problem solving method b ed

Problem-Solving Method in Teaching

The problem-solving method is a highly effective teaching strategy that is designed to help students develop critical thinking skills and problem-solving abilities . It involves providing students with real-world problems and challenges that require them to apply their knowledge, skills, and creativity to find solutions. This method encourages active learning, promotes collaboration, and allows students to take ownership of their learning.

Definition of problem-solving method.

Problem-solving is a process of identifying, analyzing, and resolving problems. The problem-solving method in teaching involves providing students with real-world problems that they must solve through collaboration and critical thinking. This method encourages students to apply their knowledge and creativity to develop solutions that are effective and practical.

Meaning of Problem-Solving Method

The meaning and Definition of problem-solving are given by different Scholars. These are-

Woodworth and Marquis(1948) : Problem-solving behavior occurs in novel or difficult situations in which a solution is not obtainable by the habitual methods of applying concepts and principles derived from past experience in very similar situations.

Skinner (1968): Problem-solving is a process of overcoming difficulties that appear to interfere with the attainment of a goal. It is the procedure of making adjustments in spite of interference

Benefits of Problem-Solving Method

The problem-solving method has several benefits for both students and teachers. These benefits include:

Encourages active learning: The problem-solving method encourages students to actively participate in their own learning by engaging them in real-world problems that require critical thinking and collaboration
Promotes collaboration: Problem-solving requires students to work together to find solutions. This promotes teamwork, communication, and cooperation.
Builds critical thinking skills: The problem-solving method helps students develop critical thinking skills by providing them with opportunities to analyze and evaluate problems
Increases motivation: When students are engaged in solving real-world problems, they are more motivated to learn and apply their knowledge.
Enhances creativity: The problem-solving method encourages students to be creative in finding solutions to problems.

Steps in Problem-Solving Method

The problem-solving method involves several steps that teachers can use to guide their students. These steps include

Identifying the problem: The first step in problem-solving is identifying the problem that needs to be solved. Teachers can present students with a real-world problem or challenge that requires critical thinking and collaboration.
Analyzing the problem: Once the problem is identified, students should analyze it to determine its scope and underlying causes.
Generating solutions: After analyzing the problem, students should generate possible solutions. This step requires creativity and critical thinking.
Evaluating solutions: The next step is to evaluate each solution based on its effectiveness and practicality
Selecting the best solution: The final step is to select the best solution and implement it.

Verification of the concluded solution or Hypothesis

The solution arrived at or the conclusion drawn must be further verified by utilizing it in solving various other likewise problems. In case, the derived solution helps in solving these problems, then and only then if one is free to agree with his finding regarding the solution. The verified solution may then become a useful product of his problem-solving behavior that can be utilized in solving further problems. The above steps can be utilized in solving various problems thereby fostering creative thinking ability in an individual.

The problem-solving method is an effective teaching strategy that promotes critical thinking, creativity, and collaboration. It provides students with real-world problems that require them to apply their knowledge and skills to find solutions. By using the problem-solving method, teachers can help their students develop the skills they need to succeed in school and in life.

Jonassen, D. (2011). Learning to solve problems: A handbook for designing problem-solving learning environments. Routledge.
Hmelo-Silver, C. E. (2004). Problem-based learning: What and how do students learn? Educational Psychology Review, 16(3), 235-266.
Mergendoller, J. R., Maxwell, N. L., & Bellisimo, Y. (2006). The effectiveness of problem-based instruction: A comparative study of instructional methods and student characteristics. Interdisciplinary Journal of Problem-based Learning, 1(2), 49-69.
Richey, R. C., Klein, J. D., & Tracey, M. W. (2011). The instructional design knowledge base: Theory, research, and practice. Routledge.
Savery, J. R., & Duffy, T. M. (2001). Problem-based learning: An instructional model and its constructivist framework. CRLT Technical Report No. 16-01, University of Michigan. Wojcikowski, J. (2013). Solving real-world problems through problem-based learning. College Teaching, 61(4), 153-156

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Teaching problem solving.

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Tips and Techniques

Expert vs. novice problem solvers, communicate.

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

Encourage Independence

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

Be sensitive

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

Encourage Thoroughness and Patience

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

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

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

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Problem-Based Learning (PBL)

What is Problem-Based Learning (PBL)? PBL is a student-centered approach to learning that involves groups of students working to solve a real-world problem, quite different from the direct teaching method of a teacher presenting facts and concepts about a specific subject to a classroom of students. Through PBL, students not only strengthen their teamwork, communication, and research skills, but they also sharpen their critical thinking and problem-solving abilities essential for life-long learning.

By working with PBL, students will:

Become engaged with open-ended situations that assimilate the world of work
Participate in groups to pinpoint what is known/ not known and the methods of finding information to help solve the given problem.
Investigate a problem; through critical thinking and problem solving, brainstorm a list of unique solutions.
Analyze the situation to see if the real problem is framed or if there are other problems that need to be solved.

How to Begin PBL

Establish the learning outcomes (i.e., what is it that you want your students to really learn and to be able to do after completing the learning project).
Find a real-world problem that is relevant to the students; often the problems are ones that students may encounter in their own life or future career.
Discuss pertinent rules for working in groups to maximize learning success.
Practice group processes: listening, involving others, assessing their work/peers.
Explore different roles for students to accomplish the work that needs to be done and/or to see the problem from various perspectives depending on the problem (e.g., for a problem about pollution, different roles may be a mayor, business owner, parent, child, neighboring city government officials, etc.).
Determine how the project will be evaluated and assessed. Most likely, both self-assessment and peer-assessment will factor into the assignment grade.

Designing Classroom Instruction

How to Orchestrate a PBL Activity

Explain Problem-Based Learning to students: its rationale, daily instruction, class expectations, grading.
Serve as a model and resource to the PBL process; work in-tandem through the first problem
Help students secure various resources when needed.
Supply ample class time for collaborative group work.
Give feedback to each group after they share via the established format; critique the solution in quality and thoroughness. Reinforce to the students that the prior thinking and reasoning process in addition to the solution are important as well.

Teacher’s Role in PBL

The Role of the Students

Anchored Instruction Model

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Definitions of Instructional Design

What is instructional design? Instructional design can be defined as the creation of instructional materials, modules or lessons. The instructional design process consists of determining the needs of the learners, defining the end…

Instructional Design

What is Instructional Design The Association for Educational Communications and Technology (AECT) defines instructional design as “the theory and practice of design, development, utilization, management, and evaluation of processes and resources for learning”…

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How to Create Effective Multiple Choice Questions

There are many advantages to using multiple choice (MC) questions as an evaluation / assessment strategy. They are easy to set up, easy to mark, and allow teachers to cover a wide range…

Center for Teaching Innovation

Resource library.

Establishing Community Agreements and Classroom Norms
Sample group work rubric
Problem-Based Learning Clearinghouse of Activities, University of Delaware

Problem-Based Learning

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

Why Use Problem-Based Learning?

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

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

Considerations for Using Problem-Based Learning

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

Students generally must:

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

Getting Started with Problem-Based Learning

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

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

Problem Solving in Mathematics Education

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First Online: 28 June 2016

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Peter Liljedahl 6 ,
Manuel Santos-Trigo 7 ,
Uldarico Malaspina 8 &
Regina Bruder 9

Part of the book series: ICME-13 Topical Surveys ((ICME13TS))

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Problem solving in mathematics education has been a prominent research field that aims at understanding and relating the processes involved in solving problems to students’ development of mathematical knowledge and problem solving competencies. The accumulated knowledge and field developments include conceptual frameworks to characterize learners’ success in problem solving activities, cognitive, metacognitive, social and affective analysis, curriculum proposals, and ways to foster problem solving approaches. In the survey, four interrelated areas are reviewed: (i) the relevance of heuristics in problem solving approaches—why are they important and what research tells us about their use? (ii) the need to characterize and foster creative problem solving approaches—what type of heuristics helps learners think of and practice creative solutions? (iii) the importance for learners to formulate and pursue their own problems; and (iv) the role played by the use of both multiple purpose and ad hoc mathematical action types of technologies in problem solving activities—what ways of reasoning do learners construct when they rely on the use of digital technologies and how technology and technology approaches can be reconciled?

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Mathematical Problem
Prospective Teacher
Creative Process
Digital Technology
Mathematical Task

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Mathematical problem solving has long been seen as an important aspect of mathematics, the teaching of mathematics, and the learning of mathematics. It has infused mathematics curricula around the world with calls for the teaching of problem solving as well as the teaching of mathematics through problem solving. And as such, it has been of interest to mathematics education researchers for as long as our field has existed. More relevant, mathematical problem solving has played a part in every ICME conference, from 1969 until the forthcoming meeting in Hamburg, wherein mathematical problem solving will reside most centrally within the work of Topic Study 19: Problem Solving in Mathematics Education. This booklet is being published on the occasion of this Topic Study Group.

To this end, we have assembled four summaries looking at four distinct, yet inter-related, dimensions of mathematical problem solving. The first summary, by Regina Bruder, is a nuanced look at heuristics for problem solving. This notion of heuristics is carried into Peter Liljedahl’s summary, which looks specifically at a progression of heuristics leading towards more and more creative aspects of problem solving. This is followed by Luz Manuel Santos Trigo’s summary introducing us to problem solving in and with digital technologies. The last summary, by Uldarico Malaspina Jurado, documents the rise of problem posing within the field of mathematics education in general and the problem solving literature in particular.

Each of these summaries references in some critical and central fashion the works of George Pólya or Alan Schoenfeld. To the initiated researchers, this is no surprise. The seminal work of these researchers lie at the roots of mathematical problem solving. What is interesting, though, is the diverse ways in which each of the four aforementioned contributions draw on, and position, these works so as to fit into the larger scheme of their respective summaries. This speaks to not only the depth and breadth of these influential works, but also the diversity with which they can be interpreted and utilized in extending our thinking about problem solving.

Taken together, what follows is a topical survey of ideas representing the diversity of views and tensions inherent in a field of research that is both a means to an end and an end onto itself and is unanimously seen as central to the activities of mathematics.

1 Survey on the State-of-the-Art

1.1 role of heuristics for problem solving—regina bruder.

The origin of the word heuristic dates back to the time of Archimedes and is said to have come out of one of the famous stories told about this great mathematician and inventor. The King of Syracuse asked Archimedes to check whether his new wreath was really made of pure gold. Archimedes struggled with this task and it was not until he was at the bathhouse that he came up with the solution. As he entered the tub he noticed that he had displaced a certain amount of water. Brilliant as he was, he transferred this insight to the issue with the wreath and knew he had solved the problem. According to the legend, he jumped out of the tub and ran from the bathhouse naked screaming, “Eureka, eureka!”. Eureka and heuristic have the same root in the ancient Greek language and so it has been claimed that this is how the academic discipline of “heuristics” dealing with effective approaches to problem solving (so-called heurisms) was given its name. Pólya ( 1964 ) describes this discipline as follows:

Heuristics deals with solving tasks. Its specific goals include highlighting in general terms the reasons for selecting those moments in a problem the examination of which could help us find a solution. (p. 5)

This discipline has grown, in part, from examining the approaches to certain problems more in detail and comparing them with each other in order to abstract similarities in approach, or so-called heurisms. Pólya ( 1949 ), but also, inter alia, Engel ( 1998 ), König ( 1984 ) and Sewerin ( 1979 ) have formulated such heurisms for mathematical problem tasks. The problem tasks examined by the authors mentioned are predominantly found in the area of talent programmes, that is, they often go back to mathematics competitions.

In 1983 Zimmermann provided an overview of heuristic approaches and tools in American literature which also offered suggestions for mathematics classes. In the German-speaking countries, an approach has established itself, going back to Sewerin ( 1979 ) and König ( 1984 ), which divides school-relevant heuristic procedures into heuristic tools, strategies and principles, see also Bruder and Collet ( 2011 ).

Below is a review of the conceptual background of heuristics, followed by a description of the effect mechanisms of heurisms in problem-solving processes.

1.1.1 Research Review on the Promotion of Problem Solving

In the 20th century, there has been an advancement of research on mathematical problem solving and findings about possibilities to promote problem solving with varying priorities (c.f. Pehkonen 1991 ). Based on a model by Pólya ( 1949 ), in a first phase of research on problem solving, particularly in the 1960s and the 1970s, a series of studies on problem-solving processes placing emphasis on the importance of heuristic strategies (heurisms) in problem solving has been carried out. It was assumed that teaching and learning heuristic strategies, principles and tools would provide students with an orientation in problem situations and that this could thus improve students’ problem-solving abilities (c.f. for instance, Schoenfeld 1979 ). This approach, mostly researched within the scope of talent programmes for problem solving, was rather successful (c.f. for instance, Sewerin 1979 ). In the 1980s, requests for promotional opportunities in everyday teaching were given more and more consideration: “ problem solving must be the focus of school mathematics in the 1980s ” (NCTM 1980 ). For the teaching and learning of problem solving in regular mathematics classes, the current view according to which cognitive, heuristic aspects were paramount, was expanded by certain student-specific aspects, such as attitudes, emotions and self-regulated behaviour (c.f. Kretschmer 1983 ; Schoenfeld 1985 , 1987 , 1992 ). Kilpatrick ( 1985 ) divided the promotional approaches described in the literature into five methods which can also be combined with each other.

Osmosis : action-oriented and implicit imparting of problem-solving techniques in a beneficial learning environment

Memorisation : formation of special techniques for particular types of problem and of the relevant questioning when problem solving

Imitation : acquisition of problem-solving abilities through imitation of an expert

Cooperation : cooperative learning of problem-solving abilities in small groups

Reflection : problem-solving abilities are acquired in an action-oriented manner and through reflection on approaches to problem solving.

Kilpatrick ( 1985 ) views as success when heuristic approaches are explained to students, clarified by means of examples and trained through the presentation of problems. The need of making students aware of heuristic approaches is by now largely accepted in didactic discussions. Differences in varying approaches to promoting problem-solving abilities rather refer to deciding which problem-solving strategies or heuristics are to imparted to students and in which way, and not whether these should be imparted at all or not.

1.1.2 Heurisms as an Expression of Mental Agility

The activity theory, particularly in its advancement by Lompscher ( 1975 , 1985 ), offers a well-suited and manageable model to describe learning activities and differences between learners with regard to processes and outcomes in problem solving (c.f. Perels et al. 2005 ). Mental activity starts with a goal and the motive of a person to perform such activity. Lompscher divides actual mental activity into content and process. Whilst the content in mathematical problem-solving consists of certain concepts, connections and procedures, the process describes the psychological processes that occur when solving a problem. This course of action is described in Lompscher by various qualities, such as systematic planning, independence, accuracy, activity and agility. Along with differences in motivation and the availability of expertise, it appears that intuitive problem solvers possess a particularly high mental agility, at least with regard to certain contents areas.

According to Lompscher, “flexibility of thought” expresses itself

… by the capacity to change more or less easily from one aspect of viewing to another one or to embed one circumstance or component into different correlations, to understand the relativity of circumstances and statements. It allows to reverse relations, to more or less easily or quickly attune to new conditions of mental activity or to simultaneously mind several objects or aspects of a given activity (Lompscher 1975 , p. 36).

These typical manifestations of mental agility can be focused on in problem solving by mathematical means and can be related to the heurisms known from the analyses of approaches by Pólya et al. (c.f. also Bruder 2000 ):

Reduction : Successful problem solvers will intuitively reduce a problem to its essentials in a sensible manner. To achieve such abstraction, they often use visualisation and structuring aids, such as informative figures, tables, solution graphs or even terms. These heuristic tools are also very well suited to document in retrospect the approach adopted by the intuitive problem solvers in a way that is comprehensible for all.

Reversibility : Successful problem solvers are able to reverse trains of thought or reproduce these in reverse. They will do this in appropriate situations automatically, for instance, when looking for a key they have mislaid. A corresponding general heuristic strategy is working in reverse.

Minding of aspects : Successful problem solvers will mind several aspects of a given problem at the same time or easily recognise any dependence on things and vary them in a targeted manner. Sometimes, this is also a matter of removing barriers in favour of an idea that appears to be sustainable, that is, by simply “hanging on” to a certain train of thought even against resistance. Corresponding heurisms are, for instance, the principle of invariance, the principle of symmetry (Engel 1998 ), the breaking down or complementing of geometric figures to calculate surface areas, or certain terms used in binomial formulas.

Change of aspects : Successful problem solvers will possibly change their assumptions, criteria or aspects minded in order to find a solution. Various aspects of a given problem will be considered intuitively or the problem be viewed from a different perspective, which will prevent “getting stuck” and allow for new insights and approaches. For instance, many elementary geometric propositions can also be proved in an elegant vectorial manner.

Transferring : Successful problem solvers will be able more easily than others to transfer a well-known procedure to another, sometimes even very different context. They recognise more easily the “framework” or pattern of a given task. Here, this is about own constructions of analogies and continual tracing back from the unknown to the known.

Intuitive, that is, untrained good problem solvers, are, however, often unable to access these flexibility qualities consciously. This is why they are also often unable to explain how they actually solved a given problem.

To be able to solve problems successfully, a certain mental agility is thus required. If this is less well pronounced in a certain area, learning how to solve problems means compensating by acquiring heurisms. In this case, insufficient mental agility is partly “offset” through the application of knowledge acquired by means of heurisms. Mathematical problem-solving competences are thus acquired through the promotion of manifestations of mental agility (reduction, reversibility, minding of aspects and change of aspects). This can be achieved by designing sub-actions of problem solving in connection with a (temporarily) conscious application of suitable heurisms. Empirical evidence for the success of the active principle of heurisms has been provided by Collet ( 2009 ).

Against such background, learning how to solve problems can be established as a long-term teaching and learning process which basically encompasses four phases (Bruder and Collet 2011 ):

Intuitive familiarisation with heuristic methods and techniques.

Making aware of special heurisms by means of prominent examples (explicit strategy acquisition).

Short conscious practice phase to use the newly acquired heurisms with differentiated task difficulties.

Expanding the context of the strategies applied.

In the first phase, students are familiarised with heurisms intuitively by means of targeted aid impulses and questions (what helped us solve this problem?) which in the following phase are substantiated on the basis of model tasks, are given names and are thus made aware of their existence. The third phase serves the purpose of a certain familiarisation with the new heurisms and the experience of competence through individualised practising at different requirement levels, including in the form of homework over longer periods. A fourth and delayed fourth phase aims at more flexibility through the transfer to other contents and contexts and the increasingly intuitive use of the newly acquired heurisms, so that students can enrich their own problem-solving models in a gradual manner. The second and third phases build upon each other in close chronological order, whilst the first phase should be used in class at all times.

All heurisms can basically be described in an action-oriented manner by means of asking the right questions. The way of asking questions can thus also establish a certain kind of personal relation. Even if the teacher presents and suggests the line of basic questions with a prototypical wording each time, students should always be given the opportunity to find “their” wording for the respective heurism and take a note of it for themselves. A possible key question for the use of a heuristic tool would be: How to illustrate and structure the problem or how to present it in a different way?

Unfortunately, for many students, applying heuristic approaches to problem solving will not ensue automatically but will require appropriate early and long-term promoting. The results of current studies, where promotion approaches to problem solving are connected with self-regulation and metacognitive aspects, demonstrate certain positive effects of such combination on students. This field of research includes, for instance, studies by Lester et al. ( 1989 ), Verschaffel et al. ( 1999 ), the studies on teaching method IMPROVE by Mevarech and Kramarski ( 1997 , 2003 ) and also the evaluation of a teaching concept on learning how to solve problems by the gradual conscious acquisition of heurisms by Collet and Bruder ( 2008 ).

1.2 Creative Problem Solving—Peter Liljedahl

There is a tension between the aforementioned story of Archimedes and the heuristics presented in the previous section. Archimedes, when submersing himself in the tub and suddenly seeing the solution to his problem, wasn’t relying on osmosis, memorisation, imitation, cooperation, or reflection (Kilpatrick 1985 ). He wasn’t drawing on reduction, reversibility, minding of aspects, change of aspect, or transfer (Bruder 2000 ). Archimedes was stuck and it was only, in fact, through insight and sudden illumination that he managed to solve his problem. In short, Archimedes was faced with a problem that the aforementioned heuristics, and their kind, would not help him to solve.

According to some, such a scenario is the definition of a problem. For example, Resnick and Glaser ( 1976 ) define a problem as being something that you do not have the experience to solve. Mathematicians, in general, agree with this (Liljedahl 2008 ).

Any problem in which you can see how to attack it by deliberate effort, is a routine problem, and cannot be an important discover. You must try and fail by deliberate efforts, and then rely on a sudden inspiration or intuition or if you prefer to call it luck. (Dan Kleitman, participant cited in Liljedahl 2008 , p. 19).

Problems, then, are tasks that cannot be solved by direct effort and will require some creative insight to solve (Liljedahl 2008 ; Mason et al. 1982 ; Pólya 1965 ).

1.2.1 A History of Creativity in Mathematics Education

In 1902, the first half of what eventually came to be a 30 question survey was published in the pages of L’Enseignement Mathématique , the journal of the French Mathematical Society. The authors, Édouard Claparède and Théodore Flournoy, were two Swiss psychologists who were deeply interested in the topics of mathematical discovery, creativity and invention. Their hope was that a widespread appeal to mathematicians at large would incite enough responses for them to begin to formulate some theories about this topic. The first half of the survey centered on the reasons for becoming a mathematician (family history, educational influences, social environment, etc.), attitudes about everyday life, and hobbies. This was eventually followed, in 1904, by the publication of the second half of the survey pertaining, in particular, to mental images during periods of creative work. The responses were sorted according to nationality and published in 1908.

During this same period Henri Poincaré (1854–1912), one of the most noteworthy mathematicians of the time, had already laid much of the groundwork for his own pursuit of this same topic and in 1908 gave a presentation to the French Psychological Society in Paris entitled L’Invention mathématique —often mistranslated to Mathematical Creativity Footnote 1 (c.f. Poincaré 1952 ). At the time of the presentation Poincaré stated that he was aware of Claparède and Flournoy’s work, as well as their results, but expressed that they would only confirm his own findings. Poincaré’s presentation, as well as the essay it spawned, stands to this day as one of the most insightful, and thorough treatments of the topic of mathematical discovery, creativity, and invention.

Just at this time, I left Caen, where I was living, to go on a geological excursion under the auspices of the School of Mines. The incident of the travel made me forget my mathematical work. Having reached Coutances, we entered an omnibus to go some place or other. At the moment when I put my foot on the step, the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuschian functions were identical with those of non-Euclidean geometry. I did not verify the idea; I should not have had the time, as, upon taking my seat in the omnibus, I went on with the conversation already commenced, but I felt a perfect certainty. On my return to Caen, for conscience’ sake, I verified the results at my leisure. (Poincaré 1952 , p. 53)

So powerful was his presentation, and so deep were his insights into his acts of invention and discovery that it could be said that he not so much described the characteristics of mathematical creativity, as defined them. From that point forth mathematical creativity, or even creativity in general, has not been discussed seriously without mention of Poincaré’s name.

Inspired by this presentation, Jacques Hadamard (1865–1963), a contemporary and a friend of Poincaré’s, began his own empirical investigation into this fascinating phenomenon. Hadamard had been critical of Claparède and Flournoy’s work in that they had not adequately treated the topic on two fronts. As exhaustive as the survey appeared to be, Hadamard felt that it failed to ask some key questions—the most important of which was with regard to the reason for failures in the creation of mathematics. This seemingly innocuous oversight, however, led directly to his second and “most important criticism” (Hadamard 1945 ). He felt that only “first-rate men would dare to speak of” (p. 10) such failures. So, inspired by his good friend Poincaré’s treatment of the subject Hadamard retooled the survey and gave it to friends of his for consideration—mathematicians such as Henri Poincaré and Albert Einstein, whose prominence were beyond reproach. Ironically, the new survey did not contain any questions that explicitly dealt with failure. In 1943 Hadamard gave a series of lectures on mathematical invention at the École Libre des Hautes Études in New York City. These talks were subsequently published as The Psychology of Mathematical Invention in the Mathematical Field (Hadameard 1945 ).

Hadamard’s classic work treats the subject of invention at the crossroads of mathematics and psychology. It provides not only an entertaining look at the eccentric nature of mathematicians and their rituals, but also outlines the beliefs of mid twentieth-century mathematicians about the means by which they arrive at new mathematics. It is an extensive exploration and extended argument for the existence of unconscious mental processes. In essence, Hadamard took the ideas that Poincaré had posed and, borrowing a conceptual framework for the characterization of the creative process from the Gestaltists of the time (Wallas 1926 ), turned them into a stage theory. This theory still stands as the most viable and reasonable description of the process of mathematical creativity.

1.2.2 Defining Mathematical Creativity

The phenomena of mathematical creativity, although marked by sudden illumination, actually consist of four separate stages stretched out over time, of which illumination is but one stage. These stages are initiation, incubation, illumination, and verification (Hadamard 1945 ). The first of these stages, the initiation phase, consists of deliberate and conscious work. This would constitute a person’s voluntary, and seemingly fruitless, engagement with a problem and be characterized by an attempt to solve the problem by trolling through a repertoire of past experiences. This is an important part of the inventive process because it creates the tension of unresolved effort that sets up the conditions necessary for the ensuing emotional release at the moment of illumination (Hadamard 1945 ; Poincaré 1952 ).

Following the initiation stage the solver, unable to come up with a solution stops working on the problem at a conscious level and begins to work on it at an unconscious level (Hadamard 1945 ; Poincaré 1952 ). This is referred to as the incubation stage of the inventive process and can last anywhere from several minutes to several years. After the period of incubation a rapid coming to mind of a solution, referred to as illumination , may occur. This is accompanied by a feeling of certainty and positive emotions (Poincaré 1952 ). Although the processes of incubation and illumination are shrouded behind the veil of the unconscious there are a number of things that can be deduced about them. First and foremost is the fact that unconscious work does, indeed, occur. Poincaré ( 1952 ), as well as Hadamard ( 1945 ), use the very real experience of illumination, a phenomenon that cannot be denied, as evidence of unconscious work, the fruits of which appear in the flash of illumination. No other theory seems viable in explaining the sudden appearance of solution during a walk, a shower, a conversation, upon waking, or at the instance of turning the conscious mind back to the problem after a period of rest (Poincaré 1952 ). Also deducible is that unconscious work is inextricably linked to the conscious and intentional effort that precedes it.

There is another remark to be made about the conditions of this unconscious work: it is possible, and of a certainty it is only fruitful, if it is on the one hand preceded and on the other hand followed by a period of conscious work. These sudden inspirations never happen except after some days of voluntary effort which has appeared absolutely fruitless and whence nothing good seems to have come … (Poincaré 1952 , p. 56)

Hence, the fruitless efforts of the initiation phase are only seemingly so. They not only set up the aforementioned tension responsible for the emotional release at the time of illumination, but also create the conditions necessary for the process to enter into the incubation phase.

Illumination is the manifestation of a bridging that occurs between the unconscious mind and the conscious mind (Poincaré 1952 ), a coming to (conscious) mind of an idea or solution. What brings the idea forward to consciousness is unclear, however. There are theories of the aesthetic qualities of the idea, effective surprise/shock of recognition, fluency of processing, or breaking functional fixedness. For reasons of brevity I will only expand on the first of these.

Poincaré proposed that ideas that were stimulated during initiation remained stimulated during incubation. However, freed from the constraints of conscious thought and deliberate calculation, these ideas would begin to come together in rapid and random unions so that “their mutual impacts may produce new combinations” (Poincaré 1952 ). These new combinations, or ideas, would then be evaluated for viability using an aesthetic sieve, which allows through to the conscious mind only the “right combinations” (Poincaré 1952 ). It is important to note, however, that good or aesthetic does not necessarily mean correct. Correctness is evaluated during the verification stage.

The purpose of verification is not only to check for correctness. It is also a method by which the solver re-engages with the problem at the level of details. That is, during the unconscious work the problem is engaged with at the level of ideas and concepts. During verification the solver can examine these ideas in closer details. Poincaré succinctly describes both of these purposes.

As for the calculations, themselves, they must be made in the second period of conscious work, that which follows the inspiration, that in which one verifies the results of this inspiration and deduces their consequences. (Poincaré 1952 , p. 62)

Aside from presenting this aforementioned theory on invention, Hadamard also engaged in a far-reaching discussion on a number of interesting, and sometimes quirky, aspects of invention and discovery that he had culled from the results of his empirical study, as well as from pertinent literature. This discussion was nicely summarized by Newman ( 2000 ) in his commentary on the elusiveness of invention.

The celebrated phrenologist Gall said mathematical ability showed itself in a bump on the head, the location of which he specified. The psychologist Souriau, we are told, maintained that invention occurs by “pure chance”, a valuable theory. It is often suggested that creative ideas are conjured up in “mathematical dreams”, but this attractive hypothesis has not been verified. Hadamard reports that mathematicians were asked whether “noises” or “meteorological circumstances” helped or hindered research [..] Claude Bernard, the great physiologist, said that in order to invent “one must think aside”. Hadamard says this is a profound insight; he also considers whether scientific invention may perhaps be improved by standing or sitting or by taking two baths in a row. Helmholtz and Poincaré worked sitting at a table; Hadamard’s practice is to pace the room (“Legs are the wheels of thought”, said Emile Angier); the chemist J. Teeple was the two-bath man. (p. 2039)

1.2.3 Discourses on Creativity

Creativity is a term that can be used both loosely and precisely. That is, while there exists a common usage of the term there also exists a tradition of academic discourse on the subject. A common usage of creative refers to a process or a person whose products are original, novel, unusual, or even abnormal (Csíkszentmihályi 1996 ). In such a usage, creativity is assessed on the basis of the external and observable products of the process, the process by which the product comes to be, or on the character traits of the person doing the ‘creating’. Each of these usages—product, process, person—is the roots of the discourses (Liljedahl and Allan 2014 ) that I summarize here, the first of which concerns products.

Consider a mother who states that her daughter is creative because she drew an original picture. The basis of such a statement can lie either in the fact that the picture is unlike any the mother has ever seen or unlike any her daughter has ever drawn before. This mother is assessing creativity on the basis of what her daughter has produced. However, the standards that form the basis of her assessment are neither consistent nor stringent. There does not exist a universal agreement as to what she is comparing the picture to (pictures by other children or other pictures by the same child). Likewise, there is no standard by which the actual quality of the picture is measured. The academic discourse that concerns assessment of products, on the other hand, is both consistent and stringent (Csíkszentmihályi 1996 ). This discourse concerns itself more with a fifth, and as yet unmentioned, stage of the creative process; elaboration . Elaboration is where inspiration becomes perspiration (Csíkszentmihályi 1996 ). It is the act of turning a good idea into a finished product, and the finished product is ultimately what determines the creativity of the process that spawned it—that is, it cannot be a creative process if nothing is created. In particular, this discourse demands that the product be assessed against other products within its field, by the members of that field, to determine if it is original AND useful (Csíkszentmihályi 1996 ; Bailin 1994 ). If it is, then the product is deemed to be creative. Note that such a use of assessment of end product pays very little attention to the actual process that brings this product forth.

The second discourse concerns the creative process. The literature pertaining to this can be separated into two categories—a prescriptive discussion of the creativity process and a descriptive discussion of the creativity process. Although both of these discussions have their roots in the four stages that Wallas ( 1926 ) proposed makes up the creative process, they make use of these stages in very different ways. The prescriptive discussion of the creative process is primarily focused on the first of the four stages, initiation , and is best summarized as a cause - and - effect discussion of creativity, where the thinking processes during the initiation stage are the cause and the creative outcome are the effects (Ghiselin 1952 ). Some of the literature claims that the seeds of creativity lie in being able to think about a problem or situation analogically. Other literature claims that utilizing specific thinking tools such as imagination, empathy, and embodiment will lead to creative products. In all of these cases, the underlying theory is that the eventual presentation of a creative idea will be precipitated by the conscious and deliberate efforts during the initiation stage. On the other hand, the literature pertaining to a descriptive discussion of the creative process is inclusive of all four stages (Kneller 1965 ; Koestler 1964 ). For example, Csíkszentmihályi ( 1996 ), in his work on flow attends to each of the stages, with much attention paid to the fluid area between conscious and unconscious work, or initiation and incubation. His claim is that the creative process is intimately connected to the enjoyment that exists during times of sincere and consuming engagement with a situation, the conditions of which he describes in great detail.

The third, and final, discourse on creativity pertains to the person. This discourse is space dominated by two distinct characteristics, habit and genius. Habit has to do with the personal habits as well as the habits of mind of people that have been deemed to be creative. However, creative people are most easily identified through their reputation for genius. Consequently, this discourse is often dominated by the analyses of the habits of geniuses as is seen in the work of Ghiselin ( 1952 ), Koestler ( 1964 ), and Kneller ( 1965 ) who draw on historical personalities such as Albert Einstein, Henri Poincaré, Vincent Van Gogh, D.H. Lawrence, Samuel Taylor Coleridge, Igor Stravinsky, and Wolfgang Amadeus Mozart to name a few. The result of this sort of treatment is that creative acts are viewed as rare mental feats, which are produced by extraordinary individuals who use extraordinary thought processes.

These different discourses on creativity can be summed up in a tension between absolutist and relativist perspectives on creativity (Liljedahl and Sriraman 2006 ). An absolutist perspective assumes that creative processes are the domain of genius and are present only as precursors to the creation of remarkably useful and universally novel products. The relativist perspective, on the other hand, allows for every individual to have moments of creativity that may, or may not, result in the creation of a product that may, or may not, be either useful or novel.

Between the work of a student who tries to solve a problem in geometry or algebra and a work of invention, one can say there is only a difference of degree. (Hadamard 1945 , p. 104).

Regardless of discourse, however, creativity is not “part of the theories of logical forms” (Dewey 1938 ). That is, creativity is not representative of the lock-step logic and deductive reasoning that mathematical problem solving is often presumed to embody (Bibby 2002 ; Burton 1999 ). Couple this with the aforementioned demanding constraints as to what constitutes a problem, where then does that leave problem solving heuristics? More specifically, are there creative problem solving heuristics that will allow us to resolve problems that require illumination to solve? The short answer to this question is yes—there does exist such problem solving heuristics. To understand these, however, we must first understand the routine problem solving heuristics they are built upon. In what follows, I walk through the work of key authors and researchers whose work offers us insights into progressively more creative problem solving heuristics for solving true problems.

1.2.4 Problem Solving by Design

In a general sense, design is defined as the algorithmic and deductive approach to solving a problem (Rusbult 2000 ). This process begins with a clearly defined goal or objective after which there is a great reliance on relevant past experience, referred to as repertoire (Bruner 1964 ; Schön 1987 ), to produce possible options that will lead towards a solution of the problem (Poincaré 1952 ). These options are then examined through a process of conscious evaluations (Dewey 1933 ) to determine their suitability for advancing the problem towards the final goal. In very simple terms, problem solving by design is the process of deducing the solution from that which is already known.

Mayer ( 1982 ), Schoenfeld ( 1982 ), and Silver ( 1982 ) state that prior knowledge is a key element in the problem solving process. Prior knowledge influences the problem solver’s understanding of the problem as well as the choice of strategies that will be called upon in trying to solve the problem. In fact, prior knowledge and prior experiences is all that a solver has to draw on when first attacking a problem. As a result, all problem solving heuristics incorporate this resource of past experiences and prior knowledge into their initial attack on a problem. Some heuristics refine these ideas, and some heuristics extend them (c.f. Kilpatrick 1985 ; Bruder 2000 ). Of the heuristics that refine, none is more influential than the one created by George Pólya (1887–1985).

1.2.5 George Pólya: How to Solve It

In his book How to Solve It (1949) Pólya lays out a problem solving heuristic that relies heavily on a repertoire of past experience. He summarizes the four-step process of his heuristic as follows:

Understanding the Problem

First. You have to understand the problem.

What is the unknown? What are the data? What is the condition?

Is it possible to satisfy the condition? Is the condition sufficient to determine the unknown? Or is it insufficient? Or redundant? Or contradictory?

Draw a figure. Introduce suitable notation.

Separate the various parts of the condition. Can you write them down?

Devising a Plan

Second. Find the connection between the data and the unknown. You may be obliged to consider auxiliary problems if an immediate connection cannot be found. You should obtain eventually a plan of the solution.

Have you seen it before? Or have you seen the same problem in a slightly different form?

Do you know a related problem? Do you know a theorem that could be useful?

Look at the unknown! And try to think of a familiar problem having the same or a similar unknown.

Here is a problem related to yours and solved before. Could you use it? Could you use its result? Could you use its method? Should you introduce some auxiliary element in order to make its use possible?

Could you restate the problem? Could you restate it still differently? Go back to definitions.

If you cannot solve the proposed problem try to solve first some related problem. Could you imagine a more accessible related problem? A more general problem? A more special problem? An analogous problem? Could you solve a part of the problem? Keep only a part of the condition, drop the other part; how far is the unknown then determined, how can it vary? Could you derive something useful from the data? Could you think of other data appropriate to determine the unknown? Could you change the unknown or data, or both if necessary, so that the new unknown and the new data are nearer to each other?

Did you use all the data? Did you use the whole condition? Have you taken into account all essential notions involved in the problem?

Carrying Out the Plan

Third. Carry out your plan.

Carrying out your plan of the solution, check each step. Can you see clearly that the step is correct? Can you prove that it is correct?

Looking Back

Fourth. Examine the solution obtained.

Can you check the result? Can you check the argument?

Can you derive the solution differently? Can you see it at a glance?

Can you use the result, or the method, for some other problem?

The emphasis on auxiliary problems, related problems, and analogous problems that are, in themselves, also familiar problems is an explicit manifestation of relying on a repertoire of past experience. This use of familiar problems also requires an ability to deduce from these related problems a recognizable and relevant attribute that will transfer to the problem at hand. The mechanism that allows for this transfer of knowledge between analogous problems is known as analogical reasoning (English 1997 , 1998 ; Novick 1988 , 1990 , 1995 ; Novick and Holyoak 1991 ) and has been shown to be an effective, but not always accessible, thinking strategy.

Step four in Pólya’s heuristic, looking back, is also a manifestation of utilizing prior knowledge to solve problems, albeit an implicit one. Looking back makes connections “in memory to previously acquired knowledge [..] and further establishes knowledge in long-term memory that may be elaborated in later problem-solving encounters” (Silver 1982 , p. 20). That is, looking back is a forward-looking investment into future problem solving encounters, it sets up connections that may later be needed.

Pólya’s heuristic is a refinement on the principles of problem solving by design. It not only makes explicit the focus on past experiences and prior knowledge, but also presents these ideas in a very succinct, digestible, and teachable manner. This heuristic has become a popular, if not the most popular, mechanism by which problem solving is taught and learned.

1.2.6 Alan Schoenfeld: Mathematical Problem Solving

The work of Alan Schoenfeld is also a refinement on the principles of problem solving by design. However, unlike Pólya ( 1949 ) who refined these principles at a theoretical level, Schoenfeld has refined them at a practical and empirical level. In addition to studying taught problem solving strategies he has also managed to identify and classify a variety of strategies, mostly ineffectual, that students invoke naturally (Schoenfeld 1985 , 1992 ). In so doing, he has created a better understanding of how students solve problems, as well as a better understanding of how problems should be solved and how problem solving should be taught.

For Schoenfeld, the problem solving process is ultimately a dialogue between the problem solver’s prior knowledge, his attempts, and his thoughts along the way (Schoenfeld 1982 ). As such, the solution path of a problem is an emerging and contextually dependent process. This is a departure from the predefined and contextually independent processes of Pólya’s ( 1949 ) heuristics. This can be seen in Schoenfeld’s ( 1982 ) description of a good problem solver.

To examine what accounts for expertise in problem solving, you would have to give the expert a problem for which he does not have access to a solution schema. His behavior in such circumstances is radically different from what you would see when he works on routine or familiar “non-routine” problems. On the surface his performance is no longer proficient; it may even seem clumsy. Without access to a solution schema, he has no clear indication of how to start. He may not fully understand the problem, and may simply “explore it for a while until he feels comfortable with it. He will probably try to “match” it to familiar problems, in the hope it can be transformed into a (nearly) schema-driven solution. He will bring up a variety of plausible things: related facts, related problems, tentative approaches, etc. All of these will have to be juggled and balanced. He may make an attempt solving it in a particular way, and then back off. He may try two or three things for a couple of minutes and then decide which to pursue. In the midst of pursuing one direction he may go back and say “that’s harder than it should be” and try something else. Or, after the comment, he may continue in the same direction. With luck, after some aborted attempts, he will solve the problem. (p. 32-33)

Aside from demonstrating the emergent nature of the problem solving process, this passage also brings forth two consequences of Schoenfeld’s work. The first of these is the existence of problems for which the solver does not have “access to a solution schema”. Unlike Pólya ( 1949 ), who’s heuristic is a ‘one size fits all (problems)’ heuristic, Schoenfeld acknowledges that problem solving heuristics are, in fact, personal entities that are dependent on the solver’s prior knowledge as well as their understanding of the problem at hand. Hence, the problems that a person can solve through his or her personal heuristic are finite and limited.

The second consequence that emerges from the above passage is that if a person lacks the solution schema to solve a given problem s/he may still solve the problem with the help of luck . This is an acknowledgement, if only indirectly so, of the difference between problem solving in an intentional and mechanical fashion verses problem solving in a more creative fashion, which is neither intentional nor mechanical (Pehkonen 1997 ).

1.2.7 David Perkins: Breakthrough Thinking

As mentioned, many consider a problem that can be solved by intentional and mechanical means to not be worthy of the title ‘problem’. As such, a repertoire of past experiences sufficient for dealing with such a ‘problem’ would disqualify it from the ranks of ‘problems’ and relegate it to that of ‘exercises’. For a problem to be classified as a ‘problem’, then, it must be ‘problematic’. Although such an argument is circular it is also effective in expressing the ontology of mathematical ‘problems’.

Perkins ( 2000 ) also requires problems to be problematic. His book Archimedes’ Bathtub: The Art and Logic of Breakthrough Thinking (2000) deals with situations in which the solver has gotten stuck and no amount of intentional or mechanical adherence to the principles of past experience and prior knowledge is going to get them unstuck. That is, he deals with problems that, by definition, cannot be solved through a process of design [or through the heuristics proposed by Pólya ( 1949 ) and Schoenfeld ( 1985 )]. Instead, the solver must rely on the extra-logical process of what Perkins ( 2000 ) calls breakthrough thinking .

Perkins ( 2000 ) begins by distinguishing between reasonable and unreasonable problems. Although both are solvable, only reasonable problems are solvable through reasoning. Unreasonable problems require a breakthrough in order to solve them. The problem, however, is itself inert. It is neither reasonable nor unreasonable. That quality is brought to the problem by the solver. That is, if a student cannot solve a problem by direct effort then that problem is deemed to be unreasonable for that student. Perkins ( 2000 ) also acknowledges that what is an unreasonable problem for one person is a perfectly reasonable problem for another person; reasonableness is dependent on the person.

This is not to say that, once found, the solution cannot be seen as accessible through reason. During the actual process of solving, however, direct and deductive reasoning does not work. Perkins ( 2000 ) uses several classic examples to demonstrate this, the most famous being the problem of connecting nine dots in a 3 × 3 array with four straight lines without removing pencil from paper, the solution to which is presented in Fig. 1 .

Nine dots—four lines problem and solution

To solve this problem, Perkins ( 2000 ) claims that the solver must recognize that the constraint of staying within the square created by the 3 × 3 array is a self-imposed constraint. He further claims that until this is recognized no amount of reasoning is going to solve the problem. That is, at this point in the problem solving process the problem is unreasonable. However, once this self-imposed constraint is recognized the problem, and the solution, are perfectly reasonable. Thus, the solution of an, initially, unreasonable problem is reasonable.

The problem solving heuristic that Perkins ( 2000 ) has constructed to deal with solvable, but unreasonable, problems revolves around the idea of breakthrough thinking and what he calls breakthrough problems . A breakthrough problem is a solvable problem in which the solver has gotten stuck and will require an AHA! to get unstuck and solve the problem. Perkins ( 2000 ) poses that there are only four types of solvable unreasonable problems, which he has named wilderness of possibilities , the clueless plateau , narrow canyon of exploration , and oasis of false promise . The names for the first three of these types of problems are related to the Klondike gold rush in Alaska, a time and place in which gold was found more by luck than by direct and systematic searching.

The wilderness of possibilities is a term given to a problem that has many tempting directions but few actual solutions. This is akin to a prospector searching for gold in the Klondike. There is a great wilderness in which to search, but very little gold to be found. The clueless plateau is given to problems that present the solver with few, if any, clues as to how to solve it. The narrow canyon of exploration is used to describe a problem that has become constrained in such a way that no solution now exists. The nine-dot problem presented above is such a problem. The imposed constraint that the lines must lie within the square created by the array makes a solution impossible. This is identical to the metaphor of a prospector searching for gold within a canyon where no gold exists. The final type of problem gets its name from the desert. An oasis of false promise is a problem that allows the solver to quickly get a solution that is close to the desired outcome; thereby tempting them to remain fixed on the strategy that they used to get this almost-answer. The problem is, that like the canyon, the solution does not exist at the oasis; the solution strategy that produced an almost-answer is incapable of producing a complete answer. Likewise, a desert oasis is a false promise in that it is only a reprieve from the desolation of the dessert and not a final destination.

Believing that there are only four ways to get stuck, Perkins ( 2000 ) has designed a problem solving heuristic that will “up the chances” of getting unstuck. This heuristic is based on what he refers to as “the logic of lucking out” (p. 44) and is built on the idea of introspection. By first recognizing that they are stuck, and then recognizing that the reason they are stuck can only be attributed to one of four reasons, the solver can access four strategies for getting unstuck, one each for the type of problem they are dealing with. If the reason they are stuck is because they are faced with a wilderness of possibilities they are to begin roaming far, wide, and systematically in the hope of reducing the possible solution space to one that is more manageable. If they find themselves on a clueless plateau they are to begin looking for clues, often in the wording of the problem. When stuck in a narrow canyon of possibilities they need to re-examine the problem and see if they have imposed any constraints. Finally, when in an oasis of false promise they need to re-attack the problem in such a way that they stay away from the oasis.

Of course, there are nuances and details associated with each of these types of problems and the strategies for dealing with them. However, nowhere within these details is there mention of the main difficulty inherent in introspection; that it is much easier for the solver to get stuck than it is for them to recognize that they are stuck. Once recognized, however, the details of Perkins’ ( 2000 ) heuristic offer the solver some ways for recognizing why they are stuck.

1.2.8 John Mason, Leone Burton, and Kaye Stacey: Thinking Mathematically

The work of Mason et al. in their book Thinking Mathematically ( 1982 ) also recognizes the fact that for each individual there exists problems that will not yield to their intentional and mechanical attack. The heuristic that they present for dealing with this has two main processes with a number of smaller phases, rubrics, and states. The main processes are what they refer to as specializing and generalizing. Specializing is the process of getting to know the problem and how it behaves through the examination of special instances of the problem. This process is synonymous with problem solving by design and involves the repeated oscillation between the entry and attack phases of Mason et al. ( 1982 ) heuristic. The entry phase is comprised of ‘getting started’ and ‘getting involved’ with the problem by using what is immediately known about it. Attacking the problem involves conjecturing and testing a number of hypotheses in an attempt to gain greater understanding of the problem and to move towards a solution.

At some point within this process of oscillating between entry and attack the solver will get stuck, which Mason et al. ( 1982 ) refer to as “an honourable and positive state, from which much can be learned” (p. 55). The authors dedicate an entire chapter to this state in which they acknowledge that getting stuck occurs long before an awareness of being stuck develops. They proposes that the first step to dealing with being stuck is the simple act of writing STUCK!

The act of expressing my feelings helps to distance me from my state of being stuck. It frees me from incapacitating emotions and reminds me of actions that I can take. (p. 56)

The next step is to reengage the problem by examining the details of what is known, what is wanted, what can be introduced into the problem, and what has been introduced into the problem (imposed assumptions). This process is engaged in until an AHA!, which advances the problem towards a solution, is encountered. If, at this point, the problem is not completely solved the oscillation is then resumed.

At some point in this process an attack on the problem will yield a solution and generalizing can begin. Generalizing is the process by which the specifics of a solution are examined and questions as to why it worked are investigated. This process is synonymous with the verification and elaboration stages of invention and creativity. Generalization may also include a phase of review that is similar to Pólya’s ( 1949 ) looking back.

1.2.9 Gestalt: The Psychology of Problem Solving

The Gestalt psychology of learning believes that all learning is based on insights (Koestler 1964 ). This psychology emerged as a response to behaviourism, which claimed that all learning was a response to external stimuli. Gestalt psychologists, on the other hand, believed that there was a cognitive process involved in learning as well. With regards to problem solving, the Gestalt school stands firm on the belief that problem solving, like learning, is a product of insight and as such, cannot be taught. In fact, the theory is that not only can problem solving not be taught, but also that attempting to adhere to any sort of heuristic will impede the working out of a correct solution (Krutestkii 1976 ). Thus, there exists no Gestalt problem solving heuristic. Instead, the practice is to focus on the problem and the solution rather than on the process of coming up with a solution. Problems are solved by turning them over and over in the mind until an insight, a viable avenue of attack, presents itself. At the same time, however, there is a great reliance on prior knowledge and past experiences. The Gestalt method of problem solving, then, is at the same time very different and very similar to the process of design.

Gestalt psychology has not fared well during the evolution of cognitive psychology. Although it honours the work of the unconscious mind it does so at the expense of practicality. If learning is, indeed, entirely based on insight then there is little point in continuing to study learning. “When one begins by assuming that the most important cognitive phenomena are inaccessible, there really is not much left to talk about” (Schoenfeld 1985 , p. 273). However, of interest here is the Gestalt psychologists’ claim that focus on problem solving methods creates functional fixedness (Ashcraft 1989 ). Mason et al. ( 1982 ), as well as Perkins ( 2000 ) deal with this in their work on getting unstuck.

1.2.10 Final Comments

Mathematics has often been characterized as the most precise of all sciences. Lost in such a misconception is the fact that mathematics often has its roots in the fires of creativity, being born of the extra-logical processes of illumination and intuition. Problem solving heuristics that are based solely on the processes of logical and deductive reasoning distort the true nature of problem solving. Certainly, there are problems in which logical deductive reasoning is sufficient for finding a solution. But these are not true problems. True problems need the extra-logical processes of creativity, insight, and illumination, in order to produce solutions.

Fortunately, as elusive as such processes are, there does exist problem solving heuristics that incorporate them into their strategies. Heuristics such as those by Perkins ( 2000 ) and Mason et al. ( 1982 ) have found a way of combining the intentional and mechanical processes of problem solving by design with the extra-logical processes of creativity, illumination, and the AHA!. Furthermore, they have managed to do so without having to fully comprehend the inner workings of this mysterious process.

1.3 Digital Technologies and Mathematical Problem Solving—Luz Manuel Santos-Trigo

Mathematical problem solving is a field of research that focuses on analysing the extent to which problem solving activities play a crucial role in learners’ understanding and use of mathematical knowledge. Mathematical problems are central in mathematical practice to develop the discipline and to foster students learning (Pólya 1945 ; Halmos 1994 ). Mason and Johnston-Wilder ( 2006 ) pointed out that “The purpose of a task is to initiate mathematically fruitful activity that leads to a transformation in what learners are sensitized to notice and competent to carry out” (p. 25). Tasks are essential for learners to elicit their ideas and to engage them in mathematical thinking. In a problem solving approach, what matters is the learners’ goals and ways to interact with the tasks. That is, even routine tasks can be a departure point for learners to extend initial conditions and transform them into some challenging activities.

Thus, analysing and characterizing ways in which mathematical problems are formulated (Singer et al. 2015 ) and the process involved in pursuing and solving those problems generate important information to frame and structure learning environments to guide and foster learners’ construction of mathematical concepts and problem solving competences (Santos-Trigo 2014 ). Furthermore, mathematicians or discipline practitioners have often been interested in unveiling and sharing their own experience while developing the discipline. As a results, they have provided valuable information to characterize mathematical practices and their relations to what learning processes of the discipline entails. It is recognized that the work of Pólya ( 1945 ) offered not only bases to launch several research programs in problem solving (Schoenfeld 1992 ; Mason et al. 1982 ); but also it became an essential resource for teachers to orient and structure their mathematical lessons (Krulik and Reys 1980 ).

1.3.1 Research Agenda

A salient feature of a problem solving approach to learn mathematics is that teachers and students develop and apply an enquiry or inquisitive method to delve into mathematical concepts and tasks. How are mathematical problems or concepts formulated? What types of problems are important for teachers/learners to discuss and engage in mathematical reasoning? What mathematical processes and ways of reasoning are involved in understanding mathematical concepts and solving problems? What are the features that distinguish an instructional environment that fosters problem-solving activities? How can learners’ problem solving competencies be assessed? How can learners’ problem solving competencies be characterized and explained? How can learners use digital technologies to understand mathematics and to develop problem-solving competencies? What ways of reasoning do learners construct when they use digital technologies in problem solving approaches? These types of questions have been important in the problem solving research agenda and delving into them has led researchers to generate information and results to support and frame curriculum proposals and learning scenarios. The purpose of this section is to present and discuss important themes that emerged in problem solving approaches that rely on the systematic use of several digital technologies.

In the last 40 years, the accumulated knowledge in the problem solving field has shed lights on both a characterization of what mathematical thinking involves and how learners can construct a robust knowledge in problem solving environments (Schoenfeld 1992 ). In this process, the field has contributed to identify what types of transformations traditional learning scenarios might consider when teachers and students incorporate the use of digital technologies in mathematical classrooms. In this context, it is important to briefly review what main themes and developments the field has addressed and achieved during the last 40 years.

1.3.2 Problem Solving Developments

There are traces of mathematical problems and solutions throughout the history of civilization that explain the humankind interest for identifying and exploring mathematical relations (Kline 1972 ). Pólya ( 1945 ) reflects on his own practice as a mathematician to characterize the process of solving mathematical problems through four main phases: Understanding the problem, devising a plan, carrying out the plan, and looking back. Likewise, Pólya ( 1945 ) presents and discusses the role played by heuristic methods throughout all problem solving phases. Schoenfeld ( 1985 ) presents a problem solving research program based on Pólya’s ( 1945 ) ideas to investigate the extent to which problem solving heuristics help university students to solve mathematical problems and to develop a way of thinking that shows consistently features of mathematical practices. As a result, he explains the learners’ success or failure in problem solving activities can be characterized in terms their mathematical resources and ways to access them, cognitive and metacognitive strategies used to represent and explore mathematical tasks, and systems of beliefs about mathematics and solving problems. In addition, Schoenfeld ( 1992 ) documented that heuristics methods as illustrated in Pólya’s ( 1945 ) book are ample and general and do not include clear information and directions about how learners could assimilate, learn, and use them in their problem solving experiences. He suggested that students need to discuss what it means, for example, to think of and examining special cases (one important heuristic) in finding a closed formula for series or sequences, analysing relationships of roots of polynomials, or focusing on regular polygons or equilateral/right triangles to find general relations about these figures. That is, learners need to work on examples that lead them to recognize that the use of a particular heuristic often involves thinking of different type of cases depending on the domain or content involved. Lester and Kehle ( 2003 ) summarize themes and methodological shifts in problem solving research up to 1995. Themes include what makes a problem difficult for students and what it means to be successful problem solvers; studying and contrasting experts and novices’ problem solving approaches; learners’ metacognitive, beliefs systems and the influence of affective behaviours; and the role of context; and social interactions in problem solving environments. Research methods in problem solving studies have gone from emphasizing quantitative or statistical design to the use of cases studies and ethnographic methods (Krutestkii ( 1976 ). Teaching strategies also evolved from being centred on teachers to the active students’ engagement and collaboration approaches (NCTM 2000 ). Lesh and Zawojewski ( 2007 ) propose to extend problem solving approaches beyond class setting and they introduce the construct “model eliciting activities” to delve into the learners’ ideas and thinking as a way to engage them in the development of problem solving experiences. To this end, learners develop and constantly refine problem-solving competencies as a part of a learning community that promotes and values modelling construction activities. Recently, English and Gainsburg ( 2016 ) have discussed the importance of modeling eliciting activities to prepare and develop students’ problem solving experiences for 21st Century challenges and demands.

Törner et al. ( 2007 ) invited mathematics educators worldwide to elaborate on the influence and developments of problem solving in their countries. Their contributions show a close relationship between countries mathematical education traditions and ways to frame and implement problem solving approaches. In Chinese classrooms, for example, three instructional strategies are used to structure problem solving lessons: one problem multiple solutions , multiple problems one solution , and one problem multiple changes . In the Netherlands, the realistic mathematical approach permeates the students’ development of problem solving competencies; while in France, problem solving activities are structured in terms of two influential frameworks: The theory of didactical situations and anthropological theory of didactics.

In general, problem solving frameworks and instructional approaches came from analysing students’ problem solving experiences that involve or rely mainly on the use of paper and pencil work. Thus, there is a need to re-examined principles and frameworks to explain what learners develop in learning environments that incorporate systematically the coordinated use of digital technologies (Hoyles and Lagrange 2010 ). In this perspective, it becomes important to briefly describe and identify what both multiple purpose and ad hoc technologies can offer to the students in terms of extending learning environments and representing and exploring mathematical tasks. Specifically, a task is used to identify features of mathematical reasoning that emerge through the use digital technologies that include both mathematical action and multiple purpose types of technologies.

1.3.3 Background

Digital technologies are omnipresent and their use permeates and shapes several social and academic events. Mobile devices such as tablets or smart phones are transforming the way people communicate, interact and carry out daily activities. Churchill et al. ( 2016 ) pointed out that mobile technologies provide a set of tools and affordances to structure and support learning environments in which learners continuously interact to construct knowledge and solve problems. The tools include resources or online materials, efficient connectivity to collaborate and discuss problems, ways to represent, explore and store information, and analytical and administration tools to management learning activities. Schmidt and Cohen ( 2013 ) stated that nowadays it is difficult to imagine a life without mobile devices, and communication technologies are playing a crucial role in generating both cultural and technical breakthroughs. In education, the use of mobile artefacts and computers offers learners the possibility of continuing and extending peers and groups’ mathematical discussions beyond formal settings. In this process, learners can also consult online materials and interact with experts, peers or more experienced students while working on mathematical tasks. In addition, dynamic geometry systems (GeoGebra) provide learners a set of affordances to represent and explore dynamically mathematical problems. Leung and Bolite-Frant ( 2015 ) pointed out that tools help activate an interactive environment in which teachers and students’ mathematical experiences get enriched. Thus, the digital age brings new challenges to the mathematics education community related to the changes that technologies produce to curriculum, learning scenarios, and ways to represent, explore mathematical situations. In particular, it is important to characterize the type of reasoning that learners can develop as a result of using digital technologies in their process of learning concepts and solving mathematical problems.

1.3.4 A Focus on Mathematical Tasks

Mathematical tasks are essential elements for engaging learners in mathematical reasoning which involves representing objects, identifying and exploring their properties in order to detect invariants or relationships and ways to support them. Watson and Ohtani ( 2015 ) stated that task design involves discussions about mathematical content and students’ learning (cognitive perspective), about the students’ experiences to understand the nature of mathematical activities; and about the role that tasks played in teaching practices. In this context, tasks are the vehicle to present and discuss theoretical frameworks for supporting the use of digital technology, to analyse the importance of using digital technologies in extending learners’ mathematical discussions beyond formal settings, and to design ways to foster and assess the use of technologies in learners’ problem solving environments. In addition, it is important to discuss contents, concepts, representations and strategies involved in the process of using digital technologies in approaching the tasks. Similarly, it becomes essential to discuss what types of activities students will do to learn and solve the problems in an environment where the use of technologies fosters and values the participation and collaboration of all students. What digital technologies are important to incorporate in problem solving approaches? Dynamic Geometry Systems can be considered as a milestone in the development of digital technologies. Objects or mathematical situations can be represented dynamically through the use of a Dynamic Geometry System and learners or problem solvers can identify and examine mathematical relations that emerge from moving objects within the dynamic model (Moreno-Armella and Santos-Trigo 2016 ).

Leung and Bolite-Frant ( 2015 ) stated that “dynamic geometry software can be used in task design to cover a large epistemic spectrum from drawing precise robust geometrical figures to exploration of new geometric theorems and development of argumentation discourse” (p. 195). As a result, learners not only need to develop skills and strategies to construct dynamic configuration of problems; but also ways of relying on the tool’s affordances (quantifying parameters or objects attributes, generating loci, graphing objects behaviours, using sliders, or dragging particular elements within the configuration) in order to identify and support mathematical relations. What does it mean to represent and explore an object or mathematical situation dynamically?

A simple task that involves a rhombus and its inscribed circle is used to illustrate how a dynamic representation of these objects and embedded elements can lead learners to identify and examine mathematical properties of those objects in the construction of the configuration. To this end, learners are encouraged to pose and pursue questions to explain the behaviours of parameters or attributes of the family of objects that is generated as a result of moving a particular element within the configuration.

1.3.5 A Task: A Dynamic Rhombus

Figure 2 represents a rhombus APDB and its inscribed circle (O is intersection of diagonals AD and BP and the radius of the inscribed circle is the perpendicular segment from any side of the rhombus to point O), vertex P lies on a circle c centred at point A. Circle c is only a heuristic to generate a family of rhombuses. Thus, point P can be moved along circle c to generate a family of rhombuses. Indeed, based on the symmetry of the circle it is sufficient to move P on the semicircle B’CA to draw such a family of rhombuses.

A dynamic construction of a rhombus

1.3.6 Posing Questions

A goal in constructing a dynamic model or configuration of problems is always to identify and explore mathematical properties and relations that might result from moving objects within the model. How do the areas of both the rhombus and the inscribed circle behave when point P is moved along the arc B’CB? At what position of point P does the area of the rhombus or inscribed circle reach the maximum value? The coordinates of points S and Q (Fig. 3 ) are the x -value of point P and as y -value the corresponding area values of rhombus ABDP and the inscribed circle respectively. Figure 2 shows the loci of points S and Q when point P is moved along arc B’CB. Here, finding the locus via the use of GeoGebra is another heuristic to graph the area behaviour without making explicit the algebraic model of the area.

Graphic representation of the area variation of the family of rhombuses and inscribed circles generated when P is moved through arc B’CB

The area graphs provide information to visualize that in that family of generated rhombuses the maximum area value of the inscribed circle and rhombus is reached when the rhombus becomes a square (Fig. 4 ). That is, the controlled movement of particular objects is an important strategy to analyse the area variation of the family of rhombuses and their inscribed circles.

Visualizing the rhombus and the inscribed circle with maximum area

It is important to observe the identification of points P and Q in terms of the position of point P and the corresponding areas and the movement of point P was sufficient to generate both area loci. That is, the graph representation of the areas is achieved without having an explicit algebraic expression of the area variation. Clearly, the graphic representations provide information regarding the increasing or decreasing interval of both areas; it is also important to explore what properties both graphic representations hold. The goal is to argue that the area variation of the rhombus represents an ellipse and the area of the inscribed circle represents a parabola. An initial argument might involve selecting five points on each locus and using the tool to draw the corresponding conic section (Fig. 5 ). In this case, the tool affordances play an important role in generating the graphic representation of the areas’ behaviours and in identifying properties of those representations. In this context, the use of the tool can offer learners the opportunity to problematize (Santos-Trigo 2007 ) a simple mathematical object (rhombus) as a means to search for mathematical relations and ways to support them.

Drawing the conic section that passes through five points

1.3.7 Looking for Different Solutions Methods

Another line of exploration might involve asking for ways to construct a rhombus and its inscribed circle: Suppose that the side of the rhombus and the circle are given, how can you construct the rhombus that has that circle inscribed? Figure 6 shows the given data, segment A 1 B 1 and circle centred at O and radius OD. The initial goal is to draw the circle tangent to the given segment. To this end, segment AB is congruent to segment A 1 B 1 and on this segment a point P is chosen and a perpendicular to segment AB that passes through point P is drawn. Point C is on this perpendicular and the centre of a circle with radius OD and h is the perpendicular to line PC that passes through point C. Angle ACB changes when point P is moved along segment AB and point E and F are the intersection of line h and the circle with centre M the midpoint of AB and radius MA (Fig. 6 ).

Drawing segment AB tangent to the given circle

Figure 7 a shows the right triangle AFB as the base to construct the rhombus and the inscribed circle and Fig. 7 b shows the second solution based on triangle AEB.

a Drawing the rhombus and the inscribed circle. b Drawing the second solution

Another approach might involve drawing the given circle centred at the origin and the segment as EF with point E on the y-axis. Line OC is perpendicular to segment EF and the locus of point C when point E moves along the y-axis intersects the given circle (Fig. 8 a, b). Both figures show two solutions to draw the rhombus that circumscribe the given circle.

a and b Another solution that involves finding a locus of point C

In this example, the GeoGebra affordances not only are important to construct a dynamic model of the task; but also offer learners and opportunity to explore relations that emerge from moving objects within the model. As a result, learners can rely on different concepts and strategies to solve the tasks. The idea in presenting this rhombus task is to illustrate that the use of a Dynamic Geometry System provides affordances for learners to construct dynamic representation of mathematical objects or problems, to move elements within the representation to pose questions or conjectures to explain invariants or patterns among involved parameters; to search for arguments to support emerging conjectures, and to develop a proper language to communicate results.

1.3.8 Looking Back

Conceptual frameworks used to explain learners’ construction of mathematical knowledge need to capture or take into account the different ways of reasoning that students might develop as a result of using a set of tools during the learning experiences. Figure 9 show some digital technologies that learners can use for specific purpose at the different stages of problem solving activities.

The coordinated use of digital tools to engage learners in problem solving experiences

The use of a dynamic system (GeoGebra) provides a set of affordances for learners to conceptualize and represent mathematical objects and tasks dynamically. In this process, affordances such as moving objects orderly (dragging), finding loci of objects, quantifying objects attributes (lengths, areas, angles, etc.), using sliders to vary parameters, and examining family of objects became important to look for invariance or objects relationships. Likewise, analysing the parameters or objects behaviours within the configuration might lead learners to identify properties to support emerging mathematical relations. Thus, with the use of the tool, learners might conceptualize mathematical tasks as an opportunity for them to engage in mathematical activities that include constructing dynamic models of tasks, formulating conjectures, and always looking for different arguments to support them. Similarly, learners can use an online platform to share their ideas, problem solutions or questions in a digital wall and others students can also share ideas or solution methods and engaged in mathematical discussions that extend mathematical classroom activities.

1.4 Problem Posing: An Overview for Further Progress—Uldarico Malaspina Jurado

Problem posing and problem solving are two essential aspects of the mathematical activity; however, researchers in mathematics education have not emphasized their attention on problem posing as much as problem solving. In that sense, due to its importance in the development of mathematical thinking in students since the first grades, we agree with Ellerton’s statement ( 2013 ): “for too long, successful problem solving has been lauded as the goal; the time has come for problem posing to be given a prominent but natural place in mathematics curricula and classrooms” (pp. 100–101); and due to its importance in teacher training, with Abu-Elwan’s statement ( 1999 ):

While teacher educators generally recognize that prospective teachers require guidance in mastering the ability to confront and solve problems, what is often overlooked is the critical fact that, as teachers, they must be able to go beyond the role as problem solvers. That is, in order to promote a classroom situation where creative problem solving is the central focus, the practitioner must become skillful in discovering and correctly posing problems that need solutions. (p. 1)

Scientists like Einstein and Infeld ( 1938 ), recognized not only for their notable contributions in the fields they worked, but also for their reflections on the scientific activity, pointed out the importance of problem posing; thus it is worthwhile to highlight their statement once again:

The formulation of a problem is often more essential than its solution, which may be merely a matter of mathematical or experimental skills. To raise new questions, new possibilities, to regard old questions from a new angle, requires creative imagination and marks real advance in science. (p. 92)

Certainly, it is also relevant to remember mathematician Halmos’s statement ( 1980 ): “I do believe that problems are the heart of mathematics, and I hope that as teachers (…) we will train our students to be better problem posers and problem solvers than we are” (p. 524).

An important number of researchers in mathematics education has focused on the importance of problem posing, and we currently have numerous, very important publications that deal with different aspects of problem posing related to the mathematics education of students in all educational levels and to teacher training.

1.4.1 A Retrospective Look

Kilpatrick ( 1987 ) marked a historical milestone in research related to problem posing and points out that “problem formulating should be viewed not only as a goal of instruction but also as a means of instruction” (Kilpatrick 1987 , p. 123); and he also emphasizes that, as part of students’ education, all of them should be given opportunities to live the experience of discovering and posing their own problems. Drawing attention to the few systematic studies on problem posing performed until then, Kilpatrick contributes defining some aspects that required studying and investigating as steps prior to a theoretical building, though he warns, “attempts to teach problem-formulating skills, of course, need not await a theory” (p. 124).

Kilpatrick refers to the “Source of problems” and points out how virtually all problems students solve have been posed by another person; however, in real life “many problems, if not most, must be created or discovered by the solver, who gives the problem an initial formulation” (p. 124). He also points out that problems are reformulated as they are being solved, and he relates this to investigation, reminding us what Davis ( 1985 ) states that, “what typically happens in a prolonged investigation is that problem formulation and problem solution go hand in hand, each eliciting the other as the investigation progresses” (p. 23). He also relates it to the experiences of software designers, who formulate an appropriate sequence of sub-problems to solve a problem. He poses that a subject to be examined by teachers and researchers “is whether, by drawing students’ attention to the reformulating process and given them practice in it, we can improve their problem solving performance” (p. 130). He also points out that problems may be a mathematical formulation as a result of exploring a situation and, in that sense, “school exercises in constructing mathematical models of a situation presented by the teacher are intended to provide students with experiences in formulating problems.” (p. 131).

Another important section of Kilpatrick’s work ( 1987 ) is Processes of Problem Formulating , in which he considers association, analogy, generalization and contradiction. He believes the use of concept maps to represent concept organization, as cognitive scientists Novak and Gowin suggest, might help to comprehend such concepts, stimulate creative thinking about them, and complement the ideas Brown and Walter ( 1983 ) give for problem posing by association. Further, in the section “Understanding and developing problem formulating abilities”, he poses several questions, which have not been completely answered yet, like “Perhaps the central issue from the point of view of cognitive science is what happens when someone formulates the problem? (…) What is the relation between problem formulating, problem solving and structured knowledge base? How rich a knowledge base is needed for problem formulating? (…) How does experience in problem formulating add to knowledge base? (…) What metacognitive processes are needed for problem formulating?”

It is interesting to realize that some of these questions are among the unanswered questions proposed and analyzed by Cai et al. ( 2015 ) in Chap. 1 of the book Mathematical Problem Posing (Singer et al. 2015 ). It is worth stressing the emphasis on the need to know the cognitive processes in problem posing, an aspect that Kilpatrick had already posed in 1987, as we just saw.

1.4.2 Researches and Didactic Experiences

Currently, there are a great number of publications related to problem posing, many of which are research and didactic experiences that gather the questions posed by Kilpatrick, which we just commented. Others came up naturally as reflections raised in the framework of problem solving, facing the natural requirement of having appropriate problems to use results and suggestions of researches on problem solving, or as a response to a thoughtful attitude not to resign to solving and asking students to solve problems that are always created by others. Why not learn and teach mathematics posing one’s own problems?

1.4.3 New Directions of Research

Singer et al. ( 2013 ) provides a broad view about problem posing that links problem posing experiences to general mathematics education; to the development of abilities, attitudes and creativity; and also to its interrelation with problem solving, and studies on when and how problem-solving sessions should take place. Likewise, it provides information about research done regarding ways to pose new problems and about the need for teachers to develop abilities to handle complex situations in problem posing contexts.

Singer et al. ( 2013 ) identify new directions in problem posing research that go from problem-posing task design to the development of problem-posing frameworks to structure and guide teachers and students’ problem posing experiences. In a chapter of this book, Leikin refers to three different types of problem posing activities, associated with school mathematics research: (a) problem posing through proving; (b) problem posing for investigation; and (c) problem posing through investigation. This classification becomes evident in the problems posed in a course for prospective secondary school mathematics teachers by using a dynamic geometry environment. Prospective teachers posed over 25 new problems, several of which are discussed in the article. The author considers that, by developing this type of problem posing activities, prospective mathematics teachers may pose different problems related to a geometric object, prepare more interesting lessons for their students, and thus gradually develop their mathematical competence and their creativity.

1.4.4 Final Comments

This overview, though incomplete, allows us to see a part of what problem posing experiences involve and the importance of this area in students mathematical learning. An important task is to continue reflecting on the questions posed by Kilpatrick ( 1987 ), as well as on the ones that come up in the different researches aforementioned. To continue progressing in research on problem posing and contribute to a greater consolidation of this research line, it will be really important that all mathematics educators pay more attention to problem posing, seek to integrate approaches and results, and promote joint and interdisciplinary works. As Singer et al. ( 2013 ) say, going back to Kilpatrick’s proposal ( 1987 ),

Problem posing is an old issue. What is new is the awareness that problem posing needs to pervade the education systems around the world, both as a means of instruction (…) and as an object of instruction (…) with important targets in real-life situations. (p. 5)

Although it can be argued that there is a difference between creativity, discovery, and invention (see Liljedahl and Allan 2014 ) for the purposes of this book these will be assumed to be interchangeable.

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Uldarico Malaspina

Technical University Darmstadt, Darmstadt, Germany

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Liljedahl, P., Santos-Trigo, M., Malaspina, U., Bruder, R. (2016). Problem Solving in Mathematics Education. In: Problem Solving in Mathematics Education. ICME-13 Topical Surveys. Springer, Cham. https://doi.org/10.1007/978-3-319-40730-2_1

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35 problem-solving techniques and methods for solving complex problems

Design your next session with SessionLab

Join the 150,000+ facilitators  using SessionLab.

How do you identify problems?

How do you identify the right solution.

Tips for more effective problem-solving

Complete problem-solving methods

Problem-solving techniques to identify and analyze problems
Problem-solving techniques for developing solutions

Problem-solving warm-up activities

Closing activities for a problem-solving process.

Before you can move towards finding the right solution for a given problem, you first need to identify and define the problem you wish to solve.

Here, you want to clearly articulate what the problem is and allow your group to do the same. Remember that everyone in a group is likely to have differing perspectives and alignment is necessary in order to help the group move forward.

Identifying a problem accurately also requires that all members of a group are able to contribute their views in an open and safe manner. It can be scary for people to stand up and contribute, especially if the problems or challenges are emotive or personal in nature. Be sure to try and create a psychologically safe space for these kinds of discussions.

Remember that problem analysis and further discussion are also important. Not taking the time to fully analyze and discuss a challenge can result in the development of solutions that are not fit for purpose or do not address the underlying issue.

Successfully identifying and then analyzing a problem means facilitating a group through activities designed to help them clearly and honestly articulate their thoughts and produce usable insight.

With this data, you might then produce a problem statement that clearly describes the problem you wish to be addressed and also state the goal of any process you undertake to tackle this issue.

Finding solutions is the end goal of any process. Complex organizational challenges can only be solved with an appropriate solution but discovering them requires using the right problem-solving tool.

After you’ve explored a problem and discussed ideas, you need to help a team discuss and choose the right solution. Consensus tools and methods such as those below help a group explore possible solutions before then voting for the best. They’re a great way to tap into the collective intelligence of the group for great results!

Remember that the process is often iterative. Great problem solvers often roadtest a viable solution in a measured way to see what works too. While you might not get the right solution on your first try, the methods below help teams land on the most likely to succeed solution while also holding space for improvement.

Every effective problem solving process begins with an agenda . A well-structured workshop is one of the best methods for successfully guiding a group from exploring a problem to implementing a solution.

In SessionLab, it’s easy to go from an idea to a complete agenda . Start by dragging and dropping your core problem solving activities into place . Add timings, breaks and necessary materials before sharing your agenda with your colleagues.

The resulting agenda will be your guide to an effective and productive problem solving session that will also help you stay organized on the day!

Tips for more effective problem solving

Problem-solving activities are only one part of the puzzle. While a great method can help unlock your team’s ability to solve problems, without a thoughtful approach and strong facilitation the solutions may not be fit for purpose.

Let’s take a look at some problem-solving tips you can apply to any process to help it be a success!

Clearly define the problem

Jumping straight to solutions can be tempting, though without first clearly articulating a problem, the solution might not be the right one. Many of the problem-solving activities below include sections where the problem is explored and clearly defined before moving on.

This is a vital part of the problem-solving process and taking the time to fully define an issue can save time and effort later. A clear definition helps identify irrelevant information and it also ensures that your team sets off on the right track.

Don’t jump to conclusions

It’s easy for groups to exhibit cognitive bias or have preconceived ideas about both problems and potential solutions. Be sure to back up any problem statements or potential solutions with facts, research, and adequate forethought.

The best techniques ask participants to be methodical and challenge preconceived notions. Make sure you give the group enough time and space to collect relevant information and consider the problem in a new way. By approaching the process with a clear, rational mindset, you’ll often find that better solutions are more forthcoming.

Try different approaches

Problems come in all shapes and sizes and so too should the methods you use to solve them. If you find that one approach isn’t yielding results and your team isn’t finding different solutions, try mixing it up. You’ll be surprised at how using a new creative activity can unblock your team and generate great solutions.

Don’t take it personally

Depending on the nature of your team or organizational problems, it’s easy for conversations to get heated. While it’s good for participants to be engaged in the discussions, ensure that emotions don’t run too high and that blame isn’t thrown around while finding solutions.

You’re all in it together, and even if your team or area is seeing problems, that isn’t necessarily a disparagement of you personally. Using facilitation skills to manage group dynamics is one effective method of helping conversations be more constructive.

Get the right people in the room

Your problem-solving method is often only as effective as the group using it. Getting the right people on the job and managing the number of people present is important too!

If the group is too small, you may not get enough different perspectives to effectively solve a problem. If the group is too large, you can go round and round during the ideation stages.

Creating the right group makeup is also important in ensuring you have the necessary expertise and skillset to both identify and follow up on potential solutions. Carefully consider who to include at each stage to help ensure your problem-solving method is followed and positioned for success.

Document everything

The best solutions can take refinement, iteration, and reflection to come out. Get into a habit of documenting your process in order to keep all the learnings from the session and to allow ideas to mature and develop. Many of the methods below involve the creation of documents or shared resources. Be sure to keep and share these so everyone can benefit from the work done!

Bring a facilitator

Facilitation is all about making group processes easier. With a subject as potentially emotive and important as problem-solving, having an impartial third party in the form of a facilitator can make all the difference in finding great solutions and keeping the process moving. Consider bringing a facilitator to your problem-solving session to get better results and generate meaningful solutions!

Develop your problem-solving skills

It takes time and practice to be an effective problem solver. While some roles or participants might more naturally gravitate towards problem-solving, it can take development and planning to help everyone create better solutions.

You might develop a training program, run a problem-solving workshop or simply ask your team to practice using the techniques below. Check out our post on problem-solving skills to see how you and your group can develop the right mental process and be more resilient to issues too!

Design a great agenda

Workshops are a great format for solving problems. With the right approach, you can focus a group and help them find the solutions to their own problems. But designing a process can be time-consuming and finding the right activities can be difficult.

Check out our workshop planning guide to level-up your agenda design and start running more effective workshops. Need inspiration? Check out templates designed by expert facilitators to help you kickstart your process!

In this section, we’ll look at in-depth problem-solving methods that provide a complete end-to-end process for developing effective solutions. These will help guide your team from the discovery and definition of a problem through to delivering the right solution.

If you’re looking for an all-encompassing method or problem-solving model, these processes are a great place to start. They’ll ask your team to challenge preconceived ideas and adopt a mindset for solving problems more effectively.

Six Thinking Hats
Lightning Decision Jam
Problem Definition Process
Discovery & Action Dialogue

Design Sprint 2.0

Open Space Technology

1. Six Thinking Hats

Individual approaches to solving a problem can be very different based on what team or role an individual holds. It can be easy for existing biases or perspectives to find their way into the mix, or for internal politics to direct a conversation.

Six Thinking Hats is a classic method for identifying the problems that need to be solved and enables your team to consider them from different angles, whether that is by focusing on facts and data, creative solutions, or by considering why a particular solution might not work.

Like all problem-solving frameworks, Six Thinking Hats is effective at helping teams remove roadblocks from a conversation or discussion and come to terms with all the aspects necessary to solve complex problems.

2. Lightning Decision Jam

Featured courtesy of Jonathan Courtney of AJ&Smart Berlin, Lightning Decision Jam is one of those strategies that should be in every facilitation toolbox. Exploring problems and finding solutions is often creative in nature, though as with any creative process, there is the potential to lose focus and get lost.

Unstructured discussions might get you there in the end, but it’s much more effective to use a method that creates a clear process and team focus.

In Lightning Decision Jam, participants are invited to begin by writing challenges, concerns, or mistakes on post-its without discussing them before then being invited by the moderator to present them to the group.

From there, the team vote on which problems to solve and are guided through steps that will allow them to reframe those problems, create solutions and then decide what to execute on.

By deciding the problems that need to be solved as a team before moving on, this group process is great for ensuring the whole team is aligned and can take ownership over the next stages.

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

3. Problem Definition Process

While problems can be complex, the problem-solving methods you use to identify and solve those problems can often be simple in design.

By taking the time to truly identify and define a problem before asking the group to reframe the challenge as an opportunity, this method is a great way to enable change.

Begin by identifying a focus question and exploring the ways in which it manifests before splitting into five teams who will each consider the problem using a different method: escape, reversal, exaggeration, distortion or wishful. Teams develop a problem objective and create ideas in line with their method before then feeding them back to the group.

This method is great for enabling in-depth discussions while also creating space for finding creative solutions too!

Problem Definition #problem solving #idea generation #creativity #online #remote-friendly A problem solving technique to define a problem, challenge or opportunity and to generate ideas.

4. The 5 Whys

Sometimes, a group needs to go further with their strategies and analyze the root cause at the heart of organizational issues. An RCA or root cause analysis is the process of identifying what is at the heart of business problems or recurring challenges.

The 5 Whys is a simple and effective method of helping a group go find the root cause of any problem or challenge and conduct analysis that will deliver results.

By beginning with the creation of a problem statement and going through five stages to refine it, The 5 Whys provides everything you need to truly discover the cause of an issue.

The 5 Whys #hyperisland #innovation This simple and powerful method is useful for getting to the core of a problem or challenge. As the title suggests, the group defines a problems, then asks the question “why” five times, often using the resulting explanation as a starting point for creative problem solving.

5. World Cafe

World Cafe is a simple but powerful facilitation technique to help bigger groups to focus their energy and attention on solving complex problems.

World Cafe enables this approach by creating a relaxed atmosphere where participants are able to self-organize and explore topics relevant and important to them which are themed around a central problem-solving purpose. Create the right atmosphere by modeling your space after a cafe and after guiding the group through the method, let them take the lead!

Making problem-solving a part of your organization’s culture in the long term can be a difficult undertaking. More approachable formats like World Cafe can be especially effective in bringing people unfamiliar with workshops into the fold.

World Cafe #hyperisland #innovation #issue analysis World Café is a simple yet powerful method, originated by Juanita Brown, for enabling meaningful conversations driven completely by participants and the topics that are relevant and important to them. Facilitators create a cafe-style space and provide simple guidelines. Participants then self-organize and explore a set of relevant topics or questions for conversation.

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. It’s great at helping remove resistance to change and can help get buy-in at every level too!

This process of enabling frontline ownership is great in ensuring follow-through and is one of the methods you will want in your toolbox as a facilitator.

Discovery & Action Dialogue (DAD) #idea generation #liberating structures #action #issue analysis #remote-friendly DADs make it easy for a group or community to discover practices and behaviors that enable some individuals (without access to special resources and facing the same constraints) to find better solutions than their peers to common problems. These are called positive deviant (PD) behaviors and practices. DADs make it possible for people in the group, unit, or community to discover by themselves these PD practices. DADs also create favorable conditions for stimulating participants’ creativity in spaces where they can feel safe to invent new and more effective practices. Resistance to change evaporates as participants are unleashed to choose freely which practices they will adopt or try and which problems they will tackle. DADs make it possible to achieve frontline ownership of solutions.

7. Design Sprint 2.0

Want to see how a team can solve big problems and move forward with prototyping and testing solutions in a few days? The Design Sprint 2.0 template from Jake Knapp, author of Sprint, is a complete agenda for a with proven results.

Developing the right agenda can involve difficult but necessary planning. Ensuring all the correct steps are followed can also be stressful or time-consuming depending on your level of experience.

Use this complete 4-day workshop template if you are finding there is no obvious solution to your challenge and want to focus your team around a specific problem that might require a shortcut to launching a minimum viable product or waiting for the organization-wide implementation of a solution.

8. Open space technology

Open space technology- developed by Harrison Owen – creates a space where large groups are invited to take ownership of their problem solving and lead individual sessions. Open space technology is a great format when you have a great deal of expertise and insight in the room and want to allow for different takes and approaches on a particular theme or problem you need to be solved.

Start by bringing your participants together to align around a central theme and focus their efforts. Explain the ground rules to help guide the problem-solving process and then invite members to identify any issue connecting to the central theme that they are interested in and are prepared to take responsibility for.

Once participants have decided on their approach to the core theme, they write their issue on a piece of paper, announce it to the group, pick a session time and place, and post the paper on the wall. As the wall fills up with sessions, the group is then invited to join the sessions that interest them the most and which they can contribute to, then you’re ready to begin!

Everyone joins the problem-solving group they’ve signed up to, record the discussion and if appropriate, findings can then be shared with the rest of the group afterward.

Open Space Technology #action plan #idea generation #problem solving #issue analysis #large group #online #remote-friendly Open Space is a methodology for large groups to create their agenda discerning important topics for discussion, suitable for conferences, community gatherings and whole system facilitation

Techniques to identify and analyze problems

Using a problem-solving method to help a team identify and analyze a problem can be a quick and effective addition to any workshop or meeting.

While further actions are always necessary, you can generate momentum and alignment easily, and these activities are a great place to get started.

We’ve put together this list of techniques to help you and your team with problem identification, analysis, and discussion that sets the foundation for developing effective solutions.

Let’s take a look!

The Creativity Dice
Fishbone Analysis
Problem Tree
SWOT Analysis
Agreement-Certainty Matrix
The Journalistic Six
LEGO Challenge
What, So What, Now What?
Journalists

Individual and group perspectives are incredibly important, but what happens if people are set in their minds and need a change of perspective in order to approach a problem more effectively?

Flip It is a method we love because it is both simple to understand and run, and allows groups to understand how their perspectives and biases are formed.

Participants in Flip It are first invited to consider concerns, issues, or problems from a perspective of fear and write them on a flip chart. Then, the group is asked to consider those same issues from a perspective of hope and flip their understanding.

No problem and solution is free from existing bias and by changing perspectives with Flip It, you can then develop a problem solving model quickly and effectively.

Flip It! #gamestorming #problem solving #action Often, a change in a problem or situation comes simply from a change in our perspectives. Flip It! is a quick game designed to show players that perspectives are made, not born.

10. The Creativity Dice

One of the most useful problem solving skills you can teach your team is of approaching challenges with creativity, flexibility, and openness. Games like The Creativity Dice allow teams to overcome the potential hurdle of too much linear thinking and approach the process with a sense of fun and speed.

In The Creativity Dice, participants are organized around a topic and roll a dice to determine what they will work on for a period of 3 minutes at a time. They might roll a 3 and work on investigating factual information on the chosen topic. They might roll a 1 and work on identifying the specific goals, standards, or criteria for the session.

Encouraging rapid work and iteration while asking participants to be flexible are great skills to cultivate. Having a stage for idea incubation in this game is also important. Moments of pause can help ensure the ideas that are put forward are the most suitable.

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

11. Fishbone Analysis

Organizational or team challenges are rarely simple, and it’s important to remember that one problem can be an indication of something that goes deeper and may require further consideration to be solved.

Fishbone Analysis helps groups to dig deeper and understand the origins of a problem. It’s a great example of a root cause analysis method that is simple for everyone on a team to get their head around.

Participants in this activity are asked to annotate a diagram of a fish, first adding the problem or issue to be worked on at the head of a fish before then brainstorming the root causes of the problem and adding them as bones on the fish.

Using abstractions such as a diagram of a fish can really help a team break out of their regular thinking and develop a creative approach.

Fishbone Analysis #problem solving ##root cause analysis #decision making #online facilitation A process to help identify and understand the origins of problems, issues or observations.

12. Problem Tree

Encouraging visual thinking can be an essential part of many strategies. By simply reframing and clarifying problems, a group can move towards developing a problem solving model that works for them.

In Problem Tree, groups are asked to first brainstorm a list of problems – these can be design problems, team problems or larger business problems – and then organize them into a hierarchy. The hierarchy could be from most important to least important or abstract to practical, though the key thing with problem solving games that involve this aspect is that your group has some way of managing and sorting all the issues that are raised.

Once you have a list of problems that need to be solved and have organized them accordingly, you’re then well-positioned for the next problem solving steps.

Problem tree #define intentions #create #design #issue analysis A problem tree is a tool to clarify the hierarchy of problems addressed by the team within a design project; it represents high level problems or related sublevel problems.

13. SWOT Analysis

Chances are you’ve heard of the SWOT Analysis before. This problem-solving method focuses on identifying strengths, weaknesses, opportunities, and threats is a tried and tested method for both individuals and teams.

Start by creating a desired end state or outcome and bare this in mind – any process solving model is made more effective by knowing what you are moving towards. Create a quadrant made up of the four categories of a SWOT analysis and ask participants to generate ideas based on each of those quadrants.

Once you have those ideas assembled in their quadrants, cluster them together based on their affinity with other ideas. These clusters are then used to facilitate group conversations and move things forward.

SWOT analysis #gamestorming #problem solving #action #meeting facilitation The SWOT Analysis is a long-standing technique of looking at what we have, with respect to the desired end state, as well as what we could improve on. It gives us an opportunity to gauge approaching opportunities and dangers, and assess the seriousness of the conditions that affect our future. When we understand those conditions, we can influence what comes next.

14. Agreement-Certainty Matrix

Not every problem-solving approach is right for every challenge, and deciding on the right method for the challenge at hand is a key part of being an effective team.

The Agreement Certainty matrix helps teams align on the nature of the challenges facing them. By sorting problems from simple to chaotic, your team can understand what methods are suitable for each problem and what they can do to ensure effective results.

If you are already using Liberating Structures techniques as part of your problem-solving strategy, the Agreement-Certainty Matrix can be an invaluable addition to your process. We’ve found it particularly if you are having issues with recurring problems in your organization and want to go deeper in understanding the root cause.

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

Organizing and charting a team’s progress can be important in ensuring its success. SQUID (Sequential Question and Insight Diagram) is a great model that allows a team to effectively switch between giving questions and answers and develop the skills they need to stay on track throughout the process.

Begin with two different colored sticky notes – one for questions and one for answers – and with your central topic (the head of the squid) on the board. Ask the group to first come up with a series of questions connected to their best guess of how to approach the topic. Ask the group to come up with answers to those questions, fix them to the board and connect them with a line. After some discussion, go back to question mode by responding to the generated answers or other points on the board.

It’s rewarding to see a diagram grow throughout the exercise, and a completed SQUID can provide a visual resource for future effort and as an example for other teams.

SQUID #gamestorming #project planning #issue analysis #problem solving When exploring an information space, it’s important for a group to know where they are at any given time. By using SQUID, a group charts out the territory as they go and can navigate accordingly. SQUID stands for Sequential Question and Insight Diagram.

16. Speed Boat

To continue with our nautical theme, Speed Boat is a short and sweet activity that can help a team quickly identify what employees, clients or service users might have a problem with and analyze what might be standing in the way of achieving a solution.

Methods that allow for a group to make observations, have insights and obtain those eureka moments quickly are invaluable when trying to solve complex problems.

In Speed Boat, the approach is to first consider what anchors and challenges might be holding an organization (or boat) back. Bonus points if you are able to identify any sharks in the water and develop ideas that can also deal with competitors!

Speed Boat #gamestorming #problem solving #action Speedboat is a short and sweet way to identify what your employees or clients don’t like about your product/service or what’s standing in the way of a desired goal.

17. The Journalistic Six

Some of the most effective ways of solving problems is by encouraging teams to be more inclusive and diverse in their thinking.

Based on the six key questions journalism students are taught to answer in articles and news stories, The Journalistic Six helps create teams to see the whole picture. By using who, what, when, where, why, and how to facilitate the conversation and encourage creative thinking, your team can make sure that the problem identification and problem analysis stages of the are covered exhaustively and thoughtfully. Reporter’s notebook and dictaphone optional.

The Journalistic Six – Who What When Where Why How #idea generation #issue analysis #problem solving #online #creative thinking #remote-friendly A questioning method for generating, explaining, investigating ideas.

18. LEGO Challenge

Now for an activity that is a little out of the (toy) box. LEGO Serious Play is a facilitation methodology that can be used to improve creative thinking and problem-solving skills.

The LEGO Challenge includes giving each member of the team an assignment that is hidden from the rest of the group while they create a structure without speaking.

What the LEGO challenge brings to the table is a fun working example of working with stakeholders who might not be on the same page to solve problems. Also, it’s LEGO! Who doesn’t love LEGO!

LEGO Challenge #hyperisland #team A team-building activity in which groups must work together to build a structure out of LEGO, but each individual has a secret “assignment” which makes the collaborative process more challenging. It emphasizes group communication, leadership dynamics, conflict, cooperation, patience and problem solving strategy.

19. What, So What, Now What?

If not carefully managed, the problem identification and problem analysis stages of the problem-solving process can actually create more problems and misunderstandings.

The What, So What, Now What? problem-solving activity is designed to help collect insights and move forward while also eliminating the possibility of disagreement when it comes to identifying, clarifying, and analyzing organizational or work problems.

Facilitation is all about bringing groups together so that might work on a shared goal and the best problem-solving strategies ensure that teams are aligned in purpose, if not initially in opinion or insight.

Throughout the three steps of this game, you give everyone on a team to reflect on a problem by asking what happened, why it is important, and what actions should then be taken.

This can be a great activity for bringing our individual perceptions about a problem or challenge and contextualizing it in a larger group setting. This is one of the most important problem-solving skills you can bring to your organization.

W³ – What, So What, Now What? #issue analysis #innovation #liberating structures You can help groups reflect on a shared experience in a way that builds understanding and spurs coordinated action while avoiding unproductive conflict. It is possible for every voice to be heard while simultaneously sifting for insights and shaping new direction. Progressing in stages makes this practical—from collecting facts about What Happened to making sense of these facts with So What and finally to what actions logically follow with Now What . The shared progression eliminates most of the misunderstandings that otherwise fuel disagreements about what to do. Voila!

20. Journalists

Problem analysis can be one of the most important and decisive stages of all problem-solving tools. Sometimes, a team can become bogged down in the details and are unable to move forward.

Journalists is an activity that can avoid a group from getting stuck in the problem identification or problem analysis stages of the process.

In Journalists, the group is invited to draft the front page of a fictional newspaper and figure out what stories deserve to be on the cover and what headlines those stories will have. By reframing how your problems and challenges are approached, you can help a team move productively through the process and be better prepared for the steps to follow.

Journalists #vision #big picture #issue analysis #remote-friendly This is an exercise to use when the group gets stuck in details and struggles to see the big picture. Also good for defining a vision.

Problem-solving techniques for developing solutions

The success of any problem-solving process can be measured by the solutions it produces. After you’ve defined the issue, explored existing ideas, and ideated, it’s time to narrow down to the correct solution.

Use these problem-solving techniques when you want to help your team find consensus, compare possible solutions, and move towards taking action on a particular problem.

Improved Solutions
Four-Step Sketch
15% Solutions
How-Now-Wow matrix
Impact Effort Matrix

21. Mindspin

Brainstorming is part of the bread and butter of the problem-solving process and all problem-solving strategies benefit from getting ideas out and challenging a team to generate solutions quickly.

With Mindspin, participants are encouraged not only to generate ideas but to do so under time constraints and by slamming down cards and passing them on. By doing multiple rounds, your team can begin with a free generation of possible solutions before moving on to developing those solutions and encouraging further ideation.

This is one of our favorite problem-solving activities and can be great for keeping the energy up throughout the workshop. Remember the importance of helping people become engaged in the process – energizing problem-solving techniques like Mindspin can help ensure your team stays engaged and happy, even when the problems they’re coming together to solve are complex.

MindSpin #teampedia #idea generation #problem solving #action A fast and loud method to enhance brainstorming within a team. Since this activity has more than round ideas that are repetitive can be ruled out leaving more creative and innovative answers to the challenge.

22. Improved Solutions

After a team has successfully identified a problem and come up with a few solutions, it can be tempting to call the work of the problem-solving process complete. That said, the first solution is not necessarily the best, and by including a further review and reflection activity into your problem-solving model, you can ensure your group reaches the best possible result.

One of a number of problem-solving games from Thiagi Group, Improved Solutions helps you go the extra mile and develop suggested solutions with close consideration and peer review. By supporting the discussion of several problems at once and by shifting team roles throughout, this problem-solving technique is a dynamic way of finding the best solution.

Improved Solutions #creativity #thiagi #problem solving #action #team You can improve any solution by objectively reviewing its strengths and weaknesses and making suitable adjustments. In this creativity framegame, you improve the solutions to several problems. To maintain objective detachment, you deal with a different problem during each of six rounds and assume different roles (problem owner, consultant, basher, booster, enhancer, and evaluator) during each round. At the conclusion of the activity, each player ends up with two solutions to her problem.

23. Four Step Sketch

Creative thinking and visual ideation does not need to be confined to the opening stages of your problem-solving strategies. Exercises that include sketching and prototyping on paper can be effective at the solution finding and development stage of the process, and can be great for keeping a team engaged.

By going from simple notes to a crazy 8s round that involves rapidly sketching 8 variations on their ideas before then producing a final solution sketch, the group is able to iterate quickly and visually. Problem-solving techniques like Four-Step Sketch are great if you have a group of different thinkers and want to change things up from a more textual or discussion-based approach.

Four-Step Sketch #design sprint #innovation #idea generation #remote-friendly The four-step sketch is an exercise that helps people to create well-formed concepts through a structured process that includes: Review key information Start design work on paper, Consider multiple variations , Create a detailed solution . This exercise is preceded by a set of other activities allowing the group to clarify the challenge they want to solve. See how the Four Step Sketch exercise fits into a Design Sprint

24. 15% Solutions

Some problems are simpler than others and with the right problem-solving activities, you can empower people to take immediate actions that can help create organizational change.

Part of the liberating structures toolkit, 15% solutions is a problem-solving technique that focuses on finding and implementing solutions quickly. A process of iterating and making small changes quickly can help generate momentum and an appetite for solving complex problems.

Problem-solving strategies can live and die on whether people are onboard. Getting some quick wins is a great way of getting people behind the process.

It can be extremely empowering for a team to realize that problem-solving techniques can be deployed quickly and easily and delineate between things they can positively impact and those things they cannot change.

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

25. How-Now-Wow Matrix

The problem-solving process is often creative, as complex problems usually require a change of thinking and creative response in order to find the best solutions. While it’s common for the first stages to encourage creative thinking, groups can often gravitate to familiar solutions when it comes to the end of the process.

When selecting solutions, you don’t want to lose your creative energy! The How-Now-Wow Matrix from Gamestorming is a great problem-solving activity that enables a group to stay creative and think out of the box when it comes to selecting the right solution for a given problem.

Problem-solving techniques that encourage creative thinking and the ideation and selection of new solutions can be the most effective in organisational change. Give the How-Now-Wow Matrix a go, and not just for how pleasant it is to say out loud.

How-Now-Wow Matrix #gamestorming #idea generation #remote-friendly When people want to develop new ideas, they most often think out of the box in the brainstorming or divergent phase. However, when it comes to convergence, people often end up picking ideas that are most familiar to them. This is called a ‘creative paradox’ or a ‘creadox’. The How-Now-Wow matrix is an idea selection tool that breaks the creadox by forcing people to weigh each idea on 2 parameters.

26. Impact and Effort Matrix

All problem-solving techniques hope to not only find solutions to a given problem or challenge but to find the best solution. When it comes to finding a solution, groups are invited to put on their decision-making hats and really think about how a proposed idea would work in practice.

The Impact and Effort Matrix is one of the problem-solving techniques that fall into this camp, empowering participants to first generate ideas and then categorize them into a 2×2 matrix based on impact and effort.

Activities that invite critical thinking while remaining simple are invaluable. Use the Impact and Effort Matrix to move from ideation and towards evaluating potential solutions before then committing to them.

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

27. Dotmocracy

If you’ve followed each of the problem-solving steps with your group successfully, you should move towards the end of your process with heaps of possible solutions developed with a specific problem in mind. But how do you help a group go from ideation to putting a solution into action?

Dotmocracy – or Dot Voting -is a tried and tested method of helping a team in the problem-solving process make decisions and put actions in place with a degree of oversight and consensus.

One of the problem-solving techniques that should be in every facilitator’s toolbox, Dot Voting is fast and effective and can help identify the most popular and best solutions and help bring a group to a decision effectively.

Dotmocracy #action #decision making #group prioritization #hyperisland #remote-friendly Dotmocracy is a simple method for group prioritization or decision-making. It is not an activity on its own, but a method to use in processes where prioritization or decision-making is the aim. The method supports a group to quickly see which options are most popular or relevant. The options or ideas are written on post-its and stuck up on a wall for the whole group to see. Each person votes for the options they think are the strongest, and that information is used to inform a decision.

All facilitators know that warm-ups and icebreakers are useful for any workshop or group process. Problem-solving workshops are no different.

Use these problem-solving techniques to warm up a group and prepare them for the rest of the process. Activating your group by tapping into some of the top problem-solving skills can be one of the best ways to see great outcomes from your session.

Check-in/Check-out
Doodling Together
Show and Tell
Constellations
Draw a Tree

28. Check-in / Check-out

Solid processes are planned from beginning to end, and the best facilitators know that setting the tone and establishing a safe, open environment can be integral to a successful problem-solving process.

Check-in / Check-out is a great way to begin and/or bookend a problem-solving workshop. Checking in to a session emphasizes that everyone will be seen, heard, and expected to contribute.

If you are running a series of meetings, setting a consistent pattern of checking in and checking out can really help your team get into a groove. We recommend this opening-closing activity for small to medium-sized groups though it can work with large groups if they’re disciplined!

Check-in / Check-out #team #opening #closing #hyperisland #remote-friendly Either checking-in or checking-out is a simple way for a team to open or close a process, symbolically and in a collaborative way. Checking-in/out invites each member in a group to be present, seen and heard, and to express a reflection or a feeling. Checking-in emphasizes presence, focus and group commitment; checking-out emphasizes reflection and symbolic closure.

29. Doodling Together

Thinking creatively and not being afraid to make suggestions are important problem-solving skills for any group or team, and warming up by encouraging these behaviors is a great way to start.

Doodling Together is one of our favorite creative ice breaker games – it’s quick, effective, and fun and can make all following problem-solving steps easier by encouraging a group to collaborate visually. By passing cards and adding additional items as they go, the workshop group gets into a groove of co-creation and idea development that is crucial to finding solutions to problems.

Doodling Together #collaboration #creativity #teamwork #fun #team #visual methods #energiser #icebreaker #remote-friendly Create wild, weird and often funny postcards together & establish a group’s creative confidence.

30. Show and Tell

You might remember some version of Show and Tell from being a kid in school and it’s a great problem-solving activity to kick off a session.

Asking participants to prepare a little something before a workshop by bringing an object for show and tell can help them warm up before the session has even begun! Games that include a physical object can also help encourage early engagement before moving onto more big-picture thinking.

By asking your participants to tell stories about why they chose to bring a particular item to the group, you can help teams see things from new perspectives and see both differences and similarities in the way they approach a topic. Great groundwork for approaching a problem-solving process as a team!

Show and Tell #gamestorming #action #opening #meeting facilitation Show and Tell taps into the power of metaphors to reveal players’ underlying assumptions and associations around a topic The aim of the game is to get a deeper understanding of stakeholders’ perspectives on anything—a new project, an organizational restructuring, a shift in the company’s vision or team dynamic.

31. Constellations

Who doesn’t love stars? Constellations is a great warm-up activity for any workshop as it gets people up off their feet, energized, and ready to engage in new ways with established topics. It’s also great for showing existing beliefs, biases, and patterns that can come into play as part of your session.

Using warm-up games that help build trust and connection while also allowing for non-verbal responses can be great for easing people into the problem-solving process and encouraging engagement from everyone in the group. Constellations is great in large spaces that allow for movement and is definitely a practical exercise to allow the group to see patterns that are otherwise invisible.

Constellations #trust #connection #opening #coaching #patterns #system Individuals express their response to a statement or idea by standing closer or further from a central object. Used with teams to reveal system, hidden patterns, perspectives.

32. Draw a Tree

Problem-solving games that help raise group awareness through a central, unifying metaphor can be effective ways to warm-up a group in any problem-solving model.

Draw a Tree is a simple warm-up activity you can use in any group and which can provide a quick jolt of energy. Start by asking your participants to draw a tree in just 45 seconds – they can choose whether it will be abstract or realistic.

Once the timer is up, ask the group how many people included the roots of the tree and use this as a means to discuss how we can ignore important parts of any system simply because they are not visible.

All problem-solving strategies are made more effective by thinking of problems critically and by exposing things that may not normally come to light. Warm-up games like Draw a Tree are great in that they quickly demonstrate some key problem-solving skills in an accessible and effective way.

Draw a Tree #thiagi #opening #perspectives #remote-friendly With this game you can raise awarness about being more mindful, and aware of the environment we live in.

Each step of the problem-solving workshop benefits from an intelligent deployment of activities, games, and techniques. Bringing your session to an effective close helps ensure that solutions are followed through on and that you also celebrate what has been achieved.

Here are some problem-solving activities you can use to effectively close a workshop or meeting and ensure the great work you’ve done can continue afterward.

One Breath Feedback
Who What When Matrix
Response Cards

How do I conclude a problem-solving process?

All good things must come to an end. With the bulk of the work done, it can be tempting to conclude your workshop swiftly and without a moment to debrief and align. This can be problematic in that it doesn’t allow your team to fully process the results or reflect on the process.

At the end of an effective session, your team will have gone through a process that, while productive, can be exhausting. It’s important to give your group a moment to take a breath, ensure that they are clear on future actions, and provide short feedback before leaving the space.

The primary purpose of any problem-solving method is to generate solutions and then implement them. Be sure to take the opportunity to ensure everyone is aligned and ready to effectively implement the solutions you produced in the workshop.

Remember that every process can be improved and by giving a short moment to collect feedback in the session, you can further refine your problem-solving methods and see further success in the future too.

33. One Breath Feedback

Maintaining attention and focus during the closing stages of a problem-solving workshop can be tricky and so being concise when giving feedback can be important. It’s easy to incur “death by feedback” should some team members go on for too long sharing their perspectives in a quick feedback round.

One Breath Feedback is a great closing activity for workshops. You give everyone an opportunity to provide feedback on what they’ve done but only in the space of a single breath. This keeps feedback short and to the point and means that everyone is encouraged to provide the most important piece of feedback to them.

One breath feedback #closing #feedback #action This is a feedback round in just one breath that excels in maintaining attention: each participants is able to speak during just one breath … for most people that’s around 20 to 25 seconds … unless of course you’ve been a deep sea diver in which case you’ll be able to do it for longer.

34. Who What When Matrix

Matrices feature as part of many effective problem-solving strategies and with good reason. They are easily recognizable, simple to use, and generate results.

The Who What When Matrix is a great tool to use when closing your problem-solving session by attributing a who, what and when to the actions and solutions you have decided upon. The resulting matrix is a simple, easy-to-follow way of ensuring your team can move forward.

Great solutions can’t be enacted without action and ownership. Your problem-solving process should include a stage for allocating tasks to individuals or teams and creating a realistic timeframe for those solutions to be implemented or checked out. Use this method to keep the solution implementation process clear and simple for all involved.

Who/What/When Matrix #gamestorming #action #project planning With Who/What/When matrix, you can connect people with clear actions they have defined and have committed to.

35. Response cards

Group discussion can comprise the bulk of most problem-solving activities and by the end of the process, you might find that your team is talked out!

Providing a means for your team to give feedback with short written notes can ensure everyone is head and can contribute without the need to stand up and talk. Depending on the needs of the group, giving an alternative can help ensure everyone can contribute to your problem-solving model in the way that makes the most sense for them.

Response Cards is a great way to close a workshop if you are looking for a gentle warm-down and want to get some swift discussion around some of the feedback that is raised.

Response Cards #debriefing #closing #structured sharing #questions and answers #thiagi #action It can be hard to involve everyone during a closing of a session. Some might stay in the background or get unheard because of louder participants. However, with the use of Response Cards, everyone will be involved in providing feedback or clarify questions at the end of a session.

Save time and effort discovering the right solutions

A structured problem solving process is a surefire way of solving tough problems, discovering creative solutions and driving organizational change. But how can you design for successful outcomes?

With SessionLab, it’s easy to design engaging workshops that deliver results. Drag, drop and reorder blocks to build your agenda. When you make changes or update your agenda, your session timing adjusts automatically , saving you time on manual adjustments.

Collaborating with stakeholders or clients? Share your agenda with a single click and collaborate in real-time. No more sending documents back and forth over email.

Explore how to use SessionLab to design effective problem solving workshops or watch this five minute video to see the planner in action!

Over to you

The problem-solving process can often be as complicated and multifaceted as the problems they are set-up to solve. With the right problem-solving techniques and a mix of creative exercises designed to guide discussion and generate purposeful ideas, we hope we’ve given you the tools to find the best solutions as simply and easily as possible.

Is there a problem-solving technique that you are missing here? Do you have a favorite activity or method you use when facilitating? Let us know in the comments below, we’d love to hear from you!

thank you very much for these excellent techniques

Certainly wonderful article, very detailed. Shared!

Your list of techniques for problem solving can be helpfully extended by adding TRIZ to the list of techniques. TRIZ has 40 problem solving techniques derived from methods inventros and patent holders used to get new patents. About 10-12 are general approaches. many organization sponsor classes in TRIZ that are used to solve business problems or general organiztational problems. You can take a look at TRIZ and dwonload a free internet booklet to see if you feel it shound be included per your selection process.

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PEDAGOGY OF COMMERCE [Teaching of Commerce] - B.Ed Notes

Pedagogy of commerce | teaching of commerce | pedagogy of business studies and accounting | teaching of accounts and business studies | pedagogy of commerce notes for b.ed, what do you mean by pedagogy of commerce is it science or art or both discuss the nature and scope of pedagogy of commerce/ business studies.

Pedagogy of Commerce Short Notes
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Pedagogy of Commerce or Teaching of Commerce, Accounting, business studies subject B.Ed, b ed, bed, b-ed, 1st, 2nd,3rd, 4th, 5th, 6th, first, second, third, fourth, fifth, sixth semester year student teachers teaching notes FOR CLASS 11 AND 12, study material, pdf, ppt,book,exam texbook,ebook handmade last minute examination passing marks short and easy to understand notes in English Medium download free

MEANING OF COMMERCE

The meaning of the term ‘commerce’ represents that particular domain of knowledge which deals with the concepts, principles, theories, processes, and skill that could be applied in the proper conduct of all tasks and transactions related to commercial activities .

According to Dr. Evelyn Thomas, “commerce is a term that embraces all those functions involved in the making, buying, selling and transport of goods.”

All the activities which are undertaken in connection with the production of things and carrying them from the place of production to the place of consumption are called business. In short,

The activities connected with the production are called industry,
And the activities connected with the transport of goods produced from the factories to the consumers are called commerce.

Business Activities:

Commerce (Trade, Auxiliaries to Trade)

There is a long way which has to be covered in taking the finished products from the place of production to the place of consumption. It is a misconception that commerce is only the purchase and sale of goods. In commerce, all those activities are included whose objective is to remove the obstacles in the distribution of goods. This process begins in the following way.

Definitions of Commerce

Various scholars have defined the word in the following way.

In the words of Evelyn Thomas , “Commercial occupations deal with the buying and selling of goods, the exchange of commodities, and the distribution of the finished products.”

According to James Stephenson , “Commerce is the sum total of those processes which are engaged in the removal of the hindrances of persons (trade), place (transport and insurance), and time (warehousing) in the exchange (banking) of commodities.”

In the words of William R. Spreegal , “Commerce is mainly concerned with the transfer of goods. The activities related to classification, integration, storing, finance-management and transportation and insurance are to be performed in it.”

Characteristics / Features of Commerce

On the basis of the above definitions following are the characteristics of commerce:

Commerce is the distribution system of goods.
In commerce, all the activities related to the sale and purchase of goods and those which help in sales and purchase of goods are included.
Commerce creates utility.
It brings industry and consumers close to each other.
Commerce helps in providing the good at a relevant price, at a proper time at the proper place, in proper form to the proper person.
It is a study of human beings as producers and consumers.
It is a study of distribution and exchange.

Components of Commerce

On the basis of the definition of commerce we can divide it into two parts:

In general words, trade means buying and selling goods with the purpose of getting profit. In other words, trade means the exchange of goods and services for the purpose of material profit of both the buyer and the sellers. We can carry it on the smallest as well as largest scale. Trade can be classified into two categories:

Home trade : Home trade means buying and selling things within the national boundaries. It includes the following:

Local trade: It has limits to some villages, towns, or districts.
State trade: It is a trade among different districts of a state.
National trade: It is a trade among different states of the country.

Foreign trade : When trade crosses the national boundaries and reaches the foreign land then we call it foreign trade. It includes:

Import Trade : When a businessman of a country buys some goods from the businessman of another country.
Export trade: When the trader of our country sells something to the trader of another country.
Entrepot trade: When a country imports some goods from another country and then exports the same goods to some third country.

Auxiliaries to Trade

Commerce includes not only trade but also various other subsidiary activities which are undertaken to overcome the hurdle appearing in the way of trade. The following activities are included in it:

There are fewer producers and a large number of consumers. Hence, it is not possible for all the consumers to purchase goods directly from the producers. So, there is a need for middlemen to establish contacts between producers and consumers. They consist of wholesalers, retailers,s, etc.

Money is required from the moment when raw material is purchased till the sale of the finished product. Hurdles of finance in the form of cash, letter of credit, the need for foreign exchange, to expand the business arises and these can be eliminated with the help of banking. Other means than banks as finance companies, stock exchanges are also utilized in the present-day economy to raise funds.

Advertising

Nowadays a producer, to attract the customer towards his product, has to provide full knowledge of it to the customer. Advertising helps in this activity. It enhances the knowledge of the customers and eliminates the hindrance of information.

Whether goods are sold on a small scale or large scale, at the time of storage and transporting of these goods from one place to another, the risk is always there as damage of goods, theft of goods, the effect of natural calamities, etc. Insurance can solve this problem.

Warehousing

There remains a time gap between the manufacturing of any product and the consumption of that product. So we have to store finished products. hence we can solve this problem by warehousing.

Transportation

Today, the trade has been extended from the local level to the national as well as international level. Hence, there is a need for transportation. So we can eliminate the hindrance of a trade by transport i.e. by carrying commodities from the production place to place where there is need.

Communication

There is a large distance between the producers, middlemen, and consumers. The means of communication like telephone, telegram, letter, telex, etc. help in finalizing the business deals. Due to the availability of modern means of communication as a computer, internet, etc., we perform business on an international basis.

In the present-day economy, the importance of packaging is increasing day by day. It is needed to maintain the quality of the product and to send it safely. It also provides an attractive look to the product.

Stock and produce exchange

Stock and produce exchange can also be called a measure of the economic growth of a country. These are the important activities of the business. Stock exchange provides a platform for the sale and purchase of shares and debentures whereas produce exchange is an effective means of providing the goods. The stock exchange can have a good impact on disinvestment and providing the proper environment for trade and industries but it can destroy it also.

Thus it is clear that during the sale and purchase of goods there arise many hindrances and these hindrances can be eliminated with the help of commerce.

Nature of Commerce

In the nature of commerce, we have to indicate whether it is a science or an art, a positive science or normative science, whether it can pass moral judgments in business. In order to decide whether commerce is a science or an art firstly we have to understand the meaning of ‘science’ and ‘art’.

Commerce – A Science

Science is a systematized body of knowledge, which traces the relationship between cause and effect. Science is not a mere collection of facts, because a mere collection of facts can never constitute a science. To understand the meaning of science, we should know about the characteristics of science which are as follows:

The attitude of science is objective.
It has a fixed explanation.
It has the power to predict.
The important characteristic is that is systematic.

Thus we find that commerce is that branch of knowledge where the various facts relevant to it have been systematically collected, classified, and analyzed. Judged from this point of view, commerce is a full-fledged science.

Some writers seem to imagine that commerce cannot be given the dignified status of science because:

Lack of exactness in commerce.
Scientific laws are completely true and fixed, whereas in commerce there is no such type of laws because it is based on human efforts which change according to the different situations.
The data on which scientific laws based are correct and true but the data in commerce can’t be so exact and true.
There is the inability to predict the future course of events as accurately as the physical sciences can.

It is no doubt, true that commerce cannot predict the future course of events as accurately as natural sciences and it very often happens that business prophecies are falsified by subsequent events. But on the basis of this only we can’t deny the scientific nature of commerce studies. The only reason for this lack of predictability is that commerce deals with highly complex and various forces, some of which are not amenable to correct prediction. Commerce deals with men endowed with freedom of will and there are no guarantee that they will act in a preconceived manner.

Commerce – An Art

An ‘art’ like science is also a systematized body of knowledge. The object of art is the formulation of percepts immediately applicable to policy. The practical aspect of art distinguishes it from science, which may be merely theoretical. Commerce, in certain respects, is an art as well.

There are several branches of commerce, which offer us practical guidance in the development of trade, in managing an institution. It provides occupational training to the students like clerical, accountancy, etc. It develops human values into the students like patience, honesty, cooperation, brotherhood, etc.

These human values help the students in later life as a businessman. It teaches how to accumulate money and how to invest that.

Thus we can say that commerce is both a science and an art because as a science, it formulates certain rules, principles, and theories while as an art, it gives them a practical shape.

It is both an academic discipline and a vocational discipline. In the words of Proof. Cossa, “Science provides us theoretical knowledge whereas art provides training in practical practices.

Scope of Commerce

Commerce today covers a vast field and comprises many branches of scholarship in its fold. Like the bee it sucks honey from every flavor: The subject matter of commerce is very wide because it includes all the commercial activities performed by the man in the economy.

Commerce is a body of organized knowledge and thoughts about trade affairs. It is a science of arts, trade, and aids to trade; aids to trade are insurance, transport, communication, advertisements, etc.

The scope of commerce has to be broad enough to acquaint the pupils with a wide range of trade activities that are meaningful to them. We generally study the following facts to have a knowledge of the scope of commerce.

The subject matter of commerce

Commerce has its own subject matter. The Commerce area is both a knowledge subject and a skill subject. The objectives of the study of commerce are both preparatory to further studies in colleges and terminal to enter into the careers of middle-level lives of employment. Its subject matter is very vast.

The subject matter of commerce includes the study of general commerce, economics, geography, commercial laws, book-keeping, business management, accountancy, advertising and salesmanship, office practices, etc. Most of the subject matter serves to introduce the students to the activities of business enterprises.

According to University Education Commission 1948-49. Professional business education should include mathematics, statistics, theory of organization, business structure; finance, including management and budgeting of assets and of expenses; philosophy, history, and theory of law and organization of work including economic; process analysis and procedures, standardization of skills, cost analysis and the like.

Form of trade organization and types of trade

If we want to get organized knowledge about the problems of commerce, firstly we have to get knowledge about the nature and types of trade. When the trade is carried on within a country it is known as internal trade. The scope of this type of trade limits to the country and its problems can be solved by keeping the situations of the country in mind. But the problems related to international or foreign trade are different.

The problems related to it may be transportation, exchange control, imperial preference, tax, and toll, etc. thus internal and foreign trade is two different parts of commerce and there is a need to think about the problems related to it from different angles.

It is necessary to understand the organization of the trade to understand its problems. In an ancient period when the field of exchange was limited, the trade was in the hands of individual traders.

So in the present-day economy, there is planning to abolish the role of these middlemen, but their role is so specific and important that it is difficult to finish their complete role. The wholesaler buys the goods in large quantities from the producer or manufacturer or their authorized dealer and sells the same to the retailer in small quantities. Later on, the retailer sells these things to the customers as per their need.

Limitations of Commerce

While studying the scope of commerce, we come to know about the limitations as follows:

It studies only business activities.
Commerce is completely a positive science.
It includes the study of activities related to business.
It studies the activities which are undertaken to overcome the hurdle appearing in the way of trade.

Thus commerce is a vast discipline comprised of different branches of management. We can summarize it as:

Social intercourse: The dealing of one person or class in society with another, familiarity.
The exchange or buying and selling of commodities, especially the exchange of merchandise, on a large scale, between different places and communities, extended trade or traffic.
To carry on trade, to traffic.
It deals with goods and services.
Deals with the creation of form, place, and time utilities.
Deals with a profit motive.

COMMERCE EDUCATION

Commerce education is directly concerned with the day to day life of the students.

According to Herrick, “commerce education is that form of instruction that both directly and indirectly prepares the businessman for his calling”.

In Herrick’s view, commerce education is the preparation of a businessman. It includes all types of education which make one person become a great businessman.

Lyon stated commerce education as “Any education which a businessman has and which makes him a better businessman is for him a commerce, no matter whether it was obtained in the walls of a school or not”.

According to the Lyon curriculum for commerce, education is all the activities of businessmen and the ways by which he became a great businessman. To him that education is not restricted to the schools only since the student can even get his education outside the school i.e., from society.

Importance of Commerce Education

The main purpose of commerce education is to provide knowledge about commerce and to prepare the student for vocational competency
Commerce education is useful for the students to understand the various aspects of changing the ownership of goods
Commerce education is aimed at giving adequate knowledge about the wholesale trade, retail, export trade, import trade, and entire-port trade .
It provides some knowledge about the movement of goods etc., Transport, Communication Insurance, Ware-housing, Money, Banking & Finance, and Mercantile Agencies.

Importance of Accountancy Education

The main purpose of teaching Accountancy is to make the students understand the importance of bookkeeping .
This helps the students to know how to prepare and interprets simple financial statements and reports.
It aids the students to understand the posting of business transactions in the various forms of Accounts books such as journals, ledger, and other subsidiaries' books, etc.
The commerce education helps the students to draw conclusions about the financial position of the organizations.

CORRELATION OF COMMERCE AND ACCOUNTANCY WITH OTHER SUBJECTS

Correlation of Commerce and Accountancy with reference to Economics: These commercial activities complete the full cycle of economic activities. In short, we can say that Economics is the mother of commerce.

Correlation of Commerce and Accountancy with reference to Mathematics: An accountant applies the fundamental arithmetical processes in preparing the accounts. Further in the field of sale tax, income tax, etc. Knowledge of mathematics is essential.

Correlation of Commerce and Accountancy with reference to Geography: The raw materials required for any commercially significant commodity have to be collected from various places and made available in the Centre of production. The availability of such materials is always based upon geographical conditions.

Correlation of Commerce with Business Management:

All activities ensuring the free flow of goods from the producer to the consumer are considered as elements of commerce.
These include transportation, insurance, and advertising banning warehousing. Etc.
Each of these elements demands effective management.
The future of any business that involves the production and sale of goods and services depends on efficient management.
So, it may be said that there is an inseparable relation between commerce and management.

VALUES OF TEACHING COMMERCE AND ACCOUNTANCY

Practical or Utilitarian value
Social value
Cultural value
Moral value
Disciplinary value
Vocational value

RECENT DEVELOPMENTS IN COMMERCE

In Banking: Electronic Funds Transfer System (EFTS), Automated Teller Machines (ATMs), Debit Cards, Credit Cards, Core Banking, Tele Banking, and Internet Banking.
In Marketing: E-commerce
In Insurance Sector: Private Companies
In Communication: Fax, Internet, E-Mail, Extranet, Video Conferencing, and Teleconferencing
In Trade: Online Trading

Planning is the basic or primary function of management. In simple words, planning is deciding in advance what to do, when to do it, how to do it and who is to do it.

According to Y.Dror, ‘Planning is the process of preparing a set of decisions for action in future directed at achieving goals by optional means’.

NEED AND IMPORTANCE OF PLANNING

The plan determines what will be learned by the students.
Planning transforms the available curriculum into activities, assignments, and tasks for students.

The teachers are expected to plan a variety of planning activities. Such planning activities maybe some of the following.

Year (Annual) Plan
Unit (Resources) Plan
Lesson (Daily) Plan

INSTRUCTIONAL PLANNING

Instructional planning is a process of the teacher using appropriate curricula, instructional strategies, and resources during the planning process to address the diverse needs of students.

After getting the work allotted, the teacher’s first task is to plan for the year’s work. This plan is known as Year Plan.

The planning for a unit is known as the unit plan. If the teacher knows the subject matter of each and every unit very well, he can prepare the unit plan after preparing the year plan.

Some examples for units in Commerce and Accountancy are,

Commerce: Trade, Sole trader, partnership firms, Joint Stock Companies, Banking, Insurance
Accountancy: Introduction to accounting, Journal, Ledger, Subsidiary books, Trail Balance, Final account, Rectification of Errors.

LESSON PLAN

Stands (1949) define,” Lesson plan is actually a plan of action. It therefore, includes the working philosophy of the teacher, his/her knowledge of philosophy, his/her information about and understanding of his/her pupils, his/her comprehension of the objectives of education, his/her knowledge of the material to be taught, and his/her ability to utilize effective methods”.

The lesson plan is considered as the teachers’ mental and emotional visualization of the classroom experiences. Lesson plans are prepared based on the objectives.

In simple, it provides an opportunity to become an efficient and excellent teacher.

Steps in a Lesson Plan

Introduction
Presentation
Generalization
Application
Recapitulation
Blackboard Summary
Home Assignment

Basic Principles of a Good Lesson Plan

Clarity of Objectives
Knowledge of the Subject
General knowledge of all subjects
Knowledge of the principles and strategies of teaching
Knowledge of student’s Nature
Clarity about previous knowledge
Knowledge of the class level
Division of Units
Use of Material Aids
Flexibility
Time Conscious
Learn More About Lesson Planning

BLOOM’S TAXONOMY

Education is chiefly concerned with producing or modifying the patterns of behavior in human beings.

After considerable research and investigation, the eminent Educational Psychologist, Benjamin S. Bloom and his associates of USA have come to the conclusion that educational objectives describing student behavior can be classified into three broad categories or domains and these categories are in hierarchical order.

Psychomotor
Learn More about Bloom’s Taxonomy and Revised Boom Taxonomy

MICRO TEACHING

Micro-teaching was introduced in India in 1967. In India, the first book on Microteaching was written by N.L. Dosajh under the Caption ‘Modification of Teacher Behaviour through Micro Teaching (1977).

Micro teaching is a scaled-down sample of teaching in which a teacher teaches a small unit to a small group of 5 to 10 pupils for a small period of 5 to 10 minutes. Such a situation offers a helpful setting for a teacher to acquire new teaching skills and to refine old ones.

D.W. Allen (1966): "Micro Teaching is a scaled down teaching encounter in class size and time".

Learn More About Micro Teaching and All Microteaching Skills

IMPORTANT SKILLS OF MICRO-TEACHING

Skill of Introducing A Lesson
Skill of Reinforcement
Skill of Stimulus Variation
Skill of Explaining
Skill of Illustrating with Examples
Skill of Using Black Board
Skill of Probing Questions
Learn More about Teaching Skills

QUALITIES OF COMMERCE TEACHER

The following are the important qualities for successful commerce and accountancy teacher.

1. Scholarship: This means a sound knowledge of the subject matter.

2. Professional Training: The commerce teacher must have up-to-date knowledge and a thorough understanding of the present banking system, commerce, industry, etc. since he has to teach all these ever-changing aspects to the students.

3. Personality: His personality is the third essential requirement for the commerce teacher to be successful in his profession.

Personality Aspects includes:

Personal appearance
Recognition of the amenities of life
Good language
Sympathy and Understanding
Self-Control
Adaptability and Resourcefulness
Organizing Ability
Directive Ability

Other Qualities:

Good Character
Aptitude in the teaching profession
Use a variety of effective teaching-learning procedures.
Trained by using various techniques
Able to develop and use instructional materials
Organize subject matter for instructional purposes
Appreciate the value of learning
Use appropriate equipment and machines
Conducting carrier guidance programs
Organize and supervise the co-curricular activities

RESPONSIBILITIES OF COMMERCE TEACHER

The following are the responsibilities of the commerce teacher:

Character development
Effective teaching and learning
Adjusting individual difference
Classroom management
Evaluation of pupil performance
Curriculum development and implementation
Developing good family and community relationships
Total school effectiveness
Professional growth and ethics

PROBLEMS FACED BY THE COMMERCE TEACHER

High student low teacher ratio.
Lack of proper infrastructure
Inadequate teaching aids
Untrained and ill-equipped teachers.
It is more content-oriented rather than skill and practice-oriented.
Lack of practical exposure both to the teacher and teaching methods
The content (syllabus) is not up-to-date with the latest scenario
Commerce teacher is a jack of all trades : perhaps he is the only person who is expected to teach all the subjects. Like commerce, banking, entrepreneurship, business management, or sometimes economics as a compulsory subject even if he or she may be interested in accountancy.

CURRICULUM OF COMMERCE

The term curriculum is derived from the Latin word "currere" which means path. In this sense curriculum is the path through which the student has to go forward in order to reach the goal envisaged by education.

The curriculum should be considered as a broad-based term encompassing every aspect concerning the study of the course. It is now considered on the totality of experiences to which a pupil is exposed within the boundaries of the school and outside.

Principles of Curriculum Construction

The Principle of Child-Centeredness
The Principle of Community-Centeredness
The Principle of Activity-Centeredness
The Principle of Integration
Forward-looking Principle
Conservative Principle
Renewal Principle
Creative Principle
Motivation Principle
Maturity Principle
The Principle of Preparation for life
The Principle of Elasticity and Flexibility
The Principle of Comprehensiveness

OBJECTIVES OF TEACHING COMMERCE AT THE CLASS 11th AND 12th

To develop in the students an interest in the theory and practice in business, trade, and industry
To acquaint students with the theoretical foundations and practices of organizing, managing, and handling routine operations of a business firm .
To inculcate attitudes and values leading to the integration of business with the social system with a positive approach
To enable the students to apply the principles and functions of management to specific aspects of the business.
To equip the students with essential fundamental knowledge for setting-up, organizing, and handling routine operations of a small-scale factory.
To equip the students with basic information on modern methods of office operations for effectively carrying out paperwork in a business office.
To generate and promote awareness of students in modern techniques of maintaining accounting records with the help of computers.
To enable the students to analyze financial statements and interpret the result for decision making.
To acquaint the students with practice and procedure of determination of cost from the point of its elements.
To create an awareness of the necessity of auditing the detection/rectification of errors/frauds in the process of accounting.

METHODS OF TEACHING COMMERCE

Lecture Method
Demonstration Method
Team Teaching Method
Problem Solving Method
Inductive and Deductive Method
Project Method
Discussion Method

Panel Discussion

Brain storming, heuristic method.

Surveys and Market Studies

LECTURE METHOD

In the field of any theory subject, it has great significance.
Nowadays in Colleges and higher education institutions most of the teachers are using the lecture methods.
A competent teacher can make the lecture meaningful and interesting by posing problematic situations and by using interesting and illustrative mediators.

DEMONSTRATION METHOD

The demonstration is a useful instructional method which is employed in teaching Commerce .
Demonstration means showing how something is to be done or not be done.
Through demonstration, a teacher presents a skill before the students.
The student’s role is that of the observer and recorder of information and skills.
In a higher secondary class, the commerce teacher can adopt this method related to the development of skill is being taught.

TEAM TEACHING METHOD

Team teaching is one of the most interesting and significant recent development in education.
It is an innovation in a school organization in which two or more teachers teach a group of students.
The group is benefited from the expertise of different teachers.

David Warwick, “A team teaching is a form of organization, in which individual teachers decide to pool resources, interest and expertise, in order to devise and implement scheme of work suitable to the needs of their pupils and the facilities of their school".

PROBLEM-SOLVING METHOD

Problem-solving is an instructional method or technique whereby the teacher and pupils attempt in a conscious, planned, and purposeful effort to arrive at some explanation or solution to some educationally significant difficulty.
It is a planned attack upon a difficulty or perplexity for the purpose of finding a solution.

According to Gates, "a problem exists for an individual when he has a definite goal, he cannot reach by the behavior pattern which he already has available."

Problem-solving is not merely a method of teaching. It is more a method of organization of subject matter in such a way that it can be dealt with through the study of problems.

INDUCTIVE AND DEDUCTIVE METHOD

INDUCTIVE METHOD:

The inductive method makes the students arrive at general conclusions or establish laws through observation of particular and concrete them.
Rules discovered are more likely to be grasped well than rules explained.
Therefore, the inductive method is more effective in learning.
This approach is mainly developmental.
It is easy to understand bookkeeping principles because the doubts about how and why of the formula are clarified in the very beginning.
It gives an opportunity for active participation for the students in the discovery of a formula.
This reduces the dependence on memorization.
It is the best method to introduce the new rule.
For example , the commerce teacher can teach the way of preparing a trial balance under this method. Instead of explaining the rules for trial balance the teacher can ask the students to prepare a ledger and find out the balances.

DEDUCTIVE METHOD:

The deductive method is the opposite of the inductive approach .
In this method - the learner proceeds from general to particular, abstract to the concrete, and formula to examples .
The pre-constructed formula or definition is told to the students and they are asked to solve or face the new situation with the help of that formula.
For example, the teacher can also teach the trail balance by way of this deductive method; instead of asking the students to prepare trial balance by way of an inductive method that is the first ledger then finding the nature of balance and the trail balance.
The teacher can first explain the rule for preparing a trial balance.
That is all the assets, expenditure and losses come under the debit side of the trial balances.
All the liabilities, profits, and receipts come under the credit side of the trial balance.
Then he can give a problem and ask them to prepare a trial balance.
Here the learner proceeds from the general rule to solve a particular problem.

PROJECT METHOD

This method is the direct outcome of John Dewey’s pragmatic philosophy . It is based on the idea that true knowledge is acquired not merely by reading books nor by attending lectures but by purposive planning and doing by the learners themselves for the purpose of handling problematic life situations.

‘Learning by doing’, ‘Learning by living’, ’Problem orientation’, and ‘working in natural settings’ are the four cardinal principles of this method.

Steps in the Project Method

Providing a situation
Choosing and purposing
Executing the project
Evaluating the project

DISCUSSION METHOD

A group discussion means an exchange of ideas accompanied by active learning, with all the members of the group participating in it. It is a free discussion regarding a topic by a group.

Mc Bumey and Hance have defined group discussion as, "the co-operative deliberation of problems by persons thinking and conversing together in face to face co-acting in group under the direction of the leader."

The seminar technique is usually practicable in higher education programs.
In this technique, a person presents a readymade paper or lecture on a specific subject before a group.
Nowadays audio-visual aids are also used while presenting the matter.
The paper presenter can either be an expert or one of the members of the group.
Sometimes, the copies of the paper being presented are distributed to the audience in advance.
After the presentation, there is a general discussion in which all participants can participate.
Here, the participants get an opportunity to clear their doubts.

Dressel defines the term seminar as, "the structured group discussion that may proceed or follow a formal lecture, often in the form of an essay or a paper presentation".

A symposium is a discussion by different speakers on the same topic emphasizing different aspects .
Selected speakers present prepared speeches.
Generally, the chairman and the speakers discuss the various aspects of a theme in advance and allot to each one a particular aspect so that each speaker limits his presentation to that aspect.
The term workshop has been borrowed from 'engineering'.
In a workshop, a person has to engage in some productive task to produce something tangible.
In an educational workshop also something tangible has to be produced by the participants.
The product maybe some equipment, instructional material, an action plan. etc.

According to R.A. Sharma. "Workshop is an assembled group of ten to twenty-five persons who share a common interest or problem. They meet together to improve their individual proficiency to solve a problem or to externalize knowledge and skill of a subject through intensive practical work and discussions."

Objectives of the workshop

To develop the psychomotor skill of the learner.
To make the subject matter interesting to the student.
To motivate the students for a particular topic.
To give training to teachers in specific areas.

The panel discussion is one of the socialized procedures. This is a procedure in which a small group of persons or pupils discuss the assigned problem creatively among themselves in front of an audience.

Brainstorming is basically an activity designed to promote creativity.
It is a form of discussion which enables the group to do collective creative thinking.
Brainstorming in the class situation invariably leads to the generation of new ideas and approaches to the study of the topics.
This technique is very useful for enhancing the contribution and involvement of students in the teaching-learning processes.
Under this method, pupils are led to discover the facts for themselves with the help of experiments, apparatus, or books.
The method emphasizes the process of the growth of mind by one’s own effort rather than pouring cooked material into empty vessels.
Simulation is the presenting of a problem or an event presented in artificially created situations similar to the real one.
The presentation is made as near as possible to the real situation or event.
A mini working model of an airplane being used in training pilots to learn and practice the working of an aircraft is an example of simulation
Role-playing is a teaching technique in which students assume an identity other than their own and play the role of others with whom the new identity has been assumed.
The role played may be that of a teacher, a parent, a salesman, a manager, a banker, and even inanimate things familiar in the course of interaction with the society.

SURVEYS AND MARKET STUDIES

In this method, information is obtained by asking questions to the selected respondents.
A commerce teacher can use the market survey as a method of teaching a complex concept or a process involving a variety of ideas.

EDUCATIONAL TECHNOLOGY IN LEARNING COMMERCE AND ACCOUNTANCY

Educational Technology is concerned with the systematic application of science and technology in the field of education.

Some of the educational technologies that are used in the learning of commerce are:

PROGRAMMED INSTRUCTION

The learning performed or instruction provided by a teaching machine or programmed textbook is referred to as programmed learning or instruction.

PERSONALISED SYSTEM OF INSTRUCTION (PSI)

The personalized system of instruction as the name suggests stands for a system of instruction totally personalized or individualized. Here the person or individual who receives instruction is a key figure. He dominates the entire scene of the teaching-learning process

COMPUTER ASSISTED INSTRUCTION (CAI)

An instructional technique based on the two-way interaction of a learner and computer with the objective of human learning and retention is known as Computer Assisted Instruction.
Here the computer actually assists the student in the learning process with the help of stored instructional programs designed to serve a variety of purposes such as informing, guiding, and testing the student until a prescribed level of proficiency is reached.

VIDEO CONFERENCING

In video conferencing, the resource persons at the teaching end may use mainly television cameras to show demonstrations, activities, discussions, etc. (television-based video conferencing) or may transmit the visuals generated through the computer

INTERACTIVE WHITEBOARD

An interactive whiteboard is an instructional tool that allows computer images to be displayed onto a board using a digital projector.

SMART CLASSROOM

Smart classrooms are the classrooms enhanced with technological equipment for the purpose of better learning and teaching .

WEB RESOURCES

Web-based education has become a cheaper and superior printed book of the modern era.

SOCIAL MEDIA

Social networking has become one of the most important communication tools among people nowadays.
The most famous in the world of social networks are Facebook (Facebook.com) and Twitter (Twitter.com) and others.
On the whole, one of the biggest assets of each social media tool lies in bringing together students of all ages to help them with all types of assignments, starting with the homework and finishing with different researches.

COMMERCE CLUB

Commerce club or association has started the leadership of the commerce teacher.
The commerce teacher should take all possible steps to run the club effectively by gaining adequate support from the administration students and the community.

COMMUNITY RESOURCES

The main aim of using the community resources is to give equal opportunity to all the students to take part in such activities and to enrich their interest and understanding of the contributions made by other streams to the teaching of commerce.

TYPES OF COMMUNITY RESOURCES:

Commerce Association or Forum
Exhibitions
Debates and competitions
Commerce Magazine
Social service
Vacation work

FIELD TRIPS

Educational visits to banks, insurance offices, factories, business houses, stock exchange markets, supermarkets, production centers, and exhibitions help students to explore their environment.
It helps the teacher to teach lessons with suitable practical examples.
Experiences gained by these visits are not easily forgotten.
Since it is a practical experience it provides an opportunity to acquire knowledge and understand the subject.
It links not only the classroom subject but also provides general education.
It provides useful contacts with the real world.

Excursions to industrial Centers:

It is very difficult to explain in details about the actual working of various industries in India.
Whatever explanations are given by the teachers are theoretical in nature.
Students may be taken to the place, where raw materials are kept.
They may be shown the various processes through which the raw material passes.
Ultimately, they should be shown the finished products.
In these processes’ students will actually observe the working of each section.
The working of each section should be explained by the person in charge of the section.

The important reason why students should be taken to the factories and other industrial concerns is that they may be able to see the factories and also see how goods are produced from raw materials.

Thus, excursions to industrial centers will benefit students and enrich their experiences about the working of the industries in India.

Excursion to Places of Geographical importance:

Excursions to places of Geographical importance should be arranged to explain to students the need and importance of locality and regions of the country.
Only by visiting the places of geographical importance, students can have permanent contrived real experiences.

VISIT TO INDUSTRIES

Visits to important industries and big business houses may be arranged at regular intervals.

The students of commerce are able to get real knowledge as to know how the business work, they may be lead to the different sections of the business and should actually watch how the papers of business transactions are actually prepared.

For example,

They may see how debit and credit notes are prepared,
How the invoice prepared,
How the bill of exchange is prepared and
How the accounting books maintained in that firm.

Author Remarks:

PEDAGOGY OF COMMERCE Is A Subject Taught In B.Ed And In Some Other Teaching Courses Also. On This Page, You Will Find Teaching of Commerce Short Examination Notes And Downloadable Free PDF Book In English Medium For B.Ed First Year And Second Year and Semester 1, 2, 3, 4, 5, and 6. Here We Have Covered Some of The Main Topics and Important MCQ Questions of Pedagogy of Business Studies and Accounting Which Will Really Help in Your Exam Preparation and Also You Can Make Your Assignment Report and File for BEd Very Easily with The Help of These Notes. These Notes and Free PDF Book on Teaching of Accounts and Business Studies Subject Will Be Helpful for All the Students and Teachers of Any College or University. We Have Also Suggested Some of the Best Reference Books and Study Material PDF for PEDAGOGY OF COMMERCE [Teaching of Commerce] That you can Also Go Through. Students and Teachers Preparing for All The Teaching Exams Like CTET, TET, UPTET, HTET Can Also Learn With The Notes Provided Above.

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All Subject B.Ed Books and Notes
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Lesson Plans for Teachers

The Problem –solving Method in Education

Dr. V.K.Maheshwari, M.A (Socio, Phil) B.Sc. M. Ed, Ph.D.

Former Principal, K.L.D.A.V. (P.G) College, Roorkee, India

Science subject is one of the important subjects in school education. However, really the traditional teaching methods are challenged for their inability to foster critical thinking, holistic learning environment among children. The science subject must develop science process skills where children, observe, measure, classify, process information, interpret think on solving problems, analyze, synthesize, formulate conclusions, etc. but, it should be kept in mind that, creativity in an essential element of science.

Problem-solving is, and should be, a very real part of the curriculum. It presupposes that students can take on some of the responsibility for their own learning and can take personal action to solve problems, resolve conflicts, discuss alternatives, and focus on thinking as a vital element of the curriculum. It provides students with opportunities to use their newly acquired knowledge in meaningful, real-life activities and assists them in working at higher levels of thinking

Meaning and Definition of Problem solving method

In a problem solving method, children learn by working on problems. This enables the students to learn new knowledge by facing the problems to be solved. The students are expected to observe, understand, analyze, interpret find solutions, and perform applications that lead to a holistic understanding of the concept. This method develops scientific process skills. This method helps in developing brainstorming approach to learning concepts.

The students thinking on problem and their understanding of the science behind it is based on common sense. It does not start from textual knowledge. Rather it proceeds from experiencing to gradually forming concepts through books at later stage. It is a process from practice to theory not vice versa. Knowledge here is not a goal but a natural out came of working on tasks. Students live in the real world and like to deal with concrete things where they can touch, feel manipulate things then the method is useful in igniting the process of science learning

A problem is a task for which Problem–solving may be a purely mental difficulty or it may be physical and involve manipulation of data. , the person confronting it wants or needs to find a solution because the person has no readily available procedure for finding the solution. The person must make an attempt to find a solution. Problem solving is the act of defining a problem; determining the cause of the problem; identifying, prioritizing and selecting alternatives for a solution; and implementing a solution.

Problem-solving method aims at presenting the knowledge to be learnt in the form of a problem. It begins with a problematic situation and consists of continuous, meaningful, well-integrated activity. The problems are test to the students in a natural way and it is ensured that the students are genuinely interested to solve them.

Problem–solving may be a purely mental difficulty or it may be physical and involve manipulation of data. Problem-solving is the ability to identify and solve problems by applying appropriate skills systematically.

Problem-solving is a process—an ongoing activity in which we take what we know to discover what we don’t know. It involves overcoming obstacles by generating hypo-theses, testing those predictions, and arriving at satisfactory solutions.

Objectives of Problem-Solving: The specific objectives of problem solving in science are :

Willingness to try problems and improve their perseverance when solving problems.
Improve pupils’ self-concepts with respect to the abilities to solve problems.
Make pupils aware of the problem-solving strategies.
Make pupils aware of the value of approaching problems in a systematic manner.
Make pupils aware that many problems can be solved in more than one way.
Improve pupils’ abilities to select appropriate solution strategies.
Improve pupils’ abilities to implement solution strategies accurately.
Improve pupils’ abilities to get more correct answers to problems
The appreciation of the existence of a problems and a desire to solve it
The accumulation of the facts and data which are pertinent to the problem.
Logical interpretation of the data supported by adequate valid experience.

Tips for effective use of Problem solving method

Ask questions and make suggestions . Ask students to predict “what would happen if …” or explain why something happened. This will help them to develop analytical and deductive thinking skills. . Do this by providing positive reinforcement to let students know when they have mastered a new concept or skill.
Don’t fear group work. Students can frequently help each other, and talking about a problem helps them think more critically about the steps needed to solve the problem.
Help students understand the problem . In order to solve problems, students need to define the end goal. If you succeed at helping students answer the questions “what?” and “why?”, finding the answer to “how?” will be easier. Have students identify specific problems, difficulties, or confusions. Don’t waste time working through problems that students already understand?
If students are unable to articulate their concerns, determine where they are having trouble. Identify the specific concepts or principles associated with the problem. Make students articulate their problem solving process. In a one-on-one tutoring session, ask the student to work his/her problem out loud. This slows down the thinking process, making it more accurate and allowing you to access understanding.
Link errors to misconceptions . Use errors as evidence of misconceptions, not carelessness or random guessing. Make an effort to isolate the misconception and correct it, then teach students to do this by themselves. We can all learn from mistakes. Try to communicate that the process is more important than the answer so that the student learns that it is OK to not have an instant solution.
Model the problem solving process rather than just giving students the answer . As you work through the problem, consider how a novice might struggle with the concepts and make your thinking clear .Provide only minimal assistance and only when needed to overcome obstacles.
Take enough time . Budget enough time for: understanding the problem and defining the goal, both individually and as a class; dealing with questions from you and your students; making, finding, and fixing mistakes; and solving entire problems in a single session.
Teach within a specific context . Teach problem-solving skills in the context in which they will be used .Use real-life problems in explanations, examples, and exams. Do not teach problem solving as an independent, abstract skill.
Work as a facilitator. Teacher must keep in mind that if in a child-directed learning not teacher-directed. He must be alert and active to arouse interest among students. Must provide democratic atmosphere. Teacher must provide situation for all students to come formed and contribute towards the success of the activity.

Procedural steps of Problem solving method

Problem-based learning is a method of educating adult learners that combines theoretical knowledge with practical activities. The process engages participants in considering complex and challenging issues and encourages them towards finding an appropriate solution. The expectation is that participants will have the motivation to learn because the problem scenarios are based on real-life situations found in the workplace. The expectation is that participants will have the motivation to learn because the problem scenarios are based on real-life situations.

The procedural steps can be devided in two phases

a- Pre-active /Planning phase

b- Active / Execution phase

c- Post-active/Evaluation phase

The Pre-active / planning phase includes :

Defining the problem:

The system . Have students identify the system under study by interpreting the information provided in the problem statement. Drawing a diagram is a great way to do this.
Known(s) and concepts . List what is known about the problem, and identify the knowledge needed to understand (and eventually) solve it.
Unknown(s) . identifying the unknown(s) becomes simpler. One unknown is generally the answer to the problem, but there may be other unknowns. Be sure that students understand what they are expected to find.
Units and symbols . Select, interpret, and use units and symbols. Emphasize the use of units whenever applicable. Develop a habit of using appropriate units and symbols yourself at all times.
Constraints . Teach students to look for the words only, must, neglect, or assume to help identify the constraints.

Active / Execution phase

The following are the general procedural steps in Execution phase:

Selection of the Problem and Presentation of the Problem : A number of problems are confronted by the students in the class or outside. They are made to select a problem as per their capacity and interest.

Each student is made to feel responsible for presenting the problem in front of the teacher and class as per his insight. The students are free to give their suggestions on the problem. Diagnose the situation so that your focus is on the problem .

Generation of alternative solutions: All the facts related to problem are collected either by a students or group. As a number of facts will be collected, it will help the students to keep the most pertinent facts and discard rest.

Generate alternative solutions: Postpone the selection of one solution until several alternatives have been proposed. Having a standard with which to compare the characteristics of the final solution is not the same as defining the desired outcome. Considering multiple alternatives can significantly enhance the value of final solution. Many alternative solutions should be generated before evaluating any of them. A common mistake in problem solving is that alternatives are evaluated as they are proposed, so the first desired solution is chosen, even if it’s not the best fit.

Evaluate and select an alternative: Skilled problem solvers use a series of considerations when selecting the best alternative. They consider the extent to which:

A particular alternative will solve the problem without causing other unanticipated problems.
All the individuals involved will accept the alternative.
Implementation of the alternative is likely.
The alternative fits within the organizational constraints.

Implement and follow up on the solution: This is most important phase as a proper outline at this stage will lead to purposeful activity. The teacher will guide students to draw exact plan and follow it properly so that the solution to problem is reached. It is more or less like planning stage, where in a clear indication of outline leads to better result. Feedback channels must be built into the implementation of the solution, to produce continuous monitoring and testing of actual events against expectations. Problem solving, and the techniques used to derive elucidation, can only be effective in an organization if the solution remains in place and is updated to respond to future changes.

Post-active/Evaluation phase .

The following steps comes under this phase;

Reaching the Inferences and conclusions : The tentative solutions which are offered by students are properly noted down. A good number of arrangements, discussion, brainstorming results in reaching a satisfactory conclusion. The teacher has to be very careful at this stage as, if may lead to wrong conclusions. The discussions must be healthy and conducive atmosphere must be provided in the classroom for it.

The students review the entire process and find out each and every stage where in they have made any mistakes. Self-criticism and Self-realization will give training of self confidence. The teacher must see that objective have been achieved.

Writing the Report:

A complete report should be written by the students. This will include, how they planned, what discussions were held, how duties were assigned, how satisfactory conclusion was reached etc. It’s vitally important that students have multiple opportunities to assess their own problem-solving skills and the solutions they generate from using those skills. Frequently, students are overly dependent upon teachers to evaluate their performance in the classroom. The process of self-assessment is not easy, however. It involves risk-taking, self-assurance, and a certain level of independence. But it can be effectively promoted by asking students questions such as “How do you feel about your progress so far?” “Are you satisfied with the results you obtained?” and “Why do you believe this is an appropriate response to the problem?

Limitations of the problem-solving method.

Difficult to teach all topics of curriculum- Difficult to organise e- contents of syllabus according to this method. All topics and areas cannot be covered by this method. There is a lack of suitable books and references for the students. This is not suitable for all level students. Method does not suit students of lower classes.Mental activity dominates this method. Hence there is neglect of physical and practical experiences.

Can encourage dirty competition- Most people working in a group unconsciously perceive the situation as competitive. This generates behaviour which is destructive and drains the creative energy of the group. The natural reaction is to regain self-esteem, often by trying to sabotage the ideas of those who disagreed with us. Instead of looking for ways to improve on their ideas we choose to destroy them. These types of behaviour create an atmosphere which is incompatible with effective problem solving.

Possible lack of effective direction- Sometimes there is no effective teacher to give direction to the discussion, with the result that it wanders aimlessly. There is short of talented teachers to practice this method. There is always a doubt of drawing wrong conclusions.

Time and resource constraints- Problem solving is a relatively slow process. It is not economical from time and money point of view. Time consuming method.

Merits of Problem solving method

Knowledge Retention- Problem-based learning is practical and it requires participants to use their reasoning and problem-solving skills to resolve the scenarios they are presented with. As a result, the learning process is more effective because participants are not trying to memorize large volumes of information .

Develops Competencies- This method follows the principle of learning by doing. Problem-based learning is a collaborative method that fosters teamwork, diversity and mutual respect, which are invaluable competencies in the workplace. Participants also develop their abilities to think strategically.

Context Specific - In schools the problem-based learning may be limited in its effectiveness because it is highly context specific. During the learning process, participants are given a specific problem that is based on a foreseeable work scenario. They learn to use old facts in new references.

Method is scientific in nature - Develops good study habits and reasoning power. Helps to improve and apply knowledge and experiences. Stimulates thinking of the child. Develops desirable study habits in the students.

Develops qualities of initiative and self-dependence in the students- Students learn virtues such as patience, cooperation, and self-confidence. Learning becomes more interesting and purposeful. Develops qualities of initiative and self-dependence in the students, as they have to face similar problematic situations in real life too. Shared responsibility makes individuals more willing to take risks. The discussion of different points of view also helps the group to be more realistic in assessing the risks associated with particular courses of action.

Reduced bias- The shared responsibility of a group in arriving at decisions can. encourage individuals to explore seemingly unrealistic ideas and to challenge accepted ways of doing things. Individual biases and prejudices can be challenged by the ,group, forcing the individual to recognise them. Group pressure can also encourage individuals to accept that change is needed.

Better solutions- Groups of individuals can bring a broad range of ideas, knowledge and skills to bear on a problem. This creates a stimulating interaction of diverse ideas which results in a wider range and better quality of solutions. . They become capable to generalize. Students learn to find solution to their problem. When people who are affected by a problem or who will be involved in implementation are involved in finding a solution, they will know how and why that particular solution was chosen. Also, people with knowledge relevant to the problem can communicate that knowledge directly if they participate in solving the problem.

“There are many problems throughout the world, some that are very simplistic while others are very complicated with many details. In order to be an effective problem solver, a person has to have the ability to use prior problem solving skills on problems in the existing future”

(Ormrod, 2008).

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Problem Solving Method of Teaching

Problem solving method.

Life is full of problems and man is able to solve them successfully only if he has proper experience and the art of overcoming difficulties in life. This habit may be acquired from early stages of education. As the child grows up he learns new methods of tackling problems. This habit of making efforts and solving independently the various problems prove useful in learning the various facts related with different curricular areas and also helps the child in solving actual life problems at the later stage.

If we try to solve a problem with logic, then surely we reach some goal and solve the problem. Solving the obstacles of the problem in a logical way and achieving a goal comes under the problem solving method .

Definitions of Problem Solving Method

(i) According to Skinner , “Problem solving is the process of overcoming difficulties that hinder the achievement of a goal."

(ii) According to John Dewey , "Problem solving is woven into the fabric of logical thinking. The problem determines the goal and the goal controls the thinking process."

(iii) According to Risk , “Problem solving is a planned act with the aim of finding a satisfactory solution to a difficulty or complexity. It does not involve mere gathering of facts or irrational acceptance of the views of a scholar, but it is thoughtful thinking.” process."

It is clear from the above definitions that when a person deviates from the objectives or goals based on knowledge facts, then a state of tension arises in him and this tension is reduced only when its end comes in the form of a solution to that problem. In Levin's definition, the term place of life has been used. Levin's place of life refers to the environment around a person. When any difficulty arises in this area, then the problem arises in front of the person and the difficulties of the problem motivate him to solve the problem. Reaching this solution state while trying is called problem solving.

Criteria for problem selection

The problem should be intellectually challenging to children..
The problem should not be entirely unfamiliar to the learners it should be related to their previous experience.
The problem should be related to a basic human activity.
The problem should have practical relevance.
The problem should have the potential to create interest among in the specific problem in particular and problem solving in general.

The Process Adopted

In problem solving method a systematic and orderly process is adopted for carrying out the teaching learning process. The process begins with the felt difficulty or problem. The student is then made to think out all the possible situations of the confronted problem on the basis of what does he know. Inability of finding out the solution with the help of his previous knowledge and experiences makes him to engage in serious exploration with the help of self-study, mutual discussion and independent practical work.

He tries to test one by one the possible alternatives and solutions of his problem and then by his continuous efforts get success in finding out the best of this solution may be further verified on the basis of its applicability and reliability in the solution of similar problems in other identical situations.

Steps in Problem Solving

The problem solving method has the following steps-

1. Worry:- The first step of problem solving method is worry. In this stage, a situation is presented to the students in such a way that they feel difficulty and worried about it and they also realize that they will not be able to solve this difficulty through any predetermined method. In such a situation they will try to solve this problem or situation in a difficult way. Will be compelled to think rationally.

2. Definition:- In this second stage of problem solving method, the difficulty related to the problem is defined and it is clearly explained. There are also small problems associated with each problem. These problems are also explained in detail to the students and then the method to solve them is also prescribed. This ends the second step of the problem solving method.

3. Solution Efforts:- The third stage of problem solving is the step of the efforts made to solve the problem. In this, the facts related to the problem are studied, experimented and discussed. An attempt is made to solve the problem by categorizing and analyzing them. Pre-determined principles are also re-examined. During this, various types of tools and instruments etc. have to be resorted to. If the size of the problem is very large, then an attempt is made to solve the problem by dividing it into smaller parts.

4. Conjecture or Hypothesis:- In the third stage, the facts related to the solution of the problem are collected, they are analyzed in this step. In this activity all the students of the class give their support. A hypothesis is formulated about the problem solving and most of the questions that this hypothesis has been put together corroborate it. He is given the final approval and it is understood that only through this it is possible to solve the problem. This is called hypothesis. After this an attempt is made to solve the problem through this hypothesis.

5. Evaluation:- In this last step of problem solving method, its veracity is tested again by reusing the hypothesis created. To do this, the hypothesis is correlated with what has been learned and its veracity is judged and tested on the basis of prior experiences. After this comes the decision position and the problem is solved..

Keep in mind that out of these five steps, the first four steps are of the induction method and the fifth and last step is of the subtraction method. These five verses are completely intertwined and related to each other. They cannot be separated from each other.

It serves as a preparation for adult life.
It develops the power of critical thinking.
It makes pupil active recipient of knowledge.
It develops values of tolerance and open mindedness.
It helps for the easy assimilation of knowledge.
It helps to establish harmonious relations between teacher and pupils.
This method will become monotonous if used to frequently.
The problem solving method can easily lead to the selection of trivial and untimely topics.
This is appropriate for developing cognitive competencies, but not for bringing about affective changes.

Teacher's Place in Problem Solving Method

Due to this method being student-centred, more emphasis is placed on the individual work of the students. For this reason, it is often a misconception that teachers have no special role in this method. But this thinking is absolutely meaningless and not correct. In fact, the teacher is that important link in the teaching process, in the absence of which it is not possible to complete the teaching process. In this method also the teacher has an important place. It is the teacher who effectively presents the problems to the students and creates such situations in which the student is motivated and compelled to solve the problem. The teacher also has to take care at every step that the interest of the students remains in it. The students also need the teacher's guidance while collecting the material related to the problem.

In the absence of teacher's guidance, students collect unsuitable material which is not helpful in solving the problem in any way. It is also the responsibility of the teacher to save the students from jumping to conclusions too soon on the basis of guesses. The teacher has to completely observe that the students are working in the right direction and if their direction is wrong then the teacher has to guide the students. In short, step by step teacher's guidance is very essential for the students. Therefore, it can be said that to think that there is no importance and role of teacher in problem solving method is a misconception.

Small Group Instruction Method of Teaching
Flipped Classroom Teaching Method
Programmed Instruction Method of Teaching