Do these tasks right away.
This simple matrix can help you visualize your tasks. You can then consider the long-term outcome of these tasks and focus on those that will make you not only productive but efficient.
The ABC Method involves assigning a priority status of A, B, or C to each of the items on your task list.
Consider the following while making your task list:
When you know what you need to do, you can quickly complete those tasks.
You can use the first 30 minutes of your day to create a to-do list that aligns with your weekly plans.
However, remember to be flexible with your plans and account for unexpected tasks.
You can then revisit your task list at the end of the day to see which task or activity took longer to complete and identify any issues you might be facing at work.
Additionally, you consider blocking off specific brackets of time on your schedule, so you are guaranteed to have time in your schedule without distraction or meetings.
Goal-setting can help you clearly understand what you want to achieve.
To achieve a long-term goal, you need to identify smaller tasks and set goals along the way.
For example, if you’ve set a long-term goal to take on more job responsibility, you need to set smaller goals like improving certain skills.
However, more importantly, you should set SMART goals , i.e., your goals should be Specific, Measurable, Achievable, Relevant, and Time-based. These parameters will help you set realistic goals and avoid demotivating situations that can arise due to unmet targets.
Sure, all these tips can help you with time management and organization, but you need the right tools and apps to help you follow them.
Here are three types of tools to help you make the most of your skills:
Online calendars are an excellent tool for time management.
You can pick online calendars like Google Calendar, Outlook Calendar, Apple Calendar, etc.
Whether you’re a project manager or an employee, project management tools are a must to track work progress.
A project management tool can help you:
You can use tools like Trello , Basecamp , Asana , and more.
A time management tool can help you record your work hours, get auto-generated timesheets, and have measurable data to maximize productivity.
Time Doctor is one such time tracking and employee productivity management tool. It’s used by SMBs like Thrive Market and large companies like RE/MAX to boost productivity.
Time Doctor tracks all workday activities, giving you real-time, actionable insights to improve your time management and productivity.
Explore all the useful features this tool provides .
While creating your schedule, you should also set time limits for each task.
This can help:
Leverage these time management skills to boost your profile. Learn how you can do this in the next section.
An individual with excellent time management skills can adapt to new problems and readjust as needed to complete a task.
And traits like planning, scheduling, strategy, delegation, and adaptability make an employee dependable, and employers look out for that.
Here are some time management and organizational skills you can put on your resume:
Wrapping up
Good time management and organization is an essential soft skill. It ensures timely delivery of quality work, preventing stress and work conflicts.
You can check out the tips mentioned here and implement them to hone your skills.
Once done, you’ll be able to master time management and boost your productivity through the roof!
Vaishali Badgujar is a seasoned Content and SEO specialist who provides ROI-focused managed SEO services. She is dedicated to helping businesses connect with their audience online and see real growth through her work.
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Problem-solving skills are skills that help you identify and solve problems effectively and efficiently . Your ability to solve problems is one of the main ways that hiring managers and recruiters assess candidates, as those with excellent problem-solving skills are more likely to autonomously carry out their responsibilities.
A true problem solver can look at a situation, find the cause of the problem (or causes, because there are often many issues at play), and then come up with a reasonable solution that effectively fixes the problem or at least remedies most of it.
The ability to solve problems is considered a soft skill , meaning that it’s more of a personality trait than a skill you’ve learned at school, on the job, or through technical training.
That being said, your proficiency with various hard skills will have a direct bearing on your ability to solve problems. For example, it doesn’t matter if you’re a great problem-solver; if you have no experience with astrophysics, you probably won’t be hired as a space station technician .
Problem-solving is considered a skill on its own, but it’s supported by many other skills that can help you be a better problem solver. These skills fall into a few different categories of problem-solving skills.
Problem recognition and analysis. The first step is to recognize that there is a problem and discover what it is or what the root cause of it is.
You can’t begin to solve a problem unless you’re aware of it. Sometimes you’ll see the problem yourself and other times you’ll be told about the problem. Both methods of discovery are very important, but they can require some different skills. The following can be an important part of the process:
Active listening
Data analysis
Historical analysis
Communication
Create possible solutions. You know what the problem is, and you might even know the why of it, but then what? Your next step is the come up with some solutions.
Most of the time, the first solution you come up with won’t be the right one. Don’t fall victim to knee-jerk reactions; try some of the following methods to give you solution options.
Brainstorming
Forecasting
Decision-making
Topic knowledge/understanding
Process flow
Evaluation of solution options. Now that you have a lot of solution options, it’s time to weed through them and start casting some aside. There might be some ridiculous ones, bad ones, and ones you know could never be implemented. Throw them away and focus on the potentially winning ideas.
This step is probably the one where a true, natural problem solver will shine. They intuitively can put together mental scenarios and try out solutions to see their plusses and minuses. If you’re still working on your skill set — try listing the pros and cons on a sheet of paper.
Prioritizing
Evaluating and weighing
Solution implementation. This is your “take action” step. Once you’ve decided which way to go, it’s time to head down that path and see if you were right. This step takes a lot of people and management skills to make it work for you.
Dependability
Teambuilding
Troubleshooting
Follow-Through
Believability
Trustworthiness
Project management
Evaluation of the solution. Was it a good solution? Did your plan work or did it fail miserably? Sometimes the evaluation step takes a lot of work and review to accurately determine effectiveness. The following skills might be essential for a thorough evaluation.
Customer service
Feedback responses
Flexibility
You now have a ton of skills in front of you. Some of them you have naturally and some — not so much. If you want to solve a problem, and you want to be known for doing that well and consistently, then it’s time to sharpen those skills.
Develop industry knowledge. Whether it’s broad-based industry knowledge, on-the-job training , or very specific knowledge about a small sector — knowing all that you can and feeling very confident in your knowledge goes a long way to learning how to solve problems.
Be a part of a solution. Step up and become involved in the problem-solving process. Don’t lead — but follow. Watch an expert solve the problem and, if you pay attention, you’ll learn how to solve a problem, too. Pay attention to the steps and the skills that a person uses.
Practice solving problems. Do some role-playing with a mentor , a professor , co-workers, other students — just start throwing problems out there and coming up with solutions and then detail how those solutions may play out.
Go a step further, find some real-world problems and create your solutions, then find out what they did to solve the problem in actuality.
Identify your weaknesses. If you could easily point out a few of your weaknesses in the list of skills above, then those are the areas you need to focus on improving. How you do it is incredibly varied, so find a method that works for you.
Solve some problems — for real. If the opportunity arises, step in and use your problem-solving skills. You’ll never really know how good (or bad) you are at it until you fail.
That’s right, failing will teach you so much more than succeeding will. You’ll learn how to go back and readdress the problem, find out where you went wrong, learn more from listening even better. Failure will be your best teacher ; it might not make you feel good, but it’ll make you a better problem-solver in the long run.
Once you’ve impressed a hiring manager with top-notch problem-solving skills on your resume and cover letter , you’ll need to continue selling yourself as a problem-solver in the job interview.
There are three main ways that employers can assess your problem-solving skills during an interview:
By asking questions that relate to your past experiences solving problems
Posing hypothetical problems for you to solve
By administering problem-solving tests and exercises
The third method varies wildly depending on what job you’re applying for, so we won’t attempt to cover all the possible problem-solving tests and exercises that may be a part of your application process.
Luckily, interview questions focused on problem-solving are pretty well-known, and most can be answered using the STAR method . STAR stands for situation, task, action, result, and it’s a great way to organize your answers to behavioral interview questions .
Let’s take a look at how to answer some common interview questions built to assess your problem-solving capabilities:
At my current job as an operations analyst at XYZ Inc., my boss set a quarterly goal to cut contractor spending by 25% while maintaining the same level of production and moving more processes in-house. It turned out that achieving this goal required hiring an additional 6 full-time employees, which got stalled due to the pandemic. I suggested that we widen our net and hire remote employees after our initial applicant pool had no solid candidates. I ran the analysis on overhead costs and found that if even 4 of the 6 employees were remote, we’d save 16% annually compared to the contractors’ rates. In the end, all 6 employees we hired were fully remote, and we cut costs by 26% while production rose by a modest amount.
I try to step back and gather research as my first step. For instance, I had a client who needed a graphic designer to work with Crello, which I had never seen before, let alone used. After getting the project details straight, I began meticulously studying the program the YouTube tutorials, and the quick course Crello provides. I also reached out to coworkers who had worked on projects for this same client in the past. Once I felt comfortable with the software, I started work immediately. It was a slower process because I had to be more methodical in my approach, but by putting in some extra hours, I turned in the project ahead of schedule. The client was thrilled with my work and was shocked to hear me joke afterward that it was my first time using Crello.
As a digital marketer , website traffic and conversion rates are my ultimate metrics. However, I also track less visible metrics that can illuminate the story behind the results. For instance, using Google Analytics, I found that 78% of our referral traffic was coming from one affiliate, but that these referrals were only accounting for 5% of our conversions. Another affiliate, who only accounted for about 10% of our referral traffic, was responsible for upwards of 30% of our conversions. I investigated further and found that the second, more effective affiliate was essentially qualifying our leads for us before sending them our way, which made it easier for us to close. I figured out exactly how they were sending us better customers, and reached out to the first, more prolific but less effective affiliate with my understanding of the results. They were able to change their pages that were referring us traffic, and our conversions from that source tripled in just a month. It showed me the importance of digging below the “big picture” metrics to see the mechanics of how revenue was really being generated through digital marketing.
You can bring up your problem-solving skills in your resume summary statement , in your work experience , and under your education section , if you’re a recent graduate. The key is to include items on your resume that speak direclty to your ability to solve problems and generate results.
If you can, quantify your problem-solving accomplishments on your your resume . Hiring managers and recruiters are always more impressed with results that include numbers because they provide much-needed context.
This sample resume for a Customer Service Representative will give you an idea of how you can work problem solving into your resume.
Michelle Beattle 111 Millennial Parkway Chicago, IL 60007 (555) 987-6543 [email protected] Professional Summary Qualified Customer Services Representative with 3 years in a high-pressure customer service environment. Professional, personable, and a true problem solver. Work History ABC Store — Customer Service Representative 01/2015 — 12/2017 Managed in-person and phone relations with customers coming in to pick up purchases, return purchased products, helped find and order items not on store shelves, and explained details and care of merchandise. Became a key player in the customer service department and was promoted to team lead. XYZ Store — Customer Service Representative/Night Manager 01/2018 — 03/2020, released due to Covid-19 layoffs Worked as the night manager of the customer service department and filled in daytime hours when needed. Streamlined a process of moving customers to the right department through an app to ease the burden on the phone lines and reduce customer wait time by 50%. Was working on additional wait time problems when the Covid-19 pandemic caused our stores to close permanently. Education Chicago Tech 2014-2016 Earned an Associate’s Degree in Principles of Customer Care Skills Strong customer service skills Excellent customer complaint resolution Stock record management Order fulfillment New product information Cash register skills and proficiency Leader in problem solving initiatives
You can see how the resume gives you a chance to point out your problem-solving skills and to show where you used them a few times. Your cover letter is your chance to introduce yourself and list a few things that make you stand out from the crowd.
Michelle Beattle 111 Millennial Parkway Chicago, IL 60007 (555) 987-6543 [email protected] Dear Mary McDonald, I am writing in response to your ad on Zippia for a Customer Service Representative . Thank you for taking the time to consider me for this position. Many people believe that a job in customer service is simply listening to people complain all day. I see the job as much more than that. It’s an opportunity to help people solve problems, make their experience with your company more enjoyable, and turn them into life-long advocates of your brand. Through my years of experience and my educational background at Chicago Tech, where I earned an Associate’s Degree in the Principles of Customer Care, I have learned that the customers are the lifeline of the business and without good customer service representatives, a business will falter. I see it as my mission to make each and every customer I come in contact with a fan. I have more than five years of experience in the Customer Services industry and had advanced my role at my last job to Night Manager. I am eager to again prove myself as a hard worker, a dedicated people person, and a problem solver that can be relied upon. I have built a professional reputation as an employee that respects all other employees and customers, as a manager who gets the job done and finds solutions when necessary, and a worker who dives in to learn all she can about the business. Most of my customers have been very satisfied with my resolution ideas and have returned to do business with us again. I believe my expertise would make me a great match for LMNO Store. I have enclosed my resume for your review, and I would appreciate having the opportunity to meet with you to further discuss my qualifications. Thank you again for your time and consideration. Sincerely, Michelle Beattle
You’ve no doubt noticed that many of the skills listed in the problem-solving process are repeated. This is because having these abilities or talents is so important to the entire course of getting a problem solved.
In fact, they’re worthy of a little more attention. Many of them are similar, so we’ll pull them together and discuss how they’re important and how they work together.
Communication, active listening, and customer service skills. No matter where you are in the process of problem-solving, you need to be able to show that you’re listening and engaged and really hearing what the problem is or what a solution may be.
Obviously, the other part of this is being able to communicate effectively so people understand what you’re saying without confusion. Rolled into this are customer service skills , which really are all about listening and responding appropriately — it’s the ultimate in interpersonal communications.
Analysis (data and historical), research, and topic knowledge/understanding. This is how you intellectually grasp the issue and approach it. This can come from studying the topic and the process or it can come from knowledge you’ve gained after years in the business. But the best solutions come from people who thoroughly understand the problem.
Creativity, brainstorming, troubleshooting, and flexibility. All of you creative thinkers will like this area because it’s when your brain is at its best.
Coming up with ideas, collaborating with others, leaping over hurdles, and then being able to change courses immediately, if need be, are all essential. If you’re not creative by nature, then having a team of diverse thinkers can help you in this area.
Dependability, believability, trustworthiness, and follow-through. Think about it, these are all traits a person needs to have to make change happen and to make you comfortable taking that next step with them. Someone who is shifty and shady and never follows through, well, you’re simply not going to do what they ask, are you?
Leadership, teambuilding, decision-making, and project management. These are the skills that someone who is in charge is brimming with. These are the leaders you enjoy working for because you know they’re doing what they can to keep everything in working order. These skills can be learned but they’re often innate.
Prioritizing, prediction, forecasting, evaluating and weighing, and process flow. If you love flow charts, data analysis, prediction modeling, and all of that part of the equation, then you might have some great problem-solving abilities.
These are all great skills because they can help you weed out bad ideas, see flaws, and save massive amounts of time in trial and error.
What is a good example of problem-solving skills?
Good examples of porblem-solving skills include research, analysis, creativity, communciation, and decision-making. Each of these skills build off one another to contribute to the problem solving process. Research and analysis allow you to identify a problem.
Creativity and analysis help you consider different solutions. Meanwhile, communication and decision-making are key to working with others to solve a problem on a large scale.
What are 3 key attributes of a good problem solver?
3 key attributes of a good problem solver are persistence, intellegince, and empathy. Persistence is crucial to remain motivated to work through challenges. Inellegince is needed to make smart, informed choices. Empathy is crucial to maintain positive relationships with others as well as yourself.
What can I say instead of problem-solving skills?
Instead of saying problem-solving skills, you can say the following:
Critical thinker
Solutions-oriented
Engineering
Using different words is helpful, especially when writing your resume and cover letter.
What is problem-solving in the workplace?
Problem-solving in the workplace is the ability to work through any sort of challenge, conflict, or unexpected situation and still achieve business goals. Though it varies by profession, roblem-solving in the workplace is very important for almost any job, because probelms are inevitable. You need to have the appropriate level of problem-solving skills if you want to succeed in your career, whatever it may be.
Department of Labor – Problem Solving and Critical Thinking
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Kristin Kizer is an award-winning writer, television and documentary producer, and content specialist who has worked on a wide variety of written, broadcast, and electronic publications. A former writer/producer for The Discovery Channel, she is now a freelance writer and delighted to be sharing her talents and time with the wonderful Zippia audience.
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Published: February 13, 2023
Interview Questions and Answers
Actionable advice from real experts:
Biron Clark
Former Recruiter
Contributor
Dr. Kyle Elliott
Career Coach
Hayley Jukes
Editor-in-Chief
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 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:
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.
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
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.
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.”
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.
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.
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.
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.
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.
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.
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.
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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What’s life without its challenges? All of us will at some point encounter professional and personal hurdles. That might mean resolving a conflict with coworkers or making a big life decision. With effective problem solving skills, you’ll find tricky situations easier to navigate, and welcome challenges as opportunities to learn, grow and thrive.
In this guide, we dive into the importance of problem solving skills and look at examples that show how relevant they are to different areas of your life. We cover how to find creative solutions and implement them, as well as ways to refine your skills in communication and critical thinking. Ready to start solving problems? Read on.
Before we cover strategies for improving problem solving skills, it’s important to first have a clear understanding of the problem solving process. Here are the steps in solving a problem:
There’s more to problem solving than coming up with a quick fix. Effective problem solving requires wide range of skills and abilities, such as:
Problem solving skills in the workplace are invaluable, whether you need them for managing a team, dealing with clients or juggling deadlines. To get a better understanding of how you might use these skills in real-life scenarios, here are some problem solving examples that are common in the workplace.
Analytical thinking is something that comes naturally to some, while others have to work a little harder. It involves being able to look at problem solving from a logical perspective, breaking down the issues into manageable parts.
Quality control: in a manufacturing facility, analytical thinking helps identify the causes of product defects in order to pinpoint solutions.
Market research: marketing teams rely on analytical thinking to examine consumer data, identify market trends and make informed decisions on ad campaigns.
Critical thinkers are able to approach problems objectively, looking at different viewpoints without rushing to a decision. Critical thinking is an important aspect of problem solving, helping to uncover biases and assumptions and weigh up the quality of the information before making any decisions.
Making decisions is often the hardest part of problem solving. How do you know which solution is the right one? It involves evaluating information, considering potential outcomes and choosing the most suitable option. Effective problem solving relies on making well-informed decisions.
Research skills are pivotal when it comes to problem solving, to ensure you have all the information you need to make an informed decision. These skills involve searching for relevant data, critically evaluating information sources, and drawing meaningful conclusions.
A little creative flair goes a long way. By thinking outside the box, you can approach problems from different angles. Creative thinking involves combining existing knowledge, experiences and perspectives in new and innovative ways to come up with inventive solutions.
It’s not always easy to work with other people, but collaboration is a key element in problem solving, allowing you to make use of different perspectives and areas of expertise to find solutions.
Being able to mediate conflicts is a great skill to have. It involves facilitating open communication, understanding different perspectives and finding solutions that work for everyone. Conflict resolution is essential for managing any differences in opinion that arise.
Risk management is essential across many workplaces. It involves analysing potential threats and opportunities, evaluating their impact and implementing strategies to minimise negative consequences. Risk management is closely tied to problem solving, as it addresses potential obstacles and challenges that may arise during the problem solving process.
Effective communication is a skill that will get you far in all areas of life. When it comes to problem solving, communication plays an important role in facilitating collaboration, sharing insights and ensuring that all stakeholders have the same expectations.
Ready to improve your problem solving skills? In this section we explore strategies and techniques that will give you a head start in developing better problem solving skills.
Developing a problem solving mindset will help you tackle challenges effectively . Start by accepting problems as opportunities for growth and learning, rather than as obstacles or setbacks. This will allow you to approach every challenge with a can-do attitude.
Patience is also essential, because it will allow you to work through the problem and its various solutions mindfully. Persistence is also important, so you can keep adapting your approach until you find the right solution.
Finally, don’t forget to ask questions. What do you need to know? What assumptions are you making? What can you learn from previous attempts? Approach problem solving as an opportunity to acquire new skills . Stay curious, seek out solutions, explore new possibilities and remain open to different problem solving approaches.
There’s no point trying to solve a problem you don’t understand. To analyse a problem effectively, you need to be able to define it. This allows you to break it down into smaller parts, making it easier to find causes and potential solutions. Start with a well-defined problem statement that is precise and specific. This will help you focus your efforts on the core issue, so you don’t waste time and resources on the wrong concerns.
Critical thinking and creativity are both important when it comes to looking at the problem objectively and thinking outside the box. Critical thinking encourages you to question assumptions, recognise biases and seek evidence to support your conclusions. Creative thinking allows you to look at the problem from different angles to reveal new insights and opportunities.
Research and decision-making skills are pivotal in problem solving as they enable you to gather relevant information, analyse options and choose the best course of action. Research provides the information and data needed, and ensures that you have a comprehensive understanding of the problem and its context. Effective decision-making is about selecting the solution that best addresses the problem.
Being able to work with others is one of the most important skills to have at work. Collaboration skills enable everyone to work effectively as a team, share their perspectives and collectively find solutions.
Once you’ve overcome a challenge, take the time to look back with a critical eye. How effective was the outcome? Could you have tweaked anything in your process? Learning from past experiences is important when it comes to problem solving. It involves reflecting on both successes and failures to gain insights, refine strategies and make more informed decisions in the future.
Problem-solving tools and resources are a great help when it comes to navigating complex challenges. These tools offer structured approaches, methodologies and resources that can streamline the process.
Practice makes perfect! Using your skills in real life allows you to refine them, adapt to new challenges and build confidence in your problem solving capabilities. Make sure to try out these skills whenever you can.
Effectively showcasing your problem solving skills on your resumé is a great way to demonstrate your ability to address challenges and add value to a workplace. We'll explore how to demonstrate problem solving skills on your resumé, so you stand out from the crowd.
A resumé summary is your introduction to potential employers and provides an opportunity to succinctly showcase your skills. The resumé summary is often the first section employers read. It offers a snapshot of your qualifications and sets the tone for the rest of your resumé.
Your resumé summary should be customised for different job applications, ensuring that you highlight the specific problem solving skills relevant to the position you’re applying for.
Example 1: Project manager with a proven track record of solving complex operational challenges. Skilled in identifying root causes, developing innovative solutions and leading teams to successful project completion.
Example 2: Detail-oriented data analyst with strong problem solving skills. Proficient in data-driven decision-making, quantitative analysis and using statistical tools to solve business problems.
The experience section of your resumé presents the perfect opportunity to demonstrate your problem solving skills in action.
The skills section of your resumé should showcase your top abilities, including problem solving skills. Here are some tips for including these skills.
Including a dedicated section for projects or case studies in your resumé allows you to provide specific examples of your problem solving skills in action. It goes beyond simply listing skills, to demonstrate how you are able to apply those skills to real-world challenges.
Example – Data Analysis
Case Study: Market Expansion Strategy
A well-crafted cover letter is your first impression on any potential employer. Integrating problem solving skills can support your job application by showcasing your ability to address challenges and contribute effectively to their team. Here’s a quick run-down on what to include:
Problem solving skills are essential in all areas of life, enabling you to overcome challenges, make informed decisions, settle conflicts and drive innovation. We've explored the significance of problem solving skills and how to improve, demonstrate and leverage them effectively. It’s an ever-evolving skill set that can be refined over time.
By actively incorporating problem solving skills into your day-to-day, you can become a more effective problem solver at work and in your personal life as well.
Common problem solving techniques include brainstorming, root cause analysis, SWOT analysis, decision matrices, the scientific method and the PDCA (Plan-Do-Check-Act) cycle. These techniques offer structured approaches to identify, analyse and address problems effectively.
Improving critical thinking involves practising skills such as analysis, evaluation and problem solving. It helps to engage in activities like reading, solving puzzles, debating and self-reflection.
Common obstacles to problem solving include biases, lack of information or resources, and resistance to change. Recognising and addressing these obstacles is essential for effective problem solving.
To overcome resistance to change, it's essential to communicate the benefits of the proposed solution clearly, involve stakeholders in the decision-making process, address concerns and monitor the implementation's progress to demonstrate its effectiveness.
Problem solving skills are highly valuable in a career as they enable you to navigate challenges, make informed decisions, adapt to change and contribute to innovation and efficiency. These skills enhance your professional effectiveness and can lead to career advancement and increased job satisfaction.
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Problem-solving skills are important not just for work. In the words of Karl Popper , “All life is problem-solving.”
What on earth does that mean? Simply that being alive means facing challenges. With problem-solving skills, you can navigate issues with greater ease, making hard times, well, less hard.
But what are problem-solving skills? How do you know if you have them or not? Why do they matter to your job search? And what should you do if you don’t feel yours are up to snuff? Luckily, we’re about to get into all of that.
If you’re curious about the world of problem-solving skills, here’s what you need to know.
Before we dig into any examples, let’s focus first on an important question: what are problem-solving skills.
To answer that question, let’s start with the barebones basics. According to Merriam-Webster , problem-solving is “the process or act of finding a solution to a problem.” Why does that matter? Well, because it gives you insight into what problem-solving skills are.
Any skill that helps you find solutions to problems can qualify. And that means problem-solving skills aren’t just one capability, but a toolbox filled with soft skills and hard skills that come together during your time of need.
The ability to solve problems is relevant to any part of your life. Whether your writing a grocery list or dealing with a car that won’t start, you’re actually problem-solving.
The same is true at work, too. Most tasks actually involve a degree of problem-solving. Really? Really.
Think about it this way; when you’re given an assignment, you’re being asked, “Can you do this thing?” Doing that thing is the problem.
Then, you have to find a path that lets you accomplish what you need to do. That is problem-solving.
Yes, sometimes what you need to handle isn’t “challenging” in the difficulty sense. But that doesn’t mean it doesn’t count.
Besides, some of what you need to do will legitimately be hard. Maybe you’re given a new responsibility, or something goes wrong during a project. When that happens, you’ll have to navigate unfamiliar territory, gather new information, and think outside of the box. That’s problem-solving, too.
That’s why hiring managers favor candidates with problem-solving skills. They make you more effective in your role, increasing the odds that you can find solutions whenever the need arises.
Alright, you probably have a good idea of what problem-solving skills are. Now, it’s time to talk about why they matter to your job search.
We’ve already touched on one major point: hiring managers prefer candidates with strong problem-solving skills. That alone makes these capabilities a relevant part of the equation. If you don’t show the hiring manager you’ve got what it takes to excel, you may struggle to land a position.
But that isn’t the only reason these skills matter. Problem-solving skills can help you during the entire job search process. After all, what’s a job search but a problem – or a series of problems – that needs an answer.
You need a new job; that’s the core problem you’re solving. But every step is its own unique challenge. Finding an opening that matches your skills, creating a resume that resonates with the hiring manager, nailing the interview, and negotiating a salary … those are all smaller problems that are part of the bigger one.
So, problem-solving skills really are at the core of the job search experience. By having strong capabilities in this area, you may find a new position faster than you’d expect.
Okay, you may be thinking, “If hiring managers prefer candidates with problem-solving skills, which ones are they after? Are certain problem-solving capabilities more important today? Is there something I should be going out of my way to showcase?”
While any related skills are worth highlighting, some may get you further than others. Analysis, research, creativity, collaboration , organization, and decision-making are all biggies. With those skills, you can work through the entire problem-solving process, making them worthwhile additions to your resume.
But that doesn’t mean you have to focus there solely. Don’t shy away from showcasing everything you bring to the table. That way, if a particular hiring manager is looking for a certain capability, you’re more likely to tap on what they’re after.
At this point, it’s ridiculously clear that problem-solving skills are valuable in the eyes of hiring managers. So, how do you show them that you’ve got all of the capabilities they are after? By using the right approach.
When you’re writing your resume or cover letter , your best bet is to highlight achievements that let you put your problem-solving skills to work. That way, you can “show” the hiring manager you have what it takes.
Showing is always better than telling. Anyone can write down, “I have awesome problem-solving skills.” The thing is, that doesn’t really prove that you do. With a great example, you offer up some context, and that makes a difference.
How do you decide on which skills to highlight on your resume or cover letter? By having a great strategy. With the Tailoring Method , it’s all about relevancy. The technique helps you identify skills that matter to that particular hiring manager, allowing you to speak directly to their needs.
Plus, you can use the Tailoring Method when you answer job interview questions . With that approach, you’re making sure those responses are on-point, too.
But when do you talk about your problem-solving capabilities during an interview? Well, there’s a good chance you’ll get asked problem-solving interview questions during your meeting. Take a look at those to see the kinds of questions that are perfect for mentioning these skills.
However, you don’t have to stop there. If you’re asked about your greatest achievement or your strengths, those could be opportunities, too. Nearly any open-ended question could be the right time to discuss those skills, so keep that in mind as you practice for your interview.
Developing problem-solving skills may seem a bit tricky on the surface, especially if you think you don’t have them. The thing is, it doesn’t actually have to be hard. You simply need to use the right strategy.
First, understand that you probably do have problem-solving skills; you simply may not have realized it. After all, life is full of challenges that you have to tackle, so there’s a good chance you’ve developed some abilities along the way.
Now, let’s reframe the question and focus on how to improve your problem-solving skills. Here’s how to go about it.
In many cases, problem-solving is all about the process. You:
By understanding the core process, you can apply it more effectively. That way, when you encounter an issue, you’ll know how to approach it, increasing the odds you’ll handle the situation effectively.
Any activity that lets you take the steps listed above could help you hone your problem-solving skills. For example, brainteasers, puzzles, and logic-based games can be great places to start.
Whether it’s something as straightforward – but nonetheless challenging – as Sudoku or a Rubik’s Cube, or something as complex as Settlers of Catan, it puts your problem-solving skills to work. Plus, if you enjoy the activity, it makes skill-building fun, making it a win-win.
If you’re looking for a practical approach, you’re in luck. You can also look at the various challenges you face during the day and think about how to overcome them.
For example, if you always experience a mid-day energy slump that hurts your productivity, take a deep dive into that problem. Define what’s happening, think about why it occurs, consider various solutions, pick one to try, and analyze the results.
By using the problem-solving approach more often in your life, you’ll develop those skills further and make using these capabilities a habit. Plus, you may find ways to improve your day-to-day living, which is a nice bonus.
If you’re currently employed, volunteering for projects that push you slightly outside of your comfort zone can help you develop problem-solving skills, too. You’ll encounter the unknown and have to think outside of the box, both of which can boost critical problem-solving-related skills.
Plus, you may gain other capabilities along the way, like experience with new technologies or tools. That makes the project an even bigger career booster, which is pretty awesome.
Alright, we’ve taken a pretty deep dive into what problem-solving skills are. Now, it’s time for some problem-solving skills examples.
As we mentioned above, there are a ton of capabilities and traits that can support better problem-solving. By understanding what they are, you can showcase the right abilities during your job search.
So, without further ado, here is a quick list of problem-solving skill examples:
Do you have to showcase all of those skills during your job search individually? No, not necessarily. Instead, you want to highlight a range of capabilities based on what the hiring manager is after. If you’re using the Tailoring Method, you’ll know which ones need to make their way into your resume, cover letter, and interview answers.
Now, are there other skills that support problem-solving? Yes, there certainly can be.
Essentially any skill that helps you go from the problem to the solution can, in its own right, be a problem-solving skill.
All of the skills above can be part of the equation. But, if you have another capability that helps you flourish when you encounter an obstacle, it can count, too.
Reflect on your past experience and consider how you’ve navigated challenges in the past. If a particular skill helped you do that, then it’s worth highlighting during a job search.
If you would like to find out more about skills to put on a resume , we’ve taken a close look at the topic before. Along with problem-solving skills, we dig into a variety of other areas, helping you choose what to highlight so that you can increase your odds of landing your perfect job.
Ultimately, problem-solving skills are essential for professionals in any kind of field. By honing your capabilities and showcasing them during your job search, you can become a stronger candidate and employee. In the end, that’s all good stuff, making it easier for you to keep your career on track today, tomorrow, and well into the future.
Co-Founder and CEO of TheInterviewGuys.com. Mike is a job interview and career expert and the head writer at TheInterviewGuys.com.
His advice and insights have been shared and featured by publications such as Forbes , Entrepreneur , CNBC and more as well as educational institutions such as the University of Michigan , Penn State , Northeastern and others.
Learn more about The Interview Guys on our About Us page .
Mike simpson.
Co-Founder and CEO of TheInterviewGuys.com. Mike is a job interview and career expert and the head writer at TheInterviewGuys.com. His advice and insights have been shared and featured by publications such as Forbes , Entrepreneur , CNBC and more as well as educational institutions such as the University of Michigan , Penn State , Northeastern and others. Learn more about The Interview Guys on our About Us page .
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The importance of cognitive load and levels of expertise, scaffolding problem structure, impact of a diagram and levels of learner expertise, the present study.
According to cognitive load theory, an expert learner can access prior knowledge to interpret a diagram with minimum effort. A novice learner who lacks prior knowledge, in contrast, would experience a high level of cognitive load when processing a diagram, which would interfere with learning. On learning to solve percentage problems, we investigated the effect of approach (non-algebra vs. algebra), with and without diagram support for two levels of learner expertise. Participants were 218 Asian students whose mean age was 15.00 ( SD = 0.18). Contrary to our hypothesis, the presence or absence of a diagram may bore little, if any, benefit for students who were more knowledgeable. Considering two levels of learner expertise together, as hypothesized, performance outcomes favoring more knowledgeable students were greater for the unitary-pictorial approach than the equation-pictorial approach. Interestingly, the impact of a diagram was more pronounced for more knowledgeable students; however, with relevant prior algebra knowledge, the presence of a diagram (e.g., equation-pictorial approach) can also be helpful for less knowledgeable students. Overall, then, our research undertaking has yielded important implications for teaching and research purposes.
Do we need to incorporate a diagram for effective learning of word problems? To what extent does incorporation of a diagram affect learners who have varying levels of expertise? Would the type of instructional approach (non-algebra vs. algebra) differentially affect learning outcomes for learners with varying levels of expertise? Consistent with prior research (Jitendra et al., Citation 2011 ; Mayer & Gallini, Citation 1990 ; Ngu et al., Citation 2018 ), we define a diagram or a picture as a visual representation that depicts the relationship between values and a variable cited in the problem text, which reflects the ‘problem structure’ of a word problem. A percentage problem such as “ Mary pays $350 in tax per month, which is 8% of her monthly income. What is her monthly income?” poses a challenge not only to middle school students (Baratta et al., Citation 2010 ) but also to the pre-service teachers (Koay, Citation 1998 ). Our examination of the literature indicates that the Australian Curriculum actually recommends the use of what is known as ‘the unitary approach’. The unitary approach emphasizes the unit ‘percentage’ concept for learning to solve such percentage problems ( Table 1 )—for example: “… calculate 1% of the monthly income, and then multiply this by 100% to solve the problem”. Our examination also showcases an interesting contention: mathematics textbooks in Singapore (e.g., Chow, Citation 2007 ) advocate the use of the ‘algebra approach’ (8% x = $350, solve for x ). What is poignant, though, is that neither the unitary approach nor the algebra approach involves the use of diagrams as visual representations to assist learners to uncover the problem structure that is embedded in the problem text.
Researchers have acknowledged the benefits of using diagrams to facilitate learning of word problems (Booth & Koedinger, Citation 2012 ; Jitendra et al., Citation 2011 ; Nathan et al., Citation 1992 ; Ngu et al., Citation 2009 ; Citation 2018 ); Schwonke et al., Citation 2009 ). However, the treatment condition differs from the control condition not only with respect to the diagram only (Jitendra et al., Citation 2011 ); thus, the presence of a diagram to facilitate learning of word problems is unclear. Moreover, from our understanding, the relationship between diagram support and levels of expertise is inconclusive. Existing research indicates that diagram support benefits both low prior knowledge students (Booth & Koedinger, Citation 2012 ) and high prior knowledge students (Schwonke et al., Citation 2009 ). Having said this, however, very little is known about the relationship between type of approach (i.e., non-algebra vs. algebra) and levels of expertise when one learns a specific type of word problems.
The significance of the present research undertaking lies in our attempt to address the limitations of the literature. Using an experimental study approach, our research investigation ( N = 218 students) in regular classrooms seeks to examine the relationship between type of approach (i.e., unitary vs. unitary-pictorial [non-algebra]; equation vs. equation-pictorial [algebra]) and levels of learner expertise (i.e., less knowledgeable vs. more knowledgeable) for learning to solve percentage problems. We argue that a comparison of a dichotomy between two versions of a specific approach (e.g., unitary vs. unitary-pictorial [non-algebra]) in which we examine one variable only (i.e., without a diagram vs. a diagram) would provide more accurate assessment of the benefit of incorporating a diagram to facilitate learning of percentage problems across varying levels of learner expertise. Moreover, we also want to investigate whether the type of approach (non-algebra vs. algebra) would differentially affect performance outcomes across two levels of learner expertise on learning to solve percentage problems. Given that the present study focuses on instructional designs and variations in learner expertise, we choose to situate our study within the framework of cognitive load theory (Sweller et al., Citation 2011 , Citation 2019 ).
There are different theories or theoretical premises in education (e.g., behaviorism , constructivism ) that we may use to explain quality teaching and/or to facilitate effective learning experiences (Kolb et al., Citation 2001 ; Schunk, Citation 2008 ; Wiest, Citation 1967 ). One notable learning theory that has received considerable interest from scholars worldwide is John Sweller’s ( Citation 1994 , Citation 2005 , Citation 2012 ) cognitive load theory , which details the intricacy of what is known as the ‘human cognitive architecture’ that affects how people learn. Our human cognitive architecture consists of a working memory , which is constrained by its capacity to process new information (Cowan, Citation 2001 ; Peterson & Peterson, Citation 1959 ). In contrast, though, a person’s long-term memory has an unlimited capacity to store information in the form of what is known as ‘schemas’ (Tricot & Sweller, Citation 2014 ). Owing to the limitation of a person’s working memory, an instructional design that has multiple interrelated elements (e.g., integration of a diagram and solution procedure) would impose a high level of cognitive load for a novice learner, which in turn would interfere with learning. An element is anything that requires learning (e.g., a number, a symbol, a procedure) (Chen et al., Citation 2017 ). An expert learner who has prior knowledge, in contrast, would experience a low level of cognitive load. In this analysis, he or she can retrieve prior knowledge from the long-term memory and use it to process multiple interrelated elements as a single unit of element with minimum working memory resources (Kalyuga et al., Citation 2003 ). In other words, unlike novice learners, learners who are more knowledgeable may treat multiple interrelated elements as a single unit. For example, consider the following percentage-increase problem: “ Katie works at Hungry Jack and she earns $20 an hour. She will receive a pay rise of 5% starting next week. What will be Katie new pay for an hour?” A knowledgeable student who has relevant prior knowledge would solve this problem as follows: $20 + ($20 × 5%) = $21. As such, having a diagram that specifically aims to scaffold the problem structure (i.e., “new pay = original pay + increased amount”) would not add value and/or useful information for the knowledgeable student to solve the percentage-increase problem.
According to Mayer ( Citation 1985 ), the greatest hurdle to master a word problem lies in the problem representation, which requires a diagram to act as a visual aid to represent the underlying problem structure. Considering Mayer’s ( Citation 1985 ) acknowledgment, how do we design a diagram that could accurately capture the problem structure of a word problem? In a study involving elementary school students in Singapore, Ng & Lee ( Citation 2009 ) advocated the use of what is known as the ‘model method’ in order to assist with the representation of a word problem. A key aspect of the model method is to use the relative size of say, rectangles, to represent different quantities in problem text, resulting in the generation of different types of problem solution. For example, two rectangles that have identical dimensions are regarded as two units, and each of which can assume the role of a variable (e.g., x ).
The theoretical tenet of the model method (Ng & Lee, Citation 2009 ) has been explored by a number of researchers. For example, following on from Ng & Lee’s ( Citation 2009 ) premise, Chu et al. ( Citation 2017 ) adapted the model method to assist 7 th grade students to learn how to solve linear equations. Researchers have also recognized the importance of designing a diagram that reveals the problem structure of a word problem, as opposed to using a pictorial representation to display the storyline of a word problem (Hegarty & Kozhevnikov, Citation 1999 ). Designing a diagram to highlight a problem structure is more beneficial for learning than using it to indicate a storyline of a word problem. For example, designing a tree diagram to highlight the concept of probability may, in fact, help to enhance students’ understanding of probability (e.g., a tree diagram shows the probability of ¼ and ¾ as two branches originating from a node, which will then add up to 1 or 100%)(Schwonke et al., Citation 2009 ).
Researchers such as Jitendra et al. ( Citation 2011 ) have advocated the benefit of using schema-based instructions for learning word problems. The main premise here is to use a schematic diagram to capture the quantitative relationship between values and a variable that are embedded in the problem contexts. Jitendra et al.’s ( Citation 2011 ) research has noted that schema-based instructions are beneficial to both main stream students and students with learning difficulties (Jitendra et al., Citation 2016 ). In an earlier study, Dole ( Citation 2000 ) used a dual-scale number line and found that this successfully improved students’ performance outcomes for three types of percentage problem that are interrelated: (i) find part (25% of 60 = x ), (ii) find percent ( x % of 60 = 15), and (iii) find-whole (25% of x = 15). A dual-scale number line illustrates a dichotomy where one side denotes the percentage (%) and the other side, in contrast, denotes an unknown quantity is similar to the schematic diagrams that Jitendra et al. ( Citation 2011 ) proposed. Moreover, Dole’s ( Citation 2000 ) use of a dual-scale number line specifically portrays the problem structures of the three types of percentage problem in terms of the proportion concept.
Other studies including our own undertakings (e.g., Ngu et al., Citation 2014 ; Citation 2016 ) have also used diagrams to scaffold the problem structures of word problems (e.g., percentage-change problems). For example, in one of our earlier studies (Ngu et al., Citation 2014 ), we provided a diagram that consisted of a horizontal line divided into two portions by a vertical line, depicting the increased or decreased quantity as a fraction of the original quantity. The incorporation of such a diagram has helped students with their learning of percentage-change problems across different cultural settings (Ngu et al., Citation 2014 ; Citation 2016 ). Furthermore, our research has yielded evidence to showcase that the presence of a diagram assisted students with their learning of challenging percentage-change problems that have an unknown variable (e.g., x ) appearing twice in the equation (e.g., 72 = x + 15% x , solve for x )(Koedinger et al., Citation 2008 ).
Overall, then, as the preceding sections have shown, there is support for the inclusion and use of diagrams when learning to solve word problems (e.g., Jitendra et al., Citation 2011 ). In particular, the information portrayed in the diagrams have assisted students to formulate equations to solve word problems. In fact, this is the algebra approach which has been shown to be a powerful means for solving word problems (Kieran, Citation 1992 ). However, the merit of incorporating a diagram for learning to solve word problems goes beyond the context of algebra problem-solving. For example, Ngu et al. ( Citation 2018 ) have advanced this line of inquiry by incorporating a diagram in the unitary-pictorial approach for learning to solve challenging percentage-change problems. The diagram depicts the proportion concept that aligns percentage (%) and quantity—provides a clue to calculate the sub-goal of the unit percentage, which is central to the unitary-pictorial approach. In the present study, we used this unitary-pictorial approach (Ngu et al., Citation 2018 ) for a different type of percentage problems - the find-whole percentage problems.
While researchers have acknowledged the benefit of using diagrams to understand problem structures, the relation between the impact of a diagram and levels of learner expertise is somewhat less clear. For example, how do students with expertise knowledge differ from their novice counterparts in terms of utilization of relevant diagrams? A study by Mayer & Gallini ( Citation 1990 ) indicated that pictorial support aided students’ comprehension and understanding of science instructional texts, and that the benefit was greater for low prior knowledge learners. In a similar vein, Mayer ( Citation 1997 ) noted that pictorial support assisted students with low domain knowledge to understand the significance of multimedia materials. A study by Booth & Koedinger ( Citation 2012 ) further supports the beneficial effect(s) of incorporating diagrams for learning to solve word problems, especially for students of lower ability. In particular, exposure to diagrams enabled students of lower ability to perform better and to make fewer conceptual errors than those students who received instructions without the aid of diagrams.
An important issue for consideration relates to the challenge of building referential connections between textual material and its corresponding pictorial representation. This is particularly the case for novice learners whose domain knowledge is somewhat fragmented (Cox & Brna, Citation 1995 ; Hannus & Hyönä, Citation 1999 ; Stern et al., Citation 2003 ). They have limited ability to connect specific structural elements and their relation portrayed in a graph to corresponding information in the problem text. On learning to solve probability word problems, Schwonke et al. ( Citation 2009 ) found that the presence of diagrams was more helpful for students who had high rather than low prior knowledge. Apparently, as assessed by gaze data, active processing of diagrams increases conceptual understanding of the problem structure. Schwonke et al.’s ( Citation 2009 ) findings are similar to those of Hannus & Hyönä’s ( Citation 1999 ) findings, which showed that high-ability students spent more time on relevant aspects of not only the text but also the diagram than low-ability students.
Overall, research has obtained mixed findings in regard to the impact of a diagram and levels of learner expertise upon learning to solve word problems. Some studies show promising results for low prior knowledge students (e.g., Booth & Koedinger, Citation 2012 ), whereas other studies are in favor of high prior knowledge students (e.g., Schwonke et al., Citation 2009 )—for example, high-ability students experience greater learning gains following more effortful processing of texts that are coupled with diagram support. Such discrepancies, in fact, form logical grounding for our current research undertaking. Moreover, in line with existing research development, our design of diagrams emphasizes the proportion concept and seeks to capture the problem structure of percentage problems (e.g., Jitendra et al., Citation 2011 ). In the next section, we outline the design of the four instructional approaches: unitary approach , unitary-pictorial approach , equation approach , equation-pictorial approach . As we discuss later, we wish to explore the algebra approach and the non-algebra approach for learning, with and without diagram support. Table 1 contains examples of all four mentioned instructional approaches.
Step 1: identify the values cited in the problem context.
Step 2: calculate a unit percentage, which is the sub-goal of the solution procedure.
Step 3: multiply the sub-goal by 100% to obtain the answer.
This unitary approach does not have a diagram to scaffold the problem structure, which is the proportion concept. Therefore, irrespective of a student’s level of expertise (e.g., the student could be a novice), he may in fact struggle to use the unitary approach to solve percentage problems.
The main difference between the unitary-pictorial approach and the unitary approach is the fact that the former has a diagram. The diagram scaffolds the proportion concept, which is central to the problem structure of percentage problems ( Table 1 ). Specifically, the diagram aligns the percentage with the quantity: 8% with $350, and 100% with monthly income (i.e., the solution). As a result, the diagram provides a clue for students to calculate the sub-goal, which is the unit percentage. Having calculated the sub-goal, the monthly income will be: sub-goal × 100. Similar to the model designed by Ng & Lee ( Citation 2009 ), we expect to find that a diagram would better assist those students that are more knowledgeable to gain insight into the relative size of the quantities in relation to differential percentage (i.e., 8% aligns with a smaller quantity than that of 100%).
Step1: the use of a variable (e.g., x ) to represent, say, the monthly income.
Step 2: the integration of values and the variable in an equation – for example: 8% × x = $350.
Step 3: demonstrating how to solve for the unknown (e.g., x ).
Again, regardless of a student’s level of expertise, he may struggle to identify the problem structure of the percentage problem in the problem context without diagram support.
Consider two levels of learner expertise separately, for less knowledgeable students, differential performance outcomes would not occur: (i) between the unitary approach and the unitary-pictorial approach, and (ii) between the equation approach and the equation-pictorial approach.
Consider two levels of learner expertise separately, for more knowledgeable students, performance outcomes would: (i) favor the unitary-pictorial approach over the unitary approach, and (ii) favor the equation-pictorial approach over the equation approach.
Consider two levels of learner expertise together, performance outcomes favoring students who are more knowledgeable would be more pronounced for the unitary-pictorial approach than the unitary approach.
Consider two levels of learner expertise together, performance outcomes favoring students who are more knowledgeable would be more pronounced for the equation-pictorial approach than the equation approach.
Overall, then, the present study is novel for its use of comparable and contrasting instructional approaches to examine students’ learning of percentage problems. Significant in this case is our proposed premise, contextualized within the framework of cognitive load theory (Sweller, Citation 2012 ; Sweller et al., Citation 2011 ), which consists of the use of visualizations to scaffold the problem structure of percentage problems. What makes the proposed research inquiry unique is our focus on variations of existing knowledge of the subject matter—for example, what would be the relative efficiency of the unitary approach and the unitary-pictorial approach for a student who has limited knowledge of percentage problems?
Drawing from two private secondary schools, 218 Asian students (boys = 49%, girls = 51%) whose mean age was 15.00 ( SD = 0.18) consented to participate in the study. English language was the medium of instruction and students followed National Curriculum for Secondary School Mathematics. The mathematics teachers indicated that students had learned percentage problems in previous year via the unitary approach and the equation approach.
The materials comprise a pretest that has identical content as the post-test, an instruction sheet, and acquisition problems. The pretest (or post-test) consists of 10 simple problems and two complex problems. The simple problems resemble the acquisition problems given that they share similar problem structure (Reed, Citation 1987 ). There are two complex problems where the first complex problem has two parts, and the second complex problem has three parts. Students needed to adapt the solution procedure of the simple problems in order to solve these complex problems. We assigned one mark for a practice problem, a simple problem, and each part of a complex problem solved correctly irrespective of whether students provided solution steps. We disregarded computational errors. We assigned zero mark if students made errors in the procedural steps.
The instruction sheet for each approach comprises the definition of percentage, which is common across the four approaches, and review of prior knowledge and a worked example ( Table 1 ). The review of prior knowledge emphasizes specific concepts pertaining to a particular approach. For the unitary approach and the unitary-pictorial approach, the focus is on the unit percentage in relation to proportion concept (e.g., If 6 kg of apples cost $30, what is the cost of 1 kg of apples? ). For the equation approach and the equation-pictorial approach, the review highlights the concept of variable and equation solving skills. For example, placing 2 x = 6 and 10% x = 200 side-by-side so that students could compare and identify the same method to solve both linear equations.
The acquisition problems for each approach consists of six worked example—practice problem pairs. The implementation of multiple worked example—practice problem pairs to facilitate learning was in line with prior studies (e.g., van Gog et al., Citation 2011 ). Each pair consists of a worked example pairs with a practice problem that shares a similar problem structure. Students needed to study each worked example and solved a practice problem.
Two researchers together with three mathematics teachers in each school implemented the data collection. We obtained ethics clearance (Approval No. HE15-314) prior to data collection. We randomly assigned students from each school to four groups and they completed the intervention in a specific venue in the school. We did not include five students who failed to complete all test materials in the final data analysis.
We applied the same experimental procedure across the two schools. We informed the students that they were going to learn percentage problems. We further informed students that the procedure consisted of a few written tasks: a pretest (10 min), an acquisition phase that comprised an instruction sheet (5 min) and acquisition problems (15 min), and a post-test (10 min). We advised students to read the written instruction in the first page of each task before they began, and to seek help if they had trouble understanding the materials in the acquisition phase. Furthermore, we instructed students to complete each task individually and not to discuss the tasks with their classmates. We asked students to try their best to complete the written tasks and informed them that the written tasks would not contribute toward their school mathematics assessment. We distributed and collected each task after the time had expired with one exception—we collected the instruction sheet after the acquisition phase.
Firstly, students across the four approaches sat for a pretest. Secondly, students in the respective approach completed the acquisition phase in which they studied an instruction sheet and completed six pairs of worked example—practice problem. During the acquisition phase, students were allowed to access the instruction sheet while solving the practice problems. We did not assist students to solve the practice problems. Lastly, all students completed a post-test. In sum, students across the four groups were matched with same time and materials to complete the intervention. The only difference between the four groups was the design of the instructional approach.
The pretest had a Cronbach’s alpha of .91. The practice problems had a Cronbach’s alpha of .69 after deleting the 6 th practice problem, because only a few students attempted this practice problem. The post-test had respective Cronbach’s alpha of .83 and .87 for the simple problems and the complex problems. Prior studies have used students who studied at different year levels (e.g., Year 8 vs. Year 9)(Bokosmaty et al., Citation 2015 ) and mean scores of pretests (Blayney et al., Citation 2016 ; Ngu et al., Citation 2023 ) as relevant indexes to differentiate students’ levels of prior knowledge. Accordingly, we used the mean scores of a pretest as a point of reference to classify students into two levels of expertise (Blayney et al., Citation 2016 ). In this case, to ensure a sufficient gap between less and more knowledgeable students, we assigned those students whose mean scores were less than .15 as ‘less knowledgeable’ students ( N = 71), and the rest as ‘more knowledgeable’ students ( N = 142).
Table 2 presents the means and standard deviations of the practice problems, and the post-test that comprised both the simple problems and the complex problems. We used a 2 × 2 ANOVA factorial design to examine the effect of instructional approach (i.e., unitary vs. unitary-pictorial; equation vs. equation-pictorial) and learner expertise (i.e., less knowledgeable vs. more knowledgeable) on learning to solve percentage problems. The independent variables are instructional approach and learner expertise, whereas the dependent variables are the practice problems, and the simple problems and the complex problems in the post-test. We used pairwise comparison to examine performance outcomes within an individual level of learner expertise (i.e., less knowledgeable or more knowledgeable) and between two levels of expertise (i.e., less knowledgeable vs. more knowledgeable).
The unitary approach and the unitary-pictorial approach.
We performed 2 (approach) × 2 (learner expertise) ANOVAs to analyze the mean scores of the practice problems, and the simple problems and the complex problems in the post-test. As shown in Table 3 , a statistically significant main effect of learner expertise was found; however, neither the main effect of the approach nor the approach × learner expertise interaction effect was statistically significant.
When we consider the two levels of learner expertise separately:
Differential performance outcomes between the unitary approach and the unitary-pictorial approach were not observed across the practice problems, and the simple problems and thecomplex problems, irrespective of students’ level of knowledge (i.e., less knowledgeable or more knowledgeable students). Such results support hypothesis 1 but not support hypothesis 2, which stated that performance outcomes would favor the unitary-pictorial approach for more knowledgeable students.
For the practice problems, performance outcomes favored more knowledgeable students for the unitary-pictorial approach ( p = 0.01) but not the unitary approach ( p = 0.06). As shown in Figure 1 , differential performance outcomes between the less and more knowledgeable students were greater for the unitary-pictorial approach ( M = 0.73 vs. 0.89, Cohen’s d = 0.63) than the unitary approach ( M = 0.78 vs. 0.89, Cohen’s d = 0.53). Such results, in this case, support hypothesis 3.
For the simple problems, a similar pattern of results emerged. In support of hypothesis 3, more knowledgeable students outperformed less knowledgeable students for the unitary approach ( p <.001) as well as the unitary-pictorial approach ( p <.001). As indicated in Figure 1 , performance outcomes favoring more knowledgeable students on simple problems were greater for the unitary-pictorial approach ( M = 0.64 vs. M = 0.89, Cohen’s d = 0.94) than the unitary approach M = 0.72 vs. M = 0.92, Cohen’s d = 0.75). However, differences in effect sizes for the unitary-pictorial approach vs. the unitary approach were rather small.
For the complex problems, again, a similar pattern of results occurred. The more knowledgeable students outperformed the less knowledgeable students for the unitary approach ( p = 0.04) and the unitary-pictorial approach ( p <.001), thus supporting hypothesis 3. As revealed in Figure 1 , performance outcomes for the complex problems in favor of more knowledgeable students were greater for the unitary-pictorial approach ( M = 0.01 vs. M = 0.37, Cohen’s d = 1.53) than the unitary approach ( M = 0.09 vs. M = 0.27, Cohen’s d = 0.65).
Figure 1. 2 (approach) × 2 (learner expertise) ANOVA.
Note . Regarding the practice problems, simple problems and complex problems in the post-test: (i) the unitary-pictorial approach was not better than the unitary approach irrespective of less knowledgeable students (support hypothesis) or more knowledgeable students (not support hypothesis), and (ii) differential performance outcomes favored more knowledgeable were greater for the unitary-pictorial approach than the unitary approach (support hypothesis).
As shown in Table 3 , the two-way ANOVA results for the practice problems, the simple problems and the complex problems of the equation approach and the equation-pictorial approach mirror that of the unitary approach and the unitary-pictorial approach. A significant main effect of the learner expertise was found. In contrast, both the main effect of approach and the approach × learner expertise interaction effect were nonsignificant.
Differential performance outcomes between the equation approach and the equation-pictorial approach were not observed for the practice problems, the simple problems, and the complex problems, irrespective of students’ level of knowledge (i.e., less knowledgeable or more knowledgeable students). Thus, the results support hypothesis 1 (ii) but not hypothesis 2 (ii), which stated that performance outcomes would favor the equation-pictorial approach for more knowledgeable students.
For the practice problems, as shown in Figure 2 , differential performance outcomes between less and more knowledgeable students were relatively small across the equation approach and the equation-pictorial approach. Thus, from this account, hypothesis 4 is not supported.
For the simple problems, contrary to hypothesis 4, as displayed in Figure 2 , differential performance outcomes between less knowledgeable and more knowledgeable students were greater for the equation approach ( M = 0.74 vs. M = 0.91, Cohen’s d = 0.62) than the equation-pictorial approach ( M = 0.86 vs. M = 0.95, Cohen’s d = 0.48). Indeed, more knowledgeable students outperformed less knowledgeable students for the equation approach ( p = 0.01) but not the equation-pictorial approach ( p = 0.14). It appears that the presence of a diagram in the equation-pictorial approach provided equivalent learning benefit for both less knowledgeable and more knowledgeable students. This may be due to the algebra foundation of less knowledgeable students, which appeared to be on par with more knowledgeable students.
For the complex problems, the more knowledgeable students outperformed less knowledgeable students for the equation approach ( p = 0.01) as well as the equation-pictorial approach ( p = 0.01). As revealed in Figure 2 , differential performance outcomes between less and more knowledgeable students for the complex problems were almost the same across the equation approach ( M = 0.14 vs. M = 0.36, Cohen’s d = 0.84) and the equation-pictorial approach ( M = 0.11 vs. M = 0.32, Cohen’s d = 0.81). Thus, the results partially support hypothesis 4, which stated that performance outcomes favoring more knowledgeable students would be greater for the equation-pictorial approach.
Figure 2. 2 (approach) × 2 (learner expertise) ANOVA.
Note . (i) For the practice problems, simple problems and complex problems, the equation-pictorial approach was not better than the equation approach irrespective of less knowledgeable students (support hypothesis) or more knowledgeable students (not support hypothesis), and (ii) differential performance outcomes did not favor more knowledgeable students across the equation approach and the equation-pictorial approach for the practice problems (not support hypothesis), differential performance outcomes on simple problems favored more knowledgeable students for the equation approach (contradict hypothesis), and differential performance outcomes between less and more knowledgeable students for the complex problems were almost the same across the equation approach and the equation-pictorial approach (partially support hypothesis).
The uniqueness of the present study lies in our seeking to elucidate theoretical understanding of the relative effectiveness of four instructional approaches with the inclusion or exclusion of diagrams. Does the provision of a diagram help students (e.g., those with limited prior knowledge) to successfully solve percentage problems? Our focus here is to consider a comparison between students of less knowledgeable and those of more knowledgeable when engaging in two different types of instructional design: the non-algebra approach (i.e., unitary, unitary-pictorial) vs. the algebra approach (i.e., equation, equation-pictorial). Overall, then, as reported in the preceding sections, findings that we have ascertained make meaningful theoretical and practical contributions to the study of instructional practice and level of learner expertise.
On learning to solve word problems, as the extant literatures have shown that diagram support (e.g., the use of visual representations) may provide useful scaffold to help students uncover problem structures that often are embedded in problem texts. Our research undertaking, as reported earlier, indicates some interesting evidence and theoretical insights for consideration. Considering two levels of learner expertise separately, irrespective of students’ level of knowledge (i.e., less knowledgeable vs. more knowledgeable), we find that the unitary-pictorial approach and the equation-pictorial approach were not ‘better’ than the unitary approach and the equation approach, respectively. This finding is unexpected and somewhat of a surprise, contradicting one of our original hypotheses, which stated that incorporation of a diagram as a visual aid would have a greater impact on students who are more knowledgeable (Hannus & Hyönä, Citation 1999 ; Schwonke et al., Citation 2009 ).
In line with the design principles of cognitive load theory (Sweller et al., Citation 2011 ; Citation 2019 ), it is plausible for more knowledgeable students to take advantage of their prior knowledge of a unit concept to successfully solve percentage problems, using the unitary approach with or without the aid of diagrams (Schwonke et al., Citation 2009 ). Similarly, prior algebra problem-solving experience could have enabled more knowledgeable students to use either the equation approach or the equation-pictorial approach to solve the percentage problems efficiently. As a result, visual aids such as diagrams do not make discernible learning effects for more knowledgeable students when learning to solve percentage problems. Again, in line with the premise of cognitive load theory, students who lack relevant prior knowledge are less likely to know how to interpret diagrams for the purpose of generating solutions for the percentage problems (Cox & Brna, Citation 1995 ; Hannus & Hyönä, Citation 1999 ; Stern et al., Citation 2003 ).
Interestingly, viewing the two levels of learner expertise together, we find that students who were more knowledgeable actually outperformed students who were less knowledgeable for the non-algebra approaches (i.e., unitary, unitary-pictorial) and, to a lesser extent, the algebra approaches (i.e., equation, equation-pictorial). As to why this is the case, we need to consider the Asian learning-sociocultural context as a contributing factor. Asian students are more familiar with algebra problem-solving as opposed to non-algebra problem-solving strategies (Cai, Citation 2000 ; Ngu et al., Citation 2018 ). Thus, we reason that Asian students in this study, irrespective of their knowledge levels (e.g., level of prior knowledge) were somewhat grounded in ‘comparable’ algebra foundation, which indeed may help them to coordinate the relation between values and a variable in the diagram with the solution procedure. With this in mind, we purport that less knowledgeable students may also benefit from the equation-pictorial approach.
When viewing the two levels of learner expertise together, it is plausible to advocate that there is benefit for educators to use instructional designs that are accompanied by diagrams to assist students of different levels of expertise. For example, from our findings, the positive effect for such discourse is evident with students of high expertise for the non-algebra approach (unitary-pictorial) as well as the algebra approach (equation-pictorial), although somewhat of a lesser extent. We also encourage the provision of diagram support for students who are less knowledgeable. There is evidence that less knowledgeable students were not inferior to more knowledgeable students with respect to the equation-pictorial approach. (e.g., performance outcomes for the practice problems). It appears that prior algebra knowledge may enable all students to benefit from the diagram in the equation-pictorial approach. It is advisable therefore for educators to improve students’ prior algebra knowledge and incorporate diagrams to help students overcome the obstacle of identifying the problem structure of word problems.
We acknowledge that our research undertaking has several important caveats for consideration. First, we note that for both the unitary-pictorial approach and the equation-pictorial approach, we used a worked example that consisted of problem text, a diagram, and a solution procedure. What is problematic or contentious is whether and/or the extent to which the diagram itself actually aided students to understand the solution procedure. As a result, we encourage researchers to use what is known as ‘gaze data analysis’ (Schwonke et al., Citation 2009 ) to determine whether participants process information in the diagram and the solution procedure simultaneously or independently of each other. This may help to tease out what we term as ‘cognitive convolution’ (i.e., is there too much information (e.g., diagram + text) to process at any moment in time, confounding a student’s comprehension and understanding?).
Second, the current study investigated the relationship between levels of learner expertise and: (i) the dichotomy between two versions of a specific approach (e.g., without a diagram vs. a diagram), and (ii) type of approach (i.e., non-algebra vs. algebra). We firmly believe it is appropriate to use a 2 (approach) × 2 (level of expertise) ANOVA to elucidate the potent role of a diagram and type of approach upon performance outcomes across two levels of students’ knowledge (e.g., students who are more knowledgeable). Nonetheless, additional research could consider a 4 (approach) × 2 (level of expertise) methodological design to examine a comparison between different instructional approaches (e.g., unitary approach vs. equation approach) and two levels of student knowledge (e.g., less knowledgeable students vs. more knowledgeable students) with a focus to determine which approach is most effective for a given level of student prior knowledge for learning to solve percentage problems.
Third, the diagram in the equation-pictorial approach scaffolds the proportion concept for subsequent generation of solution procedure. It is plausible for researchers to expand on the equation-pictorial approach and consider alternatives. For example, in terms of diagram support, consider the use of a pie chart to represent, say, 8% x = $350, where x represents the whole circle, and 8% x represents a fraction of the whole, x. As a result, then, a pie chart may be used to scaffold a fraction of the whole quantity. A follow-up study is needed to compare these two versions of the equation-pictorial approach.
Fourth, considering Stern et al.’s ( Citation 2003 ) earlier design and, more recently, Rellensmann et al.’s ( Citation 2022 ) innovative design, educators could ask students to draw a diagram (e.g., a pie chart) rather than having them passively read information from a given diagram. Opportunities to construct (e.g., drawing a diagram) rather than to simply read and passively interpret a given diagram may assist students to actively seek out crucial information for meaningful understanding. This recommendation, in particular, aligns with the tenets of constructivist teaching (Almala, Citation 2006 ; Jonassen, Citation 1991 ; White, Citation 2002 ), which emphasize the importance of participation , active construction , individual initiative and volition , etc. Therefore, examining the construction of diagrams for students of varying knowledge levels would contribute empirically to the literature—for example, consider the following: would it be more helpful for a less knowledgeable student to construct a diagram to uncover the problem structure that is embedded in the problem text of a word problem?
Fifth, the complexity of a word problem may determine how we design a diagram to accurately capture the problem structure of the word problem. In the present study, both the unitary-pictorial approach (i.e., non-algebra approach) and the equation-pictorial approach (i.e., algebra approach) share a similar diagram that portrays the proportion concept needed to solve the percentage problems. Specifically, the percentage problems in this study do not have irrelevant information in the cover stories, neither do they require the use of objects (such as in the trigonometry problems) to construct a spatial relationship between objects in order to capture the underlying problem structure (Rellensmann et al., Citation 2022 ). Having said this, however, future research could examine the relationship between the complexity of word problems and the design of diagrams to scaffold - underlying problem structures.
Sixth, Mathematics educators tend to regard problem solving in the context of solving ill-structured word problems that usually have multiple solution paths (Jee Yun & Kim, Citation 2016 ). Cognitive science researchers, in contrast, are more inclined to study problem-solving within the context of well-structured word problems that have fixed solution paths and, to a lesser extent, ill-structured problems (Sweller et al., Citation 2011 ). In the present study, we investigated the percentage problems, which are a type of well-structured word problems that have fixed solution paths (Sweller et al., Citation 2011 ). In terms of clarity and potential empirical contribution then, future inquiry may wish to examine the relationship between ill-structured word problems that have multiple solution paths, different types of instructional approach (e.g., non-algebra vs. algebra), with or without diagram support, and levels of learner expertise.
Seventh, in line with existing research (e.g., Mayer & Gallini, Citation 1990 ) and our research pertaining to percentage problems (Ngu et al., Citation 2023 ; Ngu et al., Citation 2018 ), the present study used the term ‘pictorial’ in the unitary-pictorial approach and the equation-pictorial approach to denote the visual representations, scaffolding the underlying problem structures of percentage problems. Nonetheless, as suggested by one of the reviewers, it may be more appropriate to use ‘diagram’ instead of ‘pictorial’ to represent a problem structure of a word problem. Thus, researchers may wish to consider the alternative term ‘unitary-diagram approach’ instead of ‘unitary-pictorial approach’ so as to avoid misinterpretation of using the term ‘pictorial’ to denote a storyline of a word problem (Hegarty & Kozhevnikov, Citation 1999 ).
Finally, central to the present study is our attempt to develop appropriate instructional designs that could help facilitate quality learning experiences in mathematics. Quality learning experiences, in this case, may espouse personal enjoyment (Hagenauer & Hascher, Citation 2014 ; Koops, Citation 2017 ), engagement in mastery and deep learning (O’Grady & Choy, Citation 2008 ; Senko & Miles, Citation 2008 ), and personal experience of flow (Asakawa, Citation 2010 ; Csíkszentmihályi, Citation 2014 ). A follow-up study could examine the relationship between appropriate instructional designs (without a diagram vs. a diagram, non-algebra vs. algebra) and quality learning experiences and other adaptive-related outcomes in mathematics. For example, aside from effective cognitive processing (e.g., the processing of visual representations), how does the utilization of visualizations serve to motivate novice learners?
The present study is unique for its emphasis on examining the dichotomy of a specific type of instructional approach (without a diagram vs. a diagram) across two types of instructional approach (non-algebra vs. algebra) and two levels of learner expertise, which has not been investigated by prior researchers. Can the presence of a diagram improve the effectiveness of a specific instructional approach (e.g., equation vs. equation-pictorial)? Our research interest to identify the relative effectiveness of the dichotomy of a specific type of instructional approach (unitary vs. unitary-pictorial) lends itself to students of varying levels of expertise. For example, students who lack prior knowledge may find it more difficult to identify and/or to interpret the problem structure that is embedded in the diagram. It is also plausible that the presence or absence of a diagram bears little, if any, benefit for more knowledgeable students. Overall, considering two levels of learner expertise together, our research undertaking has yielded a number of interesting findings—for example: (i) performance outcomes favoring more knowledgeable students were more pronounced for the unitary-pictorial approach than the equation-pictorial approach, and (ii) with relevant prior algebra knowledge, less knowledgeable students also benefited from the equation-pictorial approach. From an educational point of view, the current results can help mathematics educators to make informed decision regarding when to incorporate diagrams in order to cater for students’ diverse learning needs.
The authors would like to thank the teachers and students of the schools who participated in this study. A special thank is extended to the reviewers for their insights and critical comments, which helped to shape the final version of the article.
The authors have no conflict of interest to declare.
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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The importance of cognitive load and levels of expertise. There are different theories or theoretical premises in education (e.g., behaviorism, constructivism) that we may use to explain quality teaching and/or to facilitate effective learning experiences (Kolb et al., Citation 2001; Schunk, Citation 2008; Wiest, Citation 1967).One notable learning theory that has received considerable ...
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