A Guide to Problem Solving

Age 16 to 18.

When confronted with a problem, in which the solution is not clear, you need to be a skilled problem-solver to know how to proceed. When you look at STEP problems for the first time, it may seem like this problem-solving skill is out of your reach, but like any skill, you can improve your problem-solving with practice. How do I become a better problem-solver? First and foremost, the best way to become better at problem-solving is to try solving lots of problems! If you are preparing for STEP, it makes sense that some of these problems should be STEP questions, but to start off with it's worth spending time looking at problems from other sources. This collection of NRICH problems  is designed for younger students, but it's very worthwhile having a go at a few to practise the problem-solving technique in a context where the mathematics should be straightforward to you. Then as you become a more confident problem-solver you can try more past STEP questions. One student who worked with NRICH said: "From personal experience, I was disastrous at STEP to start with. Yet as I persisted with it for a long time it eventually started to click - 'it' referring to being able to solve problems much more easily. This happens because your brain starts to recognise that problems fall into various categories and you subconsciously remember successes and pitfalls of previous 'similar' problems." A Problem-solving Heuristic for STEP Below you will find some questions you can ask yourself while you are solving a problem. The questions are divided into four phases, based loosely on those found in George Pólya's 1945 book "How to Solve It". Understanding the problem

• What area of mathematics is this?
• What exactly am I being asked to do?
• What do I know?
• What do I need to find out?
• What am I uncertain about?
• Can I put the problem into my own words?

Devising a plan

• Work out the first few steps before leaping in!
• Have I seen something like it before?
• Is there a diagram I could draw to help?
• Is there another way of representing?
• Would it be useful to try some suitable numbers first?
• Is there some notation that will help?

Carrying out the plan STUCK!

• Try special cases or a simpler problem
• Work backwards
• Guess and check
• Be systematic
• Work towards subgoals
• Imagine your way through the problem
• Has the plan failed? Know when it's time to abandon the plan and move on.

Looking back

• Have I answered the question?
• Sanity check for sense and consistency
• Check the problem has been fully solved
• Read through the solution and check the flow of the logic.

Throughout the problem solving process it's important to keep an eye on how you're feeling and making sure you're in control:

• Am I getting stressed?
• Is my plan working?
• Am I spending too long on this?
• Could I move on to something else and come back to this later?
• Am I focussing on the problem?
• Is my work becoming chaotic, do I need to slow down, go back and tidy up?
• Do I need to STOP, PEN DOWN, THINK?

Finally, don't forget that STEP questions are designed to take at least 30-45 minutes to solve, and to start with they will take you longer than that. As a last resort, read the solution, but not until you have spent a long time just thinking about the problem, making notes, trying things out and looking at resources that can help you. If you do end up reading the solution, then come back to the same problem a few days or weeks later to have another go at it.

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6 Steps to Solve Math Problems

The ability to solve math problems not only boosts one’s abstract thinking, it is also a marketable skill in the workplace as many employers require that their employees have taken several math courses in college. This skills are in high demand when it comes to dissertation editing  and writing or some essay writing tasks.

Problem solving is a process of finding the solutions to difficult issues. Whether or not a student major in math, either at the college or university level, being able to solve math problems is very beneficial. The ability to solve math problems without computer usage not only boosts one’s abstract thinking, it is also a marketable skill in the workplace as many employers require that their employees have taken several math courses in college.

THE ART OF EFFECTIVE PROBLEM SOLVING

Though solving math problems seems awfully tedious in nature or even overly difficult to the novice, the practice is essentially solving a problem finding couse and effect . And whenever a problem emerges, there is at least one solution to that problem. There is a multitude of ways to solve a math problem. It involves visualizing, approaching and solving math problems in a detailed set of instructions the student should refer to in the event a math problem seems insurmountable.

These Are the Best Steps to Follow:

STEP 1. Determine the kind of math the problem is calling for. Does this particular math problem call for multiplying fractions? Solving algebraic equations? Solving quadratic equations? Knowing where to start and what school of math is being incorporated is key in helping the student solve their problem.

STEP 2. Review what has already been covered in the math course for which this particular assignment, or math problem, has been given. If it’s a specific formula, or set of formulas, that the problem calls for, more than likely the student can find the formula in the chapters or sections their professor has already covered over the course of the term or semester. Most academic institutions offer numerous resources for students struggling with math problems.

STEP 3. Begin to solve the problem, apply knowledge and skills already learned in the course. Identify what the problem is calling for and read the directions, if they are present, very carefully. At times, the system of “guess and check” may help; in other cases, use objects and other such tools to model the problem – sometimes a visual illustration of the problem may serve the student best. Look for patterns, use logical reasoning, and work backward, if possible.

STEP 4. Write down and show each step. Sometimes, by writing down their work, the student who may be a visual learner may best solve their problem – or this may overcome a set of obstacles standing in their way of solving the problem. This tactic allows the student to track and even double-check their approach to the problem as well as their mental process of getting the needed results. The student who is struggling with a math assignment must never attempt to solve the entire problem in their head.

STEP 5. Verify that the answer is correct and makes sense to the student if they are in the future tested on solving such math problems. Often in most textbooks, in math courses, especially, the answer is in the back of the book – that is if an assignment is taken directly from the textbook.

STEP 6. The student must always remember that their professor’s job is to help them understand the math that the course calls for. So, the student should approach their professor in the event a math problem presents difficulties. Most academic institutions employ the student body’s most skilled math students as tutors for students whose strengths are not math-oriented. This kind of service is often incorporated in the student’s tuition, so they should certainly take advantage of it.

PROBLEM SOLVING SKILLS

PROBLEM SOLVING STRATEGIES

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• Problem Solving Information

What is Problem Solving?

On this page we discuss "What is Problem Solving?" under three headings: introduction, four stages of problem solving, and the scientific approach.

Introduction

Naturally enough, problem solving is about solving problems. And we’ll restrict ourselves to thinking about mathematical problems here even though problem solving in school has a wider goal. When you think about it, the whole aim of education is to equip students to solve problems.

But problem solving also contributes to mathematics itself. Mathematics consists of skills and processes. The skills are things that we are all familiar with. These include the basic arithmetical processes and the algorithms that go with them. They include algebra in all its levels as well as sophisticated areas such as the calculus. This is the side of the subject that is largely represented in the Strands of Number and Algebra, Geometry and Measurement and Statistics.

On the other hand, the processes of mathematics are the ways of using the skills creatively in new situations. Mathematical processes include problem solving, logic and reasoning, and communicating ideas. These are the parts of mathematics that enable us to use the skills in a wide variety of situations.

It is worth starting by distinguishing between the three words "method", "answer" and "solution". By "method" we mean the means used to get an answer. This will generally involve one or more Problem Solving Strategies . On the other hand, we use "answer" to mean a number, quantity or some other entity that the problem is asking for. Finally, a "solution" is the whole process of solving a problem, including the method of obtaining an answer and the answer itself.

method + answer = solution

But how do we do Problem Solving? There are four basic steps. Pólya enunciated these in 1945 but all of them were known and used well before then. Pólya’s four stages of problem solving are listed below.

Four Stages of Problem Solving

1. Understand and explore the problem 2. Find a strategy 3. Use the strategy to solve the problem 4. Look back and reflect on the solution.

Although we have listed the four stages in order, for difficult problems it may not be possible to simply move through them consecutively to produce an answer. It is frequently the case that students move backwards and forwards between and across the steps.

You can't solve a problem unless you can first understand it. This requires not only knowing what you have to find but also the key pieces of information that need to be put together to obtain the answer.

Students will often not be able to absorb all the important information of a problem in one go. It will almost always be necessary to read a problem several times, both at the start and while working on it. With younger students it is worth repeating the problem and then asking them to put the question in their own words. Older students might use a highlighter to mark the important parts of the problem.

Finding a strategy tends to suggest that it is a simple matter to think of an appropriate strategy. However, for many problems students may find it necessary to play around with the information before they are able to think of a strategy that might produce a solution. This exploratory phase will also help them to understand the problem better and may make them aware of some piece of information that they had neglected after the first reading.

Having explored the problem and decided on a strategy, the third step, solve the problem , can be attempted. Hopefully now the problem will be solved and an answer obtained. During this phase it is important for the students to keep a track of what they are doing. This is useful to show others what they have done and it is also helpful in finding errors should the right answer not be found.

At this point many students, especially mathematically able ones, will stop. But it is worth getting them into the habit of looking back over what they have done. There are several good reasons for this. First of all it is good practice for them to check their working and make sure that they have not made any errors. Second, it is vital to make sure that the answer they obtained is in fact the answer to the problem. Third, in looking back and thinking a little more about the problem, students are often able to see another way of solving the problem. This new solution may be a nicer solution than the original and may give more insight into what is really going on. Finally, students may be able to generalise or extend the problem.

Generalising a problem means creating a problem that has the original problem as a special case. So a problem about three pigs may be changed into one which has any number of pigs.

In Problem 4 of What is a Problem? , there is a problem on towers. The last part of that problem asks how many towers can be built for any particular height. The answer to this problem will contain the answer to the previous three questions. There we were asked for the number of towers of height one, two and three. If we have some sort of formula, or expression, for any height, then we can substitute into that formula to get the answer for height three, for instance. So the "any" height formula is a generalisation of the height three case. It contains the height three case as a special example.

Extending a problem is a related idea. Here though, we are looking at a new problem that is somehow related to the first one. For instance, a problem that involves addition might be looked at to see if it makes any sense with multiplication. A rather nice problem is to take any whole number and divide it by two if it’s even and multiply it by three and add one if it’s odd. Keep repeating this manipulation. Is the answer you get eventually 1? We’ll do an example. Let’s start with 34. Then we get

34 → 17 → 52 → 26 → 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1

We certainly got to 1 then. Now it turns out that no one in the world knows if you will always get to 1 this way, no matter where you start. That’s something for you to worry about. But where does the extension come in? Well we can extend this problem, by just changing the 3 to 5. So this time instead of dividing by 2 if the number is even and multiplying it by three and adding one if it’s odd, try dividing by 2 if the number is even and multiplying it by 5 and adding one if it’s odd. This new problem doesn’t contain the first one as a special case, so it’s not a generalisation. It is an extension though – it’s a problem that is closely related to the original.

It is by this method of generalisation and extension that mathematics makes great strides forward. Up until Pythagoras’ time, many right-angled triangles were known. For instance, it was known that a triangle with sides 3, 4 and 5 was a right-angled triangle. Similarly people knew that triangles with sides 5, 12 and 13, and 7, 24 and 25 were right angled. Pythagoras’ generalisation was to show that EVERY triangle with sides a, b, c was a right-angled triangle if and only if a 2 + b 2 = c 2 .

This brings us to an aspect of problem solving that we haven’t mentioned so far. That is justification (or proof). Your students may often be able to guess what the answer to a problem is but their solution is not complete until they can justify their answer.

Now in some problems it is hard to find a justification. Indeed you may believe that it is not something that any of the class can do. So you may be happy that the students can find an answer. However, bear in mind that this justification is what sets mathematics apart from every other discipline. Consequently the justification step is an important one that shouldn’t be missed too often.

Scientific Approach

Another way of looking at the Problem Solving process is what might be called the scientific approach. We show this in the diagram below.

Here the problem is given and initially the idea is to experiment with it or explore it in order to get some feeling as to how to proceed. After a while it is hoped that the solver is able to make a conjecture or guess what the answer might be. If the conjecture is true it might be possible to prove or justify it. In that case the looking back process sets in and an effort is made to generalise or extend the problem. In this case you have essentially chosen a new problem and so the whole process starts over again.

Sometimes, however, the conjecture is wrong and so a counter-example is found. This is an example that contradicts the conjecture. In that case another conjecture is sought and you have to look for a proof or another counterexample.

Some problems are too hard so it is necessary to give up. Now you may give up so that you can take a rest, in which case it is a ‘for now’ giving up. Actually this is a good problem solving strategy. Often when you give up for a while your subconscious takes over and comes up with a good idea that you can follow. On the other hand, some problems are so hard that you eventually have to give up ‘for ever’. There have been many difficult problems throughout history that mathematicians have had to give up on.

Study Smarter

17 maths problem solving strategies to boost your learning.

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

Friday, 3rd June 2022

• What are strategies?

Understand the problem

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

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

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

What are maths problem-solving strategies?

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

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

Polya's 4-step process for solving problems

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

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

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

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

Do I understand all the words used in the problem?

What am I asked to find or show?

Can I restate the problem in my own words?

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

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

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

What information is extra or irrelevant?

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

1. Read the problem aloud

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

2. Summarise the information

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

3. Create a picture or diagram

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

4. Act it out

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

5. Use keyword analysis

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

How many more?

How many left?

Equal parts

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

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

6. Look for a pattern

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

7. Guess and check

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

8. Working backwards

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

9. Use a formula

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

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

10. Eliminate the possibilities

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

11. Use direct reasoning

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

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

12. Solve a simpler problem

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

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

13. Solve an equation

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

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

14. Be systematic

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

15. Check your work

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

16. Be flexible

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

17. Don't give up

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

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

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

What was the most challenging part of the problem?

Was one method more effective than another?

Would you do something differently next time?

What have you learned from this experience?

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

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

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

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

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

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

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