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Module 1: Problem Solving Strategies

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Unlike exercises, there is never a simple recipe for solving a problem. You can get better and better at solving problems, both by building up your background knowledge and by simply practicing. As you solve more problems (and learn how other people solved them), you learn strategies and techniques that can be useful. But no single strategy works every time.

Pólya’s How to Solve It

George Pólya was a great champion in the field of teaching effective problem solving skills. He was born in Hungary in 1887, received his Ph.D. at the University of Budapest, and was a professor at Stanford University (among other universities). He wrote many mathematical papers along with three books, most famously, “How to Solve it.” Pólya died at the age 98 in 1985.1

1. Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY

In 1945, Pólya published the short book How to Solve It , which gave a four-step method for solving mathematical problems:

First, you have to understand the problem.

After understanding, then make a plan.

Carry out the plan.

Look back on your work. How could it be better?

This is all well and good, but how do you actually do these steps?!?! Steps 1. and 2. are particularly mysterious! How do you “make a plan?” That is where you need some tools in your toolbox, and some experience to draw upon.

Much has been written since 1945 to explain these steps in more detail, but the truth is that they are more art than science. This is where math becomes a creative endeavor (and where it becomes so much fun). We will articulate some useful problem solving strategies, but no such list will ever be complete. This is really just a start to help you on your way. The best way to become a skilled problem solver is to learn the background material well, and then to solve a lot of problems!

Problem Solving Strategy 1 (Guess and Test)

Make a guess and test to see if it satisfies the demands of the problem. If it doesn't, alter the guess appropriately and check again. Keep doing this until you find a solution.

Mr. Jones has a total of 25 chickens and cows on his farm. How many of each does he have if all together there are 76 feet?

Step 1: Understanding the problem

We are given in the problem that there are 25 chickens and cows.

All together there are 76 feet.

Chickens have 2 feet and cows have 4 feet.

We are trying to determine how many cows and how many chickens Mr. Jones has on his farm.

Step 2: Devise a plan

Going to use Guess and test along with making a tab

Many times the strategy below is used with guess and test.

Make a table and look for a pattern:

Procedure: Make a table reflecting the data in the problem. If done in an orderly way, such a table will often reveal patterns and relationships that suggest how the problem can be solved.

Step 3: Carry out the plan:

Notice we are going in the wrong direction! The total number of feet is decreasing!

Better! The total number of feet are increasing!

Step 4: Looking back:

Check: 12 + 13 = 25 heads

24 + 52 = 76 feet.

We have found the solution to this problem. I could use this strategy when there are a limited number of possible answers and when two items are the same but they have one characteristic that is different.

Videos to watch:

1. Click on this link to see an example of “Guess and Test”

http://www.mathstories.com/strategies.htm

2. Click on this link to see another example of Guess and Test.

http://www.mathinaction.org/problem-solving-strategies.html

Check in question 1:

Place the digits 8, 10, 11, 12, and 13 in the circles to make the sums across and vertically equal 31. (5 points)

Check in question 2:

Old McDonald has 250 chickens and goats in the barnyard. Altogether there are 760 feet . How many of each animal does he have? Make sure you use Polya’s 4 problem solving steps. (12 points)

Problem Solving Strategy 2 (Draw a Picture). Some problems are obviously about a geometric situation, and it is clear you want to draw a picture and mark down all of the given information before you try to solve it. But even for a problem that is not geometric thinking visually can help!

Videos to watch demonstrating how to use "Draw a Picture".

1. Click on this link to see an example of “Draw a Picture”

2. Click on this link to see another example of Draw a Picture.

Problem Solving Strategy 3 ( Using a variable to find the sum of a sequence.)

Gauss's strategy for sequences.

last term = fixed number ( n -1) + first term

The fix number is the the amount each term is increasing or decreasing by. "n" is the number of terms you have. You can use this formula to find the last term in the sequence or the number of terms you have in a sequence.

Ex: 2, 5, 8, ... Find the 200th term.

Last term = 3(200-1) +2

Last term is 599.

To find the sum of a sequence: sum = [(first term + last term) (number of terms)]/ 2

Sum = (2 + 599) (200) then divide by 2

Sum = 60,100

Check in question 3: (10 points)

Find the 320 th term of 7, 10, 13, 16 …

Then find the sum of the first 320 terms.

Problem Solving Strategy 4 (Working Backwards)

This is considered a strategy in many schools. If you are given an answer, and the steps that were taken to arrive at that answer, you should be able to determine the starting point.

Videos to watch demonstrating of “Working Backwards”

https://www.youtube.com/watch?v=5FFWTsMEeJw

Karen is thinking of a number. If you double it, and subtract 7, you obtain 11. What is Karen’s number?

1. We start with 11 and work backwards.

2. The opposite of subtraction is addition. We will add 7 to 11. We are now at 18.

3. The opposite of doubling something is dividing by 2. 18/2 = 9

4. This should be our answer. Looking back:

9 x 2 = 18 -7 = 11

5. We have the right answer.

Check in question 4:

Christina is thinking of a number.

If you multiply her number by 93, add 6, and divide by 3, you obtain 436. What is her number? Solve this problem by working backwards. (5 points)

Problem Solving Strategy 5 (Looking for a Pattern)

Definition: A sequence is a pattern involving an ordered arrangement of numbers.

We first need to find a pattern.

Ask yourself as you search for a pattern – are the numbers growing steadily larger? Steadily smaller? How is each number related?

Example 1: 1, 4, 7, 10, 13…

Find the next 2 numbers. The pattern is each number is increasing by 3. The next two numbers would be 16 and 19.

Example 2: 1, 4, 9, 16 … find the next 2 numbers. It looks like each successive number is increase by the next odd number. 1 + 3 = 4.

So the next number would be

25 + 11 = 36

Example 3: 10, 7, 4, 1, -2… find the next 2 numbers.

In this sequence, the numbers are decreasing by 3. So the next 2 numbers would be -2 -3 = -5

-5 – 3 = -8

Example 4: 1, 2, 4, 8 … find the next two numbers.

This example is a little bit harder. The numbers are increasing but not by a constant. Maybe a factor?

So each number is being multiplied by 2.

16 x 2 = 32

1. Click on this link to see an example of “Looking for a Pattern”

2. Click on this link to see another example of Looking for a Pattern.

Problem Solving Strategy 6 (Make a List)

Example 1 : Can perfect squares end in a 2 or a 3?

List all the squares of the numbers 1 to 20.

1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400.

Now look at the number in the ones digits. Notice they are 0, 1, 4, 5, 6, or 9. Notice none of the perfect squares end in 2, 3, 7, or 8. This list suggests that perfect squares cannot end in a 2, 3, 7 or 8.

How many different amounts of money can you have in your pocket if you have only three coins including only dimes and quarters?

Quarter’s dimes

0 3 30 cents

1 2 45 cents

2 1 60 cents

3 0 75 cents

Videos demonstrating "Make a List"

Check in question 5:

How many ways can you make change for 23 cents using only pennies, nickels, and dimes? (10 points)

Problem Solving Strategy 7 (Solve a Simpler Problem)

Geometric Sequences:

How would we find the nth term?

Solve a simpler problem:

1, 3, 9, 27.

1. To get from 1 to 3 what did we do?

2. To get from 3 to 9 what did we do?

Let’s set up a table:

Term Number what did we do

Looking back: How would you find the nth term?

Find the 10 th term of the above sequence.

Let L = the tenth term

Problem Solving Strategy 8 (Process of Elimination)

This strategy can be used when there is only one possible solution.

I’m thinking of a number.

The number is odd.

It is more than 1 but less than 100.

It is greater than 20.

It is less than 5 times 7.

The sum of the digits is 7.

It is evenly divisible by 5.

a. We know it is an odd number between 1 and 100.

b. It is greater than 20 but less than 35

21, 23, 25, 27, 29, 31, 33, 35. These are the possibilities.

c. The sum of the digits is 7

21 (2+1=3) No 23 (2+3 = 5) No 25 (2 + 5= 7) Yes Using the same process we see there are no other numbers that meet this criteria. Also we notice 25 is divisible by 5. By using the strategy elimination, we have found our answer.

Check in question 6: (8 points)

Jose is thinking of a number.

The number is not odd.

The sum of the digits is divisible by 2.

The number is a multiple of 11.

It is greater than 5 times 4.

It is a multiple of 6

It is less than 7 times 8 +23

What is the number?

Click on this link for a quick review of the problem solving strategies.

https://garyhall.org.uk/maths-problem-solving-strategies.html

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How to Solve Math Problems Faster: 15 Techniques to Show Students

Written by Marcus Guido

• Teaching Strategies

“Test time. No calculators.”

You’ll intimidate many students by saying this, but teaching techniques to solve math problems with ease and speed can make it less daunting.

This can also  make math more rewarding . Instead of relying on calculators, students learn strategies that can improve their concentration and estimation skills while building number sense. And, while there are educators who  oppose math “tricks”  for valid reasons, proponents point to benefits such as increased confidence to handle difficult problems.

Here are 15 techniques to show students,  helping them solve math problems faster:

Addition and Subtraction

1. two-step addition.

Many students struggle when learning to add integers of three digits or higher together, but changing the process’s steps can make it easier.

The first step is to  add what’s easy.  The second step is to  add the rest.

Let’s say students must find the sum of 393 and 89. They should quickly see that adding 7 onto 393 will equal 400 — an easier number to work with. To balance the equation, they can then subtract 7 from 89.

Broken down, the process is:

• (393 + 7) + (89 – 7)

With this fast technique, big numbers won’t look as scary now.

2. Two-Step Subtraction

There’s a similar method for subtraction.

Remove what’s easy. Then remove what’s left.

Suppose students must find the difference of 567 and 153. Most will feel that 500 is a simpler number than 567. So, they just have to take away 67 from the minuend — 567 — and the subtrahend — 153 — before solving the equation.

Here’s the process:

• (567 – 67) – (153 – 67)

Instead of two complex numbers, students will only have to tackle one.

3. Subtracting from 1,000

You can  give students confidence  to handle four-digit integers with this fast technique.

To subtract a number from 1,000, subtract that number’s first two digits from 9. Then, subtract the final digit from 10.

Let’s say students must solve 1,000 – 438.  Here are the steps:

This also applies to 10,000, 100,000 and other integers that follow this pattern.

Multiplication and Division

4. doubling and halving.

When students have to multiply two integers, they can speed up the process when one is an even number. They just need to  halve the even number and double the other number.

Students can stop the process when they can no longer halve the even integer, or when the equation becomes manageable.

Using 33 x 48 as an example,  here’s the process:

The only prerequisite is understanding the 2 times table.

5. Multiplying by Powers of 2

This tactic is a speedy variation of doubling and halving.

It simplifies multiplication if a number in the equation is a power of 2, meaning it works for 2, 4, 8, 16 and so on.

Here’s what to do:  For each power of 2 that makes up that number, double the other number.

For example, 9 x 16 is the same thing as 9 x (2 x 2 x 2 x 2) or 9 x 24. Students can therefore double 9 four times to reach the answer:

Unlike doubling and halving, this technique demands an understanding of exponents along with a strong command of the 2 times table.

6. Multiplying by 9

For most students, multiplying by 9 — or 99, 999 and any number that follows this pattern — is difficult compared with multiplying by a power of 10.

But there’s an easy tactic to solve this issue, and  it has two parts.

First, students round up the 9  to 10. Second, after solving the new equation, they subtract the number they just multiplied by 10 from the answer.

For example, 67 x 9 will lead to the same answer as 67 x 10 – 67. Following the order of operations will give a result of 603. Similarly, 67 x 99 is the same as 67 x 100 – 67.

Despite more steps, altering the equation this way is usually faster.

7. Multiplying by 11

There’s an easier way for multiplying two-digit integers by 11.

Let’s say students must find the product of 11 x 34.

The idea is to put a space between the digits, making it 3_4. Then, add the two digits together and put the sum in the space.

The answer is 374.

What happens if the sum is two digits? Students would put the second digit in the space and add 1 to the digit to the left of the space.  For example:

It’s multiplication without having to multiply.

8. Multiplying Even Numbers by 5

This technique only requires basic division skills.

There are two steps,  and 5 x 6 serves as an example. First, divide the number being multiplied by 5 — which is 6 — in half. Second, add 0 to the right of number.

The result is 30, which is the correct answer.

It’s an ideal, easy technique for students mastering the 5 times table.

9. Multiplying Odd Numbers by 5

This is another time-saving tactic that works well when teaching students the 5 times table.

This one has three steps,  which 5 x 7 exemplifies.

First, subtract 1 from the number being multiplied by 5, making it an even number. Second, cut that number in half — from 6 to 3 in this instance. Third, add 5 to the right of the number.

The answer is 35.

Who needs a calculator?

10. Squaring a Two-Digit Number that Ends with 1

Squaring a high two-digit number can be tedious, but there’s a shortcut if 1 is the second digit.

There are four steps to this shortcut,  which 812 exemplifies:

• Subtract 1 from the integer: 81 – 1 = 80
• Square the integer, which is now an easier number: 80 x 80 = 6,400
• Add the integer with the resulting square twice: 6,400 + 80 + 80 = 6,560
• Add 1: 6,560 + 1 = 6,561

This work-around eliminates the difficulty surrounding the second digit, allowing students to work with multiples of 10.

11. Squaring a Two-Digit Numbers that Ends with 5

Squaring numbers ending in 5 is easier, as there are  only two parts of the process.

First, students will always make 25 the product’s last digits.

Second, to determine the product’s first digits, students must multiply the number’s first digit — 9, for example — by the integer that’s one higher — 10, in this case.

So, students would solve 952 by designating 25 as the last two digits. They would then multiply 9 x 10 to receive 90. Putting these numbers together, the  result is 9,025.

Just like that, a hard problem becomes easy multiplication for many students.

12. Calculating Percentages

Cross-multiplication is an  important skill  to develop, but there’s an easier way to calculate percentages.

For example, if students want to know what 65% of 175 is, they can multiply the numbers together and move the decimal place two digits to the left.

The result is 113.75, which is indeed the correct answer.

This shortcut is a useful timesaver on tests and quizzes.

13. Balancing Averages

To determine the average among a set of numbers, students can balance them instead of using a complex formula.

Suppose a student wants to volunteer for an average of 10 hours a week over a period of four weeks. In the first three weeks, the student worked for 10, 12 and 14 hours.

To determine the number of hours required in the fourth week, the student must  add how much he or she surpassed or missed the target average  in the other weeks:

• 14 hours – 10 hours = 4 hours
• 12 – 10 = 2
• 10 – 10 = 0
• 4 hours + 2 hours + 0 hours = 6 hours

To learn the number of hours for the final week, the student must  subtract the sum from the target average:

• 10 hours – 6 hours = 4 hours

With practice, this method may not even require pencil and paper. That’s how easy it is. 

Word Problems

14. identifying buzzwords.

Students who struggle to translate  word problems  into equations will benefit from learning how to spot buzzwords — phrases that indicate specific actions.

This isn’t a trick. It’s a tactic.

Teach students to look for these buzzwords,  and what skill they align with in most contexts:

Be sure to include buzzwords that typically appear in their textbooks (or other classroom  math books ), as well as ones you use on tests and assignments.

As a result, they should have an  easier time processing word problems .

15. Creating Sub-Questions

For complex word problems, show students how to dissect the question by answering three specific sub-questions.

Each student should ask him or herself:

• What am I looking for?  — Students should read the question over and over, looking for buzzwords and identifying important details.
• What information do I need?  — Students should determine which facts, figures and variables they need to solve the question. For example, if they determine the question is rooted in subtraction, they need the minuend and subtrahend.
• What information do I have?  — Students should be able to create the core equation using the information in the word problem, after determining which details are important.

These sub-questions help students avoid overload.

Instead of writing and analyzing each detail of the question, they’ll be able to identify key information. If you identify students who are struggling with these, you can use  peer learning  as needed.  

For more fresh approaches to teaching math in your classroom, consider treating your students to a range of  fun math activities .

Final Thoughts About these Ways to Solve Math Problems Faster

Showing these 15 techniques to students can give them the  confidence to tackle tough questions .

They’re also  mental math  exercises, helping them build skills related to focus, logic and critical thinking.

A rewarding class equals an  engaging class . That’s an easy equation to remember.

> Create or log into your teacher account on Prodigy  — a free, adaptive math game that adjusts content to accommodate player trouble spots and learning speeds. Aligned to US and Canadian curricula, it’s loved by more than 500,000 teachers and 15 million students.

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How to Solve Math Problems

Last Updated: May 16, 2023 Fact Checked

This article was co-authored by Daron Cam . Daron Cam is an Academic Tutor and the Founder of Bay Area Tutors, Inc., a San Francisco Bay Area-based tutoring service that provides tutoring in mathematics, science, and overall academic confidence building. Daron has over eight years of teaching math in classrooms and over nine years of one-on-one tutoring experience. He teaches all levels of math including calculus, pre-algebra, algebra I, geometry, and SAT/ACT math prep. Daron holds a BA from the University of California, Berkeley and a math teaching credential from St. Mary's College. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 588,754 times.

Although math problems may be solved in different ways, there is a general method of visualizing, approaching and solving math problems that may help you to solve even the most difficult problem. Using these strategies can also help you to improve your math skills overall. Keep reading to learn about some of these math problem solving strategies.

Understanding the Problem

• Draw a Venn diagram. A Venn diagram shows the relationships among the numbers in your problem. Venn diagrams can be especially helpful with word problems.
• Draw a graph or chart.
• Arrange the components of the problem on a line.
• Draw simple shapes to represent more complex features of the problem.

Developing a Plan

Solving the Problem

Expert Q&A

• Seek help from your teacher or a math tutor if you get stuck or if you have tried multiple strategies without success. Your teacher or a math tutor may be able to easily identify what is wrong and help you to understand how to correct it. Thanks Helpful 1 Not Helpful 1
• Keep practicing sums and diagrams. Go through the concept your class notes regularly. Write down your understanding of the methods and utilize it. Thanks Helpful 2 Not Helpful 0

You Might Also Like

• ↑ Daron Cam. Math Tutor. Expert Interview. 29 May 2020.
• ↑ http://www.interventioncentral.org/academic-interventions/math/math-problem-solving-combining-cognitive-metacognitive-strategies
• ↑ http://tutorial.math.lamar.edu/Extras/StudyMath/ProblemSolving.aspx
• ↑ https://math.berkeley.edu/~gmelvin/polya.pdf

About This Article

To solve a math problem, try rewriting the problem in your own words so it's easier to solve. You can also make a drawing of the problem to help you figure out what it's asking you to do. If you're still completely stuck, try solving a different problem that's similar but easier and then use the same steps to solve the harder problem. Even if you can't figure out how to solve it, try to make an educated guess instead of leaving the question blank. To learn how to come up with a solid plan to use to help you solve a math problem, scroll down! Did this summary help you? Yes No

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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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17 maths problem solving strategies to boost your learning.

Worded problems getting the best of you? With this list of maths problem-solving strategies , you'll overcome any maths hurdle that comes your way.

Friday, 3rd June 2022

• What are strategies?

Understand the problem

Devise a plan, carry out the plan, look back and reflect, practise makes progress.

Problem-solving is a critical life skill that everyone needs. Whether you're dealing with everyday issues or complex challenges, being able to solve problems effectively can make a big difference to your quality of life.

While there is no one 'right' way to solve a problem, having a toolkit of different techniques that you can draw upon will give you the best chance of success. In this article, we'll explore 17 different math problem-solving strategies you can start using immediately to deepen your learning and improve your skills.

What are maths problem-solving strategies?

Before we get into the strategies themselves, let's take a step back and answer the question: what are these strategies? In simple terms, these are methods we use to solve mathematical problems—essential for anyone learning how to study maths . These can be anything from asking open-ended questions to more complex concepts like the use of algebraic equations.

The beauty of these techniques is they go beyond strictly mathematical application. It's more about understanding a given problem, thinking critically about it and using a variety of methods to find a solution.

Polya's 4-step process for solving problems

We're going to use Polya's 4-step model as the framework for our discussion of problem-solving activities . This was developed by Hungarian mathematician George Polya and outlined in his 1945 book How to Solve It. The steps are as follows:

We'll go into more detail on each of these steps as well as take a look at some specific problem-solving strategies that can be used at each stage.

This may seem like an obvious one, but it's crucial that you take the time to understand what the problem is asking before trying to solve it. Especially with a math word problem , in which the question is often disguised in language, it's easy for children to misinterpret what's being asked.

Here are some questions you can ask to help you understand the problem:

Do I understand all the words used in the problem?

What am I asked to find or show?

Can I restate the problem in my own words?

Can I think of a picture or diagram that might help me understand the problem?

Is there enough information to enable me to find a solution?

Is there anything I need to find out first in order to find the answer?

What information is extra or irrelevant?

Once you've gone through these questions, you should have a good understanding of what the problem is asking. Now let's take a look at some specific strategies that can be used at this stage.

1. Read the problem aloud

This is a great strategy for younger students who are still learning to read. By reading the problem aloud, they can help to clarify any confusion and better understand what's being asked. Teaching older students to read aloud slowly is also beneficial as it encourages them to internalise each word carefully.

2. Summarise the information

Using dot points or a short sentence, list out all the information given in the problem. You can even underline the keywords to focus on the important information. This will help to organise your thoughts and make it easier to see what's given, what's missing, what's relevant and what isn't.

3. Create a picture or diagram

This is a no-brainer for visual learners. By drawing a picture, let's say with division problems, you can better understand what's being asked and identify any information that's missing. It could be a simple sketch or a more detailed picture, depending on the problem.

4. Act it out

Visualising a scenario can also be helpful. It can enable students to see the problem in a different way and develop a more intuitive understanding of it. This is especially useful for math word problems that are set in a particular context. For example, if a problem is about two friends sharing candy, kids can act out the problem with real candy to help them understand what's happening.

5. Use keyword analysis

What does this word tell me? Which operations do I need to use? Keyword analysis involves asking questions about the words in a problem in order to work out what needs to be done. There are certain key words that can hint at what operation you need to use.

How many more?

How many left?

Equal parts

Once you understand the problem, it's time to start thinking about how you're going to solve it. This is where having a plan is vital. By taking the time to think about your approach, you can save yourself a lot of time and frustration later on.

There are many methods that can be used to figure out a pathway forward, but the key is choosing an appropriate one that will work for the specific problem you're trying to solve. Not all students understand what it means to plan a problem so we've outlined some popular problem-solving techniques during this stage.

6. Look for a pattern

Sometimes, the best way to solve a problem is to look for a pattern. This could be a number, a shape pattern or even just a general trend that you can see in the information given. Once you've found it, you can use it to help you solve the problem.

7. Guess and check

While not the most efficient method, guess and check can be helpful when you're struggling to think of an answer or when you're dealing with multiple possible solutions. To do this, you simply make a guess at the answer and then check to see if it works. If it doesn't, you make another systematic guess and keep going until you find a solution that works.

8. Working backwards

Regressive reasoning, or working backwards, involves starting with a potential answer and working your way back to figure out how you would get there. This is often used when trying to solve problems that have multiple steps. By starting with the end in mind, you can work out what each previous step would need to be in order to arrive at the answer.

9. Use a formula

There will be some problems where a specific formula needs to be used in order to solve it. Let's say we're calculating the cost of flooring panels in a rectangular room (6m x 9m) and we know that the panels cost $15 per sq. metre.

There is no mention of the word 'area', and yet that is exactly what we need to calculate. The problem requires us to use the formula for the area of a rectangle (A = l x w) in order to find the total cost of the flooring panels.

10. Eliminate the possibilities

When there are a lot of possibilities, one approach could be to start by eliminating the answers that don't work. This can be done by using a process of elimination or by plugging in different values to see what works and what doesn't.

11. Use direct reasoning

Direct reasoning, also known as top-down or forward reasoning, involves starting with what you know and then using that information to try and solve the problem . This is often used when there is a lot of information given in the problem.

By breaking the problem down into smaller chunks, you can start to see how the different pieces fit together and eventually work out a solution.

12. Solve a simpler problem

One of the most effective methods for solving a difficult problem is to start by solving a simpler version of it. For example, in order to solve a 4-step linear equation with variables on both sides, you could start by solving a 2-step one. Or if you're struggling with the addition of algebraic fractions, go back to solving regular fraction addition first.

Once you've mastered the easier problem, you can then apply the same knowledge to the challenging one and see if it works.

13. Solve an equation

Another common problem-solving technique is setting up and solving an equation. For instance, let's say we need to find a number. We know that after it was doubled, subtracted from 32, and then divided by 4, it gave us an answer of 6. One method could be to assign this number a variable, set up an equation, and solve the equation by 'backtracking and balancing the equation'.

Now that you have a plan, it's time to implement it. This is where you'll put your problem-solving skills to the test and see if your solution actually works. There are a few things to keep in mind as you execute your plan:

14. Be systematic

When trying different methods or strategies, it's important to be systematic in your approach. This means trying one problem-solving strategy at a time and not moving on until you've exhausted all possibilities with that particular approach.

15. Check your work

Once you think you've found a solution, it's important to check your work to make sure that it actually works. This could involve plugging in different values or doing a test run to see if your solution works in all cases.

16. Be flexible

If your initial plan isn't working, don't be afraid to change it. There is no one 'right' way to solve a problem, so feel free to try different things, seek help from different resources and continue until you find a more efficient strategy or one that works.

17. Don't give up

It's important to persevere when trying to solve a difficult problem. Just because you can't see a solution right away doesn't mean that there isn't one. If you get stuck, take a break and come back to the problem later with fresh eyes. You might be surprised at what you're able to see after taking some time away from it.

Once you've solved the problem, take a step back and reflect on the process that you went through. Most middle school students forget this fundamental step. This will help you to understand what worked well and what could be improved upon next time.

Whether you do this after a math test or after an individual problem, here are some questions to ask yourself:

What was the most challenging part of the problem?

Was one method more effective than another?

Would you do something differently next time?

What have you learned from this experience?

By taking the time to reflect on your process you'll be able to improve upon it in future and become an even better problem solver. Make sure you write down any insights so that you can refer back to them later.

There is never only one way to solve math problems. But the best way to become a better problem solver is to practise, practise, practise! The more you do it, the better you'll become at identifying different strategies, and the more confident you'll feel when faced with a challenging problem.

The list we've covered is by no means exhaustive, but it's a good starting point for you to begin your journey. When you get stuck, remember to keep an open mind. Experiment with different approaches. Different word problems. Be prepared to go back and try something new. And most importantly, don't forget to have fun!

The essence and beauty of mathematics lies in its freedom. So while these strategies provide nice frameworks, the best work is done by those who are comfortable with exploration outside the rules, and of course, failure! So go forth, make mistakes and learn from them. After all, that's how we improve our problem-solving skills and ability.

Lastly, don't be afraid to ask for help. If you're struggling to solve math word problems, there's no shame in seeking assistance from a certified Melbourne maths tutor . In every lesson at Math Minds, our expert teachers encourage students to think creatively, confidently and courageously.

If you're looking for a mentor who can guide you through these methods, introduce you to other problem-solving activities and help you to understand Mathematics in a deeper way - get in touch with our team today. Sign up for your free online maths assessment and discover a world of new possibilities.

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Problem Solving, Using and Applying and Functional Mathematics

Problem solving.

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

Comprehension

Representation.

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

Planning, analysis and synthesis

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

Execution and communication

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

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

Using and Applying Mathematics

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

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

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

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

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

Problem solvers need to reason mathematically including through:

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

Functional Mathematics

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

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

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You don’t have to be a math teacher to know that a lot of students—and likely a lot of parents (it’s been awhile!)—are intimidated by math problems, especially if they involve large numbers. Learning techniques on how to do math quickly can help students develop greater confidence in math , improve math skills and understanding, and excel in advanced courses.

If it’s your job to teach those, here’s a great refresher.

Fast math tricks infographic

10 tricks for doing fast math

Here are 10 fast math strategies students (and adults!) can use to do math in their heads. Once these strategies are mastered, students should be able to accurately and confidently solve math problems that they once feared solving.

1. Adding large numbers

Adding large numbers just in your head can be difficult. This method shows how to simplify this process by making all the numbers a multiple of 10. Here is an example:

While these numbers are hard to contend with, rounding them up will make them more manageable. So, 644 becomes 650 and 238 becomes 240.

Now, add 650 and 240 together. The total is 890. To find the answer to the original equation, it must be determined how much we added to the numbers to round them up.

650 – 644 = 6 and 240 – 238 = 2

Now, add 6 and 2 together for a total of 8

To find the answer to the original equation, 8 must be subtracted from the 890.

890 – 8 = 882

So the answer to 644 +238 is 882.

2. Subtracting from 1,000

Here’s a basic rule to subtract a large number from 1,000: Subtract every number except the last from 9 and subtract the final number from 10

For example:

1,000 – 556

Step 1: Subtract 5 from 9 = 4

Step 2: Subtract 5 from 9 = 4

Step 3: Subtract 6 from 10 = 4

The answer is 444.

3. Multiplying 5 times any number

When multiplying the number 5 by an even number, there is a quick way to find the answer.

For example, 5 x 4 =

• Step 1: Take the number being multiplied by 5 and cut it in half, this makes the number 4 become the number 2.
• Step 2: Add a zero to the number to find the answer. In this case, the answer is 20.

When multiplying an odd number times 5, the formula is a bit different.

For instance, consider 5 x 3.

• Step 1: Subtract one from the number being multiplied by 5, in this instance the number 3 becomes the number 2.
• Step 2: Now halve the number 2, which makes it the number 1. Make 5 the last digit. The number produced is 15, which is the answer.

4. Division tricks

Here’s a quick way to know when a number can be evenly divided by these certain numbers:

• 10 if the number ends in 0
• 9 when the digits are added together and the total is evenly divisible by 9
• 8 if the last three digits are evenly divisible by 8 or are 000
• 6 if it is an even number and when the digits are added together the answer is evenly divisible by 3
• 5 if it ends in a 0 or 5
• 4 if it ends in 00 or a two digit number that is evenly divisible by 4
• 3 when the digits are added together and the result is evenly divisible by the number 3
• 2 if it ends in 0, 2, 4, 6, or 8

5. Multiplying by 9

This is an easy method that is helpful for multiplying any number by 9. Here is how it works:

Let’s use the example of 9 x 3.

Step 1 : Subtract 1 from the number that is being multiplied by 9.

3 – 1 = 2

The number 2 is the first number in the answer to the equation.

Step 2 : Subtract that number from the number 9.

9 – 2 = 7

The number 7 is the second number in the answer to the equation.

So, 9 x 3 = 27

6. 10 and 11 times tricks

The trick to multiplying any number by 10 is to add a zero to the end of the number. For example, 62 x 10 = 620.

There is also an easy trick for multiplying any two-digit number by 11. Here it is:

Take the original two-digit number and put a space between the digits. In this example, that number is 25.

Now add those two numbers together and put the result in the center:

2_(2 + 5)_5

The answer to 11 x 25 is 275.

If the numbers in the center add up to a number with two digits, insert the second number and add 1 to the first one. Here is an example for the equation 11 x 88

(8 + 1)_6_8

There is the answer to 11 x 88: 968

7. Percentage

Finding a percentage of a number can be somewhat tricky, but thinking about it in the right terms makes it much easier to understand. For instance, to find out what 5% of 235 is, follow this method:

• Step 1: Move the decimal point over by one place, 235 becomes 23.5.
• Step 2: Divide 23.5 by the number 2, the answer is 11.75. That is also the answer to the original equation.

8. Quickly square a two-digit number that ends in 5

Let’s use the number 35 as an example.

• Step 1: Multiply the first digit by itself plus 1.
• Step 2: Put a 25 at the end.

35 squared = [3 x (3 + 1)] & 25

[3 x (3 + 1)] = 12

12 & 25 = 1225

35 squared = 1225

9. Tough multiplication

When multiplying large numbers, if one of the numbers is even, divide the first number in half, and then double the second number. This method will solve the problem quickly. For instance, consider

Step 1: Divide the 20 by 2, which equals 10. Double 120, which equals 240.

Then multiply your two answers together.

10 x 240 = 2400

The answer to 20 x 120 is 2,400.

10. Multiplying numbers that end in zero

Multiplying numbers that end in zero is actually quite simple. It involves multiplying the other numbers together and then adding the zeros at the end. For instance, consider:

Step 1: Multiply the 2 times the 4

Step 2: Put all four of the zeros after the 8

200 x 400= 80,000

Practicing these fast math tricks can help both students and teachers improve their math skills and become secure in their knowledge of mathematics—and unafraid to work with numbers in the future.

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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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20 best maths problem solving strategies & step-by-step math problem solver.

If you find yourself struggling with maths problem solving , we understand your concerns. Math problems can be challenging and overwhelming, especially when you’re unsure how to approach them. But worry no more! We are here to provide you with effective solutions. In this article, we will guide you through 20 step-by-step strategies and introduce a valuable math problem solver to help you conquer any maths problem with confidence. From understanding the problem to employing strategic approaches and utilizing visual aids, our comprehensive guidance will empower you to excel in maths problem solving. Say goodbye to frustration and embrace a new level of mathematical proficiency with our expert strategies and reliable math problem solver.

Part1. Common Issues in Math Problem-Solving

As students tread the path of math learning, they often encounter stumbling blocks that hinder effective problem-solving. These challenges manifest as misconceptions, calculation errors, and a lack of systematic approaches. One prevailing adversary is the constraint of time, which can lead to rushed problem-solving attempts and consequently, errors.

It’s paramount to recognize these stumbling blocks to effectively address them. Acknowledging that errors are an inherent part of the learning process is the first step toward achieving accuracy and mastery. Through our exploration of efficient math problem-solving methods, we will unravel techniques to navigate these challenges skillfully. By fostering a proactive attitude toward errors and setbacks, students can refine their mathematical skills and bolster their confidence in tackling even the most intricate problems.

Part 2. Strategies for Time-Saving and Error-Reducing Math Problem-Solving

In the realm of math learning, the adage “time is of the essence” holds true. Efficient time management is not just a convenience; it’s a cornerstone of successful math problem-solving. Many students, in their quest to solve problems swiftly, inadvertently compromise accuracy. However, with a strategic and well-paced approach, it’s entirely possible to strike a balance between speed and precision.

Through this blog, we will introduce you to proven time-saving techniques tailored specifically for math problem-solving. By dissecting complex problems into manageable components and leveraging mental math strategies, students can optimize their time utilization without sacrificing the accuracy that’s essential for grasping mathematical concepts. Join us in discovering how a systematic approach to time management can revolutionize your child’s math learning experience.

Enhancing Accuracy: Methods to Reduce Errors In this segment, we delve deeper into the art of enhancing accuracy and reducing errors, highlighting strategies that foster a profound understanding of concepts while refining calculation techniques and cognitive approaches.

1）Embracing Errors as Catalysts for Growth

Mistakes are an intrinsic part of learning. Instead of fearing errors, we should embrace them as opportunities for growth. By analyzing mistakes, students gain insights into their thought processes, identifying areas where concepts might be hazy or calculations flawed. This introspection sharpens critical thinking and transforms errors into catalysts for improvement.

2）The Precision of Double-Checking

A powerful tool in the arsenal of accuracy is the practice of double-checking. Encouraging students to review their work systematically cultivates an environment of meticulousness. When solving math problems, an initial solution can be followed by a second verification pass. This deliberate process not only minimizes errors but also instills the habit of thoroughness—a skill that resonates in both mathematical pursuits and beyond.

3）Identifying and Overcoming Common Mistakes

By identifying recurring calculation errors and misconceptions, students can proactively address these pitfalls. Whether it’s recognizing common factors leading to miscalculations or dissecting misconceptions that hinder problem-solving, understanding the root causes of errors empowers students to rectify and prevent them. This strategic approach transforms errors from setbacks into stepping stones toward accuracy.

4）Precision Through Self-Correction

Promoting self-correction encourages students to take ownership of their learning. When an error is identified, students are encouraged to analyze where and why it occurred, subsequently correcting it themselves. This process deepens comprehension and reinforces memory, ensuring that mistakes do not recur. The act of self-correction instills a sense of responsibility and accountability in one’s problem-solving journey.

This journey to accuracy not only refines mathematical skills but also nurtures a mindset of lifelong learning—one that embraces challenges and mistakes as stepping stones toward becoming masterful problem solvers.

Part3. Strategies for Time-Saving in Homework

The synergy between efficiency and accuracy in math problem-solving is undeniable. In this segment, we delve deeper into strategies that intricately weave these two facets together, offering students a roadmap to streamline their work while maintaining the integrity of solutions. As students navigate the challenge of managing limited time without sacrificing thorough analysis, a strategic blend of techniques emerges to guide them.

1）Decoding the Systematic Approach

The art of time-saving begins with a systematic problem-solving framework. By deconstructing problems into their core components, students gain a clearer understanding of the task at hand. This process enables them to devise a structured plan, breaking down the problem into manageable steps. A systematic approach not only optimizes time usage but also enhances clarity and precision in problem-solving.

2）The Mental Math Advantage

In the pursuit of efficiency, mental math techniques shine as valuable assets. Armed with shortcuts, mental math empowers students to perform calculations swiftly without relying solely on pen and paper. These techniques, from rapid multiplication tricks to quick estimation methods, liberate precious moments for addressing more complex facets of a problem. The result is a time-efficient yet accurate problem-solving process.

3）Prioritization: Navigating the Essential

Time constraints often demand a selective approach to problem-solving. Prioritization comes to the forefront as students identify critical elements of a problem that warrant deeper analysis. By focusing efforts on intricate aspects and avoiding unnecessary detours, students can allocate time effectively, ensuring that their solutions are comprehensive while adhering to deadlines.

With systematic problem-solving, mental math mastery, and skillful prioritization, students emerge not just as problem solvers, but as architects of their own mathematical journey, where every minute counts towards mathematical excellence.

Part4. Holistic Strategies for Enhancing Math Problem-Solving

In this comprehensive chapter, we delve into a trio of dynamic approaches that collectively enrich the world of math learning and problem-solving. From fostering a growth mindset to harnessing technological tools and tailored strategies, our exploration promises to equip learners with a diverse toolkit for mathematical excellence.

1）Developing a Growth Mindset in Math Problem-solving

Beyond the realm of skills and strategies, an often overlooked yet transformative facet of math learning is the mindset with which students approach challenges. A growth mindset, characterized by resilience, adaptability, and a hunger for learning, is a cornerstone of successful math problem-solving.

Mathematics, at its essence, is a dynamic landscape of patterns, puzzles, and discoveries. By reframing challenges as opportunities for growth and embracing mistakes as valuable feedback, students can transcend limitations and reach new heights in math learning.

We should explore tangible methods to foster a growth mindset within students, creating an environment that encourages exploration, risk-taking, and a passion for uncovering the beauty of mathematics. Mindset cultivation is an integral component of math problem-solving that fuels the flame of lifelong learning.

2）Technological Tools for Math Problem-Solving

In the ever-evolving landscape of education, technology has emerged as a powerful ally in enhancing math learning experiences. This segment sheds light on the symbiotic relationship between math problem-solving and digital resources. While traditional methods remain invaluable, technology can amplify learning through interactive platforms, engaging applications, and dynamic visualizations.

When it comes to solving mathematical problems, there are numerous technological tools available to provide support and assistance to students. Here are some examples of commonly used technological tools for mathematical problem-solving:

• Equation Solvers : Equation solver apps can automatically solve various types of equations, from linear equations to higher-order algebraic equations. Students can input an equation, and the app will demonstrate the solution process step by step.
• Math Problem-Solving Apps : Math problem-solving apps offer exercises of varying difficulty levels, covering areas such as algebra, geometry, trigonometry, and more. These apps typically provide hints and solutions for each problem to help students overcome challenges.
• WuKong Math : This online platform offers a plethora of online math lessons, math instructional videos, practice exercises, and challenges suitable for students of different ages and difficulty levels. It helps students solidify their math knowledge and problem-solving skills.
• GeoGebra : This is free math software that combines geometry, algebra, and computation tools for exploring mathematical concepts, graphing functions, and investigating relationships.
• Virtual Manipulatives : Virtual manipulatives simulate physical teaching aids used in education, such as counting rods and geometric models. These tools help students understand mathematical concepts interactively and intuitively.
• Online Math Communities : Online math communities (such as Math StackExchange) allow students to ask questions and interact with math enthusiasts and experts to receive advice and guidance on mathematical problem-solving.

The combination of these tools can provide students with a rich learning experience, from real-time feedback to the exploration of visual concepts, all contributing to building confidence and competence in mathematical problem-solving. However, when using these tools, students should maintain moderation while continuing to develop traditional mathematical problem-solving skills.

Part5. Tailored Problem-solving Approaches for Different Learning Styles

Education is a combination of multiple learning styles, and each student possesses unique cognitive characteristics that can be personalized to optimize their path to solving mathematical problems.

Visual learners, auditory learners, and kinesthetic learners each have different approaches to processing and retaining information. We can develop different math problem-solving strategies for different types of learners to ensure that students with different tendencies are effectively catered for. For example, visual aids, mnemonics, interactive activities, and hands-on experiments can be used to meet the different needs of math learners.

By recognizing and embracing each student’s individual learning style, parents and educators can work together to create an environment where math problem-solving becomes an intuitive and engaging process.

Part6. Parental Involvement in Math Problem-Solving

Education is a collaborative endeavor, and parents play an instrumental role in shaping their child’s math learning trajectory. This segment emphasizes the significance of parental involvement and offers practical strategies for creating a supportive math learning environment at home.

1）Clarifying Concepts and Providing Support

Homework often presents opportunities for students to apply concepts learned in class. Parents can assist by clarifying doubts and explaining concepts in a simplified manner. By providing clear explanations and examples, you contribute to your child’s comprehension and ensure they have a solid foundation to tackle more complex math problems.

2）Promoting Problem-Solving Strategies

Rather than solely providing answers, encourage your child to explore different problem-solving strategies. Engage in discussions that guide them to think critically about how to approach a problem. By asking open-ended questions, you stimulate their analytical skills and help them develop a deeper understanding of mathematical principles.

3）Creating a Structured Routine

Establishing a consistent homework routine can enhance your child’s focus and time management skills. Set aside a designated time and quiet workspace for math homework. Your guidance in organizing tasks and breaking them into manageable steps teaches your child valuable organizational skills that extend beyond math.

4）Balancing Independence and Support

Striking a balance between providing assistance and promoting independence is crucial. Offer guidance when needed, but encourage your child to tackle problems on their own. Gradually stepping back allows them to develop problem-solving autonomy and a sense of accomplishment.

5）Celebrating Achievements

Acknowledge your child’s accomplishments in math problem-solving. Celebrate their progress, whether it’s mastering a new concept or persevering through a challenging problem. Your recognition boosts their confidence and reinforces their dedication to learning.

6）Effective Communication with Teachers

Maintaining open communication with your child’s math teacher is invaluable. Regularly discuss your child’s progress, strengths, and areas needing improvement. Collaboratively working with educators ensures a cohesive approach to your child’s math learning.

By embracing the techniques outlined in this series, you can empower your child with more than just mathematical skills. It’s a way to foster a mindset of resilience, adaptability, and curiosity, a mindset that will serve them well in all aspects of life. Math learning is not confined to classrooms; it’s a journey of discovery, a quest for understanding the world around us. Remember that each challenge encountered is an opportunity for growth, each error a stepping stone towards mastery. By embracing the journey, celebrating successes, and learning from setbacks, you can nurture a generation of lifelong math problem-solvers who approach challenges with confidence and grace.

Graduated from Columbia University in the United States and has rich practical experience in mathematics competitions’ teaching, including Math Kangaroo, AMC… He teaches students the ways to flexible thinking and quick thinking in sloving math questions, and he is good at inspiring and guiding students to think about mathematical problems and find solutions.

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Graduated from Columbia University in the United States and has rich practical experience in mathematics competitions' teaching, including Math Kangaroo, AMC... He teaches students the ways to flexible thinking and quick thinking in sloving math questions, and he is good at inspiring and guiding students to think about mathematical problems and find solutions.

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3 Simple Strategies to Improve Students’ Problem-Solving Skills

These strategies are designed to make sure students have a good understanding of problems before attempting to solve them.

Research provides a striking revelation about problem solvers. The best problem solvers approach problems much differently than novices. For instance, one meta-study showed that when experts evaluate graphs , they tend to spend less time on tasks and answer choices and more time on evaluating the axes’ labels and the relationships of variables within the graphs. In other words, they spend more time up front making sense of the data before moving to addressing the task.

While slower in solving problems, experts use this additional up-front time to more efficiently and effectively solve the problem. In one study, researchers found that experts were much better at “information extraction” or pulling the information they needed to solve the problem later in the problem than novices. This was due to the fact that they started a problem-solving process by evaluating specific assumptions within problems, asking predictive questions, and then comparing and contrasting their predictions with results. For example, expert problem solvers look at the problem context and ask a number of questions:

• What do we know about the context of the problem?
• What assumptions are underlying the problem? What’s the story here?
• What qualitative and quantitative information is pertinent?
• What might the problem context be telling us? What questions arise from the information we are reading or reviewing?
• What are important trends and patterns?

As such, expert problem solvers don’t jump to the presented problem or rush to solutions. They invest the time necessary to make sense of the problem.

Now, think about your own students: Do they immediately jump to the question, or do they take time to understand the problem context? Do they identify the relevant variables, look for patterns, and then focus on the specific tasks?

If your students are struggling to develop the habit of sense-making in a problem- solving context, this is a perfect time to incorporate a few short and sharp strategies to support them.

3 Ways to Improve Student Problem-Solving

1. Slow reveal graphs: The brilliant strategy crafted by K–8 math specialist Jenna Laib and her colleagues provides teachers with an opportunity to gradually display complex graphical information and build students’ questioning, sense-making, and evaluating predictions.

For instance, in one third-grade class, students are given a bar graph without any labels or identifying information except for bars emerging from a horizontal line on the bottom of the slide. Over time, students learn about the categories on the x -axis (types of animals) and the quantities specified on the y -axis (number of baby teeth).

The graphs and the topics range in complexity from studying the standard deviation of temperatures in Antarctica to the use of scatterplots to compare working hours across OECD (Organization for Economic Cooperation and Development) countries. The website offers a number of graphs on Google Slides and suggests questions that teachers may ask students. Furthermore, this site allows teachers to search by type of graph (e.g., scatterplot) or topic (e.g., social justice).

2. Three reads: The three-reads strategy tasks students with evaluating a word problem in three different ways . First, students encounter a problem without having access to the question—for instance, “There are 20 kangaroos on the grassland. Three hop away.” Students are expected to discuss the context of the problem without emphasizing the quantities. For instance, a student may say, “We know that there are a total amount of kangaroos, and the total shrinks because some kangaroos hop away.”

Next, students discuss the important quantities and what questions may be generated. Finally, students receive and address the actual problem. Here they can both evaluate how close their predicted questions were from the actual questions and solve the actual problem.

To get started, consider using the numberless word problems on educator Brian Bushart’s site . For those teaching high school, consider using your own textbook word problems for this activity. Simply create three slides to present to students that include context (e.g., on the first slide state, “A salesman sold twice as much pears in the afternoon as in the morning”). The second slide would include quantities (e.g., “He sold 360 kilograms of pears”), and the third slide would include the actual question (e.g., “How many kilograms did he sell in the morning and how many in the afternoon?”). One additional suggestion for teams to consider is to have students solve the questions they generated before revealing the actual question.

3. Three-Act Tasks: Originally created by Dan Meyer, three-act tasks follow the three acts of a story . The first act is typically called the “setup,” followed by the “confrontation” and then the “resolution.”

This storyline process can be used in mathematics in which students encounter a contextual problem (e.g., a pool is being filled with soda). Here students work to identify the important aspects of the problem. During the second act, students build knowledge and skill to solve the problem (e.g., they learn how to calculate the volume of particular spaces). Finally, students solve the problem and evaluate their answers (e.g., how close were their calculations to the actual specifications of the pool and the amount of liquid that filled it).

Often, teachers add a fourth act (i.e., “the sequel”), in which students encounter a similar problem but in a different context (e.g., they have to estimate the volume of a lava lamp). There are also a number of elementary examples that have been developed by math teachers including GFletchy , which offers pre-kindergarten to middle school activities including counting squares , peas in a pod , and shark bait .

Students need to learn how to slow down and think through a problem context. The aforementioned strategies are quick ways teachers can begin to support students in developing the habits needed to effectively and efficiently tackle complex problem-solving.

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Generalized fuzzy difference method for solving fuzzy initial value problem

• Open access
• Published: 27 March 2024
• Volume 43 , article number  129 , ( 2024 )

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• S. Soroush 1 ,
• T. Allahviranloo   ORCID: orcid.org/0000-0002-6673-3560 1 , 2 ,
• H. Azari 3 &
• M. Rostamy-Malkhalifeh 3  

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We are going to explain the fuzzy Adams–Bashforth methods for solving fuzzy differential equations focusing on the concept of g -differentiability. Considering the analysis of normal, convex, upper semicontinuous, compactly supported fuzzy sets in \(R^n\) and also convergence of the methods, the general expression of solutions is obtained. Finally, we demonstrate the importance of our method with some illustrative examples. These examples are provided aiming to solve the fuzzy differential equations.

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Avoid common mistakes on your manuscript.

1 Introduction

According to the most recent published papers, the fuzzy differential equation was introduced in 1978. Moreover, Kandel ( 1980 ) and Byatt and Kandel ( 1978 ) present the fuzzy differential equation and have rapidly expanded literature. First-order linear fuzzy differential equations emerge in modeling the uncertainty of dynamical systems. The solutions of first-order linear fuzzy differential equations have been widely considered (e. g., see Chalco-Cano and Roman-Flores 2008 ; Buckley and Feuring 2000 ; Seikkala 1987 ; Diamond 2002 ; Song and Wu 2000 ; Allahviranloo et al. 2009 ; Zabihi et al. 2023 ; Allahviranloo and Pedrycz 2020 ).

The most famous numerical solutions of order fuzzy differential equations are investigated and analyzed under the Hukuhara and gH -differentiability (Safikhani et al. 2023 ). It is widely believed that the common Hukuhara difference and so Hukuhara derivative between two fuzzy numbers are accessible under special circumstances (Kaleva 1987 ; Diamond 1999 , 2000 ). The gH -derivative, however, is available in less restrictive conditions, even though this is not always the case (Dubois et al. 2008 ). To overcome these serious defects of the concepts mentioned above, Bede and Stefanini (Dubois et al. 2008 ) describe g -derivative. In 2007, Allahviranloo used the predictor–corrector under the Seikkala-derivative method to propose a numerical solution of fuzzy differential equations (Allahviranloo et al. 2007 ).

Here, we investigate the Adams–Bashforth method to solve fuzzy differential equations focusing on g -differentiability. We restrict our study on normal, convex, upper semicontinuous, and compactly supported fuzzy sets in \(\mathbb {R}^n\) .

This paper has been arranged as mentioned below: firstly, in Sect.  2 , we recall the necessary definitions to be used in the rest of the article, after a preliminary section in Sect.  3 , which is dedicated to the description of the Adams–Bashforth method to fix the purposed equation. The convergence theorem is formulated and proved in Sect.  4 . For checking the accuracy of the method, three examples are presented. In Sect.  5 , their solutions are compared with the exact solutions. In the last section, some conclusions are given.

2 Preliminaries

Definition 2.1.

(Mehrkanoon et al. 2009 ) A fuzzy subset of the real line with a normal, convex, and upper semicontinuous membership function of bounded support is a fuzzy number \(\tilde{w}\) . The family of fuzzy numbers is indicated by F .

We show an arbitrary fuzzy number with an ordered pair of functions \((\underline{w}(\gamma ),\overline{w}(\gamma ))\) , \(0\le \gamma \le 1\) which provides the following:

\(\underline{w}(\gamma )\) is a bounded left continuous non-decreasing function over [0, 1], corresponding to any \(\gamma \) .

\(\overline{w}(\gamma )\) is a bounded left continuous non-decreasing function over [0, 1], corresponding to any \(\gamma \) .

Then, the \(\gamma \) -level set

is a closed bounded interval, which is denoted by:

Definition 2.2

(Bede and Stefanini 2013 ) The g -difference is defined as follows:

In Bede and Stefanini ( 2013 ), the difference between g -derivative and q -derivative has been fully investigated.

Definition 2.3

(Bede and Stefanini 2013 ; Diamond 1999 , The Hausdorff distance ) The Hausdorff distance is defined as follows:

where \(|| \cdot ||=D(\cdot , \cdot )\) and the gH -difference \(\circleddash _{gH}\) is with interval operands \([u]^\gamma \) and \([v]^\gamma \)

By definition, D is a metric in \(R_F\) which has the subsequent properties:

\(D(w+t, z+t)=D(w,z ) \qquad \forall w, z, t \in R_F\) ,

\(D(rw,rz)=|r|D(w,z)\qquad \forall w, z\in \ R_F, r\in R\) ,

\(D(w+t,z+d)\le D(w,z)+ D(t, d)\qquad \forall w, z, t, d \in R_F\) .

Then, \((D, R_F)\) is called a complete metric space.

Definition 2.4

(Bede and Stefanini 2013 ) Neumann’s integral of \(k{:}\, [m, n] \rightarrow R_F\) is defined level-wise by the fuzzy

Definition 2.5

(Bede and Stefanini 2013 ) Suppose \(k{:}\, [m,n] \rightarrow R_F\) is a function with \([k(y)]^{\gamma }=[\underline{k}_{\gamma }(y), \overline{k}_{\gamma }(y)]\) . If \(\underline{k}_{\gamma }(y)\) and \(\overline{k}_{\gamma }(y)\) are differentiable real-valued functions with respect to y , uniformly for \(\gamma \in [0, 1]\) , then k ( y ) is g -differentiable and we have

Definition 2.6

(Bede and Stefanini 2013 ) Let \(y_0 \in [m, n]\) and t be such that \(y_0+t \in ]m, n[\) , then the g -derivative of a function \(k{:}\, ]m, n[ \rightarrow R_F\) at \(y_0\) is defined as

If there exists \(k'_g(y_0)\in R_F\) satisfying ( 7 ), we call it generalized differentiable ( g -differentiable for short) at \(y_0\) . This relation depends on the existence of \(\circleddash _g\) , and there exists no such guarantee for this desire.

Theorem 2.7

Suppose \(k{:}\,[m,n]\rightarrow R_F\) is a continuous function with \([k(y)]^{\gamma }=[k^{-}_{\gamma }(y), k^{+}_{\gamma }(y)]\) and g -differentiable in [ m ,  n ]. In this case, we obtain

To show the assertion, it is enough to show their equality in level-wise form, suppose k is g -differentiable, so we have

\(\square \)

Definition 2.8

(Kaleva 1990 , fuzzy Cauchy problem ) Suppose \(x'_g(s)=k(s,x(s))\) is the first-order fuzzy differential equation, where y is a fuzzy function of s , k ( s ,  x ( s )) is a fuzzy function of the crisp variable s , and the fuzzy variable x and \(x'\) is the g -fuzzy derivative of x . By the initial value \(x(s_0)=\gamma _0\) , we define the first-order fuzzy Cauchy problem:

Proposition 2.9

Suppose \(\textit{k, h}{:}\, [\textit{a}, \textit{b}] {\rightarrow } R_F\) are two bounded functions, then

Since \(k(y)\le \textrm{sup}_A k\) and \(k(y)\le \textrm{sup}_A k\) for every \(y \in [m,n]\) , one can obtain \(k(y)+h(y)\le \textrm{sup}_A k+\textrm{sup}_A h\) . Thus, \(k+h\) is bounded from above by \(\textrm{sup}_A k+\textrm{sup}_A h\) , so \(\textrm{sup}_A( k+h) \le \textrm{sup}_A k+ \textrm{sup}_A h\) . The proof for the infimum is similar. \(\square \)

Definition 2.10

Let \(\{\widetilde{q}_m\}^\infty _{m=0}\) be a fuzzy sequence. Then, we define the backward g -difference \(\nabla _g \widetilde{q}_m\) as follows

So, we have

Consequently,

Proposition 2.11

For a given fuzzy sequence \({\left\{ {\widetilde{q}}_m\right\} }^{\infty }_{m=0}\) , by supposing backward g -difference, we have

We prove proposition by induction that, for all \(n \in \mathbb {Z}^+\) ,

Using Definition 2.10 , for base case, \(n = 1\) , we have

Induction step : Let \( k \in \mathbb {Z}^+\) be given and suppose ( 14 ) is true for \(n=k\) . Then,

Conclusion : For all \(m\in \mathbb {Z}^+\) , ( 14 ) is correct, by the principle of induction. \(\square \)

Definition 2.12

( Switching Point ) The concept of switching point refers to an interval where fuzzy differentiability of type-(i) turns into type-(ii) and also vice versa.

3 Fuzzy Adams–Bashforth method

To derivative of a fuzzy multistep method, we consider the solution of the initial-value problem:

To obtain the approximation \(t_{j+1}\) at the mesh point \(s_{j+1}\) , where initial values

are assumed.

If integrated over the interval \([s_j,s_{j+1}]\) , we get

but, without knowing \(\widetilde{x}(s)\) , we cannot integrate \(\widetilde{k}(s,\widetilde{x}(s))\) , one can apply an interpolating polynomial \(\widetilde{q}(s)\) to \(\widetilde{k}(s, \widetilde{x}(s))\) , which is computed by the data points \(\left( s_0, {\widetilde{t}}_0\right) , \left( s_1,{\widetilde{t}}_1\right) ,\ldots \left( s_j,{\widetilde{t}}_j\right) \) . These data were obtained in Sect.  2 .

Indeed, by supposing that \(\widetilde{x}\left( s_j\right) \approx \ {\widetilde{t}}_j\ \) , Eq. ( 17 ) is rewritten as

To take a fuzzy Adams–Bashforth explicit m -step method under the notion of g -difference, we construct the backward difference polynomial \({\widetilde{q}}_{n-1}(s)\) ,

We assume that the n th derivatives of the fuzzy function k exist. This means that all derivatives are g -differentiable. As \({\widetilde{q}}_{n-1}(s)\) is an interpolation polynomial of degree \(n-1\) , some number \({\xi }_j\) in \(\left( s_{j+1-n}, s_j\right) \) exists with

where the corresponding notation \({\widetilde{k}}^{(n)}_g (s, \widetilde{x}(s)),n\in \mathbb {N},\) exists. Moreover, it can be mentioned that the existence of this corresponding formula based on the existence of \({\circleddash }_g\) , and while \({\circleddash }_g\) exist this relation always exists.

We introduce the \(s=s_j+\beta h\) , with \(\textrm{d}s=h \textrm{d}\beta \) , substituting these variable into \({\widetilde{q}}_{n-1}(s)\) and the error term indicates

So, we will get

Obviously, the product of \(\beta \ \left( \beta +1\right) \cdots \left( \beta +n-1\right) \) does not change sign on [0, 1], so the Weighted Mean Value Theorem for some number \({\mu }_j\) , where \(s_{j+1-n}< {\mu }_j< s_{j+1}\) , can be applied to the last term in Eq. ( 22 ), hence it becomes

So, it simplifies to

So, Eq. ( 20 ) is written as

It is also worth mentioning that the notions \(\Delta _{g}\) and \(\oplus \) are extensively utilized solving the problems of sup and inf existence.

To illustrate this method, we discuss solving the fuzzy initial value problem \({\widetilde{x}}'\left( s\right) =\widetilde{k}(s,\widetilde{x}\left( s\right) )\) by Adams–Bashforth’s three-step method. To derive the three-step Adams–Bashforth technique, with \(n= 3\) , We have

For \(m=2, 3,\ldots , N-1.\) So

Here, we also describe our model as introduced models \(\Delta _{g}\) and \(\oplus \) .

By considering

As a consequence

from which we obtain

From ( 24 ) and ( 29 ), we get

if we suppose that

Then, we have

Similarly, we have

Then, we can say

4 Convergence

We begin our dissection with definitions of the convergence of multistep difference equation and consistency before discussing methods to solve the differential equation.

Definition 4.1

The differential fuzzy equation with initial condition

and similarly, the other models can be derived as

is the \(\left( j+1\right) \) st step in a multistep method. At this step, it has a fuzzy local truncation error as follows

Exists N that for all \(j= n-1, n, \ldots N-1\) , and \(h=\frac{b-a}{N}\) , where

And \(\widetilde{x}\left( s_j\right) \) indicates the exact value of the solution of the differential equation. The approximation \({\widetilde{t}}_j\) is taken from the different methods at the j th step.

Definition 4.2

A multistep method with local truncation error \({\widetilde{\nu }}_{j+1}\left( h\right) \) at the \((j+1)\) th step is called consistent with the differential equation approximation if

Theorem 4.3

Let the initial-value problem

be approximated by a multistep difference method:

Let a number \(h_0>0\) exist, and \(\phi \left( s_j,\widetilde{t}\left( s_j\right) \right) \) be continuous, with meets the constant Lipschitz T

Then, the difference method is convergent if and only if it is consistent. It is equal to

We are aware of the concept of convergence for the multistep method. As the step size approaches zero, the solution of the difference equation approaches the solution to the differential equation. In other words

For the multistep fuzzy Adams–Bashforth method, we have seen that

using Proposition 2.11 , \(\nabla ^l_g{\widetilde{k}}_m=h^l\widetilde{k}^{(l)}_{m_g}\) , and substituting it in Eq. ( 66 ), we have

under the hypotheses of paper, \(\widetilde{k}({(s}_j,\widetilde{x}(s_j))\in R_F\) , and by definition g -differentiability \(\widetilde{k}^{(n)}({(s}_j,\widetilde{x}(s_j))\in R_F\) so by Definition 2.1 \(\ {\widetilde{k}}^{(n)}\left( {(s}_j,\widetilde{x}\left( s_j\right) \right) \in R_F\) for \(j\ge 0\) are bounded, thus exists M such that

When \(h\rightarrow 0\) , we will have \(Z\rightarrow 0\) so

So, we see that it satisfied the first condition of Definition 4.2 . The concept of the second part is that if the one-step method generating the starting values is also consistent, then the multistep method is consistent. So our method is consistent; therefore according to Theorem 4.3 , this difference method is convergent.

Example 5.1

Consider the initial-value problem

Obviously, one can check the exact solution as follows:

Indeed, the solution is a triangular number

So, the exact solution in mesh point \(s=0.01\) is

On the other hand with the proposed method, the approximated solution in \(s=0.01\) is as follows:

where \(\tilde{t}^\gamma \) is a approximated of \(\tilde{x}\) .

The maximum error in \(s=0.1\) , \(s=0.2, \ldots , s=1\) , also shows the errors (Table 1 ).

Thus, we have

where ( 90 ) are real values. Suppose

By ( 85 ), ( 86 ), ( 92 ), we obtain

According to the previous sections, this example has been solved by the two-step Adams–Bashforth method with \(t=0.1\) and \(N=10\) . We use the following relations to solve it.

Example 5.2

First, we solve the problem with the gH -differentiability. The initial-value problem on [0, 1] is \([(i)-gH]\) -differentiable and \([(i)-gH]\) -differentiable on (1, 2]. By solving the following system, the \([(i)-gH]\) -differentiable solution will be achieved

By solving the following system, the \([(ii)-gH]\) -differentiable solution will be achieved

If we apply the Euler method to the approximate solution of the initial-value problem by

The results are presented in Table  2 .

In the calculations of this method, we need to consider the \(i-gh\) -differentiability and \(ii-gH\) -differentiability. But when we use g -differentiability, we do not need to check the different states of the differentiability. To solve using the method mentioned in the article, we have:

Or we have \(x^\gamma (0)-[\gamma , 2-\gamma ]\) . The exact solution is as follows:

The results of the solution using the Adams–Bashforth two-step method for \(h = 1\) and calculating the approximate value of the solution and the error of the method can be seen in the Table  3 .

Consider the initial-value problem \(\tilde{x'}=(s\ominus 1)\odot \tilde{x}^2\) , where \(s\in [-1,1]\)

the exact solution is

6 Conclusion

In the present paper, the proposed method, which is based on the concept of g-differentiability, provides a fuzzy solution. This solution is related to a set of equations from the family of Adams-Bashforth differential equations, which coincide with the solutions derived by fuzzy differential equations.

The gH -difference is a powerful and versatile fuzzy differential operator that is more flexible, robust, and computationally efficient, making it a good choice for solving a wide range of fuzzy differential equations. It does not need i and ii -differentiability. In Examples, we compare g -differentiability and gH -differentiability.

G-differentiability allows for capturing gradual changes in a fuzzy-valued function. G-differentiable functions exhibit certain degrees of smoothness and continuity, which can be useful in modeling and analyzing fuzzy systems. The choice of the parameter g in g -differentiability is crucial and depends on the specific problem. Determining an appropriate value for g requires careful consideration and analysis. H -differentiability combines the gradual reduction of fuzziness (via the parameter g ) with the Hukuhara difference ( H -difference). It provides a more refined analysis of fuzzy-valued functions. gH -differentiability offers enhanced modeling capabilities by considering both the gradual reduction of fuzziness and the separation between fuzzy numbers or fuzzy sets. But gH -differentiability introduces an additional level of complexity compared to g -differentiability or H -differentiability alone. The combination of gradual reduction and H -difference requires careful understanding and analysis to ensure proper application.

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The authors are thankful to the area editor and referees for giving valuable comments and suggestions.

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Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran

S. Soroush & T. Allahviranloo

Research Center of Performance and Productivity Analysis, Istinye University, Istanbul, Turkey

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Department of Applied Mathematics, Faculty of Mathematics Sciences, Shahid Beheshti University, Tehran, Iran

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Soroush, S., Allahviranloo, T., Azari, H. et al. Generalized fuzzy difference method for solving fuzzy initial value problem. Comp. Appl. Math. 43 , 129 (2024). https://doi.org/10.1007/s40314-024-02645-2

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Received : 30 March 2023

Revised : 12 January 2024

Accepted : 14 February 2024

Published : 27 March 2024

DOI : https://doi.org/10.1007/s40314-024-02645-2

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