core math example 5

When solving a complex geometry problem, students should recognise the effectiveness of using a protractor and ruler to accurately measure angles and lengths. 

For example, Given an angle ABC where point B is the vertex of the angle, construct an angle bisector of angle ABC using a ruler and a protractor. Then, using the angle bisector you have constructed, draw a line segment from point B to the bisected angle’s line that is exactly 5 cm long. Measure and report the angle sizes of the two new angles created by the angle bisector.

Calculations need to be carried out precisely. To do this, students need to be aware of key mathematical terminology for the Common Core Standards they are studying. This involves using appropriate units and labels and stating mathematical results clearly.

Common Core math example 6

A student ensures that their work is clear and organized. They pay attention to detail, avoiding errors and inaccuracies in their calculations. Below is a worked example of a student showing how to solve a word problem involving multiple percentages.

Solving math problems accurately means students need to recognize and use mathematical patterns and structure. They should be able to identify relationships between mathematical ideas and make connections between different mathematical representations.

Common Core math example 7

When solving a multiplication of decimals problem, a student recognizes that breaking down the whole numbers and decimal parts into their factors makes the problem easier to solve. They identify the underlying structure of the problem and use it to their advantage.

Identifying and generalizing patterns and regularities in mathematical situations is key to proficiency in problem soving and reasoning . Students should be able to notice repeated reasoning and use it to solve math problems efficiently. 

Common Core math example 8

Example: A student identifies similarities between a problem they’re working on and a previous math problem. They utilize the patterns in the prior example to complete the new problem. This also helps them to solve similar problems in the future.

Teachers need to understand Common Core math standards to recognize the appropriate instructional strategies and promote a growth mindset in the classroom .  

Here are 8 tips for maximising student progress when teaching the Common Core State Standards:

1. Understand the Common Core State Standards

Familiarize yourself with the math Common Core State Standards for your specific grade level. Take the time to understand the mathematical practices, domains and teaching strategies required for your grade.

2. Focus on conceptual understanding

Prioritize conceptual understanding over rote memorization. You can achieve this by helping students understand the “why” behind math concepts and skills. Always encourage them to explain and justify their reasoning.

3. Promote multiple approaches

Offer your students a range of math strategies and approaches to problem-solving. The more methods in their math bank, the better equipped they are to find a solution. Asking students to share their thinking process helps those who are not grasping the content from the math instruction.

4. Real-world connections

Connecting mathematical concepts to real-world situations makes learning more meaningful and relevant. You can do this by implementing math problems where students work collaboratively to solve complex, open-ended word problems with real-world relevance. 

5. Use visual representations

Diagrams, models, and manipulatives support students’ understanding of mathematical concepts by making abstract concepts more concrete and accessible. For example, you could use algebra tiles when students are first learning how to solve algebraic equations and inequalities to help them contexutalize the abstract nature of algebra.

6. Encourage discourse and collaboration

Promote a classroom environment where students feel comfortable sharing their ideas, asking questions and engaging in mathematical discourse. 

Encourage discourse by using techniques such as turn and talk, or the 3 reads method for word problems.

7. Assess progress

Use formative and summative assessments to monitor students’ progress and understanding of mathematical concepts and adjust instruction accordingly based on assessment data.  

Some examples of a formative assessment are: 

• Exit tickets
• Rating scales
• Thumbs up or thumbs down 

Summative assessments include: 

• Check for understanding quizzes
• End-of-topic quizzes

Assessment resources:

• Practice state assessments  

8. Professional development

Continuously seek professional development opportunities to deepen your understanding of Common Core math and improve your teaching practices. Collaborate with colleagues and participate in workshops, conferences, and online courses.

Embracing Common Core principles can help equip students for future challenges

Educators’ commitment to teaching Common Core math goes beyond math instruction. It’s about nurturing critical thinking and problem solving, ensuring students are prepared for the wider world. 

Math lessons are no longer simply giving students math worksheets and grading them on the correct answer. The American education system has developed a math curriculum that anchors mathematical concepts in real-world relevance, promotes diverse problem-solving strategies, and encourages a collaborative learning environment.  

Educators have a responsibility to ensure students have the tools and mathematical literacy they need to succeed.

Common Core math examples FAQ

1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning

1. Focus: Emphasizes focusing deeply on a smaller number of key topics at each grade level. This is done to ensure students develop a deep understanding of foundational mathematical ideas. 2. Coherence: Emphasizes the importance of coherence in mathematical instruction. This is done to support students in making meaningful connections between different mathematical ideas, helping them see how concepts are related and reinforcing their understanding over time 3. Rigor: Focuses on increasing the rigor of mathematical instruction by demanding that students engage in conceptual understanding, procedural fluency, and application of mathematical concepts in real-world contexts. In this context, rigor means ensuring that students develop a deep understanding of mathematical concepts, are able to apply their knowledge in various contexts, and can solve complex problems through reasoning and critical thinking.

Forty states have fully adopted Common Core math, while Minnesota partially embraces it. South Carolina, Oklahoma, Indiana, Florida, and Arizona initially adopted but later repealed Common Core. Alaska, Nebraska, Texas, and Virginia never adopted it.

Do you have students who need extra support in math? Give your students more opportunities to consolidate learning and practice skills through personalized math tutoring with their own dedicated online math tutor. Each student receives differentiated instruction designed to close their individual learning gaps, and scaffolded learning ensures every student learns at the right pace. Lessons are aligned with your state’s standards and assessments, plus you’ll receive regular reports every step of the way. Personalized one-on-one math tutoring programs are available for: – 2nd grade tutoring – 3rd grade tutoring – 4th grade tutoring – 5th grade tutoring – 6th grade tutoring – 7th grade tutoring – 8th grade tutoring Why not learn more about how it works ?

Ultimate Guide to Problem Solving Techniques [FREE]

Are you trying to build problem solving and reasoning skills in the classroom?

Here are 9 ready-to-go printable problem solving techniques that all your students should know, including challenges, short explanations and questioning prompts.

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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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Building Problem-solving Skills for 7th-Grade Math

Mathematics is a subject that requires problem-solving skills to excel. In 7th grade, students begin to encounter more complex math concepts, and the ability to analyze and solve problems becomes increasingly important. Building problem-solving skills for math not only helps students to master math concepts but also prepares them for success in higher-level math courses and in life beyond academics. 

In this article, we will several key skills that are needed for success in 7th-grade math, and also explore how they can benefit students both academically and personally. We will also provide tips and strategies to help students develop and improve their problem-solving skills. Let’s dive in!

Building Analytical Skills

The first of seven important skills to build is that of analytical skills. These allow students to analyze a problem and break it down into smaller parts. From there, they’re able to identify the key components that need to be addressed. Analytical skills also hone students’ abilities to identify patterns. Students should be able to identify patterns in mathematical data, such as in number sequences, geometric shapes, and graphs. Importantly, students should not just be able to recognize the pattens, but they should be able to describe them (more on that in communication) and use them to make predictions and solve problems.

We alluded to this earlier, but breaking down problems is an essential component of analytical skills. Students with strong analytical skills can break problems down into smaller and more manageable parts. They are then able to identify key components of a problem and use this information to develop a strategy for solving it. 

Along with identifying patterns comes identifying relationships. Students with good mathematical analytical skills can identify relationships between different mathematical concepts, such as the relationship between addition and subtraction, or the relationship between angles and shapes. Through strengthening this skill, students will be able to describe these relationships and use them to solve problems. 

An important part of analytical skills is the ability to analyze data. Students should be able to analyzeand interpret data presented in a variety of formats, such as graphs, charts, and tables. They should be able to use this data to make predictions, draw conclusions, and solve problems.

Speaking of conclusions, reaching sound conclusions based on mathematical data is a fundamental skill needed for making predictions based on trends in a graph, or drawing inferences from a set of data.

Another skill students should master is the ability to compare and contrast mathematical concepts, such as the properties of different shapes or the strategies for solving different types of problems. Through this, they’ll be able to use the information they gather to solve problems. 

With all these skills at play comes arguably the most important: Critical thinking. This is an indicator that a student really grasps the concepts and it’s just repeating them back to you on command. Critical thinking is the ability to evaluate information and arguments, and make judgements and decisions based on evidence, and apply logic and reasoning to solve problems.

Building Creative Thinking

This is the ability for students (or anyone, really) to think outside the box and come up with innovative solutions to problems. This involves being able to approach problems from different angles and to consider multiple perspectives. For a 7th-grader, this skill can be exercised through the following:

• Thinking Outside the Box: Students should be encouraged to think creatively and come up with innovative solutions to problems. This involves thinking outside the box and considering multiple perspectives.
• Finding Multiple Solutions: Students should be able to come up with multiple solutions to a problem and evaluate each one to determine which is the most effective.
• Developing Original Ideas: Students should be able to develop original ideas and approaches to solving problems. This involves being able to come up with unique and innovative solutions that may not have been tried before.
• Making Connections: Students should be able to make connections between different mathematical concepts and apply these connections to solve problems. This involves looking for similarities and differences between concepts and using this information to make new connections.
• Visualizing Solutions : Students should be able to visualize solutions to problems and use diagrams, charts, and other visual aids to help them solve problems.
• Using Metaphors and Analogies: Students should be able to use metaphors and analogies to help them understand complex mathematical concepts. This involves using familiar concepts to explain unfamiliar ones and making connections between different ideas.

Building Problem-Solving Strategies

It may sound like the same thing, but building problem-solving strategies is not the same as building problem-solving skills. Building strategies for problem-solving lends itself to actual problem-solving. Let’s expand on this: Say your student is presented a problem that they’re struggling with, these are some of the problem-solving strategies they may use in order to solve the puzzle.

• Identify the problem: The first step in problem-solving is to identify the problem and understand what is being asked. Students should carefully read the problem and make sure they understand the question before attempting to solve it.
• Draw a diagram: Students can draw a diagram to help visualize the problem and better understand the relationships between different parts of the problem.
• Use logic: Students can use logic to identify patterns and relationships in the problem. They can use this information to develop a plan to solve the problem.
• Break the problem down: Students can break a complex problem down into smaller, more manageable parts. They can then solve each part of the problem individually before combining the solutions to get the final answer.
• Guess and check: Students can guess and check different solutions to the problem until they find the correct answer. This method involves trying different solutions and evaluating the results until the correct answer is found.
• Use algebra: Algebraic equations can be used to solve a variety of mathematical problems. Students can use algebraic equations to represent the problem and solve for the unknown variable.
• Work backward: Students can work backward from the final answer to determine the steps required to solve the problem. This method involves starting with the end goal and working backward to determine the steps needed to get there.

Building Persistence and Perseverance

In an increasingly instant-gratification world with apps, searches and AI chatbots just a click away, this is an important skill not just in the math classroom, but for life in general. Problem-solving, whether that’s a math problem or a life challenge, often requires persistence and perseverance. Student need to learn to be able to stick with a problem even when it seems challenging, difficult, or seemingly impossible. Here are ways you can encourage your students to stick it out when working on problems:

• Trying multiple approaches: When faced with a challenging problem, students can demonstrate persistence by trying multiple approaches until they find one that works. They don’t give up after one attempt but keep trying until they find a solution.
• Reframing the problem: If a problem seems particularly difficult, students can demonstrate perseverance by reframing the problem in a different way. This can help them see the problem from a new perspective and come up with a different approach to solve it.
• Asking for help: Sometimes, even with persistence, a problem may still be difficult to solve. In these cases, students can demonstrate perseverance by asking for help from their teacher or classmates. This shows that they are willing to put in the effort to find a solution, even if it means seeking assistance.
• Learning from mistakes: Making mistakes is a natural part of the problem-solving process, but students can demonstrate persistence by learning from their mistakes and using them to improve their problem-solving skills. They don’t get discouraged by their mistakes, but instead, they use them as an opportunity to learn and grow.
• Staying focused: In order to solve complex math problems, it’s important for students to stay focused and avoid distractions. Students can demonstrate perseverance by staying focused on the problem at hand and not getting distracted by other things.

Building Communication Skills

We alluded to this earlier, but a central part of building problem-solving skills is building the ability to articulate a problem or a solution. This isn’t just for the sake of personal understanding, but critical for collaboration. Students need to be able to explain their thinking, ask questions, and work with others to solve problems. Here are some examples of communication skills that can be used to build problem-solving skills:

• Clarifying understanding: Students can ask questions to clarify their understanding of the problem. They can seek clarification from their teacher or classmates to ensure they are interpreting the problem correctly.
• Explaining their reasoning: When solving a math problem, students can explain their reasoning to show how they arrived at a particular solution. This can help others understand their thought process and can also help students identify errors in their own work.
• Collaborating with peers: Problem-solving can be a collaborative effort. Students can work together in groups to solve problems and communicate their ideas and solutions with each other. This can lead to a better understanding of the problem and can also help students learn from each other.
• Writing clear explanations: When presenting their solutions to a math problem, students can write clear and concise explanations that are easy to understand. This can help others follow their thought process and can also help them communicate their ideas more effectively.
• Using math vocabulary: Math has its own language and using math vocabulary correctly is essential for effective communication. Students can demonstrate their understanding of math concepts by using correct mathematical terms and symbols when explaining their solutions.

Building Mathematical Knowledge

This would seem like a no-brainer, since you’re a math educator clicking on an article about building math problem-solving skills. However, it’s worth being explicit that problem-solving in math requires a solid understanding of mathematical concepts, including arithmetic, algebra, geometry, and data analysis. Students need to be able to apply these concepts to solve problems in real-world contexts.

7th-grade math covers a wide range of mathematical concepts and skills. Here are some examples of mathematical knowledge that 7th-grade math students should have:

• Algebraic expressions and equations: Students should be able to write and simplify algebraic expressions and solve one-step and two-step equations.
• Proportional relationships: Students should be able to understand and apply proportional relationships, including identifying proportional relationships in tables, graphs, and equations.
• Geometry: Students should have a solid understanding of geometry concepts such as angles, triangles, quadrilaterals, circles, and transformations.
• Statistics and probability: Students should be able to analyze and interpret data using measures of central tendency and variability, and understand basic probability concepts.
• Rational numbers: Students should have a solid understanding of rational numbers, including ordering, adding, subtracting, multiplying, and dividing fractions and decimals.
• Integers: Students should be able to perform operations with integers, including adding, subtracting, multiplying, and dividing.
• Ratios and proportions: Students should be able to understand and use ratios and proportions in a variety of contexts, including scale drawings and maps.

In conclusion, problem-solving skills are essential for success in 7th grade math. Analytical skills, critical and creative thinking, problem-solving strategies, persistence, communication skills, and mathematical knowledge are all important components of effective problem-solving. By developing these skills, students can approach math problems with confidence and achieve their full potential.

If you enjoyed this read, be sure to browse more of our articles . More importantly, if you want to save yourself hours of preparation time by having full math curriculums, review guides and tests available at the click of a button, be sure to sign up to our 7th Grade Newsletter . You’ll receive loads of free lesson resources, tips and advice and exclusive subscription offers!

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5 Teaching Mathematics Through Problem Solving

Janet Stramel

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

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

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

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

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

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

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

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

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

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

Key features of a good mathematics problem includes:

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

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

But look at the b ack.

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

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

Mathematics Tasks and Activities that Promote Teaching through Problem Solving

Choosing the Right Task

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

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

Low Floor High Ceiling Tasks

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

The strengths of using Low Floor High Ceiling Tasks:

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

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

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

Math in 3-Acts

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

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

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

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

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

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

Number Talks

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

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

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

• Inside Mathematics Number Talks
• Number Talks Build Numerical Reasoning

Saying “This is Easy”

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

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

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

Instead, you and your students could say the following:

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

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

Using “Worksheets”

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

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

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

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

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

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

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

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

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

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

when we teach students how to problem solve

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

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

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

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

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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.

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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7 Problem Solving Skills That Aren’t Just Buzzwords (+ Resume Example)

• Julia Mlcuchova , 
• Updated April 8, 2024 9 min read

Problem-solving skills are something everybody should include on their resume, yet only a few seem to understand what these skills actually are. If you've always felt that the term "problem-solving skills" is rather vague and wanted to know more, you've come to the right place.

In this article, we're going to explain what problem-solving skills really mean. We'll talk about what makes up good problem-solving skills and give you tips on how to get better at them. You'll also find out how to make your problem-solving abilities look more impressive to those who might want to hire you.

Sounds good, right? Curious to learn more? 

In this article we’ll show you:

• What are problem solving skills;
• Why are they important; 
• Specific problem solving skills examples;
• How to develop your problem solving skills;
• And, how to showcase them on your resume.

Table of Contents

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What are problem solving skills?

Why are problem solving skills important, the best 7 problem solving skills examples, how to develop problem solving skills, problem solving skills resume example, key takeaways: problem solving skills.

First of all, they're more than just a buzzword!

Problem-solving skills are a set of specific abilities that allow you to deal with unexpected situations in the workplace, whether it be job related or team related. 

It's a complex process that involves several “sub skills” or “sub steps,” namely:

• Recognizing and identifying the issue at hand.
• Breaking the problem down into smaller parts and analyzing how they relate to one another. 
• Creating potential solutions to the problem, evaluating them and picking the best one.  
• Applying the chosen solution and assessing its outcome. 
• Learning from the whole process to deal with future problems more effectively. 

As you can see, it's not just about solving problems that are right in front of us, but also about predicting potential issues and being prepared to deal with them before they arise.  

Despite what you may believe, problem-solving skills aren't just for managers . 

Think about it this way: Why do employers hire employees in the first place? To solve problems for them!

And, as we all know, problems don't discriminate. In other words, it doesn't matter whether you're just an intern, an entry-level professional, or a seasoned veteran, you'll constantly face some kind of challenges. And the only difference is in how complex they will get.

This is also reflected in the way employers assess suitability of potential job candidates. 

In fact, research shows that the ability to deal with unexpected complications is prioritized by an overwhelming 60% of employers across all industries, making it one of the most compelling skills on your resume.

So, regardless of your job description or your career level, you're always expected to find solutions for problems, either independently or as a part of a team. 

And that's precisely what makes problem-solving skills so invaluable and universal ! 

Wondering how good is your resume?

Find out with our AI Resume Checker! Just upload your resume and see what can be improved.

As we've said before, problem-solving isn't really just one single skill. 

Instead, your ability to handle workplace issues with composure depends on several different “sub-skills”. 

So, which specific skills make an employee desirable even for the most demanding of recruiters? 

In no particular order, you should focus on these 7 skills : 

• Analytical skills
• Research skills
• Critical thinking 
• Decision-making
• Collaboration
• Having a growth mindset

Let's have a look at each of them in greater detail!

#1 Analytical skills

Firstly, to truly understand complex problems, you need to break them down into more manageable parts . Then, you observe them closely and ask yourself: “ Which parts work and which don't,” How do these parts contribute to the problem as a whole,” and "What exactly needs to be fixed?” In other words, you gather data , you study it, and compare it - all to pinpoint the cause of the issue as closely as possible.

#2 Research skills

Another priceless tool is your research skills (sometimes relying on just one source of information isn't enough). Besides, to make a truly informed decision , you'll have to dig a little deeper. Being a good researcher means looking for potential solutions to a problem in a wider context. For example: going through team reports, customer feedback, quarterly sales or current market trends.  

#3 Critical thinking

Every employer wants to hire people who can think critically. Yet, the ability to evaluate situations objectively and from different perspectives , is actually pretty hard to come by. But as long as you stay open-minded, inquisitive, and with a healthy dose of skepticism, you'll be able to assess situations based on facts and evidence more successfully. Plus, critical thinking comes in especially handy when you need to examine your own actions and processes. 

 #4 Creativity

Instead of following the old established processes that don't work anymore, you should feel comfortable thinking outside the box. The thing is, problems have a nasty habit of popping up unexpectedly and rapidly. And sometimes, you have to get creative in order to solve them fast. Especially those that have no precedence. But this requires a blend of intuition, industry knowledge, and quick thinking - a truly rare combination. 

#5 Decision-making

The analysis, research, and brainstorming are done. Now, you need to look at the possible solutions, and make the final decision (informed, of course). And not only that, you also have to stand by it ! Because once the train gets moving, there's no room for second guessing. Also, keep in mind that you need to be prepared to take responsibility for all decisions you make. That's no small feat! 

#6 Collaboration

Not every problem you encounter can be solved by yourself alone. And this is especially true when it comes to complex projects. So, being able to actively listen to your colleagues, take their ideas into account, and being respectful of their opinions enables you to solve problems together. Because every individual can offer a unique perspective and skill set. Yes, democracy is hard, but at the end of the day, it's teamwork that makes the corporate world go round. 

#7 Having a growth mindset

Let's be honest, no one wants their work to be riddled with problems. But facing constant challenges and changes is inevitable. And that can be scary! However, when you're able to see these situations as opportunities to grow instead of issues that hold you back, your problem solving skills reach new heights. And the employers know that too!

Now that we've shown you the value problem-solving skills can add to your resume, let's ask the all-important question: “How can I learn them?”

Well…you can't. At least not in the traditional sense of the word. 

Let us explain: Since problem-solving skills fall under the umbrella of soft skills , they can't be taught through formal education, unlike computer skills for example. There's no university course that you can take and graduate as a professional problem solver. 

But, just like other interpersonal skills, they can be nurtured and refined over time through practice and experience. 

Unfortunately, there's no one-size-fits-all approach, but the following tips can offer you inspiration on how to improve your problem solving skills:

• Cultivate a growth mindset. Remember what we've said before? Your attitude towards obstacles is the first step to unlocking your problem-solving potential. 
• Gain further knowledge in your specialized field. Secondly, it's a good idea to delve a little deeper into your chosen profession. Because the more you read on a subject, the easier it becomes to spot certain patterns and relations.  
• Start with small steps. Don't attack the big questions straight away — you'll only set yourself up for failure. Instead, start with more straightforward tasks and work your way up to more complex problems. 
• Break problems down into more digestible pieces. Complex issues are made up of smaller problems. And those can be further divided into even smaller problems, and so on. Until you're left with only the basics. 
• Don't settle for a single solution. Instead, keep on exploring other possible answers.
• Accept failure as a part of the learning process. Finally, don't let your failures discourage you. After all, you're bound to misstep a couple of times before you find your footing. Just keep on practicing. 

How to improve problem solving skills with online courses

While it’s true that formal education won’t turn you into a master problem solver, you can still hone your skills with courses and certifications offered by online learning platforms :

• Analytical skills. You can sharpen your analytical skills with Data Analytics Basics for Everyone from IBM provided by edX (Free); or Decision Making and Analytical Thinking: Fortune 500 provided by Udemy (＄21,74).
• Creativity. And, to unlock your inner creative mind, you can try Creative Thinking: Techniques and Tools for Success from the Imperial College London provided by Coursera (Free).
• Critical thinking. Try Introduction to Logic and Critical Thinking Specialization from Duke University provided by Coursera (Free); or Logical and Critical Thinking offered by The University of Auckland via FutureLearn.  
• Decision-making. Or, you can learn how to become more confident when it's time to make a decision with Decision-Making Strategies and Executive Decision-Making both offered by LinkedIn Learning (1 month free trial).
• Communication skills . Lastly, to improve your collaborative skills, check out Communicating for Influence and Impact online at University of Cambridge. 

The fact that everybody and their grandmothers put “ problem-solving skills ” on their CVs has turned the phrase into a cliche. 

But there's a way to incorporate these skills into your resume without sounding pretentious and empty. Below, we've prepared a mock-up resume that manages to do just that.

FYI, if you like this design, you can use the template to create your very own resume. Just click the red button and fill in your information (or let the AI do it for you).

Problem solving skills on resume example

This resume was written by our experienced resume writers specifically for this profession.

Why this example works?

• Firstly, the job description itself is neatly organized into bullet points .  
• Instead of simply listing soft skills in a skills section , you can incorporate them into the description of your work experience entry.  
• Also, the language here isn't vague . This resume puts each problem-solving skill into a real-life context by detailing specific situations and obstacles. 
• And, to highlight the impact of each skill on your previous job position, we recommend quantifying your results whenever possible. 
• Finally, starting each bullet point with an action verb (in bold) makes you look more dynamic and proactive.

To sum it all up, problem-solving skills continue gaining popularity among employers and employees alike. And for a good reason!

Because of them, you can overcome any obstacles that stand in the way of your professional life more efficiently and systematically. 

In essence, problem-solving skills refer to the ability to recognize a challenge, identify its root cause, think of possible solutions , and then implement the most effective one. 

Believing that these skills are all the same would be a serious misconception. In reality, this term encompasses a variety of different abilities , including:

In short, understanding, developing, and showcasing these skills, can greatly boost your chances at getting noticed by the hiring managers. So, don't hesitate and start working on your problem-solving skills right now!

Julia has recently joined Kickresume as a career writer. From helping people with their English to get admitted to the uni of their dreams to advising them on how to succeed in the job market. It would seem that her career is on a steadfast trajectory. Julia holds a degree in Anglophone studies from Metropolitan University in Prague, where she also resides. Apart from creative writing and languages, she takes a keen interest in literature and theatre.

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How to Develop Foundational Math Skills for Career Success

Developing strong foundational math skills is crucial for career success in today’s rapidly evolving world. This article explores what foundational math skills entail, their importance, and provides guidance on assessing current math skills, setting clear goals, seeking additional resources, and utilizing effective study techniques. The article also emphasizes the practical application of math skills in real-life scenarios and offers techniques to strengthen mental math abilities.

Defining Foundational Math Skills

Foundational math skills refer to the fundamental knowledge and abilities required to understand and apply mathematical concepts effectively. These skills form the basis for more advanced mathematical learning and are crucial for success in various academic and professional pursuits. Let’s explore what foundational math skills entail and why they are essential:

• Numerical Operations: Foundational math skills encompass a solid understanding of numerical operations, including addition, subtraction, multiplication, and division. Proficiency in these operations lays the groundwork for solving complex mathematical problems.
• Fraction, Decimal, and Percentage Understanding: A strong grasp of fractions, decimals, and percentages is integral to foundational math skills. Proficiency in working with these forms of numbers enables individuals to handle real-life situations involving measurements, ratios, and proportions.
• Algebraic Concepts: Foundational math skills involve familiarity with algebraic concepts such as equations, variables, and functions. Proficiency in algebra provides the tools to solve equations, analyze patterns, and express relationships between variables.
• Geometry and Spatial Reasoning: Understanding geometric principles and spatial relationships is another aspect of foundational math skills. Proficiency in geometry helps individuals visualize and analyze shapes, sizes, angles, and distances.
• Data Analysis and Statistics: Foundational math skills include the ability to interpret and analyze data, as well as apply statistical concepts. Proficiency in data analysis and statistics enables individuals to make informed decisions based on quantitative information.

The Importance of Foundational Math Skills

Developing strong foundational math skills is of paramount importance in today’s rapidly evolving world. Here are some key reasons why these skills hold tremendous value:

• Gateway to Success: Foundational math skills act as a gateway to various academic and professional opportunities. They provide the necessary foundation for pursuing careers in STEM fields, finance, data analysis, and other math-intensive disciplines.
• Critical Thinking and Problem-Solving: Proficiency in math fosters critical thinking abilities, enabling individuals to approach problems analytically and devise logical solutions. It enhances problem-solving skills, promotes logical reasoning, and encourages systematic thinking.
• Career Advancement: Studies have consistently shown a strong correlation between math skills and career advancement. According to the U.S. Bureau of Labor Statistics, occupations that require advanced math skills tend to offer higher salaries and better growth prospects.
• Versatility: Foundational math skills have broad applications across industries. From analyzing financial data to designing algorithms, from conducting scientific experiments to predicting market trends, math plays a crucial role in diverse professional settings.
• Data Literacy: In our modern, data-driven society, having a solid grasp of foundational math skills is crucial. These skills empower individuals to comprehend and analyze numerical information, enabling them to make informed choices and extract valuable insights.
• Adaptability to Technological Advancements: As technology continues to advance, proficiency in math becomes even more crucial. Emerging fields such as artificial intelligence, machine learning, and data science rely heavily on mathematical foundations, making math skills a valuable asset in the job market.

A study published in the Journal of Educational Psychology revealed a strong link between a solid math foundation and improved scores in both math and reading comprehension. The research, conducted with a large sample size, demonstrated that students with strong math skills not only excelled in problem-solving but also showed enhanced reading abilities. These findings highlight the important cognitive benefits of developing math proficiency, including improved critical thinking and logical reasoning skills. This research has significant implications for educators and policymakers, emphasizing the need to prioritize comprehensive math instruction to foster well-rounded individuals prepared for success in today’s world.

Assessing Your Current Math Skills

Before you embark on the journey of developing and strengthening your foundational math skills, it is essential to assess your current level of mathematical proficiency. By evaluating your abilities, you can identify areas of strength and pinpoint specific areas that require improvement.

Begin by reflecting on your past experiences with math and gauge your overall confidence in the subject. Consider how you have performed in previous math courses or any mathematical tasks you have encountered in your academic or professional life.

Familiarize yourself with the math curriculum standards or syllabus that aligns with your current educational level or desired career path. This will give you a clear understanding of the concepts and skills you should ideally be proficient in.

To gauge your understanding of different mathematical concepts, solve a variety of sample math problems or take online assessments designed to evaluate math skills. These resources can provide insights into your strengths and weaknesses, helping you focus on areas that need improvement.

It can also be beneficial to seek feedback from a math teacher, tutor, or mentor who can assess your abilities and provide guidance. Their expertise can help you identify specific areas for improvement and suggest appropriate resources or study materials.

Remember, assessing your current math skills is not meant to discourage you, but rather to provide a starting point for growth. With a clear understanding of your strengths and weaknesses, you can create a targeted plan to enhance your foundational math skills and build a solid mathematical foundation for future success.

Setting Clear Goals

Setting clear goals is a vital step in achieving success in any endeavor, including developing foundational math skills. By defining specific objectives, you provide yourself with direction and motivation, helping you stay focused and track your progress. Here are some guidelines for setting clear goals:

• Be Specific: Clearly define what you want to accomplish in terms of math skills. For example, your goal could be to master multiplication tables or improve problem-solving abilities.
• Set Measurable Targets: Create goals that can be quantified or measured. For instance, you could aim to solve a certain number of math problems correctly within a specified time frame.
• Make Goals Attainable: Ensure that your goals are realistic and achievable. Set challenging targets, but also consider your current skill level and available resources.
• Relevance Matters: Align your goals with your broader academic or career aspirations. This connection will enhance your motivation and sense of purpose.
• Create a Timeline: Begin by establishing a timeline or deadline for accomplishing each of your goals. By breaking down larger objectives into smaller milestones, you can maintain motivation and make steady progress.
• Monitor Your Progress: Consistently track your advancement toward your goals. This enables you to identify areas where you might need to make adjustments or seek additional assistance.

Keep in mind that goal setting is an ongoing process. As you make headway, you may find it necessary to revise or establish new goals to continually challenge yourself. Regularly reassess your goals to ensure they remain relevant and inspiring as you continue to enhance your foundational math skills.

Seeking Additional Resources

In your journey to develop foundational math skills, seeking additional resources can provide valuable support and enhance your learning experience. Explore math apps like Khan Academy, Photomath, or Mathway, which offer interactive lessons, practice exercises, and step-by-step solutions.

Online platforms such as Coursera and edX provide a wide range of math courses designed to cater to different skill levels and learning objectives.

Mathematics-focused websites like MathisFun and Math Playground offer engaging activities, games, and puzzles that can reinforce key math concepts in an enjoyable way.

Textbooks and workbooks are valuable resources for self-study and practice in math. Look for reputable publishers like McGraw-Hill, Pearson, Oxford University Press, and Effortless Math. They offer comprehensive math materials aligned with educational standards, ensuring clarity, accuracy, and effective learning. Effortless Math is a trusted provider of high-quality resources, including textbooks and workbooks. Combined with materials from renowned publishers, students can enhance their math skills and achieve success.

Publications such as “Mathematics Teacher” and “Teaching Children Mathematics” are scholarly journals that provide insights into effective teaching strategies, innovative approaches, and research-based practices for math education.

Lastly, don’t underestimate the power of connecting with math communities and forums. Platforms like Math Stack Exchange and Reddit’s r/learnmath allow you to ask questions, seek guidance, and engage in discussions with fellow learners and math enthusiasts.

By incorporating these additional resources into your learning journey, you can access diverse perspectives, interactive materials, and expert insights, ultimately enhancing your foundational math skills. Remember to adapt your resource choices based on your individual learning style and preferences to optimize your progress.

Effective Study Techniques

Utilizing effective study techniques is crucial for maximizing your learning potential and developing solid foundational math skills. Research has shown that employing specific strategies can significantly enhance retention and understanding of mathematical concepts. Consider implementing the following techniques:

According to a study published in the Psychology of Addictive Behaviors, regular practice is key to improving math skills. Students who dedicated at least 30 minutes per day to math practice demonstrated significant gains in their mathematical abilities.

Setting aside dedicated study time and creating a consistent study routine helps build discipline and focus. Allocate regular time slots for math practice to establish a habit and make the most of your study sessions.

Breaking down complex math problems into smaller, manageable steps allows for easier comprehension and problem-solving. This strategy, known as chunking, helps to reduce cognitive load and facilitates deeper understanding of mathematical concepts.

Utilize a variety of resources such as textbooks, online tutorials, practice worksheets, and interactive tools to reinforce your learning. Research has shown that diverse learning materials and approaches contribute to better understanding and retention of math skills.

Engaging in active learning methods, such as solving practice problems, working on real-life math applications, and teaching concepts to others, promotes active engagement and enhances comprehension and retention.

Collaborating with peers through group study sessions or joining math study groups can foster a supportive learning environment, facilitate discussion, and provide opportunities for sharing different problem-solving approaches.

By incorporating these effective study techniques into your learning routine, you can optimize your efforts and accelerate your progress in developing foundational math skills. Remember that consistency, active engagement, and utilizing a variety of resources are key to achieving long-lasting mastery.

Applying Foundational Math Skills in Real-Life Scenarios

Foundational math skills play a crucial role in our everyday lives, extending beyond the classroom or workplace. These skills are essential for navigating various real-life situations and making informed decisions. Let’s explore some examples of how foundational math skills are applied in practical contexts:

• Budgeting and Personal Finance: Understanding math concepts such as addition, subtraction, percentages, and interest rates enables individuals to create and manage budgets effectively. Whether it’s calculating monthly expenses, determining savings goals, or comparing prices to find the best deals, math skills are essential for making sound financial decisions.
• Cooking and Baking: In the kitchen, math skills are put to work when following recipes, adjusting ingredient quantities, and converting measurements. Being able to calculate ingredient proportions, scale recipes up or down, and understand cooking times and temperatures contribute to successful culinary endeavors.
• Home Improvement and DIY Projects: From measuring and estimating materials needed for renovations to calculating dimensions for furniture placement, foundational math skills are invaluable when working on home improvement projects. Understanding concepts like area, volume, and angles helps ensure accurate measurements and successful project outcomes.
• Travel and Navigation: When planning trips or navigating unfamiliar places, math skills come into play. Calculating distances, converting between units of measurement (e.g., miles to kilometers), and budgeting travel expenses all rely on foundational math skills. Additionally, understanding concepts like time zones and calculating travel times contribute to efficient travel planning.
• Data Analysis and Interpretation: In our data-driven world, being able to interpret and analyze numerical information is essential. Foundational math skills help in understanding statistics, interpreting graphs and charts, and making informed decisions based on data. Whether it’s analyzing market trends, evaluating research findings, or interpreting financial reports, math skills are vital for data literacy.

These examples demonstrate how foundational math skills are applied in practical, everyday scenarios. By having a solid grasp of mathematical concepts, individuals can navigate various aspects of life with confidence and make well-informed decisions in diverse contexts.

Strengthening Mental Math

Developing strong mental math skills is essential for performing quick and accurate calculations without relying on external aids. By practicing mental math regularly, you can enhance your computational abilities and improve your overall mathematical fluency. Here are some effective techniques to strengthen your mental math skills along with accompanying exercises:

Practice Daily: Dedicate a few minutes each day to practicing mental math exercises. Start with simple calculations and gradually increase the complexity as you progress. Regular practice trains your brain to perform calculations more efficiently.

Exercise : Calculate the following mentally:

b) 98 – 43

Break Down Numbers: When faced with larger numbers, break them down into smaller, more manageable parts. This technique simplifies calculations and makes them easier to solve mentally.

Exercise : Multiply the following mentally:

Utilize Number Patterns: Recognize and utilize number patterns to simplify mental calculations. Understanding and applying such patterns can significantly speed up your mental math processes.

Exercise : Use number patterns to find the missing numbers:

a) 2, 4, 6, __, 10

b) 5, 10, 15, __, 25

Estimation Skills: Develop the ability to estimate quantities and approximate results. Estimation helps you quickly assess the reasonableness of answers and make informed decisions.

Exercise : Estimate the following:

a) The sum of 345 and 221

b) The product of 18 and 27

Real-Life Applications: Look for opportunities in your daily life to apply mental math skills. Calculate the tip at a restaurant, mentally determine the change you should receive, or estimate the time it takes to complete a task.

Exercise : Estimate the total cost of the items in your grocery cart before reaching the checkout counter.

Expanding Beyond the Basics

While foundational math skills provide a solid framework, it’s important to expand your knowledge beyond the basics to develop a more comprehensive understanding of mathematics. Here are some ways to explore and expand your math skills:

• Advanced Topics: Dive into more advanced mathematical topics that align with your interests or career aspirations. Explore subjects such as algebra, geometry, trigonometry, calculus, or statistics. These areas of study provide deeper insights into mathematical concepts and their applications in various fields.
• Problem-Solving Challenges: Engage in challenging problem-solving exercises that require critical thinking and creativity. Solve puzzles, tackle math competitions, or participate in mathematical problem-solving communities. These activities push you to think outside the box and develop innovative problem-solving strategies.
• Technology and Mathematics: Embrace technology as a tool to enhance your math skills. Utilize graphing calculators, math software, or online resources to visualize concepts, perform complex calculations, and explore mathematical simulations. Technology can deepen your understanding and open up new avenues for exploration.
• Real-World Applications: Seek opportunities to apply your math skills to real-world situations. Look for connections between mathematics and other disciplines like physics, engineering, economics, or computer science. By understanding how math is applied in various contexts, you gain a more practical and holistic perspective.
• Mathematical Literature: Explore mathematical literature, textbooks, and research papers to expand your knowledge. Read about the history of mathematics, famous mathematicians, or contemporary mathematical breakthroughs. These resources expose you to different mathematical perspectives and foster a deeper appreciation for the subject.

By venturing beyond the basics, you unlock new dimensions of mathematics and nurture a curiosity that fuels continuous learning and exploration. Embrace the challenge of delving deeper into the subject, and let your passion for math lead you towards exciting and rewarding discoveries.

Final Words

Strong foundational math skills are essential for career success in various fields. By understanding the importance of these skills and following effective strategies for improvement, individuals can enhance their math proficiency and unlock numerous opportunities. Whether it’s through setting clear goals, seeking additional resources, or applying math skills in practical contexts, developing a solid mathematical foundation is key to achieving long-term success.

by: Effortless Math Team about 9 months ago (category: Blog )

Effortless Math Team

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Interview Questions

Comprehensive Interview Guide: 60+ Professions Explored in Detail

26 Good Examples of Problem Solving (Interview Answers)

By Biron Clark

Published: November 15, 2023

Employers like to hire people who can solve problems and work well under pressure. A job rarely goes 100% according to plan, so hiring managers will be more likely to hire you if you seem like you can handle unexpected challenges while staying calm and logical in your approach.

But how do they measure this?

They’re going to ask you interview questions about these problem solving skills, and they might also look for examples of problem solving on your resume and cover letter. So coming up, I’m going to share a list of examples of problem solving, whether you’re an experienced job seeker or recent graduate.

Then I’ll share sample interview answers to, “Give an example of a time you used logic to solve a problem?”

Problem-Solving Defined

It is the ability to identify the problem, prioritize based on gravity and urgency, analyze the root cause, gather relevant information, develop and evaluate viable solutions, decide on the most effective and logical solution, and plan and execute implementation. 

Problem-solving also involves critical thinking, communication, listening, creativity, research, data gathering, risk assessment, continuous learning, decision-making, and other soft and technical skills.

Solving problems not only prevent losses or damages but also boosts self-confidence and reputation when you successfully execute it. The spotlight shines on you when people see you handle issues with ease and savvy despite the challenges. Your ability and potential to be a future leader that can take on more significant roles and tackle bigger setbacks shine through. Problem-solving is a skill you can master by learning from others and acquiring wisdom from their and your own experiences. 

It takes a village to come up with solutions, but a good problem solver can steer the team towards the best choice and implement it to achieve the desired result.

Watch: 26 Good Examples of Problem Solving

Examples of problem solving scenarios in the workplace.

• Correcting a mistake at work, whether it was made by you or someone else
• Overcoming a delay at work through problem solving and communication
• Resolving an issue with a difficult or upset customer
• Overcoming issues related to a limited budget, and still delivering good work through the use of creative problem solving
• Overcoming a scheduling/staffing shortage in the department to still deliver excellent work
• Troubleshooting and resolving technical issues
• Handling and resolving a conflict with a coworker
• Solving any problems related to money, customer billing, accounting and bookkeeping, etc.
• Taking initiative when another team member overlooked or missed something important
• Taking initiative to meet with your superior to discuss a problem before it became potentially worse
• Solving a safety issue at work or reporting the issue to those who could solve it
• Using problem solving abilities to reduce/eliminate a company expense
• Finding a way to make the company more profitable through new service or product offerings, new pricing ideas, promotion and sale ideas, etc.
• Changing how a process, team, or task is organized to make it more efficient
• Using creative thinking to come up with a solution that the company hasn’t used before
• Performing research to collect data and information to find a new solution to a problem
• Boosting a company or team’s performance by improving some aspect of communication among employees
• Finding a new piece of data that can guide a company’s decisions or strategy better in a certain area

Problem Solving Examples for Recent Grads/Entry Level Job Seekers

• Coordinating work between team members in a class project
• Reassigning a missing team member’s work to other group members in a class project
• Adjusting your workflow on a project to accommodate a tight deadline
• Speaking to your professor to get help when you were struggling or unsure about a project
• Asking classmates, peers, or professors for help in an area of struggle
• Talking to your academic advisor to brainstorm solutions to a problem you were facing
• Researching solutions to an academic problem online, via Google or other methods
• Using problem solving and creative thinking to obtain an internship or other work opportunity during school after struggling at first

You can share all of the examples above when you’re asked questions about problem solving in your interview. As you can see, even if you have no professional work experience, it’s possible to think back to problems and unexpected challenges that you faced in your studies and discuss how you solved them.

Interview Answers to “Give an Example of an Occasion When You Used Logic to Solve a Problem”

Now, let’s look at some sample interview answers to, “Give me an example of a time you used logic to solve a problem,” since you’re likely to hear this interview question in all sorts of industries.

Example Answer 1:

At my current job, I recently solved a problem where a client was upset about our software pricing. They had misunderstood the sales representative who explained pricing originally, and when their package renewed for its second month, they called to complain about the invoice. I apologized for the confusion and then spoke to our billing team to see what type of solution we could come up with. We decided that the best course of action was to offer a long-term pricing package that would provide a discount. This not only solved the problem but got the customer to agree to a longer-term contract, which means we’ll keep their business for at least one year now, and they’re happy with the pricing. I feel I got the best possible outcome and the way I chose to solve the problem was effective.

Example Answer 2:

In my last job, I had to do quite a bit of problem solving related to our shift scheduling. We had four people quit within a week and the department was severely understaffed. I coordinated a ramp-up of our hiring efforts, I got approval from the department head to offer bonuses for overtime work, and then I found eight employees who were willing to do overtime this month. I think the key problem solving skills here were taking initiative, communicating clearly, and reacting quickly to solve this problem before it became an even bigger issue.

Example Answer 3:

In my current marketing role, my manager asked me to come up with a solution to our declining social media engagement. I assessed our current strategy and recent results, analyzed what some of our top competitors were doing, and then came up with an exact blueprint we could follow this year to emulate our best competitors but also stand out and develop a unique voice as a brand. I feel this is a good example of using logic to solve a problem because it was based on analysis and observation of competitors, rather than guessing or quickly reacting to the situation without reliable data. I always use logic and data to solve problems when possible. The project turned out to be a success and we increased our social media engagement by an average of 82% by the end of the year.

Answering Questions About Problem Solving with the STAR Method

When you answer interview questions about problem solving scenarios, or if you decide to demonstrate your problem solving skills in a cover letter (which is a good idea any time the job description mention problem solving as a necessary skill), I recommend using the STAR method to tell your story.

STAR stands for:

It’s a simple way of walking the listener or reader through the story in a way that will make sense to them. So before jumping in and talking about the problem that needed solving, make sure to describe the general situation. What job/company were you working at? When was this? Then, you can describe the task at hand and the problem that needed solving. After this, describe the course of action you chose and why. Ideally, show that you evaluated all the information you could given the time you had, and made a decision based on logic and fact.

Finally, describe a positive result you got.

Whether you’re answering interview questions about problem solving or writing a cover letter, you should only choose examples where you got a positive result and successfully solved the issue.

Example answer:

Situation : We had an irate client who was a social media influencer and had impossible delivery time demands we could not meet. She spoke negatively about us in her vlog and asked her followers to boycott our products. (Task : To develop an official statement to explain our company’s side, clarify the issue, and prevent it from getting out of hand). Action : I drafted a statement that balanced empathy, understanding, and utmost customer service with facts, logic, and fairness. It was direct, simple, succinct, and phrased to highlight our brand values while addressing the issue in a logical yet sensitive way.   We also tapped our influencer partners to subtly and indirectly share their positive experiences with our brand so we could counter the negative content being shared online.  Result : We got the results we worked for through proper communication and a positive and strategic campaign. The irate client agreed to have a dialogue with us. She apologized to us, and we reaffirmed our commitment to delivering quality service to all. We assured her that she can reach out to us anytime regarding her purchases and that we’d gladly accommodate her requests whenever possible. She also retracted her negative statements in her vlog and urged her followers to keep supporting our brand.

What Are Good Outcomes of Problem Solving?

Whenever you answer interview questions about problem solving or share examples of problem solving in a cover letter, you want to be sure you’re sharing a positive outcome.

Below are good outcomes of problem solving:

• Saving the company time or money
• Making the company money
• Pleasing/keeping a customer
• Obtaining new customers
• Solving a safety issue
• Solving a staffing/scheduling issue
• Solving a logistical issue
• Solving a company hiring issue
• Solving a technical/software issue
• Making a process more efficient and faster for the company
• Creating a new business process to make the company more profitable
• Improving the company’s brand/image/reputation
• Getting the company positive reviews from customers/clients

Every employer wants to make more money, save money, and save time. If you can assess your problem solving experience and think about how you’ve helped past employers in those three areas, then that’s a great start. That’s where I recommend you begin looking for stories of times you had to solve problems.

Tips to Improve Your Problem Solving Skills

Throughout your career, you’re going to get hired for better jobs and earn more money if you can show employers that you’re a problem solver. So to improve your problem solving skills, I recommend always analyzing a problem and situation before acting. When discussing problem solving with employers, you never want to sound like you rush or make impulsive decisions. They want to see fact-based or data-based decisions when you solve problems.

Next, to get better at solving problems, analyze the outcomes of past solutions you came up with. You can recognize what works and what doesn’t. Think about how you can get better at researching and analyzing a situation, but also how you can get better at communicating, deciding the right people in the organization to talk to and “pull in” to help you if needed, etc.

Finally, practice staying calm even in stressful situations. Take a few minutes to walk outside if needed. Step away from your phone and computer to clear your head. A work problem is rarely so urgent that you cannot take five minutes to think (with the possible exception of safety problems), and you’ll get better outcomes if you solve problems by acting logically instead of rushing to react in a panic.

You can use all of the ideas above to describe your problem solving skills when asked interview questions about the topic. If you say that you do the things above, employers will be impressed when they assess your problem solving ability.

If you practice the tips above, you’ll be ready to share detailed, impressive stories and problem solving examples that will make hiring managers want to offer you the job. Every employer appreciates a problem solver, whether solving problems is a requirement listed on the job description or not. And you never know which hiring manager or interviewer will ask you about a time you solved a problem, so you should always be ready to discuss this when applying for a job.

Related interview questions & answers:

• How do you handle stress?
• How do you handle conflict?
• Tell me about a time when you failed

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ORIGINAL RESEARCH article

This article is part of the research topic.

The important role of the early school years for reading, writing and math development: Assessment and Intervention at school entry

I have three more than you, you have three less than me? Levels of flexibility in dealing with additive situations Provisionally Accepted

• 1 Ludwig Maximilian University of Munich, Germany

The final, formatted version of the article will be published soon.

Assessment and intervention in the early years should ideally be based on evidence-based models describing the structure and development of students' skills. Mathematical word problems have been identified as a challenge for mathematics learners for a long time and in many countries. We investigate flexibility in dealing with additive situations as a construct that develops during grades 1 through 3 and contributes to the development of students' word problem solving skills. We introduce the construct based on prior research on the difficulty of different situation structures entailed in word problems. We use data from three prior empirical studies with N = 383 German grade 2 and 3 students to develop a model of discrete levels of students' flexibility in dealing with additive situations. We use this model to investigate how the learners in our sample distribute across the different levels. Moreover, we apply it to describe students' development over several weeks in one study comprising three measurements. We derive conclusions about the construct in terms of determinants of task complexity, and about students' development and then provide an outlook on potential uses of the model in research and practice.

Keywords: flexibility in dealing with additive situations, Level model, Mathematics, word problem solving, primary school, assessment, assessment-based intervention

Received: 17 Nov 2023; Accepted: 09 Apr 2024.

Copyright: © 2024 Ufer, Kaiser, Gabler and Niklas. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY) . The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

* Correspondence: Prof. Stefan Ufer, Ludwig Maximilian University of Munich, Munich, Germany

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