how to improve problem solving speed

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How can I get faster at doing math? [closed]

In general, my level of mathematics is good with respect to the competitions that I need to appear in.

However, no matter what the competition is, whether it is an easier level or a more difficult level of Olympiad, or JEE, or just some internal math test, I am always facing a time issue. I get almost all my answers correct, but I’m simply not able to get score beyond a threshold because of the time crunch. Even simple advices like how to get faster at calculations help.

What is it that you advice I should do to get faster at solving problems? Even simple suggestions on how to get faster at calculations help.

• contest-math

• $\begingroup$ It's possible that you have a form of dyscalculia (that I believe I suffer from) which is poorly understood. That is, you have high conceptual mathematical abilities but low calculation abilities. Consider computer science as a field. Computers are really good at calculations. $\endgroup$ –  JimmyJames Commented Jul 19, 2023 at 21:57
• $\begingroup$ The what-you-want question seems important here. I mean, if you truly just want to win competitions, then that'd seem to call for an aggressively simplistic mindset -- though that might be at-odds with other goals you might have. $\endgroup$ –  Nat Commented Jul 20, 2023 at 13:03
• $\begingroup$ @PM2Ring, OP mentions JEE. It is, by definition, a competition, but it is also the entrance exam for the top engineering universities in the country. Many students feel like they need to write the JEE for this reason. $\endgroup$ –  aqualubix Commented Jul 23, 2023 at 4:39

2 Answers 2

In my view, paradoxically you have to be slower in order to be faster. To be fast at maths you have to be less focused on getting the job in front of you right now done, and more focused during the learning stage on learning deeper insights into what you're doing. When you attain a high level of MASTERY , problems are faster to solve.

I'm slow at learning maths because I need deep insights in order to remember things, and I'm never satisfied until I know something inside out.

Taking a really simple example, to calculate $5 \times (10+4)$ you might say that's $5\times14=70$ and be done with it and move on.

But if you're focused on learning deeper insights you might think about it in multiple different ways:

Multiplication distributes over addition so I can multiply before I add, or after:

$(5\times10) + (5\times 4)=5\times(10+4)=70$

You might think about how multiplying is equivalent to adding the exponents of the prime factors:

$10+4$ has the prime factors $\{2^1,7^1\}$ and $5$ has the prime factor $\{5^1\}$ so the product is $2^{1+0}\times5^{1+0}\times7^{0+1}=70$

Since we write numbers in base $10$ you might think about multiplying by five as multiplying by $\frac{10}2$ so $5 \times (10+4) = \frac{10}2\times14$ then cancel the twos to get $10\times7$

You might think of $14$ as $20-6$ giving you $100-30=70$

Then you might think about the fact that this last example is

$5\times (20-6)=(5\times20-5\times6)$ and ask yourself whether this means that multiplication distributes over subtraction.

Then you might deduce that this is true because subtraction is simply the addition of a negative number.

Then you might think about how addition is an operation on the monoid of non-negative integers, and how subtraction is the extension of rightward transformations on the real line to leftward transformations on the line which are their inverse transformations. And by introducing subtraction you extend the closure of your algebraic operations to include the negatives, and this makes the integers a group according to the group axioms. Multiplying a negative number scales to the left rather than scaling to the right.

A good exercise, is to pay attention to how you solved a problem . Sometimes you do it automatically, and the process is subconscious. Once you become conscious of your method, ask yourself what other methods you might have used, and solve the same problem by those methods. You will learn shortcuts this way and multiple ways of understanding the same thing.

The more insight you build, the more visualisation tools you have at hand on which to pin memories. Then when you have to apply quickly e.g. in an exam, it's not a case of finding your one way to solve a problem. You can see multiple ways and one may instantly jump out as a quick solution. And you can move on with confidence, knowing you've arrived at your answer by multiple different methods.

This of course, requires an investment of more time in the learning stage. But by building deeper insights and a higher level of mastery, you can progress faster when it comes to applying what you learnt.

Improving your speed in solving mathematical problems requires a combination of practice, strategy, and efficiency. Here are some suggestions that may help you get faster at solving problems:

Practice regularly: Consistent practice is crucial for developing speed. Solve a variety of problems regularly, including both easier and more challenging ones. This will help you build familiarity with different problem types and increase your overall speed.

Develop mental math skills: Strengthen your mental math abilities by practicing mental calculations, such as addition, subtraction, multiplication, and division. Learn techniques like estimation, rounding, and simplification to quickly approximate and simplify calculations.

Focus on key concepts: Identify the key concepts and strategies relevant to the types of problems you frequently encounter. Mastering these core concepts will enable you to solve problems more efficiently and quickly.

Improve problem-solving techniques: Learn and practice various problem-solving techniques, such as visualization, pattern recognition, and logical reasoning. These techniques can help you approach problems in a more organized and efficient manner, saving you time in the process.

Time yourself: Set time limits for solving practice problems and exams to simulate the pressure of real test conditions. This will help you develop a sense of time management and train yourself to work efficiently within specific time constraints.

Utilize shortcuts and tricks: Learn and utilize shortcuts, tricks, and formulas specific to the types of problems you are solving. Familiarize yourself with commonly used formulas, properties, and identities in areas such as algebra, geometry, and trigonometry. This will help you save time by avoiding unnecessary calculations.

Analyze your mistakes: Review and analyze your mistakes and the solutions to the problems you have solved. Identify the areas where you are consistently making errors or spending too much time. Understanding your weaknesses will allow you to focus on those areas during your practice and improve your efficiency.

Build problem-solving intuition: Work on developing an intuition for problem-solving by solving a wide range of problems. With practice, you will start recognizing common problem-solving patterns and become more adept at choosing the most efficient approach for each problem.

Remember, improving speed takes time and effort. Be patient and persistent in your practice, and gradually you will see improvement.

• $\begingroup$ Upvoted for mentioning pattern matching . Generally speaking, humans are primarily pattern-matching machines, even to the point where we think we recognize a pattern where none exists. $\endgroup$ –  njuffa Commented Jul 19, 2023 at 22:13

Not the answer you're looking for? Browse other questions tagged contest-math advice .

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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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How to Improve Mental Math Skills

Last Updated: August 2, 2024 Approved

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. There are 8 references cited in this article, which can be found at the bottom of the page. wikiHow marks an article as reader-approved once it receives enough positive feedback. In this case, 83% of readers who voted found the article helpful, earning it our reader-approved status. This article has been viewed 325,539 times.

Eventually, you'll find yourself in a situation where you'll have to solve a math problem without a calculator. Trying to imagine a pen and paper in your head often doesn't help much. Fortunately there are faster and easier ways to do calculations in your head—and they often break down a problem in a way that makes more sense than what you learned in school. Whether you're a stressed-out student or a math wizard looking for even faster tricks, there's something for everyone to learn.

Break addition and subtraction problems into parts.

• 712 + 281 → "700 + 200," "10 + 80," and "2 + 1"
• 700 + 200 = 9 00, then 10 + 80 = 9 0, then 2 + 1 = 3
• 900 + 90 + 3 = 993 .
• Thinking in "hundreds" or "tens" instead of single digits will make it easier to keep track when digits sum to more than ten. For example, for 37 + 45, think "30 + 40 = 70" and "7 + 5 = 12". Then add 70 + 12 to get 82.

Change the problem to make round numbers.

• Addition : For 596 + 380 , realize that you can add 4 to 596 to round it to 600, then add 600 + 380 to get 980. Undo the rounding by subtracting 4 from 980 to get 976 .
• Subtraction : For 815 - 521 , break it up into 800 - 500, 10 - 20, and 5 - 1. To turn the awkward "10 - 20" into "20 - 20", add 10 to 815 to get 825. Now solve to get 304, then undo the rounding by subtracting 10 to get 294 .
• Multiplication : For 38 x 3 , you can add 2 to 38 to make the problem 40 x 3, which is 120. Since the 2 you added got multiplied by three, you need to undo the rounding by subtracting 2 x 3 = 6 at the end to get 120 - 6 = 114 .

Learn to add many numbers at once.

• For example, 7 + 4 + 9 + 13 + 6 + 51 can be reorganized to (7 + 13) + (9 + 51) + (6 + 4) = 20 + 60 + 10 = 90.

Multiply from left to right.

• For 453 x 4 , start with 400 x 4 = 1600, then 50 x 4 = 200, then 3 x 4 = 12. Add them all together to get 1812 .
• If both numbers have more than one digit, you can break it into parts. Each digit has to multiply with each other digit, so it can be tough to keep track of it all. 34 x 12 = (34 x 10) + (34 x 2) , which you can break down further into (30 x 10) + (4 x 10) + (30 x 2) + (4 x 2) = 300 + 40 + 60 + 8 = 408 .

Try a fast multiplication trick best for numbers 11 through 19.

• Let's look at numbers close to 10, like 13 x 15 . Subtract 10 from the second number, then add your answer to the first: 15 - 10 = 5, and 13 + 5 = 18.
• Multiply your answer by ten: 18 x 10 = 180.
• Next, subtract ten from both sides and multiply the results: 3 x 5 = 15.
• Add your two answers together to get the final answer: 180 + 15 = 195 .
• Careful with smaller numbers! For 13 x 8, you start with "8 - 10 = -2", then "13 + -2 = 11". If it's hard to work with negative numbers in your head, try a different method for problems like this.
• For larger numbers, it will be easier to use a "base number" like 20 or 30 instead of 10. If you try this, make sure you use that number everywhere that 10 is used above. [3] X Research source For example, for 21 x 24, you start by adding 21 + 4 to get 25. Now multiply 25 by 20 (instead of ten) to get 500, and add 1 x 4 = 4 to get 504.

Simplify problems with numbers ending in zero.

• Addition : If all numbers have zeroes at the end, you can ignore the zeroes they have in common and restore them at the end. 85 0 + 12 0 → 85 + 12 = 97, then restore the shared zero: 97 0 .
• Subtraction works the same way: 10 00 - 7 00 → 10 - 7 = 3, then restore the two shared zeroes to get 3 00 . Notice that you can only remove the two zeroes the numbers have in common, and must keep the third zero in 1000.
• Multiplication : ignore all the zeroes, then restore each one individually. 3 000 x 5 0 → 3 x 5 = 15, then restore all four zeroes to get 15 0 , 00 0 .
• Division : you can remove all shared zeroes and the answer will be the same. 60, 000 ÷ 12, 000 = 60 ÷ 12 = 5 . Don't add any zeroes back on.

Easily multiply by 4, 5, 8, or 16.

• To multiply by 5, instead multiply by 10, then divide by 2.
• To multiply by 4, instead double the number, then double it again.
• For 8, 16, 32, or even higher powers of two, just keep doubling. For example, 13 x 8 = 13 x 2 x 2 x 2, so double 13 three times: 13 → 26 → 52 → 104 .

Memorize the 11s trick.

• What is 7 2 x 11?
• Add the two digits together: 7 + 2 = 9.
• Put the answer in between the original digits: 7 2 x 11 = 7 9 2 .
• If the sum is more than 10, place only the final digit and carry the one: 5 7 x 11 = 6 2 7 , because 5 + 7 = 12. The 2 goes in the middle and the 1 gets added to the 5 to make 6.

Turn percentages into easier problems.

• 79% of 10 is the same as 10% of 79. This is true of any two numbers. If you can't find the answer to a percentage problem, try switching it around.
• To find 10% of a number, move the decimal one place to the left (10% of 65 is 6.5). To find 1% of a number, move the decimal two places to the left (1% of 65 is 0.65).
• Use these rules for 10% and 1% to help you with more difficult percentages. For example, 5% is ½ of 10%, so 5% of 80 = (10% of 80) x ½ = 8 x ½ = 4 .
• Break percentages into easier parts: 30% of 900 = (10% of 900) x 3 = 90 x 3 = 270 .

Memorize advanced multiplication shortcuts for specific problems.

• For problems like 84 x 86 , where the tens place is the same and the ones place digits sum to exactly 10, the first digits of the answer are (8 + 1) x 8 = 72 and the last digits are 4 x 6 = 24, for an answer of 7224 . That is, for a problem AB x AC, if B + C = 10, the answer starts with A(A+1) and ends with BC. This also works for larger numbers if all digits besides the ones place are identical. [6] X Research source
• You can rewrite the powers of five (5, 25, 125, 625, ...) as powers of 10 divided by an integer (10 / 2, 100 / 4, 1000 / 8, 10000 / 16, ...). [7] X Research source So 88 x 125 becomes 88 x 1000 ÷ 8 = 88000 ÷ 8 = 11000 .

Memorize squares charts.

• Memorize the squares from 1 to 20 (or higher, if you're ambitious). (That is, 1 x 1 = 1; 2 x 2 = 4; 3 x 3 = 9, and so on.)
• To multiply two numbers, first find their average (the number exactly between them). For example, the average of 18 and 14 is 16.
• Square this answer. Once you've memorized the squares chart, you'll know that 16 x 16 is 256.
• Next, look at the difference between the original numbers and their average: 18 - 16 = 2. (Always use a positive number here.)
• Square this number as well: 2 x 2 = 4.
• To get your final answer, take the first square and subtract the second: 256 - 4 = 252 .

Find useful ways to practice your mental math.

• Flashcards are great for memorizing multiplication and division tables, or for getting used to tricks for specific kinds of problems. Write the problem on one side and the answer on the other, and quiz yourself daily until you get them all right.
• Online math quizzes are another way to test your ability. Look for a well-reviewed app or website made by an educational program.
• Practice in everyday situations. You could add together the total of items you buy as you shop, or multiply the gas cost per volume by your car's tank size to find the total cost. The more of a habit this becomes, the easier it will be.

Joseph Meyer

Exercise your mental math muscles. Improve your math skills by solving daily math problems without using calculators, paper, or counting aids. By solely using your mind and getting into math discussions with your classmates, you will refine your skills and discover new approaches to problem-solving.

Practice Problems and Answers

Community Q&A

• In the real world, you don't always need to know the exact answer. If you're at the grocery store and trying to add 7.07 + 8.95 + 10.09, you could round to the closest whole numbers and estimate that the total is roughly 7 + 9 + 10 = 26. Thanks Helpful 12 Not Helpful 3
• Some people find it easier to think in money than abstract numbers. Instead of 100 - 55, try thinking of a dollar minus a 50¢ coin and a 5¢ coin. Thanks Helpful 6 Not Helpful 9

You Might Also Like

• ↑ http://gizmodo.com/10-tips-to-improve-your-mental-math-ability-1792597814
• ↑ https://www.youtube.com/watch?v=Rgw9Ik5ZGaY
• ↑ https://www.youtube.com/watch?v=SV1dC1KAl_U
• ↑ https://www.youtube.com/watch?v=1JW9BA57aR8
• ↑ http://www.wired.co.uk/article/master-mental-maths
• ↑ https://www.youtube.com/watch?v=YCBTw8KAqkw
• ↑ https://www.scientificamerican.com/article/5-tips-faster-mental-multiplication/
• ↑ Daron Cam. Academic Tutor. Expert Interview. 29 May 2020.

About This Article

One way to improve your mental math skills is to memorize your multiplication and division tables, so you always have the answer to those problems instantly. If you have trouble memorizing the numbers, try creating your own flash cards with blank notecards and asking a friend to help you practice. Another good way to practice your mental math skills is to add up the prices of your items when you’re at the store, and check to make sure you added correctly once the cashier rings you up. You can also try downloading a mental math app like Luminosity to keep your math skills sharp. To learn how to visualize an equation in your head, read on! Did this summary help you? Yes No

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Improving your problem-solving skills with learning games for adults

We face problems every day . Whether it's a complex problem at work or a personal issue that needs solving, good problem-solving skills are essential for success in both your personal and professional life. 

If you’re feeling a little rusty in the problem-solving department, there are many ways to enhance your problem-solving abilities, like cognitive training techniques and brain games. That’s right: Games can help improve your cognitive abilities like processing speed, reasoning, and working memory , which are essential for effective problem-solving. 

So if you’re ready to learn how to improve your problem-solving skills with some of our recommended cognitive training techniques and tips, keep reading. And you’ll be making quicker, more confident decisions in no time. 

What is problem solving, and why is it important?

Problem-solving is, well, the process of identifying, defining, and finding a solution to challenges or difficulties. It involves several steps, including recognizing the existence of a problem, understanding its nature, generating potential solutions, evaluating those solutions, and then implementing the best one. 

Problem-solving is an essential skill that enables you to navigate various aspects of your personal and professional lives effectively. In your workplace, for example, you can quickly identify issues and implement appropriate solutions, contributing to increased productivity and efficiency. In your personal life, good problem-solving skills can help you navigate relationships, make informed decisions, and cope with unexpected situations.

Good problem-solving skills not only help you make better decisions but also improve your critical thinking abilities, allowing you to find effective solutions to complex problems. And by developing and honing your problem-solving skills through cognitive training, you can become more adaptable and resourceful, capable of tackling a wide range of challenges that life throws your way. 

The science behind cognitive training for improving problem solving

So, what is cognitive training? And what does science have to say about it? 

Cognitive training involves a range of activities and exercises that target different cognitive functions. These may include puzzles, memory exercises, or brain games that require strategic thinking. The goal is to stimulate your brain and enhance its ability to process information, reason effectively, and retain information. By engaging in cognitive training , you can boost your mental capabilities and improve your overall problem-solving skills.

As you engage in cognitive training exercises, you’ll experience improvements in processing speed (the ability to absorb and process information quickly), reasoning (logical thinking and decision-making), and working memory (the capacity to hold and manipulate information over short periods). These enhanced cognitive abilities directly contribute to more effective problem-solving skills.

By understanding the principles behind cognitive training and consistently practicing these types of exercises, you can enhance your problem-solving abilities and apply these skills in various aspects of your lives. But not before you learn how to identify problems, which is a key first step to finding effective solutions. 

The problem-solving process

Effective problem identification is a crucial first step in the problem-solving process. Here’s how to do it: 

• Define the Problem: Clearly articulating the issue at hand is essential for understanding its scope and complexity. So take time to describe the problem in detail, considering the context, constraints, and possible repercussions.
• Gather Information: Collect relevant data and information about the problem. This may involve research, consulting with experts, or seeking input from those affected by the issue. Having accurate and comprehensive information is critical for informed decision-making during the problem-solving process.
• Involve Others: Collaborate with your team or other people to ensure diverse perspectives and insights are considered. A good idea can come from everywhere, and a collective approach can lead to more innovative and effective solutions.
• Identify Root Causes: Once the problem is defined, delve deeper to identify its underlying causes. Use techniques such as the "5 Whys" method or cause-and-effect analysis to pinpoint the factors contributing to the issue. Addressing these root causes is crucial for developing long-term, sustainable solutions.
• Select a Problem-Solving Strategy: Employ various problem-solving methods to devise a solution that tackles the root causes effectively. These may include brainstorming, evaluating pros and cons, or implementing a trial-and-error approach. The strategy you ultimately choose should be adaptable and considerate of potential challenges or obstacles.

By following these tips for problem identification and employing problem-solving techniques, you can increase your chances of finding effective and lasting solutions to the issues you face.

6 ways to improve your critical thinking skills

Here’s the truth: You can’t effectively solve a problem without using your critical thinking skills. 

Critical thinking is the process of objectively analyzing information, evaluating the credibility of arguments, and making informed decisions based on logic and reasoning. It involves things like questioning assumptions, considering multiple perspectives, and weighing evidence before reaching a conclusion.

Think about it: Having the ability to analyze information, evaluate arguments, and make reasoned decisions allows you to approach problems logically —and we have a few tips to help you improve your ability to do just that: 

• Break Down Information: To sharpen your critical thinking abilities, practice breaking down complex information into smaller components. Identify patterns, relationships, and underlying principles that can help you better understand the situation.
• Evaluate Arguments: Develop the habit of assessing the credibility and relevance of arguments presented to you. Consider the source of the information, identify any potential biases, and scrutinize the validity of the evidence provided.
• Make Reasoned Decisions: When faced with a decision, take time to gather all relevant information and consider possible outcomes. Weigh the pros and cons before arriving at a well-reasoned conclusion that takes into account both short-term and long-term consequences. (We love a good pros and cons list.) 
• Play Brain Games: Regularly engaging in brain games such as Sudoku, crosswords, chess, or logic puzzles can be an effective way to enhance critical thinking skills. These games require you to analyze information, evaluate potential moves or solutions, and make strategic decisions based on reasoning. We’ll go into more detail about this later, so hang tight. 
• Try Mindfulness Meditation: Mindfulness meditation involves focusing on the present moment while calmly acknowledging and accepting your thoughts and feelings. Practicing mindfulness can enhance attention, concentration, and emotional regulation, all of which are critical for effective problem-solving. And if you're interested, you can try it for free for an entire year with the Balance app .
• Consider Cognitive-Behavioral Therapy (CBT): CBT is a psychotherapy technique that helps you identify and change negative thought patterns and behaviors. By learning to recognize unproductive thinking habits, you can develop more constructive approaches to problem-solving.

By exploring these various cognitive training techniques and consistently incorporating them into your daily life, you’ll be well on your way to enhancing your problem-solving skills and tackling life's challenges more logically and effectively.

How to approach problems with a critical mindset

Approaching problems with a critical mindset is a great way to turn critical thinking into a habit. But what does that mean, and how do you do it? Let’s break it down:  

• Embrace Critical Thinking: Develop the habit of questioning assumptions and challenging conventional wisdom when faced with a problem. This will help to uncover hidden biases or overlooked factors that may influence the issue at hand.
• Consider Multiple Perspectives: Explore different viewpoints and perspectives when assessing a problem. This allows for a more comprehensive understanding of the situation and can lead to innovative solutions that might not have been apparent from a single viewpoint.
• Evaluate Evidence: Gather relevant information and carefully evaluate its credibility and reliability. Assess the strength of the evidence supporting various arguments or positions before making a decision.

By following these tips, you can develop a critical mindset that habitually enables you to approach problems more effectively, leading to well-informed decisions and lasting solutions.

Problem solving methods and techniques

Now that you know a bit about how to approach a problem, here’s how you can implement these problem-solving techniques in your daily life:

• Understand the Context: When applying problem-solving techniques in different settings, it's essential to consider the unique context and constraints of each situation. The approach that works well in a professional environment may not be suitable for a personal issue, so tailor your strategies accordingly.
• Adapt and Be Flexible: Effective problem-solving requires adaptability and flexibility. Be open to changing your approach if circumstances shift or new information emerges. This willingness to adapt will help you find solutions that are relevant and sustainable in the long term.
• Communicate and Collaborate: In both workplace and personal settings, communication and collaboration are key to successful problem-solving. Share your thoughts, ideas, and concerns with team members or stakeholders, and actively seek their input. A diverse array of perspectives can lead to more innovative and effective solutions.
• Learn from Experience: Reflect on past problem-solving experiences and learn from both successes and failures. Apply these lessons to future situations to continuously improve your problem-solving skills.
• Practice Regularly: To develop strong problem-solving abilities, practice regularly by tackling problems in various aspects of your life. The more you practice, the more adept you'll become at identifying problems, generating solutions, and making well-informed decisions.

How to practice effective decision-making

By now, you know how to approach a problem. But how do you solve one? 

Effective decision-making skills are closely related to problem-solving skills, and the two can work together to help you achieve better results. So the next time you have to make a decision, give these steps a try: 

• Gather Information: Just as you need to gather information to understand a problem, you also need to gather information to make informed decisions. This may involve conducting research into various options, consulting with experts, or seeking input from those affected by the issue. Comprehensive and accurate information is crucial for evaluating potential solutions.
• Evaluate Options: Once you have gathered enough information, carefully assess the different options available to address the problem. Consider factors such as feasibility, impact, costs, and potential risks when weighing the pros and cons of each alternative.
• Make a Decision: After evaluating the options, select a solution based on the available information and your assessment of its effectiveness in addressing the root causes of the problem. Ensure that your chosen solution is sustainable in the long term and takes into account any potential challenges or obstacles that may arise.
• Monitor Outcomes: Track the outcomes of your decision to gauge its effectiveness and learn from the results. Be prepared to reassess and adjust your approach if necessary, based on feedback or changing circumstances.
• Refine Your Decision-Making Skills: Continuously work on improving your decision-making abilities by reflecting on past decisions, learning from both successes and failures, and seeking opportunities to practice these skills in various aspects of your life.

The result of putting this into action? Better outcomes and greater success. That’s a win-win if we ever saw one. 

Benefits of brain games for improving problem-solving skills

One fun way to improve all of these problem-solving and decision-making skills we’ve discussed is by playing brain games. 

Brain games stimulate your mind and foster the development of various cognitive abilities like processing speed, reasoning, and working memory, which are all essential for effective problem-solving. 

These games challenge you to think critically and make decisions based on logic and strategy. And as a result, they help cultivate a more agile and adaptable mindset that is valuable for tackling real-life problems. (Did we mention they’re also fun?) 

One popular brain training app that incorporates a wide variety of games is Elevate. 

With more than 40 games spread across math , reading , writing , speaking , and memory skills , the Elevate app offers personalized training programs based on your goals, and it adapts to your skill level and performance over time. 

By incorporating brain games into daily routines or cognitive training programs, you’ll be able to make big improvements in your critical thinking and problem-solving skills, making it easier to tackle challenges in both personal and professional aspects of your life. Oh, and did we mention they’re also fun to play?

Improve your problem-solving skills with learning games for adults

By knowing how to identify a problem, approach it with a critical mindset, and implement a few key problem-solving techniques, you’ll be able to tackle your next challenge with ease. 

And if you’re ready to up-level your overall problem-solving skills with the help of brain training games, download the Elevate app on iOS or Android today and discover 40+ brain training games, personalized training programs, and expert guidance to help you optimize your cognitive abilities and improve your overall performance in daily life. 

With the Elevate app, you can take control of your cognitive function and become a more effective problem solver. It’s what we like to call a no-brainer decision! 

Enhancing your cognitive abilities

• Learn what brain training is, its benefits, and how you can easily get started training your brain. 

The science of cognitive training

• Training cognitive skills can improve brain function. Think of it like a workout for your mind. Read on to learn how brain games can help. 

The importance of mental fitness

• Mental fitness refers to your ability to sustain your overall well-being. Learn tips to improve yours. 

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Struggling to overcome challenges in your life? We all face problems, big and small, on a regular basis.

So how do you tackle them effectively? What are some key problem-solving strategies and skills that can guide you?

Effective problem-solving requires breaking issues down logically, generating solutions creatively, weighing choices critically, and adapting plans flexibly based on outcomes. Useful strategies range from leveraging past solutions that have worked to visualizing problems through diagrams. Core skills include analytical abilities, innovative thinking, and collaboration.

Want to improve your problem-solving skills? Keep reading to find out 17 effective problem-solving strategies, key skills, common obstacles to watch for, and tips on improving your overall problem-solving skills.

Key Takeaways:

• Effective problem-solving requires breaking down issues logically, generating multiple solutions creatively, weighing choices critically, and adapting plans based on outcomes.
• Useful problem-solving strategies range from leveraging past solutions to brainstorming with groups to visualizing problems through diagrams and models.
• Core skills include analytical abilities, innovative thinking, decision-making, and team collaboration to solve problems.
• Common obstacles include fear of failure, information gaps, fixed mindsets, confirmation bias, and groupthink.
• Boosting problem-solving skills involves learning from experts, actively practicing, soliciting feedback, and analyzing others’ success.
• Onethread’s project management capabilities align with effective problem-solving tenets – facilitating structured solutions, tracking progress, and capturing lessons learned.

What Is Problem-Solving?

Problem-solving is the process of understanding an issue, situation, or challenge that needs to be addressed and then systematically working through possible solutions to arrive at the best outcome.

It involves critical thinking, analysis, logic, creativity, research, planning, reflection, and patience in order to overcome obstacles and find effective answers to complex questions or problems.

The ultimate goal is to implement the chosen solution successfully.

What Are Problem-Solving Strategies?

Problem-solving strategies are like frameworks or methodologies that help us solve tricky puzzles or problems we face in the workplace, at home, or with friends.

Imagine you have a big jigsaw puzzle. One strategy might be to start with the corner pieces. Another could be looking for pieces with the same colors. 

Just like in puzzles, in real life, we use different plans or steps to find solutions to problems. These strategies help us think clearly, make good choices, and find the best answers without getting too stressed or giving up.

Why Is It Important To Know Different Problem-Solving Strategies?

Knowing different problem-solving strategies is important because different types of problems often require different approaches to solve them effectively. Having a variety of strategies to choose from allows you to select the best method for the specific problem you are trying to solve.

This improves your ability to analyze issues thoroughly, develop solutions creatively, and tackle problems from multiple angles. Knowing multiple strategies also aids in overcoming roadblocks if your initial approach is not working.

Here are some reasons why you need to know different problem-solving strategies:

• Different Problems Require Different Tools: Just like you can’t use a hammer to fix everything, some problems need specific strategies to solve them.
• Improves Creativity: Knowing various strategies helps you think outside the box and come up with creative solutions.
• Saves Time: With the right strategy, you can solve problems faster instead of trying things that don’t work.
• Reduces Stress: When you know how to tackle a problem, it feels less scary and you feel more confident.
• Better Outcomes: Using the right strategy can lead to better solutions, making things work out better in the end.
• Learning and Growth: Each time you solve a problem, you learn something new, which makes you smarter and better at solving future problems.

Knowing different ways to solve problems helps you tackle anything that comes your way, making life a bit easier and more fun!

17 Effective Problem-Solving Strategies

Effective problem-solving strategies include breaking the problem into smaller parts, brainstorming multiple solutions, evaluating the pros and cons of each, and choosing the most viable option. 

Critical thinking and creativity are essential in developing innovative solutions. Collaboration with others can also provide diverse perspectives and ideas. 

By applying these strategies, you can tackle complex issues more effectively.

Now, consider a challenge you’re dealing with. Which strategy could help you find a solution? Here we will discuss key problem strategies in detail.

1. Use a Past Solution That Worked

This strategy involves looking back at previous similar problems you have faced and the solutions that were effective in solving them.

It is useful when you are facing a problem that is very similar to something you have already solved. The main benefit is that you don’t have to come up with a brand new solution – you already know the method that worked before will likely work again.

However, the limitation is that the current problem may have some unique aspects or differences that mean your old solution is not fully applicable.

The ideal process is to thoroughly analyze the new challenge, identify the key similarities and differences versus the past case, adapt the old solution as needed to align with the current context, and then pilot it carefully before full implementation.

An example is using the same negotiation tactics from purchasing your previous home when putting in an offer on a new house. Key terms would be adjusted but overall it can save significant time versus developing a brand new strategy.

2. Brainstorm Solutions

This involves gathering a group of people together to generate as many potential solutions to a problem as possible.

It is effective when you need creative ideas to solve a complex or challenging issue. By getting input from multiple people with diverse perspectives, you increase the likelihood of finding an innovative solution.

The main limitation is that brainstorming sessions can sometimes turn into unproductive gripe sessions or discussions rather than focusing on productive ideation —so they need to be properly facilitated.

The key to an effective brainstorming session is setting some basic ground rules upfront and having an experienced facilitator guide the discussion. Rules often include encouraging wild ideas, avoiding criticism of ideas during the ideation phase, and building on others’ ideas.

For instance, a struggling startup might hold a session where ideas for turnaround plans are generated and then formalized with financials and metrics.

3. Work Backward from the Solution

This technique involves envisioning that the problem has already been solved and then working step-by-step backward toward the current state.

This strategy is particularly helpful for long-term, multi-step problems. By starting from the imagined solution and identifying all the steps required to reach it, you can systematically determine the actions needed. It lets you tackle a big hairy problem through smaller, reversible steps.

A limitation is that this approach may not be possible if you cannot accurately envision the solution state to start with.

The approach helps drive logical systematic thinking for complex problem-solving, but should still be combined with creative brainstorming of alternative scenarios and solutions.

An example is planning for an event – you would imagine the successful event occurring, then determine the tasks needed the week before, two weeks before, etc. all the way back to the present.

4. Use the Kipling Method

This method, named after author Rudyard Kipling, provides a framework for thoroughly analyzing a problem before jumping into solutions.

It consists of answering six fundamental questions: What, Where, When, How, Who, and Why about the challenge. Clearly defining these core elements of the problem sets the stage for generating targeted solutions.

The Kipling method enables a deep understanding of problem parameters and root causes before solution identification. By jumping to brainstorm solutions too early, critical information can be missed or the problem is loosely defined, reducing solution quality.

Answering the six fundamental questions illuminates all angles of the issue. This takes time but pays dividends in generating optimal solutions later tuned precisely to the true underlying problem.

The limitation is that meticulously working through numerous questions before addressing solutions can slow progress.

The best approach blends structured problem decomposition techniques like the Kipling method with spurring innovative solution ideation from a diverse team. 

An example is using this technique after a technical process failure – the team would systematically detail What failed, Where/When did it fail, How it failed (sequence of events), Who was involved, and Why it likely failed before exploring preventative solutions.

5. Try Different Solutions Until One Works (Trial and Error)

This technique involves attempting various potential solutions sequentially until finding one that successfully solves the problem.

Trial and error works best when facing a concrete, bounded challenge with clear solution criteria and a small number of discrete options to try. By methodically testing solutions, you can determine the faulty component.

A limitation is that it can be time-intensive if the working solution set is large.

The key is limiting the variable set first. For technical problems, this boundary is inherent and each element can be iteratively tested. But for business issues, artificial constraints may be required – setting decision rules upfront to reduce options before testing.

Furthermore, hypothesis-driven experimentation is far superior to blind trial and error – have logic for why Option A may outperform Option B.

Examples include fixing printer jams by testing different paper tray and cable configurations or resolving website errors by tweaking CSS/HTML line-by-line until the code functions properly.

6. Use Proven Formulas or Frameworks (Heuristics)

Heuristics refers to applying existing problem-solving formulas or frameworks rather than addressing issues completely from scratch.

This allows leveraging established best practices rather than reinventing the wheel each time.

It is effective when facing recurrent, common challenges where proven structured approaches exist.

However, heuristics may force-fit solutions to non-standard problems.

For example, a cost-benefit analysis can be used instead of custom weighting schemes to analyze potential process improvements.

Onethread allows teams to define, save, and replicate configurable project templates so proven workflows can be reliably applied across problems with some consistency rather than fully custom one-off approaches each time.

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7. Trust Your Instincts (Insight Problem-Solving)

Insight is a problem-solving technique that involves waiting patiently for an unexpected “aha moment” when the solution pops into your mind.

It works well for personal challenges that require intuitive realizations over calculated logic. The unconscious mind makes connections leading to flashes of insight when relaxing or doing mundane tasks unrelated to the actual problem.

Benefits include out-of-the-box creative solutions. However, the limitations are that insights can’t be forced and may never come at all if too complex. Critical analysis is still required after initial insights.

A real-life example would be a writer struggling with how to end a novel. Despite extensive brainstorming, they feel stuck. Eventually while gardening one day, a perfect unexpected plot twist sparks an ideal conclusion. However, once written they still carefully review if the ending flows logically from the rest of the story.

8. Reverse Engineer the Problem

This approach involves deconstructing a problem in reverse sequential order from the current undesirable outcome back to the initial root causes.

By mapping the chain of events backward, you can identify the origin of where things went wrong and establish the critical junctures for solving it moving ahead. Reverse engineering provides diagnostic clarity on multi-step problems.

However, the limitation is that it focuses heavily on autopsying the past versus innovating improved future solutions.

An example is tracing back from a server outage, through the cascade of infrastructure failures that led to it finally terminating at the initial script error that triggered the crisis. This root cause would then inform the preventative measure.

9. Break Down Obstacles Between Current and Goal State (Means-End Analysis)

This technique defines the current problem state and the desired end goal state, then systematically identifies obstacles in the way of getting from one to the other.

By mapping the barriers or gaps, you can then develop solutions to address each one. This methodically connects the problem to solutions.

A limitation is that some obstacles may be unknown upfront and only emerge later.

For example, you can list down all the steps required for a new product launch – current state through production, marketing, sales, distribution, etc. to full launch (goal state) – to highlight where resource constraints or other blocks exist so they can be addressed.

Onethread allows dividing big-picture projects into discrete, manageable phases, milestones, and tasks to simplify execution just as problems can be decomposed into more achievable components. Features like dependency mapping further reinforce interconnections.

Using Onethread’s issues and subtasks feature, messy problems can be decomposed into manageable chunks.

10. Ask “Why” Five Times to Identify the Root Cause (The 5 Whys)

This technique involves asking “Why did this problem occur?” and then responding with an answer that is again met with asking “Why?” This process repeats five times until the root cause is revealed.

Continually asking why digs deeper from surface symptoms to underlying systemic issues.

It is effective for getting to the source of problems originating from human error or process breakdowns.

However, some complex issues may have multiple tangled root causes not solvable through this approach alone.

An example is a retail store experiencing a sudden decline in customers. Successively asking why five times may trace an initial drop to parking challenges, stemming from a city construction project – the true starting point to address.

11. Evaluate Strengths, Weaknesses, Opportunities, and Threats (SWOT Analysis)

This involves analyzing a problem or proposed solution by categorizing internal and external factors into a 2×2 matrix: Strengths, Weaknesses as the internal rows; Opportunities and Threats as the external columns.

Systematically identifying these elements provides balanced insight to evaluate options and risks. It is impactful when evaluating alternative solutions or developing strategy amid complexity or uncertainty.

The key benefit of SWOT analysis is enabling multi-dimensional thinking when rationally evaluating options. Rather than getting anchored on just the upsides or the existing way of operating, it urges a systematic assessment through four different lenses:

• Internal Strengths: Our core competencies/advantages able to deliver success
• Internal Weaknesses: Gaps/vulnerabilities we need to manage
• External Opportunities: Ways we can differentiate/drive additional value
• External Threats: Risks we must navigate or mitigate

Multiperspective analysis provides the needed holistic view of the balanced risk vs. reward equation for strategic decision making amid uncertainty.

However, SWOT can feel restrictive if not tailored and evolved for different issue types.

Teams should view SWOT analysis as a starting point, augmenting it further for distinct scenarios.

An example is performing a SWOT analysis on whether a small business should expand into a new market – evaluating internal capabilities to execute vs. risks in the external competitive and demand environment to inform the growth decision with eyes wide open.

12. Compare Current vs Expected Performance (Gap Analysis)

This technique involves comparing the current state of performance, output, or results to the desired or expected levels to highlight shortfalls.

By quantifying the gaps, you can identify problem areas and prioritize address solutions.

Gap analysis is based on the simple principle – “you can’t improve what you don’t measure.” It enables facts-driven problem diagnosis by highlighting delta to goals, not just vague dissatisfaction that something seems wrong. And measurement immediately suggests improvement opportunities – address the biggest gaps first.

This data orientation also supports ROI analysis on fixing issues – the return from closing larger gaps outweighs narrowly targeting smaller performance deficiencies.

However, the approach is only effective if robust standards and metrics exist as the benchmark to evaluate against. Organizations should invest upfront in establishing performance frameworks.

Furthermore, while numbers are invaluable, the human context behind problems should not be ignored – quantitative versus qualitative gap assessment is optimally blended.

For example, if usage declines are noted during software gap analysis, this could be used as a signal to improve user experience through design.

13. Observe Processes from the Frontline (Gemba Walk)

A Gemba walk involves going to the actual place where work is done, directly observing the process, engaging with employees, and finding areas for improvement.

By experiencing firsthand rather than solely reviewing abstract reports, practical problems and ideas emerge.

The limitation is Gemba walks provide anecdotes not statistically significant data. It complements but does not replace comprehensive performance measurement.

An example is a factory manager inspecting the production line to spot jam areas based on direct reality rather than relying on throughput dashboards alone back in her office. Frontline insights prove invaluable.

14. Analyze Competitive Forces (Porter’s Five Forces)

This involves assessing the marketplace around a problem or business situation via five key factors: competitors, new entrants, substitute offerings, suppliers, and customer power.

Evaluating these forces illuminates risks and opportunities for strategy development and issue resolution. It is effective for understanding dynamic external threats and opportunities when operating in a contested space.

However, over-indexing on only external factors can overlook the internal capabilities needed to execute solutions.

A startup CEO, for example, may analyze market entry barriers, whitespace opportunities, and disruption risks across these five forces to shape new product rollout strategies and marketing approaches.

15. Think from Different Perspectives (Six Thinking Hats)

The Six Thinking Hats is a technique developed by Edward de Bono that encourages people to think about a problem from six different perspectives, each represented by a colored “thinking hat.”

The key benefit of this strategy is that it pushes team members to move outside their usual thinking style and consider new angles. This brings more diverse ideas and solutions to the table.

It works best for complex problems that require innovative solutions and when a team is stuck in an unproductive debate. The structured framework keeps the conversation flowing in a positive direction.

Limitations are that it requires training on the method itself and may feel unnatural at first. Team dynamics can also influence success – some members may dominate certain “hats” while others remain quiet.

A real-life example is a software company debating whether to build a new feature. The white hat focuses on facts, red on gut feelings, black on potential risks, yellow on benefits, green on new ideas, and blue on process. This exposes more balanced perspectives before deciding.

Onethread centralizes diverse stakeholder communication onto one platform, ensuring all voices are incorporated when evaluating project tradeoffs, just as problem-solving should consider multifaceted solutions.

16. Visualize the Problem (Draw it Out)

Drawing out a problem involves creating visual representations like diagrams, flowcharts, and maps to work through challenging issues.

This strategy is helpful when dealing with complex situations with lots of interconnected components. The visuals simplify the complexity so you can thoroughly understand the problem and all its nuances.

Key benefits are that it allows more stakeholders to get on the same page regarding root causes and it sparks new creative solutions as connections are made visually.

However, simple problems with few variables don’t require extensive diagrams. Additionally, some challenges are so multidimensional that fully capturing every aspect is difficult.

A real-life example would be mapping out all the possible causes leading to decreased client satisfaction at a law firm. An intricate fishbone diagram with branches for issues like service delivery, technology, facilities, culture, and vendor partnerships allows the team to trace problems back to their origins and brainstorm targeted fixes.

17. Follow a Step-by-Step Procedure (Algorithms)

An algorithm is a predefined step-by-step process that is guaranteed to produce the correct solution if implemented properly.

Using algorithms is effective when facing problems that have clear, binary right and wrong answers. Algorithms work for mathematical calculations, computer code, manufacturing assembly lines, and scientific experiments.

Key benefits are consistency, accuracy, and efficiency. However, they require extensive upfront development and only apply to scenarios with strict parameters. Additionally, human error can lead to mistakes.

For example, crew members of fast food chains like McDonald’s follow specific algorithms for food prep – from grill times to ingredient amounts in sandwiches, to order fulfillment procedures. This ensures uniform quality and service across all locations. However, if a step is missed, errors occur.

The Problem-Solving Process

The problem-solving process typically includes defining the issue, analyzing details, creating solutions, weighing choices, acting, and reviewing results.

In the above, we have discussed several problem-solving strategies. For every problem-solving strategy, you have to follow these processes. Here’s a detailed step-by-step process of effective problem-solving:

Step 1: Identify the Problem

The problem-solving process starts with identifying the problem. This step involves understanding the issue’s nature, its scope, and its impact. Once the problem is clearly defined, it sets the foundation for finding effective solutions.

Identifying the problem is crucial. It means figuring out exactly what needs fixing. This involves looking at the situation closely, understanding what’s wrong, and knowing how it affects things. It’s about asking the right questions to get a clear picture of the issue. 

This step is important because it guides the rest of the problem-solving process. Without a clear understanding of the problem, finding a solution is much harder. It’s like diagnosing an illness before treating it. Once the problem is identified accurately, you can move on to exploring possible solutions and deciding on the best course of action.

Step 2: Break Down the Problem

Breaking down the problem is a key step in the problem-solving process. It involves dividing the main issue into smaller, more manageable parts. This makes it easier to understand and tackle each component one by one.

After identifying the problem, the next step is to break it down. This means splitting the big issue into smaller pieces. It’s like solving a puzzle by handling one piece at a time. 

By doing this, you can focus on each part without feeling overwhelmed. It also helps in identifying the root causes of the problem. Breaking down the problem allows for a clearer analysis and makes finding solutions more straightforward. 

Each smaller problem can be addressed individually, leading to an effective resolution of the overall issue. This approach not only simplifies complex problems but also aids in developing a systematic plan to solve them.

Step 3: Come up with potential solutions

Coming up with potential solutions is the third step in the problem-solving process. It involves brainstorming various options to address the problem, considering creativity and feasibility to find the best approach.

After breaking down the problem, it’s time to think of ways to solve it. This stage is about brainstorming different solutions. You look at the smaller issues you’ve identified and start thinking of ways to fix them. This is where creativity comes in. 

You want to come up with as many ideas as possible, no matter how out-of-the-box they seem. It’s important to consider all options and evaluate their pros and cons. This process allows you to gather a range of possible solutions. 

Later, you can narrow these down to the most practical and effective ones. This step is crucial because it sets the stage for deciding on the best solution to implement. It’s about being open-minded and innovative to tackle the problem effectively.

Step 4: Analyze the possible solutions

Analyzing the possible solutions is the fourth step in the problem-solving process. It involves evaluating each proposed solution’s advantages and disadvantages to determine the most effective and feasible option.

After coming up with potential solutions, the next step is to analyze them. This means looking closely at each idea to see how well it solves the problem. You weigh the pros and cons of every solution.

Consider factors like cost, time, resources, and potential outcomes. This analysis helps in understanding the implications of each option. It’s about being critical and objective, ensuring that the chosen solution is not only effective but also practical.

This step is vital because it guides you towards making an informed decision. It involves comparing the solutions against each other and selecting the one that best addresses the problem.

By thoroughly analyzing the options, you can move forward with confidence, knowing you’ve chosen the best path to solve the issue.

Step 5: Implement and Monitor the Solutions

Implementing and monitoring the solutions is the final step in the problem-solving process. It involves putting the chosen solution into action and observing its effectiveness, making adjustments as necessary.

Once you’ve selected the best solution, it’s time to put it into practice. This step is about action. You implement the chosen solution and then keep an eye on how it works. Monitoring is crucial because it tells you if the solution is solving the problem as expected. 

If things don’t go as planned, you may need to make some changes. This could mean tweaking the current solution or trying a different one. The goal is to ensure the problem is fully resolved. 

This step is critical because it involves real-world application. It’s not just about planning; it’s about doing and adjusting based on results. By effectively implementing and monitoring the solutions, you can achieve the desired outcome and solve the problem successfully.

Why This Process is Important

Following a defined process to solve problems is important because it provides a systematic, structured approach instead of a haphazard one. Having clear steps guides logical thinking, analysis, and decision-making to increase effectiveness. Key reasons it helps are:

• Clear Direction: This process gives you a clear path to follow, which can make solving problems less overwhelming.
• Better Solutions: Thoughtful analysis of root causes, iterative testing of solutions, and learning orientation lead to addressing the heart of issues rather than just symptoms.
• Saves Time and Energy: Instead of guessing or trying random things, this process helps you find a solution more efficiently.
• Improves Skills: The more you use this process, the better you get at solving problems. It’s like practicing a sport. The more you practice, the better you play.
• Maximizes collaboration: Involving various stakeholders in the process enables broader inputs. Their communication and coordination are streamlined through organized brainstorming and evaluation.
• Provides consistency: Standard methodology across problems enables building institutional problem-solving capabilities over time. Patterns emerge on effective techniques to apply to different situations.

The problem-solving process is a powerful tool that can help us tackle any challenge we face. By following these steps, we can find solutions that work and learn important skills along the way.

Key Skills for Efficient Problem Solving

Efficient problem-solving requires breaking down issues logically, evaluating options, and implementing practical solutions.

Key skills include critical thinking to understand root causes, creativity to brainstorm innovative ideas, communication abilities to collaborate with others, and decision-making to select the best way forward. Staying adaptable, reflecting on outcomes, and applying lessons learned are also essential.

With practice, these capacities will lead to increased personal and team effectiveness in systematically addressing any problem.

 Let’s explore the powers you need to become a problem-solving hero!

Critical Thinking and Analytical Skills

Critical thinking and analytical skills are vital for efficient problem-solving as they enable individuals to objectively evaluate information, identify key issues, and generate effective solutions. 

These skills facilitate a deeper understanding of problems, leading to logical, well-reasoned decisions. By systematically breaking down complex issues and considering various perspectives, individuals can develop more innovative and practical solutions, enhancing their problem-solving effectiveness.

Communication Skills

Effective communication skills are essential for efficient problem-solving as they facilitate clear sharing of information, ensuring all team members understand the problem and proposed solutions. 

These skills enable individuals to articulate issues, listen actively, and collaborate effectively, fostering a productive environment where diverse ideas can be exchanged and refined. By enhancing mutual understanding, communication skills contribute significantly to identifying and implementing the most viable solutions.

Decision-Making

Strong decision-making skills are crucial for efficient problem-solving, as they enable individuals to choose the best course of action from multiple alternatives. 

These skills involve evaluating the potential outcomes of different solutions, considering the risks and benefits, and making informed choices. Effective decision-making leads to the implementation of solutions that are likely to resolve problems effectively, ensuring resources are used efficiently and goals are achieved.

Planning and Prioritization

Planning and prioritization are key for efficient problem-solving, ensuring resources are allocated effectively to address the most critical issues first. This approach helps in organizing tasks according to their urgency and impact, streamlining efforts towards achieving the desired outcome efficiently.

Emotional Intelligence

Emotional intelligence enhances problem-solving by allowing individuals to manage emotions, understand others, and navigate social complexities. It fosters a positive, collaborative environment, essential for generating creative solutions and making informed, empathetic decisions.

Leadership skills drive efficient problem-solving by inspiring and guiding teams toward common goals. Effective leaders motivate their teams, foster innovation, and navigate challenges, ensuring collective efforts are focused and productive in addressing problems.

Time Management

Time management is crucial in problem-solving, enabling individuals to allocate appropriate time to each task. By efficiently managing time, one can ensure that critical problems are addressed promptly without neglecting other responsibilities.

Data Analysis

Data analysis skills are essential for problem-solving, as they enable individuals to sift through data, identify trends, and extract actionable insights. This analytical approach supports evidence-based decision-making, leading to more accurate and effective solutions.

Research Skills

Research skills are vital for efficient problem-solving, allowing individuals to gather relevant information, explore various solutions, and understand the problem’s context. This thorough exploration aids in developing well-informed, innovative solutions.

Becoming a great problem solver takes practice, but with these skills, you’re on your way to becoming a problem-solving hero. 

How to Improve Your Problem-Solving Skills?

Improving your problem-solving skills can make you a master at overcoming challenges. Learn from experts, practice regularly, welcome feedback, try new methods, experiment, and study others’ success to become better.

Learning from Experts

Improving problem-solving skills by learning from experts involves seeking mentorship, attending workshops, and studying case studies. Experts provide insights and techniques that refine your approach, enhancing your ability to tackle complex problems effectively.

To enhance your problem-solving skills, learning from experts can be incredibly beneficial. Engaging with mentors, participating in specialized workshops, and analyzing case studies from seasoned professionals can offer valuable perspectives and strategies. 

Experts share their experiences, mistakes, and successes, providing practical knowledge that can be applied to your own problem-solving process. This exposure not only broadens your understanding but also introduces you to diverse methods and approaches, enabling you to tackle challenges more efficiently and creatively.

Improving problem-solving skills through practice involves tackling a variety of challenges regularly. This hands-on approach helps in refining techniques and strategies, making you more adept at identifying and solving problems efficiently.

One of the most effective ways to enhance your problem-solving skills is through consistent practice. By engaging with different types of problems on a regular basis, you develop a deeper understanding of various strategies and how they can be applied. 

This hands-on experience allows you to experiment with different approaches, learn from mistakes, and build confidence in your ability to tackle challenges.

Regular practice not only sharpens your analytical and critical thinking skills but also encourages adaptability and innovation, key components of effective problem-solving.

Openness to Feedback

Being open to feedback is like unlocking a secret level in a game. It helps you boost your problem-solving skills. Improving problem-solving skills through openness to feedback involves actively seeking and constructively responding to critiques. 

This receptivity enables you to refine your strategies and approaches based on insights from others, leading to more effective solutions. 

Learning New Approaches and Methodologies

Learning new approaches and methodologies is like adding new tools to your toolbox. It makes you a smarter problem-solver. Enhancing problem-solving skills by learning new approaches and methodologies involves staying updated with the latest trends and techniques in your field. 

This continuous learning expands your toolkit, enabling innovative solutions and a fresh perspective on challenges.

Experimentation

Experimentation is like being a scientist of your own problems. It’s a powerful way to improve your problem-solving skills. Boosting problem-solving skills through experimentation means trying out different solutions to see what works best. This trial-and-error approach fosters creativity and can lead to unique solutions that wouldn’t have been considered otherwise.

Analyzing Competitors’ Success

Analyzing competitors’ success is like being a detective. It’s a smart way to boost your problem-solving skills. Improving problem-solving skills by analyzing competitors’ success involves studying their strategies and outcomes. Understanding what worked for them can provide valuable insights and inspire effective solutions for your own challenges. 

Challenges in Problem-Solving

Facing obstacles when solving problems is common. Recognizing these barriers, like fear of failure or lack of information, helps us find ways around them for better solutions.

Fear of Failure

Fear of failure is like a big, scary monster that stops us from solving problems. It’s a challenge many face. Because being afraid of making mistakes can make us too scared to try new solutions. 

How can we overcome this? First, understand that it’s okay to fail. Failure is not the opposite of success; it’s part of learning. Every time we fail, we discover one more way not to solve a problem, getting us closer to the right solution. Treat each attempt like an experiment. It’s not about failing; it’s about testing and learning.

Lack of Information

Lack of information is like trying to solve a puzzle with missing pieces. It’s a big challenge in problem-solving. Because without all the necessary details, finding a solution is much harder. 

How can we fix this? Start by gathering as much information as you can. Ask questions, do research, or talk to experts. Think of yourself as a detective looking for clues. The more information you collect, the clearer the picture becomes. Then, use what you’ve learned to think of solutions. 

Fixed Mindset

A fixed mindset is like being stuck in quicksand; it makes solving problems harder. It means thinking you can’t improve or learn new ways to solve issues. 

How can we change this? First, believe that you can grow and learn from challenges. Think of your brain as a muscle that gets stronger every time you use it. When you face a problem, instead of saying “I can’t do this,” try thinking, “I can’t do this yet.” Look for lessons in every challenge and celebrate small wins. 

Everyone starts somewhere, and mistakes are just steps on the path to getting better. By shifting to a growth mindset, you’ll see problems as opportunities to grow. Keep trying, keep learning, and your problem-solving skills will soar!

Jumping to Conclusions

Jumping to conclusions is like trying to finish a race before it starts. It’s a challenge in problem-solving. That means making a decision too quickly without looking at all the facts. 

How can we avoid this? First, take a deep breath and slow down. Think about the problem like a puzzle. You need to see all the pieces before you know where they go. Ask questions, gather information, and consider different possibilities. Don’t choose the first solution that comes to mind. Instead, compare a few options. 

Feeling Overwhelmed

Feeling overwhelmed is like being buried under a mountain of puzzles. It’s a big challenge in problem-solving. When we’re overwhelmed, everything seems too hard to handle. 

How can we deal with this? Start by taking a step back. Breathe deeply and focus on one thing at a time. Break the big problem into smaller pieces, like sorting puzzle pieces by color. Tackle each small piece one by one. It’s also okay to ask for help. Sometimes, talking to someone else can give you a new perspective. 

Confirmation Bias

Confirmation bias is like wearing glasses that only let you see what you want to see. It’s a challenge in problem-solving. Because it makes us focus only on information that agrees with what we already believe, ignoring anything that doesn’t. 

How can we overcome this? First, be aware that you might be doing it. It’s like checking if your glasses are on right. Then, purposely look for information that challenges your views. It’s like trying on a different pair of glasses to see a new perspective. Ask questions and listen to answers, even if they don’t fit what you thought before.

Groupthink is like everyone in a group deciding to wear the same outfit without asking why. It’s a challenge in problem-solving. It means making decisions just because everyone else agrees, without really thinking it through. 

How can we avoid this? First, encourage everyone in the group to share their ideas, even if they’re different. It’s like inviting everyone to show their unique style of clothes. 

Listen to all opinions and discuss them. It’s okay to disagree; it helps us think of better solutions. Also, sometimes, ask someone outside the group for their thoughts. They might see something everyone in the group missed.

Overcoming obstacles in problem-solving requires patience, openness, and a willingness to learn from mistakes. By recognizing these barriers, we can develop strategies to navigate around them, leading to more effective and creative solutions.

What are the most common problem-solving techniques?

The most common techniques include brainstorming, the 5 Whys, mind mapping, SWOT analysis, and using algorithms or heuristics. Each approach has its strengths, suitable for different types of problems.

What’s the best problem-solving strategy for every situation?

There’s no one-size-fits-all strategy. The best approach depends on the problem’s complexity, available resources, and time constraints. Combining multiple techniques often yields the best results.

How can I improve my problem-solving skills?

Improve your problem-solving skills by practicing regularly, learning from experts, staying open to feedback, and continuously updating your knowledge on new approaches and methodologies.

Are there any tools or resources to help with problem-solving?

Yes, tools like mind mapping software, online courses on critical thinking, and books on problem-solving techniques can be very helpful. Joining forums or groups focused on problem-solving can also provide support and insights.

What are some common mistakes people make when solving problems?

Common mistakes include jumping to conclusions without fully understanding the problem, ignoring valuable feedback, sticking to familiar solutions without considering alternatives, and not breaking down complex problems into manageable parts.

Final Words

Mastering problem-solving strategies equips us with the tools to tackle challenges across all areas of life. By understanding and applying these techniques, embracing a growth mindset, and learning from both successes and obstacles, we can transform problems into opportunities for growth. Continuously improving these skills ensures we’re prepared to face and solve future challenges more effectively.

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How to Improve Problem Solving Skills [10 Ways]

While it might seem like some people are just born with stronger problem-solving skills, there are strategies that anyone can use to improve them.

That’s right, it’s possible to significantly enhance your abilities in this area — and the best part is, most of these activities are also pretty fun!

What Are Problem Solving Skills?

Before we get to the engaging activities, let’s refine our understanding of problem-solving skills, which are any techniques that help you consistently:

• Understand the causes of problems
• Overcome short-term crises
• Create strategies to solve longer-term problems
• Turn problems into opportunities

What Problem Solving Skills Should I Have?

You’ll be able to solve problems in your role better as you grow in your industry-specific knowledge. But there are also a few universal problem solving skills we all need:

• Defining the Problem: Deeply understanding a problem through research , leading to better solutions. Research can include interviewing, reading books and emails, analyzing financial data, searching your organization’s intranet, and organizing your findings.
• Brainstorming: Creating a myriad of new solutions quickly. In group brainstorms, allow everyone to state ideas. Appreciate all input, and avoid criticism. Then, organize solutions into groups around common themes.
• Analyzing: Using disciplined thought processes to evaluate each possible solution. Besides listing their costs and benefits, you might apply deductive reasoning, game theory, and the rules of logic (including fallacies) to them.
• Managing Risk: Anticipating and trying to avoid the downsides of key solutions. Your team can list potential risks, rate how likely each is, predict a date by which each might either happen or no longer be an issue, and devise ways to reduce those risks.
• Deciding: The ability to decide on a solution and move forward with it. After an appropriate amount of time, an analysis of possible solutions, and feedback from team members, a designated decider must choose and implement a solution.
• Managing Emotions: Applying emotional intelligence in order to improve your and your team members’ ability to think clearly. This requires you to recognize emotions in yourself and others, manage feelings, and channel emotions into useful work.

10 Exciting Ways to Improve Problem Solving Skills

Use these ten creative ways to improve problem solving skills, develop more strategic ways of thinking , and train your brain to do more.

1. Dance Your Heart Out

Did you know that dancing has a positive impact on neural processing, possibly developing new neural pathways to go around dopamine-depleted blockages in the brain?

This means that if you engage in ballet or another form of structured dance, doing so may facilitate convergent thinking . In other words, it may help you find a single, appropriate answer to a problem. If you need help with divergent thinking (finding multiple answers to a problem), engaging in more improvised types of dance such as hip-hop or tap might just do the trick.

2. Work out Your Brain with Logic Puzzles or Games

The winning strategy when playing chess, Sudoku, a Rubik’s Cube, or other brain-boosting games is actually to work the problem backward, not forward. The same strategy can apply to realistic strategic-thinking situations.

To build up your brain muscle and develop new problem-solving techniques, practice some logic puzzles and other games .

3. Get a Good Night’s Sleep

More than any other sleeping or awake state, Rapid Eye Movement (REM) sleep directly enhances creative processing in the brain. REM sleep helps “stimulate associative networks, allowing the brain to make new and useful associations between unrelated ideas” and are “not due to selective memory enhancements” such as memory consolidation, which occurs when awake.

4. Work out to Some Tunes

A study of cardiac rehabilitation patients tested verbal fluency after exercising with and without music. Results showed that when they listened to music while working out, participants more than doubled their scores on verbal fluency tests in contrast to when they worked out in silence. According to the study’s lead author, “The combination of music and exercise may stimulate and increase cognitive arousal while helping to organize the cognitive output.”

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5. keep an “idea journal” with you, 6. participate in yoga.

The powerful combination of body awareness, breathing, and meditation that is required during yoga practice has been shown to significantly raise cognitive test scores. Other results from a University of Illinois study include shorter reaction times, more accuracy, and increased attention.

7. Eat Some Cheerios (And Then Think About It)

The Cheerios Effect is the name physicists have given to the event that happens when the last few cheerios in a bowl always cling to each other. The cause of this occurrence is surface tension.

The takeaway is that when it comes to experiencing tension while trying to solve a problem, cling to those around you. Rely on others’ experiences and ideas, even those from different career fields. Draw connections. Brainstorm. Work together to get the job done.

8. Use Mind Maps to Help Visualize the Problem

Mind Maps , a visual snapshot of a problem and its possible solutions, can help focus the mind, stimulate the brain, increase the capacity for creative thinking, and generate more ideas for solutions.

Make a Mind Map by drawing your problem as the central idea. Add “main branches” consisting of all the reasons for the problem. Use “sub-branches” to explore further details.

Next, make a separate Mind Map of all possible solutions to the central problem. Add “main branches” showing all the ways that your problem can be solved, such as colleagues that can help, techniques you can apply, and other resources you can use. Add “sub-branches” to further explore the details. Make a final branch with the most suitable solution for the main problem. Use “sub-branches” for details.

Through this exercise, you should be able to see which “branch” or option is the most practical, time-saving, and cost-effective problem solving method .

9. Create “Psychological Distance”

What is psychological distance? According to the construal level theory (CLT), it’s “anything that we do not experience as occurring now, here, and to ourselves.” Some examples include taking another person’s perspective or thinking of the problem as unlikely.

Scientists have shown that by increasing the mental distance between us and our problem, we’ll have an increase in creative solutions. This happens because thinking more abstractly helps us form unexpected connections between seemingly unrelated concepts, thus allowing our minds to increase its problem solving capacity.

10. Play Some Soccer

A link has been found between our brain’s “executive functions” and sports success . When in action, our brains are quickly multitasking between moving, anticipating, strategizing, reacting, and performing. Doing all these things at once requires an enormous amount of brain activity.

This can be related to our working world when we plan, reason, monitor our actions and problem solve all at once. Therefore, it may be concluded that when you play soccer or any other fast-moving sport, you’re rewiring your brain to be quicker at thinking, processing, and reacting to problems.

To learn more about how to develop your problem-solving and decision making capabilities or to receive training on applied strategic thinking skills , contact CMOE today!

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9 Ways to Improve Math Skills Quickly & Effectively

Written by Ashley Crowe

• Parent Resources

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

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

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

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

But first, the basics.

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

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

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

What are considered basic math skills?

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

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

How to improve math skills 

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

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

1. Wrap your head around the concepts

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

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

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

2. Try game-based learning

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

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

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

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

3. Bring math into daily life

You use basic math every day. 

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

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

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

4. Implement daily practice

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

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

5. Sketch word problems

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

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

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

6. Set realistic goals

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

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

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

Set Goals and Rewards in Prodigy Math

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

7. Engage with a math tutor

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

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

8. Focus on one concept at a time

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

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

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

9. Teach others math you already know

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

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

Embracing technology to improve math skills

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

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

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

Look online for more math help

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

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

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

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How to improve your problem solving skills and build effective problem solving strategies

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Effective problem solving is all about using the right process and following a plan tailored to the issue at hand. Recognizing your team or organization has an issue isn’t enough to come up with effective problem solving strategies. 

To truly understand a problem and develop appropriate solutions, you will want to follow a solid process, follow the necessary problem solving steps, and bring all of your problem solving skills to the table.   We’ll forst look at what problem solving strategies you can employ with your team when looking for a way to approach the process. We’ll then discuss the problem solving skills you need to be more effective at solving problems, complete with an activity from the SessionLab library you can use to develop that skill in your team.

Let’s get to it! 

Problem solving strategies

What skills do i need to be an effective problem solver, how can i improve my problem solving skills.

Problem solving strategies are methods of approaching and facilitating the process of problem-solving with a set of techniques , actions, and processes. Different strategies are more effective if you are trying to solve broad problems such as achieving higher growth versus more focused problems like, how do we improve our customer onboarding process?

Broadly, the problem solving steps outlined above should be included in any problem solving strategy though choosing where to focus your time and what approaches should be taken is where they begin to differ. You might find that some strategies ask for the problem identification to be done prior to the session or that everything happens in the course of a one day workshop.

The key similarity is that all good problem solving strategies are structured and designed. Four hours of open discussion is never going to be as productive as a four-hour workshop designed to lead a group through a problem solving process.

Good problem solving strategies are tailored to the team, organization and problem you will be attempting to solve. Here are some example problem solving strategies you can learn from or use to get started.

Use a workshop to lead a team through a group process

Often, the first step to solving problems or organizational challenges is bringing a group together effectively. Most teams have the tools, knowledge, and expertise necessary to solve their challenges – they just need some guidance in how to use leverage those skills and a structure and format that allows people to focus their energies.

Facilitated workshops are one of the most effective ways of solving problems of any scale. By designing and planning your workshop carefully, you can tailor the approach and scope to best fit the needs of your team and organization. 

Problem solving workshop

• Creating a bespoke, tailored process
• Tackling problems of any size
• Building in-house workshop ability and encouraging their use

Workshops are an effective strategy for solving problems. By using tried and test facilitation techniques and methods, you can design and deliver a workshop that is perfectly suited to the unique variables of your organization. You may only have the capacity for a half-day workshop and so need a problem solving process to match. 

By using our session planner tool and importing methods from our library of 700+ facilitation techniques, you can create the right problem solving workshop for your team. It might be that you want to encourage creative thinking or look at things from a new angle to unblock your groups approach to problem solving. By tailoring your workshop design to the purpose, you can help ensure great results.

One of the main benefits of a workshop is the structured approach to problem solving. Not only does this mean that the workshop itself will be successful, but many of the methods and techniques will help your team improve their working processes outside of the workshop. 

We believe that workshops are one of the best tools you can use to improve the way your team works together. Start with a problem solving workshop and then see what team building, culture or design workshops can do for your organization!

Run a design sprint

Great for: 

• aligning large, multi-discipline teams
• quickly designing and testing solutions
• tackling large, complex organizational challenges and breaking them down into smaller tasks

By using design thinking principles and methods, a design sprint is a great way of identifying, prioritizing and prototyping solutions to long term challenges that can help solve major organizational problems with quick action and measurable results.

Some familiarity with design thinking is useful, though not integral, and this strategy can really help a team align if there is some discussion around which problems should be approached first. 

The stage-based structure of the design sprint is also very useful for teams new to design thinking.  The inspiration phase, where you look to competitors that have solved your problem, and the rapid prototyping and testing phases are great for introducing new concepts that will benefit a team in all their future work. 

It can be common for teams to look inward for solutions and so looking to the market for solutions you can iterate on can be very productive. Instilling an agile prototyping and testing mindset can also be great when helping teams move forwards – generating and testing solutions quickly can help save time in the long run and is also pretty exciting!

Break problems down into smaller issues

Organizational challenges and problems are often complicated and large scale in nature. Sometimes, trying to resolve such an issue in one swoop is simply unachievable or overwhelming. Try breaking down such problems into smaller issues that you can work on step by step. You may not be able to solve the problem of churning customers off the bat, but you can work with your team to identify smaller effort but high impact elements and work on those first.

This problem solving strategy can help a team generate momentum, prioritize and get some easy wins. It’s also a great strategy to employ with teams who are just beginning to learn how to approach the problem solving process. If you want some insight into a way to employ this strategy, we recommend looking at our design sprint template below!

Use guiding frameworks or try new methodologies

Some problems are best solved by introducing a major shift in perspective or by using new methodologies that encourage your team to think differently.

Props and tools such as Methodkit , which uses a card-based toolkit for facilitation, or Lego Serious Play can be great ways to engage your team and find an inclusive, democratic problem solving strategy. Remember that play and creativity are great tools for achieving change and whatever the challenge, engaging your participants can be very effective where other strategies may have failed.

LEGO Serious Play

• Improving core problem solving skills
• Thinking outside of the box
• Encouraging creative solutions

LEGO Serious Play is a problem solving methodology designed to get participants thinking differently by using 3D models and kinesthetic learning styles. By physically building LEGO models based on questions and exercises, participants are encouraged to think outside of the box and create their own responses. 

Collaborate LEGO Serious Play exercises are also used to encourage communication and build problem solving skills in a group. By using this problem solving process, you can often help different kinds of learners and personality types contribute and unblock organizational problems with creative thinking. 

Problem solving strategies like LEGO Serious Play are super effective at helping a team solve more skills-based problems such as communication between teams or a lack of creative thinking. Some problems are not suited to LEGO Serious Play and require a different problem solving strategy.

Card Decks and Method Kits

• New facilitators or non-facilitators 
• Approaching difficult subjects with a simple, creative framework
• Engaging those with varied learning styles

Card decks and method kids are great tools for those new to facilitation or for whom facilitation is not the primary role. Card decks such as the emotional culture deck can be used for complete workshops and in many cases, can be used right out of the box. Methodkit has a variety of kits designed for scenarios ranging from personal development through to personas and global challenges so you can find the right deck for your particular needs.

Having an easy to use framework that encourages creativity or a new approach can take some of the friction or planning difficulties out of the workshop process and energize a team in any setting. Simplicity is the key with these methods. By ensuring everyone on your team can get involved and engage with the process as quickly as possible can really contribute to the success of your problem solving strategy.

Source external advice

Looking to peers, experts and external facilitators can be a great way of approaching the problem solving process. Your team may not have the necessary expertise, insights of experience to tackle some issues, or you might simply benefit from a fresh perspective. Some problems may require bringing together an entire team, and coaching managers or team members individually might be the right approach. Remember that not all problems are best resolved in the same manner.

If you’re a solo entrepreneur, peer groups, coaches and mentors can also be invaluable at not only solving specific business problems, but in providing a support network for resolving future challenges. One great approach is to join a Mastermind Group and link up with like-minded individuals and all grow together. Remember that however you approach the sourcing of external advice, do so thoughtfully, respectfully and honestly. Reciprocate where you can and prepare to be surprised by just how kind and helpful your peers can be!

Mastermind Group

• Solo entrepreneurs or small teams with low capacity
• Peer learning and gaining outside expertise
• Getting multiple external points of view quickly

Problem solving in large organizations with lots of skilled team members is one thing, but how about if you work for yourself or in a very small team without the capacity to get the most from a design sprint or LEGO Serious Play session? 

A mastermind group – sometimes known as a peer advisory board – is where a group of people come together to support one another in their own goals, challenges, and businesses. Each participant comes to the group with their own purpose and the other members of the group will help them create solutions, brainstorm ideas, and support one another. 

Mastermind groups are very effective in creating an energized, supportive atmosphere that can deliver meaningful results. Learning from peers from outside of your organization or industry can really help unlock new ways of thinking and drive growth. Access to the experience and skills of your peers can be invaluable in helping fill the gaps in your own ability, particularly in young companies.

A mastermind group is a great solution for solo entrepreneurs, small teams, or for organizations that feel that external expertise or fresh perspectives will be beneficial for them. It is worth noting that Mastermind groups are often only as good as the participants and what they can bring to the group. Participants need to be committed, engaged and understand how to work in this context. 

Coaching and mentoring

• Focused learning and development
• Filling skills gaps
• Working on a range of challenges over time

Receiving advice from a business coach or building a mentor/mentee relationship can be an effective way of resolving certain challenges. The one-to-one format of most coaching and mentor relationships can really help solve the challenges those individuals are having and benefit the organization as a result.

A great mentor can be invaluable when it comes to spotting potential problems before they arise and coming to understand a mentee very well has a host of other business benefits. You might run an internal mentorship program to help develop your team’s problem solving skills and strategies or as part of a large learning and development program. External coaches can also be an important part of your problem solving strategy, filling skills gaps for your management team or helping with specific business issues. 

Now we’ve explored the problem solving process and the steps you will want to go through in order to have an effective session, let’s look at the skills you and your team need to be more effective problem solvers.

Problem solving skills are highly sought after, whatever industry or team you work in. Organizations are keen to employ people who are able to approach problems thoughtfully and find strong, realistic solutions. Whether you are a facilitator , a team leader or a developer, being an effective problem solver is a skill you’ll want to develop.

Problem solving skills form a whole suite of techniques and approaches that an individual uses to not only identify problems but to discuss them productively before then developing appropriate solutions.

Here are some of the most important problem solving skills everyone from executives to junior staff members should learn. We’ve also included an activity or exercise from the SessionLab library that can help you and your team develop that skill. 

If you’re running a workshop or training session to try and improve problem solving skills in your team, try using these methods to supercharge your process!

Active listening

Active listening is one of the most important skills anyone who works with people can possess. In short, active listening is a technique used to not only better understand what is being said by an individual, but also to be more aware of the underlying message the speaker is trying to convey. When it comes to problem solving, active listening is integral for understanding the position of every participant and to clarify the challenges, ideas and solutions they bring to the table.

Some active listening skills include:

• Paying complete attention to the speaker.
• Removing distractions.
• Avoid interruption.
• Taking the time to fully understand before preparing a rebuttal.
• Responding respectfully and appropriately.
• Demonstrate attentiveness and positivity with an open posture, making eye contact with the speaker, smiling and nodding if appropriate. Show that you are listening and encourage them to continue.
• Be aware of and respectful of feelings. Judge the situation and respond appropriately. You can disagree without being disrespectful.   
• Observe body language. 
• Paraphrase what was said in your own words, either mentally or verbally.
• Remain neutral. 
• Reflect and take a moment before responding.
• Ask deeper questions based on what is said and clarify points where necessary.   

Active Listening   #hyperisland   #skills   #active listening   #remote-friendly   This activity supports participants to reflect on a question and generate their own solutions using simple principles of active listening and peer coaching. It’s an excellent introduction to active listening but can also be used with groups that are already familiar with it. Participants work in groups of three and take turns being: “the subject”, the listener, and the observer.

Analytical skills

All problem solving models require strong analytical skills, particularly during the beginning of the process and when it comes to analyzing how solutions have performed.

Analytical skills are primarily focused on performing an effective analysis by collecting, studying and parsing data related to a problem or opportunity. 

It often involves spotting patterns, being able to see things from different perspectives and using observable facts and data to make suggestions or produce insight. 

Analytical skills are also important at every stage of the problem solving process and by having these skills, you can ensure that any ideas or solutions you create or backed up analytically and have been sufficiently thought out.

Nine Whys   #innovation   #issue analysis   #liberating structures   With breathtaking simplicity, you can rapidly clarify for individuals and a group what is essentially important in their work. You can quickly reveal when a compelling purpose is missing in a gathering and avoid moving forward without clarity. When a group discovers an unambiguous shared purpose, more freedom and more responsibility are unleashed. You have laid the foundation for spreading and scaling innovations with fidelity.

Collaboration

Trying to solve problems on your own is difficult. Being able to collaborate effectively, with a free exchange of ideas, to delegate and be a productive member of a team is hugely important to all problem solving strategies.

Remember that whatever your role, collaboration is integral, and in a problem solving process, you are all working together to find the best solution for everyone. 

Marshmallow challenge with debriefing   #teamwork   #team   #leadership   #collaboration   In eighteen minutes, teams must build the tallest free-standing structure out of 20 sticks of spaghetti, one yard of tape, one yard of string, and one marshmallow. The marshmallow needs to be on top. The Marshmallow Challenge was developed by Tom Wujec, who has done the activity with hundreds of groups around the world. Visit the Marshmallow Challenge website for more information. This version has an extra debriefing question added with sample questions focusing on roles within the team.

Communication  

Being an effective communicator means being empathetic, clear and succinct, asking the right questions, and demonstrating active listening skills throughout any discussion or meeting. 

In a problem solving setting, you need to communicate well in order to progress through each stage of the process effectively. As a team leader, it may also fall to you to facilitate communication between parties who may not see eye to eye. Effective communication also means helping others to express themselves and be heard in a group.

Bus Trip   #feedback   #communication   #appreciation   #closing   #thiagi   #team   This is one of my favourite feedback games. I use Bus Trip at the end of a training session or a meeting, and I use it all the time. The game creates a massive amount of energy with lots of smiles, laughs, and sometimes even a teardrop or two.

Creative problem solving skills can be some of the best tools in your arsenal. Thinking creatively, being able to generate lots of ideas and come up with out of the box solutions is useful at every step of the process. 

The kinds of problems you will likely discuss in a problem solving workshop are often difficult to solve, and by approaching things in a fresh, creative manner, you can often create more innovative solutions.

Having practical creative skills is also a boon when it comes to problem solving. If you can help create quality design sketches and prototypes in record time, it can help bring a team to alignment more quickly or provide a base for further iteration.

The paper clip method   #sharing   #creativity   #warm up   #idea generation   #brainstorming   The power of brainstorming. A training for project leaders, creativity training, and to catalyse getting new solutions.

Critical thinking

Critical thinking is one of the fundamental problem solving skills you’ll want to develop when working on developing solutions. Critical thinking is the ability to analyze, rationalize and evaluate while being aware of personal bias, outlying factors and remaining open-minded.

Defining and analyzing problems without deploying critical thinking skills can mean you and your team go down the wrong path. Developing solutions to complex issues requires critical thinking too – ensuring your team considers all possibilities and rationally evaluating them. 

Agreement-Certainty Matrix   #issue analysis   #liberating structures   #problem solving   You can help individuals or groups avoid the frequent mistake of trying to solve a problem with methods that are not adapted to the nature of their challenge. The combination of two questions makes it possible to easily sort challenges into four categories: simple, complicated, complex , and chaotic .  A problem is simple when it can be solved reliably with practices that are easy to duplicate.  It is complicated when experts are required to devise a sophisticated solution that will yield the desired results predictably.  A problem is complex when there are several valid ways to proceed but outcomes are not predictable in detail.  Chaotic is when the context is too turbulent to identify a path forward.  A loose analogy may be used to describe these differences: simple is like following a recipe, complicated like sending a rocket to the moon, complex like raising a child, and chaotic is like the game “Pin the Tail on the Donkey.”  The Liberating Structures Matching Matrix in Chapter 5 can be used as the first step to clarify the nature of a challenge and avoid the mismatches between problems and solutions that are frequently at the root of chronic, recurring problems.

Data analysis 

Though it shares lots of space with general analytical skills, data analysis skills are something you want to cultivate in their own right in order to be an effective problem solver.

Being good at data analysis doesn’t just mean being able to find insights from data, but also selecting the appropriate data for a given issue, interpreting it effectively and knowing how to model and present that data. Depending on the problem at hand, it might also include a working knowledge of specific data analysis tools and procedures. 

Having a solid grasp of data analysis techniques is useful if you’re leading a problem solving workshop but if you’re not an expert, don’t worry. Bring people into the group who has this skill set and help your team be more effective as a result.

Decision making

All problems need a solution and all solutions require that someone make the decision to implement them. Without strong decision making skills, teams can become bogged down in discussion and less effective as a result. 

Making decisions is a key part of the problem solving process. It’s important to remember that decision making is not restricted to the leadership team. Every staff member makes decisions every day and developing these skills ensures that your team is able to solve problems at any scale. Remember that making decisions does not mean leaping to the first solution but weighing up the options and coming to an informed, well thought out solution to any given problem that works for the whole team.

Lightning Decision Jam (LDJ)   #action   #decision making   #problem solving   #issue analysis   #innovation   #design   #remote-friendly   The problem with anything that requires creative thinking is that it’s easy to get lost—lose focus and fall into the trap of having useless, open-ended, unstructured discussions. Here’s the most effective solution I’ve found: Replace all open, unstructured discussion with a clear process. What to use this exercise for: Anything which requires a group of people to make decisions, solve problems or discuss challenges. It’s always good to frame an LDJ session with a broad topic, here are some examples: The conversion flow of our checkout Our internal design process How we organise events Keeping up with our competition Improving sales flow

Dependability

Most complex organizational problems require multiple people to be involved in delivering the solution. Ensuring that the team and organization can depend on you to take the necessary actions and communicate where necessary is key to ensuring problems are solved effectively.

Being dependable also means working to deadlines and to brief. It is often a matter of creating trust in a team so that everyone can depend on one another to complete the agreed actions in the agreed time frame so that the team can move forward together. Being undependable can create problems of friction and can limit the effectiveness of your solutions so be sure to bear this in mind throughout a project. 

Team Purpose & Culture   #team   #hyperisland   #culture   #remote-friendly   This is an essential process designed to help teams define their purpose (why they exist) and their culture (how they work together to achieve that purpose). Defining these two things will help any team to be more focused and aligned. With support of tangible examples from other companies, the team members work as individuals and a group to codify the way they work together. The goal is a visual manifestation of both the purpose and culture that can be put up in the team’s work space.

Emotional intelligence

Emotional intelligence is an important skill for any successful team member, whether communicating internally or with clients or users. In the problem solving process, emotional intelligence means being attuned to how people are feeling and thinking, communicating effectively and being self-aware of what you bring to a room. 

There are often differences of opinion when working through problem solving processes, and it can be easy to let things become impassioned or combative. Developing your emotional intelligence means being empathetic to your colleagues and managing your own emotions throughout the problem and solution process. Be kind, be thoughtful and put your points across care and attention. 

Being emotionally intelligent is a skill for life and by deploying it at work, you can not only work efficiently but empathetically. Check out the emotional culture workshop template for more!

Facilitation

As we’ve clarified in our facilitation skills post, facilitation is the art of leading people through processes towards agreed-upon objectives in a manner that encourages participation, ownership, and creativity by all those involved. While facilitation is a set of interrelated skills in itself, the broad definition of facilitation can be invaluable when it comes to problem solving. Leading a team through a problem solving process is made more effective if you improve and utilize facilitation skills – whether you’re a manager, team leader or external stakeholder.

The Six Thinking Hats   #creative thinking   #meeting facilitation   #problem solving   #issue resolution   #idea generation   #conflict resolution   The Six Thinking Hats are used by individuals and groups to separate out conflicting styles of thinking. They enable and encourage a group of people to think constructively together in exploring and implementing change, rather than using argument to fight over who is right and who is wrong.

Flexibility 

Being flexible is a vital skill when it comes to problem solving. This does not mean immediately bowing to pressure or changing your opinion quickly: instead, being flexible is all about seeing things from new perspectives, receiving new information and factoring it into your thought process.

Flexibility is also important when it comes to rolling out solutions. It might be that other organizational projects have greater priority or require the same resources as your chosen solution. Being flexible means understanding needs and challenges across the team and being open to shifting or arranging your own schedule as necessary. Again, this does not mean immediately making way for other projects. It’s about articulating your own needs, understanding the needs of others and being able to come to a meaningful compromise.

The Creativity Dice   #creativity   #problem solving   #thiagi   #issue analysis   Too much linear thinking is hazardous to creative problem solving. To be creative, you should approach the problem (or the opportunity) from different points of view. You should leave a thought hanging in mid-air and move to another. This skipping around prevents premature closure and lets your brain incubate one line of thought while you consciously pursue another.

Working in any group can lead to unconscious elements of groupthink or situations in which you may not wish to be entirely honest. Disagreeing with the opinions of the executive team or wishing to save the feelings of a coworker can be tricky to navigate, but being honest is absolutely vital when to comes to developing effective solutions and ensuring your voice is heard. 

Remember that being honest does not mean being brutally candid. You can deliver your honest feedback and opinions thoughtfully and without creating friction by using other skills such as emotional intelligence. 

Explore your Values   #hyperisland   #skills   #values   #remote-friendly   Your Values is an exercise for participants to explore what their most important values are. It’s done in an intuitive and rapid way to encourage participants to follow their intuitive feeling rather than over-thinking and finding the “correct” values. It is a good exercise to use to initiate reflection and dialogue around personal values.

Initiative 

The problem solving process is multi-faceted and requires different approaches at certain points of the process. Taking initiative to bring problems to the attention of the team, collect data or lead the solution creating process is always valuable. You might even roadtest your own small scale solutions or brainstorm before a session. Taking initiative is particularly effective if you have good deal of knowledge in that area or have ownership of a particular project and want to get things kickstarted.

That said, be sure to remember to honor the process and work in service of the team. If you are asked to own one part of the problem solving process and you don’t complete that task because your initiative leads you to work on something else, that’s not an effective method of solving business challenges.

15% Solutions   #action   #liberating structures   #remote-friendly   You can reveal the actions, however small, that everyone can do immediately. At a minimum, these will create momentum, and that may make a BIG difference.  15% Solutions show that there is no reason to wait around, feel powerless, or fearful. They help people pick it up a level. They get individuals and the group to focus on what is within their discretion instead of what they cannot change.  With a very simple question, you can flip the conversation to what can be done and find solutions to big problems that are often distributed widely in places not known in advance. Shifting a few grains of sand may trigger a landslide and change the whole landscape.

Impartiality

A particularly useful problem solving skill for product owners or managers is the ability to remain impartial throughout much of the process. In practice, this means treating all points of view and ideas brought forward in a meeting equally and ensuring that your own areas of interest or ownership are not favored over others. 

There may be a stage in the process where a decision maker has to weigh the cost and ROI of possible solutions against the company roadmap though even then, ensuring that the decision made is based on merit and not personal opinion. 

Empathy map   #frame insights   #create   #design   #issue analysis   An empathy map is a tool to help a design team to empathize with the people they are designing for. You can make an empathy map for a group of people or for a persona. To be used after doing personas when more insights are needed.

Being a good leader means getting a team aligned, energized and focused around a common goal. In the problem solving process, strong leadership helps ensure that the process is efficient, that any conflicts are resolved and that a team is managed in the direction of success.

It’s common for managers or executives to assume this role in a problem solving workshop, though it’s important that the leader maintains impartiality and does not bulldoze the group in a particular direction. Remember that good leadership means working in service of the purpose and team and ensuring the workshop is a safe space for employees of any level to contribute. Take a look at our leadership games and activities post for more exercises and methods to help improve leadership in your organization.

Leadership Pizza   #leadership   #team   #remote-friendly   This leadership development activity offers a self-assessment framework for people to first identify what skills, attributes and attitudes they find important for effective leadership, and then assess their own development and initiate goal setting.

In the context of problem solving, mediation is important in keeping a team engaged, happy and free of conflict. When leading or facilitating a problem solving workshop, you are likely to run into differences of opinion. Depending on the nature of the problem, certain issues may be brought up that are emotive in nature. 

Being an effective mediator means helping those people on either side of such a divide are heard, listen to one another and encouraged to find common ground and a resolution. Mediating skills are useful for leaders and managers in many situations and the problem solving process is no different.

Conflict Responses   #hyperisland   #team   #issue resolution   A workshop for a team to reflect on past conflicts, and use them to generate guidelines for effective conflict handling. The workshop uses the Thomas-Killman model of conflict responses to frame a reflective discussion. Use it to open up a discussion around conflict with a team.

Planning 

Solving organizational problems is much more effective when following a process or problem solving model. Planning skills are vital in order to structure, deliver and follow-through on a problem solving workshop and ensure your solutions are intelligently deployed.

Planning skills include the ability to organize tasks and a team, plan and design the process and take into account any potential challenges. Taking the time to plan carefully can save time and frustration later in the process and is valuable for ensuring a team is positioned for success.

3 Action Steps   #hyperisland   #action   #remote-friendly   This is a small-scale strategic planning session that helps groups and individuals to take action toward a desired change. It is often used at the end of a workshop or programme. The group discusses and agrees on a vision, then creates some action steps that will lead them towards that vision. The scope of the challenge is also defined, through discussion of the helpful and harmful factors influencing the group.

Prioritization

As organisations grow, the scale and variation of problems they face multiplies. Your team or is likely to face numerous challenges in different areas and so having the skills to analyze and prioritize becomes very important, particularly for those in leadership roles.

A thorough problem solving process is likely to deliver multiple solutions and you may have several different problems you wish to solve simultaneously. Prioritization is the ability to measure the importance, value, and effectiveness of those possible solutions and choose which to enact and in what order. The process of prioritization is integral in ensuring the biggest challenges are addressed with the most impactful solutions.

Impact and Effort Matrix   #gamestorming   #decision making   #action   #remote-friendly   In this decision-making exercise, possible actions are mapped based on two factors: effort required to implement and potential impact. Categorizing ideas along these lines is a useful technique in decision making, as it obliges contributors to balance and evaluate suggested actions before committing to them.

Project management

Some problem solving skills are utilized in a workshop or ideation phases, while others come in useful when it comes to decision making. Overseeing an entire problem solving process and ensuring its success requires strong project management skills. 

While project management incorporates many of the other skills listed here, it is important to note the distinction of considering all of the factors of a project and managing them successfully. Being able to negotiate with stakeholders, manage tasks, time and people, consider costs and ROI, and tie everything together is massively helpful when going through the problem solving process. 

Record keeping

Working out meaningful solutions to organizational challenges is only one part of the process.  Thoughtfully documenting and keeping records of each problem solving step for future consultation is important in ensuring efficiency and meaningful change. 

For example, some problems may be lower priority than others but can be revisited in the future. If the team has ideated on solutions and found some are not up to the task, record those so you can rule them out and avoiding repeating work. Keeping records of the process also helps you improve and refine your problem solving model next time around!

Personal Kanban   #gamestorming   #action   #agile   #project planning   Personal Kanban is a tool for organizing your work to be more efficient and productive. It is based on agile methods and principles.

Research skills

Conducting research to support both the identification of problems and the development of appropriate solutions is important for an effective process. Knowing where to go to collect research, how to conduct research efficiently, and identifying pieces of research are relevant are all things a good researcher can do well. 

In larger groups, not everyone has to demonstrate this ability in order for a problem solving workshop to be effective. That said, having people with research skills involved in the process, particularly if they have existing area knowledge, can help ensure the solutions that are developed with data that supports their intention. Remember that being able to deliver the results of research efficiently and in a way the team can easily understand is also important. The best data in the world is only as effective as how it is delivered and interpreted.

Customer experience map   #ideation   #concepts   #research   #design   #issue analysis   #remote-friendly   Customer experience mapping is a method of documenting and visualizing the experience a customer has as they use the product or service. It also maps out their responses to their experiences. To be used when there is a solution (even in a conceptual stage) that can be analyzed.

Risk management

Managing risk is an often overlooked part of the problem solving process. Solutions are often developed with the intention of reducing exposure to risk or solving issues that create risk but sometimes, great solutions are more experimental in nature and as such, deploying them needs to be carefully considered. 

Managing risk means acknowledging that there may be risks associated with more out of the box solutions or trying new things, but that this must be measured against the possible benefits and other organizational factors. 

Be informed, get the right data and stakeholders in the room and you can appropriately factor risk into your decision making process. 

Decisions, Decisions…   #communication   #decision making   #thiagi   #action   #issue analysis   When it comes to decision-making, why are some of us more prone to take risks while others are risk-averse? One explanation might be the way the decision and options were presented.  This exercise, based on Kahneman and Tversky’s classic study , illustrates how the framing effect influences our judgement and our ability to make decisions . The participants are divided into two groups. Both groups are presented with the same problem and two alternative programs for solving them. The two programs both have the same consequences but are presented differently. The debriefing discussion examines how the framing of the program impacted the participant’s decision.

Team-building 

No single person is as good at problem solving as a team. Building an effective team and helping them come together around a common purpose is one of the most important problem solving skills, doubly so for leaders. By bringing a team together and helping them work efficiently, you pave the way for team ownership of a problem and the development of effective solutions. 

In a problem solving workshop, it can be tempting to jump right into the deep end, though taking the time to break the ice, energize the team and align them with a game or exercise will pay off over the course of the day.

Remember that you will likely go through the problem solving process multiple times over an organization’s lifespan and building a strong team culture will make future problem solving more effective. It’s also great to work with people you know, trust and have fun with. Working on team building in and out of the problem solving process is a hallmark of successful teams that can work together to solve business problems.

9 Dimensions Team Building Activity   #ice breaker   #teambuilding   #team   #remote-friendly   9 Dimensions is a powerful activity designed to build relationships and trust among team members. There are 2 variations of this icebreaker. The first version is for teams who want to get to know each other better. The second version is for teams who want to explore how they are working together as a team.

Time management 

The problem solving process is designed to lead a team from identifying a problem through to delivering a solution and evaluating its effectiveness. Without effective time management skills or timeboxing of tasks, it can be easy for a team to get bogged down or be inefficient.

By using a problem solving model and carefully designing your workshop, you can allocate time efficiently and trust that the process will deliver the results you need in a good timeframe.

Time management also comes into play when it comes to rolling out solutions, particularly those that are experimental in nature. Having a clear timeframe for implementing and evaluating solutions is vital for ensuring their success and being able to pivot if necessary.

Improving your skills at problem solving is often a career-long pursuit though there are methods you can use to make the learning process more efficient and to supercharge your problem solving skillset.

Remember that the skills you need to be a great problem solver have a large overlap with those skills you need to be effective in any role. Investing time and effort to develop your active listening or critical thinking skills is valuable in any context. Here are 7 ways to improve your problem solving skills.

Share best practices

Remember that your team is an excellent source of skills, wisdom, and techniques and that you should all take advantage of one another where possible. Best practices that one team has for solving problems, conducting research or making decisions should be shared across the organization. If you have in-house staff that have done active listening training or are data analysis pros, have them lead a training session. 

Your team is one of your best resources. Create space and internal processes for the sharing of skills so that you can all grow together. 

Ask for help and attend training

Once you’ve figured out you have a skills gap, the next step is to take action to fill that skills gap. That might be by asking your superior for training or coaching, or liaising with team members with that skill set. You might even attend specialized training for certain skills – active listening or critical thinking, for example, are business-critical skills that are regularly offered as part of a training scheme.

Whatever method you choose, remember that taking action of some description is necessary for growth. Whether that means practicing, getting help, attending training or doing some background reading, taking active steps to improve your skills is the way to go.

Learn a process 

Problem solving can be complicated, particularly when attempting to solve large problems for the first time. Using a problem solving process helps give structure to your problem solving efforts and focus on creating outcomes, rather than worrying about the format. 

Tools such as the seven-step problem solving process above are effective because not only do they feature steps that will help a team solve problems, they also develop skills along the way. Each step asks for people to engage with the process using different skills and in doing so, helps the team learn and grow together. Group processes of varying complexity and purpose can also be found in the SessionLab library of facilitation techniques . Using a tried and tested process and really help ease the learning curve for both those leading such a process, as well as those undergoing the purpose.

Effective teams make decisions about where they should and shouldn’t expend additional effort. By using a problem solving process, you can focus on the things that matter, rather than stumbling towards a solution haphazardly. 

Create a feedback loop

Some skills gaps are more obvious than others. It’s possible that your perception of your active listening skills differs from those of your colleagues. 

It’s valuable to create a system where team members can provide feedback in an ordered and friendly manner so they can all learn from one another. Only by identifying areas of improvement can you then work to improve them. 

Remember that feedback systems require oversight and consideration so that they don’t turn into a place to complain about colleagues. Design the system intelligently so that you encourage the creation of learning opportunities, rather than encouraging people to list their pet peeves.

While practice might not make perfect, it does make the problem solving process easier. If you are having trouble with critical thinking, don’t shy away from doing it. Get involved where you can and stretch those muscles as regularly as possible. 

Problem solving skills come more naturally to some than to others and that’s okay. Take opportunities to get involved and see where you can practice your skills in situations outside of a workshop context. Try collaborating in other circumstances at work or conduct data analysis on your own projects. You can often develop those skills you need for problem solving simply by doing them. Get involved!

Use expert exercises and methods

Learn from the best. Our library of 700+ facilitation techniques is full of activities and methods that help develop the skills you need to be an effective problem solver. Check out our templates to see how to approach problem solving and other organizational challenges in a structured and intelligent manner.

There is no single approach to improving problem solving skills, but by using the techniques employed by others you can learn from their example and develop processes that have seen proven results. 

Try new ways of thinking and change your mindset

Using tried and tested exercises that you know well can help deliver results, but you do run the risk of missing out on the learning opportunities offered by new approaches. As with the problem solving process, changing your mindset can remove blockages and be used to develop your problem solving skills.

Most teams have members with mixed skill sets and specialties. Mix people from different teams and share skills and different points of view. Teach your customer support team how to use design thinking methods or help your developers with conflict resolution techniques. Try switching perspectives with facilitation techniques like Flip It! or by using new problem solving methodologies or models. Give design thinking, liberating structures or lego serious play a try if you want to try a new approach. You will find that framing problems in new ways and using existing skills in new contexts can be hugely useful for personal development and improving your skillset. It’s also a lot of fun to try new things. Give it a go!

Encountering business challenges and needing to find appropriate solutions is not unique to your organization. Lots of very smart people have developed methods, theories and approaches to help develop problem solving skills and create effective solutions. Learn from them!

Books like The Art of Thinking Clearly , Think Smarter, or Thinking Fast, Thinking Slow are great places to start, though it’s also worth looking at blogs related to organizations facing similar problems to yours, or browsing for success stories. Seeing how Dropbox massively increased growth and working backward can help you see the skills or approach you might be lacking to solve that same problem. Learning from others by reading their stories or approaches can be time-consuming but ultimately rewarding.

A tired, distracted mind is not in the best position to learn new skills. It can be tempted to burn the candle at both ends and develop problem solving skills outside of work. Absolutely use your time effectively and take opportunities for self-improvement, though remember that rest is hugely important and that without letting your brain rest, you cannot be at your most effective. 

Creating distance between yourself and the problem you might be facing can also be useful. By letting an idea sit, you can find that a better one presents itself or you can develop it further. Take regular breaks when working and create a space for downtime. Remember that working smarter is preferable to working harder and that self-care is important for any effective learning or improvement process.

Want to design better group processes?

Over to you

Now we’ve explored some of the key problem solving skills and the problem solving steps necessary for an effective process, you’re ready to begin developing more effective solutions and leading problem solving workshops.

Need more inspiration? Check out our post on problem solving activities you can use when guiding a group towards a great solution in your next workshop or meeting. Have questions? Did you have a great problem solving technique you use with your team? Get in touch in the comments below. We’d love to chat!

James Smart is Head of Content at SessionLab. He’s also a creative facilitator who has run workshops and designed courses for establishments like the National Centre for Writing, UK. He especially enjoys working with young people and empowering others in their creative practice.

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How To Increase Solving Speed In Quantitative Aptitude

Quantitative Aptitude is one of the most important aspects of placement examination. It helps candidates to demonstrate their thinking, problem-solving and quick decision-making abilities to the employers.

But many candidates struggle to complete the Quantitative Aptitude sections in time due to the length and complexity of calculations. This led to many candidates trying and searching for methods to improve their speed in the Quantitative Aptitude section.

That’s why we have written this article to help students identify and understand the approaches and techniques on how to increase solving speed in Quantitative Aptitude.

How To Increase Solving Speed In Quantitative Aptitude Section

These are the methods you can use to increase solving speed in Quantitative Aptitude:

Step 1 – Understand the Complete Syllabus

The initial stage in any significant endeavour has always been the most crucial. Because the first one determines the overall trajectory of the excursion. The route of the Quantitative Aptitude examination is equivalent. Without question, it is a challenging task packed with peaks and troughs as well as significant difficulties.

The Quantitative Aptitude syllabus is comprehensive, diversified, and, in some instances, open-ended. It is also essential to understand that the syllabus is neither limited nor extensive. If you understand the syllabus completely, you can devise a reasonable plan for your preparation that will require fewer hours, explore more topics, and practice more to improve your solving speed.

Step 2 – Improve Your Weaker Areas

It is natural for students to have topics in which they are weaker in Quantitative Aptitude. After all, various problem-solving approaches or abilities might well be required for different topics. The dilemma then becomes whether students should concentrate on refining a topic or on improving weaker concepts.

Students, in our opinion, should concentrate on their weakest subjects. This is due to the fact that having a topic in which the student feels inadequate would induce unnecessary tension during the examination. As a result, they may be able to perform at a degree beyond their greatest or in other words it can hamper their ability to solve questions with a good calculation speed.

It is advised that a student must starts from essential questions for their weaker topics. The objective is to guarantee that students have a clear understanding of their basics and that they acquire confidence as they answer each subsequent question while improving their calculation speed.

Step 3 – Learn Speed Maths & Vedic Mathematics Tricks

The term ‘Vedic’ is derived from the Sanskrit word ‘Veda,’ which denotes ‘Knowledge.’ Vedic Maths is a remarkable compilation of sutras for solving maths problems quickly and efficiently. If you go through any placement exam paper, you will see that there are many questions that can be answered efficiently and quickly by utilising these Vedic Maths Tricks.

Vedic Maths is no piece of cake as it requires extensive practice. These approaches may appear complex or challenging at first, but they will execute excellently when you begin your calculations with practice. Using Vedic Maths, problems are simplified to one-line answers. It can be learned and developed quickly and effectively. Compared to traditional methods, the calculations are quicker and more reliable.

Step 4 – Be Thorough with Tables, Squares & Cubes

Tables, squares, cubes, and so on are typically used to speed up calculations. Memorising and practising will help you remember them. You must sometimes uncover relationships between numbers and, on other occasions, just memorise.

Tables, squares, and cubes can be thought of as tools, similar to a screwdriver used to twist a screw. Understanding the principles and practising them leads to mastery of tables, squares, and cubes. Experts recommend memorising tables up to 20, squares up to 25, and cubes up to 10 to improve speed in your calculations.

Step 5 – Observe & Understand the Question

Quantitative Aptitude is a challenging concept to master. The most crucial facet that might make it more difficult for students is that they do not properly understand the question before answering it. To not fall into the trap of solving without understanding the question, follow the process mentioned underneath.

Pay close attention to what the question is asking. Understand it and attempt to picture it in your mind. After that, figure out which concepts you can apply. Is it necessary to use addition or the Remainder theorem, for example? You can go to the following step if you understand the problem and know which concept to use.

Attempting to tackle it in a single step might be a daunting task. Instead, seek an approach that breaks down the problem into smaller chunks and arrives at a solution.

Step 6 – Incorporating the Right Shortcuts/Methods

Tricks and shortcut methods can help you solve the Quantitative Aptitude section quickly.

These methods give students the confidence that they are approaching the solution to a problem faster. Today’s students have a multitude of shortcuts and techniques at their disposal, and understanding which to employ is half the battle. Students might utilise several shortcut methods to answer the problem depending on circumstances. Students acquire problem-solving abilities and grow more comfortable exploring for new answers when they understand how to get what they want.

Step 7 – Practice

When you see or read anything only once, you don’t learn it, at least not quite enough to retain it permanently. It may engage you for a few more encounters, but you easily forget about it and move on to something else.

While the ageing process has an impact on our recollection, there is still a lot we can really do to help us remember more when we want to study. For generations, repetition has been utilised as a memorisation technique. The right kind of repetition could assist your memory significantly. To study Quantitative Aptitude for placement exams , you must practice all you have learnt.

If knowledge is repeated or revisited on a frequent basis but at gradually increasing intervals, it gets transferred to another brain region to be retained in long-term memory. That becomes entrenched with duration. Every time you discover something new and practice it, you strengthen your memory’s specific learning or behaviour and make it simpler to remember or recall.

Final Words

We hope this article helps students identify and understand the approaches and techniques on how to increase solving speed in Quantitative Aptitude. If you have any queries regarding the techniques, feel free to drop a comment in the comments section. We wish you all the best in your preparations.

Frequently Asked Questions

These are the frequently asked questions regarding how to increase solving speed in Quantitative Aptitude:

1. Why does solving speed matter in acing the quantitative aptitude section?

Solving speed matters in acing the Quantitative Aptitude section because most of the Quantitative Aptitude section is set in a time-constrained environment, making it absolutely necessary to solve all questions in time.

2. How can I get started in improving the solving speed?

You can start by understanding the whole syllabus, then identifying your strong and weak points, and then improving them by learning different shortcut methods and tables to improve your calculation speed.

3. What are the factors that affect the solving speed?

The factors that affect the solving speed include the ability to break down a problem into smaller parts, approaching each part as a separate entity and identifying which shortcut method or technique can help get the desired results.

4. How does concept clarification go hand in hand with solving speed?

Concept clarification goes hand in hand with solving speed because understanding a concept thoroughly can help students approach the problem without any doubt, improving the solving speed.

Explore Quantitative Aptitude Guides

• Quantitative Aptitude Preparation Guide
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• Quantitative Aptitude Easiest Topics

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Thirumoorthy

Thirumoorthy serves as a teacher and coach. He obtained a 99 percentile on the CAT. He cleared numerous IT jobs and public sector job interviews, but he still decided to pursue a career in education. He desires to elevate the underprivileged sections of society through education

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How to Improve Problem Solving Skills

Quick Summary

• Problem-solving skills are essential for personal growth, career advancement, and tackling life’s challenges. 62% of recruiters seek people who can solve complex problems.
• Learning how to improve problem solving skills will help you applying these skills to various situations from debugging to organizing schedules.
• Core techniques include breaking problems down, analyzing systematically, and applying creative thinking are essential strategies in different fields. Continuous practice, embracing challenges, and learning from mistakes helps to improve problem-solving abilities.

Table of Contents

Picture this situation

“You’re faced with a tricky situation at work, a challenging coding problem, or a complex personal decision. Your heart races, your palms get sweaty, and your mind goes completely blank. Sound familiar?”

That’s where solid problem-solving skills come in handy. They’re not just for math whzzes or tech gurus – they’re essential tools for everyone, every day.

According to Abraham Lincoln “ Give me six hours to chop down a tree and I will spend the first four sharpening the axe. “  

This famous quote depicts the importance of problem solving skills. 

How to improve problem solving skills isn’t just about acing tests or impressing your boss (though those are nice perks). It’s about growing as a person, boosting your confidence, and opening doors to new opportunities you might never have imagined. Currently, 62% of recruiters are seeking people who can solve complex things. 

Whether you’re debugging code that refuses to cooperate, engineering the next big thing that could change the world, or just figuring out how to organise your chaotic schedule, these skills are your trusty sidekick.

So, let’s roll up our sleeves and get into the nitty-gritty of becoming a problem-solving pro.

Core Problem Solving Techniques

At the heart of how to improve problem solving skills lies a set of core techniques. These are your go-to strategies, applicable across various fields and situations. Think of them as the Swiss Army knife in your mental toolkit – versatile, reliable, and always ready when you need them.

• Break it down

When faced with a big, scary problem, your first instinct might be to run for the hills. Instead, take a deep breath and start slicing that monster into smaller, more manageable chunks. It’s like eating an elephant – one bite at a time. This approach makes even the most daunting tasks seem doable.

Here’s where you let your imagination run wild. Let your ideas flow freely, no matter how crasy they might seem. No judgment, just pure creativity. You never know which wild idea might lead to the perfect solution. Remember, some of the world’s greatest inventions started as “crasy” ideas.

Once you have a list of potential solutions, it’s time to put on your critic’s hat. Weigh the pros and cons of each option. Consider factors like feasibility, resources required, potential outcomes, and possible obstacles. This step helps you separate the wheat from the chaff.

Choose the best solution and put it into action. Remember, a good plan today is better than a perfect plan tomorrow. Don’t get stuck in analysis paralysis – sometimes, you need to take the plunge and learn as you go.

After implementing your solution, take a step back and assess the results. What worked? What didn’t? This reflection is crucial for continuous improvement. It’s not just about solving the current problem, but also about becoming better at problem-solving in general.

Problem-solving is rarely a one-and-done deal. Use what you’ve learned to refine your approach and tackle similar problems more effectively in the future. Each problem you solve is a stepping stone to becoming a better problem solver.

Improving Problem Solving Skills in Different Fields

Now, let’s explore how to improve problem solving skills in specific areas. Whether you’re a budding programmer dreaming of creating the next big app, an aspiring engineer with visions of innovative designs, or a student preparing for competitive exams, we’ve got you covered.

Programming Problem Solving Skills

In the fast-paced world of technology, knowing how to improve problem solving skills in programming is like having a superpower. Here’s how you can level up your coding game:

• Code regularly: Practice makes perfect, and coding is no exception. Set aside time each day to write code, even if it’s just for fun. The more you code, the more natural it becomes.
• Take on challenges: Platforms like LeetCode, HackerRank, and CodeWars offer coding pussles that will put your skills to the test. Start with easier problems and gradually work your way up to more complex ones.
• Learn algorithms: Understanding different algorithms and data structures is like adding new tools to your programming toolkit. They help you solve problems more efficiently and elegantly.
• Pair program: Two heads are better than one. Collaborate with fellow coders to tackle problems together. You’ll learn new approaches and perspectives while improving your communication skills .
• Review and refactor: Look back at your old code. Can you make it more efficient? Cleaner? This process will sharpen your skills over time and help you develop a keen eye for quality code.

Engineering Problem Solving Skills

For those wondering how to improve problem solving skills in engineering , here are some targeted strategies:

• Think analytically: Break down complex engineering problems into smaller, solvable components. This approach helps you tackle even the most daunting projects step by step.
• Use simulations: Leverage software tools to model and test your solutions before implementation. This can save time, resources, and prevent costly mistakes.
• Stay updated: Engineering practices evolve rapidly. Keep learning to stay ahead of the curve. Attend workshops, read journals, and engage with the engineering community.
• Cross-disciplinary approach: Don’t limit yourself to one field. Often, the best engineering solutions come from combining knowledge from different areas. Biology might inspire a mechanical design, or psychology could inform a user interface.

Tips to Improve General Problem-Solving Skills

Wondering how to improve solving problem skills in general? Here are some universal tips that apply across all fields:

• Identify and define the problem clearly: Start by pinpointing the exact issue at hand. Ask yourself, “What’s the real problem here?” Often, what seems to be the problem is just a symptom of a deeper issue. Take time to articulate the problem in clear, specific terms. This clarity will guide your entire problem-solving process.
• Gather all relevant information and data: Before jumping to solutions, collect as much pertinent information as possible. This might involve research, asking questions, or analysing data. The more informed you are, the better equipped you’ll be to find an effective solution.
• Brainstorm multiple solutions without judgment: Let your creativity flow freely. Generate as many potential solutions as you can, no matter how outlandish they might seem at first. This divergent thinking can lead to innovative approaches you might not have considered otherwise.
• Evaluate and compare potential solutions: Once you have a list of possible solutions, critically assess each one. Consider factors such as feasibility, resources required, potential outcomes, and possible obstacles. This analytical approach helps you narrow down your options to the most promising ones.
• Break the problem down into smaller, manageable steps: Large, complex problems can be overwhelming. By breaking them down into smaller components, you make them more approachable and easier to tackle. This method also helps you identify specific areas that might need more attention or resources.
• Develop a step-by-step action plan: Once you’ve chosen a solution, create a detailed plan for implementation. Outline the specific steps you’ll take, set deadlines, and allocate resources. This roadmap will keep you focused and organised throughout the problem-solving process.
• Implement the chosen solution with confidence: With your plan in place, it’s time to take action. Move forward decisively, trusting in the thought process that led you to this solution. Remember, even if things don’t go perfectly, you can always adjust your approach.
• Monitor progress and make adjustments as needed: Regularly assess how well your solution is working. Be prepared to make tweaks or even significant changes if you encounter unexpected challenges. Flexibility is key in effective problem-solving.
• Reflect on the outcome to learn from the experience: Once you’ve resolved the problem, take time to review the entire process. What worked well? What could have been done differently? This reflection helps you refine your problem-solving skills for future challenges.
• Practice problem-solving regularly to build skills and confidence: Like any skill, problem-solving improves with practice. Seek out opportunities to solve problems in your daily life, work, or even through pussles and brain teasers. The more you practice, the more natural and effective your problem-solving abilities will become.

Specific Techniques for Enhancing Problem Solving Skills

Let’s dive deeper into how to improve analytical and problem solving skills, how to improve complex problem solving skills, and more.

Analytical and Logical Reasoning

To learn how to improve logical reasoning and problem solving skills, and boost your analytical prowess, follow the tips below:

• Play strategy games: Chess, Sudoku, and similar games can sharpen your analytical skills. They force you to think several steps ahead and consider multiple possibilities.
• Practice logical pussles: Engage in logic problems regularly to strengthen your reasoning abilities. Crosswords, riddles, and brain teasers are great for this.
• Study mathematics: Math is the language of logic. Improving your math skills will naturally enhance your analytical thinking. Even if you’re not a “math person,” basic mathematical concepts can significantly boost your problem-solving abilities.

Creative Problem Solving

Wondering how to improve creative problem solving skills? Try these techniques:

• Brainstorm without limits: Let your imagination run wild. The crasiest ideas often lead to innovative solutions. Use techniques like mind mapping or free writing to get your creative juices flowing.
• Use mind mapping: Visualise problems and potential solutions to spark creativity. This technique helps you see connections you might have missed otherwise.
• Take breaks: Sometimes, stepping away from a problem allows your subconscious to work its magic. Ever noticed how great ideas often come to you in the shower or while taking a walk? That’s your subconscious at work.

Critical Thinking and Decision Making

For those pondering how to improve critical thinking and problem solving skills or how to improve decision making and problem solving skills, consider these strategies:

• Question assumptions: Don’t take things at face value. Always ask “why?” Challenging assumptions can lead to breakthrough solutions.
• Consider multiple perspectives: Look at problems from different angles to develop a well-rounded view. Try to put yourself in others’ shoes to gain new insights.
• Use decision-making frameworks: Tools like SWOT analysis, decision matrices, or the Eisenhower Box can help structure your thinking and lead to better decisions.

Enhancing Problem Solving Skills for Specific Exams

Preparing for exams requires a targeted approach. Here’s how to fine-tune your skills for specific tests:

If you’re wondering how to improve problem solving skills for JEE, try these tips:

• Understand the syllabus: Know what topics are covered and focus your efforts accordingly. This will help you prioritise your study time effectively.
• Practice time management: JEE is as much about speed as it is about accuracy. Learn to pace yourself and know when to move on from a difficult question.
• Join study groups: Collaborative learning can expose you to different problem-solving approaches. Explaining concepts to others can also reinforce your own understanding.

For those wondering how to improve problem solving skills in physics:

• Master the fundamentals: A strong grasp of basic principles will help you tackle complex problems. Make sure you have a solid foundation before moving on to advanced topics.
• Use mnemonics: Create memory aids to recall important formulas and concepts quickly. This can be a lifesaver during exams when time is of the essence.
• Solve problems daily: Consistent practice is key to improving your physics problem-solving skills. Set aside time each day to work on physics problems, gradually increasing the difficulty level.

Mastering Problem-Solving Skills: A Lifelong Journey

Mastering how to improve problem solving skills is a lifelong journey. It’s not just about acing exams or excelling at work – it’s about equipping yourself with the tools to navigate life’s challenges with confidence and creativity.

Remember, every problem you face is an opportunity to grow. Whether you’re debugging stubborn code, tackling a tough engineering problem, or just figuring out your daily schedule, each challenge helps you build your problem-solving muscles.

So, keep practicing, stay curious, and don’t be afraid to make mistakes. Embrace challenges as opportunities to learn and grow. After all, some of the world’s greatest discoveries came from problem-solving gone “wrong.” Who knows? Your next “failed” solution might just lead to an incredible breakthrough that changes the world.

As you continue on your journey to become a master problem solver, remember that the skills you’re developing are invaluable in every aspect of life. They’ll help you in your career, in your personal relationships, and in achieving your goals. So keep pushing yourself, keep learning, and never stop asking “How can I solve this?”

Frequently Added Questions (FAQs)

What are the key techniques to improve problem-solving skills.

The core techniques include breaking down problems into manageable parts, brainstorming a wide range of solutions, carefully evaluating options, implementing the best solution, reviewing the outcomes, and iterating based on what you’ve learned. Regular practice and exposure to diverse problems also play a crucial role.

How can I enhance my problem-solving skills in programming?

To improve your programming problem-solving skills, practice coding regularly, tackle coding challenges on platforms like LeetCode or HackerRank, learn and apply various algorithms and data structures, engage in pair programming, and regularly review and refactor your code. Additionally, working on personal projects can provide real-world problem-solving experience.

What role do problem-solving skills play in the workplace?

Problem-solving skills are crucial in the workplace for handling daily tasks, managing projects, resolving conflicts, and driving innovation. They help employees navigate challenges, make informed decisions, and contribute to the overall success of the organisation. Strong problem-solving skills can also lead to career advancement opportunities.

How can I improve my analytical and logical reasoning abilities?

To boost analytical and logical reasoning skills, engage in activities like solving pussles (e.g., Sudoku, crosswords), playing strategy games (e.g., chess), practicing logical reasoning problems, and studying mathematics. Reading books on logic and critical thinking can also be beneficial. Regular practice and challenging yourself with increasingly difficult problems is key.

What are some ways to boost creative problem-solving skills?

To enhance creative problem-solving, engage in open-ended brainstorming sessions, use mind mapping techniques to visualise problems and solutions, practice lateral thinking exercises, and allow time for ideas to incubate. Exposing yourself to diverse experiences and perspectives can also stimulate creativity. Remember, sometimes the most innovative solutions come from combining ideas from different fields.

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How to Effectively Practice CP + Problem Solving Guide

This is a slight tweak of a practice guide I wrote a while ago on USACO reddit since I thought it could be helpful to people here. Some USACO specific sections or extra clutter I left out here that aren't needed for a general audience. This should cover all general cp advice I have so I never have to retype.

Introduction

This is a post on how I believe is the best method to practice modern day competitive programming based on my experiences. I assume you already have some knowledge and know simple things like binary search and dfs/bfs, but read the footnote if you are complete beginner (never code, solved <50 problems, div2 A/B too difficult, grey or stuck low pupil).

First, a quick tl;dr of the practice strategy before a bunch of specifics and explanation:

In short, mostly you only need to use codeforces (no matter what contest you're training for), find a rating range where you can solve around ~30-40% of the time on your own, and just grind down the problem set tab in reverse order of id (the default sorting). Also take part in every live contest you can, and virtual any live contests you miss. Also, if your primary goal is some goal outside of codeforces (let's say USACO, but could replace with any OI or if ICPC replace instance of OI with ICPC) Approximately once per week (probably on each weekend), I recommend you virtual an OI contest then upsolve the ones you understand the editorial for after. This should be old USACO contests until you finish all in the past 5 or so years, then use OI checklist to find new contests. Make sure you go for subtasks just as you would in real contests when doing so.

Some parts of this method may seem strange to you, so I'll explain in more detail and comment on why I believe it is the best method, and give some proof. If you're too lazy to read all of it, the most important parts of this article are bolded . Also, I am assuming you are able to practice somewhat regularly (at least a few days of practice done each week for multiple months), and this practice is unlikely to work if you don't. However if you really want to improve fast, ideally practice should be daily, no breaks. .

Goal of Practice

First off, what is the main goal in practicing efficiently? I would argue you want to come across as many subtle ideas and concepts as quickly as possible and learn to intuitively realize when to apply them. This is what my practice method is centered around.

Another important goal is you should also feel discomfort in effort of trying to think new ideas as much as possible, but don't mistake this as time being confused with discomfort having no idea what to do. Actively making new insights as fast as possible is the state you should be in a lot during live contests and need to endure actively thinking new ideas while trying to not repeat same ideas in your mind. But when you have no clue how to approach/understand a solution to a problem, you are more likely to lose focus and are not helping yourself, so you want to minimize this.

Why Codeforces?

So, why only codeforces? Well, recent codeforces problems do a decently good job of introducing a large variety of concepts, particularly in the 2000+ rating range. Thanks to the large standards of wanting non-standard problems each contests, many small math tricks and greedy techniques are introduced, along with standard algorithms and data structure appearing decently enough. This is why I think they are the best collection of problems, as opposed to many other judges that are more standard and less diverse or innovative. Recent codeforces contests are by far better than old contests however, so that is why you should grind down the problems from most to least recent in the problem set tab. If you have done all contests later than contest 450, you should probably start using another judge and start doing more virtual contests, but at that point you probably don't need this guide.

How to Approach Problems in Practice

Alright, so codeforces seems good. Why only a rating range where you can solve ~30-40% of the time? Shouldn't you be practicing coming up with solutions on your own? Well, like I said earlier, you want to come across as many concepts as quickly as possible. If you're able to solve ~80%+ of the problems you're doing on your own, even if it takes a while, or in fact especially if it takes a while, you are not using your time most effectively, as you were already able to come up with the concept on your own. It is OK to read editorials often , that is where you actually learn new things. Binary search on the problem set tab to find a rating range of problems that fits the ~30-40% specification, and I recommend the rating range to a few hundred points wide. You can just shift range upward whenever lower end feels easier and you're solving more.

Well, the next natural question is how long should you take before reading editorials? I will argue only spend 15m thinking, after that if you're still having ideas keep thinking, but if you're just stuck read the editorial. However, if reading the editorial gives you new ideas continue thinking again. Sure, you may discover a trick you came up with yourself you can use later after a long time thinking, but was it worth spending 3h coming up with the solution on your own when you could've gone through 2 or 3 more problems if you read the editorial instead. However, going through too hard problems is just as bad is going through too easy problems. It is not worth spending 4h understanding a 3000 rated problem when you could learn much more concepts from 4 2300 rated problems in the same amount of time (if that's good for your skill level). That's why I say ~30-40%, this is usually the point where you can understand the editorial relatively quickly but aren't able to see the concepts on your own. Also, this is another reason to use codeforces instead of other sources, the problems are shorter so you can get through more faster and it is easier to find many problems of similar skill level .

Some important notes, however, are to take the 15m of thinking very seriously and implement every problem . This is extremely important!!! you should only be looking at editorial when you are really out of ideas and trying to think longer will just make you unfocused or reiterate old ideas. In other words you should feel mentally exhausted!! (or you're not working thinking hard enough). Don't be lazier than you would be in a contest, don't give up because you don't want to think harder on details, don't think/implement leisurely. It is important to practice making observations on your own, and you should be solving problems in the range more and more often as you go down the problem list, that's how you know you're improving. If you're not improving, you are likely not exhausting yourself thoroughly. You may think you can get through more concepts earlier without implement too, and this would fit the main goal of practice better, however, it's important to always implement every problem that isn't completely trivial, even if you mind solve it on your own, as you will remember it better and often you will realize you didn't understand the details as well as you thought before implementing. Always implement before reading editorial if you think you have idea, even when not sure, and don't look at others implementation before you solve even if you read editorial except for last resort.

I also recommend timing yourself when doing problems, at least while implementing. This will help you stay focused and improve your implement speed (which is important so you don't waste time implementing in contest). If you record your times you should hopefully see yourself getting faster for a fixed problem difficulty :).

When you finish a problem, make sure you reflect on techniques and mindset used and how you could generalize thought process to solve other problems more efficiently (imagine you were teaching someone else best way to approach similar problems). Similarly do this when you learn new algorithms or tricks and imagine how you would come up with on your own. Try to come up with your own list similar to one I have in "extra advice how to think" section. The goal is to find short fundamental list of questions to ask yourself that will always lead to the solution, not just categorizing by algorithms or techniques used. Similarly reflect on what can go wrong and how to consciously avoid mental traps. Also, it can be good to look at others solutions after you finish a problem quickly to see if there are any implementation tricks you don't know, and similarly reflect how you could make your code more concise.

When to Learn Algorithms/Data structures

Next thing to come up is when in this am I supposed to learn new standard algorithms and data structures? I advise when you come across an algorithm or any other concept (maybe math idea) in an editorial you don't know about to immediately find and read an article about it, implement in the context of this problem, and then continue just moving down the problem set tab. You can usually find an article on USACO guide, cp-algorithms, or a codeforces blog. The idea behind this is that algorithms should come up at a rate according to their relevance, so if the algorithm really is important you should see it in more problems soon, and you don't need to go looking for more problems with the topic. Similarly, it is important to see algorithms in context, which is why you should not practice by topic , as you will likely miss out on many more subtle techniques and tricks not in a topics list and get too used to knowing the algorithm used ahead time when you should be trying to figure that out in the 15m thinking time.

However, if you want a break or have other extra time when you can't do problems, reading through random algorithm articles in the locations listed above is a good way to expose you to some new ideas. But it is still more important to be actively solving problems when you can.

Live Contests

The number one thing that probably looks wrong with this practice method, despite the reasonings I gave earlier, is that you seem like you are not practicing solving problems on your own often enough. This is where live contests come in. It is important to take part in as many live contests as possible from every judge you can (except ones where every problem feels too easy) . This is where you practice thinking on your own, and if you look enough there are tons of contests all the time, particularly high quality ones from atcoder and codeforces. You should also upsolve the hardest problem you didn't solve during the contest, however, after that you should just go back to the codeforces problem set grind unless there are more problems from the contest within your practice rating range on codeforces. Lastly, to make sure you're taking enough contest, take every codeforces contest you miss that would be rated for you as a virtual contest.

Also, if your primary objective is some other contest (say USACO/OI but can replace with ICPC), you should do OI virtual about once per week as subtasks are becoming more important in USACO plus probably good to have more extended focus practice anyway. You also want to shift practice to doing mostly OI virtuals the week before a USACO contest begins. Make sure for these virtuals you are going for maximum points like in a real contest which may mean implementing subtasks, not just implementing full solves (or whatever other contest specific traits that differ from codeforces). If you aren't practicing a ton or you feel virtuals are taking too much time away from doing codeforces practice maybe do every other week instead of every week.

Scheduling Practice

This is less important but more just some pointers on scheduling time to practice consistently. I think it is obviously best to practice daily, and it isn't as hard as you may think it is if you build up good habits. I think it is good to have a regularly scheduled time where you can practice each day , as this makes it more of a consistent habit. Similarly, if you can set aside a specific location to practice as well that would be good , as this can give your mindset the habit that a specific time and place is for practicing only, and you build focus**.** Try to practice at least 90m for your scheduled time , but preferably longer. And get off discord!!! when you're practicing in the designated time :clown:.

Besides scheduled practice time, you can probably fit in more practice time in some or many days in different ways as well if you are serious. For example, I think it is good to memorize some problems at the beginning of each day, maybe a bit harder than you'd normally practice, and think about them all day during school, shower, eating, etc., or maybe the same problems for a few days. This helps you practice thinking more on your own. Also, when you have free time in class or while in car and someone else is driving or something, this is a good time to read algorithm articles. When I went to public school I also bought a portable keyboard to practice in class and spent most school lunch days in the library doing problems, but this might be overkill. Point is find all times of day to practice any way possible when you can, but most import is the scheduled practice time.

Adjustments Closer to Big Contest

If you are training for some main goal (hopefully for the past several months at least, following above methods), when you are within a few weeks away of big contest, start spending more practice time on vc's for that contest, and look over the syllabus/relevant ideas for that contest if list exists . Also consider if you are in these pitfalls:

You are too slow at working out ideas or implementation => do more fast-paced contest vc's, time yourself in other practice.

You are bad at allocating time in OI/ICPC style => focus on more relevant vc's and practice subtask allocation, figuring out which problems to work on, and when to move on like in real contest.

Still not able to make big insights that seem to come out of nowhere => try more guessing and some atcoder lol.

Hopefully this was somewhat useful to some of you, and gives you a comprehensive guide on how to practice for USACO and competitive programming in general. Please share this with others if you think it is useful.

For any more experienced people, let me know if there is anything you strongly disagree with what I said, I'd be interested to hear your viewpoint, though you're unlikely to change my mind :).

**I recommend the beginning of the usaco training page to complete beginners . I think it is a good way to start out as it guides you on the basics, and you should be able to start as soon as you know the very basics to a programming language, preferably c++ (you can use codeacademy to learn basics, it should take only a couple days max. you learn other parts about the language as you solve more problems and googling as needed). However, as soon as you finish chapter 1 or the problems feel easy (or if codeforces is still too intimidating maybe hard max finish chapter 2), that is when I recommend you start using this practice method, and perhaps also try some problems from the cses sorting and searching section. However most people reading this should already have some experience.

Sources mentioned: USACO — http://www.usaco.org Codeforces — https://codeforces.com Atcoder — https://atcoder.jp CSES — https://cses.fi/problemset/ Training gate — https://usaco.training OI Checklist — https://oichecklist.pythonanywhere.com Cp-Algorithms — https://cp-algorithms.com USACO Guide — https://usaco.guide Codeacademy — https://www.codecademy.com/catalog/language/c-plus-plus

Extra Advice How to Think to Solve Problems

Overall, just make sure you are always thinking new ideas and repeatedly combining old observations to make new ones. Don't worry about solving all at once, just think one small step at a time! Usually this means think what do you know for sure, then use to guess ideas on properties and direction and check if you can prove, combine your previous observations, then repeat. When really stuck, guess more extreme (it is another thing people who aren't improving don't do enough). Actually write down you're observations and make sure you're writing new things as fast as possible, even when seems small or irrelevant. But for some more direct tips, try going through the following checklist when approaching a problem:

• look at everything from perspective of binary (both bit representation and splitting things in halves) and graph (pairs in input), or sometimes as geometry coordinates
• think how information you have can be reused (like dp but more general, eg 2ptr or extending construction, sweepline, split query into reusable known parts), ask what is dependent and how, order by dependency. also try making one choice then and get same problem then induct (eg greedy, mst, dp, decision tree like trie, ask "what do i know for sure"), or combine smaller problems to get answer (eg range dp, d&q, mitm), so can reuse info of smaller problem.
• reduce things to as simple as possible, compact representation of info, get rid of redundant transitions/states/etc. what is minimum needed for condition to be true? when something changes or decision made what is minimum that actually matters? sometimes combine operations into simpler one (eg try turning operation into something can binary exponentiate). bound everything as tight as possible and use to reduce states to consider. is answer/construction equivalent to bounds/minimum conditions (guess when stuck)?
• make formulas out of everything, expand/rewrite as many ways possible (even simple like |x| -> +-x). think about related formulas to transform (eg combo) and other representations (don't forget matrix/polynomial).
• visualize everything, draw things out
• look for structures like montonicity, concavity, etc. (eg bsearch/dp opt) along with new conditions/constraints implied (eg sqrtn distinct of n total), and do this for every part of problem, whether specific part or entire structure of solution
• go through testcases by hand (both initially with brute force and with your current best ideas), maybe also make generator/brute force checker if stuck to further look for patterns.
• don't think same things over and over, write down everything you think and try to always write down new ideas, every small new observation is progress and may be able to be combined with other ideas eventually, but rethinking same things will not help
• think of simplified cases then extend/reduce to them (reduce a[i] = 0/1, array->tree->graph, 2^x->k) or imagine assuming something you wish exists already exists (like data structure often range query, constraint eg for bsearch, previous knowledge, etc.) and solving from there, chances are thing then does exist if helps
• reverse/change ordering of process (eg change order to simpler like change general add/delete to add/[delete most recent op] offline) or look at inverse (especially for counting) or just view problem in different way in general, restate problem/conditions in as many different ways as you can to get new perspectives. nice transformation usually means right direction (eg difference array).
• if something reminds you of standard algorithm, or you find too slow solution for some part, think of every way you know how to do that standard thing and see if any modification relates to what you are doing, and think deeply what parts can be changed for specific problem
• if something seems random in statement like any abnormal constraint or is similar to known problem but different in some way, is probably key to solving so consider why it is put in statement
• don't forget sometimes can brute force small choices or if too many choices can pick random one or something that stands out (like max/min, only closest on left/right, etc.), extremals is often key. think carefully and guess what not matter if problem seems too hard initially. in constructive/interactive with many options can likely solve with only small subset of options.
• don't overcomplicate. try multiple directions, if too many steps or edge cases probably not right direction. almost always a nice easily provable solution. guess nice things (eg simplest greedy/construction), hope they work, then check but don't get stuck forcing path. take step back when clowning on small details even if you know it is right general direction
• try focusing on answer for one element at time instead of entire process (like in counting or creating bsearch condition, local easier to update for queries), or sometimes opposite (eg graph out all solutions, know ahead of time offline). in general change scope of thinking
• believe you can solve every problem, but also treat every problem as a challenge that you take one step at a time. even most standard ideas you can learn on your own if you treat same way as any other problem
• if something you remember very vaguely seems similar but you don't remember source and barely remember details, don't waste time trying to remember old thing, just start resolving from scratch
• as stated by Perpetually_Purple in comment below, sometimes can try to cheese with random/heuristic if running out of time. especially true for OI contests with subtasks
• sometimes can split problem into parts which can be solved differently based on constraints (eg sqrt decomp, small to large, upd and qry compute different parts, even/odd).
• also break into independent problems (eg intervals that don't affect each other, solve x and y coordinate separately). when dependent on multiple things, process in order that gets rid of thinking about one and only worry about others (eg sweep one dimension, query other).
• map things to a canonical form (eg lexographically minimal) or map representations that are equivalent to help with counting or alternate way to view solution. (eg think of greedy idea to get specific configuration then have counting dp mimic the greedy method to not overcount, find simple idea for single query then speed up multiple queries by precomputing conditions when add during greedy to speed up).
• imagine assuming you know solution ahead of time and analyze or fix choices ahead of time and solve rest, can use this to prove things equivalent, choices not to consider, or properties of optimal configuration.
• try only computing minimum necessary at each point of time, especially for update/query. can sometimes use amortized/lazy arguments (eg keep track of covered intervals in set, lazy prop on segtree).
• ask what stays the same and what changes. how does a single operation affect properties of a problem (sum/difference of elements, always progress towards goal, reversable, etc.)? when doing testcases by hand guess these types of things then prove/disprove. use these properties to prove things like which choices are optimal or what is bottleneck to bound on answer.
• Similar to 3 and 10, try compressing groups of things and solve over those group when relations within them are irrelevant, and keep updating when you can simplify further throughout process (eg compress cycles, scc, biconnected components, directed mst).'
• When guessing idea, make sure you are listing through all assumptions being made and that those assumptions you know for sure hold true and completely encompass the problem. Also make samples around idea of what you think could go wrong, and use that to help you prove or disprove idea. If you're taking too long disproving wrong ideas, you likely need to go more one step at a time, don't guess extreme until more stuck.
• If stuck working out details when have main idea, work out more testcases by hand and/or write detailed pseudocode and find what steps you are not entirely sure what they work and think harder. Don't be lazy about writing details!

Also it is good to use problem constraints to guide your initial direction of thinking, but don't let it constrain you to specific ideas. And whatever you do don't misread the problem , better to spend slightly longer reading and understanding correctly than solve wrong thing.

Implementation Tips

First check briefly that you are not missing easier idea/method to implement. That will save most time.

Try to have clear idea of each segment of code you will write, then write as fast as possible. Sometimes you don't have clear idea of entire code you write and only general outline, and that's ok, but in your mind have different parts of code in small chunks and have each small chunk planned out clearly before you write then think if needed before writing next part. Try to keep plan your code to be as concise as possible while still easily readable and make it where you are not rewriting same thing multiple times. If you keep rewriting, you need to step back and plan out better, check your ideas.

Also for debugging, just make a bunch of print statements in code and look for problems. Try to binary search and figure out where in the code the outputs are first not what you'd expect. If you realize some part is not right, don't get stuck making small edits trying to fix, go back to planning and rewrite when clear. Also try working through some examples by hand following steps of code, and read through every single line of code. It is likely the mistake is somewhere where you were sure you couldn't mess up lol.

Also adding one sentence comment to code on main idea of every problem might be nice if you ever need to refer back.

Allocating Time in OI Contest

I'm assuming 3 problems in 4 hours (adjust scale as needed). I usually read all 3 problems in first 15 minutes, then spend about 15 minutes each to think about each problem and decide order of difficulty I find easiest. If I fully mind solve one in that time I immediately implement, otherwise I do as follows. I then try to divide the next three hours to be roughly even among the three problems, and try the problems in order from easiest to hardest.

While focusing on a problem, it is very important to stay focused on only that problem. For most of hour on problem should implement as soon as you full solve but only implement subtasks to test ideas, if you think it help you towards full solution, or you are completely out of new ideas (in which case move on after implementing subtask if u still don't have new ideas). However, if you already use up ~50min for that problem and still don't know full solution and won't reach in next 5min, even if you think you could make more progress, just implement what subtasks you know and move on. It is important to actually move on as you may have wrongly assessed which problem was easiest so you want to have time to try all the problems (this has been my downfall multiple times in past). This means once you move on don't have more lingering thoughts usually and fully focus on next problem.

Math + CS Practice

If you are practicing math olympiad and cs olympiad, or just want some reading material that might help you, try reading some of and doing some problems from this combo book . Overall it will be better for you to just do be actively solving more problems for cp practice, but if you have some other free time it is a pretty good read and cp is basically olympiad combo + data structures + implementation anyway.

Practicing for math olympiad in general will also help you with competitive programming, but if you are only focusing on cp it is better to just work on cp problems.

Extra Motivation

In everything in life, the key to success is learning to find fulfillment in every small step you make towards progress. Related to cp, every problem solved and every day of practice is one step closer to your competitive programming goals. When solving a problem every new observation is one step closer to finding the solution.

Also, make sure you know your priorities and what you really want out of life, don't have regrets. If you really want to be good in cp, stop wasting time, stop taking days off, start solving problems as much as you can and you will find success. Obsess over what you want most until you achieve it.

How To Improve Speed And Accuracy in JEE

Speed and accuracy have a really complex relationship. Perhaps they are the most vital things to take care of while attending a highly competitive exam like JEE Main or Advanced. If you don't have the required pace, you will be snatched off your opportunity and left with the regret of how you could not attempt an easy question. On the other hand, accuracy is quite subtle when it comes to making you regret. You are going to realize your mistake after leaving the hall, or sometimes after the result comes out. But speed and accuracy -- they just do not get along. Precision requires time, and speed is all about reducing the amount of time. So let me be honest here, it's not easy to manage both of them at once, in fact, it's one of the hardest jobs - but maybe not as hard as cracking the exam with an acceptance rate of 2% and 12 lakh candidates.

Did I Make You Nervous? Don't Be.

Nervousness and stress are the arch-enemies of speed and accuracy. When you are nervous, your brain cannot function it its expected ability. On the other hand, when you are confident and stress-free, your brain produces Dopamine, which improves your concentration and problem-solving skills, eventually improving your speed and accuracy.

The Art Of Time Management

Time management is necessary for learning or revising as well as for acing the exam. When done correctly, it is nothing short of an art. During revision, you need to make a compact plan for how you are going to handle each and every topic. You have to stick to your own routine. Procrastination is not an option. When you have a specific deadline for finishing a topic, your brain will start working faster. Your subconscious mind will be trained to respond to the urgency, and nothing will be able to distract you at that time.

During your exam, you have to take a similar approach, only less aggressive. This is not a training for your brain, so precision matters. Your strategy should be well-planned, but cautious. Start with the easier problems and make your way up to the harder ones. If you practiced time management during your mock tests and revisions, speed should not be an issue for you, and the extra cautiousness would take care of the accuracy.

The Cognitive Approach To Problem-solving

During your revision, gradually increase the level of problems' difficulty, and try to reduce the usage of pen and paper. Initially, you might face problems, the answers might not be accurate, but as you continue, the improvement in speed and accuracy will be noticeable.

Soon you will be confident enough to do short mind calculations and skip steps to solve the problems quickly. But don't be overconfident. Always be extra cautious during your exam.

Creative Thinking Makes A Difference

No one will check if you solved the problem in a traditional step by step method, so you have the chance to be creative while solving a problem. During revision and mock tests, always try to find a better and quicker way to solve a question. You will discover a lot of reliable shortcut methods that were never taught to you. Your problem-solving speed would increase as you keep thinking creatively.

Now, here is the important part, don't try this during the exam, unless really necessary. Stick to the methods that you are 100% sure about. Your brain has already been trained to function faster, don't be deliberate to prove that during your exam.

Smartness Is As Important As Knowledge

You have to be smart while solving the problems. Unconventional ways are not discouraged at all. Learn some calculations beforehand. Memorise some square roots, cube roots, derivatives, and integrations. Skim through the shortcut techniques, memorize multiplication table up to 20, apply elimination technique when necessary. These techniques might not be good for learning, but they will prove really helpful in JEE. So practice these during your revision and apply only when they are the most viable options.

Don't Get Stuck

Never, and I repeat, never get stuck in a single problem during the exam. Once you make a time management strategy at the beginning, you would have an idea how much time you can spend on a single problem. So stick to the plan and avoid being stuck at a particularly difficult problem. Move on, solve the other problems, and come back to it later if you are left with enough time for a revision.

Practice - The Ultimate Advice

Okay, don't skip the paragraph - I know you are tired of hearing people advising you to “practice, practice and practice”. But trust me, this reminder is necessary, as practicing problems on the same concepts for 2 years is a really frustrating job. Just think about it for a moment, how much have your problem-solving speed improved since the first day you started studying for JEE? No matter how much boring this advice may sound, it always proves to be the most important one.

Prepare through mock tests with proper timing, and every time try to finish quicker than the last time. If you are taking offline mock tests, and your plan is to appear for the online JEE exam, be careful with the time management. The shading process does eat up a lot of time. Be careful about silly mistakes. Try to identify if there is a pattern in what kinds of mistakes you are making, and rectify them.

Double-checking, Something We All Love To Hate

After answering the 90 questions, you will not feel too enthusiastic about the idea of going through all of those questions and rough works once again, searching for some silly mistakes which you are sure you didn't make. But double checking is the key to achieving accuracy. The options given for an MCQ can often be misleading and too close to reason out. Keep your rough work organized to make the process less frustrating. You will thank yourself later for saving at least 8 to 12 marks, which is enough to make a significant difference in your rank.

So, that was all for this article. Do your best to establish a friendship between speed and accuracy; the result will be worth the effort. Best of luck.

Harshita Srivastava IIT Kanpur 13 September 2019

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• Published: 17 August 2024

Evaluating single multiplicative neuron models in physics-informed neural networks for differential equations

• Melih Agraz 1  

Scientific Reports volume  14 , Article number:  19073 ( 2024 ) Cite this article

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• Applied mathematics
• Applied physics

Machine learning is a prominent and highly effective field of study, renowned for its ability to yield favorable outcomes in estimation and classification tasks. Within this domain, artificial neural networks (ANNs) have emerged as one of the most powerful methodologies. Physics-informed neural networks (PINNs) have proven particularly adept at solving physics problems formulated as differential equations, incorporating boundary and initial conditions into the ANN’s loss function. However, a critical challenge in ANNs lies in determining the optimal architecture, encompassing the selection of the appropriate number of neurons and layers. Traditionally, the Single Multiplicative Neuron Model (SMNM) has been explored as a solution to this issue, utilizing a single neuron with a multiplication function in the hidden layer to enhance computational efficiency. This study initially aimed to apply the SMNM within the PINNs framework, targeting the differential equation \(y'-y=0\) with boundary conditions \(y(0) = 1\) and \(y(1) = e\) . Upon implementation, however, it was discovered that while the conventional SMNM approach was theorized to offer significant advantages, multiplicative aggregate function led to a failure in convergence. Consequently, we introduced a “mimic single multiplicative neuron model (mimic-SMNM)” employing an architecture with a single neuron, designed to simulate the SMNM’s conceptual advantages while ensuring convergence and computational efficiency. Comparative analysis revealed that the real-PINNs accurately solved the equation, the true SMNM failed to converge, and the mimic model was highlighted for its architectural simplicity and computational feasibility, directly implying it is faster and more efficient than real PINNs for the solution of simple differential equations. Furthermore, our findings demonstrated that our proposed mimic-SMNM model achieves a five-times increase in computational speed compared to real PINNs after 30,000 epochs.

Introduction

Artificial Neural Networks (ANNs), the cornerstone of modern machine learning techniques, are designed through mathematical modeling to emulate the functionality of biological neurons. Demonstrating effectiveness across a myriad of scientific domains, ANNs have been applied to challenges in regression, classification, time series prediction, and function approximation, showcasing their versatility and computational power 1 , 2 , 3 , 4 . Among the significant advancements in this field, Physics-Informed Neural Networks (PINNs) stand out as a innovative approach that leverages ANNs to approximate solutions to differential equations pivotal to physical sciences 5 . Rooted in the power of deep neural networks to approximate virtually any function-given adequate training data-PINNs distinguish themselves by integrating physical laws directly into the learning process. This integration allows PINNs to not only process empirical data but also to encode knowledge of the system’s governing physical principles, making them particularly adept at solving mathematical problems defined by partial differential equations (PDEs). The capability of PINNs to infer from data and approximate solutions to PDEs, without the need for explicit forms of the equations, marks a development in computational methods. This approach extends problem-solving possibilities within various fields including physics, engineering, and computer science.

A common PINNs utilizes the structure of a Multi-Layer Perceptron (MLP) for its representation, recognizing the foundational advancements in neural network architectures, including the widely utilized MLP 6 . In our study, we introduce an innovative approach by integrating the Single Multiplicative Neuron Model (SMNM), characterized by a hidden layer consisting of a single neuron with a multiplicative aggregate function, into the PINNs framework. This integration is particularly novel, as the conventional architectures employed in PINNs, such as MLPs, have not previously been adapted to incorporate SMNM characteristics. The SMNM model, first introduced by Yadav et al. 7 , has catalyzed subsequent research aimed at refining and expanding its predictive proficiency. For example, Egrioglu et al. introduced an adaptation of the SMNM model for forecasting, integrating both moving average and autoregressive terms 8 . Subsequent research by Gundogdu et al. refined the model further by incorporating a Gaussian activation function, streamlining the optimization of both the activation function and the network’s weights in a unified step 9 . Cui et al., have significantly broadened its applicability, especially in time series forecasting 10 . Innovative training methodologies, like the differential evolution process introduced by Bas 11 and the sine cosine algorithm by Kolay 12 , have further optimized SMNM’s performance. The application of SMNM to various domains, from classification problems tackled by Kandpal et al. 13 to energy consumption prediction by Wu et al. 14 , illustrates the model’s adaptability and potential for broader applications.

In this study, we sought to address the architectural selection challenges in PINNs through the use of the SMNM. Initially, our approach aimed to leverage the unique capabilities of SMNM, characterized by its single neuron, to enhance computational efficiency and solution accuracy within the PINNs framework. This application was intended to solve differential equations, such as \(y'-y=0\) with specific boundary conditions, promising rapid and precise approximations that could mark an advancement in applied mathematics and machine learning. However, during our investigation, we found that the true SMNM, when accurately applied, failed to converge to the solution. This unexpected outcome led us to develop a ‘mimic multiplicative neuron model (mimic-SMNM)’ adapted to PINNs (mimic-SMNM-PINNs). This new model, designed to simulate the SMNM’s theoretical benefits while ensuring practical applicability and convergence, employs an architecture consisting of 2 hidden layers and a single neuron. This model not only addresses the architectural selection challenge but also achieves a notable increase in computational speed compared to traditional PINNs, without sacrificing solution accuracy. This evolution of our study underscores the importance of adaptability in research and opens new avenues for future innovations in neural network models for complex problem-solving.

The paper is organized as follows: In the “ Methods ” section, we detail the theoretical underpinnings and practical implementations of PINNs and SMNM, providing a comprehensive overview of our approach. The “ Results ” section presents the findings from our experiments, showcasing the effectiveness of our proposed models. Finally, the Conclusion summarizes the key outcomes of our study, reflecting on the implications and potential future directions.

There are numerous challenges and problems that can arise during the design and training of ANNs. Some of the most significant issues include 15 : overfitting, underfitting, vanishing gradients, exploding gradients, getting stuck in local minima, imbalanced data, noise in the data, poor choice of hyperparameters, and deciding the number of neurons. Selecting the number of neurons in the hidden layer is a crucial issue in ANNs, often referred to as the architecture selection problem 8 , 11 . Yadav et al. 7 proposed the SMNM as a potential solution to this problem. PINNs employ feed-forward neural networks to solve PDEs and ordinary differential equations (ODEs). However, they face challenges in terms of time approximation and architecture selection. This study initially set out to integrate the SMNM concept within the PINNs framework to address both the challenges of time approximation and architecture optimization in solving differential equations. However, our exploration took an unexpected turn upon the realization that the true implementation of SMNM, when correctly applied within PINNs, did not converge for differential equations such as \(y'-y=0\) . This showed that a reevaluation of our approach and led to the development of a new strategy: the mimic SMNM model. This revised model seeks to encapsulate the theoretical advantages of the original SMNM while ensuring practical applicability, convergence, and enhanced computational speed within the PINNs framework, thus addressing the initially identified architectural selection challenge.

Physics informed neural networks (PINNs)

PINNs were first proposed by Karniadakis et al. 5 , 16 , 17 as an ANNs-based solution for solving partial differential equations (PDEs). The introduction of a residual network in PINNs has been widely recognized as a significant breakthrough. This residual network encodes the governing physics equations and utilizes the output of a deep learning network to derive a residual value within the PINNs network’s output 18 .

The general design of the PDEs solved by PINNs can be demonstrated by the following equation:

In this context, \({\textbf {u}} (\text{ i.e., } \mathbf { u(t,x)})\) represents the hidden solution with boundary ( \(g(t,{\textbf {x}})\) ) and initial conditions ( h( x ) ), where u is the target variable (for example, a wave), and \({\textbf {u}}_t\) is the derivative of u with respect to time t within the interval [0,  T ]. The variable \({\textbf {x}}\) is an independent variable within the domain \(\Omega \) . In this scenario, \({\textbf {x}}\) and t are the given inputs (for instance, time t and location \({\textbf {x}}\) ). \(N(u; \lambda )\) signifies a non-linear or linear differential operator with a set of PDE parameters \(\lambda \) .

In the process of solving the differential equation, \({\textbf {u}}\) is approximated using a fully connected deep neural network, with \((t, {\textbf {x}})\) as the input and \(u_{NN}(t,{\textbf {x}},)\) as the output. The left side of Eq. ( 1 ) ( \(u_t + N(u; \lambda )\) ) can be defined as f ( t ,  x ), i.e.,

The ANNs is created by hidden layers, where the inputs and outputs of layers are propagated over the network as

The shared parameters between u ( t ,  x ) and f ( t ,  x ) within the neural networks are learned by minimizing the loss function 19 or others. Figure 1 depicts the working diagram of the PINNs approach. According to Fig. 1 , a fully-connected deep feed-forward neural network is employed to approximate u(t, x). This approximation is then used to compute the loss of the initial conditions ( \(MSE_0\) ), boundary conditions loss ( \(MSE_b\) ), and residual loss ( \(MSE_r\) ) 20 .

In this context, \(MSE_0\) , \(MSE_b\) , and \(MSE_r\) correspond to the loss of initial conditions, boundary conditions, and the penalties on the residuals of the governing equations, respectively. To calculate residuals for MSEr, derivatives of the outputs (i.e., ut and Nxu) with respect to the inputs are required. Automatic differentiation 21 is used for this derivative calculation. The total loss value is optimized using an optimizer such as stochastic gradient descent 22 , the ADAM optimizer 20 or other similar techniques. The working diagram of the PINNs approach is shown in Fig. 1 . As per Fig. 1 , a fully-connected deep feed-forward neural network is applied to approximate u ( t ,  x ) ( \(\hat{u}\) is the prediction of u ). This approximation is then used to compute the loss of the initial conditions \(MSE_0\) , boundary conditions loss \(MSE_b\) , and residual loss \(MSE_r\) ,.

Physics-informed neural networks (PINNs) working simple schema.

Since the inception of PINNs, they have garnered significant attention within the fluid mechanics and scientific computing communities, leading to substantial progress. Karniadakis and his colleagues have extended these methods, resulting in a plethora of new variants. These include stochastic PINNs 23 , fractional PINNs (fPINNs) 24 , conservative physics-informed neural networks (CPINNs) 25 , parareal physics-informed neural networks (PPINNs) 26 , extended physics-informed neural networks (XPINNs) 27 , non-local PINNs (nPINNs) 28 , PINNs’ variational formulation based on the Galerkin method (hp-VPINN) 29 , parallel PINNs 30 , Bayesian PINNs 31 , and learning non-linear operators via DeepONet 32 . Additionally, Chen et al. 33 developed DFS-Net, an improved data-free surrogate model leveraging deep neural networks for solving PDEs within the framework of PINNs. They introduced an attention-based mechanism to enhance prediction stability and accuracy, addressing the limitations of traditional PINNs.

Single multiplicative neuron model (SMNM)

Artificial neural networks (ANNs) are computational models inspired by the architecture and functioning of the human brain. They consist of a significant number of interconnected processing units, known as artificial neurons, which mimic the function and structure of neurons in the human brain. ANNs are utilized to model complex patterns and relationships in data and are commonly employed in tasks related to machine learning and artificial intelligence. These tasks include image classification, natural language processing, decision-making, and more. In Fig. 2 , the operations performed by each neuron are demonstrated to be straightforward. Initially, the inputs are weighted and summed. Following this, the outputs are passed through a process known as the activation function, resulting in the generation of the final outputs, as depicted in Fig. 2 .

Architecture of fully connected classical neural network.

Where \(\Sigma \) in Fig. 2 refers to sum of product for the current neuron in as shown in Eq. ( 6 ) and \(\Phi \) is the activation function.

where \(x_1, x_2, \ldots , x_n\) are the inputs, \(w_1, w_2, \ldots , w_n\) are weights, \(\Phi \) is the activation function with u linear combiner and \(y=\Phi (u+b)\) is the output.

The concept of a single multiplicative neuron was first introduced by Yadav 7 . This novel approach employed a multiplicative function, marking a departure from the traditional additive-joining function used in the neural networks initially proposed by McCulloch and Pitts 34 . The architecture of a single multiplicative neural network is illustrated in Fig. 3 .

Architecture of a single multiplicative neuron model where \(\Omega (x, \theta )\) is a multiplicative function given in Eq. ( 7 ).

We can define \(\Omega \) in Fig. 3 as the in Eq. ( 7 ) below.

where \(x_i (i=1,2,...,n)\) input features, Y is the output, \(\theta \) is parameters for weights \(w_i\) and biases \(b_i\) and \(\Phi \) is an activation function.

mimic-single multiplicative neuron model (mimic-SMNM)

The mimic-SMNM integrates the concept of layer idea in the SMNM within PINNs (mimic-SMNM-PINNs), offering a perspective on solving differential equations through a neural network architecture that significantly apply the traditional additive models with single neuron.

Layers configuration

The architecture of mimic-SMNM is defined by an input layer and a hidden layer containing a single neuron. This minimalist configuration underscores the model’s departure from conventional designs, aiming to explore the computational and predictive efficiencies of such a structure.

Loss function

At the heart of the mimic-SMNM’s design is its composite loss function, which integrates boundary loss and residual loss. The boundary loss quantifies discrepancies between the network’s predictions and actual boundary conditions. In contrast, the residual loss assesses the network’s adherence to the governing differential equations by evaluating the residuals at specified points. The total loss is a summation of these components, guiding the optimization process towards solutions that respect the physical boundaries and the dynamics of differential equation’s.

Optimization strategy

The optimization of mimic-SMNM employs the Adam optimizer, chosen for its effectiveness in handling sparse gradients and adapting learning rates. This strategy is crucial for the convergence of models with highly constrained architectures and complex loss landscapes.

Training and prediction process

Training involves iteratively minimizing the composite loss over a specified number of iterations, with periodic logging to monitor convergence. This process refines the model’s parameters, enabling it to predict accurately the behavior governed by the differential equation. Predictive validation is conducted through comparisons against exact solutions, showcasing the model’s efficacy.

Aggregate function

Contrary to the initial portrayal, the mimic-SMNM incorporates a summation-based aggregate function rather than a multiplicative approach, aligning more closely with traditional neural network methodologies. This aggregate function computes the weighted sum of inputs, plus biases, a foundational aspect distinguishing it from the envisioned multiplicative aggregate concept. The significance of this correction lies in its impact on the model’s operational dynamics, where the essence of the aggregation shifts from a product-based to a sum-based mechanism. This adaptation ensures that the model retains the conventional approach of input integration, crucial for maintaining compatibility with established neural network practices, while aspiring to encapsulate the simplicity and efficiency benefits attributed to the original SMNM inspiration.

Experimental setup

In the experimental setup of the PINNs model, weights are initialized using the Xavier method, which is designed to keep the activation outputs and backpropagated gradients at a reasonable scale throughout the training process. This approach involves setting the initial weights according to a truncated normal distribution, with the standard deviation calculated based on the number of input and output neurons to the layer. Biases are initialized to zero. For training, the Adam optimizer is employed with a learning rate of 0.001, optimizing a loss function that includes both boundary and residual losses. The model architecture is configured to evaluate both the model’s prediction of the function y and the physics-informed residual f , ensuring that the trained model adheres to the underlying differential equations and boundary conditions specified for the problem. This setup facilitates an efficient training process, while also maintaining the fidelity of the model to the physical laws governing the system.

The PINNs method is an approach that leverages ANNs to approximate solutions to PDEs and ODEs. In this study, we propose the SMNM as a solution to the architecture selection problem within PINNs. However, SMNM integration did not converge; for this reason, we proposed mimic-SMNM-PINNs. We evaluated the accuracy of the methods by solving the equation \( y' - y = 0 \) , given the boundary conditions \( y(0) = 1 \) and \( y(1) = e \) . Figure 4 shows the approximation results of the three methods: real-PINNs, SMNM-PINNs, and mimic-SMNM-PINNs. The results show that SMNM-PINNs did not converge to the real solution, but mimic-SMNM-PINNs converged perfectly.

Comparative approximation results for the differential equation y’-y=0 with boundary conditions y(0) = 1 and y(1) = e, showcasing the performances of three distinct methods. Panel ( a ) presents the results for real-PINNs (real-PINNs with 5 layers of 20 neurons), illustrating their accurate solution of the equation. Panel ( b ) depicts the outcomes for SMNM-PINNs, which, in contrast, did not converge to the real solution. Panel ( c ) displays the successful convergence achieved by mimic-SMNM-PINNs.

Figures 5 and 6 showcase the loss functions for the real-PINNs and mimic-SMNM-PINNs, respectively, over 30,000 epochs. Upon close examination, both figures suggest that each model achieves a comparable level of loss reduction over the training process. Specifically, Fig. 5 , corresponding to real-PINNs, displays an initial rapid decrease in total loss followed by a phase of fluctuation, while boundary and residual losses exhibit notable variance. Despite these oscillations, the real-PINNs’ losses eventually align closely with those observed in Fig. 6 for mimic-SMNM-PINNs, which presents a steadier descent in all loss components. The mimic-SMNM-PINNs achieve a consistent and monotonic reduction in total, boundary, and residual losses, hinting at a more stable convergence. The final loss values for both models are remarkably similar, indicating that, both real-PINNs and mimic-SMNM-PINNs effectively approximate the solution with a comparable level of accuracy. The Figs. 5 and 6 display a consistent vertical axis, enhancing the visual comparability between the two methods and supporting the statement that the final loss values are indeed similar.

Results of ( a ) Total Loss, ( b ) Boundary Loss (Loss u bd), and ( c ) Residual Loss (Loss res) for 30,000 epochs of real-PINNs with 5 layers of 20 neurons, aimed at solving the differential equation \(y'-y=0\) with boundary conditions y(0) = 1, y(1) = e, showcase the network’s progression in reducing each type of loss over the course of training.

Additionally, the mean squared error (MSE) for mimic-SMNM-PINNs and real-PINNs is depicted in Fig. 7 . In Fig. 7 , we are presented with a graphical representation of the pointwise errors emanating from the application of real-PINNs and mimic-SMNM-PINNs to a specified differential equation. Figure 7 a elucidates a pointwise error distribution for the real-PINNs, revealing a Relative MSE valued at 4.141e−05, which denotes a high degree of model precision in its predictive capacity relative to the true solutions of the differential equation. Conversely, Fig. 7 b illustrates the pointwise error for the mimic-SMNM-PINNs, where the Relative MSE is quantified to be 1.403e−03-substantially higher by an order of magnitude compared to that of the real-PINNs. Despite this elevated error metric, the resulting MSE still falls within an acceptable range in the context of numerical solution approximation for differential equations. A comparative analysis of these two sub-figures discloses that both modeling approaches proficiently approximate the solution of the differential equation; however, the real-PINNs demonstrate a notably enhanced performance with a lower MSE. The difference in MSE between the two models is calculated to be 0.00136159, further accentuating the superior accuracy of the real-PINNs over the mimic-SMNM-PINNs.

Results of ( a ) Total Loss, ( b ) Boundary Loss (Loss u bd), and ( c ) Residual Loss (Loss res) for 30,000 epochs of mimic-SMNM-PINNs, applied to solve the differential equation \(y'-y=0\) with boundary conditions y(0) = 1, y(1) = e, illustrates the efficacy of the mimic-SMNM-PINNs model in systematically reducing all three types of loss across the training duration.

MSE (Mean Squared Error) comparison between ( a ) real-PINNs and ( b ) mimic-SMNM-PINNs, highlights the differences in performance accuracy between the two models when applied to solving the differential equation \(y'-y=0\) boundary conditions y(0) = 1, y(1) = e, illustrating their respective capabilities in accurately approximating the solution to the given problem.

Figure 8 depicts a time comparison between two distinct types of PINNs: mimic-SMNM-PINNs and real-PINNs, over the course of training for 30,000 epochs. From the graph, it is evident that the training time for mimic-SMNM-PINNs increases at a significantly faster rate compared to that of real-PINNs as the number of epochs grows. The disparity in time efficiency is particularly noteworthy at the 30,000th epoch, where the mimic-SMNM-PINNs’ training time far exceeds that of the real-PINNs. This comparison starkly illustrates the computational load differences between the two methods, with real-PINNs offering a time-efficient alternative to the more resource-intensive mimic-SMNM-PINNs within the evaluated epoch range.

Comparative Training Time Efficiency of mimic-SMNM-PINNs versus real-PINNs (with 5 layers of 20 neurons). This figure illustrates the training duration for both models over 30,000 epochs, highlighting the enhanced time efficiency of the mimic-SMNM-PINNs. The training process was repeated five times to ensure the achieved level of measurement precision.

After implementing the standard mimic-SMNM-PINNs in the PINNs and observing its performance limitations, we further explored an adapted version known as the mimic-SMNM-PINNs method on the 1D harmonic oscillator. The differential equations representing the model are presented below:

with the initial conditions:

We focused on solving the problem for the under-damped state, i.e., when

We received Fig. 9 a–d. Figure 9 a shows the PINNs solution for a 1D harmonic oscillator after 3000 iterations. Figure 9 b shows the PINNs solution for a 1D harmonic oscillator after 50,000 iterations. Figure 9 c shows the mimic-SMNM-PINNs solution for a 1D harmonic oscillator after 3000 iterations. Figure 9 d shows the mimic-SMNM-PINNs solution for a 1D harmonic oscillator after 50,000 iterations. It is evident from the figures that mimic-SMNM-PINNs can approximate very slowly. Therefore, we can say that the proposed mimic-SMNM-PINNs can be effective for solving simple differential equations.

1D harmonic oscillator: ( a ) PINNs solution for 3000 iterations, ( b ) PINNs solution for 50,000 iterations, ( c ) mimic-PINNs solution for 3000 iterations, ( d ) mimic-PINNs solution for 50,000 iterations.

Statistical analysis

In this study, comprehensive statistical analyses were conducted to ensure the robustness and reliability of our findings. The primary objective was to validate the effectiveness of the mimic-SMNM within the PINNs framework, compared to the traditional PINNs and real-SMNM approaches.

Error metrics

Mean Squared Error (MSE) was calculated for each model to assess the accuracy of the solutions provided by the different neural network architectures. This metric is crucial for evaluating the performance of the models in accurately solving the differential equations.

Computational efficiency

We measured the computational time for each model across 30,000 epochs to compare their efficiency for the simple model and 50,000 epochs for the complex model. These analysis helps in determining the practical applicability of the models in real-world scenarios. These tests were chosen to directly address our research questions concerning the computational efficiency and solution accuracy of the proposed mimic-SMNM model. The convergence analysis and error metrics are particularly pertinent in validating the model’s capability to efficiently and accurately solve differential equations within the PINNs framework. Additionally, in this study, ‘computational speed’ refers to the total training time required for the model to achieve convergence.

Software and tools

All statistical analyses were conducted using Python. Python’s TensorFlow and NumPy libraries were primarily utilized for implementing and evaluating the neural network models.

This study embarked on exploring the efficacy of PINNs integrated with a SMNM to address the architecture selection challenge in solving differential equations. Despite initial anticipation, the conventional SMNM did not converge, leading to the development of a mimic-SMNM model. This adaptation, employing a dual-layer structure with a single neuron, aimed to retain the conceptual benefits of SMNM while ensuring convergence and computational efficiency. Our experimental results demonstrated that while the real-PINNs successfully solved the differential equation \(y'-y=0\) boundary conditions y(0) = 1, y(1) = e, the true SMNM failed to converge. However, the mimic-SMNM-PINNs, despite not reaching the hypothesized efficiency, offered a valuable compromise by achieving convergence and a significant increase in computational speed compared to real PINNs after 30,000 epochs. Additionally, mimic-SMNM-PINNs did not perform well for the 1D harmonic oscillator. Therefore, we can say that the proposed method can be fast and effective for the solution of simple differential equations, but it is slow and may not converge for complex problems.

Our findings underscore the importance of flexibility in neural network architectures, particularly within the PINNs framework, for solving differential equations. The mimic-SMNM-PINNs model serves as a practical solution, balancing computational efficiency with the complexity of architectural design for simple differential equations. The comparative analysis revealed that the convergence behaviors and loss reduction patterns for the mimic-SMNM-PINNs are similar to those of the real-PINNs for simple differential equations. Additionally, the computational speeds between the real-PINNs and mimic-SMNM-PINNs models show a fivefold gain for the mimic-SMNM-PINNs for simple differential equations.

For future research, integrating more complex neuron models, such as dendritic neuron models 35 , 36 , 37 , into the PINNs framework presents an exciting frontier. These models hold the promise of adding a new layer of sophistication and adaptability to PINNs architectures, potentially enhancing the solution accuracy and computational efficiency for a broader spectrum of PDEs.

Data availability

The dataset used in this study is synthetic and can be generated using the code available on GitHub.

Code availability

All codes are available on GitHub at https://github.com/melihagraz .

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Acknowledgements

The authors extend my deepest gratitude to Professor Geroge Karniadakis, whose guidance represents one of the most significant opportunities in my academic career.

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Melih Agraz

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M.A. wrote the code of the analysis and the main manuscript text, and prepared the figures and tables.

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Agraz, M. Evaluating single multiplicative neuron models in physics-informed neural networks for differential equations. Sci Rep 14 , 19073 (2024). https://doi.org/10.1038/s41598-024-67483-y

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DOI : https://doi.org/10.1038/s41598-024-67483-y

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40 politely-worded templates to get invoices paid

The most common accounts receivable problems and how to solve them

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Late payments cost business in the UK £1.6 billion per year and the crushing financial pressure shows no sign of letting up. Data collected by Xero shows that the cost of unpaid invoices has doubled in the last two years alone. The average payment delay is also 6.1 days past the deadline, more than double what it was in 2021.

One of the best solutions to these rising late payment costs is effective and streamlined accounts receivable (AR) management. Putting an efficient and holistic AR management system in place reduces payment times, prevents bad debt, and minimises the chance of common payment roadblocks such as disputes and errors.

This comprehensive guide will highlight the common AR management problems found in businesses of all scales and provide simple solutions to help you overcome the current late payment crisis . 

Late payments

The first and most impactful of the issues commonly found in inefficient AR processes is consistent late payment. Doing nothing to solve late payment issues results in 50,000 businesses going bankrupt every year, so the stakes could not be higher.

Late payments have a significant impact on your business’s cash flow. The resulting drop in liquidy stifles growth, reduces your ability to grasp opportunities, and can drive you into further debt.

Common causes of late invoice payments

Two of the most common causes for late payments include:

• Inefficient invoicing - Invoices that are sent late, incomplete, or with errors in the details slow the entire payment process down. More work hours then need to be spent fixing these issues, increasing the total cost of late payment.
• Lack of reminders - A consistent schedule of payment reminders is critical to getting your invoices paid on time. Assuming your invoice will make it to the correct person and will get paid without a well placed reminder ignores the fact that 87% of businesses are consistently paid late.

Solutions to late payments

There are several easy-to-implement solutions that can help your business avoid being one of the 50,000 businesses going bankrupt every year.

• Automated invoicing - Automation is a critical part of a streamlined and effective AR management system. Employing an automated invoicing system ensures that invoices are sent on time while also reducing human error. Sending an accurate and timely invoice is the best way to get paid and automated invoicing allows AR teams to do so without adding to their workload.
• Regular reminders - There are a huge range of reasons why your invoice might be going ignored and while you’re waiting to get paid, your cash flow is suffering. Automated reminder systems take most of the work out of chasing your invoices, allowing you to send regular polite reminders that keep your invoice at the top of the pile.
• Clear payment terms - Your payment terms are the foundation of your AR process. They let your customers know when and how they should pay and what the consequences of non-payment are. Having unclear, or worse, no payment terms muddies the waters, leading to inconsistencies, errors and delays. 

Disputed invoices

Errors in your invoices lead to disputes. Disputes with your customers impact your carefully cultivated relationship, while also leading to more payment delays. 

Ensuring basic accuracy and timeliness in your invoicing can solve both of these problems, reducing payment delays and helping to promote repeat business by enhancing your customer relationships.

Common causes of invoice disputes 

The most common causes of disputed invoices are:

• Human error - While some human error is to be expected in business, smaller or overworked teams can often lead to compounding errors in your AR processes. These errors lead to delays, disputes, and further losses.
• Unclear terms - Clear payment terms ensure you and your customers are always on the same page when it comes to payment details and timelines. Unclear payment terms lead to misunderstandings and miscommunications and, at worst, can sour your relationship with a potentially valuable customer.

Solutions to prevent invoice disputes

Thankfully, there are some very easy to adopt solutions that can prevent errors from creeping into your AR processes and causing disputes.

• Accurate invoicing - Accuracy in your invoicing is critical. All invoices should, at least, be double checked before sending. Automated invoicing systems can help to reduce human error and increase accuracy while not negatively impacting your AR team’s workload. 
• Clear communication -  Implement transparent payment terms and clear communication to prevent disputes from ever happening. If you and your customer both understand a clear set of payment terms, then any minor errors can be caught and rectified easily with the minimum of misunderstandings.
• An effective dispute resolution process -  Not only does a robust dispute resolution process help to quickly resolve issues, but it can also help to build a positive relationship with your customers. Just having a dispute resolution process in place demonstrates to your customers that you take any issues they raise seriously and will work with them to resolve any disputes.

Inefficient accounts receivables processes

All systems are vulnerable to creeping inefficiencies. Without regular review, your AR processes will gradually become less efficient and more time consuming, leading to costly errors and higher spend but reduced results.

Monitoring, reviewing and revising your systems when inefficiencies start to appear can reduce errors, eliminate delays and ensure your cash flow remains healthy. 

Common causes of inefficient receivables processes

The most commonly reported causes of inefficient accounts receivables processes include:

• Outdated systems - Outdated legacy systems are a common culprit when it comes to inefficient accounts receivables processes. Many companies and teams are slow or reluctant to embrace new systems or technologies due to a bias towards systems they already understand.
• Lack of automation -  Automated systems can’t replace skilled workers, but they can reduce the number of repetitive tasks they have to perform, freeing them up to apply their skills in more productive ways.

The solution: how to make your receivables process more efficient 

As AR technology advances, businesses can choose from a wide range of AR systems to make their accounts receivable processes more efficient.

• Modern AR software - Regularly update or replace legacy systems to make your AR system more efficient while allowing you to take advantage of the features and benefits newer software offers.
• Staff training - Provide effective and supportive training to make staff more comfortable with new systems and adapt to more efficient ways of working. Confident and well trained staff are better empowered to work to the best of their abilities.
• Process reviews - Regular process audits can help to expose and solve errors and inefficiencies, while highlighting areas for improvement that can further reduce payment delays.

Poor credit management

On average, companies in the UK write off £5.8bn in bad debt every year, with one in 10 scrapping bills worth more than £100,000. For some, unrecoverable debt is responsible for driving them into insolvency.

Effective credit management mitigates the risk of bad debt while providing you with a wider range of opportunities to recover outstanding payments. 

Common causes of poor credit management

Some of the most common causes of substandard credit management are:

• Inadequate credit checks - Limited or non-existent credit checking is surprisingly common , with most companies only credit checking customers during the onboarding process, if at all.
• Inconsistent terms -  Without obvious guidelines on when and who to offer credit to, businesses can find themselves offering lines of credit to customers who have a history of not making payments. Even with credit-worthy customers, inconsistent terms can lead to misunderstandings, delays and disputes.

The solution: how to improve your credit management

Mitigating risk is essential to ensuring consistent cash flow and the overall safety of your business. Thankfully, there are several steps you can take to enhance your credit management processes:

• Thorough credit checks - Credit checking during on boarding and at regular intervals during your business's relationship with a customer is the best way to ensure you don’t accrue bad debt.
• Monitor client behavior -   Monitor your client’s payment behavior and how it changes over time, as this allows you to adapt to their changing creditworthiness and minimize risk.
• A comprehensive credit policy - Put in place a comprehensive credit policy to ensure your staff fully understand when to offer credit. It also means your customers know under what circumstances they’ll be offered credit and why that might change.

Lack of communication with debtors

Open communication is the bedrock of a productive customer relationship. Failing to communicate effectively with your customers can lead to disputes or worsen an existing situation.

Without clear communication, it becomes significantly harder to resolve any payment disputes and keep your customers happy while ensuring cash flow.

Common causes of lack of debtor communication

Some of the most common causes of poor communication include:

• Infrequent follow-ups - Follow ups include everything from chasing a late invoice to responding with a thank you when payment is made. Neglecting your follows up can lead to forgotten and late payments while denying you the opportunity to build on your customer relationships.
• Unclear instructions - Unclear or contradictory payment instructions slow the entire payment process down and can frustrate your customers.

The solution: how to improve debtor communications

Improving your communications with your client is as easy as taking the following steps:

• Regular communication - Maintain regular contact with your customer to give them a channel through which they can ask questions or clarify issues. Consistent communication is the best way to head off any potential disputes.
• Client education - Give your customers everything they need to understand and adhere to your payment processes to maximize your chances of getting paid on time while demonstrating that you also value your customer’s time.

Prevent common accounts receivables problems with AI and automation

Chaser’s comprehensive suite of automation options give you all the tools you need to streamline and enhance your AR processes. From error-free automated invoice to consistent, multi-channel automated reminders, Chaser frees your AR staff up from repetitive tasks and lets them apply their skills to something more productive. 

With Chaser, you can monitor your client’s payment behavior and use Chaser’s integrated credit checking facility to determine credit-worthiness and reduce the risk of bad debt. Chaser’s AI-powered late payment predictor and recommended chasing times give you the tools you need to anticipate late payments and ensure your reminders arrive at exactly the right time. 

By using a combination of AI and automation features, Chaser helps you chase and collect payments in the fastest, friendliest way possible.

Streamline your AR system to improve your cash flow

While AR problems such as invoice errors, unclear payment terms, a lack of follow ups, and spotty credit checking are both common and can lead to insolvency, fixing your AR system isn’t as hard as it sounds.

By implementing the solutions above you’ll be able to eliminate costly errors and delays from your AR management system, ensuring greater liquidity and avoiding bad debt.

For more expert tips and advice from financial professionals, as well as a range of accounts receivable resources, check out the other articles on the Chaser blog and news page.

To automate and streamline your accounts receivables process and start eliminating problems today, try Chaser for free , or speak to an expert .

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