Identify Goal
Define Problem
Define Problem
Gather Data
Define Causes
Identify Options
Clarify Problem
Generate Ideas
Evaluate Options
Generate Ideas
Choose the Best Solution
Implement Solution
Select Solution
Take Action
MacLeod offers her own problem solving procedure, which echoes the above steps:
“1. Recognize the Problem: State what you see. Sometimes the problem is covert. 2. Identify: Get the facts — What exactly happened? What is the issue? 3. and 4. Explore and Connect: Dig deeper and encourage group members to relate their similar experiences. Now you're getting more into the feelings and background [of the situation], not just the facts. 5. Possible Solutions: Consider and brainstorm ideas for resolution. 6. Implement: Choose a solution and try it out — this could be role play and/or a discussion of how the solution would be put in place. 7. Evaluate: Revisit to see if the solution was successful or not.”
Many of these problem solving techniques can be used in concert with one another, or multiple can be appropriate for any given problem. It’s less about facilitating a perfect CPS session, and more about encouraging team members to continually think outside the box and push beyond personal boundaries that inhibit their innovative thinking. So, try out several methods, find those that resonate best with your team, and continue adopting new techniques and adapting your processes along the way.
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What are the 5 steps to problem-solving, 10 effective problem-solving strategies, what skills do efficient problem solvers have, how to improve your problem-solving skills.
Problems come in all shapes and sizes — from workplace conflict to budget cuts.
Creative problem-solving is one of the most in-demand skills in all roles and industries. It can boost an organization’s human capital and give it a competitive edge.
Problem-solving strategies are ways of approaching problems that can help you look beyond the obvious answers and find the best solution to your problem .
Let’s take a look at a five-step problem-solving process and how to combine it with proven problem-solving strategies. This will give you the tools and skills to solve even your most complex problems.
Good problem-solving is an essential part of the decision-making process . To see what a problem-solving process might look like in real life, let’s take a common problem for SaaS brands — decreasing customer churn rates.
To solve this problem, the company must first identify it. In this case, the problem is that the churn rate is too high.
Next, they need to identify the root causes of the problem. This could be anything from their customer service experience to their email marketing campaigns. If there are several problems, they will need a separate problem-solving process for each one.
Let’s say the problem is with email marketing — they’re not nurturing existing customers. Now that they’ve identified the problem, they can start using problem-solving strategies to look for solutions.
This might look like coming up with special offers, discounts, or bonuses for existing customers. They need to find ways to remind them to use their products and services while providing added value. This will encourage customers to keep paying their monthly subscriptions.
They might also want to add incentives, such as access to a premium service at no extra cost after 12 months of membership. They could publish blog posts that help their customers solve common problems and share them as an email newsletter.
The company should set targets and a time frame in which to achieve them. This will allow leaders to measure progress and identify which actions yield the best results.
Perhaps you’ve got a problem you need to tackle. Or maybe you want to be prepared the next time one arises. Either way, it’s a good idea to get familiar with the five steps of problem-solving.
Use this step-by-step problem-solving method with the strategies in the following section to find possible solutions to your problem.
The first step is to know which problem you need to solve. Then, you need to find the root cause of the problem.
The best course of action is to gather as much data as possible, speak to the people involved, and separate facts from opinions.
Once this is done, formulate a statement that describes the problem. Use rational persuasion to make sure your team agrees .
Identifying the problem allows you to see which steps need to be taken to solve it.
First, break the problem down into achievable blocks. Then, use strategic planning to set a time frame in which to solve the problem and establish a timeline for the completion of each stage.
At this stage, the aim isn’t to evaluate possible solutions but to generate as many ideas as possible.
Encourage your team to use creative thinking and be patient — the best solution may not be the first or most obvious one.
Use one or more of the different strategies in the following section to help come up with solutions — the more creative, the better.
Once you’ve generated potential solutions, narrow them down to a shortlist. Then, evaluate the options on your shortlist.
There are usually many factors to consider. So when evaluating a solution, ask yourself the following questions:
Once you’ve identified your solution and got buy-in from your team, it’s time to implement it.
But the work doesn’t stop there. You need to monitor your solution to see whether it actually solves your problem.
Request regular feedback from the team members involved and have a monitoring and evaluation plan in place to measure progress.
If the solution doesn’t achieve your desired results, start this step-by-step process again.
There are many different ways to approach problem-solving. Each is suitable for different types of problems.
The most appropriate problem-solving techniques will depend on your specific problem. You may need to experiment with several strategies before you find a workable solution.
Here are 10 effective problem-solving strategies for you to try:
Let’s break each of these down.
It might seem obvious, but if you’ve faced similar problems in the past, look back to what worked then. See if any of the solutions could apply to your current situation and, if so, replicate them.
The more people you enlist to help solve the problem, the more potential solutions you can come up with.
Use different brainstorming techniques to workshop potential solutions with your team. They’ll likely bring something you haven’t thought of to the table.
Working backward is a way to reverse engineer your problem. Imagine your problem has been solved, and make that the starting point.
Then, retrace your steps back to where you are now. This can help you see which course of action may be most effective.
This is a method that poses six questions based on Rudyard Kipling’s poem, “ I Keep Six Honest Serving Men .”
Answering these questions can help you identify possible solutions.
Sometimes it can be difficult to visualize all the components and moving parts of a problem and its solution. Drawing a diagram can help.
This technique is particularly helpful for solving process-related problems. For example, a product development team might want to decrease the time they take to fix bugs and create new iterations. Drawing the processes involved can help you see where improvements can be made.
A trial-and-error approach can be useful when you have several possible solutions and want to test them to see which one works best.
Finding the best solution to a problem is a process. Remember to take breaks and get enough rest . Sometimes, a walk around the block can bring inspiration, but you should sleep on it if possible.
A good night’s sleep helps us find creative solutions to problems. This is because when you sleep, your brain sorts through the day’s events and stores them as memories. This enables you to process your ideas at a subconscious level.
If possible, give yourself a few days to develop and analyze possible solutions. You may find you have greater clarity after sleeping on it. Your mind will also be fresh, so you’ll be able to make better decisions.
Getting input from a group of people can help you find solutions you may not have thought of on your own.
For solo entrepreneurs or freelancers, this might look like hiring a coach or mentor or joining a mastermind group.
For leaders , it might be consulting other members of the leadership team or working with a business coach .
It’s important to recognize you might not have all the skills, experience, or knowledge necessary to find a solution alone.
The Pareto principle — also known as the 80/20 rule — can help you identify possible root causes and potential solutions for your problems.
Although it’s not a mathematical law, it’s a principle found throughout many aspects of business and life. For example, 20% of the sales reps in a company might close 80% of the sales.
You may be able to narrow down the causes of your problem by applying the Pareto principle. This can also help you identify the most appropriate solutions.
Every situation is different, and the same solutions might not always work. But by keeping a record of successful problem-solving strategies, you can build up a solutions toolkit.
These solutions may be applicable to future problems. Even if not, they may save you some of the time and work needed to come up with a new solution.
Improving problem-solving skills is essential for professional development — both yours and your team’s. Here are some of the key skills of effective problem solvers:
And they see problems as opportunities. Everyone is born with problem-solving skills. But accessing these abilities depends on how we view problems. Effective problem-solvers see problems as opportunities to learn and improve.
Ready to work on your problem-solving abilities? Get started with these seven tips.
One of the best ways to improve your problem-solving skills is to learn from experts. Consider enrolling in organizational training , shadowing a mentor , or working with a coach .
Practice using your new problem-solving skills by applying them to smaller problems you might encounter in your daily life.
Alternatively, imagine problematic scenarios that might arise at work and use problem-solving strategies to find hypothetical solutions.
Often, the first solution you think of to solve a problem isn’t the most appropriate or effective.
Instead of thinking on the spot, give yourself time and use one or more of the problem-solving strategies above to activate your creative thinking.
Receiving feedback is always important for learning and growth. Your perception of your problem-solving skills may be different from that of your colleagues. They can provide insights that help you improve.
There are entire books written about problem-solving methodologies if you want to take a deep dive into the subject.
We recommend starting with “ Fixed — How to Perfect the Fine Art of Problem Solving ” by Amy E. Herman.
Tried-and-tested problem-solving techniques can be useful. However, they don’t teach you how to innovate and develop your own problem-solving approaches.
Sometimes, an unconventional approach can lead to the development of a brilliant new idea or strategy. So don’t be afraid to suggest your most “out there” ideas.
Do you have competitors who have already solved the problem you’re facing? Look at what they did, and work backward to solve your own problem.
For example, Netflix started in the 1990s as a DVD mail-rental company. Its main competitor at the time was Blockbuster.
But when streaming became the norm in the early 2000s, both companies faced a crisis. Netflix innovated, unveiling its streaming service in 2007.
If Blockbuster had followed Netflix’s example, it might have survived. Instead, it declared bankruptcy in 2010.
When facing a problem, it’s worth taking the time to find the right solution.
Otherwise, we risk either running away from our problems or headlong into solutions. When we do this, we might miss out on other, better options.
Use the problem-solving strategies outlined above to find innovative solutions to your business’ most perplexing problems.
If you’re ready to take problem-solving to the next level, request a demo with BetterUp . Our expert coaches specialize in helping teams develop and implement strategies that work.
Maximize your time and productivity with strategies from our expert coaches.
Elizabeth Perry is a Coach Community Manager at BetterUp. She uses strategic engagement strategies to cultivate a learning community across a global network of Coaches through in-person and virtual experiences, technology-enabled platforms, and strategic coaching industry partnerships. With over 3 years of coaching experience and a certification in transformative leadership and life coaching from Sofia University, Elizabeth leverages transpersonal psychology expertise to help coaches and clients gain awareness of their behavioral and thought patterns, discover their purpose and passions, and elevate their potential. She is a lifelong student of psychology, personal growth, and human potential as well as an ICF-certified ACC transpersonal life and leadership Coach.
5 problem-solving questions to prepare you for your next interview, what are metacognitive skills examples in everyday life, 31 examples of problem solving performance review phrases, what is lateral thinking 7 techniques to encourage creative ideas, leadership activities that encourage employee engagement, learn what process mapping is and how to create one (+ examples), how much do distractions cost 8 effects of lack of focus, can dreams help you solve problems 6 ways to try, similar articles, the pareto principle: how the 80/20 rule can help you do more with less, thinking outside the box: 8 ways to become a creative problem solver, 3 problem statement examples and steps to write your own, contingency planning: 4 steps to prepare for the unexpected, learn to sweat the small stuff: how to improve attention to detail, stay connected with betterup, get our newsletter, event invites, plus product insights and research..
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In this section we examine problem-solving strategies. People face problems every day—usually, multiple problems throughout the day. Sometimes these problems are straightforward: To double a recipe for pizza dough, for example, all that is required is that each ingredient in the recipe be doubled. Sometimes, however, the problems we encounter are more complex. For example, say you have a work deadline, and you must mail a printed copy of a report to your supervisor by the end of the business day. The report is time-sensitive and must be sent overnight. You finished the report last night, but your printer will not work today. What should you do? First, you need to identify the problem and then apply a strategy, usually a set of steps, for solving the problem.
We begin this module on Problem Solving by giving a short description of what psychologists regard as a problem. Afterwards we are going to present different approaches towards problem solving, starting with gestalt psychologists and ending with modern search strategies connected to artificial intelligence. In addition we will also consider how experts do solve problems and finally we will have a closer look at two topics: The neurophysiological background on the one hand and the question what kind of role can be assigned to evolution regarding problem solving on the other.
The most basic definition is “A problem is any given situation that differs from a desired goal”. This definition is very useful for discussing problem solving in terms of evolutionary adaptation, as it allows to understand every aspect of (human or animal) life as a problem. This includes issues like finding food in harsh winters, remembering where you left your provisions, making decisions about which way to go, repeating and varying all kinds of complex movements by learning, and so on. Though all these problems were of crucial importance during the evolutionary process that created us the way we are, they are by no means solved exclusively by humans. We find a most amazing variety of different solutions for these problems of adaptation in animals as well (just consider, e.g., by which means a bat hunts its prey, compared to a spider ).
However, for this module, we will mainly focus on abstract problems that humans may encounter (e.g. playing chess or doing an assignment in college). Furthermore, we will not consider those situations as abstract problems that have an obvious solution: Imagine a college student, let's call him Knut. Knut decides to take a sip of coffee from the mug next to his right hand. He does not even have to think about how to do this. This is not because the situation itself is trivial (a robot capable of recognizing the mug, deciding whether it is full, then grabbing it and moving it to Knut’s mouth would be a highly complex machine) but because in the context of all possible situations it is so trivial that it no longer is a problem our consciousness needs to be bothered with. The problems we will discuss in the following all need some conscious effort, though some seem to be solved without us being able to say how exactly we got to the solution. Still we will find that often the strategies we use to solve these problems are applicable to more basic problems, as well as the more abstract ones such as completing a reading or writing assignment for a college class.
Non-trivial, abstract problems can be divided into two groups:
For many abstract problems it is possible to find an algorithmic solution. We call all those problems well-defined that can be properly formalized, which comes along with the following properties:
Not surprisingly, a problem that fulfills these requirements can be implemented algorithmically (also see convergent thinking ). Therefore many well-defined problems can be very effectively solved by computers, like playing chess.
Though many problems can be properly formalized (sometimes only if we accept an enormous complexity) there are still others where this is not the case. Good examples for this are all kinds of tasks that involve creativity , and, generally speaking, all problems for which it is not possible to clearly define a given state and a goal state: Formalizing a problem of the kind “Please paint a beautiful picture” may be impossible. Still this is a problem most people would be able to access in one way or the other, even if the result may be totally different from person to person. And while Knut might judge that picture X is gorgeous, you might completely disagree.
Nevertheless ill-defined problems often involve sub-problems that can be totally well-defined. On the other hand, many every-day problems that seem to be completely well-defined involve a great deal of creativity and many ambiguities. For example, suppose Knut has to read some technical material and then write an essay about it.
If we think of Knut's fairly ill-defined task of writing an essay, he will not be able to complete this task without first understanding the text he has to write about. This step is the first sub-goal Knut has to solve. Interestingly, ill-defined problems often involve subproblems that are well-defined.
Knut’s situation could be explained as a classical example of problem solving: He needs to get from his present state – an unfinished assignment – to a goal state - a completed assignment - and has certain operators to achieve that goal. Both Knut’s short and long term memory are active. He needs his short term memory to integrate what he is reading with the information from earlier passages of the paper. His long term memory helps him remember what he learned in the lectures he took and what he read in other books. And of course Knut’s ability to comprehend language enables him to make sense of the letters printed on the paper and to relate the sentences in a proper way.
Same place, different day. Knut is sitting at his desk again, staring at a blank paper in front of him, while nervously playing with a pen in his right hand. Just a few hours left to hand in his essay and he has not written a word. All of a sudden he smashes his fist on the table and cries out: "I need a plan!
Generally speaking, problem representations are models of the situation as experienced by the agent. Representing a problem means to analyze it and split it into separate components:
Therefore the efficiency of Problem Solving depends on the underlying representations in a person’s mind. Analyzing the problem domain according to different dimensions, i.e., changing from one representation to another, results in arriving at a new understanding of a problem. This is basically what is described as restructuring.
There are two very different ways of approaching a goal-oriented situation . In one case an organism readily reproduces the response to the given problem from past experience. This is called reproductive thinking .
The second way requires something new and different to achieve the goal, prior learning is of little help here. Such productive thinking is (sometimes) argued to involve insight . Gestalt psychologists even state that insight problems are a separate category of problems in their own right.
Tasks that might involve insight usually have certain features – they require something new and non-obvious to be done and in most cases they are difficult enough to predict that the initial solution attempt will be unsuccessful. When you solve a problem of this kind you often have a so called "AHA-experience" – the solution pops up all of a sudden. At one time you do not have any ideas of the answer to the problem, you do not even feel to make any progress trying out different ideas, but in the next second the problem is solved.
Sometimes, previous experience or familiarity can even make problem solving more difficult. This is the case whenever habitual directions get in the way of finding new directions – an effect called fixation .
Functional fixedness concerns the solution of object-use problems . The basic idea is that when the usual way of using an object is emphasised, it will be far more difficult for a person to use that object in a novel manner.
An example is the two-string problem : Knut is left in a room with a chair and a pair of pliers given the task to bind two strings together that are hanging from the ceiling. The problem he faces is that he can never reach both strings at a time because they are just too far away from each other. What can Knut do?
Figure \(\PageIndex{1}\): Put the two strings together by tying the pliers to one of the strings and then swing it toward the other one.
Functional fixedness as involved in the examples above illustrates a mental set – a person’s tendency to respond to a given task in a manner based on past experience. Because Knut maps an object to a particular function he has difficulties to vary the way of use (pliers as pendulum's weight).
When you are presented with a problem—whether it is a complex mathematical problem or a broken printer, how do you solve it? Before finding a solution to the problem, the problem must first be clearly identified. After that, one of many problem solving strategies can be applied, hopefully resulting in a solution. Regardless of strategy, you will likely be guided, consciously or unconsciously, by your knowledge of cause-effect relations among the elements of the problem and the similarity of the problem to previous problems you have solved before. As discussed in earlier sections of this chapter, innate dispositions of the brain to look for and represent causal and similarity relations are key components of general intelligence (Koenigshofer, 2017).
A problem-solving strategy is a plan of action used to find a solution. Different strategies have different action plans associated with them. For example, a well-known strategy is trial and error. The old adage, “If at first you don’t succeed, try, try again” describes trial and error. In terms of your broken printer, you could try checking the ink levels, and if that doesn’t work, you could check to make sure the paper tray isn’t jammed. Or maybe the printer isn’t actually connected to your laptop. When using trial and error, you would continue to try different solutions until you solved your problem. Although trial and error is not typically one of the most time-efficient strategies, it is a commonly used one.
Method | Description | Example |
---|---|---|
Trial and error | Continue trying different solutions until problem is solved | Restarting phone, turning off WiFi, turning off bluetooth in order to determine why your phone is malfunctioning |
Algorithm | Step-by-step problem-solving formula | Instruction manual for installing new software on your computer |
Heuristic | General problem-solving framework | Working backwards; breaking a task into steps |
Another type of strategy is an algorithm. An algorithm is a problem-solving formula that provides you with step-by-step instructions used to achieve a desired outcome (Kahneman, 2011). You can think of an algorithm as a recipe with highly detailed instructions that produce the same result every time they are performed. Algorithms are used frequently in our everyday lives, especially in computer science. When you run a search on the Internet, search engines like Google use algorithms to decide which entries will appear first in your list of results. Facebook also uses algorithms to decide which posts to display on your newsfeed. Can you identify other situations in which algorithms are used?
A heuristic is another type of problem solving strategy. While an algorithm must be followed exactly to produce a correct result, a heuristic is a general problem-solving framework (Tversky & Kahneman, 1974). You can think of these as mental shortcuts that are used to solve problems. A “rule of thumb” is an example of a heuristic. Such a rule saves the person time and energy when making a decision, but despite its time-saving characteristics, it is not always the best method for making a rational decision. Different types of heuristics are used in different types of situations, but the impulse to use a heuristic occurs when one of five conditions is met (Pratkanis, 1989):
Working backwards is a useful heuristic in which you begin solving the problem by focusing on the end result. Consider this example: You live in Washington, D.C. and have been invited to a wedding at 4 PM on Saturday in Philadelphia. Knowing that Interstate 95 tends to back up any day of the week, you need to plan your route and time your departure accordingly. If you want to be at the wedding service by 3:30 PM, and it takes 2.5 hours to get to Philadelphia without traffic, what time should you leave your house? You use the working backwards heuristic to plan the events of your day on a regular basis, probably without even thinking about it.
Another useful heuristic is the practice of accomplishing a large goal or task by breaking it into a series of smaller steps. Students often use this common method to complete a large research project or long essay for school. For example, students typically brainstorm, develop a thesis or main topic, research the chosen topic, organize their information into an outline, write a rough draft, revise and edit the rough draft, develop a final draft, organize the references list, and proofread their work before turning in the project. The large task becomes less overwhelming when it is broken down into a series of small steps.
The idea of regarding problem solving as a search problem originated from Alan Newell and Herbert Simon while trying to design computer programs which could solve certain problems. This led them to develop a program called General Problem Solver which was able to solve any well-defined problem by creating heuristics on the basis of the user's input. This input consisted of objects and operations that could be done on them.
As we already know, every problem is composed of an initial state, intermediate states and a goal state (also: desired or final state), while the initial and goal states characterise the situations before and after solving the problem. The intermediate states describe any possible situation between initial and goal state. The set of operators builds up the transitions between the states. A solution is defined as the sequence of operators which leads from the initial state across intermediate states to the goal state.
The simplest method to solve a problem, defined in these terms, is to search for a solution by just trying one possibility after another (also called trial and error ).
As already mentioned above, an organised search, following a specific strategy, might not be helpful for finding a solution to some ill-defined problem, since it is impossible to formalise such problems in a way that a search algorithm can find a solution.
As an example we could just take Knut and his essay: he has to find out about his own opinion and formulate it and he has to make sure he understands the sources texts. But there are no predefined operators he can use, there is no panacea how to get to an opinion and even not how to write it down.
In Means-End Analysis you try to reduce the difference between initial state and goal state by creating sub-goals until a sub-goal can be reached directly (in computer science, what is called recursion works on this basis).
An example of a problem that can be solved by Means-End Analysis is the " Towers of Hanoi "
Figure \(\PageIndex{2}\): Towers of Hanoi with 8 discs – A well defined problem (image from Wikimedia Commons; https://commons.wikimedia.org/wiki/F..._of_Hanoi.jpeg , by User:Evanherk .licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license).
The initial state of this problem is described by the different sized discs being stacked in order of size on the first of three pegs (the “start-peg“). The goal state is described by these discs being stacked on the third pegs (the “end-peg“) in exactly the same order.
Figure \(\PageIndex{3}\): This animation shows the solution of the game "Tower of Hanoi" with four discs. (image from Wikimedia Commons; https://commons.wikimedia.org/wiki/F...of_Hanoi_4.gif ; by André Karwath aka Aka ; licensed under the Creative Commons Attribution-Share Alike 2.5 Generic license).
There are three operators:
In order to use Means-End Analysis we have to create sub-goals. One possible way of doing this is described in the picture:
1. Moving the discs lying on the biggest one onto the second peg.
2. Shifting the biggest disc to the third peg.
3. Moving the other ones onto the third peg, too
You can apply this strategy again and again in order to reduce the problem to the case where you only have to move a single disc – which is then something you are allowed to do.
Strategies of this kind can easily be formulated for a computer; the respective algorithm for the Towers of Hanoi would look like this:
1. move n-1 discs from A to B
2. move disc #n from A to C
3. move n-1 discs from B to C
where n is the total number of discs, A is the first peg, B the second, C the third one. Now the problem is reduced by one with each recursive loop.
Means-end analysis is important to solve everyday-problems – like getting the right train connection: You have to figure out where you catch the first train and where you want to arrive, first of all. Then you have to look for possible changes just in case you do not get a direct connection. Third, you have to figure out what are the best times of departure and arrival, on which platforms you leave and arrive and make it all fit together.
Analogies describe similar structures and interconnect them to clarify and explain certain relations. In a recent study, for example, a song that got stuck in your head is compared to an itching of the brain that can only be scratched by repeating the song over and over again. Useful analogies appears to be based on a psychological mapping of relations between two very disparate types of problems that have abstract relations in common. Applied to STEM problems, Gray and Holyoak (2021) state: "Analogy is a powerful tool for fostering conceptual understanding and transfer in STEM and other fields. Well-constructed analogical comparisons focus attention on the causal-relational structure of STEM concepts, and provide a powerful capability to draw inferences based on a well-understood source domain that can be applied to a novel target domain." Note that similarity between problems of different types in their abstract relations, such as causation, is a key feature of reasoning, problem-solving and inference when forming and using analogies. Recall the discussion of general intelligence in module 14.2. There, similarity relations, causal relations, and predictive relations between events were identified as key components of general intelligence, along with ability to visualize in imagination possible future actions and their probable outcomes prior to commiting to actual behavior in the physical world (Koenigshofer, 2017).
One special kind of restructuring, the way already mentioned during the discussion of the Gestalt approach, is analogical problem solving. Here, to find a solution to one problem – the so called target problem, an analogous solution to another problem – the source problem, is presented.
An example for this kind of strategy is the radiation problem posed by K. Duncker in 1945:
As a doctor you have to treat a patient with a malignant, inoperable tumour, buried deep inside the body. There exists a special kind of ray, which is perfectly harmless at a low intensity, but at the sufficient high intensity is able to destroy the tumour – as well as the healthy tissue on his way to it. What can be done to avoid the latter?
When this question was asked to participants in an experiment, most of them couldn't come up with the appropriate answer to the problem. Then they were told a story that went something like this:
A General wanted to capture his enemy's fortress. He gathered a large army to launch a full-scale direct attack, but then learned, that all the roads leading directly towards the fortress were blocked by mines. These roadblocks were designed in such a way, that it was possible for small groups of the fortress-owner's men to pass them safely, but every large group of men would initially set them off. Now the General figured out the following plan: He divided his troops into several smaller groups and made each of them march down a different road, timed in such a way, that the entire army would reunite exactly when reaching the fortress and could hit with full strength.
Here, the story about the General is the source problem, and the radiation problem is the target problem. The fortress is analogous to the tumour and the big army corresponds to the highly intensive ray. Consequently a small group of soldiers represents a ray at low intensity. The solution to the problem is to split the ray up, as the general did with his army, and send the now harmless rays towards the tumour from different angles in such a way that they all meet when reaching it. No healthy tissue is damaged but the tumour itself gets destroyed by the ray at its full intensity.
M. Gick and K. Holyoak presented Duncker's radiation problem to a group of participants in 1980 and 1983. Only 10 percent of them were able to solve the problem right away, 30 percent could solve it when they read the story of the general before. After given an additional hint – to use the story as help – 75 percent of them solved the problem.
With this results, Gick and Holyoak concluded, that analogical problem solving depends on three steps:
1. Noticing that an analogical connection exists between the source and the target problem. 2. Mapping corresponding parts of the two problems onto each other (fortress → tumour, army → ray, etc.) 3. Applying the mapping to generate a parallel solution to the target problem (using little groups of soldiers approaching from different directions → sending several weaker rays from different directions)
The concept that links the target problem with the analogy (the “source problem“) is called problem schema. Gick and Holyoak obtained the activation of a schema on their participants by giving them two stories and asking them to compare and summarize them. This activation of problem schemata is called “schema induction“.
The two presented texts were picked out of six stories which describe analogical problems and their solution. One of these stories was "The General."
After solving the task the participants were asked to solve the radiation problem. The experiment showed that in order to solve the target problem reading of two stories with analogical problems is more helpful than reading only one story: After reading two stories 52% of the participants were able to solve the radiation problem (only 30% were able to solve it after reading only one story, namely: “The General“).
The process of using a schema or analogy, i.e. applying it to a novel situation, is called transduction . One can use a common strategy to solve problems of a new kind.
To create a good schema and finally get to a solution using the schema is a problem-solving skill that requires practice and some background knowledge.
With the term expert we describe someone who devotes large amounts of his or her time and energy to one specific field of interest in which he, subsequently, reaches a certain level of mastery. It should not be of surprise that experts tend to be better in solving problems in their field than novices (people who are beginners or not as well trained in a field as experts) are. They are faster in coming up with solutions and have a higher success rate of right solutions. But what is the difference between the way experts and non-experts solve problems? Research on the nature of expertise has come up with the following conclusions:
When it comes to problems that are situated outside the experts' field, their performance often does not differ from that of novices.
Knowledge: An experiment by Chase and Simon (1973a, b) dealt with the question how well experts and novices are able to reproduce positions of chess pieces on chessboards when these are presented to them only briefly. The results showed that experts were far better in reproducing actual game positions, but that their performance was comparable with that of novices when the chess pieces were arranged randomly on the board. Chase and Simon concluded that the superior performance on actual game positions was due to the ability to recognize familiar patterns: A chess expert has up to 50,000 patterns stored in his memory. In comparison, a good player might know about 1,000 patterns by heart and a novice only few to none at all. This very detailed knowledge is of crucial help when an expert is confronted with a new problem in his field. Still, it is not pure size of knowledge that makes an expert more successful. Experts also organise their knowledge quite differently from novices.
Organization: In 1982 M. Chi and her co-workers took a set of 24 physics problems and presented them to a group of physics professors as well as to a group of students with only one semester of physics. The task was to group the problems based on their similarities. As it turned out the students tended to group the problems based on their surface structure (similarities of objects used in the problem, e.g. on sketches illustrating the problem), whereas the professors used their deep structure (the general physical principles that underlay the problems) as criteria. By recognizing the actual structure of a problem experts are able to connect the given task to the relevant knowledge they already have (e.g. another problem they solved earlier which required the same strategy).
Analysis: Experts often spend more time analyzing a problem before actually trying to solve it. This way of approaching a problem may often result in what appears to be a slow start, but in the long run this strategy is much more effective. A novice, on the other hand, might start working on the problem right away, but often has to realise that he reaches dead ends as he chose a wrong path in the very beginning.
Divergent thinking.
The term divergent thinking describes a way of thinking that does not lead to one goal, but is open-ended. Problems that are solved this way can have a large number of potential 'solutions' of which none is exactly 'right' or 'wrong', though some might be more suitable than others.
Solving a problem like this involves indirect and productive thinking and is mostly very helpful when somebody faces an ill-defined problem , i.e. when either initial state or goal state cannot be stated clearly and operators are either insufficient or not given at all.
The process of divergent thinking is often associated with creativity, and it undoubtedly leads to many creative ideas. Nevertheless, researches have shown that there is only modest correlation between performance on divergent thinking tasks and other measures of creativity. Additionally it was found that in processes resulting in original and practical inventions things like searching for solutions, being aware of structures and looking for analogies are heavily involved, too.
Figure \(\PageIndex{4}\): functional MRI images of the brains of musicians playing improvised jazz revealed that a large brain region involved in monitoring one's performance shuts down during creative improvisation, while a small region involved in organizing self-initiated thoughts and behaviors is highly activated (Image and modified caption from Wikimedia Commons. File:Creative Improvisation (24130148711).jpg; https://commons.wikimedia.org/wiki/F...130148711).jpg ; by NIH Image Gallery ; As a work of the U.S. federal government , the image is in the public domain .
Convergent thinking patterns are problem solving techniques that unite different ideas or fields to find a solution. The focus of this mindset is speed, logic and accuracy, also identification of facts, reapplying existing techniques, gathering information. The most important factor of this mindset is: there is only one correct answer. You only think of two answers, namely right or wrong. This type of thinking is associated with certain science or standard procedures. People with this type of thinking have logical thinking, are able to memorize patterns, solve problems and work on scientific tests. Most school subjects sharpen this type of thinking ability.
Research shows that the creative process involves both types of thought processes.
Presenting Neurophysiology in its entirety would be enough to fill several books. Instead, let's focus only on the aspects that are especially relevant to problem solving. Still, this topic is quite complex and problem solving cannot be attributed to one single brain area. Rather there are systems or networks of several brain areas working together to perform a specific problem solving task. This is best shown by an example, playing chess:
Task | Location of Brain activity |
---|---|
(also called the "what"-pathway of visual processing) (also called the "where"-pathway of visual processing) (forming new memories) |
Table 2: Brain areas involved in a complex cognitive task.
One of the key tasks, namely planning and executing strategies , is performed by the prefrontal cortex (PFC) , which also plays an important role for several other tasks correlated with problem solving. This can be made clear from the effects of damage to the PFC on ability to solve problems.
Patients with a lesion in this brain area have difficulty switching from one behavioral pattern to another. A well known example is the wisconsin card-sorting task . A patient with a PFC lesion who is told to separate all blue cards from a deck, would continue sorting out the blue ones, even if the experimenter next told him to sort out all brown cards. Transferred to a more complex problem, this person would most likely fail, because he is not flexible enough to change his strategy after running into a dead end or when the problem changes.
Another example comes from a young homemaker, who had a tumour in the frontal lobe. Even though she was able to cook individual dishes, preparing a whole family meal was an impossible task for her.
Mushiake et al. (2009) note that to achieve a goal in a complex environment, such as problem‐solving situations like those above, we must plan multiple steps of action. When planning a series of actions, we have to anticipate future outcomes that will occur as a result of each action, and, in addition, we must mentally organize the temporal sequence of events in order to achieve the goal. These researchers investigated the role of lateral prefrontal cortex (PFC) in problem solving in monkeys. They found that "PFC neurons reflected final goals and immediate goals during the preparatory period. [They] also found some PFC neurons reflected each of all the forthcoming steps of actions during the preparatory period and they increased their [neural] activity step by step during the execution period. [Furthermore, they] found that the transient increase in synchronous activity of PFC neurons was involved in goal subgoal transformations. [They concluded] that the PFC is involved primarily in the dynamic representation of multiple future events that occur as a consequence of behavioral actions in problem‐solving situations" (Mushiake et al., 2009, p. 1). In other words, the prefrontal cortex represents in our imagination the sequence of events following each step that we take in solving a particular problem, guiding us step by step to the solution.
As the examples above illustrate, the structure of our brain is of great importance regarding problem solving, i.e. cognitive life. But how was our cognitive apparatus designed? How did perception-action integration as a central species-specific property of humans come about? The answer, as argued extensively in earlier sections of this book, is, of course, natural selection and other forces of genetic evolution. Clearly, animals and humans with genes facilitating brain organization that led to good problem solving skills would be favored by natural selection over genes responsible for brain organization less adept at solving problems. We became equipped with brains organized for effective problem solving because flexible abilities to solve a wide range of problems presented by the environment enhanced ability to survive, to compete for resources, to escape predators, and to reproduce (see chapter on Evolution and Genetics in this text).
In short, good problem solving mechanisms in brains designed for the real world gave a competitive advantage and increased biological fitness. Consequently, humans (and many other animals to a lesser degree) have "innate ability to problem-solve in the real world. Solving real world problems in real time given constraints posed by one's environment is crucial for survival . . . Real world problem solving (RWPS) is different from those that occur in a classroom or in a laboratory during an experiment. They are often dynamic and discontinuous, accompanied by many starts and stops . . . Real world problems are typically ill-defined, and even when they are well-defined, often have open-ended solutions . . . RWPS is quite messy and involves a tight interplay between problem solving, creativity, and insight . . . In psychology and neuroscience, problem-solving broadly refers to the inferential steps taken by an agent [human, animal, or computer] that leads from a given state of affairs to a desired goal state" (Sarathy, 2018, p. 261-2). According to Sarathy (2018), the initial stage of RWPS requires defining the problem and generating a representation of it in working memory. This stage involves activation of parts of the " prefrontal cortex (PFC) , default mode network (DMN) , and the dorsal anterior cingulate cortex (dACC) ." The DMN includes the medial prefrontal cortex , posterior cingulate cortex , and the inferior parietal lobule . Other structures sometimes considered part of the network are the lateral temporal cortex , hippocampal formation , and the precuneus . This network of structures is called "default mode" because these structures show increased activity when one is not engaged in focused, attentive, goal-directed actions, but rather a "resting state" (a baseline default state) and show decreased neural activity when one is focused and attentive to a particular goal-directed behavior (Raichle, et al., 2001).
Jeurissen, et al., (2014) examined a special type of reasoning, moral reasoning, using TMS (Transcranial Magnetic Stimulation). The dorsolateral prefrontal cortex (DLPFC) and temporal-parietal junction (TPJ) have both been shown to be involved in moral judgments, but this study by Jeurissen, et al., (2014) uses TMS to tease out the different roles these brain areas play in different scenarios involving moral dilemmas.
Moral dilemmas have been categorized by researchers as moral-impersonal (e.g., trolley or switch dilemma-- save the lives of five workmen at the expense of the life of one by switching train to another track) and moral-personal dilemmas (e.g., footbridge dilemma-- push a strange r in front of a train to save the lives of five others). In the first scenario, the person just pulls a switch resulting in death of one person to save five, but in the second, the person pushes the victim to their death to save five others.
Dual-process theory proposes that moral decision-making involves two components: an automatic emotional response and a voluntary application of a utilitarian decision-rule (in this case, one death to save five people is worth it). The thought of being responsible for the death of another person elicits an aversive emotional response, but at the same time, cognitive reasoning favors the utilitarian option. Decision making and social cognition are often associated with the DLPFC. Neurons in the prefrontal cortex have been found to be involved in cost-benefit analysis and categorize stimuli based on the predicted consequences.
Theory-of-mind (TOM) is a cognitive mechanism which is used when one tries to understand and explain the knowledge, beliefs, and intention of others. TOM and empathy are often associated with TPJ functioning .
In the article by Jeurissen, et al., (2014), brain activity is measured by BOLD. BOLD refers to Blood-oxygen-level-dependent imaging , or BOLD-contrast imaging, which is a way to measure neural activity in different brain areas in MRI images .
Greene et al., 2001 (Links to an external site.) , 2004 (Links to an external site.) reported that activity in the prefrontal cortex is thought to be important for the cognitive reasoning process , which can counteract the automatic emotional response that occurs in moral dilemmas like the one in Jeurissen, et al., (2014). Greene et al. (2001) (Links to an external site.) found that the medial portions of the medial frontal gyrus, the posterior cingulate gyrus, and the bilateral angular gyrus showed a higher BOLD response in the moral-personal condition than the moral-impersonal condition. The right middle frontal gyrus and the bilateral parietal lobes showed a lower BOLD response in the moral-personal condition than in the moral impersonal. Furthermore, Greene et al. (2004) (Links to an external site.) showed an increased BOLD response for the bilateral amygdale in personal compared to the impersonal dilemmas.
Given the role of the prefrontal cortex in moral decision-making, Jeurissen, et al., (2014) hypothesized that when magnetically stimulating prefrontal cortex , they will selectively influence the decision process of the moral personal dilemmas because the cognitive reasoning for which the DLPFC is important is disrupted , thereby releasing the emotional component making it more influential in the resolution of the dilemma. Because the activity in the TPJ is related to emotional processing and theory of mind ( Saxe and Kanwisher, 2003 (Links to an external site.) ; Young et al., 2010 (Links to an external site.) ), Jeurissen, et al., (2014) hypothesized that when magnetically stimulating this area, the TPJ, during a moral decision, this will selectively influence the decision process of moral-impersonal dilemmas.
Results of this study by Jeurissen, et al., (2014) showed an important role of the TPJ in moral judgment . Experiments using fMRI ( Greene et al., 2004 (Links to an external site.) ), have found the cingulate cortex to be involved in moral judgment . In earlier studies, the cingulate cortex was found to be involved in the emotional response. Since the moral-personal dilemmas are more emotional ly salient, the higher activity observed for TPJ in the moral-personal condition (more emotional) is consistent with this view. Another area that is hypothesized to be associated with the emotional response is the temporal cortex . In this study by Jeurissen, et al., (2014) , magnetic stimulation of the right DLPFC and right TPJ shows roles for right DLPFC (reasoning and utilitarian) and right TPJ (emotion) in moral impersonal and moral personal dilemmas respectively. TMS over the right DLPFC (disrupting neural activity here) leads to behavior changes consistent with less cognitive control over emotion . After right DLPFC stimulation, participants show less feelings of regret than after magnetic stimulation of the right TPJ. This last finding indicates that the right DLPFC is involved in evaluating the outcome of the decision process. In summary, this experiment by Jeurissen, et al., (2014) adds to evidence of a critical role of right DLPFC and right TPJ in moral decision-making and supports that hypothesis that the former is involved in judgments based on cognitive reasoning and anticipation of outcomes, whereas the latter is involved in emotional processing related to moral dilemmas.
Many different strategies exist for solving problems. Typical strategies include trial and error, applying algorithms, and using heuristics. To solve a large, complicated problem, it often helps to break the problem into smaller steps that can be accomplished individually, leading to an overall solution. The brain mechanisms involved in problem solving vary to some degree depending upon the sensory modalities involved in the problem and its solution, however, the prefrontal cortex is one brain region that appears to be centrally involved in all problem-solving. The prefrontal cortex is required for flexible shifts in attention, for representing the problem in working memory, and for holding steps in problem solving in working memory along with representations of future consequences of those actions permitting planning and execution of plans. Also implicated is the Default Mode Network (DMN) including medial prefrontal cortex, posterior cingulate cortex, and the inferior parietal module, and sometimes the lateral temporal cortex, hippocampus, and the precuneus. Moral reasoning involves a different set of brain areas, primarily the dorsolateral prefrontal cortex (DLPFC) and temporal-parietal junction (TPJ).
Review Questions
Gray, M. E., & Holyoak, K. J. (2021). Teaching by analogy: From theory to practice. Mind, Brain, and Education , 15 (3), 250-263.
Hunt, L. T., Behrens, T. E., Hosokawa, T., Wallis, J. D., & Kennerley, S. W. (2015). Capturing the temporal evolution of choice across prefrontal cortex. Elife , 4 , e11945.
Mushiake, H., Sakamoto, K., Saito, N., Inui, T., Aihara, K., & Tanji, J. (2009). Involvement of the prefrontal cortex in problem solving. International review of neurobiology , 85 , 1-11.
Jeurissen, D., Sack, A. T., Roebroeck, A., Russ, B. E., & Pascual-Leone, A. (2014). TMS affects moral judgment, showing the role of DLPFC and TPJ in cognitive and emotional processing. Frontiers in neuroscience , 8 , 18.
Kahneman, D. (2011). Thinking, fast and slow. New York: Farrar, Straus, and Giroux.
Koenigshofer, K. A. (2017). General Intelligence: Adaptation to Evolutionarily Familiar Abstract Relational Invariants, Not to Environmental or Evolutionary Novelty. The Journal of Mind and Behavior , 119-153.
Pratkanis, A. (1989). The cognitive representation of attitudes. In A. R. Pratkanis, S. J. Breckler, & A. G. Greenwald (Eds.), Attitude structure and function (pp. 71–98). Hillsdale, NJ: Erlbaum.
Raichle, M. E., MacLeod, A. M., Snyder, A. Z., Powers, W. J., Gusnard, D. A., & Shulman, G. L. (2001). A default mode of brain function. Proceedings of the National Academy of Sciences , 98 (2), 676-682.
Sawyer, K. (2011). The cognitive neuroscience of creativity: a critical review. Creat. Res. J. 23, 137–154. doi: 10.1080/10400419.2011.571191
Tversky, A., & Kahneman, D. (1974). Judgment under uncertainty: Heuristics and biases. Science , 185 (4157), 1124–1131.
Mushiake, H., Sakamoto, K., Saito, N., Inui, T., Aihara, K., & Tanji, J. (2009). Involvement of the prefrontal cortex in problem solving. International review of neurobiology , 85 , 1-11. https://www.sciencedirect.com/scienc...74774209850010
"Overview," "Problem Solving Strategies," adapted from Problem Solving by OpenStax Colleg licensed CC BY-NC 4.0 via OER Commons
"Defining Problems," "Problem Solving as a Search Problem," "Creative Cognition," "Brain Mechanisms in Problem-Solving" adapted by Kenneth A. Koenigshofer, Ph.D., from 2.1, 2.2, 2.3, 2.4, 2.5, 2.6 in Cognitive Psychology and Cognitive Neuroscience (Wikibooks) https://en.wikibooks.org/wiki/Cognit...e_Neuroscience ; unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0 . Legal ; the LibreTexts libraries are Powered by MindTouch
Moral Reasoning was written by Kenneth A. Koenigshofer, Ph.D, Chaffey College.
People form mental concepts of categories of objects, which permit them to respond appropriately to new objects they encounter. Most concepts cannot be strictly defined but are organized around the “best” examples or prototypes, which have the properties most common in the category. Objects fall into many different categories, but there is usually a most salient one, called the basic-level category, which is at an intermediate level of specificity (e.g., chairs, rather than furniture or desk chairs). Concepts are closely related to our knowledge of the world, and people can more easily learn concepts that are consistent with their knowledge. Theories of concepts argue either that people learn a summary description of a whole category or else that they learn exemplars of the category. Recent research suggests that there are different ways to learn and represent concepts and that they are accomplished by different neural systems.
Consider the following set of objects: some dust, papers, a computer monitor, two pens, a cup, and an orange. What do these things have in common? Only that they all happen to be on my desk as I write this. This set of things can be considered a category , a set of objects that can be treated as equivalent in some way. But, most of our categories seem much more informative—they share many properties. For example, consider the following categories: trucks, wireless devices, weddings, psychopaths, and trout. Although the objects in a given category are different from one another, they have many commonalities. When you know something is a truck, you know quite a bit about it. The psychology of categories concerns how people learn, remember, and use informative categories such as trucks or psychopaths.
The mental representations we form of categories are called concepts . There is a category of trucks in the actual physical world, and I also have a concept of trucks in my head. We assume that people’s concepts correspond more or less closely to the actual category, but it can be useful to distinguish the two, as when someone’s concept is not really correct.
Concepts are at the core of intelligent behavior . We expect people to be able to know what to do in new situations and when confronting new objects. If you go into a new classroom and see chairs, a blackboard, a projector, and a screen, you know what these things are and how they will be used. You’ll sit on one of the chairs and expect the instructor to write on the blackboard or project something onto the screen. You do this even if you have never seen any of these particular objects before , because you have concepts of classrooms, chairs, projectors, and so forth, that tell you what they are and what you’re supposed to do with them. Furthermore, if someone tells you a new fact about the projector—for example, that it has a halogen bulb—you are likely to extend this fact to other projectors you encounter. In short, concepts allow you to extend what you have learned about a limited number of objects to a potentially infinite set of entities (i.e. generalization ). Notice how categories and concepts arise from similarity, one of the abstract features of the world that has been genetically internalized into the brain during evolution , creating an innate disposition of brains to search for and to represent groupings of similar things, forming one component of general intelligence. One property of the human brain that distinguishes us from other animals is the high degrees of abstraction in similarity relations that the human brain is capable of encoding compared to the brains of non-human animals (Koenigshofer, 2017).
Simpler organisms, such as animals and human infants, also have concepts ( Mareschal, Quinn, & Lea, 2010 ). Squirrels may have a concept of predators, for example, that is specific to their own lives and experiences. However, animals likely have many fewer concepts and cannot understand complex concepts such as mortgages or musical instruments.
You know thousands of categories, most of which you have learned without careful study or instruction. Although this accomplishment may seem simple, we know that it isn’t, because it is difficult to program computers to solve such intellectual tasks. If you teach a learning program that a robin, a swallow, and a duck are all birds, it may not recognize a cardinal or peacock as a bird. However, this shortcoming in computers may be at least partially overcome when the type of processing used is parallel distributed processing as employed in artificial neural networks (Koenigshofer, 2017), discussed in this chapter. As we’ll shortly see, the problem for computers is that objects in categories are often surprisingly diverse.
Traditionally, it has been assumed that categories are well-defined . This means that you can give a definition that specifies what is in and out of the category. Such a definition has two parts. First, it provides the necessary features for category membership: What must objects have in order to be in it? Second, those features must be jointly sufficient for membership: If an object has those features, then it is in the category. For example, if I defined a dog as a four-legged animal that barks, this would mean that every dog is four-legged, an animal, and barks, and also that anything that has all those properties is a dog.
Unfortunately, it has not been possible to find definitions for many familiar categories. Definitions are neat and clear-cut; the world is messy and often unclear. For example, consider our definition of dogs. In reality, not all dogs have four legs; not all dogs bark. I knew a dog that lost her bark with age (this was an improvement); no one doubted that she was still a dog. It is often possible to find some necessary features (e.g., all dogs have blood and breathe), but these features are generally not sufficient to determine category membership (you also have blood and breathe but are not a dog).
Even in domains where one might expect to find clear-cut definitions, such as science and law, there are often problems. For example, many people were upset when Pluto was downgraded from its status as a planet to a dwarf planet in 2006. Upset turned to outrage when they discovered that there was no hard-and-fast definition of planethood: “Aren’t these astronomers scientists? Can’t they make a simple definition?” In fact, they couldn’t. After an astronomical organization tried to make a definition for planets, a number of astronomers complained that it might not include accepted planets such as Neptune and refused to use it. If everything looked like our Earth, our moon, and our sun, it would be easy to give definitions of planets, moons, and stars, but the universe has not conformed to this ideal.
Borderline items.
Experiments also showed that the psychological assumptions of well-defined categories were not correct. Hampton ( 1979 ) asked subjects to judge whether a number of items were in different categories. He did not find that items were either clear members or clear nonmembers. Instead, he found many items that were just barely considered category members and others that were just barely not members, with much disagreement among subjects. Sinks were barely considered as members of the kitchen utensil category, and sponges were barely excluded. People just included seaweed as a vegetable and just barely excluded tomatoes and gourds. Hampton found that members and nonmembers formed a continuum, with no obvious break in people’s membership judgments. If categories were well defined, such examples should be very rare. Many studies since then have found such borderline members that are not clearly in or clearly out of the category.
McCloskey and Glucksberg ( 1978 ) found further evidence for borderline membership by asking people to judge category membership twice, separated by two weeks. They found that when people made repeated category judgments such as “Is an olive a fruit?” or “Is a sponge a kitchen utensil?” they changed their minds about borderline items—up to 22 percent of the time. So, not only do people disagree with one another about borderline items, they disagree with themselves! As a result, researchers often say that categories are fuzzy , that is, they have unclear boundaries that can shift over time.
A related finding that turns out to be most important is that even among items that clearly are in a category, some seem to be “better” members than others ( Rosch, 1973 ). Among birds, for example, robins and sparrows are very typical . In contrast, ostriches and penguins are very atypical (meaning not typical). If someone says, “There’s a bird in my yard,” the image you have will be of a smallish passerine bird such as a robin, not an eagle or hummingbird or turkey.
You can find out which category members are typical merely by asking people. Table 1 shows a list of category members in order of their rated typicality. Typicality is perhaps the most important variable in predicting how people interact with categories. The following text box is a partial list of what typicality influences.
We can understand the two phenomena of borderline members and typicality as two sides of the same coin. Think of the most typical category member: This is often called the category prototype . Items that are less and less similar to the prototype become less and less typical. At some point, these less typical items become so atypical that you start to doubt whether they are in the category at all. Is a rug really an example of furniture? It’s in the home like chairs and tables, but it’s also different from most furniture in its structure and use. From day to day, you might change your mind as to whether this atypical example is in or out of the category. So, changes in typicality ultimately lead to borderline members.
Intuitively, it is not surprising that robins are better examples of birds than penguins are, or that a table is a more typical kind of furniture than is a rug. But given that robins and penguins are known to be birds, why should one be more typical than the other? One possible answer is the frequency with which we encounter the object: We see a lot more robins than penguins, so they must be more typical. Frequency does have some effect, but it is actually not the most important variable ( Rosch, Simpson, & Miller, 1976 ). For example, I see both rugs and tables every single day, but one of them is much more typical as furniture than the other.
The best account of what makes something typical comes from Rosch and Mervis’s ( 1975 ) family resemblance theory . They proposed that items are likely to be typical if they (a) have the features that are frequent in the category and (b) do not have features frequent in other categories. Let’s compare two extremes, robins and penguins. Robins are small flying birds that sing, live in nests in trees, migrate in winter, hop around on your lawn, and so on. Most of these properties are found in many other birds. In contrast, penguins do not fly, do not sing, do not live in nests or in trees, do not hop around on your lawn. Furthermore, they have properties that are common in other categories, such as swimming expertly and having wings that look and act like fins. These properties are more often found in fish than in birds.
According to Rosch and Mervis, then, it is not because a robin is a very common bird that makes it typical. Rather, it is because the robin has the shape, size, body parts, and behaviors that are very common (i.e. most similar) among birds—and not common among fish, mammals, bugs, and so forth.
In a classic experiment, Rosch and Mervis ( 1975 ) made up two new categories, with arbitrary features. Subjects viewed example after example and had to learn which example was in which category. Rosch and Mervis constructed some items that had features that were common in the category and other items that had features less common in the category. The subjects learned the first type of item before they learned the second type. Furthermore, they then rated the items with common features as more typical. In another experiment, Rosch and Mervis constructed items that differed in how many features were shared with a different category. The more features were shared, the longer it took subjects to learn which category the item was in. These experiments, and many later studies, support both parts of the family resemblance theory.
Many important categories fall into hierarchies , in which more concrete categories are nested inside larger, abstract categories. For example, consider the categories: brown bear, bear, mammal, vertebrate, animal, entity. Clearly, all brown bears are bears; all bears are mammals; all mammals are vertebrates; and so on. Any given object typically does not fall into just one category—it could be in a dozen different categories, some of which are structured in this hierarchical manner. Examples of biological categories come to mind most easily, but within the realm of human artifacts, hierarchical structures can readily be found: desk chair, chair, furniture, artifact, object.
Brown ( 1958 ), a child language researcher, was perhaps the first to note that there seems to be a preference for which category we use to label things. If your office desk chair is in the way, you’ll probably say, “Move that chair,” rather than “Move that desk chair” or “piece of furniture.” Brown thought that the use of a single, consistent name probably helped children to learn the name for things. And, indeed, children’s first labels for categories tend to be exactly those names that adults prefer to use ( Anglin, 1977 ).
This preference is referred to as a preference for the basic level of categorization , and it was first studied in detail by Eleanor Rosch and her students ( Rosch, Mervis, Gray, Johnson, & Boyes-Braem, 1976 ). The basic level represents a kind of Goldilocks effect, in which the category used for something is not too small (northern brown bear) and not too big (animal), but is just right (bear). The simplest way to identify an object’s basic-level category is to discover how it would be labeled in a neutral situation. Rosch et al. ( 1976 ) showed subjects pictures and asked them to provide the first name that came to mind. They found that 1,595 names were at the basic level, with 14 more specific names ( subordinates ) used. Only once did anyone use a more general name ( superordinate ). Furthermore, in printed text, basic-level labels are much more frequent than most subordinate or superordinate labels (e.g., Wisniewski & Murphy, 1989 ).
The preference for the basic level is not merely a matter of labeling. Basic-level categories are usually easier to learn. As Brown noted, children use these categories first in language learning, and superordinates are especially difficult for children to fully acquire. [1] People are faster at identifying objects as members of basic-level categories ( Rosch et al., 1976 ).
Rosch et al. ( 1976 ) initially proposed that basic-level categories cut the world at its joints, that is, merely reflect the big differences between categories like chairs and tables or between cats and mice that exist in the world. However, it turns out that which level is basic is not universal. North Americans are likely to use names like tree, fish , and bird to label natural objects. But people in less industrialized societies seldom use these labels and instead use more specific words, equivalent to elm, trout, and finch ( Berlin, 1992 ). Because Americans and many other people living in industrialized societies know so much less than our ancestors did about the natural world, our basic level has “moved up” to what would have been the superordinate level a century ago. Furthermore, experts in a domain often have a preferred level that is more specific than that of non-experts. Birdwatchers see sparrows rather than just birds, and carpenters see roofing hammers rather than just hammers ( Tanaka & Taylor, 1991 ). This all suggests that the preferred level is not (only) based on how different categories are in the world, but that people’s knowledge and interest in the categories has an important effect.
One explanation of the basic-level preference is that basic-level categories are more differentiated: The category members are similar to one another, but they are different from members of other categories ( Murphy & Brownell, 1985 ; Rosch et al., 1976 ). (The alert reader will note a similarity to the explanation of typicality I gave above. However, here we’re talking about the entire category and not individual members.) Chairs are pretty similar to one another, sharing a lot of features (legs, a seat, a back, similar size and shape); they also don’t share that many features with other furniture. Superordinate categories are not as useful because their members are not very similar to one another. What features are common to most furniture? There are very few. Subordinate categories are not as useful, because they’re very similar to other categories: Desk chairs are quite similar to dining room chairs and easy chairs. As a result, it can be difficult to decide which subordinate category an object is in ( Murphy & Brownell, 1985 ). Experts can differ from novices in which categories are the most differentiated, because they know different things about the categories, therefore changing how similar the categories are.
[1] This is a controversial claim, as some say that infants learn superordinates before anything else (Mandler, 2004). However, if true, then it is very puzzling that older children have great difficulty learning the correct meaning of words for superordinates, as well as in learning artificial superordinate categories (Horton & Markman, 1980; Mervis, 1987). However, it seems fair to say that the answer to this question is not yet fully known.
Now that we know these facts about the psychology of concepts, the question arises of how concepts are mentally represented. There have been two main answers. The first, somewhat confusingly called the prototype theory suggests that people have a summary representation of the category, a mental description that is meant to apply to the category as a whole. (The significance of summary will become apparent when the next theory is described.) This description can be represented as a set of weighted features ( Smith & Medin, 1981 ). The features are weighted by their frequency in the category. For the category of birds, having wings and feathers would have a very high weight; eating worms would have a lower weight; living in Antarctica would have a lower weight still, but not zero, as some birds do live there.
The idea behind prototype theory is that when you learn a category, you learn a general description that applies to the category as a whole: Birds have wings and usually fly; some eat worms; some swim underwater to catch fish. People can state these generalizations, and sometimes we learn about categories by reading or hearing such statements (“The kimodo dragon can grow to be 10 feet long”).
When you try to classify an item, you see how well it matches that weighted list of features. For example, if you saw something with wings and feathers fly onto your front lawn and eat a worm, you could (unconsciously) consult your concepts and see which ones contained the features you observed. This example possesses many of the highly weighted bird features, and so it should be easy to identify as a bird.
This theory readily explains the phenomena we discussed earlier. Typical category members have more, higher-weighted features. Therefore, it is easier to match them to your conceptual representation. Less typical items have fewer or lower-weighted features (and they may have features of other concepts). Therefore, they don’t match your representation as well (less similarity). This makes people less certain in classifying such items. Borderline items may have features in common with multiple categories or not be very close to any of them. For example, edible seaweed does not have many of the common features of vegetables but also is not close to any other food concept (meat, fish, fruit, etc.), making it hard to know what kind of food it is.
A very different account of concept representation is the exemplar theory ( exemplar being a fancy name for an example; Medin & Schaffer, 1978 ). This theory denies that there is a summary representation. Instead, the theory claims that your concept of vegetables is remembered examples of vegetables you have seen. This could of course be hundreds or thousands of exemplars over the course of your life, though we don’t know for sure how many exemplars you actually remember.
How does this theory explain classification? When you see an object, you (unconsciously) compare it to the exemplars in your memory, and you judge how similar it is to exemplars in different categories. For example, if you see some object on your plate and want to identify it, it will probably activate memories of vegetables, meats, fruit, and so on. In order to categorize this object, you calculate how similar it is to each exemplar in your memory. These similarity scores are added up for each category. Perhaps the object is very similar to a large number of vegetable exemplars, moderately similar to a few fruit, and only minimally similar to some exemplars of meat you remember. These similarity scores are compared, and the category with the highest score is chosen . [2]
Why would someone propose such a theory of concepts? One answer is that in many experiments studying concepts, people learn concepts by seeing exemplars over and over again until they learn to classify them correctly. Under such conditions, it seems likely that people eventually memorize the exemplars ( Smith & Minda, 1998 ). There is also evidence that close similarity to well-remembered objects has a large effect on classification . Allen and Brooks ( 1991 ) taught people to classify items by following a rule. However, they also had their subjects study the items, which were richly detailed. In a later test, the experimenters gave people new items that were very similar to one of the old items but were in a different category. That is, they changed one property so that the item no longer followed the rule. They discovered that people were often fooled by such items. Rather than following the category rule they had been taught, they seemed to recognize the new item as being very similar to an old one and so put it, incorrectly, into the same category.
Many experiments have been done to compare the prototype and exemplar theories. Overall, the exemplar theory seems to have won most of these comparisons . However, the experiments are somewhat limited in that they usually involve a small number of exemplars that people view over and over again. It is not so clear that exemplar theory can explain real-world classification in which people do not spend much time learning individual items (how much time do you spend studying squirrels? or chairs?). Also, given that some part of our knowledge of categories is learned through general statements we read or hear, it seems that there must be room for a summary description separate from exemplar memory.
Many researchers would now acknowledge that concepts are represented through multiple cognitive systems. For example, your knowledge of dogs may be in part through general descriptions such as “dogs have four legs.” But you probably also have strong memories of some exemplars (your family dog, Lassie) that influence your categorization. Furthermore, some categories also involve rules (e.g., a strike in baseball). How these systems work together is the subject of current study.
[2] Actually, the decision of which category is chosen is more complex than this, but the details are beyond this discussion.
The final topic has to do with how concepts fit with our broader knowledge of the world. We have been talking very generally about people learning the features of concepts. For example, they see a number of birds and then learn that birds generally have wings, or perhaps they remember bird exemplars. From this perspective, it makes no difference what those exemplars or features are—people just learn them. But consider two possible concepts of buildings and their features in Table 2.
Imagine you had to learn these two concepts by seeing exemplars of them, each exemplar having some of the features listed for the concept (as well as some idiosyncratic features). Learning the donker concept would be pretty easy. It seems to be a kind of underwater building, perhaps for deep-sea explorers. Its features seem to go together. In contrast, the blegdav doesn’t really make sense. If it’s in the desert, how can you get there by submarine, and why do they have polar bears as pets? Why would farmers live in the desert or use submarines? What good would steel windows do in such a building? This concept seems peculiar. In fact, if people are asked to learn new concepts that make sense, such as donkers, they learn them quite a bit faster than concepts such as blegdavs that don’t make sense ( Murphy & Allopenna, 1994 ). Furthermore, the features that seem connected to one another (such as being underwater and getting there by submarine) are learned better than features that don’t seem related to the others (such as being red).
Such effects demonstrate that when we learn new concepts, we try to connect them to the knowledge we already have about the world. If you were to learn about a new animal that doesn’t seem to eat or reproduce, you would be very puzzled and think that you must have gotten something wrong. By themselves, the prototype and exemplar theories don’t predict this. They simply say that you learn descriptions or exemplars, and they don’t put any constraints on what those descriptions or exemplars are. However, the knowledge approach to concepts emphasizes that concepts are meant to tell us about real things in the world, and so our knowledge of the world is used in learning and thinking about concepts.
We can see this effect of knowledge when we learn about new pieces of technology. For example, most people could easily learn about tablet computers (such as iPads) when they were first introduced by drawing on their knowledge of laptops, cell phones, and related technology. Of course, this reliance on past knowledge can also lead to errors, as when people don’t learn about features of their new tablet that weren’t present in their cell phone or expect the tablet to be able to do something it can’t.
One important aspect of people’s knowledge about categories is called psychological essentialism ( Gelman, 2003 ; Medin & Ortony, 1989 ). People tend to believe that some categories—most notably natural kinds such as animals, plants, or minerals—have an underlying property that is found only in that category and that causes its other features. Most categories don’t actually have essences, but this is sometimes a firmly held belief. For example, many people will state that there is something about dogs, perhaps some specific gene or set of genes, that all dogs have and that makes them bark, have fur, and look the way they do. Therefore, decisions about whether something is a dog do not depend only on features that you can easily see but also on the assumed presence of this cause.
Belief in an essence can be revealed through experiments describing fictional objects. Keil ( 1989 ) described to adults and children a fiendish operation in which someone took a raccoon, dyed its hair black with a white stripe down the middle, and implanted a “sac of super-smelly yucky stuff” under its tail. The subjects were shown a picture of a skunk and told that this is now what the animal looks like. What is it? Adults and children over the age of 4 all agreed that the animal is still a raccoon. It may look and even act like a skunk, but a raccoon cannot change its stripes (or whatever!)—it will always be a raccoon.
Importantly, the same effect was not found when Keil described a coffeepot that was operated on to look like and function as a bird feeder. Subjects agreed that it was now a bird feeder. Artifacts don’t have an essence.
Signs of essentialism include (a) objects are believed to be either in or out of the category, with no in-between; (b) resistance to change of category membership or of properties connected to the essence; and (c) for living things, the essence is passed on to progeny.
Essentialism is probably helpful in dealing with much of the natural world, but it may be less helpful when it is applied to humans. Considerable evidence suggests that people think of gender, racial, and ethnic groups as having essences, which serves to emphasize the difference between groups and even justify discrimination ( Hirschfeld, 1996 ). Historically, group differences were described by inheriting the blood of one’s family or group. “Bad blood” was not just an expression but a belief that negative properties were inherited and could not be changed. After all, if it is in the nature of “those people” to be dishonest (or clannish or athletic ...), then that could hardly be changed, any more than a raccoon can change into a skunk.
Research on categories of people is an exciting ongoing enterprise, and we still do not know as much as we would like to about how concepts of different kinds of people are learned in childhood and how they may (or may not) change in adulthood. Essentialism doesn’t apply only to person categories, but it is one important factor in how we think of groups.
Concepts are central to our everyday thought. When we are planning for the future or thinking about our past, we think about specific events and objects in terms of their categories. If you’re visiting a friend with a new baby, you have some expectations about what the baby will do, what gifts would be appropriate, how you should behave toward it, and so on. Knowing about the category of babies helps you to effectively plan and behave when you encounter this child you’ve never seen before. Such inferences from knowledge about a category are highly adaptive and an important component of thinking and intelligence.
Learning about those categories is a complex process that involves seeing exemplars (babies), hearing or reading general descriptions (“Babies like black-and-white pictures”), general knowledge (babies have kidneys), and learning the occasional rule (all babies have a rooting reflex). Current research is focusing on how these different processes take place in the brain. It seems likely that these different aspects of concepts are accomplished by different neural structures ( Maddox & Ashby, 2004 ). However, it is clear that the brain is genetically predisposed to seek out similarities in the environment and to represent groupings of things forming categories that can be used to make inferences about new instances of the category which have never been encountered before. In this way knowledge is organized and expectations from this knowledge allow improved adaptation to newly encountered environmental objects and situations by virtue of their similarity to a known category previously formed (Koenigshofer, 2017).
Another interesting topic is how concepts differ across cultures. As different cultures have different interests and different kinds of interactions with the world, it seems clear that their concepts will somehow reflect those differences. On the other hand, the structure of the physical world also imposes a strong constraint on what kinds of categories are actually useful. The interplay of culture, the environment, and basic cognitive processes in establishing concepts has yet to be fully investigated.
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Murphy, G. (2021). Categories and concepts. In R. Biswas-Diener & E. Diener (Eds), Noba textbook series: Psychology. Champaign, IL: DEF publishers. Retrieved from http://noba.to/6vu4cpkt
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Logical reasoning is of great societal importance and, as stressed by the twenty-first century skills framework, also seen as a key aspect for the development of critical thinking. This study aims at exploring secondary school students’ logical reasoning strategies in formal reasoning and everyday reasoning tasks. With task-based interviews among 4 16- and 17-year-old pre-university students, we explored their reasoning strategies and the reasoning difficulties they encounter. In this article, we present results from linear ordering tasks, tasks with invalid syllogisms and a task with implicit reasoning in a newspaper article. The linear ordering tasks and the tasks with invalid syllogisms are presented formally (with symbols) and non-formally in ordinary language (without symbols). In tasks that were familiar to our students, they used rule-based reasoning strategies and provided correct answers although their initial interpretation differed. In tasks that were unfamiliar to our students, they almost always used informal interpretations and their answers were influenced by their own knowledge. When working on the newspaper article task, the students did not use strong formal schemes, which could have provided a clear overview. At the end of the article, we present a scheme showing which reasoning strategies are used by students in different types of tasks. This scheme might increase teachers’ awareness of the variety in reasoning strategies and can guide classroom discourse during courses on logical reasoning. We suggest that using suitable formalisations and visualisations might structure and improve students’ reasoning as well.
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P21's Framework for twenty-first Century Learning describes critical thinking as an important skill to be successful in professional and everyday life situations in an increasingly complex world (P21, 2015 ). Of great value for critical thinking is ‘reason effectively’, which is explained in the twenty-first century skills framework as “[using] various types of reasoning (inductive, deductive, etc.) as appropriate to the situation” (P21, 2015 , p. 4). Liu, Ludu, and Holton ( 2015 ) support this view and consider valid logical reasoning as a key element for sound critical thinking. Other authors suggest that improving logical reasoning skills as part of higher order thinking skills is an important objective of education (Zohar & Dori, 2003 ).
To support the development of critical thinking, it seems essential that teachers devote attention to students’ strategies to reason logically. So far, not much is known about the reasoning processes of secondary school students in different logical reasoning tasks. Therefore, this article addresses this issue by exploring how 16- and 17-year-old students reason within formal reasoning and everyday reasoning tasks. The information provided by this study seems important to increase teachers’ awareness of reasoning strategies used by students and reasoning difficulties they encounter, as well as to be able to develop instruction materials to support and improve students’ logical reasoning skills.
Halpern ( 2014 ) describes critical thinking as “purposeful, reasoned, and goal-directed” (p. 8) and contends that many definitions of critical thinking in literature use the term “reasoning/logic” (p. 8), so being able to apply the rules of logic can be seen as a requirement for critical thinking. Many studies report difficulties with logical reasoning for different age groups (e.g. Daniel & Klaczynski, 2006 ; Galotti, 1989 ; O’Brien, Shapiro, & Reali, 1971 ; Stanovich, West, & Toplak, 2016 ). Because of those difficulties, it is by no means certain that secondary school students are able to reason logically and thus develop their critical thinking abilities autonomously.
The area of logic can be divided into formal logic and informal logic. Aristotle already differentiated between formal logic with syllogisms described in Analytica Priora and ‘dialectics’ in his combined work Topica exploring arguments and opinions (Aristotle, 2015 version). Almost 2000 years later, Gottlob Frege (1848 – 1925) studied and developed formal systems to analyse thoughts, reasoning, and inferences. Also, he developed the so-called ‘predicate logic’, inspired by Leibniz (1646 – 1716), which is more advanced than the ‘propositional logic’ (Look, 2013 ; Zalta, 2016 ). Nowadays, those types of systems are often called ‘symbolic logic’ with strict validity as a key aspect (De Pater & Vergauwen, 2005 ) in which formal deductive reasoning is applied.
In general, formal systems contain a set of rules and symbols and the reasoning within these systems will provide valid results as long as one follows the defined rules (Schoenfeld, 1991 ). The corresponding reasoning is often called formal reasoning and “characterized by rules of logic and mathematics, with fixed and unchanging premises” (Teig & Scherer, 2016 , p. 1). The same use of formal procedures can be found in definitions of logical reasoning as well. For instance “Logical reasoning involves determining what would follow from stated premises if they were true” (Franks et al., 2013 , p. 146), and “When we reason logically, we are following a set of rules that specify how we ‘ought to’ derive conclusions” (Halpern, 2014 , p. 176).
However, there is no consensus on the term reasoning and it is not exclusively used for formal deductive reasoning or mathematical situations only. Although reasoning in mathematics differs immensely from everyday reasoning (Yackel & Hanna, 2003 ), even reasoning in mathematical proof is not only a formal procedure, but involves discussion, discovery, and exploration (Lakatos, 1976 ) and shows us a need for more informal methods when approaching formal reasoning problems.
In the previous section, we indicated that, dependent on the situation, reasoning demands more than applying rules of logic. For instance, the importance of transforming information as stated by Galotti ( 1989 ): “[Reasoning is a] mental activity that consists of transforming given information … in order to reach conclusions” (p. 333) and the role of samenesses as stated by Grossen ( 1991 ): “Analogical and logical reasoning are strategies for finding and using samenesses. … logical reasoning applies these derived samenesses in order to understand and control our experience” (p. 343).
The notion of broadening formal methods with more informal methods is not new. Toulmin already discusses the limitations of formal logic for all sorts of arguments in his famous book The Uses of Argument ( 1958 ). He distinguishes different logical types to emphasise how logic is used in different fields, such as law, science, and daily-life situations. In his layout of an argument, he schematises the grounds for a claim balanced with reasons that rebut a claim. He also uses qualifiers to indicate the probability of a claim.
Philosophers and educators were also dissatisfied with the dominance of formal logic, that they considered as inappropriate for evaluating real-life arguments, and started in the 1970s an informal logic movement for another approach of analysing arguments stated in ordinary, daily-life language (Van Eemeren et al., 2014 ). One of the major textbooks still in print today is Logical Self-Defense (Johnson & Blair, 2006 ), which covers an introduction in “logical thinking, reasoning, or critical thinking … that focuses on the interpretation and assessment of ‘real life’ arguments” (p. xix). In literature, this is often indicated as informal or everyday reasoning, but this term has various meanings, from reasoning originating from formal systems to all reasoning related to everyday life events (Blair & Johnson, 2000 ; Voss, Perkins, & Segal, 1991 ). Different from formal reasoning, the reasoning and the conclusions depend on the context and can be questioned on their validity as already shown by Toulmin. Therefore, the topics are open for debate and invite to ponder on justifications and objections. The argument, as the result of the reasoning, often concerns open-ended, ill-structured real world problems without one conclusive, correct response (Cerbin, 1988 ; Kuhn, 1991 ). For this, Johnson and Blair ( 2006 ) use “acceptable premises that are relevant to the conclusion and supply sufficient evidence to justify accepting it” (p. xiii). The use of acceptable premises can arise from practical reasons to reach a certain goal and often includes presumptions or presuppositions. Walton ( 1996 ) uses the term ‘presumptive reasoning’ for this kind of arguments, which he sees as dialogues.
Although presumptive reasoning is not always conclusive or accepted by everyone, it is, in particular if full knowledge is unavailable or unobtainable, according to Walton, the best supplement to describe and discuss everyday life reasoning, for which he uses argumentation schemes. Even though Blair ( 1999 ) acknowledges the importance of presumptive reasoning for describing human reasoning and the strength of conclusions derived from the premises, he questions if all arguments are dialogues and discusses the completeness of the schemes.
To sum up, we define informal reasoning as reasoning in ordinary language to construct an argument which requires a critical review of the given premises and transforming of information, as well as finding additional or similar information provided by the problem solver or by external sources.
Now, we have seen that for well-founded reasoning, formal and informal methods are useful, we need to formulate a definition of logical reasoning for this study, which captures both aspects. A definition of logical reasoning should contain both the context and the way of reasoning, which can consist of formal and informal strategies. In other words, a definition of logical reasoning should not be synonymous with formal deductive reasoning. Important key words taken from the previous sections are ‘derive conclusions’ from Halpern and ‘transforming information’ from Galotti. That can be done with rules derived from formal systems, but that is not a necessity, so informal reasoning will also be part of our definition and thus seen as a valid reasoning process. Therefore, we conclude that logical reasoning involves several steps and define logical reasoning for this study as selecting and interpreting information from a given context , making connections and verifying and drawing conclusions based on provided and interpreted information and the associated rules and processes.
Until now, we focused on the ways of reasoning and stressed the importance of the context. If we want to study how students reason in a variety of contexts, we have to differentiate between closed tasks with one correct answer and more open tasks. For this, we will use Galotti’s ( 1989 , p. 335) division: ‘formal reasoning tasks’ and ‘everyday reasoning tasks’. Formal reasoning tasks are self-contained, in which all premises are provided. For those tasks, established procedures are often available which lead to one conclusive answer. In everyday reasoning tasks, premises might be implicit or not provided at all. For those tasks, established procedures are often not available and it depends on the situation when an answer is good enough. In daily-life situations, everyday reasoning problems “are [often] not self-contained” and “the content of the problem typically has potential personal relevance” (Galotti, 1989 , p. 335). For both types of tasks, but for everyday reasoning tasks in particular, selecting and encoding relevant information is of great importance. We will call that the interpretation of the task.
Formal reasoning tasks may be provided in different forms: with symbols and completely in ordinary language without symbols. As shown in Fig. 1 , we differentiate formal reasoning tasks in formally stated and in non-formally stated tasks. Formally stated tasks are stated with a certain set of symbols, for example a task with the premises ‘(1) All A are B. (2) All B are C.’ Non-formally stated tasks are tasks stated in ordinary language, for example a task with the premises ‘(1) All mandarins are oranges. (2) All oranges are fruits.’ For each task, students’ reasoning starts with an interpretation of the given information. That might be either a formal interpretation, in other words, an interpretation within a certain set of symbols (e.g. A ⊆ B ⊆ C ⇒ “All A are C”), or an informal interpretation in ordinary language.
Types of tasks and interpretation
Everyday reasoning tasks are not translatable to formal reasoning tasks and often contain implicit premises as, for instance, in everyday life stories. Like in formal reasoning tasks, students will need to interpret the information in everyday reasoning tasks as well. That can be done completely informally, but a formal representation, such as a schematic overview, might help students to get an overview of the given situation. In this study, we focus both on students’ interpretation and the reasoning strategies that follow from there.
From prior research among university students (e.g. Lehman, Lempert, & Nisbett, 1988 ; Stenning, 1996 ), we conjecture that reasoning in all kinds of situations will benefit from the use of formal representations or formalisations. We will use the term formalisation in its broadest sense, including all sorts of symbols, schematisations, visualisations, formal notations and (formal) reasoning schemes. Stenning ( 1996 ) gives support for the role of (elementary) formal notations and rules by mentioning that “learning elementary logic can [emphasis added] improve reasoning skills” (p. 227) and can help to understand formal thoughts and arguments. Also, Lehman et al. ( 1988 ) found support for the notion that reasoning in general can improve as a result of teaching formal rules within a particular field. Nonetheless, this does not imply that every formalisation is helpful: The chosen representation should support the thinking process for the specific context, rather than that it should capture all aspects (McKendree, Small, Stenning, & Conlon, 2002 ). In this study, we will investigate which formalisations are used by the participants and if those formalisations are beneficial.
Since little is known about the reasoning processes of 16- and 17-year-old students in logical reasoning tasks, our aim is to explore their reasoning strategies. Because of its exploratory nature, we selected, according to the division provided in Fig. 1 , three elementary types of reasoning tasks: two formal reasoning tasks, to be presented with (formally stated) and without (non-formally stated) symbols, and an everyday reasoning task. Our exploratory study was guided by the following research questions: (1) How do students reason towards a conclusion in formal reasoning and everyday reasoning tasks, whether or not by using formalisations? And (2) what kind of reasoning difficulties do they encounter when proceeding to a conclusion?
For this exploratory study, we selected closed tasks (formal reasoning tasks) concerning linear ordering and syllogisms and an open-ended newspaper comprehension task (everyday reasoning task). The formal reasoning tasks were presented formally and non-formally, of which the non-formally stated task is a counter-item of the formally stated one. A non-formally stated counter-item is a translation of the corresponding formally stated task in ordinary language and vice versa. Both tasks have similar conclusions as final answer, so that the reasoning processes can be compared. Figures 2 and 3 show these formal reasoning tasks, both formally stated and non-formally stated.
Formal reasoning tasks about linear ordering, formally and non-formally stated
Formal reasoning tasks about invalid syllogisms, formally and non-formally stated
Figure 4 shows the everyday reasoning task and this task does not have a counter-item. This newspaper task is an open-ended task with implicit premises and hidden assumptions. In this task, students have to reconstruct the line of the argument. An expert in logic validated all items by checking wording and comprehensibility of the tasks.
Everyday reasoning task, reasoning in a newspaper article
This selection of tasks captures each category shown in Fig. 1 in which we expect different reasoning strategies and contains familiar and unfamiliar tasks to our students. For each task, we provide example interpretations and solutions below. These solutions are used as reference solutions to check the correctness of students’ answers, but, of course, the reasoning towards a conclusion can differ. In the everyday reasoning task in particular, different formulations are possible.
The linear ordering tasks (see Fig. 2 ), which are formal reasoning tasks, have ‘P > S’ and ‘Peter is older than Sally’ as correct answers respectively. If taken a formal interpretation, the reasoning can be P > Q > R > S for the order of the letters. If taken an informal interpretation, you can take example ages for the four persons. For example, if Peter is 50 years old, then Quint can be 20 years old, because Peter is older than Quint. Rosie is younger than Quint, so Rosie can be 10 years old. Rosie is older than Sally, so Sally can be 5 years old. In conclusion, if Peter is 50 years old and Sally 5 years old, then Peter must be older than Sally.
The syllogism tasks (see Fig. 3 ), which are formal reasoning tasks too, should have ‘does not follow necessarily from the given premises’ as correct answer as the only valid conclusive option. For the formally stated version of the syllogism task, possible formal and informal interpretations are visualised in Fig. 5 . At the left, the given syllogism is translated into ordinary language completely and thus called an informal interpretation. In this case, it is example-based with a counterexample in ordinary language, which is, of course, a sufficient explanation why the conclusion does not necessarily follow from these premises. However, it is important to recognise that an example does not always lead to a general conclusion, in particular for valid syllogisms, so in that case, there must be a translation back to the formal setting.
Formal and informal interpretations of the formally stated syllogism task
The formal interpretation with Euler diagrams at the right of Fig. 5 shows that C does not necessarily overlap with A. In this interpretation, the original given set of letter symbols is used. Similar diagrams can be drawn for the non-formally stated version of the task.
The everyday reasoning task (Fig. 4 ) requires students to (1) identify the premises (reasons) leading to the author’s conclusion, and (2) to hypothesise how these premises might be connected to the conclusion by using general knowledge or evidence that might support the author’s conclusion. Our example solution (see Fig. 6 ) is scheme-based with phrases in ordinary language. We analyse such a scheme as a formal interpretation in which the three reasons (the identified premises) are linked directly or indirectly to the author’s conclusion. For the third reason, one needs an additional reasoning step by mentioning another hidden assumption to make the argument complete. We assume that there is sufficient general knowledge on this subject among the participants. The arrows represent if-then statements and are not only part of the formal scheme, but also formalisations in themselves.
Formal scheme for the everyday reasoning task
Nevertheless, the if-then statements in the scheme can be explained in full sentences too. For the first two reasons, that will look like ‘If people smoke or inhale particulate matter, then it will affect their health and thus shorten their life.’ Such considerations based on common knowledge still show the connection, but it is not yet formalised, neither with a scheme, nor with any symbols and thus considered as a completely informal interpretation (see Fig. 1 ). As soon as one introduces logical symbols, we will call those symbols formalisations. In combination with the if-then rule, the sentence can be represented as ‘(smoking ∨ inhaling particulate matter) ⇒ unhealthy ⇒ shorter life’.
Our participants are Dutch secondary school students in their penultimate year of pre-university secondary education (11th graders) and volunteered to participate in think-aloud sessions. The first author of this article was their teacher and they all signed an informed consent. These students did not take advanced mathematics or science, but followed a mathematics course in which logical reasoning has recently become a compulsory domain (College voor Toetsen en Examens, 2016 ). This study was conducted before the participants received teaching in logical reasoning. In this article, work is discussed from two male (Edgar, James) and two female students (Anne, Susan).
We conducted task-based interviews in which students solved logical reasoning tasks aloud (Goldin, 2000 ; Van Someren, Barnard, & Sandberg, 1994 ). The interviews were conducted in Dutch and recorded with a smartpen so that verbal and written information could be connected. The students were asked to say aloud everything they were thinking of. The interviewer, who is the first author of this article, refrained from commenting as much as possible, so that ‘free problem-solving’ was a key aspect of the sessions. If a student did not understand the task or thought it was done, the interviewer would ask additional (clarification) questions, but did never provide feedback on the given answers.
The transcripts of the interviews were analysed in Dutch and selected parts were translated to English for this article. Students’ task solving was analysed qualitatively in an interpretive way and data-driven (Cohen, Manion, & Morrison, 2007 ). To get a clear picture of the reasoning process, the data sources, interview transcripts and students’ written notes, were analysed according to our definition of logical reasoning. Our analysis included the following steps: (1) students’ understanding of the task, (2) students’ interpretation of the task, (3) students’ reasoning process and strategies used, (4) students’ use of formalisations and (5) the correctness of students’ final answers. If students switch between interpretations, we will call the predominant interpretation, their main interpretation.
Students’ reasoning in counter-items is intended as an exploration of possible variation in reasoning. Because students worked on only one of each two counter-items, we cannot analyse the differences between individual students’ strategies on alternative versions of similar closed tasks.
To judge the correctness of their final answers, students’ written notes, as well as the interview transcripts, are used and compared. Possible differences are marked and combined with their interpretations and reasoning. We have to note that the verbal explanations in itself can be seen as informal, because if students are asked to do tasks aloud, they use ordinary language, but if explained with a (given) set of symbols, the interpretation of the task can still be formal. Furthermore, the verbal explanations are linked to written notes, in which possible use of formalisations is clearly visible.
Table 1 provides an overview of the results. Thereafter, for each task students’ reasoning will be illustrated in detail.
Formal reasoning tasks with linear ordering (see Fig. 2 ) are familiar to the students because these types of tasks are common in primary and secondary education. We summarise the findings first: All four students used rule-based strategies, but their initial interpretation differed. All answers were correct and well-reasoned. Only one student came up with an additional formalisation other than the given symbols. She used a very suitable tool, a number line representation with formal letters symbols, to get a clear overview of the order. We will present a detailed description of the four students.
Edgar interprets the task in a formal way by copying the formal notation, see first three lines in Fig. 7 . After writing that down, his first statements are switching to example-based reasoning (informal interpretation) that involves filling in some numbers (line [1] in transcript). After that, he quickly weighs his two interpretations (lines [2] and [3]) and switches back to the formal situation, by comparing the given letters P, R and S with the symbol for ‘greater than’ (line [4] and Fig. 7 ). Although the verbal explanations are in words, inherent to thinking aloud, he solves the task by following mathematical rules by staying in the formal system with the corresponding formal symbols. This way of reasoning provides the correct answer quickly and using the given symbols only gives a clear structure: P > R, R > S, P > S.
Formally stated linear ordering task at the left, Edgar’s written notes at the right
Edgar: [1] well, yes, you could just fill in numbers of course as an example, [2] well oh no, let's wait[3] we are not going to do that at first [4] uhm, P is greater than Q, so P is also greater than R, …
After reading the task, Anne starts immediately with a translation of the formal symbols into expressions in ordinary language by writing down ‘greater than’ and ‘less than’ in full, thus giving an informal transformation of most of the formally stated task (see Fig. 8 ). Although she still reasons with the given formal letters, she switches to ordinary language for applying the mathematical rules. She provides the correct answer.
Anne’s written notes at the left, English translation at the right
Susan translates the non-formally stated version of this formal reasoning task immediately into a formal situation with letter abbreviations for the names and the symbols > and < for ‘older than’ and ‘younger than’. Besides these formal symbols, she puts the letters in sequential order horizontally, which can be seen as a ‘number line’ representation with formal letter symbols, starting with P-Q-R reasoning that Q must be in the middle, see Fig. 9 . We call that another formalisation. After adding S as well, she comes to the right conclusion that Peter must be older than Sally, which is a translation from her formal system to the conclusion asked for in ordinary language.
Susan’s written notes
James reasons in words within the non-formally stated version of this task leading to a correct conclusion. We call his interpretation informal with a correct application of mathematical rules. After the confirmation that he has to write his reasoning down, his written explanation is completely in ordinary language, using the given names and the phrases ‘older than’ and ‘younger than’ (see Fig. 10 ). So, James’s interpretation is completely informal without switching.
James’s written notes at the left, English translation at the right
Formal reasoning tasks with syllogisms (see Fig. 3 ) are unfamiliar tasks to these students because they are not used to reasoning within syllogisms. We summarise the findings first: Three of the four students used an informal interpretation, but only two students provided a correct answer. The formally stated version caused difficulties due to not understanding the task or due to incomplete translations to an informal example. Also, the misinterpretation of ‘are’ and the confusion between ‘all’ and ‘some’ are noteworthy. We also found that a recognisable non-formally stated context can support the reasoning, despite some hindrance of real-life experiences concerning the context as well. We present a detailed description of the four students.
Susan shows that she understands that she has to accept the two premises in this formal reasoning task, regardless of their truths by writing “true” behind it, see Fig. 11 . Her next step is formalising the given statements even further by introducing the equality sign, see first lines in her written notes in Fig. 12 , so she interprets the task completely formally.
Susan accepts the given premises, English translation at the right
Formalisations used by Susan, English translation at the right
Susan tries to reason with the given letters four times (see four sections transcript) before she gives up. Again, her verbal explanations are in ordinary language, of course, inherent to thinking aloud, but she uses the given letters and stays in the formal system, so we call that a formal interpretation. In her first try (lines [1] – [7] in transcript), she states that A and B are equal (line [5]), but she cannot connect this with C. In her second try (lines [8] – [14]), she starts with stating that A and B are equal, but cannot connect C with that although saying that some B are not C (line [10]). In her third try (lines [15] – [17]), she says, once more, that A and B are equal, but she cannot connect that with C, because she does not know which B’s are C. The fourth time she writes down the last two lines shown in Fig. 12 , connecting some with a symbol for approximately, but that does not help either (lines [18] – [25]). It is important to notice that she uses the equality sign each time as equal to which conflicts with the original premise containing an inclusion.
After underlining her conclusion ‘A ≈ C’ in the fourth try, she gives up and sighs: “I just do not understand the logic of all this” (line [25]). Susan only reasoned with the given letters and formal symbols and did not switch to an informal situation.
Susan: [1] all A are B, …, so is equal [2] but some of those are C[3] so some are not, some A are not either, some A are [4] … mmm … [5] all A are B, so A and B are equal [6] some B are C, so some B are only A [7] and some B are C … mmm … [start reasoning from the beginning again] [8] all A are B, A and B are equal [9] some of those are C and some are not [10] some B are not C [11] some A, that is also B [12] some B … some A are C [13] … but all A are B, and some B are C, some A are C [rereading given syllogism] [14] no, I don’t think so [start reasoning from the beginning again] [15] I think that, … uhm …, if all A are B, A and B are equal [16] but some B are C, so some of those B’s, that has to be the case, do not necessarily have to be A, because you do not know which B’s are C, because those are equal to C, and A and B are equal, some A are C [17] ow, I really think this is difficult … [start reasoning from the beginning once more] [18] okay, all A are B [writes down A=B] [19] some B are C, so approximately [writes down B≈C] [20] and some A are C, but A and B are equal[21] some of those B are C [writes down behind A=B: ➔ B≈C] [22] and some A are C [writes down A≈C] [23] so, my conclusion, … mmm … [underlines A≈C] [24] I really don’t know [25] I just do not understand the logic of all this
James recognises that he does not know how to solve this task in a formal way by expressing “I don’t know”, so he switches to an informal interpretation of the formally stated task: starting with searching for an example in ordinary language. This can be seen as analogy- and example-based reasoning. His explanation is closely related to our example in Fig. 5 , but James only looks for one valid example instead of a counterexample. He chooses an example in which ‘some’ represents all apes (line [3] in transcript), because the set of apes not being mammals is empty. We assume that he did not recognise that because of his incorrect conclusion. He tries to use a logical structure ‘if-then’ (lines [2] – [4]) as well, but that does not solve the problem. After the valid conclusion of his example in ordinary language, he tries to explain the validity of his conclusion in a more formal way with the given letters (lines [5] – [7]) and writes that down as well (see Fig. 13 ). For this, James also states that A and B are equal (line [5]) in the same way as Susan did, and is not able to provide a more precise explanation after clarification questions by the interviewer.
James: [1] okay, well, I am going to have a look with a similar example I think [2] if, uhm, all humans are apes [3] some apes are mammals [4] then some humans are … also mammals [5] so, I think it is correct, because A and B are equal, [6] because that is necessarily true, [7] so if that’s the case for some B, it is also the case for A
Edgar’s interpretation of the non-formally stated version of this formal reasoning task is informal. He draws the correct conclusion quite easily (line [2] in transcript). He also explains, although this is not necessarily true, the possibility that some flowers might refer to roses as well as to other flowers (line [6]), which shows a notion of the rules of logic. In his written notes (see Fig. 14 ) he also shows that the word flower could contain more than one type of flowers. This reflection is quite strong and shows insight in the generality of a syllogism.
Edgar’s written notes at the left, English translation at the right
Edgar: [1] uhm… well… let’s see… [2] yes, you would say that this does not follow logically, because some flowers does not necessarily refer to all roses [3] let’s see [4] some, yes, [5] uhm… [6] it does not have to mean that roses fade quickly since some flowers might also be daisies or, well, something, or other flowers consequently
Anne also draws the right conclusion in the non-formally stated version of this task. She uses an informal interpretation and comes up with a correct answer quickly (line [1] in transcript) and provides a more complete explanation in her next sentence (line [2]), which is similar to her written answer (see Fig. 15 ). However, Anne is not completely sure about her answer. Asked for an explanation, she says that her uncertainty comes from her knowledge about fading flowers (line [8] and [9]), although she recognises that one cannot conclude that from these premises, which shows that she understands the rules of logic.
Anne: [1] You do not know if it’s the roses that fade, so you also don’t know if some roses fade quickly. [2] All roses can still be flowers, and some flowers can still fade quickly, but that does not have to mean that roses [sighs] fade quickly. [3] Yes, I think so. [4] I am less certain about this one. … [5] because roses can still be flowers, but, ow wait, and [6] … that does not have to mean that, per se, the roses fade quickly, … Intvwr: [7] And why are you less sure than in the previous task? Anne: [8] uhm… yes, because, some flowers fade quickly, yes, I don’t know, I know, I think it’s difficult to explain, but I am just more in doubt here, because … [9] I was thinking because I know, of course, that there are many other species of flowers than only roses, only from these premises you cannot see that of course
An everyday reasoning task about the analysis of a newspaper article (see Fig. 4 ) is considered unfamiliar to these students. We summarise the findings first: Both students used informal interpretations and only some basic formalisations to sum up reasons or make connections even though one of the students (Susan) provided to some extent a schematic overview. Although not essential, they did not build a (strong) formal scheme as, for example, shown in Fig. 6 . We present a detailed description of the two students.
Susan starts this task with identifying the three premises (step 1) in a structured way by writing down the three reasons mentioned in the article behind bullets (see first three lines Fig. 16 ). Thereafter, she takes her time to reconsider these reasons, the wording of the task and the phrase ‘hidden assumption’. She writes down “the hidden assumption is that people from Rotterdam live less healthy”, which hypothesises how the premises are linked to the conclusion (step 2). She explains that “it has to do with people’s health” because of the first two reasons, smoking and worse environment, but Susan struggles with an explanation for the third reason: lower education and income level (see line [1] in transcript). This reason demands more evidence. Susan implies that poorer families are the missing connection for the lower income (line [4]). For that, she uses another formalisation, which structures her written notes: an arrow to make the connection. Verbally, she provides a further explanation for the assumption ‘poorer families’ (line [5]), but she did not write that down.
Susan’s written notes at the left, English translation at the right
Overall, in her verbal explanation, she has connected all the mentioned reasons with a hidden assumption leading to her main assumption ‘poorer health’, which she underlines as well. In her written notes, she uses formalisations at three moments: at the beginning (bullets) for the first step involving identifying the premises, and arrows at two times for connections, either with hypothesised evidence based on her own knowledge (step 2), or to emphasise the main hidden assumption. Her notes provide more or less a schematic overview, but Susan did not compile a complete formal scheme.
Susan: [1] ... mmm, so ... the lower education and income level what does that have to do with ... lower level of education, ... mmm ..., yes, the amount of smokers has to do with health and the high concentration of particulate matter in the air, so it means that the health of people from Rotterdam is worse than the health of other people in the Netherlands [2] uh, to link, explain how the reasons mentioned are linked to the shorter life, by describing the hidden assumption, uhm ... [3] The shorter life is caused by ... the ... poorer health in Rotterdam compared to other Dutch people. [4] Then, the hidden assumption is that poorer health and ... maybe, uh ... lower education and income level, so perhaps poorer families [draws an arrow to connect this with lower education and income level] [5] ... and they may not buy very expensive and organic food and everything, so they would live less healthy or something, I think the hidden assumption is that they eat less healthy, or live less healthy lives especially, yes that’s what I think [6] This is what I think, the poorer health [underlines poorer health], that’s the hidden assumption. [adds arrow]
Anne underlines the three main reasons in the text: smokers, particulate matter, lower education and income level, which shows that she identified the premises (step 1). After that, she lists the three reasons behind bullets (see Fig. 17 ). That is the only formalisation she uses. The rest of the reasoning, verbal and written, is done in ordinary language. For the second and third reason, she provides a hidden assumption: “particulate matter is bad for someone and thus shortens one’s life, and the lower education level and the lower income level leads to poorer living conditions and thus shortens one’s life”. She uses her own knowledge to state that particulate matter is bad for someone’s health and to hypothesise that a lower income level leads to poorer living conditions (step 2). However, she forgets to provide a connection for the first reason, so the interviewer asked for further explanations before she added “bad for you and thus” (see top line Fig. 17 ) for the connection between smoking and shortening one’s life.
Overal, Anne identified the premises quite quickly and provided support for the reasons easily. Probably, she assumed that the connection between smoking and a shorter life was generally known, so that she only provided additional evidence after a clarification question by the interviewer.
The purpose of this study was to gain insights into the reasoning processes of 16- and 17-year-old pre-university secondary school students on logical reasoning tasks, aimed at fostering their critical thinking skills as an important objective in the twenty-first century skills framework (P21, 2015 ). In this exploratory study, we investigated (1) the way of reasoning students used in formal reasoning and everyday reasoning tasks and their use of formalisations, and (2) the difficulties they encounter in their reasoning.
In the linear ordering tasks and in the syllogism tasks, students used rule-, analogy- and example-based reasoning strategies. In the newspaper article task, students reasoned partly scheme-based, but mainly in ordinary language only. Except for the formally stated syllogism task, students used appropriate strategies to find correct answers. Although the linear ordering tasks are familiar to the students, both formats (formally and non-formally stated) led to formal and informal interpretations.
Students do not always feel certain about their method and answer, in particular in the syllogism tasks and in the everyday reasoning task. The incomplete written answers in the everyday reasoning task show that our students probably have doubts if their answers are good enough, because they have the feeling that more answers are possible. Even though the newspaper task is taken from an everyday life context and—we expect—recognisable and meaningful, students still fulfil a task for which they expect that there should be one correct answer, as is common in mathematics tasks (e.g. Jäder, Sidenvall, & Sumpter, 2017 ). The doubt students express is in line with Galotti’s ( 1989 ) description for everyday reasoning tasks, because she states that “it is often unclear whether the current ‘best’ solution is good enough” (p. 335) in contrast to formal reasoning tasks where “it is typically unambiguous when the problem is solved” (p. 335).
In our formally stated syllogism task, the students misinterpreted the phrases ‘all…are…’ and ‘some…are…’. Consequently, they did not see that their representations, such as the use of the equality sign as a formal symbol, were not suitable. Susan was completely stuck in the formally stated version and could not find a way out. The misuse of the equality sign (=) for ‘all… are…’ is a common mistake (e.g. Galotti, 1989 , p. 336). Stenning and Van Lambalgen ( 2008 ) also describe difficulties with understanding and interpreting syllogisms.
An overview of our findings is visualised in a scheme (Fig. 18 ) as an extension of Fig. 1 . We showed that students’ initial interpretations, their first thoughts, do not always match with their later choices, so students seem to switch between formal and informal interpretations. This is visualised by the arrow in the scheme. The strategies included in this scheme are derived from our exploratory study among 16- and 17-year-old students and might not provide a complete overview. Consequently, the overview can be supplemented with argumentation schemes based on presumptive reasoning (Walton, 1996 ; Walton, Reed, & Macagno, 2008 ) in further research. Our everyday reasoning task gives only a limited view of students’ possible reasoning strategies, because students were only asked to identify the premises and to use their own knowledge to find connections with the conclusion. They were not asked to find rebuttals or further backing of the claims. The connections provided by the students should be sufficient for justifiable reasoning. This correspondents to the earlier mentioned description from Johnson and Blair ( 2006 ) about acceptable premises.
Types of tasks combined with students’ interpretations and reasoning strategies
Each of the strategies shown in Fig. 18 can be supported by the use of formalisations. For example, in the non-formally stated linear ordering task, Susan used letter abbreviations, mathematical symbols and a number line representation. For cases in which students reason in ordinary language without clearly showing causality, comparison or examples, we added the category ‘informal reasoning’. This category is based on our definition of informal reasoning in the corresponding section in the Theoretical Background. We believe it is important to present ‘informal reasoning’ as separate category in the scheme, because students still managed to construct an argument in ordinary language, but without clearly showing a visible reasoning strategy, such as rule-based, example-based, scheme-based, et cetera.. Therefore, we used a dotted line in Fig. 18 . Consequently, in that case, formalisations can only be used to a certain extent as, for example, shown by Anne in her analysis of the newspaper article where she separated her three informal arguments by bullets.
In this article, we hypothesised that suitable formalisations can support the reasoning process and summarised those tools at the right-hand side of Fig. 18 . We believe that our hypothesis is strengthened by the findings in this exploratory study. Symbols (like ‘greater than’ and ‘less than’, or the equality sign) and letter abbreviations are suitable tools to shorten notations, while other tools (like a number line representation) are strong tools to visualise information. Although not used by our students, Venn and Euler diagrams are also strong tools to visualise data. However, it is our conviction that the use of formalisations, including visualisations such as Venn and Euler diagrams, is teachable and can be linked to the strategies used by the students, also in everyday reasoning tasks.
A limitation of the study is related to our choice of tasks. In our selection of tasks, we used formally stated tasks and non-formally stated tasks as counter-items for similar reasoning problems. In our design, different students worked on one of the counter-items and therefore, we could not compare the performance of an individual student on both tasks. The non-formally stated tasks were more easily to interpret by the students and led to other strategies, because their prior knowledge was helpful. Hintikka ( 2001 ) explains that “in real-life reasoning, even when it is purely deductive, familiarity with the subject matter can be strategically helpful” (p. 46). On the other hand, sometimes our students may doubt their answers, because premises in the task (e.g. Anne in the non-formally stated syllogism task) might conflict with their prior knowledge. In general, this means that our counter-items (formally versus non-formally stated tasks) cannot be considered as equivalent.
Despite the fact that our study has a limitation in the number of participants (small and selective sample) and a limited number of tasks, the information in Fig. 18 shows a variety of reasoning strategies, which is important for teachers to understand the diversity of students’ reasoning and possible difficulties in the interpretations of tasks, in particular for tasks that are not familiar to students or lead to incorrect answers.
This study not only shows the complex matter of reasoning and everyday life reasoning in particular, it also confirms that more research is needed as already mentioned by Galotti ( 1989 , 2017 ). Our exploratory study is a first step to get insights in the reasoning process of 16- and 17-year-old pre-university students and shows a gap between their verbal and written explanations. We will continue our research for an in-depth understanding. Unfamiliar tasks, such as all sorts of non-formally stated syllogisms (formal reasoning tasks) and everyday reasoning tasks seem to be useful contexts to investigate how students solve reasoning tasks and which formalisations, including visualisations, they use. Our definition of logical reasoning, mentioned in the Theoretical Background, fits this future research.
Our results show that students do not structure everyday life contexts automatically, so it is plausible that similar difficulties occur in authentic everyday life reasoning too. In future research, we intend to show that students may be supported by learning more structured reasoning strategies and the use of formalisations and visualisations.
One of the key aspects for lessons in logical reasoning must be classroom discourse when solving reasoning tasks. Lakatos ( 1976 ) already stressed the importance of dialogue in the construction of mathematical and logical reasoning. Our research might increase teachers’ awareness of that importance and, more practically, for which Fig. 18 serves as a guideline for discussion. Different interpretations and possible strategies used by students are made explicit and can be used as input for classroom discussions. We suggest that formalisations and visualisations are part of those discussions and might establish a deeper understanding. Above all, logical reasoning tasks where several ways of reasoning are possible, are highly connected to the twenty-first century skills (P21, 2015 ), and thus with the development of critical thinking skills.
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This work is part of the research programme Doctoral Grant for Teachers with project number 023.007.043, which is (partly) financed by the Netherlands Organisation for Scientific Research (NWO).
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Bronkhorst, H., Roorda, G., Suhre, C. et al. Logical Reasoning in Formal and Everyday Reasoning Tasks. Int J of Sci and Math Educ 18 , 1673–1694 (2020). https://doi.org/10.1007/s10763-019-10039-8
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DOI : https://doi.org/10.1007/s10763-019-10039-8
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1 MTA-SZTE Digital Learning Technologies Research Group, Center for Learning and Instruction, University of Szeged, 6722 Szeged, Hungary
2 MTA-SZTE Digital Learning Technologies Research Group, Institute of Education, University of Szeged, 6722 Szeged, Hungary; uh.degezs-u.yspde@ranlomyg
The data used to support the findings cannot be shared at this time as it also forms part of an ongoing study.
Complex problem solving (CPS) is considered to be one of the most important skills for successful learning. In an effort to explore the nature of CPS, this study aims to investigate the role of inductive reasoning (IR) and combinatorial reasoning (CR) in the problem-solving process of students using statistically distinguishable exploration strategies in the CPS environment. The sample was drawn from a group of university students (N = 1343). The tests were delivered via the eDia online assessment platform. Latent class analyses were employed to seek students whose problem-solving strategies showed similar patterns. Four qualitatively different class profiles were identified: (1) 84.3% of the students were proficient strategy users, (2) 6.2% were rapid learners, (3) 3.1% were non-persistent explorers, and (4) 6.5% were non-performing explorers. Better exploration strategy users showed greater development in thinking skills, and the roles of IR and CR in the CPS process were varied for each type of strategy user. To sum up, the analysis identified students’ problem-solving behaviours in respect of exploration strategy in the CPS environment and detected a number of remarkable differences in terms of the use of thinking skills between students with different exploration strategies.
Problem solving is part and parcel of our daily activities, for instance, in determining what to wear in the morning, how to use our new electronic devices, how to reach a restaurant by public transport, how to arrange our schedule to achieve the greatest work efficiency and how to communicate with people in a foreign country. In most cases, it is essential to solve the problems that recur in our study, work and daily lives. These situations require problem solving. Generally, problem solving is the thinking that occurs if we want “to overcome barriers between a given state and a desired goal state by means of behavioural and/or cognitive, multistep activities” ( Frensch and Funke 1995, p. 18 ). It has also been considered as one of the most important skills for successful learning in the 21st century. This study focuses on one specific kind of problem solving, complex problem solving (CPS). (Numerous other terms are also used ( Funke et al. 2018 ), such as interactive problem solving ( Greiff et al. 2013 ; Wu and Molnár 2018 ), and creative problem solving ( OECD 2010 ), etc.).
CPS is a transversal skill ( Greiff et al. 2014 ), operating several mental activities and thinking skills (see Molnár et al. 2013 ). In order to explore the nature of CPS, some studies have focused on detecting its component skills ( Wu and Molnár 2018 ), whereas others have analysed students’ behaviour during the problem-solving process ( Greiff et al. 2018 ; Wu and Molnár 2021 ). This study aims to link these two fields by investigating the role of thinking skills in learning by examining students’ use of statistically distinguishable exploration strategies in the CPS environment.
According to a widely accepted definition proposed by Buchner ( 1995 ), CPS is “the successful interaction with task environments that are dynamic (i.e., change as a function of users’ intervention and/or as a function of time) and in which some, if not all, of the environment’s regularities can only be revealed by successful exploration and integration of the information gained in that process” ( Buchner 1995, p. 14 ). A CPS process is split into two phases, knowledge acquisition and knowledge application. In the knowledge acquisition (KAC) phase of CPS, the problem solver understands the problem itself and stores the acquired information ( Funke 2001 ; Novick and Bassok 2005 ). In the knowledge application (KAP) phase, the problem solver applies the acquired knowledge to bring about the transition from a given state to a goal state ( Novick and Bassok 2005 ).
Problem solving, especially CPS, has frequently been compared or linked to intelligence in previous studies (e.g., Beckmann and Guthke 1995 ; Stadler et al. 2015 ; Wenke et al. 2005 ). Lotz et al. ( 2017 ) observed that “intelligence and [CPS] are two strongly overlapping constructs” (p. 98). There are many similarities and commonalities that can be detected between CPS and intelligence. For instance, CPS and intelligence share some of the same key features, such as the integration of information ( Stadler et al. 2015 ). Furthermore, Wenke et al. ( 2005 ) stated that “the ability to solve problems has featured prominently in virtually every definition of human intelligence” (p. 9); meanwhile, from the opposite perspective, intelligence has also been considered as one of the most important predictors of the ability to solve problems ( Wenke et al. 2005 ). Moreover, the relation between CPS and intelligence has also been discussed from an empirical perspective. A meta-analysis conducted by Stadler et al. ( 2015 ) selected 47 empirical studies (total sample size N = 13,740) which focused on the correlation between CPS and intelligence. The results of their analysis confirmed that a correlation between CPS and intelligence exists with a moderate effect size of M(g) = 0.43.
Due to the strong link between CPS and intelligence, assessments of these two domains have been connected and have overlapped to a certain extent. For instance, Beckmann and Guthke ( 1995 ) observed that some of the intelligence tests “capture something akin to an individual’s general ability to solve problems (e.g., Sternberg 1982 )” (p. 184). Nowadays, some widely used CPS assessment methods are related to intelligence but still constitute a distinct construct ( Schweizer et al. 2013 ), such as the MicroDYN approach ( Greiff and Funke 2009 ; Greiff et al. 2012 ; Schweizer et al. 2013 ). This approach uses the minimal complex system to simulate simplistic, artificial but still complex problems following certain construction rules ( Greiff and Funke 2009 ; Greiff et al. 2012 ).
The MicroDYN approach has been widely employed to measure problem solving in a well-defined problem context (i.e., “problems have a clear set of means for reaching a precisely described goal state”, Dörner and Funke 2017, p. 1 ). To complete a task based on the MicroDYN approach, the problem solver engages in dynamic interaction with the task to acquire relevant knowledge. It is not possible to create this kind of test environment with the traditional paper-and-pencil-based method. Therefore, it is currently only possible to conduct a MicroDYN-based CPS assessment within the computer-based assessment framework. In the context of computer-based assessment, the problem-solvers’ operations were recorded and logged by the assessment platform. Thus, except for regular achievement-focused result data, logfile data are also available for analysis. This provides the option of exploring and monitoring problem solvers’ behaviour and thinking processes, specifically, their exploration strategies, during the problem-solving process (see, e.g., Chen et al. 2019 ; Greiff et al. 2015a ; Molnár and Csapó 2018 ; Molnár et al. 2022 ; Wu and Molnár 2021 ).
Problem solving, in the context of an ill-defined problem (i.e., “problems have no clear problem definition, their goal state is not defined clearly, and the means of moving towards the (diffusely described) goal state are not clear”, Dörner and Funke 2017, p. 1), involved a different cognitive process than that in the context of a well-defined problem ( Funke 2010 ; Schraw et al. 1995 ), and it cannot be measured with the MicroDYN approach. The nature of ill-defined problem solving has been explored and discussed in numerous studies (e.g., Dörner and Funke 2017 ; Hołda et al. 2020 ; Schraw et al. 1995 ; Welter et al. 2017 ). This will not be discussed here as this study focuses on well-defined problem solving.
Frensch and Funke ( 1995 ) constructed a theoretical framework that summarizes the basic components of CPS and the interrelations among the components. The framework contains three separate components: problem solver, task and environment. The impact of the problem solver is mainly relevant to three main categories, which are memory contents, dynamic information processing and non-cognitive variables. Some thinking skills have been reported to play an important role in dynamic information processing. We can thus describe them as component skills of CPS. Inductive reasoning (IR) and combinatorial reasoning (CR) are the two thinking skills that have been most frequently discussed as component skills of CPS.
IR is the reasoning skill that has been covered most commonly in the literature. Currently, there is no universally accepted definition. Molnár et al. ( 2013 ) described it as the cognitive process of acquiring general regularities by generalizing single and specific observations and experiences, whereas Klauer ( 1990 ) defined it as the discovery of regularities that relies upon the detection of similarities and/or dissimilarities as concerns attributes of or relations to or between objects. Sandberg and McCullough ( 2010 ) provided a general conclusion of the definitions of IR: it is the process of moving from the specific to the general.
Csapó ( 1997 ) pointed out that IR is a basic component of thinking and that it forms a central aspect of intellectual functioning. Some studies have also discussed the role of IR in a problem-solving environment. For instance, Mayer ( 1998 ) stated that IR will be applied in information processing during the process of solving general problems. Gilhooly ( 1982 ) also pointed out that IR plays a key role in some activities in the problem-solving process, such as hypothesis generation and hypothesis testing. Moreover, the influence of IR on both KAC and KAP has been analysed and demonstrated in previous studies ( Molnár et al. 2013 ).
Empirical studies have also provided evidence that IR and CPS are related. Based on the results of a large-scale assessment (N = 2769), Molnár et al. ( 2013 ) showed that IR significantly correlated with 9–17-year-old students’ domain-general problem-solving achievement (r = 0.44–0.52). Greiff et al. ( 2015b ) conducted a large-scale assessment project (N = 2021) in Finland to explore the links between fluid reasoning skills and domain-general CPS. The study measured fluid reasoning as a two-dimensional model which consisted of deductive reasoning and scientific reasoning and included inductive thinking processes ( Greiff et al. 2015b ). The results drawing on structural equation modelling indicated that fluid reasoning which was partly based on IR had significant and strong predictive effects on both KAC (β = 0.51) and KAP (β = 0.55), the two phases of problem solving. Such studies have suggested that IR is one of the component skills of CPS.
According to Adey and Csapó ’s ( 2012 ) definition, CR is the process of creating complex constructions out of a set of given elements that satisfy the conditions explicitly given in or inferred from the situation. In this process, some cognitive operations, such as combinations, arrangements, permutations, notations and formulae, will be employed ( English 2005 ). CR is one of the basic components of formal thinking ( Batanero et al. 1997 ). The relationship between CR and CPS has frequently been discussed. English ( 2005 ) demonstrated that CR has an essential meaning in several types of problem situations, such as problems requiring the systematic testing of alternative solutions. Moreover, Newell ( 1993 ) pointed out that CR is applied in some key activities of problem-solving information processing, such as strategy generation and application. Its functions include, but are not limited to, helping problem solvers to discover relationships between certain elements and concepts, promoting their fluency of thinking when they are considering different strategies ( Csapó 1999 ) and identifying all possible alternatives ( OECD 2014 ). Moreover, Wu and Molnár ’s ( 2018 ) empirical study drew on a sample (N = 187) of 11–13-year-old primary school students in China. Their study built a structural equation model between CPS, IR and CR, and the result indicated that CR showed a strong and statistically significant predictive power for CPS (β = 0.55). Thus, the results of the empirical study also support the argument that CR is one of the component skills of CPS.
Wüstenberg et al. ( 2012 ) stated that the creation and implementation of strategic exploration are core actions of the problem-solving task. Exploring and generating effective information are key to successfully solving a problem. Wittmann and Hattrup ( 2004 ) illustrated that “riskier strategies [create] a learning environment with greater opportunities to discover and master the rules and boundaries [of a problem]” (p. 406). Thus, when gathering information about a complex problem, there may be differences between exploration strategies in terms of efficacy. The MicroDYN scenarios, a simplification and simulation of the real-world problem-solving context, will also be influenced by the adoption and implementation of exploration strategies.
The effectiveness of the isolated variation strategy (or “Vary-One-Thing-At-A-Time” strategy—VOTAT; Vollmeyer et al. 1996 ) in a CPS environment has been hotly debated ( Chen et al. 2019 ; Greiff et al. 2018 ; Molnár and Csapó 2018 ; Molnár et al. 2022 ; Wu and Molnár 2021 ; Wüstenberg et al. 2014 ). To use the VOTAT strategy, a problem solver “systematically varies only one input variable, whereas the others remain unchanged. This way, the effect of the variable that has just been changed can be observed directly by monitoring the changes in the output variables” ( Molnár and Csapó 2018, p. 2 ). Understanding and using VOTAT effectively is the foundation for developing more complex strategies for coordinating multiple variables and the basis for some phases of scientific thinking (i.e., inquiry, analysis, inference and argument; Kuhn 2010 ; Kuhn et al. 1995 ).
Some previous studies have indicated that students who are able to apply VOTAT are more likely to achieve higher performance in a CPS assessment ( Greiff et al. 2018 ), especially if the problem is a well-defined minimal complex system (such as MicroDYN) ( Fischer et al. 2012 ; Molnár and Csapó 2018 ; Wu and Molnár 2021 ). For instance, Molnár and Csapó ( 2018 ) conducted an empirical study to explore how students’ exploration strategies influence their performance in an interactive problem-solving environment. They measured a group (N = 4371) of 3rd- to 12th-grade (aged 9–18) Hungarian students’ problem-solving achievement and modelled students’ exploration strategies. This result confirmed that students’ exploration strategies influence their problem-solving performance. For example, conscious VOTAT strategy users proved to be the best problem-solvers. Furthermore, other empirical studies (e.g., Molnár et al. 2022 ; Wu and Molnár 2021 ) achieved similar results, thus confirming the importance of VOTAT in a MicroDYN-based CPS environment.
Lotz et al. ( 2017 ) illustrated that effective use of VOTAT is associated with higher levels of intelligence. Their study also pointed out that intelligence has the potential to facilitate successful exploration behaviour. Reasoning skills are an important component of general intelligence. Based on Lotz et al. ’s ( 2017 ) statements, the roles IR and CR play in the CPS process might vary due to students’ different strategy usage patterns. However, there is still a lack of empirical studies in this regard.
Numerous studies have explored the nature of CPS, some of them discussing and analysing it from behavioural or cognitive perspectives. However, there have barely been any that have merged these two perspectives. From the cognitive perspective, this study explores the role of thinking skills (including IR and CR) in the cognition process of CPS. From the behavioural perspective, the study focuses on students’ behaviour (i.e., their exploration strategy) in the CPS assessment process. More specifically, the research aims to fill this gap and examine students’ use of statistically distinguishable exploration strategies in CPS environments and to detect the connection between the level of students’ thinking skills and their behaviour strategies in the CPS environment. The following research questions were thus formed.
The sample was drawn from one of the largest universities in Hungary. Participation was voluntary, but students were able to earn one course credit for taking part in the assessment. The participants were students who had just started their studies there (N = 1671). 43.4% of the first-year students took part in the assessment. 50.9% of the participants were female, and 49.1% were male. We filtered the sample and excluded those who had more than 80% missing data on any of the tests. After the data were cleaned, data from 1343 students were available for analysis. The test was designed and delivered via the eDia online assessment system ( Csapó and Molnár 2019 ). The assessment was held in the university ICT room and divided into two sessions. The first session involved the CPS test, whereas the second session entailed the IR and CR tests. Each session lasted 45 min. The language of the tests was Hungarian, the mother tongue of the students.
3.2.1. complex problem solving (cps).
The CPS assessment instrument adopted the MicroDYN approach. It contains a total of twelve scenarios, and each scenario consisted of two items (one item in the KAC phase and one item in the KAP phase in each problem scenario). Twelve KAC items and twelve KAP items were therefore delivered on the CPS test for a total of twenty-four items. Each scenario has a fictional cover story. For instance, students found a sick cat in front of their house, and they were expected to feed the cat with two different kinds of cat food to help it recover.
Each item contains up to three input and three output variables. The relations between the input and output variables were formulated with linear structural equations ( Funke 2001 ). Figure 1 shows a MicroDYN sample structure containing three input variables (A, B and C), three output variables (X, Y and Z) and a number of possible relations between the variables. The complexity of the item was defined by the number of input and output variables, and the number of relations between the variables. The test began with the item with the lowest complexity. The complexity of each item gradually increased as the test progressed.
A typical MicroDYN structure with three input variables and three output variables ( Greiff and Funke 2009 ).
The interface of each item displays the value of each variable in both numerical and figural forms (See Figure 2 ). Each of the input variables has a controller, which makes it possible to vary and set the value between +2 (+ +) and −2 (− −). To operate the system, students need to click the “+” or “−” button or use the slider directly to select the value they want to be added to or subtracted from the current value of the input variable. After clicking the “Apply” button in the interface, the input variables will add or subtract the selected value, and the output variables will show the corresponding changes. The history of the values for the input and output variables within the same problem scenario is displayed on screen. If students want to withdraw all the changes and set all the variables to their original status, they can click the “Reset” button.
Screenshot of the MicroDYN item Cat—first phase (knowledge acquisition). (The items were administered in Hungarian.)
In the first phase of the problem-solving process, the KAC phase, students are asked to interact with the system by changing the value of the input variables and observing and analysing the corresponding changes in the output variables. They are then expected to determine the relationship between the input and output variables and draw it in the form of (an) arrow(s) on the concept map at the bottom of the interface. To avoid item dependence in the second phase of the problem-solving process, the students are provided with a concept map during the KAP phase (see Figure 3 ), which shows the correct connections between the input and output variables. The students are expected to interact with the system by manipulating the input variables to make the output variables reach the given target values in four steps or less. That is, they cannot click on the “Apply” button more than four times. The first phase had a 180 s time limit, whereas the second had a 90 s time limit.
Screenshot of the MicroDYN item Cat—second phase (knowledge application). (The items were administered in Hungarian).
The IR instrument (see Figure 4 ) was originally designed and developed in Hungary ( Csapó 1997 ). In the last 25 years, the instrument has been further developed and scaled for a wide age range ( Molnár and Csapó 2011 ). In addition, figural items have been added, and the assessment method has evolved from paper-and-pencil to computer-based ( Pásztor 2016 ). Currently, the instrument is widely employed in a number of countries (see, e.g., Mousa and Molnár 2020 ; Pásztor et al. 2018 ; Wu et al. 2022 ; Wu and Molnár 2018 ). In the present study, four types of items were included after test adaptation: figural series, figural analogies, number analogies and number series. Students were expected to ascertain the correct relationship between the given figures and numbers and select a suitable figure or number as their answer. Students used the drag-and-drop operation to provide their answers. In total, 49 inductive reasoning items were delivered to the participating students.
Sample items for the IR test. (The items were administered in Hungarian.).
The CR instrument (see Figure 5 ) was originally designed by Csapó ( 1988 ). The instrument was first developed in paper-and-pencil format and then modified for computer use ( Pásztor and Csapó 2014 ). Each item contained figural or verbal elements and a clear requirement for combing through the elements. Students were asked to list every single combination based on a given rule they could find. For the figural items, students provided their answers using the drag-and-drop operation; for the verbal items, they were asked to type their answers in a text box provided on screen. The test consisted of eight combinatorial reasoning items in total.
Sample item for the CR test. (The items were administered in Hungarian).
Students’ performance was automatically scored via the eDia platform. Items on the CPS and IR tests were scored dichotomously. In the first phase (KAC) of the CPS test, if a student drew all the correct relations on the concept map provided on screen within the given timeframe, his/her performance was assigned a score of 1 or otherwise a score of 0. In the second phase (KAP) of the CPS test, if the student successfully reached the given target values of the output variables by manipulating the level of the input variables within no more than four steps and the given timeframe, then his/her performance earned a score of 1 or otherwise a score of 0. On the IR test items, if a student selected the correct figure or number as his/her answer, then he or she received a score of 1; otherwise, the score was 0.
Students’ performance on the CR test items was scored according to a special J index, which was developed by Csapó ( 1988 ). The J index ranges from 0 to 1, where 1 means that the student provided all the correct combinations without any redundant combinations on the task. The formula for computing the J index is the following:
x stands for the number of correct combinations in the student’s answer,
T stands for the number of all possible correct combinations, and
y stands for the number of redundant combinations in the student’s answer.
Furthermore, according to Csapó ’s ( 1988 ) design, if y is higher than T, then the J index will be counted as 0.
Beyond concrete answer data, students’ interaction and manipulation behaviour were also logged in the assessment system. This made it possible to analyse students’ exploration behaviour in the first phase of the CPS process (KAC phase). Toward this aim, we adopted a labelling system developed by Molnár and Csapó ( 2018 ) to transfer the raw logfile data to structured data files for analysis. Based on the system, each trial (i.e., the sum of manipulations within the same problem scenario which was applied and tested by clicking the “Apply” button) was modelled as a single data entity. The sum of these trials within the same problem was defined as a strategy. In our study, we only consider the trials which were able to provide useful and new information for the problem-solvers, whereas the redundant or operations trials were excluded.
In this study, we analysed students’ trials to determine the extent to which they used the VOTAT strategy: fully, partially or not at all. This strategy is the most successful exploration strategy for such problems; it is the easiest to interpret and provides direct information about the given variable without any mediation effects ( Fischer et al. 2012 ; Greiff et al. 2018 ; Molnár and Csapó 2018 ; Wüstenberg et al. 2014 ; Wu and Molnár 2021 ). Based on the definition of VOTAT noted in Section 1.3 , we checked students’ trials to ascertain if they systematically varied one input variable while keeping the others unchanged, or applied a different, less successful strategy. We considered the following three types of trials:
We used the numbers 0, 1 and 2 to distinguish the level of students’ use of the most effective exploration strategy (i.e., VOTAT). If a student applied one or more of the above trials for every input variable within the same scenario, we considered that they had used the full VOTAT strategy and labelled this behaviour 2. If a student had only employed VOTAT on some but not all of the input variables, we concluded that they had used a partial VOTAT strategy for that problem scenario and labelled it 1. If a student had used none of the trials noted above in their problem exploration, then we determined that they had not used VOTAT at all and thus gave them a label of 0.
We used LCA (latent class analysis) to explore students’ exploration strategy profiles. LCA is a latent variable modelling approach that can be used to identify unmeasured (latent) classes of samples with similarly observed variables. LCA has been widely used in analysing logfile data for CPS assessment and in exploring students’ behaviour patterns (see, e.g., Gnaldi et al. 2020 ; Greiff et al. 2018 ; Molnár et al. 2022 ; Molnár and Csapó 2018 ; Mustafić et al. 2019 ; Wu and Molnár 2021 ). The scores for the use of VOTAT in the KAC phase (0, 1, 2; see Section 3.4 ) were used for the LCA analysis. We used Mplus ( Muthén and Muthén 2010 ) to run the LCA analysis. Several indices were used to measure the model fit: AIC (Akaike information criterion), BIC (Bayesian information criterion) and aBIC (adjusted Bayesian information criterion). With these three indicators, lower values indicate a better model fit. Entropy (ranging from 0 to 1, with values close to 1 indicating high certainty in the classification). The Lo–Mendell–Rubin adjusted likelihood ratio was used to compare the model containing n latent classes with the model containing n − 1 latent classes, and the p value was the indicator for whether a significant difference could be detected ( Lo et al. 2001 ). The results of the Lo–Mendell–Rubin adjusted likelihood ratio analysis were used to decide the correct number of latent classes in LCA models.
ANOVA was used to analyse the performance differences for CPS, IR and CR across the students from the different class profiles. The analysis was run using SPSS. A path analysis (PA) was employed in the structural equation modelling (SEM) framework to investigate the roles of CR and IR in CPS and the similarities and differences across the students from the different exploration strategy profiles. The PA models were carried out with Mplus. The Tucker–Lewis index (TLI), the comparative fit index (CFI) and the root-mean-square error of approximation (RMSEA) were used as indicators for the model fit. A TLI and CFI larger than 0.90 paired with a RMSEA less than 0.08 are commonly considered as an acceptable model fit ( van de Schoot et al. 2012 ).
All three tests showed good reliability (Cronbach’s α: CPS: 0.89; IR: 0.87; CR: 0.79). Furthermore, the two sub-dimensions of the CPS test, KAC and KAP, also showed satisfactory reliability (Cronbach’s α: KAC: 0.86; KAP: 0.78). The tests thus proved to be reliable. The means and standard deviations of students’ performance (in percentage) on each test are provided in Table 1 .
The means and standard deviations of students’ performance on each test.
CPS | IR | CR | |||
---|---|---|---|---|---|
Overall | KAC | KAP | |||
Mean (%) | 56.21 | 62.93 | 49.50 | 65.83 | 68.46 |
S.D. (%) | 22.37 | 26.65 | 22.75 | 15.41 | 20.02 |
Based on the labelled logfile data for CPS, we applied latent class analyses to identify the behaviour patterns of the students in the exploration phase of the problem-solving process. The model fits for the LCA analysis are listed in Table 2 . Compared with the 2 or 3 latent class models, the 4 latent class model has a lower AIC, BIC and aBIC, and the likelihood ratio statistical test (the Lo–Mendell–Rubin adjusted likelihood ratio test) confirmed it has a significantly better model fit. The 5 and 6 latent class models did not show a better model fit than the 4 latent class model. Therefore, based on the results, four qualitatively different exploration strategy profiles can be distinguished, which covered 96% of the students.
Fit indices for latent class analyses.
Number of Latent Classes | AIC | BIC | aBIC | Entropy | L–M–R Test | |
---|---|---|---|---|---|---|
2 | 9078 | 9333 | 9177 | 0.987 | 4255 | <0.001 |
3 | 8520 | 8905 | 8670 | 0.939 | 604 | <0.001 |
4 | 8381 | 8897 | 8582 | 0.959 | 188 | <0.05 |
5 | 8339 | 8984 | 8591 | 0.955 | 92 | 0.93 |
6 | 8309 | 9084 | 8611 | 0.877 | 96 | 0.34 |
The patterns for the four qualitatively different exploration strategy profiles are shown in Figure 6 . In total, 84.3% of the students were proficient exploration strategy users, who were able to use VOTAT in each problem scenario independent of its difficulty level (represented by the red line in Figure 5 ). In total, 6.2% of the students were rapid learners. They were not able to apply VOTAT at the beginning of the test on the easiest problems but managed to learn quickly, and, after a rapid learning curve by the end of the test, they reached the level of proficient exploration strategy users, even though the problems became much more complex (represented by the blue line). In total, 3.1% of the students proved to be non-persistent explorers, and they employed VOTAT on the easiest problems but did not transfer this knowledge to the more complex problems. Finally, they were no longer able to apply VOTAT when the complexity of the problems increased (represented by the green line). In total, 6.5% of the students were non-performing explorers; they barely used any VOTAT strategy during the whole test (represented by the pink line) independent of problem complexity.
Four qualitatively different exploration strategy profiles.
Students with different exploration strategy profiles showed different kinds of performance in each reasoning skill under investigation. Results (see Table 3 ) showed that more proficient strategy users tended to have higher achievement in all the domains assessed as well as in the two sub-dimensions in CPS (i.e., KAC and KAP; ANOVA: CPS: F(3, 1339) = 187.28, p < 0.001; KAC: F(3, 1339) = 237.15, p < 0.001; KAP: F(3, 1339) = 74.91, p < 0.001; IR: F(3, 1339) = 48.10, p < 0.001; CR: F(3, 1339) = 28.72, p < 0.001); specifically, students identified as “proficient exploration strategy users” achieved the highest level on the reasoning skills tests independent of the domains. On average, they were followed by rapid learners, non-persistent explorers and, finally, non-performing explorers. Tukey’s post hoc tests revealed more details on the performance differences of students with different exploration profiles in each of the domains being measured. Proficient strategy users proved to be significantly more skilled in each of the reasoning domains. They were followed by rapid learners, who outperformed non-persistent explorers and non-performing explorers in CPS. In the domains of IR and CR, there were no achievement differences between rapid learners and non-persistent explorers, who significantly outperformed non-performing strategy explorers.
Students’ performance on each test—grouped according to the different exploration strategy profiles.
Class Profiles | CPS | IR | CR | |||
---|---|---|---|---|---|---|
Overall | KAC | KAP | ||||
Proficient strategy users | Mean (%) | 61.37 | 69.57 | 53.17 | 67.79 | 70.47 |
S.D. (%) | 19.67 | 22.25 | 21.90 | 14.22 | 18.96 | |
Rapid learners | Mean (%) | 35.39 | 36.65 | 34.14 | 59.23 | 62.67 |
S.D. (%) | 14.26 | 20.45 | 17.15 | 14.22 | 17.60 | |
Non-persistent explorers | Mean (%) | 27.03 | 24.59 | 29.47 | 57.29 | 56.11 |
S.D. (%) | 10.75 | 14.06 | 11.80 | 18.75 | 24.52 | |
Non-performing explorers | Mean (%) | 22.75 | 19.64 | 25.86 | 50.65 | 53.72 |
S.D. (%) | 12.67 | 15.30 | 16.38 | 16.55 | 23.99 |
Path analysis was used to explore the predictive power of IR and CR for CPS and its processes, knowledge acquisition and knowledge application, for each group of students with different exploration strategy profiles. That is, four path analysis models were built to indicate the predictive power of IR and CR for CPS (see Figure 7 ), and another four path analyses models were developed to monitor the predictive power of IR and CR for the two empirically distinguishable phases of CPS (i.e., KAC and KAP) (see Figure 8 ). All eight models had good model fits, the fit indices TLI and CFI were above 0.90, and RMSEA was less than 0.08.
Path analysis models (with CPS, IR and CR) for each type of strategy user; * significant at 0.05 ( p < 0.05); ** significant at 0.01 ( p < 0.01); N.S.: no significant effect can be found.
Path analysis models (with KAC, KAP, IR and CR) for each type of strategy user; * significant at 0.05 ( p < 0.05); ** significant at 0.01 ( p < 0.01); N.S.: no significant effect can be found.
Students’ level of IR significantly predicted their level of CPS in all four path analysis models independent of their exploration strategy profile ( Figure 7 ; proficient strategy users: β = 0.432, p < 0.01; rapid learners: β = 0.350, p < 0.01; non-persistent explorers: β = 0.309, p < 0.05; and non-performing explorers: β = 0.386, p < 0.01). This was not the case for CR, which only proved to have predictive power for CPS among proficient strategy users (β = 0.104, p < 0.01). IR and CR were significantly correlated in all four models.
After examining the roles of IR and CR in the CPS process, we went further to explore the roles of these two reasoning skills in the distinguishable phases of CPS. The path analysis models ( Figure 8 ) showed that the predictive power of IR and CR for KAC and KAP was varied in each group. Levels of IR and CR among non-persistent explorers and non-performing explorers failed to predict their achievement in the KAC phase of the CPS process. Moreover, rapid learners’ level of IR significantly predicted their achievement in the KAC phase (β = 0.327, p < 0.01), but their level of CR did not have the same predictive power. Furthermore, the proficient strategy users’ levels of both reasoning skills had significant predictive power for KAC (IR: β = 0.363, p < 0.01; CR: β = 0.132, p < 0.01). In addition, in the KAP phase of the CPS problems, IR played a significant role for all types of strategy users, although with different power (proficient strategy users: β = 0.408, p < 0.01; rapid learners: β = 0.339, p < 0.01; non-persistent explorers: β = 0.361, p < 0.01; and non-performing explorers: β = 0.447, p < 0.01); by contrast, CR did not have significant predictive power for the KAP phase in any of the models.
The study aims to investigate the role of IR and CR in CPS and its phases among students using statistically distinguishable exploration strategies in different CPS environments. We examined 1343 Hungarian university students and assessed their CPS, IR and CR skills. Both achievement data and logfile data were used in the analysis. The traditional achievement indicators formed the foundation for analysing the students’ CPS, CR and IR performance, whereas process data extracted from logfile data were used to explore students’ exploration behaviour in various CPS environments.
Four qualitatively different exploration strategy profiles were distinguished: proficient strategy users, rapid learners, non-persistent explorers and non-performing explorers (RQ1). The four profiles were consistent with the result of another study conducted at university level (see Molnár et al. 2022 ), and the frequencies of these four profiles in these two studies were very similar. The two studies therefore corroborate and validate each other’s results. The majority of the participants were identified as proficient strategy users. More than 80% of the university students were able to employ effective exploration strategies in various CPS environments. Of the remaining students, some performed poorly in exploration strategy use in the early part of the test (rapid learners), some in the last part (non-persistent explorers) and some throughout the test (non-performing explorers). However, students with these three exploration strategy profiles only constituted small portions of the total sample (with proportions ranging from 3.1% to 6.5%). The university students therefore exhibited generally good performance in terms of exploration strategy use in a CPS environment, especially compared with previous results among younger students (e.g., primary school students, see Greiff et al. 2018 ; Wu and Molnár 2021 ; primary to secondary students, see Molnár and Csapó 2018 ).
The results have indicated that better exploration strategy users achieved higher CPS performance and had better development levels of IR and CR (RQ2). First, the results have confirmed the importance of VOTAT in a CPS environment. This finding is consistent with previous studies (e.g., Greiff et al. 2015a ; Molnár and Csapó 2018 ; Mustafić et al. 2019 ; Wu and Molnár 2021 ). Second, the results have confirmed that effective use of VOTAT is strongly tied to the level of IR and CR development. Reasoning forms an important component of human intelligence, and the level of development in reasoning was an indicator of the level of intelligence ( Klauer et al. 2002 ; Sternberg and Kaufman 2011 ). Therefore, this finding has supplemented empirical evidence for the argument that effective use of VOTAT is associated with levels of intelligence to a certain extent.
The roles of IR and CR proved to be varied for each type of exploration strategy user (RQ3). For instance, the level of CPS among the best exploration strategy users (i.e., the proficient strategy users) was predicted by both the levels of IR and CR, but this was not the case for students with other profiles. In addition, the results have indicated that IR played important roles in both the KAC and KAP phases for the students with relatively good exploration strategy profiles (i.e., proficient strategy users and rapid learners) but only in the KAP phase for the rest of the students (non-persistent explorers and non-performing explorers); moreover, the predictive power of CR can only be detected in the KAC phase of the proficient strategy users. To sum up, the results suggest a general trend of IR and CR playing more important roles in the CPS process among better exploration strategy users.
Combining the answers to RQ2 and RQ3, we can gain further insights into students’ exploration strategy use in a CPS environment. Our results have confirmed that the use of VOTAT is associated with the level of IR and CR development and that the importance of IR and CR increases with proficiency in exploration strategy use. Based on these findings, we can make a reasonable argument that IR and CR are essential skills for using VOTAT and that underdeveloped IR and CR will prevent students from using effective strategies in a CPS environment. Therefore, if we want to encourage students to become better exploration strategy users, it is important to first enhance their IR and CR skills. Previous studies have suggested that establishing explicit training in using effective strategies in a CPS environment is important for students’ CPS development ( Molnár et al. 2022 ). Our findings have identified the importance of IR and CR in exploration strategy use, which has important implications for designing training programmes.
The results have also provided a basis for further studies. Future studies have been suggested to further link the behavioural and cognitive perspectives in CPS research. For instance, IR and CR were considered as component skills of CPS (see Section 1.2 ). The results of the study have indicated the possibility of not only discussing the roles of IR and CR in the cognitive process of CPS, but also exploration behaviour in a CPS environment. The results have thus provided a new perspective for exploring the component skills of CPS.
There are some limitations in the study. All the tests were low stake; therefore, students might not be sufficiently motivated to do their best. This feature might have produced the missing values detected in the sample. In addition, some students’ exploration behaviour shown in this study might theoretically be below their true level. However, considering that data cleaning was adopted in this study (see Section 3.1 ), we believe this phenomenon will not have a remarkable influence on the results. Moreover, the CPS test in this study was based on the MicroDYN approach, which is a well-established and widely used artificial model with a limited number of variables and relations. However, it does not have the power to cover all kinds of complex and dynamic problems in real life. For instance, the MicroDYN approach cannot measure ill-defined problem solving. Thus, this study can only demonstrate the influence of IR and CR on problem solving in well-defined MicroDYN-simulated problems. Furthermore, VOTAT is helpful with minimally complex problems under well-defined laboratory conditions, but it may not be that helpful with real-world, ill-defined complex problems ( Dörner and Funke 2017 ; Funke 2021 ). Therefore, the generalizability of the findings is limited.
In general, the results have shed new light on students’ problem-solving behaviours in respect of exploration strategy in a CPS environment and explored differences in terms of the use of thinking skills between students with different exploration strategies. Most studies discuss students’ problem-solving strategies from a behavioural perspective. By contrast, this paper discusses them from both behavioural and cognitive perspectives, thus expanding our understanding in this area. As for educational implications, the study contributes to designing and revising training methods for CPS by identifying the importance of IR and CR in exploration behaviour in a CPS environment. To sum up, the study has investigated the nature of CPS from a fresh angle and provided a sound basis for future studies.
This study has been conducted with support provided by the National Research, Development and Innovation Fund of Hungary, financed under the OTKA K135727 funding scheme and supported by the Research Programme for Public Education Development, Hungarian Academy of Sciences (KOZOKT2021-16).
Conceptualization, H.W. and G.M.; methodology, H.W. and G.M.; formal analysis, H.W.; writing—original draft preparation, H.W.; writing—review and editing, G.M.; project administration, G.M.; funding acquisition, G.M. All authors have read and agreed to the published version of the manuscript.
Ethical approval was not required for this study in accordance with the national and institutional guidelines. The assessments which provided data for this study were integrated parts of the educational processes of the participating university. The participation was voluntary.
All of the students in the assessment turned 18, that is, it was not required or possible to request and obtain written informed parental consent from the participants.
Conflicts of interest.
Authors declare no conflict of interest.
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.
100 Last-Day-of-School Activities Your Students Will Love!
So many ways to help students learn!
Looking for some new ways to teach and learn in your classroom? This roundup of instructional strategies examples includes methods that will appeal to all learners and work for any teacher.
In the simplest of terms, instructional strategies are the methods teachers use to achieve learning objectives. In other words, pretty much every learning activity you can think of is an example of an instructional strategy. They’re also known as teaching strategies and learning strategies.
The more instructional strategies a teacher has in their tool kit, the more they’re able to reach all of their students. Different types of learners respond better to various strategies, and some topics are best taught with one strategy over another. Usually, teachers use a wide array of strategies across a single lesson. This gives all students a chance to play to their strengths and ensures they have a deeper connection to the material.
There are a lot of different ways of looking at instructional strategies. One of the most common breaks them into five basic types. It’s important to remember that many learning activities fall into more than one of these categories, and teachers rarely use one type of strategy alone. The key is to know when a strategy can be most effective, for the learners or for the learning objective. Here’s a closer look at the five basic types, with instructional strategies examples for each.
Direct instruction can also be called “teacher-led instruction,” and it’s exactly what it sounds like. The teacher provides the information, while the students watch, listen, and learn. Students may participate by answering questions asked by the teacher or practicing a skill under their supervision. This is a very traditional form of teaching, and one that can be highly effective when you need to provide information or teach specific skills.
This method gets a lot of flack these days for being “boring” or “old-fashioned.” It’s true that you don’t want it to be your only instructional strategy, but short lectures are still very effective learning tools. This type of direct instruction is perfect for imparting specific detailed information or teaching a step-by-step process. And lectures don’t have to be boring—just look at the success of TED Talks .
These are often paired with other direct instruction methods like lecturing. The teacher asks questions to determine student understanding of the material. They’re often questions that start with “who,” “what,” “where,” and “when.”
In this direct instruction method, students watch as a teacher demonstrates an action or skill. This might be seeing a teacher solving a math problem step-by-step, or watching them demonstrate proper handwriting on the whiteboard. Usually, this is followed by having students do hands-on practice or activities in a similar manner.
If you’ve ever used flash cards to help kids practice math facts or had your whole class chant the spelling of a word out loud, you’ve used drill & practice. It’s another one of those traditional instructional strategies examples. When kids need to memorize specific information or master a step-by-step skill, drill & practice really works.
This form of instruction is learner-led and helps develop higher-order thinking skills. Teachers guide and support, but students drive the learning through reading, research, asking questions, formulating ideas and opinions, and more. This method isn’t ideal when you need to teach detailed information or a step-by-step process. Instead, use it to develop critical thinking skills , especially when more than one solution or opinion is valid.
In this indirect learning method, students work their way through a problem to find a solution. Along the way, they must develop the knowledge to understand the problem and use creative thinking to solve it. STEM challenges are terrific examples of problem-solving instructional strategies.
When kids participate in true project-based learning, they’re learning through indirect and experiential strategies. As they work to find solutions to a real-world problem, they develop critical thinking skills and learn by research, trial and error, collaboration, and other experiences.
Learn more: What Is Project-Based Learning?
Students use concept maps to break down a subject into its main points and draw connections between these points. They brainstorm the big-picture ideas, then draw lines to connect terms, details, and more to help them visualize the topic.
When you think of case studies, law school is probably the first thing that jumps to mind. But this method works at any age, for a variety of topics. This indirect learning method teaches students to use material to draw conclusions, make connections, and advance their existing knowledge.
This is different than learning to read. Instead, it’s when students use texts (print or digital) to learn about a topic. This traditional strategy works best when students already have strong reading comprehension skills. Try our free reading comprehension bundle to give students the ability to get the most out of reading for meaning.
In a flipped classroom, students read texts or watch prerecorded lectures at home. Classroom time is used for deeper learning activities, like discussions, labs, and one-on-one time for teachers and students.
Learn more: What Is a Flipped Classroom?
In experiential learning, students learn by doing. Rather than following a set of instructions or listening to a lecture, they dive right into an activity or experience. Once again, the teacher is a guide, there to answer questions and gently keep learning on track if necessary. At the end, and often throughout, the learners reflect on their experience, drawing conclusions about the skills and knowledge they’ve gained. Experiential learning values the process over the product.
This is experiential learning at its best. Hands-on experiments let kids learn to establish expectations, create sound methodology, draw conclusions, and more.
Learn more: Hundreds of science experiment ideas for kids and teens
Heading out into the real world gives kids a chance to learn indirectly, through experiences. They may see concepts they already know put into practice or learn new information or skills from the world around them.
Learn more: The Big List of PreK-12 Field Trip Ideas
Teachers have long known that playing games is a fun (and sometimes sneaky) way to get kids to learn. You can use specially designed educational games for any subject. Plus, regular board games often involve a lot of indirect learning about math, reading, critical thinking, and more.
Learn more: Classic Classroom Games and Best Online Educational Games
This is another instructional strategies example that takes students out into the real world. It often involves problem-solving skills and gives kids the opportunity for meaningful social-emotional learning.
Learn more: What Is Service Learning?
As you might guess, this strategy is all about interaction between the learners and often the teacher. The focus is on discussion and sharing. Students hear other viewpoints, talk things out, and help each other learn and understand the material. Teachers can be a part of these discussions, or they can oversee smaller groups or pairings and help guide the interactions as needed. Interactive instruction helps students develop interpersonal skills like listening and observation.
It’s often said the best way to learn something is to teach it to others. Studies into the so-called “ protégé effect ” seem to prove it too. In order to teach, you first must understand the information yourself. Then, you have to find ways to share it with others—sometimes more than one way. This deepens your connection to the material, and it sticks with you much longer. Try having peers instruct one another in your classroom, and see the magic in action.
This method is specifically used in reading instruction, as a cooperative learning strategy. Groups of students take turns acting as the teacher, helping students predict, clarify, question, and summarize. Teachers model the process initially, then observe and guide only as needed.
Some teachers shy away from debate in the classroom, afraid it will become too adversarial. But learning to discuss and defend various points of view is an important life skill. Debates teach students to research their topic, make informed choices, and argue effectively using facts instead of emotion.
Learn more: High School Debate Topics To Challenge Every Student
Class, small-group, and pair discussions are all excellent interactive instructional strategies examples. As students discuss a topic, they clarify their own thinking and learn from the experiences and opinions of others. Of course, in addition to learning about the topic itself, they’re also developing valuable active listening and collaboration skills.
Learn more: Strategies To Improve Classroom Discussions
Take your classroom discussions one step further with the fishbowl method. A small group of students sits in the middle of the class. They discuss and debate a topic, while their classmates listen silently and make notes. Eventually, the teacher opens the discussion to the whole class, who offer feedback and present their own assertions and challenges.
Learn more: How I Use Fishbowl Discussions To Engage Every Student
Rather than having a teacher provide examples to explain a topic or solve a problem, students do the work themselves. Remember the one rule of brainstorming: Every idea is welcome. Ensure everyone gets a chance to participate, and form diverse groups to generate lots of unique ideas.
Role-playing is sort of like a simulation but less intense. It’s perfect for practicing soft skills and focusing on social-emotional learning . Put a twist on this strategy by having students model bad interactions as well as good ones and then discussing the difference.
This structured discussion technique is simple: First, students think about a question posed by the teacher. Pair students up, and let them talk about their answer. Finally open it up to whole-class discussion. This helps kids participate in discussions in a low-key way and gives them a chance to “practice” before they talk in front of the whole class.
Learn more: Think-Pair-Share and Fun Alternatives
Also called independent study, this form of learning is almost entirely student-led. Teachers take a backseat role, providing materials, answering questions, and guiding or supervising. It’s an excellent way to allow students to dive deep into topics that really interest them, or to encourage learning at a pace that’s comfortable for each student.
Foster independent learning strategies with centers just for math, writing, reading, and more. Provide a variety of activities, and let kids choose how they spend their time. They often learn better from activities they enjoy.
Learn more: The Big List of K-2 Literacy Centers
Once a rarity, now a daily fact of life, computer-based instruction lets students work independently. They can go at their own pace, repeating sections without feeling like they’re holding up the class. Teach students good computer skills at a young age so you’ll feel comfortable knowing they’re focusing on the work and doing it safely.
Writing an essay encourages kids to clarify and organize their thinking. Written communication has become more important in recent years, so being able to write clearly and concisely is a skill every kid needs. This independent instructional strategy has stood the test of time for good reason.
Learn more: The Big List of Essay Topics for High School
Here’s another oldie-but-goodie! When kids work independently to research and present on a topic, their learning is all up to them. They set the pace, choose a focus, and learn how to plan and meet deadlines. This is often a chance for them to show off their creativity and personality too.
Personal journals give kids a chance to reflect and think critically on topics. Whether responding to teacher prompts or simply recording their daily thoughts and experiences, this independent learning method strengthens writing and intrapersonal skills.
Learn more: The Benefits of Journaling in the Classroom
In play-based learning programs, children learn by exploring their own interests. Teachers identify and help students pursue their interests by asking questions, creating play opportunities, and encouraging students to expand their play.
Learn more: What Is Play-Based Learning?
Don’t be afraid to try new strategies from time to time—you just might find a new favorite! Here are some of the most common instructional strategies examples.
This strategy combines experiential, interactive, and indirect learning all in one. The teacher sets up a simulation of a real-world activity or experience. Students take on roles and participate in the exercise, using existing skills and knowledge or developing new ones along the way. At the end, the class reflects separately and together on what happened and what they learned.
Ever since Aesop’s fables, we’ve been using storytelling as a way to teach. Stories grab students’ attention right from the start and keep them engaged throughout the learning process. Real-life stories and fiction both work equally well, depending on the situation.
Learn more: Teaching as Storytelling
Scaffolding is defined as breaking learning into bite-sized chunks so students can more easily tackle complex material. It builds on old ideas and connects them to new ones. An educator models or demonstrates how to solve a problem, then steps back and encourages the students to solve the problem independently. Scaffolding teaching gives students the support they need by breaking learning into achievable sizes while they progress toward understanding and independence.
Learn more: What Is Scaffolding in Education?
Often paired with direct or independent instruction, spaced repetition is a method where students are asked to recall certain information or skills at increasingly longer intervals. For instance, the day after discussing the causes of the American Civil War in class, the teacher might return to the topic and ask students to list the causes. The following week, the teacher asks them once again, and then a few weeks after that. Spaced repetition helps make knowledge stick, and it is especially useful when it’s not something students practice each day but will need to know in the long term (such as for a final exam).
Graphic organizers are a way of organizing information visually to help students understand and remember it. A good organizer simplifies complex information and lays it out in a way that makes it easier for a learner to digest. Graphic organizers may include text and images, and they help students make connections in a meaningful way.
Learn more: Graphic Organizers 101: Why and How To Use Them
Jigsaw combines group learning with peer teaching. Students are assigned to “home groups.” Within that group, each student is given a specialized topic to learn about. They join up with other students who were given the same topic, then research, discuss, and become experts. Finally, students return to their home group and teach the other members about the topic they specialized in.
As the name implies, this instructional strategy approaches a topic using techniques and aspects from multiple disciplines, helping students explore it more thoroughly from a variety of viewpoints. For instance, to learn more about a solar eclipse, students might explore scientific explanations, research the history of eclipses, read literature related to the topic, and calculate angles, temperatures, and more.
This instructional strategy takes multidisciplinary instruction a step further, using it to synthesize information and viewpoints from a variety of disciplines to tackle issues and problems. Imagine a group of students who want to come up with ways to improve multicultural relations at their school. They might approach the topic by researching statistical information about the school population, learning more about the various cultures and their history, and talking with students, teachers, and more. Then, they use the information they’ve uncovered to present possible solutions.
Differentiated instruction means tailoring your teaching so all students, regardless of their ability, can learn the classroom material. Teachers can customize the content, process, product, and learning environment to help all students succeed. There are lots of differentiated instructional strategies to help educators accommodate various learning styles, backgrounds, and more.
Learn more: What Is Differentiated Instruction?
Culturally responsive teaching is based on the understanding that we learn best when we can connect with the material. For culturally responsive teachers, that means weaving their students’ various experiences, customs, communication styles, and perspectives throughout the learning process.
Learn more: What Is Culturally Responsive Teaching?
Response to Intervention, or RTI, is a way to identify and support students who need extra academic or behavioral help to succeed in school. It’s a tiered approach with various “levels” students move through depending on how much support they need.
Learn more: What Is Response to Intervention?
Inquiry-based learning means tailoring your curriculum to what your students are interested in rather than having a set agenda that you can’t veer from—it means letting children’s curiosity take the lead and then guiding that interest to explore, research, and reflect upon their own learning.
Learn more: What Is Inquiry-Based Learning?
Growth mindset is key for learners. They must be open to new ideas and processes and believe they can learn anything with enough effort. It sounds simplistic, but when students really embrace the concept, it can be a real game-changer. Teachers can encourage a growth mindset by using instructional strategies that allow students to learn from their mistakes, rather than punishing them for those mistakes.
Learn more: Growth Mindset vs. Fixed Mindset and 25 Growth Mindset Activities
This strategy combines face-to-face classroom learning with online learning, in a mix of self-paced independent learning and direct instruction. It’s incredibly common in today’s schools, where most students spend at least part of their day completing self-paced lessons and activities via online technology. Students may also complete their online instructional time at home.
This fancy term really just describes strategies that allow each student to work at their own pace using a flexible schedule. This method became a necessity during the days of COVID lockdowns, as families did their best to let multiple children share one device. All students in an asynchronous class setting learn the same material using the same activities, but do so on their own timetable.
Learn more: Synchronous vs. Asynchronous Learning
Essential questions are the big-picture questions that inspire inquiry and discussion. Teachers give students a list of several essential questions to consider as they begin a unit or topic. As they dive deeper into the information, teachers ask more specific essential questions to help kids make connections to the “essential” points of a text or subject.
Learn more: Questions That Set a Purpose for Reading
When it comes to choosing instructional strategies, there are several things to consider:
Plus, check out the things the best instructional coaches do, according to teachers ..
What's the magic formula? Continue Reading
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Success in sustainability: two cognitive strategies for effective problem-solving.
Thomas Lim is the Vice-Dean of Centre for Systems Leadership at SIM Academy. He is an AI+Web3 practitioner & author of Think.Coach.Thrive!
Systems thinking and critical thinking are distinct yet complementary cognitive tools essential for effective problem-solving. Systems thinking allows businesses to understand and address the broad impacts of their actions on an interconnected system, while critical thinking sharpens decision-making, ensuring that outcomes are viable, ethical and based on solid reasoning.
Systems thinking provides a holistic perspective, focusing on how various components of a system interact and affect each other within a broader context. It emphasizes understanding the interconnections, dynamics, long-term impacts and patterns within systems to predict future behaviors and develop sustainable solutions.
This approach is particularly valuable in complex environments like organizational change, environmental management, and technological systems, where understanding the big picture is crucial.
On the other hand, critical thinking adopts a more analytical approach, concentrating on evaluating information and arguments, identifying logical inconsistencies, and making reasoned judgments. It involves dissecting complex problems into manageable parts, emphasizing evidence-based decision-making and rigorous evaluation of ideas and assumptions.
Critical thinking is key in activities that require clear, structured thinking, such as logical reasoning, decision-making, and solution evaluation, often focusing on scrutinizing existing solutions and preventing errors.
Best 5% interest savings accounts of 2024.
Together, these methodologies enhance decision-making and problem-solving by providing both macro and micro analytical perspectives to the challenge at hand.
Integrating both of these ways of thinking into sustainability initiatives offers organizations a robust framework for tackling complex challenges through a structured yet flexible approach. It helps organizations transform their approach to sustainability from fragmented efforts into a coherent, strategic agenda that drives real change.
Here's how organizations can implement these two ways of thinking effectively:
Begin by defining a clear sustainability vision and objectively assess the current state to identify gaps. What is the desired future and the existing barriers or deficiencies preventing its realization? Engaging in this step ensures that all stakeholders have a unified understanding of the objectives and challenges.
For instance, a government agency might aim for sustainable urban development while recognizing current inefficiencies in urban infrastructure. The systemic structure would take into consideration manpower availability, lifetime cost of building projects and green funding availability.
Detail decisions across all levels from strategic to tactical, ensuring that each decision aligns with the overarching sustainability goals.
This step often involves using decision hierarchies to maintain clarity and relevance at every level, thus preventing duplications and identifying gaps in strategies.
For example, a multinational corporation might structure decisions around reducing its carbon footprint through supplier engagement programs. Using critical thinking methodologies, they could create an analytical and evidence-based workflow and test assumptions on handoffs to ensure compliance.
Identify the most impactful sustainability challenge and focus resources and efforts on areas where they can make the most significant difference to help maximize impact.
For example, a healthcare provider may prioritize waste reduction in its facilities by improving waste segregation and processing and develop the necessary systems and processes in keeping with the new disposal methods. They may modify or eliminate altogether outdated policies, leading to new behaviors of pattern over time in personnel involved.
Use both systems and critical thinking to create comprehensive, innovative and interconnected solutions. This might involve using systems diagrams to visualize problems and how they relate and employing logical reasoning to evaluate potential solutions for effectiveness and feasibility.
Remember the government agency aiming for sustainable urban development? In this scenario, they may create a stakeholder map aligning and enabling various parties to translate purpose into strategy. This would allow them to co-create multifaceted urban plans that integrate green spaces and renewable energy solutions. As a result, corresponding tactics and activities happen in an integrated, not haphazard, way.
Develop a clear and actionable theory of success that outlines the key actions and leverage points. This theory should detail how the proposed solutions will address the identified challenges and lead to the desired change, identifying where small interventions could lead to significant systemic improvements.
In the case of the multinational corporation, their leverage was in incentivizing suppliers to adopt low-carbon technologies. Their theory of success was not in "shifting the burden" but in creating a positive reinforcement loop where they focused on the quality of relationships for long-term commitment.
Put the strategies into action while establishing mechanisms for ongoing monitoring and adaptation. This includes setting up feedback loops to continuously gather data on the effectiveness of the interventions and making necessary adjustments based on empirical evidence and changing conditions.
For the healthcare provider addressing waste management challenges, this might involve adjusting waste management procedures based on ongoing feedback and outcomes. They might realize they are oftentimes reactive in their problem solving and therefore intend to conduct an intentional analysis and internalize and operationalize key insights.
As businesses become more complex and interconnected, the ability to think both systemically and critically isn’t just an advantage; it’s essential to survival and success in an interconnected world.
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6. Discovery & Action Dialogue (DAD) One of the best approaches is to create a safe space for a group to share and discover practices and behaviors that can help them find their own solutions. With DAD, you can help a group choose which problems they wish to solve and which approaches they will take to do so.
Step 2: Analyze the Problem. During this step a group should analyze the problem and the group's relationship to the problem. Whereas the first step involved exploring the "what" related to the problem, this step focuses on the "why.". At this stage, group members can discuss the potential causes of the difficulty.
Traditional, more classic problem solving is you define the problem based on an understanding of the situation. This one almost presupposes that we don't know the problem until we go see it. The second thing is you need to come up with multiple scenarios or answers or ideas or concepts, and there's a lot of divergent thinking initially.
In insight problem-solving, the cognitive processes that help you solve a problem happen outside your conscious awareness. 4. Working backward. Working backward is a problem-solving approach often ...
A heuristic is another type of problem solving strategy. While an algorithm must be followed exactly to produce a correct result, a heuristic is a general problem-solving framework (Tversky & Kahneman, 1974). You can think of these as mental shortcuts that are used to solve problems. A "rule of thumb" is an example of a heuristic.
Heuristic methods can also play an important role in your problem-solving processes. The straw man technique, for example, is similar in approach to heuristics, and it is designed to help you to build on or refine a basic idea. Another approach is to adapt the solution to a different problem to fix yours. TRIZ is a powerful methodology for ...
Problem-solving is a mental process that involves discovering, analyzing, and solving problems. The ultimate goal of problem-solving is to overcome obstacles and find a solution that best resolves the issue. The best strategy for solving a problem depends largely on the unique situation. In some cases, people are better off learning everything ...
Finding a suitable solution for issues can be accomplished by following the basic four-step problem-solving process and methodology outlined below. Step. Characteristics. 1. Define the problem. Differentiate fact from opinion. Specify underlying causes. Consult each faction involved for information. State the problem specifically.
Algorithms. In contrast to heuristics, which can be thought of as problem-solving strategies based on educated guesses, algorithms are problem-solving strategies that use rules. Algorithms are generally a logical set of steps that, if applied correctly, should be accurate. For example, you could make a cake using heuristics — relying on your ...
A heuristic is another type of problem solving strategy. While an algorithm must be followed exactly to produce a correct result, a heuristic is a general problem-solving framework (Tversky & Kahneman, 1974). You can think of these as mental shortcuts that are used to solve problems. A "rule of thumb" is an example of a heuristic.
Problem-solving is a vital skill for coping with various challenges in life. This webpage explains the different strategies and obstacles that can affect how you solve problems, and offers tips on how to improve your problem-solving skills. Learn how to identify, analyze, and overcome problems with Verywell Mind.
Defer or suspend judgement. Focus on "Yes, and…" rather than "No, but…". According to Carella, "Creative problem solving is the mental process used for generating innovative and imaginative ideas as a solution to a problem or a challenge. Creative problem solving techniques can be pursued by individuals or groups.".
The following list of strategies, although not exhaustive, is very useful: 1. Look for a pattern. 2. Examine related problems and determine if the same technique can be applied. 3. Examine a simpler or special case of the problem to gain insight into the solution of the original problem. 4. Make a table.
Step 1: Identify the Problem. The problem-solving process starts with identifying the problem. This step involves understanding the issue's nature, its scope, and its impact. Once the problem is clearly defined, it sets the foundation for finding effective solutions.
2. Break the problem down. Identifying the problem allows you to see which steps need to be taken to solve it. First, break the problem down into achievable blocks. Then, use strategic planning to set a time frame in which to solve the problem and establish a timeline for the completion of each stage. 3.
Problem solving is the process of articulating solutions to problems. Problems have two critical attributes. First, a problem is an unknown in some context. That is, there is a situation in which there is something that is unknown (the difference between a goal state and a current state). Those situations vary from algorithmic math problems to ...
Defining the problem: phrase problem as probing questions to encourage explorative thinking; make explicit goal statement; Establish criteria for evaluating the solution: identify characteristics of a satisfactory solution; distinguish requirements from desires; Analyzing the problem: discover the root cause and extent of the problem; Considering alternate solutions: brainstorm to generate ...
A heuristic is another type of problem solving strategy. While an algorithm must be followed exactly to produce a correct result, a heuristic is a general problem-solving framework (Tversky & Kahneman, 1974). You can think of these as mental shortcuts that are used to solve problems. A "rule of thumb" is an example of a heuristic.
Problem solving is the process of achieving a goal by overcoming obstacles, a frequent part of most activities. Problems in need of solutions range from simple personal tasks (e.g. how to turn on an appliance) to complex issues in business and technical fields. The former is an example of simple problem solving (SPS) addressing one issue ...
Strategies for teaching problem solving include: 1. Teaching students an attack strategy to guide the process of problem solving 2. Teaching students to recognize and solve word problems according to the schema of the problem 3. Utilizing appropriate mathematical language to help students understand the meaning of each word in a problem.
Logical reasoning is of great societal importance and, as stressed by the twenty-first century skills framework, also seen as a key aspect for the development of critical thinking. This study aims at exploring secondary school students' logical reasoning strategies in formal reasoning and everyday reasoning tasks. With task-based interviews among 4 16- and 17-year-old pre-university students ...
Complex problem solving (CPS) is considered to be one of the most important skills for successful learning. In an effort to explore the nature of CPS, this study aims to investigate the role of inductive reasoning (IR) and combinatorial reasoning (CR) in the problem-solving process of students using statistically distinguishable exploration strategies in the CPS environment.
Problem-Solving. In this indirect learning method, students work their way through a problem to find a solution. Along the way, they must develop the knowledge to understand the problem and use creative thinking to solve it. STEM challenges are terrific examples of problem-solving instructional strategies.
Access to problem-solving tools practitioners can adopt straightaway. The opportunity to use reliable pedagogical methods such as the case method and problem-based learning to teach problem-solving skills. The Lost Art of Problem-Solving. The term strategic problems is common to the academic literature yet surprisingly under-explored.
Step 4: Developing Nested Solutions. Use both systems and critical thinking to create comprehensive, innovative and interconnected solutions. This might involve using systems diagrams to visualize ...
Part 1 - Federal Acquisition Regulations System. Part 2 - Definitions of Words and Terms. Part 3 - Improper Business Practices and Personal Conflicts of Interest. Part 4 - Administrative and Information Matters. Part 5 - Publicizing Contract Actions.