example of problem solving in university

Problem solving and decision making

 "This has shown increasing demand as employers are acknowledging that graduates are expected to think for themselves and perhaps find different ways of working and thinking creatively"

Carl Gilleard, Former Chief Executive, Association of Graduate Recruiters

How to be an effective problem solver

We all use our initiative and creativity to solve problems every day. For example, you might have to change your route due to traffic congestion, solve an IT issue, or work out what to make for dinner with the ingredients left in the fridge. The challenges you may face in your professional career are likely to be a bit more complicated than these examples, however the skills and processes you use to come up with solutions are largely the same, as they rely on your ability to analyse a situation and decide on a course of action.

Problem solving and decision making  are likely to be essential aspects of a graduate-level job, so it is important to show a recruiter that you have the personal resilience and the right skills to see problems as challenges, make the right choices and learn and develop from your experiences.

You are likely to have to apply techniques of problem solving on a daily basis in a range of working situations, for example:

• using your degree subject knowledge to resolve technical or practical issues
• diagnosing and rectifying obstacles relating to processes or systems
• thinking of new or different ways of doing your job
• dealing with emergencies involving systems or people.

You may have to use a logical, methodical approach in some circumstances, or be prepared to use creativity or lateral thinking in others; you will need to be able to draw on your academic or subject knowledge to identify solutions of a practical or technical nature; you will need to use other skills such as communication and planning and organising to influence change.

Whatever issue you are faced with, some steps are fundamental:

I - Identify the problem

D - Define the problem

E - Examine alternatives

A - Act on a plan

L - Look at the consequences

This is the  IDEAL  model of problem-solving. There are other, more complex methods, but the steps are broadly similar.

What do recruiters want?

Problem solving, decision making and initiative can be asked for in a variety of ways. Many adverts will simply ask for candidates who can “ take the initiative to get a job done " or " have the ability to resolve problems "; others, however, may not make it so obvious. Phrases such as those below also indicate that initiative and problem solving are key requirements of the role:

• “We need people who can set goals and surpass them; people who have ideas, flexibility, imagination and resilience…”
• “Take responsibility and like to use their initiative; Have the confidence and the credibility to challenge and come up with new ways of working…”
•  “An enquiring mind and the ability to understand and solve complex challenges are necessary…”
• “We are looking for fresh, innovative minds and creative spirits...”
• “Ambitious graduates who can respond with pace and energy to every issue they face…”

These quotations are all taken from graduate job adverts and they are all asking for more or less the same two things:

• The ability to use your own initiative, to think for yourself, to be creative and pro-active.
• The ability to resolve problems, to think logically or laterally, to use ingenuity to overcome difficulties and to research and implement solutions.

These are important skills which recruiters look for. They want staff who will take the personal responsibility to make sure targets are met; who can see that there might be a better way of doing something and are prepared to research and implement change; who react positively, not negatively, when things go wrong.

Gaining and developing problem solving and decision making skills

Below are some examples of how you may already have gained decision making and problem solving skills at the University of Bradford and beyond. There may also be some useful suggestions here if you are looking to develop your skills further:

• dissertation - researching and analysing a specific issue and providing recommendations
• group projects  - overcoming challenges e.g. a change of circumstances, technical problems, etc.
• part-time jobs, internships and work experience*  - dealing with challenging customers, identifying and solving issues in your role, completing projects, etc.
• organising events - deciding on date, venue, marketing; solving logistical issues, etc.
• travel -  organising trips, planning and reacting to change, etc.
• enter competitions - to   provide a solution to a challenge.

*Using your initiative in a work context is about spotting opportunities to develop the business. This can, for example, include learning new technology to make your work more productive and efficient; being willing to look at processes and systems to see if there are things you can suggest to improve workflow; recognising opportunities that will improve the business and being prepared to follow them through; volunteering to learn new tasks so you can be adaptable and help out in emergencies or at peak periods.

How can you prove to a recruiter that you have these skills?

Think of examples of when you have used these skills. See the above section for suggestions, and then to provide a full and satisfying answer you can structure it using the STAR technique:

• S - Define the Situation
• T - Identify the Task
• A - Describe your Action
• R - Explain the Result

Here is a detailed example:

Define the SITUATION:  (where were you? what was your role? what was the context?) 

I work shifts at a call centre which manages orders for several online companies. One evening I had to deal with a very irate customer who had been promised a delivery a week ago and had still not received it. 

Identify the TASK:  (what was the problem? what was your aim? what had to be achieved?) 

Whilst listening to the customer, I accessed his record. This was no help in solving the problem as it simply reiterated what the customer was saying and did not give any more up-to-date information. I promised the customer that I would do my best to help but I would need to do some research and phone him back. He reluctantly accepted this.

Describe the ACTION you took:  (be clear about what you did) 

I could not check with the office as they were closed and my supervisor had already left for the evening, so I searched for the same product code to see if I could find updated information on other records.  This confirmed that the product was now back in stock and that several deliveries were actually scheduled for the following day. There seemed to have been an error which had resulted in my customer’s record not being updated, so I reserved the item for this customer and then persuaded the Logistics Manager to include him in the schedule.

Highlight the RESULT you achieved:  (what was the outcome? Be specific and, if possible, quantify the benefits)

My shift was over but I telephoned him back and explained what I had done and hoped very much that it was convenient for him to accept delivery the following day. He was delighted with the initiative I had taken and thanked me. Two days later my supervisor told me that I had received excellent feedback from a customer and I would be nominated for Employee of the Month.

To use the STAR technique effectively, remember:

• You are the STAR of the story, so focus on your own actions.
• Tell a story and capture the interest of the reader. Include relevant details but don’t waffle.
• Move from the situation, to the task, to your actions, and finally to the result with a consistent, conversational approach.

A detailed statement like this can be used in online applications, or used at interview. It is also easy to adapt it for use in your CV, for example:

• My work experience at the call centre required me to develop good problem solving skills when dealing with difficult customers with stock and delivery issues.
• I have good customer service skills developed through resolving problems relating to stock and deliveries whilst working for a call centre.

Adapting your examples

The example above, for instance, could easily be altered to prove your  communication skills , show that you can  adapt and be flexible , and that you have great  customer service skills .  It is worthwhile spending time writing statements like this about all your experiences and then adapting them to match each recruiters’ specific requirements.

Related key words / skills

•  Leadership
• Analytical and logical thinking
• Recommending
• Creative thinking
•  Adaptability and flexibility
• Time management
• Researching
• Applying knowledge

Practical help

We run regular  workshops on employability skills , and you can book an appointment with one of our advisers to discuss how to improve your employability in relation to your career choices.

Further reading

More articles:

• Problem Solving defined on Skills You Need
• Mind Tools on Problem Solving Techniques
• Problem solving: the mark of an independent employee (via TARGETjobs)
• Problem Solving and Decision Making processes via businessballs.com

You can also check out our Assessement Centre and Psychometric Tests pages for details of the problem-solving exercises recruiters use in their selection processes.

Other relevant websites with general information on skills are:

• Prospects  – features articles on skills and how to evidence them.
• TARGETjobs  – has details on essential skills and competencies.

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• Problem Solving in STEM

Solving problems is a key component of many science, math, and engineering classes.  If a goal of a class is for students to emerge with the ability to solve new kinds of problems or to use new problem-solving techniques, then students need numerous opportunities to develop the skills necessary to approach and answer different types of problems.  Problem solving during section or class allows students to develop their confidence in these skills under your guidance, better preparing them to succeed on their homework and exams. This page offers advice about strategies for facilitating problem solving during class.

How do I decide which problems to cover in section or class?

In-class problem solving should reinforce the major concepts from the class and provide the opportunity for theoretical concepts to become more concrete. If students have a problem set for homework, then in-class problem solving should prepare students for the types of problems that they will see on their homework. You may wish to include some simpler problems both in the interest of time and to help students gain confidence, but it is ideal if the complexity of at least some of the in-class problems mirrors the level of difficulty of the homework. You may also want to ask your students ahead of time which skills or concepts they find confusing, and include some problems that are directly targeted to their concerns.

You have given your students a problem to solve in class. What are some strategies to work through it?

• Try to give your students a chance to grapple with the problems as much as possible.  Offering them the chance to do the problem themselves allows them to learn from their mistakes in the presence of your expertise as their teacher. (If time is limited, they may not be able to get all the way through multi-step problems, in which case it can help to prioritize giving them a chance to tackle the most challenging steps.)
• When you do want to teach by solving the problem yourself at the board, talk through the logic of how you choose to apply certain approaches to solve certain problems.  This way you can externalize the type of thinking you hope your students internalize when they solve similar problems themselves.
• Start by setting up the problem on the board (e.g you might write down key variables and equations; draw a figure illustrating the question).  Ask students to start solving the problem, either independently or in small groups.  As they are working on the problem, walk around to hear what they are saying and see what they are writing down. If several students seem stuck, it might be a good to collect the whole class again to clarify any confusion.  After students have made progress, bring the everyone back together and have students guide you as to what to write on the board.
• It can help to first ask students to work on the problem by themselves for a minute, and then get into small groups to work on the problem collaboratively.
• If you have ample board space, have students work in small groups at the board while solving the problem.  That way you can monitor their progress by standing back and watching what they put up on the board.
• If you have several problems you would like to have the students practice, but not enough time for everyone to do all of them, you can assign different groups of students to work on different – but related - problems.

When do you want students to work in groups to solve problems?

• Don’t ask students to work in groups for straightforward problems that most students could solve independently in a short amount of time.
• Do have students work in groups for thought-provoking problems, where students will benefit from meaningful collaboration.
• Even in cases where you plan to have students work in groups, it can be useful to give students some time to work on their own before collaborating with others.  This ensures that every student engages with the problem and is ready to contribute to a discussion.

What are some benefits of having students work in groups?

• Students bring different strengths, different knowledge, and different ideas for how to solve a problem; collaboration can help students work through problems that are more challenging than they might be able to tackle on their own.
• In working in a group, students might consider multiple ways to approach a problem, thus enriching their repertoire of strategies.
• Students who think they understand the material will gain a deeper understanding by explaining concepts to their peers.

What are some strategies for helping students to form groups?  

• Instruct students to work with the person (or people) sitting next to them.
• Count off.  (e.g. 1, 2, 3, 4; all the 1’s find each other and form a group, etc)
• Hand out playing cards; students need to find the person with the same number card. (There are many variants to this.  For example, you can print pictures of images that go together [rain and umbrella]; each person gets a card and needs to find their partner[s].)
• Based on what you know about the students, assign groups in advance. List the groups on the board.
• Note: Always have students take the time to introduce themselves to each other in a new group.

What should you do while your students are working on problems?

• Walk around and talk to students. Observing their work gives you a sense of what people understand and what they are struggling with. Answer students’ questions, and ask them questions that lead in a productive direction if they are stuck.
• If you discover that many people have the same question—or that someone has a misunderstanding that others might have—you might stop everyone and discuss a key idea with the entire class.

After students work on a problem during class, what are strategies to have them share their answers and their thinking?

• Ask for volunteers to share answers. Depending on the nature of the problem, student might provide answers verbally or by writing on the board. As a variant, for questions where a variety of answers are relevant, ask for at least three volunteers before anyone shares their ideas.
• Use online polling software for students to respond to a multiple-choice question anonymously.
• If students are working in groups, assign reporters ahead of time. For example, the person with the next birthday could be responsible for sharing their group’s work with the class.
• Cold call. To reduce student anxiety about cold calling, it can help to identify students who seem to have the correct answer as you were walking around the class and checking in on their progress solving the assigned problem. You may even want to warn the student ahead of time: "This is a great answer! Do you mind if I call on you when we come back together as a class?"
• Have students write an answer on a notecard that they turn in to you.  If your goal is to understand whether students in general solved a problem correctly, the notecards could be submitted anonymously; if you wish to assess individual students’ work, you would want to ask students to put their names on their notecard.  
• Use a jigsaw strategy, where you rearrange groups such that each new group is comprised of people who came from different initial groups and had solved different problems.  Students now are responsible for teaching the other students in their new group how to solve their problem.
• Have a representative from each group explain their problem to the class.
• Have a representative from each group draw or write the answer on the board.

What happens if a student gives a wrong answer?

• Ask for their reasoning so that you can understand where they went wrong.
• Ask if anyone else has other ideas. You can also ask this sometimes when an answer is right.
• Cultivate an environment where it’s okay to be wrong. Emphasize that you are all learning together, and that you learn through making mistakes.
• Do make sure that you clarify what the correct answer is before moving on.
• Once the correct answer is given, go through some answer-checking techniques that can distinguish between correct and incorrect answers. This can help prepare students to verify their future work.

How can you make your classroom inclusive?

• The goal is that everyone is thinking, talking, and sharing their ideas, and that everyone feels valued and respected. Use a variety of teaching strategies (independent work and group work; allow students to talk to each other before they talk to the class). Create an environment where it is normal to struggle and make mistakes.
• See Kimberly Tanner’s article on strategies to promoste student engagement and cultivate classroom equity. 

A few final notes…

• Make sure that you have worked all of the problems and also thought about alternative approaches to solving them.
• Board work matters. You should have a plan beforehand of what you will write on the board, where, when, what needs to be added, and what can be erased when. If students are going to write their answers on the board, you need to also have a plan for making sure that everyone gets to the correct answer. Students will copy what is on the board and use it as their notes for later study, so correct and logical information must be written there.

For more information...

Tipsheet: Problem Solving in STEM Sections

Tanner, K. D. (2013). Structure matters: twenty-one teaching strategies to promote student engagement and cultivate classroom equity . CBE-Life Sciences Education, 12(3), 322-331.

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Problem-Based Learning

Problem-based learning  (PBL) is a student-centered approach in which students learn about a subject by working in groups to solve an open-ended problem. This problem is what drives the motivation and the learning. 

Why Use Problem-Based Learning?

Nilson (2010) lists the following learning outcomes that are associated with PBL. A well-designed PBL project provides students with the opportunity to develop skills related to:

• Working in teams.
• Managing projects and holding leadership roles.
• Oral and written communication.
• Self-awareness and evaluation of group processes.
• Working independently.
• Critical thinking and analysis.
• Explaining concepts.
• Self-directed learning.
• Applying course content to real-world examples.
• Researching and information literacy.
• Problem solving across disciplines.

Considerations for Using Problem-Based Learning

Rather than teaching relevant material and subsequently having students apply the knowledge to solve problems, the problem is presented first. PBL assignments can be short, or they can be more involved and take a whole semester. PBL is often group-oriented, so it is beneficial to set aside classroom time to prepare students to   work in groups  and to allow them to engage in their PBL project.

Students generally must:

• Examine and define the problem.
• Explore what they already know about underlying issues related to it.
• Determine what they need to learn and where they can acquire the information and tools necessary to solve the problem.
• Evaluate possible ways to solve the problem.
• Solve the problem.
• Report on their findings.

Getting Started with Problem-Based Learning

• Articulate the learning outcomes of the project. What do you want students to know or be able to do as a result of participating in the assignment?
• Create the problem. Ideally, this will be a real-world situation that resembles something students may encounter in their future careers or lives. Cases are often the basis of PBL activities. Previously developed PBL activities can be found online through the University of Delaware’s PBL Clearinghouse of Activities .
• Establish ground rules at the beginning to prepare students to work effectively in groups.
• Introduce students to group processes and do some warm up exercises to allow them to practice assessing both their own work and that of their peers.
• Consider having students take on different roles or divide up the work up amongst themselves. Alternatively, the project might require students to assume various perspectives, such as those of government officials, local business owners, etc.
• Establish how you will evaluate and assess the assignment. Consider making the self and peer assessments a part of the assignment grade.

Nilson, L. B. (2010).  Teaching at its best: A research-based resource for college instructors  (2nd ed.).  San Francisco, CA: Jossey-Bass. 

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Teaching problem solving.

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Tips and Techniques

Expert vs. novice problem solvers, communicate.

• Have students  identify specific problems, difficulties, or confusions . Don’t waste time working through problems that students already understand.
• If students are unable to articulate their concerns, determine where they are having trouble by  asking them to identify the specific concepts or principles associated with the problem.
• In a one-on-one tutoring session, ask the student to  work his/her problem out loud . This slows down the thinking process, making it more accurate and allowing you to access understanding.
• When working with larger groups you can ask students to provide a written “two-column solution.” Have students write up their solution to a problem by putting all their calculations in one column and all of their reasoning (in complete sentences) in the other column. This helps them to think critically about their own problem solving and helps you to more easily identify where they may be having problems. Two-Column Solution (Math) Two-Column Solution (Physics)

Encourage Independence

• Model the problem solving process rather than just giving students the answer. As you work through the problem, consider how a novice might struggle with the concepts and make your thinking clear
• Have students work through problems on their own. Ask directing questions or give helpful suggestions, but  provide only minimal assistance and only when needed to overcome obstacles.
• Don’t fear  group work ! Students can frequently help each other, and talking about a problem helps them think more critically about the steps needed to solve the problem. Additionally, group work helps students realize that problems often have multiple solution strategies, some that might be more effective than others

Be sensitive

• Frequently, when working problems, students are unsure of themselves. This lack of confidence may hamper their learning. It is important to recognize this when students come to us for help, and to give each student some feeling of mastery. Do this by providing  positive reinforcement to let students know when they have mastered a new concept or skill.

Encourage Thoroughness and Patience

• Try to communicate that  the process is more important than the answer so that the student learns that it is OK to not have an instant solution. This is learned through your acceptance of his/her pace of doing things, through your refusal to let anxiety pressure you into giving the right answer, and through your example of problem solving through a step-by step process.

Experts (teachers) in a particular field are often so fluent in solving problems from that field that they can find it difficult to articulate the problem solving principles and strategies they use to novices (students) in their field because these principles and strategies are second nature to the expert. To teach students problem solving skills,  a teacher should be aware of principles and strategies of good problem solving in his or her discipline .

The mathematician George Polya captured the problem solving principles and strategies he used in his discipline in the book  How to Solve It: A New Aspect of Mathematical Method (Princeton University Press, 1957). The book includes  a summary of Polya’s problem solving heuristic as well as advice on the teaching of problem solving.

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How to develop your problem-solving skills at university

James davis.

Developing problem-solving skills can be a problem unto itself for the uninformed. Fortunately, it doesn’t have to be a mystery. We’ll highlight some methods for improving your ability to think laterally, vertically and push beyond your comfort zone. Bear in mind all of these will require continuous practice and effort, but as long as you’re mindful, they won’t be as challenging as you might initially think. Even if they are, the more you solve problems, the easier they’ll become. 

But first, what exactly is problem-solving?

In a university context, it can mean just about any necessary activity or idea for overcoming… well, problems! It’s here types of problems (and consequently the sort of solutions required) start to diverge. There are several ways to categorise problems, but for the sake of argument we’ll be using the following:

Urgent problems. These are issues for which there’s only limited time to ponder and require an immediate response, but that response is both known and simple. For instance, an exam question. 

Linear problems. These are complicated and solvable through methodology, be it predefined or created for the purpose. A mathematical proof is an example. 

Nonlinear problems. Although just as complicated, their solutions are ambiguous and have no defined methodology for arriving at them. An example would be a loosely defined research paper.

The method you use for categorising is subjective. All that matters is you’re able to identify the problem to the best of your ability so you’ve got the best shot at using the most effective tool for the job. For instance, what if you’re assigned a research paper with a recommended structure and type of content already laid out? It’s far more linear at that point. What if you’re a masochist working on a famed paradox or mathematical dilemma without a solution? Maybe you’ve got a nightmare combination of these problems demanding usage of multiple methods, imagination and haste. If you can label it, you’ve got a better shot at applying the right methods. 

So what can you actually do with this knowledge? It’s pretty unhelpful just slapping all kinds of labels onto problems. What you need to do is understand the two major methods of innovative thinking, which we’ve got an entire article on here. If you’re not ready to click off just yet, here’s the run-down. 

Lateral thinking is what people refer to when they’re ‘thinking outside the box’. It’s about weighing up problems creatively to come up with multiple potential solutions and make an educated guess at which is best. 

Vertical thinking is methodical, logical and strict. It’s about driving home the solution to a problem, working through the steps until you’ve got an answer. 

At a glance, you may not see something like “vertical thinking” as particularly innovative, but adhering to formula or rulesets often requires more than just reading an instruction manual. It can involve creating those methods in the first place, perhaps even in combination with lateral thinking too. In fact, you’ll find many problems that could benefit from both. We’d definitely recommend reading our article if you’re interested in the specifics of what each entails. 

So how on earth does one go about improving these skills? Well, there are several every-day exercises and activities just about any university student in Australia can undergo. The natural progress of your degree will kind of ‘force’ you to refine them as you blast through your assignments, but it doesn’t hurt to know the tricks. 

Spend more time reflecting

If you’ve got a commute into work or uni, it pays to just turn off your podcasts, put your phone back into your pocket (especially if you’re driving), put on some quiet music and use the trip to work through problems nagging at you, be they academic or personal. This is your chance to practice whichever method of thinking is required. If you’ve got an essay you’ve yet to start, now’s the time to work through different potential structures and the research each may require. If you’ve got some basic tute questions, try to recall the method for solving them in as much detail as possible. 

Reflecting is also valuable for reviewing how you performed. What worked? What didn’t? Where do I think I need to improve regarding my problem-solving methods? Was the execution flawed? Even with something like group assignments, it pays to consider what you could have done better rather than fall to the temptation of shifting blame. This is such a simple, yet easy to overlook practice. We’re absolutely inundated with distractions after all. If you can find plenty of moments to just disconnect and think, you’ll give yourself a significant chance to truly develop as a problem-solver. 

Be inquisitive

Just asking how your mate tackled their assignment, answered their exam or thought through their issues and why can make a lot of difference. It’s a chance to expose yourself to potentially new methods of thinking, which is in-turn a chance to evaluate your current methods. Are theirs better? If they are, why not adopt them? Did they go wrong in their reasoning? Perhaps you misunderstood how they went about it. No matter the outcome, the skill development should always be positive. Whether you adopt a new method for future problems of a similar nature or stick to your guns, you’ll have tested yourself, which is what becoming a better problem-solver is all about. 

You can also ask mentors for examples of when they overcame problems, be it your parents, professors, former teachers or just about anyone whose judgement you trust. The most important thing is being exposed to different methods of thinking. To this end, your questions can be oriented toward vertical or lateral examples depending on your current focus or interest. 

Discussing methods for coming to true conclusions is valuable for both lateral and vertical thinking. A perfectly memorised method for doing something doesn’t equate to perfect execution, after all! Perhaps there’s some refinement to your process or execution you hadn’t considered. Computer science is often like this; while there are many ways to write code and make it work, for most languages, solutions are often not created equal! In this example at least, reviewing your friend’s code or asking why they wrote it as they did can be an immense help. For something more open-ended like bedside manner if you’re learning to become a nurse, you may learn new ways to express sentiments to patients in a more comforting manner. If you’re always being inquisitive and seeking better ways to do things, you’ll really advance by leaps and bounds as a problem-solver. 

Find more responsibilities

Naturally, exposing yourself to new problems carries the benefit of practice in finding solutions. You’ll likely doing all the right things just through building your resume anyway. New internships, part-time work, volunteering and getting on the committees of student clubs are all valuable ways to expose yourself to new problems. It doesn’t matter how big they are; every one is a chance to get out of your comfort zone and exercise multiple methods of thinking. 

You’ll probably be surprised at just how many new responsibilities you can take on and still have time for study. Near the end of each semester, you’ll have to dial the notch back of course, but during semester breaks and the first four or five weeks of your first and middle years, you’ll be able to find prime opportunities for these activities. No shame in keeping third-year clear, provided you’ve done all you can in previous years! 

We’d highly recommend doing all these things, even if they weren’t useful for developing your problem-solving skills. Yet, they do! Here are just a few examples of what each can do:

Internships give you an insight into what could be your first graduate job. Your work will likely involve a combination of urgent, linear and nonlinear problems. As an intern, you’ll be able to see how professionals tackle these problems sequentially, calmly and what methods they use to arrive at sound conclusions. This is an opportunity to exercise the previous steps too; asking plenty of questions and reflecting on the answers are things you absolutely should be doing to maximise value from an internship. 

Volunteering is similar. Even something simple like serving soup at a homeless shelter can carry with it the obligation to listen to an understand struggling people. Am I listening effectively? Am I communicating effectively? How can I better hear and understand the people I’m speaking with? Am I cooperating with my fellow volunteers to the greatest extent I can? With a bit of imagination, many optimisations and questions can be come mini problem-solving exercises and chances to improve. Outside these internal exercises are the real-world opportunities that logistical hurdles can present. If you’re called upon to advertise for a dinner, book a lecture or find catering for an event, these are all problems that may not have defined solutions or a single correct answer. It’s a chance to sift through your options. 

Student clubs can offer a similar experience. If you’re part of a club’s organisational committee, you may be called upon to spread the word about an event, help organise a ball, figure out how to delegate appropriately or any number of problems that again, may not have clear-cut solutions. 

Seek additional problem solving exercises

This is for those of you wanting to go the extra mile! While other activities in this article specify the utilisation of things you should already be doing anyway, or just making use of downtime, this will take a bit of concerted effort. 

There are plenty of books on critical thinking, innovation and general problem-solving you can use to bolster your efforts. Why not pick up one or two of the classics, like Edward de Bono’s Six Thinking Hats or Decision Making and Problem Solving by John Adair? If books aren’t your thing, there are all kinds of activities and problems online. Lateral thinking puzzles are a fairly popular activity among those wanting to improve their lateral thinking skills, for instance. As for vertical thinking, just undergoing math problems appropriate to your current knowledge level are a great way to develop. This might seem wildly counterintuitive if you’re an aspiring artist or historian, but what better way to develop your adherence to strict methods than with math? Hey, even just doing increasingly difficult sudoku puzzles can suffice. If you keep at it, you may develop a real taste for this sort of thing and will find it easier to tackle problems throughout academia and work. 

You should now have a much better idea of all the ways problem-solving skills can be developed at university. Honestly, the biggest takeaway is to just be creative with all the activities you’re already doing. Reading books and putting effort exclusively into this can be effective, but it’s also time consuming. Some of the most practical things you can do include simply working problem-finding and solutions into your current schedule, which is totally doable. No matter what you’re doing, there’s likely a way to get better at solving problems while doing it. Even if you’re in a repetitive part-time job, there are bound to be opportunities for self-improvement just by treating each perceived inefficiency or opportunity as a problem that can be solved. With keen reflection, asking good questions regularly and taking on more responsibilities, you’ll be well on your way to becoming a formidable problem-solver. 

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By developing strong problem-solving skills, you'll be able to find solutions even when they aren't obvious.

During your studies, you'll encounter many different problems you need to solve. Sometimes you may be unsure how to solve a problem or you may not fully understand what the problem is — which makes it hard to find the right solution.

When solving problems, it can help to start by identifying any barriers that may stop you finding a solution. Once you've identified any barriers, you can apply strategies to help overcome these and strengthen your problem-solving skills.

Some common barriers include:

• a lack of experience or confidence in how to solve problems
• trouble identifying the knowledge required to solve the problem
• believing that if you don't know how to solve the problem you shouldn't bother trying
• waiting until you know what to do before starting to write down ideas
• going with the first solution you think of instead of coming up with different options
• trying to force a solution to work when it's clear that it won't.

Once you know what barriers you're facing, apply the following problem-solving strategies to identify and work through possible solutions:

Identify relevant knowledge

Review the concepts or theories discussed in your lectures and tutorials to see if any of the information is relevant to the problem. It can be helpful to build up a summary sheet of information you learn over the semester so you have a reference guide ready.

Try multiple solutions

There may be multiple solutions to a problem, and some may work better than others.

If you find the solution you're trying doesn't fit, you'll to need to re-evaluate the problem and try other options. Remember if you make a mistake it doesn't mean you've failed — mistakes can help guide you to the correct answer.

• identify the most important points in the question you're trying to solve and any factors that may influence the solution
• make a list of possible solutions
• use a process of elimination to select potential solutions.

Organise information

Try reviewing the knowledge you have prior to problem-solving and organising it into a table, flowchart or diagram. This can help you to:

• compare and analyse information
• identify any information gaps
• avoid being overloaded by irrelevant information.

Download a guide to organising information (PDF, 3.8 MB)

Reframe the problem

Sometimes when thinking about a problem, you can make assumptions that stop you from finding a solution.

If you become stuck trying to solve a problem, it can help to approach the problem from a different perspective.

Some ways to help you reframe a problem include:

• questioning what the 'actual' problem is
• thinking about the cause of the problem
• identifying your assumptions
• creating a list of all possible solutions.
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Download tips to help your problem-solving (PDF, 159 KB) Watch an example of a common problem-solving process

39 Best Problem-Solving Examples

Problem-solving is a process where you’re tasked with identifying an issue and coming up with the most practical and effective solution.

This indispensable skill is necessary in several aspects of life, from personal relationships to education to business decisions.

Problem-solving aptitude boosts rational thinking, creativity, and the ability to cooperate with others. It’s also considered essential in 21st Century workplaces.

If explaining your problem-solving skills in an interview, remember that the employer is trying to determine your ability to handle difficulties. Focus on explaining exactly how you solve problems, including by introducing your thoughts on some of the following frameworks and how you’ve applied them in the past.

Problem-Solving Examples

1. divergent thinking.

Divergent thinking refers to the process of coming up with multiple different answers to a single problem. It’s the opposite of convergent thinking, which would involve coming up with a singular answer .

The benefit of a divergent thinking approach is that it can help us achieve blue skies thinking – it lets us generate several possible solutions that we can then critique and analyze .

In the realm of problem-solving, divergent thinking acts as the initial spark. You’re working to create an array of potential solutions, even those that seem outwardly unrelated or unconventional, to get your brain turning and unlock out-of-the-box ideas.

This process paves the way for the decision-making stage, where the most promising ideas are selected and refined.

Go Deeper: Divervent Thinking Examples

2. Convergent Thinking

Next comes convergent thinking, the process of narrowing down multiple possibilities to arrive at a single solution.

This involves using your analytical skills to identify the best, most practical, or most economical solution from the pool of ideas that you generated in the divergent thinking stage.

In a way, convergent thinking shapes the “roadmap” to solve a problem after divergent thinking has supplied the “destinations.”

Have a think about which of these problem-solving skills you’re more adept at: divergent or convergent thinking?

Go Deeper: Convergent Thinking Examples

3. Brainstorming

Brainstorming is a group activity designed to generate a multitude of ideas regarding a specific problem. It’s divergent thinking as a group , which helps unlock even more possibilities.

A typical brainstorming session involves uninhibited and spontaneous ideation, encouraging participants to voice any possible solutions, no matter how unconventional they might appear.

It’s important in a brainstorming session to suspend judgment and be as inclusive as possible, allowing all participants to get involved.

By widening the scope of potential solutions, brainstorming allows better problem definition, more creative solutions, and helps to avoid thinking “traps” that might limit your perspective.

Go Deeper: Brainstorming Examples

4. Thinking Outside the Box

The concept of “thinking outside the box” encourages a shift in perspective, urging you to approach problems from an entirely new angle.

Rather than sticking to traditional methods and processes, it involves breaking away from conventional norms to cultivate unique solutions.

In problem-solving, this mindset can bypass established hurdles and bring you to fresh ideas that might otherwise remain undiscovered.

Think of it as going off the beaten track when regular routes present roadblocks to effective resolution.

5. Case Study Analysis

Analyzing case studies involves a detailed examination of real-life situations that bear relevance to the current problem at hand.

For example, if you’re facing a problem, you could go to another environment that has faced a similar problem and examine how they solved it. You’d then bring the insights from that case study back to your own problem.

This approach provides a practical backdrop against which theories and assumptions can be tested, offering valuable insights into how similar problems have been approached and resolved in the past.

See a Broader Range of Analysis Examples Here

6. Action Research

Action research involves a repetitive process of identifying a problem, formulating a plan to address it, implementing the plan, and then analyzing the results. It’s common in educational research contexts.

The objective is to promote continuous learning and improvement through reflection and action. You conduct research into your problem, attempt to apply a solution, then assess how well the solution worked. This becomes an iterative process of continual improvement over time.

For problem-solving, this method offers a way to test solutions in real-time and allows for changes and refinements along the way, based on feedback or observed outcomes. It’s a form of active problem-solving that integrates lessons learned into the next cycle of action.

Go Deeper: Action Research Examples

7. Information Gathering

Fundamental to solving any problem is the process of information gathering.

This involves collecting relevant data , facts, and details about the issue at hand, significantly aiding in the understanding and conceptualization of the problem.

In problem-solving, information gathering underpins every decision you make.

This process ensures your actions are based on concrete information and evidence, allowing for an informed approach to tackle the problem effectively.

8. Seeking Advice

Seeking advice implies turning to knowledgeable and experienced individuals or entities to gain insights on problem-solving.

It could include mentors, industry experts, peers, or even specialized literature.

The value in this process lies in leveraging different perspectives and proven strategies when dealing with a problem. Moreover, it aids you in avoiding pitfalls, saving time, and learning from others’ experiences.

9. Creative Thinking

Creative thinking refers to the ability to perceive a problem in a new way, identify unconventional patterns, or produce original solutions.

It encourages innovation and uniqueness, often leading to the most effective results.

When applied to problem-solving, creative thinking can help you break free from traditional constraints, ideal for potentially complex or unusual problems.

Go Deeper: Creative Thinking Examples

10. Conflict Resolution

Conflict resolution is a strategy developed to resolve disagreements and arguments, often involving communication, negotiation, and compromise.

When employed as a problem-solving technique, it can diffuse tension, clear bottlenecks, and create a collaborative environment.

Effective conflict resolution ensures that differing views or disagreements do not become roadblocks in the process of problem-solving.

Go Deeper: Conflict Resolution Examples

11. Addressing Bottlenecks

Bottlenecks refer to obstacles or hindrances that slow down or even halt a process.

In problem-solving, addressing bottlenecks involves identifying these impediments and finding ways to eliminate them.

This effort not only smooths the path to resolution but also enhances the overall efficiency of the problem-solving process.

For example, if your workflow is not working well, you’d go to the bottleneck – that one point that is most time consuming – and focus on that. Once you ‘break’ this bottleneck, the entire process will run more smoothly.

12. Market Research

Market research involves gathering and analyzing information about target markets, consumers, and competitors.

In sales and marketing, this is one of the most effective problem-solving methods. The research collected from your market (e.g. from consumer surveys) generates data that can help identify market trends, customer preferences, and competitor strategies.

In this sense, it allows a company to make informed decisions, solve existing problems, and even predict and prevent future ones.

13. Root Cause Analysis

Root cause analysis is a method used to identify the origin or the fundamental reason for a problem.

Once the root cause is determined, you can implement corrective actions to prevent the problem from recurring.

As a problem-solving procedure, root cause analysis helps you to tackle the problem at its source, rather than dealing with its surface symptoms.

Go Deeper: Root Cause Analysis Examples

14. Mind Mapping

Mind mapping is a visual tool used to structure information, helping you better analyze, comprehend and generate new ideas.

By laying out your thoughts visually, it can lead you to solutions that might not have been apparent with linear thinking.

In problem-solving, mind mapping helps in organizing ideas and identifying connections between them, providing a holistic view of the situation and potential solutions.

15. Trial and Error

The trial and error method involves attempting various solutions until you find one that resolves the problem.

It’s an empirical technique that relies on practical actions instead of theories or rules.

In the context of problem-solving, trial and error allows you the flexibility to test different strategies in real situations, gaining insights about what works and what doesn’t.

16. SWOT Analysis

SWOT is an acronym standing for Strengths, Weaknesses, Opportunities, and Threats.

It’s an analytic framework used to evaluate these aspects in relation to a particular objective or problem.

In problem-solving, SWOT Analysis helps you to identify favorable and unfavorable internal and external factors. It helps to craft strategies that make best use of your strengths and opportunities, whilst addressing weaknesses and threats.

Go Deeper: SWOT Analysis Examples

17. Scenario Planning

Scenario planning is a strategic planning method used to make flexible long-term plans.

It involves imagining, and then planning for, multiple likely future scenarios.

By forecasting various directions a problem could take, scenario planning helps manage uncertainty and is an effective tool for problem-solving in volatile conditions.

18. Six Thinking Hats

The Six Thinking Hats is a concept devised by Edward de Bono that proposes six different directions or modes of thinking, symbolized by six different hat colors.

Each hat signifies a different perspective, encouraging you to switch ‘thinking modes’ as you switch hats. This method can help remove bias and broaden perspectives when dealing with a problem.

19. Decision Matrix Analysis

Decision Matrix Analysis is a technique that allows you to weigh different factors when faced with several possible solutions.

After listing down the options and determining the factors of importance, each option is scored based on each factor.

Revealing a clear winner that both serves your objectives and reflects your values, Decision Matrix Analysis grounds your problem-solving process in objectivity and comprehensiveness.

20. Pareto Analysis

Also known as the 80/20 rule, Pareto Analysis is a decision-making technique.

It’s based on the principle that 80% of problems are typically caused by 20% of the causes, making it a handy tool for identifying the most significant issues in a situation.

Using this analysis, you’re likely to direct your problem-solving efforts more effectively, tackling the root causes producing most of the problem’s impact.

21. Critical Thinking

Critical thinking refers to the ability to analyze facts to form a judgment objectively.

It involves logical, disciplined thinking that is clear, rational, open-minded, and informed by evidence.

For problem-solving, critical thinking helps evaluate options and decide the most effective solution. It ensures your decisions are grounded in reason and facts, and not biased or irrational assumptions.

Go Deeper: Critical Thinking Examples

22. Hypothesis Testing

Hypothesis testing usually involves formulating a claim, testing it against actual data, and deciding whether to accept or reject the claim based on the results.

In problem-solving, hypotheses often represent potential solutions. Hypothesis testing provides verification, giving a statistical basis for decision-making and problem resolution.

Usually, this will require research methods and a scientific approach to see whether the hypothesis stands up or not.

Go Deeper: Types of Hypothesis Testing

23. Cost-Benefit Analysis

A cost-benefit analysis (CBA) is a systematic process of weighing the pros and cons of different solutions in terms of their potential costs and benefits.

It allows you to measure the positive effects against the negatives and informs your problem-solving strategy.

By using CBA, you can identify which solution offers the greatest benefit for the least cost, significantly improving efficacy and efficiency in your problem-solving process.

Go Deeper: Cost-Benefit Analysis Examples

24. Simulation and Modeling

Simulations and models allow you to create a simplified replica of real-world systems to test outcomes under controlled conditions.

In problem-solving, you can broadly understand potential repercussions of different solutions before implementation.

It offers a cost-effective way to predict the impacts of your decisions, minimizing potential risks associated with various solutions.

25. Delphi Method

The Delphi Method is a structured communication technique used to gather expert opinions.

The method involves a group of experts who respond to questionnaires about a problem. The responses are aggregated and shared with the group, and the process repeats until a consensus is reached.

This method of problem solving can provide a diverse range of insights and solutions, shaped by the wisdom of a collective expert group.

26. Cross-functional Team Collaboration

Cross-functional team collaboration involves individuals from different departments or areas of expertise coming together to solve a common problem or achieve a shared goal.

When you bring diverse skills, knowledge, and perspectives to a problem, it can lead to a more comprehensive and innovative solution.

In problem-solving, this promotes communal thinking and ensures that solutions are inclusive and holistic, with various aspects of the problem being addressed.

27. Benchmarking

Benchmarking involves comparing one’s business processes and performance metrics to the best practices from other companies or industries.

In problem-solving, it allows you to identify gaps in your own processes, determine how others have solved similar problems, and apply those solutions that have proven to be successful.

It also allows you to compare yourself to the best (the benchmark) and assess where you’re not as good.

28. Pros-Cons Lists

A pro-con analysis aids in problem-solving by weighing the advantages (pros) and disadvantages (cons) of various possible solutions.

This simple but powerful tool helps in making a balanced, informed decision.

When confronted with a problem, a pro-con analysis can guide you through the decision-making process, ensuring all possible outcomes and implications are scrutinized before arriving at the optimal solution. Thus, it helps to make the problem-solving process both methodical and comprehensive.

29. 5 Whys Analysis

The 5 Whys Analysis involves repeatedly asking the question ‘why’ (around five times) to peel away the layers of an issue and discover the root cause of a problem.

As a problem-solving technique, it enables you to delve into details that you might otherwise overlook and offers a simple, yet powerful, approach to uncover the origin of a problem.

For example, if your task is to find out why a product isn’t selling your first answer might be: “because customers don’t want it”, then you ask why again – “they don’t want it because it doesn’t solve their problem”, then why again – “because the product is missing a certain feature” … and so on, until you get to the root “why”.

30. Gap Analysis

Gap analysis entails comparing current performance with potential or desired performance.

You’re identifying the ‘gaps’, or the differences, between where you are and where you want to be.

In terms of problem-solving, a Gap Analysis can help identify key areas for improvement and design a roadmap of how to get from the current state to the desired one.

31. Design Thinking

Design thinking is a problem-solving approach that involves empathy, experimentation, and iteration.

The process focuses on understanding user needs, challenging assumptions , and redefining problems from a user-centric perspective.

In problem-solving, design thinking uncovers innovative solutions that may not have been initially apparent and ensures the solution is tailored to the needs of those affected by the issue.

32. Analogical Thinking

Analogical thinking involves the transfer of information from a particular subject (the analogue or source) to another particular subject (the target).

In problem-solving, you’re drawing parallels between similar situations and applying the problem-solving techniques used in one situation to the other.

Thus, it allows you to apply proven strategies to new, but related problems.

33. Lateral Thinking

Lateral thinking requires looking at a situation or problem from a unique, sometimes abstract, often non-sequential viewpoint.

Unlike traditional logical thinking methods, lateral thinking encourages you to employ creative and out-of-the-box techniques.

In solving problems, this type of thinking boosts ingenuity and drives innovation, often leading to novel and effective solutions.

Go Deeper: Lateral Thinking Examples

34. Flowcharting

Flowcharting is the process of visually mapping a process or procedure.

This form of diagram can show every step of a system, process, or workflow, enabling an easy tracking of the progress.

As a problem-solving tool, flowcharts help identify bottlenecks or inefficiencies in a process, guiding improved strategies and providing clarity on task ownership and process outcomes.

35. Multivoting

Multivoting, or N/3 voting, is a method where participants reduce a large list of ideas to a prioritized shortlist by casting multiple votes.

This voting system elevates the most preferred options for further consideration and decision-making.

As a problem-solving technique, multivoting allows a group to narrow options and focus on the most promising solutions, ensuring more effective and democratic decision-making.

36. Force Field Analysis

Force Field Analysis is a decision-making technique that identifies the forces for and against change when contemplating a decision.

The ‘forces’ represent the differing factors that can drive or hinder change.

In problem-solving, Force Field Analysis allows you to understand the entirety of the context, favoring a balanced view over a one-sided perspective. A comprehensive view of all the forces at play can lead to better-informed problem-solving decisions.

TRIZ, which stands for “The Theory of Inventive Problem Solving,” is a problem-solving, analysis, and forecasting methodology.

It focuses on finding contradictions inherent in a scenario. Then, you work toward eliminating the contraditions through finding innovative solutions.

So, when you’re tackling a problem, TRIZ provides a disciplined, systematic approach that aims for ideal solutions and not just acceptable ones. Using TRIZ, you can leverage patterns of problem-solving that have proven effective in different cases, pivoting them to solve the problem at hand.

38. A3 Problem Solving

A3 Problem Solving, derived from Lean Management, is a structured method that uses a single sheet of A3-sized paper to document knowledge from a problem-solving process.

Named after the international paper size standard of A3 (or 11-inch by 17-inch paper), it succinctly records all key details of the problem-solving process from problem description to the root cause and corrective actions.

Used in problem-solving, this provides a straightforward and logical structure for addressing the problem, facilitating communication between team members, ensuring all critical details are included, and providing a record of decisions made.

39. Scenario Analysis

Scenario Analysis is all about predicting different possible future events depending upon your decision.

To do this, you look at each course of action and try to identify the most likely outcomes or scenarios down the track if you take that course of action.

This technique helps forecast the impacts of various strategies, playing each out to their (logical or potential) end. It’s a good strategy for project managers who need to keep a firm eye on the horizon at all times.

When solving problems, Scenario Analysis assists in preparing for uncertainties, making sure your solution remains viable, regardless of changes in circumstances.

How to Answer “Demonstrate Problem-Solving Skills” in an Interview

When asked to demonstrate your problem-solving skills in an interview, the STAR method often proves useful. STAR stands for Situation, Task, Action, and Result.

Situation: Begin by describing a specific circumstance or challenge you encountered. Make sure to provide enough detail to allow the interviewer a clear understanding. You should select an event that adequately showcases your problem-solving abilities.

For instance, “In my previous role as a project manager, we faced a significant issue when our key supplier abruptly went out of business.”

Task: Explain what your responsibilities were in that situation. This serves to provide context, allowing the interviewer to understand your role and the expectations placed upon you.

For instance, “It was my task to ensure the project remained on track despite this setback. Alternative suppliers needed to be found without sacrificing quality or significantly increasing costs.”

Action: Describe the steps you took to manage the problem. Highlight your problem-solving process. Mention any creative approaches or techniques that you used.

For instance, “I conducted thorough research to identify potential new suppliers. After creating a shortlist, I initiated contact, negotiated terms, assessed samples for quality and made a selection. I also worked closely with the team to re-adjust the project timeline.”

Result: Share the outcomes of your actions. How did the situation end? Did your actions lead to success? It’s particularly effective if you can quantify these results.

For instance, “As a result of my active problem solving, we were able to secure a new supplier whose costs were actually 10% cheaper and whose quality was comparable. We adjusted the project plan and managed to complete the project just two weeks later than originally planned, despite the major vendor setback.”

Remember, when you’re explaining your problem-solving skills to an interviewer, what they’re really interested in is your approach to handling difficulties, your creativity and persistence in seeking a resolution, and your ability to carry your solution through to fruition. Tailoring your story to highlight these aspects will help exemplify your problem-solving prowess.

Go Deeper: STAR Interview Method Examples

Benefits of Problem-Solving

Problem-solving is beneficial for the following reasons (among others):

• It can help you to overcome challenges, roadblocks, and bottlenecks in your life.
• It can save a company money.
• It can help you to achieve clarity in your thinking.
• It can make procedures more efficient and save time.
• It can strengthen your decision-making capacities.
• It can lead to better risk management.

Whether for a job interview or school, problem-solving helps you to become a better thinking, solve your problems more effectively, and achieve your goals. Build up your problem-solving frameworks (I presented over 40 in this piece for you!) and work on applying them in real-life situations.

Chris Drew (PhD)

Dr. Chris Drew is the founder of the Helpful Professor. He holds a PhD in education and has published over 20 articles in scholarly journals. He is the former editor of the Journal of Learning Development in Higher Education. [Image Descriptor: Photo of Chris]

• Chris Drew (PhD) https://helpfulprofessor.com/author/chris-drew-phd/ Social-Emotional Learning (Definition, Examples, Pros & Cons)
• Chris Drew (PhD) https://helpfulprofessor.com/author/chris-drew-phd/ What is Educational Psychology?
• Chris Drew (PhD) https://helpfulprofessor.com/author/chris-drew-phd/ What is IQ? (Intelligence Quotient)
• Chris Drew (PhD) https://helpfulprofessor.com/author/chris-drew-phd/ 5 Top Tips for Succeeding at University

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Being skilled in problem solving draws on several other skill areas, including analysis, flexibility and research. These can allow you to deconstruct the problem, making it easier to think laterally and develop informed solutions. A balance of logic, creativity and perseverance can sometimes be required to demonstrate your versatility in this area.

Problem solving appears frequently in job descriptions as an essential skill. No matter which professional area you are considering, a strong ability to demonstrate your problem solving skills will be a valuable offering to a future employer, especially if you can show you have acted swiftly.

Activities where you could develop your problem solving skills

Problem solving skills can be a key learning outcome of many degree courses, but you cannot assume that everyone who reads your CV will recognise this. Take care to provide evidence of the problems encountered, their complexity and the means by which you solved them. Project or business case-based activities can form excellent examples.

• University hall Residence Association The challenges and rewards of helping to run a hall of residence will put your problem solving to the test.
• Student representation Fielding questions from both your fellow students and the responses of your academic staff can require a creative approach to problem solving.
• Peer Mentoring Develop problem solving skills by supporting others through their studies and wider university life.
• Nightline Lateral thinking, compassion and good listening will help you develop your skills while supporting others.
• Competitions and prizes Look great on your CV, even if you don't win you will have experience to talk about.

How is problem solving tested in recruitment?

Written applications may start testing your problem solving skills directly by asking for examples of problems you have solved in the past. This can be probed further by psychometric tests , particularly Situational Judgement. At interview, you can expect questions which re-examine this skill:

• Tell me about a complex problem you have faced – what were the critical issues, and the steps that you took to solve the problem?
• Describe a time when you demonstrated creativity in solving a difficult problem.
• Give me an example of when you used good judgment and logic in solving a problem.

If you are unsure how to structure an answer for either application or interview questions, visit the application and interviews section of our website and find out about the CAR (context, action, result) and STAR (situation, task, action, result) models. Our recommendations are based on feedback from employers.

During an assessment centre , real life business problems will be presented to you, either individually or as part of a group task. You will need to deploy a mix of sector knowledge and analytical skills to develop proposals to solve the problem. You will be assessed on your problem solving strategy and your creativity. You may need to think on your feet, and be able to cope well under pressure in order to succeed.

Sometimes you may be asked to complete practical problem solving activities, such as building a tower from Lego, paper or other materials. This will test other skills such as clear communication and team working skills.

Critical thinking and problem solving are the skills that go hand in hand. At Compaira, we look for someone who takes initiative and thinks laterally to challenge accepted ways of approaching or solving a problem. CEO & Founder, Compaira

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WESTERN GOVERNORS UNIVERSITY

Developing your problem-solving skills.

Problem-Solving Skills

Problem-solving skills enhance your ability to identify a difficult or unforeseen situation and determine an appropriate solution. 

Using the right problem-solving approach will empower you to offer practical solutions in your professional and personal life anytime you’re faced with a problem.

This page covers different types of problem-solving skills, why they matter, and how to acquire them.

Why Are Problem-Solving Skills Important?

Problem-solving skills are crucial in empowering individuals to handle large or small obstacles throughout various aspects of life. Here are just a few ways that these skills can help you:

• Overcoming challenges: Life will always present you with problems that may hinder your personal or professional progress. Problem-solving skills empower you to identify solutions, giving you control over your future.
• Enhancing decision-making: Problem-solving skills help you assess problems as they come, gauge all the possible solutions, and make the best decision. 
• Promoting innovation: Practical problem-solving skills encourage creative thinking, enabling you to develop innovative ideas. You can then evaluate these ideas to identify effective solutions to tackle obstacles.
• Increasing efficiency: Problem-solving skills help individuals and organizations improve efficiency and save time and resources. You are assured of increased productivity if you can identify the root cause of a problem, get the appropriate solution, and implement it promptly.
• Building resilience: Since problems are part of everyone’s day-to-day life, being equipped with problem-solving skills will enable you to respond quickly and rationally to unforeseen situations and not succumb to setbacks.

What Are the Benefits of Having Problem-Solving Skills?

When you frequently apply your problem-solving skills, you become more proficient at analyzing, resolving, and adapting to challenges.

In the workplace, people with strong problem-solving skills apply a combination of creative thinking and analytical skills to help them become more confident when making decisions in the face of challenges. 

They’re better equipped to handle the challenges their job brings Problem solvers can observe, judge, and act quickly when faced with adversity.

Problem-solving skills enable you to prioritize, plan, and execute your strategies. You also learn how to think outside the box and identify opportunities in problems. 

Examples of Problem-Solving Skills

Problem-solving skills are effective in helping you identify the source of a problem and how to solve it with a structured approach.

Here are some skills associated with problem-solving:

• Analyzing data: When addressing a number of problems, it’s important to gather relevant for a thorough understanding of the issue and its underlying causes.
• Brainstorming creative solutions: Brainstorming allows you to find the right information from the different causes of the problem and develop innovative solutions. 
• Researching to gather relevant information: Having clarity on your research goals and digging into reliable resources to get relevant information.
• Use critical thinking to evaluate options: This involves objective questioning, analyzing, and systematically assessing available options. You should be able to differentiate facts from opinions. 
• Implementing logical and systematic approaches: Exploring the key components of each option to understand their practicality and relevance to the problem. Use predefined criteria to evaluate the options and then conduct a SWOT (strengths, weaknesses, opportunities, threats) analysis to determine their viability.
• Collaborating with others to find solutions: Team members gather to offer their input, giving potential solutions and collectively analyzing them to solve the problem.
• Creative thinking: This may include generating new ideas to solve problems. It involves brainstorming, mind mapping, lateral thinking, and analogical reasoning. 
• Decision-making: This is the capability to evaluate options, weigh pros and cons, and make informed decisions based on available information. 
• Problem identification: The ability to recognize and define problems accurately and clearly.  
• Problem structuring: The skill involves taking complicated or vague problems and dissecting them into smaller subproblems that are easier to understand and solve. 

How Can I Use Problem-Solving Skills? 

Problem-solving skills will have a lasting impact on your life. In your personal and professional life, you will encounter many challenges requiring you to use problem-solving skills.

Here are some instances where you’ll use problem-solving skills: 

Career Success

Problem-solving skills will help you tackle challenges at any workplace, make informed decisions, and help the organizations you work with to succeed., personal growth, these skills equip you to overcome everyday life obstacles, like managing personal finances, resolving conflicts in relationships, and more..

Innovation and Entrepreneurship

By solving problems, you can generate creative ideas, develop innovative solutions, and even solve societal problems.

Decision-Making

Making big and small decisions by gathering alternatives, evaluating outcomes, and deciding on the best course of action.

How Can I Learn Problem-Solving Skills?

At wgu, we offer several programs and courses that focus on teaching and enhancing problem-solving skills. , here are ways you can learn problem-solving skills at wgu., school of business.

You’ll learn how to apply problem-solving skills in the world of entrepreneurship and business. For example, our Bachelor of Science in Business Administration–Human Resource Management, offers problem-solving skills as a key component. 

With business degree programs, you will learn to:

• Generate a solution to a problem.
• Conduct research to find solutions to a problem.
• Articulate findings and resolutions to a problem.         
• Apply contextual reasoning to understand problems.
• Analyze data for the nature and extent of a problem.

School of Technology

You’ll learn how to apply problem-solving skills within the ever-evolving world of IT. We have different bachelor of science degrees in Cloud Computing, such as Muilticloud Track, Amazon Web Services Track, and BS in Cloud Computing - Azure Track.

With IT-related degree programs, you will learn to: 

• Select appropriate problem-solving techniques.                    
• Resolve a challenge using a problem-solving process.
• Recommend multiple solutions for a variety of problems.
• Implement approaches to address complex problems.
• Identify problems using various common frameworks.

School of Education

You’ll learn how problem-solving skills are crucial in the education sector. For example, our Master of Science in Educational Leadership covers problem-solving in the School of Education courses.

With education degree programs, you will learn to:

• Implement problem-solving skills for issue resolution.
• Solve complex problems.   
• Identify the cause of a problem.
• Solve problems in the manner most appropriate for each problem.    

Frequently Asked Questions

What are the steps involved in the problem-solving process?

The problem-solving process involves the following steps:

• Problem identification: Clearly state the problem, what makes you view it as a problem, and how you discovered it.
• Problem analysis: Gather data to help you fully understand the problem and its root causes.
• Generating potential solutions: Look for all the possible ideas from different angles to solve the problem. 
• Evaluating alternatives: Scrutinize the available options to identify the best idea. 
• Implementing the chosen solution: Follow through on the necessary steps to resolve the problem.
• Reviewing the results: Gather feedback to test the results against your expectations.

How can I improve my critical thinking skills to enhance my problem-solving?

You can improve your critical thinking skills through the following:

• Logical reasoning: Embrace creative hobbies, learn new skills, and socialize with others.
• Evaluating evidence: Determine the relevance and quality of the evidence available to challenge or support claims.
• Considering multiple perspectives: Being open to learning from your peers and adjusting your views accordingly.

What are some effective techniques for generating creative solutions to problems?

Creative problem solving (CPS) is about innovatively solving problems. The techniques below will aid you in the CPS process:

• Brainstorming: Gather relevant parties and spontaneously contribute ideas to offer the solution.
• Mind mapping: Capture and organize any form of information and ideas in a structured way to enhance your logical and creative thinking.
• Analogical reasoning: Use analogies to simplify complex scenarios to makes them easy to comprehend.

How can problem-solving skills be applied in everyday life?

Picture this: your car breaks down while running errands. It’s important to identify the problem with your car and analyze the symptoms and potential root causes of the breakdown, such as mechanical issues or battery failure. You decide that a faulty batter is the issue so you start to identify potential solutions. 

The solutions may include calling for roadside assistance, finding a mechanic nearby, or seeking help from friends or family. You then evaluate the potential solutions by weighing in factors such as cost, time, and convenience.  The best solution you find may be to call a nearby friend to help you jump-start the car and get back on the road to complete your errands. 

This is an everyday occurrence in which problem-solving skills are applicable. Other situations may vary with the degree of difficulty, urgency, and solutions required.

Are specific problem-solving skills valuable in a team or collaborative setting?

Yes. With effective communication, effective conflict resolution strategies, active listening, and consensus building, team members can build healthy working relationships and succeed in their daily decision-making processes.

Find Your Degree

Ready to take the next step in developing your problem-solving skills? Take our degree quiz to find the perfect program for you! With our flexible online learning options and personalized support, you can achieve your educational and career goals on your own terms.

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21 Problem Solving

Miriam Bassok, Department of Psychology, University of Washington, Seattle, WA

Laura R. Novick, Department of Psychology and Human Development, Vanderbilt University, Nashville, TN

• Published: 21 November 2012
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This chapter follows the historical development of research on problem solving. It begins with a description of two research traditions that addressed different aspects of the problem-solving process: ( 1 ) research on problem representation (the Gestalt legacy) that examined how people understand the problem at hand, and ( 2 ) research on search in a problem space (the legacy of Newell and Simon) that examined how people generate the problem's solution. It then describes some developments in the field that fueled the integration of these two lines of research: work on problem isomorphs, on expertise in specific knowledge domains (e.g., chess, mathematics), and on insight solutions. Next, it presents examples of recent work on problem solving in science and mathematics that highlight the impact of visual perception and background knowledge on how people represent problems and search for problem solutions. The final section considers possible directions for future research.

People are confronted with problems on a daily basis, be it trying to extract a broken light bulb from a socket, finding a detour when the regular route is blocked, fixing dinner for unexpected guests, dealing with a medical emergency, or deciding what house to buy. Obviously, the problems people encounter differ in many ways, and their solutions require different types of knowledge and skills. Yet we have a sense that all the situations we classify as problems share a common core. Karl Duncker defined this core as follows: “A problem arises when a living creature has a goal but does not know how this goal is to be reached. Whenever one cannot go from the given situation to the desired situation simply by action [i.e., by the performance of obvious operations], then there has to be recourse to thinking” (Duncker, 1945 , p. 1). Consider the broken light bulb. The obvious operation—holding the glass part of the bulb with one's fingers while unscrewing the base from the socket—is prevented by the fact that the glass is broken. Thus, there must be “recourse to thinking” about possible ways to solve the problem. For example, one might try mounting half a potato on the broken bulb (we do not know the source of this creative solution, which is described on many “how to” Web sites).

The above definition and examples make it clear that what constitutes a problem for one person may not be a problem for another person, or for that same person at another point in time. For example, the second time one has to remove a broken light bulb from a socket, the solution likely can be retrieved from memory; there is no problem. Similarly, tying shoes may be considered a problem for 5-year-olds but not for readers of this chapter. And, of course, people may change their goal and either no longer have a problem (e.g., take the guests to a restaurant instead of fixing dinner) or attempt to solve a different problem (e.g., decide what restaurant to go to). Given the highly subjective nature of what constitutes a problem, researchers who study problem solving have often presented people with novel problems that they should be capable of solving and attempted to find regularities in the resulting problem-solving behavior. Despite the variety of possible problem situations, researchers have identified important regularities in the thinking processes by which people (a) represent , or understand, problem situations and (b) search for possible ways to get to their goal.

A problem representation is a model constructed by the solver that summarizes his or her understanding of the problem components—the initial state (e.g., a broken light bulb in a socket), the goal state (the light bulb extracted), and the set of possible operators one may apply to get from the initial state to the goal state (e.g., use pliers). According to Reitman ( 1965 ), problem components differ in the extent to which they are well defined . Some components leave little room for interpretation (e.g., the initial state in the broken light bulb example is relatively well defined), whereas other components may be ill defined and have to be defined by the solver (e.g., the possible actions one may take to extract the broken bulb). The solver's representation of the problem guides the search for a possible solution (e.g., possible attempts at extracting the light bulb). This search may, in turn, change the representation of the problem (e.g., finding that the goal cannot be achieved using pliers) and lead to a new search. Such a recursive process of representation and search continues until the problem is solved or until the solver decides to abort the goal.

Duncker ( 1945 , pp. 28–37) documented the interplay between representation and search based on his careful analysis of one person's solution to the “Radiation Problem” (later to be used extensively in research analogy, see Holyoak, Chapter 13 ). This problem requires using some rays to destroy a patient's stomach tumor without harming the patient. At sufficiently high intensity, the rays will destroy the tumor. However, at that intensity, they will also destroy the healthy tissue surrounding the tumor. At lower intensity, the rays will not harm the healthy tissue, but they also will not destroy the tumor. Duncker's analysis revealed that the solver's solution attempts were guided by three distinct problem representations. He depicted these solution attempts as an inverted search tree in which the three main branches correspond to the three general problem representations (Duncker, 1945 , p. 32). We reproduce this diagram in Figure 21.1 . The desired solution appears on the rightmost branch of the tree, within the general problem representation in which the solver aims to “lower the intensity of the rays on their way through healthy tissue.” The actual solution is to project multiple low-intensity rays at the tumor from several points around the patient “by use of lens.” The low-intensity rays will converge on the tumor, where their individual intensities will sum to a level sufficient to destroy the tumor.

A search-tree representation of one subject's solution to the radiation problem, reproduced from Duncker ( 1945 , p. 32).

Although there are inherent interactions between representation and search, some researchers focus their efforts on understanding the factors that affect how solvers represent problems, whereas others look for regularities in how they search for a solution within a particular representation. Based on their main focus of interest, researchers devise or select problems with solutions that mainly require either constructing a particular representation or finding the appropriate sequence of steps leading from the initial state to the goal state. In most cases, researchers who are interested in problem representation select problems in which one or more of the components are ill defined, whereas those who are interested in search select problems in which the components are well defined. The following examples illustrate, respectively, these two problem types.

The Bird-and-Trains problem (Posner, 1973 , pp. 150–151) is a mathematical word problem that tends to elicit two distinct problem representations (see Fig. 21.2a and b ):

Two train stations are 50 miles apart. At 2 p.m. one Saturday afternoon two trains start toward each other, one from each station. Just as the trains pull out of the stations, a bird springs into the air in front of the first train and flies ahead to the front of the second train. When the bird reaches the second train, it turns back and flies toward the first train. The bird continues to do this until the trains meet. If both trains travel at the rate of 25 miles per hour and the bird flies at 100 miles per hour, how many miles will the bird have flown before the trains meet? Fig. 21.2 Open in new tab Download slide Alternative representations of Posner's ( 1973 ) trains-and-bird problem. Adapted from Novick and Hmelo ( 1994 ).

Some solvers focus on the back-and-forth path of the bird (Fig. 21.2a ). This representation yields a problem that would be difficult for most people to solve (e.g., a series of differential equations). Other solvers focus on the paths of the trains (Fig. 21.2b ), a representation that yields a relatively easy distance-rate-time problem.

The Tower of Hanoi problem falls on the other end of the representation-search continuum. It leaves little room for differences in problem representations, and the primary work is to discover a solution path (or the best solution path) from the initial state to the goal state .

There are three pegs mounted on a base. On the leftmost peg, there are three disks of differing sizes. The disks are arranged in order of size with the largest disk on the bottom and the smallest disk on the top. The disks may be moved one at a time, but only the top disk on a peg may be moved, and at no time may a larger disk be placed on a smaller disk. The goal is to move the three-disk tower from the leftmost peg to the rightmost peg.

Figure 21.3 shows all the possible legal arrangements of disks on pegs. The arrows indicate transitions between states that result from moving a single disk, with the thicker gray arrows indicating the shortest path that connects the initial state to the goal state.

The division of labor between research on representation versus search has distinct historical antecedents and research traditions. In the next two sections, we review the main findings from these two historical traditions. Then, we describe some developments in the field that fueled the integration of these lines of research—work on problem isomorphs, on expertise in specific knowledge domains (e.g., chess, mathematics), and on insight solutions. In the fifth section, we present some examples of recent work on problem solving in science and mathematics. This work highlights the role of visual perception and background knowledge in the way people represent problems and search for problem solutions. In the final section, we consider possible directions for future research.

Our review is by no means an exhaustive one. It follows the historical development of the field and highlights findings that pertain to a wide variety of problems. Research pertaining to specific types of problems (e.g., medical problems), specific processes that are involved in problem solving (e.g., analogical inferences), and developmental changes in problem solving due to learning and maturation may be found elsewhere in this volume (e.g., Holyoak, Chapter 13 ; Smith & Ward, Chapter 23 ; van Steenburgh et al., Chapter 24 ; Simonton, Chapter 25 ; Opfer & Siegler, Chapter 30 ; Hegarty & Stull, Chapter 31 ; Dunbar & Klahr, Chapter 35 ; Patel et al., Chapter 37 ; Lowenstein, Chapter 38 ; Koedinger & Roll, Chapter 40 ).

All possible problem states for the three-disk Tower of Hanoi problem. The thicker gray arrows show the optimum solution path connecting the initial state (State #1) to the goal state (State #27).

Problem Representation: The Gestalt Legacy

Research on problem representation has its origins in Gestalt psychology, an influential approach in European psychology during the first half of the 20th century. (Behaviorism was the dominant perspective in American psychology at this time.) Karl Duncker published a book on the topic in his native German in 1935, which was translated into English and published 10 years later as the monograph On Problem-Solving (Duncker, 1945 ). Max Wertheimer also published a book on the topic in 1945, titled Productive Thinking . An enlarged edition published posthumously includes previously unpublished material (Wertheimer, 1959 ). Interestingly, 1945 seems to have been a watershed year for problem solving, as mathematician George Polya's book, How to Solve It , also appeared then (a second edition was published 12 years later; Polya, 1957 ).

The Gestalt psychologists extended the organizational principles of visual perception to the domain of problem solving. They showed that various visual aspects of the problem, as well the solver's prior knowledge, affect how people understand problems and, therefore, generate problem solutions. The principles of visual perception (e.g., proximity, closure, grouping, good continuation) are directly relevant to problem solving when the physical layout of the problem, or a diagram that accompanies the problem description, elicits inferences that solvers include in their problem representations. Such effects are nicely illustrated by Maier's ( 1930 ) nine-dot problem: Nine dots are arrayed in a 3x3 grid, and the task is to connect all the dots by drawing four straight lines without lifting one's pencil from the paper. People have difficulty solving this problem because their initial representations generally include a constraint, inferred from the configuration of the dots, that the lines should not go outside the boundary of the imaginary square formed by the outer dots. With this constraint, the problem cannot be solved (but see Adams, 1979 ). Without this constraint, the problem may be solved as shown in Figure 21.4 (though the problem is still difficult for many people; see Weisberg & Alba, 1981 ).

The nine-dot problem is a classic insight problem (see van Steenburgh et al., Chapter 24 ). According to the Gestalt view (e.g., Duncker, 1945 ; Kohler, 1925 ; Maier, 1931 ; see Ohlsson, 1984 , for a review), the solution to an insight problem appears suddenly, accompanied by an “aha!” sensation, immediately following the sudden “restructuring” of one's understanding of the problem (i.e., a change in the problem representation): “The decisive points in thought-processes, the moments of sudden comprehension, of the ‘Aha!,’ of the new, are always at the same time moments in which such a sudden restructuring of the thought-material takes place” (Duncker, 1945 , p. 29). For the nine-dot problem, one view of the required restructuring is that the solver relaxes the constraint implied by the perceptual form of the problem and realizes that the lines may, in fact, extend past the boundary of the imaginary square. Later in the chapter, we present more recent accounts of insight.

The entities that appear in a problem also tend to evoke various inferences that people incorporate into their problem representations. A classic demonstration of this is the phenomenon of functional fixedness , introduced by Duncker ( 1945 ): If an object is habitually used for a certain purpose (e.g., a box serves as a container), it is difficult to see

A solution to the nine-dot problem.

that object as having properties that would enable it to be used for a dissimilar purpose. Duncker's basic experimental paradigm involved two conditions that varied in terms of whether the object that was crucial for solution was initially used for a function other than that required for solution.

Consider the candles problem—the best known of the five “practical problems” Duncker ( 1945 ) investigated. Three candles are to be mounted at eye height on a door. On the table, for use in completing this task, are some tacks and three boxes. The solution is to tack the three boxes to the door to serve as platforms for the candles. In the control condition, the three boxes were presented to subjects empty. In the functional-fixedness condition, they were filled with candles, tacks, and matches. Thus, in the latter condition, the boxes initially served the function of container, whereas the solution requires that they serve the function of platform. The results showed that 100% of the subjects who received empty boxes solved the candles problem, compared with only 43% of subjects who received filled boxes. Every one of the five problems in this study showed a difference favoring the control condition over the functional-fixedness condition, with average solution rates across the five problems of 97% and 58%, respectively.

The function of the objects in a problem can be also “fixed” by their most recent use. For example, Birch and Rabinowitz ( 1951 ) had subjects perform two consecutive tasks. In the first task, people had to use either a switch or a relay to form an electric circuit. After completing this task, both groups of subjects were asked to solve Maier's ( 1931 ) two-ropes problem. The solution to this problem requires tying an object to one of the ropes and making the rope swing as a pendulum. Subjects could create the pendulum using either the object from the electric-circuit task or the other object. Birch and Rabinowitz found that subjects avoided using the same object for two unrelated functions. That is, those who used the switch in the first task made the pendulum using the relay, and vice versa. The explanations subjects subsequently gave for their object choices revealed that they were unaware of the functional-fixedness constraint they imposed on themselves.

In addition to investigating people's solutions to such practical problems as irradiating a tumor, mounting candles on the wall, or tying ropes, the Gestalt psychologists examined how people understand and solve mathematical problems that require domain-specific knowledge. For example, Wertheimer ( 1959 ) observed individual differences in students' learning and subsequent application of the formula for finding the area of a parallelogram (see Fig. 21.5a ). Some students understood the logic underlying the learned formula (i.e., the fact that a parallelogram can be transformed into a rectangle by cutting off a triangle from one side and pasting it onto the other side) and exhibited “productive thinking”—using the same logic to find the area of the quadrilateral in Figure 21.5b and the irregularly shaped geometric figure in Figure 21.5c . Other students memorized the formula and exhibited “reproductive thinking”—reproducing the learned solution only to novel parallelograms that were highly similar to the original one.

The psychological study of human problem solving faded into the background after the demise of the Gestalt tradition (during World War II), and problem solving was investigated only sporadically until Allen Newell and Herbert Simon's ( 1972 ) landmark book Human Problem Solving sparked a flurry of research on this topic. Newell and Simon adopted and refined Duncker's ( 1945 ) methodology of collecting and analyzing the think-aloud protocols that accompany problem solutions and extended Duncker's conceptualization of a problem solution as a search tree. However, their initial work did not aim to extend the Gestalt findings

Finding the area of ( a ) a parallelogram, ( b ) a quadrilateral, and ( c ) an irregularly shaped geometric figure. The solid lines indicate the geometric figures whose areas are desired. The dashed lines show how to convert the given figures into rectangles (i.e., they show solutions with understanding).

pertaining to problem representation. Instead, as we explain in the next section, their objective was to identify the general-purpose strategies people use in searching for a problem solution.

Search in a Problem Space: The Legacy of Newell and Simon

Newell and Simon ( 1972 ) wrote a magnum opus detailing their theory of problem solving and the supporting research they conducted with various collaborators. This theory was grounded in the information-processing approach to cognitive psychology and guided by an analogy between human and artificial intelligence (i.e., both people and computers being “Physical Symbol Systems,” Newell & Simon, 1976 ; see Doumas & Hummel, Chapter 5 ). They conceptualized problem solving as a process of search through a problem space for a path that connects the initial state to the goal state—a metaphor that alludes to the visual or spatial nature of problem solving (Simon, 1990 ). The term problem space refers to the solver's representation of the task as presented (Simon, 1978 ). It consists of ( 1 ) a set of knowledge states (the initial state, the goal state, and all possible intermediate states), ( 2 ) a set of operators that allow movement from one knowledge state to another, ( 3 ) a set of constraints, and ( 4 ) local information about the path one is taking through the space (e.g., the current knowledge state and how one got there).

We illustrate the components of a problem space for the three-disk Tower of Hanoi problem, as depicted in Figure 21.3 . The initial state appears at the top (State #1) and the goal state at the bottom right (State #27). The remaining knowledge states in the figure are possible intermediate states. The current knowledge state is the one at which the solver is located at any given point in the solution process. For example, the current state for a solver who has made three moves along the optimum solution path would be State #9. The solver presumably would know that he or she arrived at this state from State #5. This knowledge allows the solver to recognize a move that involves backtracking. The three operators in this problem are moving each of the three disks from one peg to another. These operators are subject to the constraint that a larger disk may not be placed on a smaller disk.

Newell and Simon ( 1972 ), as well as other contemporaneous researchers (e.g., Atwood & Polson, 1976 ; Greeno, 1974 ; Thomas, 1974 ), examined how people traverse the spaces of various well-defined problems (e.g., the Tower of Hanoi, Hobbits and Orcs). They discovered that solvers' search is guided by a number of shortcut strategies, or heuristics , which are likely to get the solver to the goal state without an extensive amount of search. Heuristics are often contrasted with algorithms —methods that are guaranteed to yield the correct solution. For example, one could try every possible move in the three-disk Tower of Hanoi problem and, eventually, find the correct solution. Although such an exhaustive search is a valid algorithm for this problem, for many problems its application is very time consuming and impractical (e.g., consider the game of chess).

In their attempts to identify people's search heuristics, Newell and Simon ( 1972 ) relied on two primary methodologies: think-aloud protocols and computer simulations. Their use of think-aloud protocols brought a high degree of scientific rigor to the methodology used by Duncker ( 1945 ; see Ericsson & Simon, 1980 ). Solvers were required to say out loud everything they were thinking as they solved the problem, that is, everything that went through their verbal working memory. Subjects' verbalizations—their think-aloud protocols—were tape-recorded and then transcribed verbatim for analysis. This method is extremely time consuming (e.g., a transcript of one person's solution to the cryptarithmetic problem DONALD + GERALD = ROBERT, with D = 5, generated a 17-page transcript), but it provides a detailed record of the solver's ongoing solution process.

An important caveat to keep in mind while interpreting a subject's verbalizations is that “a protocol is relatively reliable only for what it positively contains, but not for that which it omits” (Duncker, 1945 , p. 11). Ericsson and Simon ( 1980 ) provided an in-depth discussion of the conditions under which this method is valid (but see Russo, Johnson, & Stephens, 1989 , for an alternative perspective). To test their interpretation of a subject's problem solution, inferred from the subject's verbal protocol, Newell and Simon ( 1972 ) created a computer simulation program and examined whether it solved the problem the same way the subject did. To the extent that the computer simulation provided a close approximation of the solver's step-by-step solution process, it lent credence to the researcher's interpretation of the verbal protocol.

Newell and Simon's ( 1972 ) most famous simulation was the General Problem Solver or GPS (Ernst & Newell, 1969 ). GPS successfully modeled human solutions to problems as different as the Tower of Hanoi and the construction of logic proofs using a single general-purpose heuristic: means-ends analysis . This heuristic captures people's tendency to devise a solution plan by setting subgoals that could help them achieve their final goal. It consists of the following steps: ( 1 ) Identify a difference between the current state and the goal (or subgoal ) state; ( 2 ) Find an operator that will remove (or reduce) the difference; (3a) If the operator can be directly applied, do so, or (3b) If the operator cannot be directly applied, set a subgoal to remove the obstacle that is preventing execution of the desired operator; ( 4 ) Repeat steps 1–3 until the problem is solved. Next, we illustrate the implementation of this heuristic for the Tower of Hanoi problem, using the problem space in Figure 21.3 .

As can be seen in Figure 21.3 , a key difference between the initial state and the goal state is that the large disk is on the wrong peg (step 1). To remove this difference (step 2), one needs to apply the operator “move-large-disk.” However, this operator cannot be applied because of the presence of the medium and small disks on top of the large disk. Therefore, the solver may set a subgoal to move that two-disk tower to the middle peg (step 3b), leaving the rightmost peg free for the large disk. A key difference between the initial state and this new subgoal state is that the medium disk is on the wrong peg. Because application of the move-medium-disk operator is blocked, the solver sets another subgoal to move the small disk to the right peg. This subgoal can be satisfied immediately by applying the move-small-disk operator (step 3a), generating State #3. The solver then returns to the previous subgoal—moving the tower consisting of the small and medium disks to the middle peg. The differences between the current state (#3) and the subgoal state (#9) can be removed by first applying the move-medium-disk operator (yielding State #5) and then the move-small-disk operator (yielding State #9). Finally, the move-large-disk operator is no longer blocked. Hence, the solver moves the large disk to the right peg, yielding State #11.

Notice that the subgoals are stacked up in the order in which they are generated, so that they pop up in the order of last in first out. Given the first subgoal in our example, repeated application of the means-ends analysis heuristic will yield the shortest-path solution, indicated by the large gray arrows. In general, subgoals provide direction to the search and allow solvers to plan several moves ahead. By assessing progress toward a required subgoal rather than the final goal, solvers may be able to make moves that otherwise seem unwise. To take a concrete example, consider the transition from State #1 to State #3 in Figure 21.3 . Comparing the initial state to the goal state, this move seems unwise because it places the small disk on the bottom of the right peg, whereas it ultimately needs to be at the top of the tower on that peg. But comparing the initial state to the solver-generated subgoal state of having the medium disk on the middle peg, this is exactly where the small disk needs to go.

Means-ends analysis and various other heuristics (e.g., the hill-climbing heuristic that exploits the similarity, or distance, between the state generated by the next operator and the goal state; working backward from the goal state to the initial state) are flexible strategies that people often use to successfully solve a large variety of problems. However, the generality of these heuristics comes at a cost: They are relatively weak and fallible (e.g., in the means-ends solution to the problem of fixing a hole in a bucket, “Dear Liza” leads “Dear Henry” in a loop that ends back at the initial state; the lyrics of this famous song can be readily found on the Web). Hence, although people use general-purpose heuristics when they encounter novel problems, they replace them as soon as they acquire experience with and sufficient knowledge about the particular problem space (e.g., Anzai & Simon, 1979 ).

Despite the fruitfulness of this research agenda, it soon became evident that a fundamental weakness was that it minimized the importance of people's background knowledge. Of course, Newell and Simon ( 1972 ) were aware that problem solutions require relevant knowledge (e.g., the rules of logical proofs, or rules for stacking disks). Hence, in programming GPS, they supplemented every problem they modeled with the necessary background knowledge. This practice highlighted the generality and flexibility of means-ends analysis but failed to capture how people's background knowledge affects their solutions. As we discussed in the previous section, domain knowledge is likely to affect how people represent problems and, therefore, how they generate problem solutions. Moreover, as people gain experience solving problems in a particular knowledge domain (e.g., math, physics), they change their representations of these problems (e.g., Chi, Feltovich, & Glaser, 1981 ; Haverty, Koedinger, Klahr, & Alibali, 2000 ; Schoenfeld & Herrmann, 1982 ) and learn domain-specific heuristics (e.g., Polya, 1957 ; Schoenfeld, 1979 ) that trump the general-purpose strategies.

It is perhaps inevitable that the two traditions in problem-solving research—one emphasizing representation and the other emphasizing search strategies—would eventually come together. In the next section we review developments that led to this integration.

The Two Legacies Converge

Because Newell and Simon ( 1972 ) aimed to discover the strategies people use in searching for a solution, they investigated problems that minimized the impact of factors that tend to evoke differences in problem representations, of the sort documented by the Gestalt psychologists. In subsequent work, however, Simon and his collaborators showed that such factors are highly relevant to people's solutions of well-defined problems, and Simon ( 1986 ) incorporated these findings into the theoretical framework that views problem solving as search in a problem space.

In this section, we first describe illustrative examples of this work. We then describe research on insight solutions that incorporates ideas from the two legacies described in the previous sections.

Relevance of the Gestalt Ideas to the Solution of Search Problems

In this subsection we describe two lines of research by Simon and his colleagues, and by other researchers, that document the importance of perception and of background knowledge to the way people search for a problem solution. The first line of research used variants of relatively well-defined riddle problems that had the same structure (i.e., “problem isomorphs”) and, therefore, supposedly the same problem space. It documented that people's search depended on various perceptual and conceptual inferences they tended to draw from a specific instantiation of the problem's structure. The second line of research documented that people's search strategies crucially depend on their domain knowledge and on their prior experience with related problems.

Problem Isomorphs

Hayes and Simon ( 1977 ) used two variants of the Tower of Hanoi problem that, instead of disks and pegs, involved monsters and globes that differed in size (small, medium, and large). In both variants, the initial state had the small monster holding the large globe, the medium-sized monster holding the small globe, and the large monster holding the medium-sized globe. Moreover, in both variants the goal was for each monster to hold a globe proportionate to its own size. The only difference between the problems concerned the description of the operators. In one variant (“transfer”), subjects were told that the monsters could transfer the globes from one to another as long as they followed a set of rules, adapted from the rules in the original Tower of Hanoi problem (e.g., only one globe may be transferred at a time). In the other variant (“change”), subjects were told that the monsters could shrink and expand themselves according to a set of rules, which corresponded to the rules in the transfer version of the problem (e.g., only one monster may change its size at a time). Despite the isomorphism of the two variants, subjects conducted their search in two qualitatively different problem spaces, which led to solution times for the change variant being almost twice as long as those for the transfer variant. This difference arose because subjects could more readily envision and track an object that was changing its location with every move than one that was changing its size.

Recent work by Patsenko and Altmann ( 2010 ) found that, even in the standard Tower of Hanoi problem, people's solutions involve object-bound routines that depend on perception and selective attention. The subjects in their study solved various Tower of Hanoi problems on a computer. During the solution of a particular “critical” problem, the computer screen changed at various points without subjects' awareness (e.g., a disk was added, such that a subject who started with a five-disc tower ended with a six-disc tower). Patsenko and Altmann found that subjects' moves were guided by the configurations of the objects on the screen rather than by solution plans they had stored in memory (e.g., the next subgoal).

The Gestalt psychologists highlighted the role of perceptual factors in the formation of problem representations (e.g., Maier's, 1930 , nine-dot problem) but were generally silent about the corresponding implications for how the problem was solved (although they did note effects on solution accuracy). An important contribution of the work on people's solutions of the Tower of Hanoi problem and its variants was to show the relevance of perceptual factors to the application of various operators during search for a problem solution—that is, to the how of problem solving. In the next section, we describe recent work that documents the involvement of perceptual factors in how people understand and use equations and diagrams in the context of solving math and science problems.

Kotovsky, Hayes, and Simon ( 1985 ) further investigated factors that affect people's representation and search in isomorphs of the Tower of Hanoi problem. In one of their isomorphs, three disks were stacked on top of each other to form an inverted pyramid, with the smallest disc on the bottom and the largest on top. Subjects' solutions of the inverted pyramid version were similar to their solutions of the standard version that has the largest disc on the bottom and the smallest on top. However, the two versions were solved very differently when subjects were told that the discs represent acrobats. Subjects readily solved the version in which they had to place a small acrobat on the shoulders of a large one, but they refrained from letting a large acrobat stand on the shoulders of a small one. In other words, object-based inferences that draw on people's semantic knowledge affected the solution of search problems, much as they affect the solution of the ill-defined problems investigated by the Gestalt psychologists (e.g., Duncker's, 1945 , candles problem). In the next section, we describe more recent work that shows similar effects in people's solutions to mathematical word problems.

The work on differences in the representation and solution of problem isomorphs is highly relevant to research on analogical problem solving (or analogical transfer), which examines when and how people realize that two problems that differ in their cover stories have a similar structure (or a similar problem space) and, therefore, can be solved in a similar way. This research shows that minor differences between example problems, such as the use of X-rays versus ultrasound waves to fuse a broken filament of a light bulb, can elicit different problem representations that significantly affect the likelihood of subsequent transfer to novel problem analogs (Holyoak & Koh, 1987 ). Analogical transfer has played a central role in research on human problem solving, in part because it can shed light on people's understanding of a given problem and its solution and in part because it is believed to provide a window onto understanding and investigating creativity (see Smith & Ward, Chapter 23 ). We briefly mention some findings from the analogy literature in the next subsection on expertise, but we do not discuss analogical transfer in detail because this topic is covered elsewhere in this volume (Holyoak, Chapter 13 ).

Expertise and Its Development

In another line of research, Simon and his colleagues examined how people solve ecologically valid problems from various rule-governed and knowledge-rich domains. They found that people's level of expertise in such domains, be it in chess (Chase & Simon, 1973 ; Gobet & Simon, 1996 ), mathematics (Hinsley, Hayes, & Simon, 1977 ; Paige & Simon, 1966 ), or physics (Larkin, McDermott, Simon, & Simon, 1980 ; Simon & Simon, 1978 ), plays a crucial role in how they represent problems and search for solutions. This work, and the work of numerous other researchers, led to the discovery (and rediscovery, see Duncker, 1945 ) of important differences between experts and novices, and between “good” and “poor” students.

One difference between experts and novices pertains to pattern recognition. Experts' attention is quickly captured by familiar configurations within a problem situation (e.g., a familiar configuration of pieces in a chess game). In contrast, novices' attention is focused on isolated components of the problem (e.g., individual chess pieces). This difference, which has been found in numerous domains, indicates that experts have stored in memory many meaningful groups (chunks) of information: for example, chess (Chase & Simon, 1973 ), circuit diagrams (Egan & Schwartz, 1979 ), computer programs (McKeithen, Reitman, Rueter, & Hirtle, 1981 ), medicine (Coughlin & Patel, 1987 ; Myles-Worsley, Johnston, & Simons, 1988 ), basketball and field hockey (Allard & Starkes, 1991 ), and figure skating (Deakin & Allard, 1991 ).

The perceptual configurations that domain experts readily recognize are associated with stored solution plans and/or compiled procedures (Anderson, 1982 ). As a result, experts' solutions are much faster than, and often qualitatively different from, the piecemeal solutions that novice solvers tend to construct (e.g., Larkin et al., 1980 ). In effect, experts often see the solutions that novices have yet to compute (e.g., Chase & Simon, 1973 ; Novick & Sherman, 2003 , 2008 ). These findings have led to the design of various successful instructional interventions (e.g., Catrambone, 1998 ; Kellman et al., 2008 ). For example, Catrambone ( 1998 ) perceptually isolated the subgoals of a statistics problem. This perceptual chunking of meaningful components of the problem prompted novice students to self-explain the meaning of the chunks, leading to a conceptual understanding of the learned solution. In the next section, we describe some recent work that shows the beneficial effects of perceptual pattern recognition on the solution of familiar mathematics problems, as well as the potentially detrimental effects of familiar perceptual chunks to understanding and reasoning with diagrams depicting evolutionary relationships among taxa.

Another difference between experts and novices pertains to their understanding of the solution-relevant problem structure. Experts' knowledge is highly organized around domain principles, and their problem representations tend to reflect this principled understanding. In particular, they can extract the solution-relevant structure of the problems they encounter (e.g., meaningful causal relations among the objects in the problem; see Cheng & Buehner, Chapter 12 ). In contrast, novices' representations tend to be bound to surface features of the problems that may be irrelevant to solution (e.g., the particular objects in a problem). For example, Chi, Feltovich, and Glaser ( 1981 ) examined how students with different levels of physics expertise group mechanics word problems. They found that advanced graduate students grouped the problems based on the physics principles relevant to the problems' solutions (e.g., conservation of energy, Newton's second law). In contrast, undergraduates who had successfully completed an introductory course in mechanics grouped the problems based on the specific objects involved (e.g., pulley problems, inclined plane problems). Other researchers have found similar results in the domains of biology, chemistry, computer programming, and math (Adelson, 1981 ; Kindfield, 1993 / 1994 ; Kozma & Russell, 1997 ; McKeithen et al., 1981 ; Silver, 1979 , 1981 ; Weiser & Shertz, 1983 ).

The level of domain expertise and the corresponding representational differences are, of course, a matter of degree. With increasing expertise, there is a gradual change in people's focus of attention from aspects that are not relevant to solution to those that are (e.g., Deakin & Allard, 1991 ; Hardiman, Dufresne, & Mestre, 1989 ; McKeithen et al., 1981 ; Myles-Worsley et al., 1988 ; Schoenfeld & Herrmann, 1982 ; Silver, 1981 ). Interestingly, Chi, Bassok, Lewis, Reimann, and Glaser ( 1989 ) found similar differences in focus on structural versus surface features among a group of novices who studied worked-out examples of mechanics problems. These differences, which echo Wertheimer's ( 1959 ) observations of individual differences in students' learning about the area of parallelograms, suggest that individual differences in people's interests and natural abilities may affect whether, or how quickly, they acquire domain expertise.

An important benefit of experts' ability to focus their attention on solution-relevant aspects of problems is that they are more likely than novices to recognize analogous problems that involve different objects and cover stories (e.g., Chi et al., 1989 ; Novick, 1988 ; Novick & Holyoak, 1991 ; Wertheimer, 1959 ) or that come from other knowledge domains (e.g., Bassok & Holyoak, 1989 ; Dunbar, 2001 ; Goldstone & Sakamoto, 2003 ). For example, Bassok and Holyoak ( 1989 ) found that, after learning to solve arithmetic-progression problems in algebra, subjects spontaneously applied these algebraic solutions to analogous physics problems that dealt with constantly accelerated motion. Note, however, that experts and good students do not simply ignore the surface features of problems. Rather, as was the case in the problem isomorphs we described earlier (Kotovsky et al., 1985 ), they tend to use such features to infer what the problem's structure could be (e.g., Alibali, Bassok, Solomon, Syc, & Goldin-Meadow, 1999 ; Blessing & Ross, 1996 ). For example, Hinsley et al. ( 1977 ) found that, after reading no more than the first few words of an algebra word problem, expert solvers classified the problem into a likely problem category (e.g., a work problem, a distance problem) and could predict what questions they might be asked and the equations they likely would need to use.

Surface-based problem categorization has a heuristic value (Medin & Ross, 1989 ): It does not ensure a correct categorization (Blessing & Ross, 1996 ), but it does allow solvers to retrieve potentially appropriate solutions from memory and to use them, possibly with some adaptation, to solve a variety of novel problems. Indeed, although experts exploit surface-structure correlations to save cognitive effort, they have the capability to realize that a particular surface cue is misleading (Hegarty, Mayer, & Green, 1992 ; Lewis & Mayer, 1987 ; Martin & Bassok, 2005 ; Novick 1988 , 1995 ; Novick & Holyoak, 1991 ). It is not surprising, therefore, that experts may revert to novice-like heuristic methods when solving problems under pressure (e.g., Beilock, 2008 ) or in subdomains in which they have general but not specific expertise (e.g., Patel, Groen, & Arocha, 1990 ).

Relevance of Search to Insight Solutions

We introduced the notion of insight in our discussion of the nine-dot problem in the section on the Gestalt tradition. The Gestalt view (e.g., Duncker, 1945 ; Maier, 1931 ; see Ohlsson, 1984 , for a review) was that insight problem solving is characterized by an initial work period during which no progress toward solution is made (i.e., an impasse), a sudden restructuring of one's problem representation to a more suitable form, followed immediately by the sudden appearance of the solution. Thus, solving problems by insight was believed to be all about representation, with essentially no role for a step-by-step solution process (i.e., search). Subsequent and contemporary researchers have generally concurred with the Gestalt view that getting the right representation is crucial. However, research has shown that insight solutions do not necessarily arise suddenly or full blown after restructuring (e.g., Weisberg & Alba, 1981 ); and even when they do, the underlying solution process (in this case outside of awareness) may reflect incremental progress toward the goal (Bowden & Jung-Beeman, 2003 ; Durso, Rea, & Dayton, 1994 ; Novick & Sherman, 2003 ).

“Demystifying insight,” to borrow a phrase from Bowden, Jung-Beeman, Fleck, and Kounios ( 2005 ), requires explaining ( 1 ) why solvers initially reach an impasse in solving a problem for which they have the necessary knowledge to generate the solution, ( 2 ) how the restructuring occurred, and ( 3 ) how it led to the solution. A detailed discussion of these topics appears elsewhere in this volume (van Steenburgh et al., Chapter 24 ). Here, we describe briefly three recent theories that have attempted to account for various aspects of these phenomena: Knoblich, Ohlsson, Haider, and Rhenius's ( 1999 ) representational change theory, MacGregor, Ormerod, and Chronicle's ( 2001 ) progress monitoring theory, and Bowden et al.'s ( 2005 ) neurological model. We then propose the need for an integrated approach to demystifying insight that considers both representation and search.

According to Knoblich et al.'s ( 1999 ) representational change theory, problems that are solved with insight are highly likely to evoke initial representations in which solvers place inappropriate constraints on their solution attempts, leading to an impasse. An impasse can be resolved by revising one's representation of the problem. Knoblich and his colleagues tested this theory using Roman numeral matchstick arithmetic problems in which solvers must move one stick to a new location to change a false numerical statement (e.g., I = II + II ) into a statement that is true. According to representational change theory, re-representation may occur through either constraint relaxation or chunk decomposition. (The solution to the example problem is to change II + to III – , which requires both methods of re-representation, yielding I = III – II ). Good support for this theory has been found based on measures of solution rate, solution time, and eye fixation (Knoblich et al., 1999 ; Knoblich, Ohlsson, & Raney, 2001 ; Öllinger, Jones, & Knoblich, 2008 ).

Progress monitoring theory (MacGregor et al., 2001 ) was proposed to account for subjects' difficulty in solving the nine-dot problem, which has traditionally been classified as an insight problem. According to this theory, solvers use the hill-climbing search heuristic to solve this problem, just as they do for traditional search problems (e.g., Hobbits and Orcs). In particular, solvers are hypothesized to monitor their progress toward solution using a criterion generated from the problem's current state. If solvers reach criterion failure, they seek alternative solutions by trying to relax one or more problem constraints. MacGregor et al. found support for this theory using several variants of the nine-dot problem (also see Ormerod, MacGregor, & Chronicle, 2002 ). Jones ( 2003 ) suggested that progress monitoring theory provides an account of the solution process up to the point an impasse is reached and representational change is sought, at which point representational change theory picks up and explains how insight may be achieved. Hence, it appears that a complete account of insight may require an integration of concepts from the Gestalt (representation) and Newell and Simon's (search) legacies.

Bowden et al.'s ( 2005 ) neurological model emphasizes the overlap between problem solving and language comprehension, and it hinges on differential processing in the right and left hemispheres. They proposed that an impasse is reached because initial processing of the problem produces strong activation of information irrelevant to solution in the left hemisphere. At the same time, weak semantic activation of alternative semantic interpretations, critical for solution, occurs in the right hemisphere. Insight arises when the weakly activated concepts reinforce each other, eventually rising above the threshold required for conscious awareness. Several studies of problem solving using compound remote associates problems, involving both behavioral and neuroimaging data, have found support for this model (Bowden & Jung-Beeman, 1998 , 2003 ; Jung-Beeman & Bowden, 2000 ; Jung-Beeman et al., 2004 ; also see Moss, Kotovsky, & Cagan, 2011 ).

Note that these three views of insight have received support using three quite distinct types of problems (Roman numeral matchstick arithmetic problems, the nine-dot problem, and compound remote associates problems, respectively). It remains to be established, therefore, whether these accounts can be generalized across problems. Kershaw and Ohlsson ( 2004 ) argued that insight problems are difficult because the key behavior required for solution may be hindered by perceptual factors (the Gestalt view), background knowledge (so expertise may be important; e.g., see Novick & Sherman, 2003 , 2008 ), and/or process factors (e.g., those affecting search). From this perspective, solving visual problems (e.g., the nine-dot problem) with insight may call upon more general visual processes, whereas solving verbal problems (e.g., anagrams, compound remote associates) with insight may call upon general verbal/semantic processes.

The work we reviewed in this section shows the relevance of problem representation (the Gestalt legacy) to the way people search the problem space (the legacy of Newell and Simon), and the relevance of search to the solution of insight problems that require a representational change. In addition to this inevitable integration of the two legacies, the work we described here underscores the fact that problem solving crucially depends on perceptual factors and on the solvers' background knowledge. In the next section, we describe some recent work that shows the involvement of these factors in the solution of problems in math and science.

Effects of Perception and Knowledge in Problem Solving in Academic Disciplines

Although the use of puzzle problems continues in research on problem solving, especially in investigations of insight, many contemporary researchers tackle problem solving in knowledge-rich domains, often in academic disciplines (e.g., mathematics, biology, physics, chemistry, meteorology). In this section, we provide a sampling of this research that highlights the importance of visual perception and background knowledge for successful problem solving.

The Role of Visual Perception

We stated at the outset that a problem representation (e.g., the problem space) is a model of the problem constructed by solvers to summarize their understanding of the problem's essential nature. This informal definition refers to the internal representations people construct and hold in working memory. Of course, people may also construct various external representations (Markman, 1999 ) and even manipulate those representations to aid in solution (see Hegarty & Stull, Chapter 31 ). For example, solvers often use paper and pencil to write notes or draw diagrams, especially when solving problems from formal domains (e.g., Cox, 1999 ; Kindfield, 1993 / 1994 ; S. Schwartz, 1971 ). In problems that provide solvers with external representation, such as the Tower of Hanoi problem, people's planning and memory of the current state is guided by the actual configurations of disks on pegs (Garber & Goldin-Meadow, 2002 ) or by the displays they see on a computer screen (Chen & Holyoak, 2010 ; Patsenko & Altmann, 2010 ).

In STEM (science, technology, engineering, and mathematics) disciplines, it is common for problems to be accompanied by diagrams or other external representations (e.g., equations) to be used in determining the solution. Larkin and Simon ( 1987 ) examined whether isomorphic sentential and diagrammatic representations are interchangeable in terms of facilitating solution. They argued that although the two formats may be equivalent in the sense that all of the information in each format can be inferred from the other format (informational equivalence), the ease or speed of making inferences from the two formats might differ (lack of computational equivalence). Based on their analysis of several problems in physics and math, Larkin and Simon further argued for the general superiority of diagrammatic representations (but see Mayer & Gallini, 1990 , for constraints on this general conclusion).

Novick and Hurley ( 2001 , p. 221) succinctly summarized the reasons for the general superiority of diagrams (especially abstract or schematic diagrams) over verbal representations: They “(a) simplify complex situations by discarding unnecessary details (e.g., Lynch, 1990 ; Winn, 1989 ), (b) make abstract concepts more concrete by mapping them onto spatial layouts with familiar interpretational conventions (e.g., Winn, 1989 ), and (c) substitute easier perceptual inferences for more computationally intensive search processes and sentential deductive inferences (Barwise & Etchemendy, 1991 ; Larkin & Simon, 1987 ).” Despite these benefits of diagrammatic representations, there is an important caveat, noted by Larkin and Simon ( 1987 , p. 99) at the very end of their paper: “Although every diagram supports some easy perceptual inferences, nothing ensures that these inferences must be useful in the problem-solving process.” We will see evidence of this in several of the studies reviewed in this section.

Next we describe recent work on perceptual factors that are involved in people's use of two types of external representations that are provided as part of the problem in two STEM disciplines: equations in algebra and diagrams in evolutionary biology. Although we focus here on effects of perceptual factors per se, it is important to note that such factors only influence performance when subjects have background knowledge that supports differential interpretation of the alternative diagrammatic depictions presented (Hegarty, Canham, & Fabricant, 2010 ).

In the previous section, we described the work of Patsenko and Altmann ( 2010 ) that shows direct involvement of visual attention and perception in the sequential application of move operators during the solution of the Tower of Hanoi problem. A related body of work documents similar effects in tasks that require the interpretation and use of mathematical equations (Goldstone, Landy, & Son, 2010 ; Landy & Goldstone, 2007a , b). For example, Landy and Goldstone ( 2007b ) varied the spatial proximity of arguments to the addition (+) and multiplication (*) operators in algebraic equations, such that the spatial layout of the equation was either consistent or inconsistent with the order-of-operations rule that multiplication precedes addition. In consistent equations , the space was narrower around multiplication than around addition (e.g., g*m + r*w = m*g + w*r ), whereas in inconsistent equations this relative spacing was reversed (e.g., s * n+e * c = n * s+c * e ). Subjects' judgments of the validity of such equations (i.e., whether the expressions on the two sides of the equal sign are equivalent) were significantly faster and more accurate for consistent than inconsistent equations.

In discussing these findings and related work with other external representations, Goldstone et al. ( 2010 ) proposed that experience with solving domain-specific problems leads people to “rig up” their perceptual system such that it allows them to look at the problem in a way that is consistent with the correct rules. Similar logic guides the Perceptual Learning Modules developed by Kellman and his collaborators to help students interpret and use algebraic equations and graphs (Kellman et al., 2008 ; Kellman, Massey, & Son, 2009 ). These authors argued and showed that, consistent with the previously reviewed work on expertise, perceptual training with particular external representations supports the development of perceptual fluency. This fluency, in turn, supports students' subsequent use of these external representations for problem solving.

This research suggests that extensive experience with particular equations or graphs may lead to perceptual fluency that could replace the more mindful application of domain-specific rules. Fisher, Borchert, and Bassok ( 2011 ) reported results from algebraic-modeling tasks that are consistent with this hypothesis. For example, college students were asked to represent verbal statements with algebraic equations, a task that typically elicits systematic errors (e.g., Clement, Lochhead, & Monk, 1981 ). Fisher et al. found that such errors were very common when subjects were asked to construct “standard form” equations ( y = ax ), which support fluent left-to-right translation of words to equations, but were relatively rare when subjects were asked to construct nonstandard division-format equations (x = y/a) that do not afford such translation fluency.

In part because of the left-to-right order in which people process equations, which mirrors the linear order in which they process text, equations have traditionally been viewed as sentential representations. However, Landy and Goldstone ( 2007a ) have proposed that equations also share some properties with diagrammatic displays and that, in fact, in some ways they are processed like diagrams. That is, spatial information is used to represent and to support inferences about syntactic structure. This hypothesis received support from Landy and Goldstone's ( 2007b ) results, described earlier, in which subjects' judgments of the validity of equations were affected by the Gestalt principle of grouping: Subjects did better when the grouping was consistent rather than inconsistent with the underlying structure of the problem (order of operations). Moreover, Landy and Goldstone ( 2007a ) found that when subjects wrote their own equations they grouped numbers and operators (+, *, =) in a way that reflected the hierarchical structure imposed by the order-of-operations rule.

In a recent line of research, Novick and Catley ( 2007 ; Novick, Catley, & Funk, 2010 ; Novick, Shade, & Catley, 2011 ) have examined effects of the spatial layout of diagrams depicting the evolutionary history of a set of taxa on people's ability to reason about patterns of relationship among those taxa. We consider here their work that investigates the role of another Gestalt perceptual principle—good continuation—in guiding students' reasoning. According to this principle, a continuous line is perceived as a single entity (Kellman, 2000 ). Consider the diagrams shown in Figure 21.6 . Each is a cladogram, a diagram that depicts nested sets of taxa that are related in terms of levels of most recent common ancestry. For example, chimpanzees and starfish are more closely related to each other than either is to spiders. The supporting evidence for their close relationship is their most recent common ancestor, which evolved the novel character of having radial cleavage. Spiders do not share this ancestor and thus do not have this character.

Cladograms are typically drawn in two isomorphic formats, which Novick and Catley ( 2007 ) referred to as trees and ladders. Although these formats are informationally equivalent (Larkin & Simon, 1987 ), Novick and Catley's ( 2007 ) research shows that they are not computationally equivalent (Larkin & Simon, 1987 ). Imagine that you are given evolutionary relationships in the ladder format, such as in Figure 21.6a (but without the four characters—hydrostatic skeleton, bilateral symmetry, radial cleavage, and trocophore larvae—and associated short lines indicating their locations on the cladogram), and your task is to translate that diagram to the tree format. A correct translation is shown in Figure 21.6b . Novick and Catley ( 2007 ) found that college students were much more likely to get such problems correct when the presented cladogram was in the nested circles (e.g., Figure 21.6d ) rather than the ladder format. Because the Gestalt principle of good continuation makes the long slanted line at the base of the ladder appear to represent a single hierarchical level, a common translation error for the ladder to tree problems was to draw a diagram such as that shown in Figure 21.6c .

The difficulty that good continuation presents for interpreting relationships depicted in the ladder format extends to answering reasoning questions as well. Novick and Catley (unpublished data) asked comparable questions about relationships depicted in the ladder and tree formats. For example, using the cladograms depicted in Figures 21.6a and 21.6b , consider the following questions: (a) Which taxon—jellyfish or earthworm—is the closest evolutionary relation to starfish, and what evidence supports your answer? (b) Do the bracketed taxa comprise a clade (a set of taxa consisting of the most recent common ancestor and all of its descendants), and what evidence supports your answer? For both such questions, students had higher accuracy and evidence quality composite scores when the relationships were depicted in the tree than the ladder format.

Four cladograms depicting evolutionary relationships among six animal taxa. Cladogram ( a ) is in the ladder format, cladograms ( b ) and ( c ) are in the tree format, and cladogram ( d ) is in the nested circles format. Cladograms ( a ), ( b ), and ( d ) are isomorphic.

If the difficulty in extracting the hierarchical structure of the ladder format is due to good continuation (which leads problem solvers to interpret continuous lines that depict multiple hierarchical levels as depicting only a single level), then a manipulation that breaks good continuation at the points where a new hierarchical level occurs should improve understanding. Novick et al. ( 2010 ) tested this hypothesis using a translation task by manipulating whether characters that are the markers for the most recent common ancestor of each nested set of taxa were included on the ladders. Figure 21.6a shows a ladder with such characters. As predicted, translation accuracy increased dramatically simply by adding these characters to the ladders, despite the additional information subjects had to account for in their translations.

The Role of Background Knowledge

As we mentioned earlier, the specific entities in the problems people encounter evoke inferences that affect how people represent these problems (e.g., the candle problem; Duncker, 1945 ) and how they apply the operators in searching for the solution (e.g., the disks vs. acrobats versions of the Tower of Hanoi problem; Kotovsky et al., 1985 ). Such object-based inferences draw on people's knowledge about the properties of the objects (e.g., a box is a container, an acrobat is a person who can be hurt). Here, we describe the work of Bassok and her colleagues, who found that similar inferences affect how people select mathematical procedures to solve problems in various formal domains. This work shows that the objects in the texts of mathematical word problems affect how people represent the problem situation (i.e., the situation model they construct; Kintsch & Greeno, 1985 ) and, in turn, lead them to select mathematical models that have a corresponding structure. To illustrate, a word problem that describes constant change in the rate at which ice is melting off a glacier evokes a model of continuous change, whereas a word problem that describes constant change in the rate at which ice is delivered to a restaurant evokes a model of discrete change. These distinct situation models lead subjects to select corresponding visual representations (e.g., Bassok & Olseth, 1995 ) and solutions methods, such as calculating the average change over time versus adding the consecutive changes (e.g., Alibali et al., 1999 ).

In a similar manner, people draw on their general knowledge to infer how the objects in a given problem are related to each other and construct mathematical solutions that correspond to these inferred object relations. For example, a word problem that involves doctors from two hospitals elicits a situation model in which the two sets of doctors play symmetric roles (e.g., work with each other), whereas a mathematically isomorphic problem that involves mechanics and cars elicits a situation model in which the sets play asymmetric roles (e.g., mechanics fix cars). The mathematical solutions people construct to such problems reflect this difference in symmetry (Bassok, Wu, & Olseth, 1995 ). In general, people tend to add objects that belong to the same taxonomic category (e.g., doctors + doctors) but divide functionally related objects (e.g., cars ÷ mechanics). People establish this correspondence by a process of analogical alignment between semantic and arithmetic relations, which Bassok and her colleagues refer to as “semantic alignment” (Bassok, Chase, & Martin, 1998 ; Doumas, Bassok, Guthormsen, & Hummel, 2006 ; Fisher, Bassok, & Osterhout, 2010 ).

Semantic alignment occurs very early in the solution process and can prime arithmetic facts that are potentially relevant to the problem solution (Bassok, Pedigo, & Oskarsson, 2008 ). Although such alignments can lead to erroneous solutions, they have a high heuristic value because, in most textbook problems, object relations indeed correspond to analogous mathematical relations (Bassok et al., 1998 ). Interestingly, unlike in the case of reliance on specific surface-structure correlations (e.g., the keyword “more” typically appears in word problems that require addition; Lewis & Mayer, 1987 ), people are more likely to exploit semantic alignment when they have more, rather than less modeling experience. For example, Martin and Bassok ( 2005 ) found very strong semantic-alignment effects when subjects solved simple division word problems, but not when they constructed algebraic equations to represent the relational statements that appeared in the problems. Of course, these subjects had significantly more experience with solving numerical word problems than with constructing algebraic models of relational statements. In a subsequent study, Fisher and Bassok ( 2009 ) found semantic-alignment effects for subjects who constructed correct algebraic models, but not for those who committed modeling errors.

Conclusions and Future Directions

In this chapter, we examined two broad components of the problem-solving process: representation (the Gestalt legacy) and search (the legacy of Newell and Simon). Although many researchers choose to focus their investigation on one or the other of these components, both Duncker ( 1945 ) and Simon ( 1986 ) underscored the necessity to investigate their interaction, as the representation one constructs for a problem determines (or at least constrains) how one goes about trying to generate a solution, and searching the problem space may lead to a change in problem representation. Indeed, Duncker's ( 1945 ) initial account of one subject's solution to the radiation problem was followed up by extensive and experimentally sophisticated work by Simon and his colleagues and by other researchers, documenting the involvement of visual perception and background knowledge in how people represent problems and search for problem solutions.

The relevance of perception and background knowledge to problem solving illustrates the fact that, when people attempt to find or devise ways to reach their goals, they draw on a variety of cognitive resources and engage in a host of cognitive activities. According to Duncker ( 1945 ), such goal-directed activities may include (a) placing objects into categories and making inferences based on category membership, (b) making inductive inferences from multiple instances, (c) reasoning by analogy, (d) identifying the causes of events, (e) deducing logical implications of given information, (f) making legal judgments, and (g) diagnosing medical conditions from historical and laboratory data. As this list suggests, many of the chapters in the present volume describe research that is highly relevant to the understanding of problem-solving behavior. We believe that important advancements in problem-solving research would emerge by integrating it with research in other areas of thinking and reasoning, and that research in these other areas could be similarly advanced by incorporating the insights gained from research on what has more traditionally been identified as problem solving.

As we have described in this chapter, many of the important findings in the field have been established by a careful investigation of various riddle problems. Although there are good methodological reasons for using such problems, many researchers choose to investigate problem solving using ecologically valid educational materials. This choice, which is increasingly common in contemporary research, provides researchers with the opportunity to apply their basic understanding of problem solving to benefit the design of instruction and, at the same time, allows them to gain a better understanding of the processes by which domain knowledge and educational conventions affect the solution process. We believe that the trend of conducting educationally relevant research is likely to continue, and we expect a significant expansion of research on people's understanding and use of dynamic and technologically rich external representations (e.g., Kellman et al., 2008 ; Mayer, Griffith, Jurkowitz, & Rothman, 2008 ; Richland & McDonough, 2010 ; Son & Goldstone, 2009 ). Such investigations are likely to yield both practical and theoretical payoffs.

Adams, J. L. ( 1979 ). Conceptual blockbusting: A guide to better ideas (2nd ed.). New York: Norton.

Google Scholar

Google Preview

Adelson, B. ( 1981 ). Problem solving and the development of abstract categories in programming languages.   Memory and Cognition , 9 , 422–433.

Alibali, M. W., Bassok, M., Solomon, K. O., Syc, S. E., & Goldin-Meadow, S. ( 1999 ). Illuminating mental representations through speech and gesture.   Psychological Science , 10 , 327–333.

Allard, F., & Starkes, J. L. ( 1991 ). Motor-skill experts in sports, dance, and other domains. In K. A. Ericsson & J. Smith (Eds.), Toward a general theory of expertise: Prospects and limits (pp. 126–152). New York: Cambridge University Press.

Anderson, J. R. ( 1982 ). Acquisition of cognitive skill.   Psychological Review , 89 , 369–406.

Anzai, Y., & Simon, H. A. ( 1979 ). The theory of learning by doing.   Psychological Review , 86 , 124–140.

Atwood, M. E, & Polson, P.G. ( 1976 ). A process model for water jug problems.   Cognitive Psychology , 8 , 191–216.

Barwise, J., & Etchemendy, J. ( 1991 ). Visual information and valid reasoning. In W. Zimmermann & S. Cunningham (Eds.), Visualization in teaching and learning mathematics (pp. 9–24). Washington, DC: Mathematical Association of America.

Bassok, M., Chase, V. M., & Martin, S. A. ( 1998 ). Adding apples and oranges: Alignment of semantic and formal knowledge.   Cognitive Psychology , 35 , 99–134.

Bassok, M., & Holyoak, K. J. ( 1989 ). Interdomain transfer between isomorphic topics in algebra and physics.   Journal of Experimental Psychology: Learning, Memory, and Cognition , 15 , 153–166.

Bassok, M., & Olseth, K. L. ( 1995 ). Object-based representations: Transfer between cases of continuous and discrete models of change.   Journal of Experimental Psychology: Learning, Memory, and Cognition , 21 , 1522–1538.

Bassok, M., Pedigo, S. F., & Oskarsson, A. T. ( 2008 ). Priming addition facts with semantic relations.   Journal of Experimental Psychology: Learning, Memory, and Cognition , 34 , 343–352.

Bassok, M., Wu, L., & Olseth, L. K. ( 1995 ). Judging a book by its cover: Interpretative effects of content on problem solving transfer.   Memory and Cognition , 23 , 354–367.

Beilock, S. L. ( 2008 ). Math performance in stressful situations.   Current Directions in Psychological Science , 17 , 339–343.

Birch, H. G. & Rabinowitz, H. S. ( 1951 ). The negative effect of previous experience on productive thinking.   Journal of Experimental Psychology , 41 , 122–126.

Blessing, S. B., & Ross, B. H. ( 1996 ). Content effects in problem categorization and problem solving.   Journal of Experimental Psychology: Learning, Memory, and Cognition , 22 , 792–810.

Bowden, E. M., & Jung-Beeman, M. ( 1998 ). Getting the right idea: Semantic activation in the right hemisphere may help solve insight problems.   Psychological Science , 6 , 435–440.

Bowden, E. M., & Jung-Beeman, M. ( 2003 ). Aha! Insight experience correlates with solution activation in the right hemisphere.   Psychonomic Bulletin and Review , 10 , 730–737.

Bowden, E. M., Jung-Beeman, M., Fleck, J., & Kounios, J. ( 2005 ). New approaches to demystifying insight.   Trends in Cognitive Sciences , 9 , 322–328.

Catrambone, R. ( 1998 ). The subgoal-learning model: Creating better examples so that students can solve novel problems.   Journal of Experimental Psychology: General , 127 , 355–376.

Chase, W. G., & Simon, H. A. ( 1973 ). Perception in chess.   Cognitive Psychology , 4 , 55–81.

Chen, D., & Holyoak, K. J. ( 2010 ). Enhancing acquisition of intuition versus planning in problem solving. In S. Ohlsson & R. Catrambone (Eds.), Proceedings of the 32nd Annual Conference of the Cognitive Science Society (pp. 1875–1880). Austin, TX: Cognitive Science Society.

Chi, M. T. H., Bassok, M., Lewis, M. W., Reimann, P., & Glaser, R. ( 1989 ). Self-explanations: How students study and use examples in learning to solve problems.   Cognitive Science , 13 , 145–182.

Chi, M. T. H., Feltovich, P. J., & Glaser, R. ( 1981 ). Categorization and representation of physics problems by experts and novices.   Cognitive Science , 5 , 121–152.

Clement, J., Lochhead, J., & Monk, G. S. ( 1981 ). Translation difficulties in learning mathematics.   The American Mathematical Monthly , 88 , 286–290.

Coughlin, L. D., & Patel, V. L. ( 1987 ). Processing of critical information by physicians and medical students.   Journal of Medical Education , 62 , 818–828.

Cox, R. ( 1999 ). Representation construction, externalised cognition and individual differences.   Learning and Instruction , 9 , 343–363.

Deakin, J. M., & Allard, F. ( 1991 ). Skilled memory in expert figure skaters.   Memory and Cognition , 19 , 79–86.

Doumas, L. A. A., Bassok, M., Guthormsen, A., & Hummel, J. E. ( 2006 ). Theory of reflexive relational generalization. In R. Sun & N. Miyake (Eds.), Proceedings of the 28th Annual Conference of the Cognitive Science Society (pp. 1246–1250). Mahwah, NJ: Erlbaum.

Dunbar, K. ( 2001 ). The analogical paradox: Why analogy is so easy in naturalistic settings, yet so difficult in the psychological laboratory. In D. Gentner, K. J. Holyoak, & B. Kokinov (Eds.), Analogy: Perspectives from cognitive science (pp. 313–362). Cambridge, MA: MIT Press.

Duncker, K. ( 1945 ). On problem-solving (L. S. Lees, Trans.). Psychological Monographs , 58 (Whole No. 270). (Original work published 1935).

Durso, F. T., Rea, C. B., & Dayton, T. ( 1994 ). Graph-theoretic confirmation of restructuring during insight.   Psychological Science , 5 , 94–98.

Egan, D. E., & Schwartz, B. J. ( 1979 ). Chunking in the recall of symbolic drawings.   Memory and Cognition , 7 , 149–158.

Ericsson, K. A., & Simon, H. A. ( 1980 ). Verbal reports as data.   Psychological Review , 87 , 215–251.

Ernst, G. W., & Newell, A. ( 1969 ). GPS: A case study in generality and problem solving . New York: Academic Press.

Fisher, K. J., & Bassok, M. ( 2009 ). Analogical alignments in algebraic modeling. In B. Kokinov, D. Gentner, & K. J. Holyoak (Eds.), Proceedings of the 2nd International Analogy Conference (pp. 137–144). Sofia, Bulgaria: New Bulgarian University Press.

Fisher, K. J., Bassok, M., & Osterhout, L. ( 2010 ). When two plus two does not equal four: Event-related potential responses to semantically incongruous arithmetic word problems. In S. Ohlsson & R. Catrambone (Eds.), Proceedings of the 32nd Annual Conference of the Cognitive Science Society (pp. 1571–1576). Austin, TX: Cognitive Science Society.

Fisher, K. J., Borchert, K., & Bassok, M. ( 2011 ). Following the standard form: Effects of equation format on algebraic modeling.   Memory and Cognition , 39 , 502–515.

Garber, P., & Goldin-Meadow, S. ( 2002 ). Gesture offers insight into problem solving in adults and children.   Cognitive Science , 26 , 817–831.

Gobet, F., & Simon, H. ( 1996 ). Recall of rapidly presented random chess positions is a function of skill.   Psychonomic Bulletin and Review , 3 , 159–163.

Goldstone, R. L., Landy, D. H., & Son, J. Y. ( 2010 ). The education of perception.   Topics in Cognitive Science , 2 , 265–284.

Goldstone, R. L., & Sakamoto, J. Y. ( 2003 ). The transfer of abstract principles governing complex adaptive systems.   Cognitive Psychology , 46 , 414–466.

Greeno, J. G. ( 1974 ). Hobbits and orcs: Acquisition of a sequential concept.   Cognitive Psychology , 6 , 270–292.

Hardiman, P. T., Dufresne, R., & Mestre, J. P. ( 1989 ). The relation between problem categorization and problem solving among experts and novices.   Memory and Cognition , 17 , 627–638.

Haverty, L. A., Koedinger, K. R., Klahr, D., & Alibali, M. W. ( 2000 ). Solving induction problems in mathematics: Not-so-trivial Pursuit.   Cognitive Science , 24 , 249–298.

Hayes, J. R., & Simon, H. A. ( 1977 ). Psychological differences among problem isomorphs. In N. J. Castellan, D. B. Pisoni, & G. R. Potts (Eds.), Cognitive theory (Vol. 2, pp. 21–44). Hillsdale, NJ: Erlbaum.

Hegarty, M., Canham, M. S., & Fabricant, S. I. ( 2010 ). Thinking about the weather: How display salience and knowledge affect performance in a graphic inference task.   Journal of Experimental Psychology: Learning, Memory, and Cognition , 36 , 37–53.

Hegarty, M., Mayer, R. E., & Green, C. E. ( 1992 ). Comprehension of arithmetic word problems: Evidence from students' eye fixations.   Journal of Educational Psychology , 84 , 76–84.

Hinsley, D. A., Hayes, J. R., & Simon, H. A. ( 1977 ). From words to equations: Meaning and representation in algebra word problems. In D. Hinsley, M. Just., & P. Carpenter (Eds.), Cognitive processes in comprehension (pp. 89–106). Hillsdale, NJ: Erlbaum.

Holyoak, K. J., & Koh, K. ( 1987 ). Surface and structural similarity in analogical transfer.   Memory and Cognition , 15 , 332–340.

Jones, G. ( 2003 ). Testing two cognitive theories of insight.   Journal of Experimental Psychology: Learning, Memory, and Cognition , 29 , 1017–1027.

Jung-Beeman, M., & Bowden, E. M. ( 2000 ). The right hemisphere maintains solution-related activation for yet-to-be solved insight problems.   Memory and Cognition , 28 , 1231–1241.

Jung-Beeman, M., Bowden, E. M., Haberman, J., Frymiare, J. L., Arambel-Liu, S., Greenblatt, R., … Kounios, J. ( 2004 ). Neural activity when people solve verbal problems with insight.   PLOS Biology , 2 , 500–510.

Kellman, P. J. ( 2000 ). An update on Gestalt psychology. In B. Landau, J. Sabini, J. Jonides, & E. Newport (Eds.), Perception, cognition, and language: Essays in honor of Henry and Lila Gleitman (pp. 157–190). Cambridge, MA: MIT Press.

Kellman, P. J., Massey, C. M., & Son, J. Y ( 2009 ). Perceptual learning modules in mathematics: Enhancing students' pattern recognition, structure extraction, and fluency.   Topics in Cognitive Science , 1 , 1–21.

Kellman, P. J., Massey, C., Roth, Z., Burke, T., Zucker, J., Saw, A., … Wise, J. A. ( 2008 ). Perceptual learning and the technology of expertise.   Pragmatics and Cognition , 16 , 356–405.

Kershaw, T. C., & Ohlsson, S. ( 2004 ). Multiple causes of difficulty in insight: The case of the nine-dot problem.   Journal of Experimental Psychology: Learning, Memory, and Cognition , 30 , 3–13.

Kindfield, A. C. H. ( 1993 /1994). Biology diagrams: Tools to think with.   Journal of the Learning Sciences , 3 , 1–36.

Kintsch, W., & Greeno, J. G. ( 1985 ). Understanding and solving word arithmetic problems.   Psychological Review , 92 , 109–129.

Knoblich, G., Ohlsson, S., Haider, H., & Rhenius, D. ( 1999 ). Constraint relaxation and chunk decomposition in insight problem solving.   Journal of Experimental Psychology: Learning, Memory, and Cognition , 25 , 1534–1555.

Knoblich, G., Ohlsson, S., & Raney, G. E. ( 2001 ). An eye movement study of insight problem solving.   Memory and Cognition , 29 , 1000–1009.

Kohler, W. ( 1925 ). The mentality of apes . New York: Harcourt Brace.

Kotovsky, K., Hayes, J. R., & Simon, H. A. ( 1985 ). Why are some problems hard? Evidence from Tower of Hanoi.   Cognitive Psychology , 17 , 248–294.

Kozma, R. B., & Russell, J. ( 1997 ). Multimedia and understanding: Expert and novice responses to different representations of chemical phenomena.   Journal of Research in Science Teaching , 34 , 949–968.

Landy, D., & Goldstone, R. L. ( 2007 a). Formal notations are diagrams: Evidence from a production task.   Memory and Cognition , 35, 2033–2040.

Landy, D., & Goldstone, R. L. ( 2007 b). How abstract is symbolic thought?   Journal of Experimental Psychology: Learning, Memory, and Cognition , 33, 720–733.

Larkin, J. H., McDermott, J., Simon, D. P., & Simon, H. A. ( 1980 ). Models of competence in solving physics problems.   Cognitive Science , 4 , 317–345.

Larkin, J. H., & Simon, H. A. ( 1987 ). Why a diagram is (sometimes) worth ten thousand words.   Cognitive Science , 11 , 65–99.

Lewis, A. B., & Mayer, R. E. ( 1987 ). students' miscomprehension of relational statements in arithmetic word problems.   Journal of Educational Psychology , 79 , 363–371.

Lynch, M. ( 1990 ). The externalized retina: Selection and mathematization in the visual documentation of objects in the life sciences. In M. Lynch & S. Woolgar (Eds.), Representation in scientific practice (pp. 153–186). Cambridge, MA: MIT Press.

MacGregor, J. N., Ormerod, T. C., & Chronicle, E. P. ( 2001 ). Information processing and insight: A process model of performance on the nine-dot and related problems.   Journal of Experimental Psychology: Learning, Memory, and Cognition , 27 , 176–201.

Maier, N. ( 1930 ). Reasoning in humans. I. On direction.   Journal of Comparative Psychology , 10 , 15–43.

Maier, N. ( 1931 ). Reasoning in humans. II. The solution of a problem and its appearance in consciousness.   Journal of Comparative Psychology , 12 , 181–194.

Markman, A. B. ( 1999 ). Knowledge representation . Mahwah, NJ: Erlbaum.

Martin, S. A., & Bassok, M. ( 2005 ). Effects of semantic cues on mathematical modeling: Evidence from word-problem solving and equation construction tasks.   Memory and Cognition , 33 , 471–478.

Mayer, R. E., & Gallini, J. K. ( 1990 ). When is an illustration worth ten thousand words?   Journal of Educational Psychology , 82 , 715–726.

Mayer, R. E., Griffith, E., Jurkowitz, I. T. N., & Rothman, D. ( 2008 ). Increased interestingness of extraneous details in a multimedia science presentation leads to decreased learning.   Journal of Experimental Psychology: Applied , 14 , 329–339.

McKeithen, K. B., Reitman, J. S., Rueter, H. H., & Hirtle, S. C. ( 1981 ). Knowledge organization and skill differences in computer programmers.   Cognitive Psychology , 13 , 307–325.

Medin, D. L., & Ross, B. H. ( 1989 ). The specific character of abstract thought: Categorization, problem solving, and induction. In R. J. Sternberg (Ed.), Advances in the psychology of human intelligence (Vol. 5, pp. 189–223). Hillsdale, NJ: Erlbaum.

Moss, J., Kotovsky, K., & Cagan, J. ( 2011 ). The effect of incidental hints when problems are suspended before, during, and after an impasse.   Journal of Experimental Psychology: Learning, Memory, and Cognition , 37 , 140–148.

Myles-Worsley, M., Johnston, W. A., & Simons, M. A ( 1988 ). The influence of expertise on X-ray image processing.   Journal of Experimental Psychology: Learning, Memory, and Cognition , 14 , 553–557.

Newell, A., & Simon, H. A. ( 1972 ). Human problem solving . Englewood Cliffs, NJ: Prentice-Hall.

Newell, A., & Simon, H. A. ( 1976 ). Computer science as empirical enquiry: Symbols and search.   Communications of the ACM , 19 , 113–126.

Novick, L. R. ( 1988 ). Analogical transfer, problem similarity, and expertise.   Journal of Experimental Psychology: Learning, Memory, and Cognition , 14 , 510–520.

Novick, L. R. ( 1995 ). Some determinants of successful analogical transfer in the solution of algebra word problems.   Thinking and Reasoning , 1 , 5–30.

Novick, L. R., & Catley, K. M. ( 2007 ). Understanding phylogenies in biology: The influence of a Gestalt perceptual principle.   Journal of Experimental Psychology: Applied , 13 , 197–223.

Novick, L. R., Catley, K. M., & Funk, D. J. ( 2010 ). Characters are key: The effect of synapomorphies on cladogram comprehension.   Evolution: Education and Outreach , 3 , 539–547.

Novick, L. R., & Hmelo, C. E. ( 1994 ). Transferring symbolic representations across non-isomorphic problems.   Journal of Experimental Psychology: Learning, Memory, and Cognition , 20 , 1296–1321.

Novick, L. R., & Holyoak, K. J. ( 1991 ). Mathematical problem solving by analogy.   Journal of Experimental Psychology: Learning, Memory, and Cognition , 17 , 398–415.

Novick, L. R., & Hurley, S. M. ( 2001 ). To matrix, network, or hierarchy: That is the question.   Cognitive Psychology , 42 , 158–216.

Novick, L. R., Shade, C. K., & Catley, K. M. ( 2011 ). Linear versus branching depictions of evolutionary history: Implications for diagram design.   Topics in Cognitive Science , 3 (3), 536–559.

Novick, L. R., & Sherman, S. J. ( 2003 ). On the nature of insight solutions: Evidence from skill differences in anagram solution.   The Quarterly Journal of Experimental Psychology , 56A , 351–382.

Novick, L. R., & Sherman, S. J. ( 2008 ). The effects of superficial and structural information on on-line problem solving for good versus poor anagram solvers.   The Quarterly Journal of Experimental Psychology , 61 , 1098–1120.

Ohlsson, S. ( 1984 ). Restructuring revisited I. Summary and critique of the Gestalt theory of problem solving.   Scandinavian Journal of Psychology , 25 , 65–78.

Öllinger, M., Jones, G., & Knoblich, G. ( 2008 ). Investigating the effect of mental set on insight problem solving.   Experimental Psychology , 55 , 269–282.

Ormerod, T. C., MacGregor, J. N., & Chronicle, E. P. ( 2002 ). Dynamics and constraints in insight problem solving.   Journal of Experimental Psychology: Learning, Memory, and Cognition , 28 , 791–799.

Paige, J. M., & Simon, H. A. ( 1966 ). Cognitive processes in solving algebra word problems. In B. Kleinmuntz (Ed.), Problem solving: Research, method, and theory (pp. 51–119). New York: Wiley

Patel, V. L., Groen, G. J., & Arocha, J. F. ( 1990 ). Medical expertise as a function of task difficulty.   Memory and Cognition , 18 , 394–406.

Patsenko, E. G., & Altmann, E. M. ( 2010 ). How planful is routine behavior? A selective attention model of performance in the Tower of Hanoi.   Journal of Experimental Psychology: General , 139 , 95–116.

Polya, G. ( 1957 ). How to solve it (2nd ed.). Princeton, NJ: Princeton University Press.

Posner, M. I. ( 1973 ). Cognition: An introduction . Glenview, IL: Scott, Foresman and Company.

Reitman, W. R. ( 1965 ). Cognition and thought . New York: Wiley.

Richland, L. E., & McDonough, I. M. ( 2010 ), Learning by analogy: Discriminating between potential analogs.   Contemporary Educational Psychology , 35 , 28–43.

Russo, J. E., Johnson, E. J., & Stephens, D. L. ( 1989 ). The validity of verbal protocols.   Memory and Cognition , 17 , 759–769.

Schoenfeld, A. H. ( 1979 ). Explicit heuristic training as a variable in problem-solving performance.   Journal for Research in Mathematics Education , 10 , 173–187.

Schoenfeld, A. H., & Herrmann, D. J. ( 1982 ). Problem perception and knowledge structure in expert and novice mathematical problem solvers.   Journal of Experimental Psychology: Learning, Memory, and Cognition , 8 , 484–494.

Schwartz, S. H. ( 1971 ). Modes of representation and problem solving: Well evolved is half solved.   Journal of Experimental Psychology , 91 , 347–350.

Silver, E. A. ( 1979 ). Student perceptions of relatedness among mathematical verbal problems.   Journal for Research in Mathematics Education , 10 , 195–210.

Silver, E. A. ( 1981 ). Recall of mathematical problem information: Solving related problems.   Journal for Research in Mathematics Education , 12 , 54–64.

Simon, D. P., & Simon, H. A. ( 1978 ). Individual differences in solving physics problems. In R. Siegler (Ed.), Children's thinking: What develops? (pp. 325–348). Hillsdale, NJ: Erlbaum.

Simon, H. A. ( 1978 ). Information-processing theory of human problem solving. In W. K. Estes (Ed.), Handbook of learning and cognitive processes (Vol. 5, pp. 271–295). Hillsdale, NJ: Erlbaum.

Simon, H. A. ( 1986 ). The information processing explanation of Gestalt Phenomena.   Computers in Human Behavior , 2 , 241–255.

Simon, H. A. ( 1990 ). Invariants of human behavior.   Annual Review of Psychology , 41 , 1–19.

Son, J. Y., & Goldstone, R. L. ( 2009 ). Fostering general transfer with specific simulations.   Pragmatics and Cognition , 17 , 1–42.

Thomas, J. C., Jr., ( 1974 ). An analysis of behavior in the hobbits-orcs problem.   Cognitive Psychology , 6 , 257–269.

Weisberg, R. W., & Alba, J. W. ( 1981 ). An examination of the alleged role of “fixation” in the solution of several “insight” problems.   Journal of Experimental Psychology: General , 110 , 169–192.

Weiser, M., & Shertz, J. ( 1983 ). Programming problem representation in novice and expert programmers.   International Journal of Man-Machine Studies , 19 , 391–398.

Wertheimer, M. ( 1959 ). Productive thinking (Rev. ed.). Chicago, IL: University of Chicago Press.

Winn, W. ( 1989 ). The design and use of instructional graphics. In H. Mandl & J. R. Levin (Eds.), Knowledge acquisition from text and pictures (pp. 125–144). Amsterdam, Netherlands: Elsevier

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Computational Thinking for Problem Solving

Taught in English

Some content may not be translated

Financial aid available

114,337 already enrolled

Gain insight into a topic and learn the fundamentals

Instructor: Susan Davidson

Included with Coursera Plus

(1,359 reviews)

Skills you'll gain

• Simple Algorithm
• Python Programming
• Problem Solving
• Computation

Details to know

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There are 4 modules in this course

Computational thinking is the process of approaching a problem in a systematic manner and creating and expressing a solution such that it can be carried out by a computer. But you don't need to be a computer scientist to think like a computer scientist! In fact, we encourage students from any field of study to take this course. Many quantitative and data-centric problems can be solved using computational thinking and an understanding of computational thinking will give you a foundation for solving problems that have real-world, social impact.

In this course, you will learn about the pillars of computational thinking, how computer scientists develop and analyze algorithms, and how solutions can be realized on a computer using the Python programming language. By the end of the course, you will be able to develop an algorithm and express it to the computer by writing a simple Python program. This course will introduce you to people from diverse professions who use computational thinking to solve problems. You will engage with a unique community of analytical thinkers and be encouraged to consider how you can make a positive social impact through computational thinking.

Pillars of Computational Thinking

Computational thinking is an approach to solving problems using concepts and ideas from computer science, and expressing solutions to those problems so that they can be run on a computer. As computing becomes more and more prevalent in all aspects of modern society -- not just in software development and engineering, but in business, the humanities, and even everyday life -- understanding how to use computational thinking to solve real-world problems is a key skill in the 21st century. Computational thinking is built on four pillars: decomposition, pattern recognition, data representation and abstraction, and algorithms. This module introduces you to the four pillars of computational thinking and shows how they can be applied as part of the problem solving process.

What's included

6 videos 4 quizzes 1 assignment 2 peer reviews 4 discussion prompts

6 videos • Total 44 minutes

• 1.1 Introduction • 4 minutes • Preview module
• 1.2 Decomposition • 6 minutes
• 1.3 Pattern Recognition • 5 minutes
• 1.4 Data Representation and Abstraction • 7 minutes
• 1.5 Algorithms • 8 minutes
• 1.6 Case Studies • 11 minutes

4 quizzes • Total 50 minutes

• 1.2 Decomposition • 10 minutes
• 1.3 Pattern Recognition • 10 minutes
• 1.4 Data Representation and Abstraction • 15 minutes
• 1.5 Algorithms • 15 minutes

1 assignment • Total 30 minutes

• Learning Style Preference Survey • 30 minutes

2 peer reviews • Total 60 minutes

• Applying Computational Thinking in Your Life • 30 minutes
• Project Part 1: Applying the Pillars of Computational Thinking • 30 minutes

4 discussion prompts • Total 40 minutes

• Applying Decomposition in Your Life • 10 minutes
• Applying Pattern Recognition in Your Life • 10 minutes
• Applying Data Representation and Abstraction in Your Life • 10 minutes
• Applying Algorithms in Your Life • 10 minutes

Expressing and Analyzing Algorithms

When we use computational thinking to solve a problem, what we’re really doing is developing an algorithm: a step-by-step series of instructions. Whether it’s a small task like scheduling meetings, or a large task like mapping the planet, the ability to develop and describe algorithms is crucial to the problem-solving process based on computational thinking. This module will introduce you to some common algorithms, as well as some general approaches to developing algorithms yourself. These approaches will be useful when you're looking not just for any answer to a problem, but the best answer. After completing this module, you will be able to evaluate an algorithm and analyze how its performance is affected by the size of the input so that you can choose the best algorithm for the problem you’re trying to solve.

7 videos 6 quizzes 4 peer reviews

7 videos • Total 68 minutes

• 2.1 Finding the Largest Value • 8 minutes • Preview module
• 2.2 Linear Search • 5 minutes
• 2.3 Algorithmic Complexity • 8 minutes
• 2.4 Binary Search • 11 minutes
• 2.5 Brute Force Algorithms • 13 minutes
• 2.6 Greedy Algorithms • 9 minutes
• 2.7 Case Studies • 12 minutes

6 quizzes • Total 65 minutes

• 2.1 Finding the Largest Value • 10 minutes
• 2.2 Linear Search • 10 minutes
• 2.3 Algorithmic Complexity • 10 minutes
• 2.4 Binary Search • 10 minutes
• 2.5 Brute Force Algorithms • 15 minutes
• 2.6 Greedy Algorithms • 10 minutes

4 peer reviews • Total 115 minutes

• Finding Minimum Values • 30 minutes
• Binary Search • 30 minutes
• Greedy vs. Brute Force Algorithms • 30 minutes
• Project Part 2: Describing Algorithms Using a Flowchart • 25 minutes

Fundamental Operations of a Modern Computer

Computational thinking is a problem-solving process in which the last step is expressing the solution so that it can be executed on a computer. However, before we are able to write a program to implement an algorithm, we must understand what the computer is capable of doing -- in particular, how it executes instructions and how it uses data. This module describes the inner workings of a modern computer and its fundamental operations. Then it introduces you to a way of expressing algorithms known as pseudocode, which will help you implement your solution using a programming language.

6 videos 5 quizzes 5 peer reviews

6 videos • Total 45 minutes

• 3.1 A History of the Computer • 7 minutes • Preview module
• 3.2 Intro to the von Neumann Architecture • 8 minutes
• 3.3 von Neumann Architecture Data • 6 minutes
• 3.4 von Neumann Architecture Control Flow • 5 minutes
• 3.5 Expressing Algorithms in Pseudocode • 8 minutes
• 3.6 Case Studies • 10 minutes

5 quizzes • Total 50 minutes

• 3.1 A History of the Computer • 10 minutes
• 3.2 Intro to the von Neumann Architecture • 10 minutes
• 3.3 von Neumann Architecture Data • 10 minutes
• 3.4 von Neumann Architecture Control Flow • 10 minutes
• 3.5 Expressing Algorithms in Pseudocode • 10 minutes

5 peer reviews • Total 120 minutes

• (Optional) von Neumann Architecture Control Instructions • 20 minutes
• (Optional) Understanding Pseudocode • 20 minutes
• von Neumann Architecture Data & Instructions • 30 minutes
• Reading & Writing Pseudocode • 25 minutes
• Project Part 3: Writing Pseudocode • 25 minutes

Applied Computational Thinking Using Python

Writing a program is the last step of the computational thinking process. It’s the act of expressing an algorithm using a syntax that the computer can understand. This module introduces you to the Python programming language and its core features. Even if you have never written a program before -- or never even considered it -- after completing this module, you will be able to write simple Python programs that allow you to express your algorithms to a computer as part of a problem-solving process based on computational thinking.

9 videos 12 readings 12 quizzes

9 videos • Total 90 minutes

• 4.1 Introduction to Python • 6 minutes • Preview module
• 4.2 Variables • 13 minutes
• 4.3 Conditional Statements • 8 minutes
• 4.4 Lists • 7 minutes
• 4.5 Iteration • 14 minutes
• 4.6 Functions • 10 minutes
• 4.7 Classes and Objects • 9 minutes
• 4.8 Case Studies • 11 minutes
• 4.9 Course Conclusion • 8 minutes

12 readings • Total 125 minutes

• Programming on the Coursera Platform • 10 minutes
• Python Playground • 0 minutes
• Variables Programming Activity • 20 minutes
• Solution to Variables Programming Activity • 10 minutes
• Conditionals Programming Activity • 20 minutes
• Solution to Conditionals Programming Activity • 10 minutes
• Solution to Lists Programming Assignment • 5 minutes
• Solution to Loops Programming Assignment • 10 minutes
• Solution to Functions Programming Assignment • 10 minutes
• Solution to Challenge Programming Assignment • 10 minutes
• Solution to Classes and Objects Programming Assignment • 10 minutes
• Solution to Project Part 4 • 10 minutes

12 quizzes • Total 185 minutes

• (Optional) Challenge Programming Assignment • 20 minutes
• 4.2 Variables • 10 minutes
• 4.3 Conditional Statements • 5 minutes
• 4.4 Lists • 10 minutes
• Lists Programming Assignment • 15 minutes
• 4.5 Iteration • 10 minutes
• Loops Programming Assignment • 30 minutes
• Functions Programming Assignment • 20 minutes
• 4.7 Classes and Objects • 10 minutes
• Classes and Objects Programming Assignment • 20 minutes
• Project Part 4: Implementing the Solution in Python • 25 minutes

Instructor ratings

We asked all learners to give feedback on our instructors based on the quality of their teaching style.

The University of Pennsylvania (commonly referred to as Penn) is a private university, located in Philadelphia, Pennsylvania, United States. A member of the Ivy League, Penn is the fourth-oldest institution of higher education in the United States, and considers itself to be the first university in the United States with both undergraduate and graduate studies.

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Learner reviews

Showing 3 of 1359

1,359 reviews

Reviewed on Nov 17, 2018

Well taught with good examples and exercises that require thinking but still approachable. Very well laid out and taught. Definitely sparked an interest to go learn more.

Reviewed on Aug 3, 2022

Informative but the last assignment on Week 4 is misleading in that you can only submit the search function but the instructions say that you can test your code and the program will test it as well.

Reviewed on Jan 17, 2019

Useful course taught at an adequate rate. I recommend it for people who are interested in learning the basics of computational thinking, i.e. a systematic approach to problem-solving.

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Frequently asked questions

Do i need to know how to program or have studied computer science in order to take this course.

No, definitely not! This course is intended for anyone who has an interest in approaching problems more systematically, developing more efficient solutions, and understanding how computers can be used in the problem solving process. No prior computer science or programming experience is required.

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6.1 Solving Problems with Newton’s Laws

Learning objectives.

By the end of this section, you will be able to:

• Apply problem-solving techniques to solve for quantities in more complex systems of forces
• Use concepts from kinematics to solve problems using Newton’s laws of motion
• Solve more complex equilibrium problems
• Solve more complex acceleration problems
• Apply calculus to more advanced dynamics problems

Success in problem solving is necessary to understand and apply physical principles. We developed a pattern of analyzing and setting up the solutions to problems involving Newton’s laws in Newton’s Laws of Motion ; in this chapter, we continue to discuss these strategies and apply a step-by-step process.

Problem-Solving Strategies

We follow here the basics of problem solving presented earlier in this text, but we emphasize specific strategies that are useful in applying Newton’s laws of motion . Once you identify the physical principles involved in the problem and determine that they include Newton’s laws of motion, you can apply these steps to find a solution. These techniques also reinforce concepts that are useful in many other areas of physics. Many problem-solving strategies are stated outright in the worked examples, so the following techniques should reinforce skills you have already begun to develop.

Problem-Solving Strategy

Applying newton’s laws of motion.

• Identify the physical principles involved by listing the givens and the quantities to be calculated.
• Sketch the situation, using arrows to represent all forces.
• Determine the system of interest. The result is a free-body diagram that is essential to solving the problem.
• Apply Newton’s second law to solve the problem. If necessary, apply appropriate kinematic equations from the chapter on motion along a straight line.
• Check the solution to see whether it is reasonable.

Let’s apply this problem-solving strategy to the challenge of lifting a grand piano into a second-story apartment. Once we have determined that Newton’s laws of motion are involved (if the problem involves forces), it is particularly important to draw a careful sketch of the situation. Such a sketch is shown in Figure 6.2 (a). Then, as in Figure 6.2 (b), we can represent all forces with arrows. Whenever sufficient information exists, it is best to label these arrows carefully and make the length and direction of each correspond to the represented force.

As with most problems, we next need to identify what needs to be determined and what is known or can be inferred from the problem as stated, that is, make a list of knowns and unknowns. It is particularly crucial to identify the system of interest, since Newton’s second law involves only external forces. We can then determine which forces are external and which are internal, a necessary step to employ Newton’s second law. (See Figure 6.2 (c).) Newton’s third law may be used to identify whether forces are exerted between components of a system (internal) or between the system and something outside (external). As illustrated in Newton’s Laws of Motion , the system of interest depends on the question we need to answer. Only forces are shown in free-body diagrams, not acceleration or velocity. We have drawn several free-body diagrams in previous worked examples. Figure 6.2 (c) shows a free-body diagram for the system of interest. Note that no internal forces are shown in a free-body diagram.

Once a free-body diagram is drawn, we apply Newton’s second law. This is done in Figure 6.2 (d) for a particular situation. In general, once external forces are clearly identified in free-body diagrams, it should be a straightforward task to put them into equation form and solve for the unknown, as done in all previous examples. If the problem is one-dimensional—that is, if all forces are parallel—then the forces can be handled algebraically. If the problem is two-dimensional, then it must be broken down into a pair of one-dimensional problems. We do this by projecting the force vectors onto a set of axes chosen for convenience. As seen in previous examples, the choice of axes can simplify the problem. For example, when an incline is involved, a set of axes with one axis parallel to the incline and one perpendicular to it is most convenient. It is almost always convenient to make one axis parallel to the direction of motion, if this is known. Generally, just write Newton’s second law in components along the different directions. Then, you have the following equations:

(If, for example, the system is accelerating horizontally, then you can then set a y = 0 . a y = 0 . ) We need this information to determine unknown forces acting on a system.

As always, we must check the solution. In some cases, it is easy to tell whether the solution is reasonable. For example, it is reasonable to find that friction causes an object to slide down an incline more slowly than when no friction exists. In practice, intuition develops gradually through problem solving; with experience, it becomes progressively easier to judge whether an answer is reasonable. Another way to check a solution is to check the units. If we are solving for force and end up with units of millimeters per second, then we have made a mistake.

There are many interesting applications of Newton’s laws of motion, a few more of which are presented in this section. These serve also to illustrate some further subtleties of physics and to help build problem-solving skills. We look first at problems involving particle equilibrium, which make use of Newton’s first law, and then consider particle acceleration, which involves Newton’s second law.

Particle Equilibrium

Recall that a particle in equilibrium is one for which the external forces are balanced. Static equilibrium involves objects at rest, and dynamic equilibrium involves objects in motion without acceleration, but it is important to remember that these conditions are relative. For example, an object may be at rest when viewed from our frame of reference, but the same object would appear to be in motion when viewed by someone moving at a constant velocity. We now make use of the knowledge attained in Newton’s Laws of Motion , regarding the different types of forces and the use of free-body diagrams, to solve additional problems in particle equilibrium .

Example 6.1

Different tensions at different angles.

Thus, as you might expect,

This gives us the following relationship:

Note that T 1 T 1 and T 2 T 2 are not equal in this case because the angles on either side are not equal. It is reasonable that T 2 T 2 ends up being greater than T 1 T 1 because it is exerted more vertically than T 1 . T 1 .

Now consider the force components along the vertical or y -axis:

This implies

Substituting the expressions for the vertical components gives

There are two unknowns in this equation, but substituting the expression for T 2 T 2 in terms of T 1 T 1 reduces this to one equation with one unknown:

which yields

Solving this last equation gives the magnitude of T 1 T 1 to be

Finally, we find the magnitude of T 2 T 2 by using the relationship between them, T 2 = 1.225 T 1 T 2 = 1.225 T 1 , found above. Thus we obtain

Significance

Particle acceleration.

We have given a variety of examples of particles in equilibrium. We now turn our attention to particle acceleration problems, which are the result of a nonzero net force. Refer again to the steps given at the beginning of this section, and notice how they are applied to the following examples.

Example 6.2

Drag force on a barge.

The drag of the water F → D F → D is in the direction opposite to the direction of motion of the boat; this force thus works against F → app , F → app , as shown in the free-body diagram in Figure 6.4 (b). The system of interest here is the barge, since the forces on it are given as well as its acceleration. Because the applied forces are perpendicular, the x - and y -axes are in the same direction as F → 1 F → 1 and F → 2 . F → 2 . The problem quickly becomes a one-dimensional problem along the direction of F → app F → app , since friction is in the direction opposite to F → app . F → app . Our strategy is to find the magnitude and direction of the net applied force F → app F → app and then apply Newton’s second law to solve for the drag force F → D . F → D .

The angle is given by

From Newton’s first law, we know this is the same direction as the acceleration. We also know that F → D F → D is in the opposite direction of F → app , F → app , since it acts to slow down the acceleration. Therefore, the net external force is in the same direction as F → app , F → app , but its magnitude is slightly less than F → app . F → app . The problem is now one-dimensional. From the free-body diagram, we can see that

However, Newton’s second law states that

This can be solved for the magnitude of the drag force of the water F D F D in terms of known quantities:

Substituting known values gives

The direction of F → D F → D has already been determined to be in the direction opposite to F → app , F → app , or at an angle of 53 ° 53 ° south of west.

In Newton’s Laws of Motion , we discussed the normal force , which is a contact force that acts normal to the surface so that an object does not have an acceleration perpendicular to the surface. The bathroom scale is an excellent example of a normal force acting on a body. It provides a quantitative reading of how much it must push upward to support the weight of an object. But can you predict what you would see on the dial of a bathroom scale if you stood on it during an elevator ride? Will you see a value greater than your weight when the elevator starts up? What about when the elevator moves upward at a constant speed? Take a guess before reading the next example.

Example 6.3

What does the bathroom scale read in an elevator.

From the free-body diagram, we see that F → net = F → s − w → , F → net = F → s − w → , so we have

Solving for F s F s gives us an equation with only one unknown:

or, because w = m g , w = m g , simply

No assumptions were made about the acceleration, so this solution should be valid for a variety of accelerations in addition to those in this situation. ( Note: We are considering the case when the elevator is accelerating upward. If the elevator is accelerating downward, Newton’s second law becomes F s − w = − m a . F s − w = − m a . )

• We have a = 1.20 m/s 2 , a = 1.20 m/s 2 , so that F s = ( 75.0 kg ) ( 9.80 m/s 2 ) + ( 75.0 kg ) ( 1.20 m/s 2 ) F s = ( 75.0 kg ) ( 9.80 m/s 2 ) + ( 75.0 kg ) ( 1.20 m/s 2 ) yielding F s = 825 N . F s = 825 N .
• Now, what happens when the elevator reaches a constant upward velocity? Will the scale still read more than his weight? For any constant velocity—up, down, or stationary—acceleration is zero because a = Δ v Δ t a = Δ v Δ t and Δ v = 0 . Δ v = 0 . Thus, F s = m a + m g = 0 + m g F s = m a + m g = 0 + m g or F s = ( 75.0 kg ) ( 9.80 m/s 2 ) , F s = ( 75.0 kg ) ( 9.80 m/s 2 ) , which gives F s = 735 N . F s = 735 N .

Thus, the scale reading in the elevator is greater than his 735-N (165-lb.) weight. This means that the scale is pushing up on the person with a force greater than his weight, as it must in order to accelerate him upward. Clearly, the greater the acceleration of the elevator, the greater the scale reading, consistent with what you feel in rapidly accelerating versus slowly accelerating elevators. In Figure 6.5 (b), the scale reading is 735 N, which equals the person’s weight. This is the case whenever the elevator has a constant velocity—moving up, moving down, or stationary.

Check Your Understanding 6.1

Now calculate the scale reading when the elevator accelerates downward at a rate of 1.20 m/s 2 . 1.20 m/s 2 .

The solution to the previous example also applies to an elevator accelerating downward, as mentioned. When an elevator accelerates downward, a is negative, and the scale reading is less than the weight of the person. If a constant downward velocity is reached, the scale reading again becomes equal to the person’s weight. If the elevator is in free fall and accelerating downward at g , then the scale reading is zero and the person appears to be weightless.

Example 6.4

Two attached blocks.

For block 1: T → + w → 1 + N → = m 1 a → 1 T → + w → 1 + N → = m 1 a → 1

For block 2: T → + w → 2 = m 2 a → 2 . T → + w → 2 = m 2 a → 2 .

Notice that T → T → is the same for both blocks. Since the string and the pulley have negligible mass, and since there is no friction in the pulley, the tension is the same throughout the string. We can now write component equations for each block. All forces are either horizontal or vertical, so we can use the same horizontal/vertical coordinate system for both objects

When block 1 moves to the right, block 2 travels an equal distance downward; thus, a 1 x = − a 2 y . a 1 x = − a 2 y . Writing the common acceleration of the blocks as a = a 1 x = − a 2 y , a = a 1 x = − a 2 y , we now have

From these two equations, we can express a and T in terms of the masses m 1 and m 2 , and g : m 1 and m 2 , and g :

Check Your Understanding 6.2

Calculate the acceleration of the system, and the tension in the string, when the masses are m 1 = 5.00 kg m 1 = 5.00 kg and m 2 = 3.00 kg . m 2 = 3.00 kg .

Example 6.5

Atwood machine.

• We have For m 1 , ∑ F y = T − m 1 g = m 1 a . For m 2 , ∑ F y = T − m 2 g = − m 2 a . For m 1 , ∑ F y = T − m 1 g = m 1 a . For m 2 , ∑ F y = T − m 2 g = − m 2 a . (The negative sign in front of m 2 a m 2 a indicates that m 2 m 2 accelerates downward; both blocks accelerate at the same rate, but in opposite directions.) Solve the two equations simultaneously (subtract them) and the result is ( m 2 − m 1 ) g = ( m 1 + m 2 ) a . ( m 2 − m 1 ) g = ( m 1 + m 2 ) a . Solving for a : a = m 2 − m 1 m 1 + m 2 g = 4 kg − 2 kg 4 kg + 2 kg ( 9.8 m/s 2 ) = 3.27 m/s 2 . a = m 2 − m 1 m 1 + m 2 g = 4 kg − 2 kg 4 kg + 2 kg ( 9.8 m/s 2 ) = 3.27 m/s 2 .
• Observing the first block, we see that T − m 1 g = m 1 a T = m 1 ( g + a ) = ( 2 kg ) ( 9.8 m/s 2 + 3.27 m/s 2 ) = 26.1 N . T − m 1 g = m 1 a T = m 1 ( g + a ) = ( 2 kg ) ( 9.8 m/s 2 + 3.27 m/s 2 ) = 26.1 N .

Check Your Understanding 6.3

Determine a general formula in terms of m 1 , m 2 m 1 , m 2 and g for calculating the tension in the string for the Atwood machine shown above.

Newton’s Laws of Motion and Kinematics

Physics is most interesting and most powerful when applied to general situations that involve more than a narrow set of physical principles. Newton’s laws of motion can also be integrated with other concepts that have been discussed previously in this text to solve problems of motion. For example, forces produce accelerations, a topic of kinematics , and hence the relevance of earlier chapters.

When approaching problems that involve various types of forces, acceleration, velocity, and/or position, listing the givens and the quantities to be calculated will allow you to identify the principles involved. Then, you can refer to the chapters that deal with a particular topic and solve the problem using strategies outlined in the text. The following worked example illustrates how the problem-solving strategy given earlier in this chapter, as well as strategies presented in other chapters, is applied to an integrated concept problem.

Example 6.6

What force must a soccer player exert to reach top speed.

• We are given the initial and final velocities (zero and 8.00 m/s forward); thus, the change in velocity is Δ v = 8.00 m/s Δ v = 8.00 m/s . We are given the elapsed time, so Δ t = 2.50 s . Δ t = 2.50 s . The unknown is acceleration, which can be found from its definition: a = Δ v Δ t . a = Δ v Δ t . Substituting the known values yields a = 8.00 m/s 2.50 s = 3.20 m/s 2 . a = 8.00 m/s 2.50 s = 3.20 m/s 2 .
• Here we are asked to find the average force the ground exerts on the runner to produce this acceleration. (Remember that we are dealing with the force or forces acting on the object of interest.) This is the reaction force to that exerted by the player backward against the ground, by Newton’s third law. Neglecting air resistance, this would be equal in magnitude to the net external force on the player, since this force causes her acceleration. Since we now know the player’s acceleration and are given her mass, we can use Newton’s second law to find the force exerted. That is, F net = m a . F net = m a . Substituting the known values of m and a gives F net = ( 70.0 kg ) ( 3.20 m/s 2 ) = 224 N . F net = ( 70.0 kg ) ( 3.20 m/s 2 ) = 224 N .

This is a reasonable result: The acceleration is attainable for an athlete in good condition. The force is about 50 pounds, a reasonable average force.

Check Your Understanding 6.4

The soccer player stops after completing the play described above, but now notices that the ball is in position to be stolen. If she now experiences a force of 126 N to attempt to steal the ball, which is 2.00 m away from her, how long will it take her to get to the ball?

Example 6.7

What force acts on a model helicopter.

The magnitude of the force is now easily found:

Check Your Understanding 6.5

Find the direction of the resultant for the 1.50-kg model helicopter.

Example 6.8

Baggage tractor.

• ∑ F x = m system a x ∑ F x = m system a x and ∑ F x = 820.0 t , ∑ F x = 820.0 t , so 820.0 t = ( 650.0 + 250.0 + 150.0 ) a a = 0.7809 t . 820.0 t = ( 650.0 + 250.0 + 150.0 ) a a = 0.7809 t . Since acceleration is a function of time, we can determine the velocity of the tractor by using a = d v d t a = d v d t with the initial condition that v 0 = 0 v 0 = 0 at t = 0 . t = 0 . We integrate from t = 0 t = 0 to t = 3 : t = 3 : d v = a d t , ∫ 0 3 d v = ∫ 0 3.00 a d t = ∫ 0 3.00 0.7809 t d t , v = 0.3905 t 2 ] 0 3.00 = 3.51 m/s . d v = a d t , ∫ 0 3 d v = ∫ 0 3.00 a d t = ∫ 0 3.00 0.7809 t d t , v = 0.3905 t 2 ] 0 3.00 = 3.51 m/s .
• Refer to the free-body diagram in Figure 6.8 (b). ∑ F x = m tractor a x 820.0 t − T = m tractor ( 0.7805 ) t ( 820.0 ) ( 3.00 ) − T = ( 650.0 ) ( 0.7805 ) ( 3.00 ) T = 938 N . ∑ F x = m tractor a x 820.0 t − T = m tractor ( 0.7805 ) t ( 820.0 ) ( 3.00 ) − T = ( 650.0 ) ( 0.7805 ) ( 3.00 ) T = 938 N .

Recall that v = d s d t v = d s d t and a = d v d t a = d v d t . If acceleration is a function of time, we can use the calculus forms developed in Motion Along a Straight Line , as shown in this example. However, sometimes acceleration is a function of displacement. In this case, we can derive an important result from these calculus relations. Solving for dt in each, we have d t = d s v d t = d s v and d t = d v a . d t = d v a . Now, equating these expressions, we have d s v = d v a . d s v = d v a . We can rearrange this to obtain a d s = v d v . a d s = v d v .

Example 6.9

Motion of a projectile fired vertically.

The acceleration depends on v and is therefore variable. Since a = f ( v ) , a = f ( v ) , we can relate a to v using the rearrangement described above,

We replace ds with dy because we are dealing with the vertical direction,

We now separate the variables ( v ’s and dv ’s on one side; dy on the other):

Thus, h = 114 m . h = 114 m .

Check Your Understanding 6.6

If atmospheric resistance is neglected, find the maximum height for the mortar shell. Is calculus required for this solution?

Interactive

Explore the forces at work in this simulation when you try to push a filing cabinet. Create an applied force and see the resulting frictional force and total force acting on the cabinet. Charts show the forces, position, velocity, and acceleration vs. time. View a free-body diagram of all the forces (including gravitational and normal forces).

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26 Expert-Backed Problem Solving Examples – Interview Answers

Published: February 13, 2023

Interview Questions and Answers

Actionable advice from real experts:

Biron Clark

Former Recruiter

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Biron Clark , Former Recruiter

Kyle Elliott , Career Coach

Hayley Jukes , Editor

As a recruiter , I know employers like to hire people who can solve problems and work well under pressure.

 A job rarely goes 100% according to plan, so hiring managers are more likely to hire you if you seem like you can handle unexpected challenges while staying calm and logical.

But how do they measure this?

Hiring managers will ask you interview questions about your problem-solving skills, and they might also look for examples of problem-solving on your resume and cover letter. 

In this article, I’m going to share a list of problem-solving examples and sample interview answers to questions like, “Give an example of a time you used logic to solve a problem?” and “Describe a time when you had to solve a problem without managerial input. How did you handle it, and what was the result?”

• Problem-solving involves identifying, prioritizing, analyzing, and solving problems using a variety of skills like critical thinking, creativity, decision making, and communication.
• Describe the Situation, Task, Action, and Result ( STAR method ) when discussing your problem-solving experiences.
• Tailor your interview answer with the specific skills and qualifications outlined in the job description.
• Provide numerical data or metrics to demonstrate the tangible impact of your problem-solving efforts.

What are Problem Solving Skills? 

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

Problem-solving encompasses other skills that can be showcased in an interview response and your resume. Problem-solving skills examples include:

• Critical thinking
• Analytical skills
• Decision making
• Research skills
• Technical skills
• Communication skills
• Adaptability and flexibility

Why is Problem Solving Important in the Workplace?

Problem-solving is essential in the workplace because it directly impacts productivity and efficiency. Whenever you encounter a problem, tackling it head-on prevents minor issues from escalating into bigger ones that could disrupt the entire workflow. 

Beyond maintaining smooth operations, your ability to solve problems fosters innovation. It encourages you to think creatively, finding better ways to achieve goals, which keeps the business competitive and pushes the boundaries of what you can achieve. 

Effective problem-solving also contributes to a healthier work environment; it reduces stress by providing clear strategies for overcoming obstacles and builds confidence within teams. 

Examples of Problem-Solving in the Workplace

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

Problem-Solving Examples for Recent Grads/Entry-Level Job Seekers

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

How To Answer “Tell Us About a Problem You Solved”

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

STAR stands for:

It’s a simple way of walking the listener or reader through the story in a way that will make sense to them. 

Start by briefly describing the general situation and the task at hand. After this, describe the course of action you chose and why. Ideally, show that you evaluated all the information you could given the time you had, and made a decision based on logic and fact. Finally, describe the positive result you achieved.

Note: Our sample answers below are structured following the STAR formula. Be sure to check them out!

EXPERT ADVICE

Dr. Kyle Elliott , MPA, CHES Tech & Interview Career Coach caffeinatedkyle.com

How can I communicate complex problem-solving experiences clearly and succinctly?

Before answering any interview question, it’s important to understand why the interviewer is asking the question in the first place.

When it comes to questions about your complex problem-solving experiences, for example, the interviewer likely wants to know about your leadership acumen, collaboration abilities, and communication skills, not the problem itself.

Therefore, your answer should be focused on highlighting how you excelled in each of these areas, not diving into the weeds of the problem itself, which is a common mistake less-experienced interviewees often make.

Tailoring Your Answer Based on the Skills Mentioned in the Job Description

As a recruiter, one of the top tips I can give you when responding to the prompt “Tell us about a problem you solved,” is to tailor your answer to the specific skills and qualifications outlined in the job description. 

Once you’ve pinpointed the skills and key competencies the employer is seeking, craft your response to highlight experiences where you successfully utilized or developed those particular abilities. 

For instance, if the job requires strong leadership skills, focus on a problem-solving scenario where you took charge and effectively guided a team toward resolution. 

By aligning your answer with the desired skills outlined in the job description, you demonstrate your suitability for the role and show the employer that you understand their needs.

Amanda Augustine expands on this by saying:

“Showcase the specific skills you used to solve the problem. Did it require critical thinking, analytical abilities, or strong collaboration? Highlight the relevant skills the employer is seeking.”  

Interview Answers to “Tell Me About a Time You Solved a Problem”

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

The example interview responses are structured using the STAR method and are categorized into the top 5 key problem-solving skills recruiters look for in a candidate.

1. Analytical Thinking

Situation: In my previous role as a data analyst , our team encountered a significant drop in website traffic.

Task: I was tasked with identifying the root cause of the decrease.

Action: I conducted a thorough analysis of website metrics, including traffic sources, user demographics, and page performance. Through my analysis, I discovered a technical issue with our website’s loading speed, causing users to bounce. 

Result: By optimizing server response time, compressing images, and minimizing redirects, we saw a 20% increase in traffic within two weeks.

2. Critical Thinking

Situation: During a project deadline crunch, our team encountered a major technical issue that threatened to derail our progress.

Task: My task was to assess the situation and devise a solution quickly.

Action: I immediately convened a meeting with the team to brainstorm potential solutions. Instead of panicking, I encouraged everyone to think outside the box and consider unconventional approaches. We analyzed the problem from different angles and weighed the pros and cons of each solution.

Result: By devising a workaround solution, we were able to meet the project deadline, avoiding potential delays that could have cost the company $100,000 in penalties for missing contractual obligations.

3. Decision Making

Situation: As a project manager , I was faced with a dilemma when two key team members had conflicting opinions on the project direction.

Task: My task was to make a decisive choice that would align with the project goals and maintain team cohesion.

Action: I scheduled a meeting with both team members to understand their perspectives in detail. I listened actively, asked probing questions, and encouraged open dialogue. After carefully weighing the pros and cons of each approach, I made a decision that incorporated elements from both viewpoints.

Result: The decision I made not only resolved the immediate conflict but also led to a stronger sense of collaboration within the team. By valuing input from all team members and making a well-informed decision, we were able to achieve our project objectives efficiently.

4. Communication (Teamwork)

Situation: During a cross-functional project, miscommunication between departments was causing delays and misunderstandings.

Task: My task was to improve communication channels and foster better teamwork among team members.

Action: I initiated regular cross-departmental meetings to ensure that everyone was on the same page regarding project goals and timelines. I also implemented a centralized communication platform where team members could share updates, ask questions, and collaborate more effectively.

Result: Streamlining workflows and improving communication channels led to a 30% reduction in project completion time, saving the company $25,000 in operational costs.

5. Persistence 

Situation: During a challenging sales quarter, I encountered numerous rejections and setbacks while trying to close a major client deal.

Task: My task was to persistently pursue the client and overcome obstacles to secure the deal.

Action: I maintained regular communication with the client, addressing their concerns and demonstrating the value proposition of our product. Despite facing multiple rejections, I remained persistent and resilient, adjusting my approach based on feedback and market dynamics.

Result: After months of perseverance, I successfully closed the deal with the client. By closing the major client deal, I exceeded quarterly sales targets by 25%, resulting in a revenue increase of $250,000 for the company.

Tips to Improve Your Problem-Solving Skills

Throughout your career, being able to showcase and effectively communicate your problem-solving skills gives you more leverage in achieving better jobs and earning more money .

So to improve your problem-solving skills, I recommend always analyzing a problem and situation before acting.

 When discussing problem-solving with employers, you never want to sound like you rush or make impulsive decisions. They want to see fact-based or data-based decisions when you solve problems.

Don’t just say you’re good at solving problems. Show it with specifics. How much did you boost efficiency? Did you save the company money? Adding numbers can really make your achievements stand out.

To get better at solving problems, analyze the outcomes of past solutions you came up with. You can recognize what works and what doesn’t.

Think about how you can improve researching and analyzing a situation, how you can get better at communicating, and deciding on the right people in the organization to talk to and “pull in” to help you if needed, etc.

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

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

More Interview Resources

• 3 Answers to “How Do You Handle Stress?”
• How to Answer “How Do You Handle Conflict?” (Interview Question)
• Sample Answers to “Tell Me About a Time You Failed”

About the Author

Biron Clark is a former executive recruiter who has worked individually with hundreds of job seekers, reviewed thousands of resumes and LinkedIn profiles, and recruited for top venture-backed startups and Fortune 500 companies. He has been advising job seekers since 2012 to think differently in their job search and land high-paying, competitive positions. Follow on Twitter and LinkedIn .

Read more articles by Biron Clark

About the Contributor

Kyle Elliott , career coach and mental health advocate, transforms his side hustle into a notable practice, aiding Silicon Valley professionals in maximizing potential. Follow Kyle on LinkedIn .

About the Editor

Hayley Jukes is the Editor-in-Chief at CareerSidekick with five years of experience creating engaging articles, books, and transcripts for diverse platforms and audiences.

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Watch CBS News

How two high school students solved a 2,000-year-old math puzzle

By Bill Whitaker , Aliza Chasan , Sara Kuzmarov, Mariah Campbell

May 5, 2024 / 7:00 PM EDT / CBS News

A high school math teacher at St. Mary's Academy in New Orleans, Michelle Blouin Williams, was looking for ingenuity when she and her colleagues set a school-wide math contest with a challenging bonus question. That bonus question asked students to create a new proof for the Pythagorean Theorem, a fundamental principle of geometry, using trigonometry. The teachers weren't necessarily expecting anyone to solve it, as proofs of the Pythagorean Theorem using trigonometry were believed to be impossible for nearly 2,000 years.

But then, in December 2022, Calcea Johnson and Ne'Kiya Jackson, seniors at St. Mary's Academy, stepped up to the challenge. The $500 prize money was a motivating factor.

After months of work, they submitted their innovative proofs to their teachers. With the contest behind them, their teachers encouraged the students to present at a mathematics conference, and then to seek to publish their work. And even today, they're not done. Now in college, they've been working on further proofs of the Pythagorean Theorem and believe they have found five more proofs. Amazingly, despite their impressive achievements, they insist they're not math geniuses.

"I think that's a stretch," Calcea said.

The St. Mary's math contest

When the pair started working on the math contest they were familiar with the Pythagorean Theorem's equation: A² + B² = C², which explains that by knowing the length of two sides of a right triangle, it's possible to figure out the length of the third side.

When Calcea and Ne'Kiya set out to create a new Pythagorean Theorem proof, they didn't know that for thousands of years, one using trigonometry was thought to be impossible.  In 2009, mathematician Jason Zimba submitted one, and now Calcea and Ne'Kiya are adding to the canon.

Calcea and Ne'Kiya had studied geometry and some trigonometry when they started working on their proofs, but said they didn't feel math was easy. As the contest went on, they spent almost all their free time developing their ideas.

"The garbage can was full of papers, which she would, you know, work out the problems and if that didn't work, she would ball it up, throw it in the trash," Cal Johnson, Calcea's dad, said.

Neliska Jackson, Ne'Kiya's mother, says lightheartedly, that most of the time, her daughter's work was beyond her. 

To document Calcea and Ne'Kiya's work, math teachers at St. Mary's submitted their proofs to an American Mathematical Society conference in Atlanta in March 2023.

"Well, our teacher approached us and was like, 'Hey, you might be able to actually present this,'" Ne'Kiya said. "I was like, 'Are you joking?' But she wasn't. So we went. I got up there. We presented and it went well, and it blew up."

Why Calcea' and Ne'kiya's work "blew up"

The reaction was insane and unexpected, Calcea said. News of their accomplishment spread around the world. The pair got a write-up in South Korea and a shoutout from former first lady Michelle Obama. They got a commendation from the governor and keys to the city of New Orleans. 

Calcea and Ne'Kiya said they think there's several reasons why people found their work so impressive. 

"Probably because we're African American, one," Ne'Kiya said. "And we're also women. So I think-- oh, and our age. Of course our ages probably played a big part."

Ne'Kiya said she'd like their accomplishment to be celebrated for what it is: "a great mathematical achievement."

In spite of the community's celebration of the students' work, St. Mary's Academy president and interim principal Pamela Rogers said that with recognition came racist calls and comments. 

"[People said] 'they could not have done it. African Americans don't have the brains to do it.' Of course, we sheltered our girls from that," Rogers said. "But we absolutely did not expect it to come in the volume that it came."

Rogers said too often society has a vision of who can be successful.

"To some people, it is not always an African American female," Rogers said. "And to us, it's always an African American female."

Success at St. Marys 

St. Mary's, a private Catholic elementary and high school, was started for young Black women just after the Civil War. Ne'Kiya and Calcea follow a long line of barrier-breaking graduates. Leah Chase , the late queen of Creole cuisine, was an alum. So was Michelle Woodfork, the first African American female New Orleans police chief, and Dana Douglas, a judge for the Fifth Circuit Court of Appeals. 

Math teacher Michelle Blouin Williams, who initiated the math contest, said Calcea and Ne'Kiya are typical St. Mary's students. She said if they're "unicorns," then every student who's matriculated through the school is a "beautiful, Black unicorn."

Students hear that message from the moment they walk in the door, Rogers said. 

"We believe all students can succeed, all students can learn," the principal said. "It does not matter the environment that you live in."

About half the students at St. Mary's get scholarships, subsidized by fundraising to defray the $8,000 a year tuition. There's no test to get in, but expectations are high and rules are strict: cellphones are not allowed and modest skirts and hair in its natural color are required. 

Students said they appreciate the rules and rigor.

"Especially the standards that they set for us," junior Rayah Siddiq said. "They're very high. And I don't think that's ever going to change." 

What's next for Ne'Kiya and Calcea

Last year when Ne'Kiya and Calcea graduated, all their classmates were accepted into college and received scholarship offers. The school has had a 100% graduation rate and a 100% college acceptance rate for 17 years, according to Rogers.

Ne'Kiya got a full ride in the pharmacy department at Xavier University in New Orleans. Calcea, the class valedictorian, is studying environmental engineering at Louisiana State University. Neither one is pursuing a career in math, though Calcea said she may minor in math.

"People might expect too much out of me if I become a mathematician," Ne'Kiya said wryly. 

Bill Whitaker is an award-winning journalist and 60 Minutes correspondent who has covered major news stories, domestically and across the globe, for more than four decades with CBS News.

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Spring commencement 2024 – meet seoin kim.

“My internship as a Data Visualization Developer at Raytheon was a pivotal experience, enhancing my technical skills and problem-solving abilities in computer engineering. This role prepared me for a future in software engineering and data visualization. As I consider my post-graduation plans, I am weighing the options of further education against jumping straight into the tech industry. The university has equipped me with a solid foundation for either path.

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As we graduate, let us carry forward the lessons of resilience, community, and diverse perspectives. Let’s continue to learn, question, and connect as we embark on our individual journeys. Congratulations to us all!”