problem solving method in teaching and learning

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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. 

Center for Teaching

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) is a teaching method in which complex real-world problems are used as the vehicle to promote student learning of concepts and principles as opposed to direct presentation of facts and concepts. In addition to course content, PBL can promote the development of critical thinking skills, problem-solving abilities, and communication skills. It can also provide opportunities for working in groups, finding and evaluating research materials, and life-long learning (Duch et al, 2001).

PBL can be incorporated into any learning situation. In the strictest definition of PBL, the approach is used over the entire semester as the primary method of teaching. However, broader definitions and uses range from including PBL in lab and design classes, to using it simply to start a single discussion. PBL can also be used to create assessment items. The main thread connecting these various uses is the real-world problem.

Any subject area can be adapted to PBL with a little creativity. While the core problems will vary among disciplines, there are some characteristics of good PBL problems that transcend fields (Duch, Groh, and Allen, 2001):

The problem must motivate students to seek out a deeper understanding of concepts.
The problem should require students to make reasoned decisions and to defend them.
The problem should incorporate the content objectives in such a way as to connect it to previous courses/knowledge.
If used for a group project, the problem needs a level of complexity to ensure that the students must work together to solve it.
If used for a multistage project, the initial steps of the problem should be open-ended and engaging to draw students into the problem.

The problems can come from a variety of sources: newspapers, magazines, journals, books, textbooks, and television/ movies. Some are in such form that they can be used with little editing; however, others need to be rewritten to be of use. The following guidelines from The Power of Problem-Based Learning (Duch et al, 2001) are written for creating PBL problems for a class centered around the method; however, the general ideas can be applied in simpler uses of PBL:

Choose a central idea, concept, or principle that is always taught in a given course, and then think of a typical end-of-chapter problem, assignment, or homework that is usually assigned to students to help them learn that concept. List the learning objectives that students should meet when they work through the problem.
Think of a real-world context for the concept under consideration. Develop a storytelling aspect to an end-of-chapter problem, or research an actual case that can be adapted, adding some motivation for students to solve the problem. More complex problems will challenge students to go beyond simple plug-and-chug to solve it. Look at magazines, newspapers, and articles for ideas on the story line. Some PBL practitioners talk to professionals in the field, searching for ideas of realistic applications of the concept being taught.
What will the first page (or stage) look like? What open-ended questions can be asked? What learning issues will be identified?
How will the problem be structured?
How long will the problem be? How many class periods will it take to complete?
Will students be given information in subsequent pages (or stages) as they work through the problem?
What resources will the students need?
What end product will the students produce at the completion of the problem?
Write a teacher's guide detailing the instructional plans on using the problem in the course. If the course is a medium- to large-size class, a combination of mini-lectures, whole-class discussions, and small group work with regular reporting may be necessary. The teacher's guide can indicate plans or options for cycling through the pages of the problem interspersing the various modes of learning.
The final step is to identify key resources for students. Students need to learn to identify and utilize learning resources on their own, but it can be helpful if the instructor indicates a few good sources to get them started. Many students will want to limit their research to the Internet, so it will be important to guide them toward the library as well.

The method for distributing a PBL problem falls under three closely related teaching techniques: case studies, role-plays, and simulations. Case studies are presented to students in written form. Role-plays have students improvise scenes based on character descriptions given. Today, simulations often involve computer-based programs. Regardless of which technique is used, the heart of the method remains the same: the real-world problem.

Where can I learn more?

PBL through the Institute for Transforming Undergraduate Education at the University of Delaware
Duch, B. J., Groh, S. E, & Allen, D. E. (Eds.). (2001). The power of problem-based learning . Sterling, VA: Stylus.
Grasha, A. F. (1996). Teaching with style: A practical guide to enhancing learning by understanding teaching and learning styles. Pittsburgh: Alliance Publishers.

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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

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Evan Glazer (University of Georgia)

Editor’s Note: Dr. Glazer chose to use the term Problem-based Instruction and Inquiry, but my reading and other references to this chapter also use the term Problem-based Learning. The reader can assume the terms are equivalent.

Description

Problem-based inquiry is an effort to challenge students to address real-world problems and resolve realistic dilemmas.

Such problems create opportunities for meaningful activities that engage students in problem solving and higher-ordered thinking in authentic settings. Many textbooks attempt to promote these skills through contrived settings without relevance to students’ lives or interests. A notorious algebra problem concerns the time at which two railway trains will pass each other:

Two trains leave different stations headed toward each other. Station A is 500 miles west of Station B. Train A leaves station A at 12:00 pm traveling toward Station B at a rate of 60 miles per hour. Train B leaves Station B at 2:30 pm for Station A at a rate of 45 miles per hour. At what time will the trains meet?

Reading this question, one might respond, “Who cares?”, or, “Why do we need to know this?” Such questions have created substantial anxiety among students and have, perhaps, even been the cause of nightmares. Critics would argue that classic “story problems” leave a lasting impression of meaningless efforts to confuse and torment students, as if they have come from hell’s library. Problem-based inquiry, on the other hand, intends to engage students in relevant, realistic problems.

Several changes would need to be made in the above problem to promote problem-based inquiry. It would first have to be acknowledged that the trains are not, in fact, traveling at constant rates when they are in motion; negotiating curves or changing tracks at high speeds can result in accidents.

Further, all of the information about the problem cannot be presented to the learner at the outset; that is, some ambiguity must exist in the context so that students have an opportunity to engage in a problem-solving activity. In addition, the situation should involve a meaningful scenario. Suppose that a person intends to catch a connecting train at the second station and requires a time-efficient itinerary? What if we are not given data about the trains, but instead, the outcome of a particular event, such as an accident?

Why should we use problem-based inquiry to help students learn?

The American educational system has been criticized for having an underachieving curriculum that leads students to memorize and regurgitate facts that do not apply to their lives (Martin, 1987; Paul, 1993). Many claim that the traditional classroom environment, with its orderly conduct and didactic teaching methods in which the teacher dispenses information, has greatly inhibited students’ opportunities to think critically (Dossey et al., 1988; Goodlad, 1984; Wood, 1987). Problem-based inquiry is an attempt to overcome these obstacles and confront the concerns presented by the National Assessment of Educational Progress:

If an unfriendly foreign power had attempted to impose on America the mediocre educational performance that exists today, we might well have viewed it as an act of war. We have, in effect, been committing an act of unthinking, unilateral educational disarmament. (A Nation at Risk, 1983)

Problem-based inquiry emphasizes learning as a process that involves problem solving and critical thinking in situated contexts. It provides opportunities to address broader learning goals that focus on preparing students for active and responsible citizenship. Students gain experience in tackling realistic problems, and emphasis is placed on using communication, cooperation, and resources to formulate ideas and develop reasoning skills.

What is a framework for a problem-based inquiry?

Situated cognition, constructivism, social learning, and communities of practice are assumed theories of learning and cognition in problem-based inquiry environments. These theories have common themes about the context and the process of learning and are often associated.

Characteristics

Some common characteristics in problem-based learning models:

Activity is grounded in a general question about a problem that has multiple possible answers and methods for addressing the question. Each problem has a general question that guides the overall task followed by ill-structured problems or questions that are generated throughout the problem-solving process. That is, to address the larger question, students must derive and investigate smaller problems or questions that relate to the findings and implications of the broader goal. The problems or questions thus created are most likely new to the students and lack known definitive methods or answers that have been predetermined by the teacher.

Learning is student-centered; the teacher acts as facilitator. In essence, the teacher creates an environment where students take ownership in the direction and content of their learning.

Students work collaboratively towards addressing the general question . All of the students work together to attain the shared goal of producing a solution to the problem. Consequently, the groups co-depend on each other’s performance and contributions in order to make their own advances in reasoning toward answering the research questions and the overall problem.

Learning is driven by the context of the problem and is not bound by an established curriculum. In this environment, students determine what and how much they need to learn in order to accomplish a specific task. Consequently, acquired information and learned concepts and strategies are tied directly to the context of the learning situation. Learning is not confined to a preset curriculum. Creation of a final product is not a necessary requirement of all problem-based inquiry models.

Project-based learning models most often include this type of product as an integral part of the learning process, because learning is expected to occur primarily in the act of creating something. Unlike problem based inquiry models, project-based learning does not necessarily address a real-world problem, nor does it focus on providing argumentation for resolution of an issue.

In a problem-based inquiry setting, there is greater emphasis on problem-solving, analysis, resolution, and explanation of an authentic dilemma. Sometimes this analysis and explanation is represented in the form of a project, but it can also take the form of verbal debate and written summary.

Instructional models and applications

There is no single method for designing problem-based inquiry learning environments.

Various techniques have been used to generate the problem and stimulate learning. Promoting student-ownership, using a particular medium to focus attention, telling stories, simulating and recreating events, and utilizing resources and data on the Internet are among them. The instructional model, problem based learning will be discussed next with attention to instructional strategies and practical examples.

Problem-Based Learning

Problem-based learning (PBL) is an instructional strategy in which students actively resolve complex problems in realistic situations.

It can be used to teach individual lessons, units, or even entire curricula. PBL is often approached in a team environment with emphasis on building skills related to consensual decision making, dialogue and discussion, team maintenance, conflict management, and team leadership. While the fundamental approach of problem solving in situated environments has been used throughout the history of schooling, the term PBL did not appear until the 1970s and was devised as an alternative approach to medical education.

In most medical programs, students initially take a series of fact intensive courses in biology and anatomy and then participate in a field experience as a medical resident in a hospital or clinic. However, Barrows reported that, unfortunately, medical residents frequently had difficulty applying knowledge from their classroom experiences in work-related, problem-solving situations. He argued that the classical framework of learning medical knowledge first in classrooms through studying and testing was too passive and removed from context to take on meaning.

Consequently, PBL was first seen as a medical field immersion experience whereby students learned about their medical specialty through direct engagement in realistic problems and gradual apprenticeship in natural or simulated settings. Problem solving is emphasized as an initial area of learning and development in PBL medical programs more so than memorizing a series of facts outside their natural context.

In addition to the field of medicine, PBL is used in many areas of education and training. In academic courses, PBL is used as a tool to help students understand the utility of a particular concept or study. For example, students may learn about recycling and materials as they determine methods that will reduce the county landfill problem.

In addition, alternative education programs have been created with a PBL emphasis to help at-risk students learn in a different way through partnerships with local businesses and government. In vocational education, PBL experiences often emphasize participation in natural settings.

For example, students in architecture address the problem of designing homes for impoverished areas. Many of the residents need safe housing and cannot afford to purchase typical homes. Consequently, students learn about architectural design and resolving the problem as they construct homes made from recycled materials. In business and the military, simulations are used as a means of instruction in PBL. The affective and physiological stress associated with warfare can influence strategic planning, so PBL in military settings promotes the use of “war games” as a tactic for facing authentic crises.

In business settings, simulations of “what if” scenarios are used to train managers in various strategies and problem-solving approaches to conflict resolution. In both military and business settings, the simulation is a tool that provides an opportunity to not only address realistic problems but to learn from mistakes in a more forgiving way than in an authentic context.

Designing the learning environment

The following elements are commonly associated with PBL activities.

Problem generation: The problems must address concepts and principles relevant to the content domain. Problems are not investigated by students solely for problem solving experiences but as a means of understanding the subject area. Some PBL activities incorporate multidisciplinary approaches, assuming the teacher can provide and coordinate needed resources such as additional content, instructional support, and other teachers. In addition, the problems must relate to real issues that are present in society or students’ lives. Contrived scenarios detract from the perceived usefulness of a concept.

Problem presentation: Students must “own” the problem, either by creating or selecting it. Ownership also implies that their contributions affect the outcome of solving the problem. Thus, more than one solution and more than one method of achieving a solution to the problem are often possible. Furthermore, ownership means that students take responsibility for representing and communicating their work in a unique way.

Predetermined formats of problem structure and analysis towards resolution are not recommended; however, the problem should be presented such that the information in the problem does not call attention to critical factors in the case that will lead to immediate resolution. Ownership also suggests that students will ask further questions, reveal further information, and synthesize critical factors throughout the problem-solving process.

Teacher role: Teachers act primarily as cognitive coaches by facilitating learning and modeling higher order thinking and meta cognitive skills. As facilitators, teachers give students control over how they learn and provide support and structure in the direction of their learning. They help the class create a common framework of expectations using tools such as general guidelines and time lines.

As cognitive modelers, teachers think aloud about strategies and questions that influence how students manage the progress of their learning and accomplish group tasks. In addition, teachers continually question students about the concepts they are learning in the context of the problem in order to probe their understanding, challenge their thinking, and help them deepen or extend their ideas.

Student role: Students first define or select an ill-structured problem that has no obvious solution. They develop alternative hypotheses to resolve the problem and discuss and negotiate their conjectures in a group. Next, they access, evaluate, and utilize data from a variety of available sources to support or refute their hypotheses. They may alter, develop, or synthesize hypotheses in light of new information. Finally, they develop clearly stated solutions that fit the problem and its inherent conditions, based upon information and reasoning to support their arguments. Solutions can be in the form of essays, presentations, or projects.

Maine School Engages Kids With Problem-Solving Challenges (11:37)

https://youtu.be/i17F-b5GG94

[PBS NewsHour].(2013, May 6). Maine School Engages Kids with Problem Solving Challenges. [Video File]. Retrieve from https://youtu.be/i17F-b5GG94

Special correspondent John Tulenko of Leaning Matters reports on a public middle school in Portland, Maine that is taking a different approach to teaching students. Teachers have swapped traditional curriculum for an unusually comprehensive science curriculum that emphasizes problem-solving, with a little help from some robots.

Effectiveness of Problem and Inquiry-based learning.

Why does inquiry-based learning only have an effect size of 0.31 when it is an approach to learning that seems to engage students and teachers so readily in the process of learning?

When is the right and wrong time to introduce inquiry and problem based learning?

Watch video from John Hattie on inquiry and problem-based learning, (2:11 minutes).

[Corwin]. (2015, Nov. 9). John Hattie on inquiry-based learning. [Video File]. Retrieved from https://youtu.be/YUooOYbgSUg.

Glazer, E. (2010) Emerging Perspectives on Learning, Teaching, and Technology, Global Text, Michael Orey. (Chapter 14) Attribution CC 3.0. Retrieved from https://textbookequity.org/Textbooks/Orey_Emerging_Perspectives_Learning.pdf

Instructional Methods, Strategies and Technologies to Meet the Needs of All Learners Copyright © 2017 by Evan Glazer (University of Georgia) is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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Teaching Problem-Solving Skills

Many instructors design opportunities for students to solve “problems”. But are their students solving true problems or merely participating in practice exercises? The former stresses critical thinking and decision making skills whereas the latter requires only the application of previously learned procedures.

Problem solving is often broadly defined as "the ability to understand the environment, identify complex problems, review related information to develop, evaluate strategies and implement solutions to build the desired outcome" (Fissore, C. et al, 2021). True problem solving is the process of applying a method – not known in advance – to a problem that is subject to a specific set of conditions and that the problem solver has not seen before, in order to obtain a satisfactory solution.

Below you will find some basic principles for teaching problem solving and one model to implement in your classroom teaching.

Principles for teaching problem solving

Model a useful problem-solving method . Problem solving can be difficult and sometimes tedious. Show students how to be patient and persistent, and how to follow a structured method, such as Woods’ model described below. Articulate your method as you use it so students see the connections.
Teach within a specific context . Teach problem-solving skills in the context in which they will be used by students (e.g., mole fraction calculations in a chemistry course). Use real-life problems in explanations, examples, and exams. Do not teach problem solving as an independent, abstract skill.
Help students understand the problem . In order to solve problems, students need to define the end goal. This step is crucial to successful learning of problem-solving skills. If you succeed at helping students answer the questions “what?” and “why?”, finding the answer to “how?” will be easier.
Take enough time . When planning a lecture/tutorial, 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.
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. Also, ask questions and make suggestions about strategies to encourage students to reflect on the problem-solving strategies that they use.
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.

Woods’ problem-solving model

Define the problem.

The system . Have students identify the system under study (e.g., a metal bridge subject to certain forces) 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) . Once you have a list of knowns, 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 . One key aspect in problem solving is teaching students how to 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 . All problems have some stated or implied constraints. Teach students to look for the words "only", "must", "neglect", or "assume" to help identify the constraints.
Criteria for success . Help students consider, from the beginning, what a logical type of answer would be. What characteristics will it possess? For example, a quantitative problem will require an answer in some form of numerical units (e.g., $/kg product, square cm, etc.) while an optimization problem requires an answer in the form of either a numerical maximum or minimum.

Think about it

“Let it simmer”. Use this stage to ponder the problem. Ideally, students will develop a mental image of the problem at hand during this stage.
Identify specific pieces of knowledge . Students need to determine by themselves the required background knowledge from illustrations, examples and problems covered in the course.
Collect information . Encourage students to collect pertinent information such as conversion factors, constants, and tables needed to solve the problem.

Plan a solution

Consider possible strategies . Often, the type of solution will be determined by the type of problem. Some common problem-solving strategies are: compute; simplify; use an equation; make a model, diagram, table, or chart; or work backwards.
Choose the best strategy . Help students to choose the best strategy by reminding them again what they are required to find or calculate.

Carry out the plan

Be patient . Most problems are not solved quickly or on the first attempt. In other cases, executing the solution may be the easiest step.
Be persistent . If a plan does not work immediately, do not let students get discouraged. Encourage them to try a different strategy and keep trying.

Encourage students to reflect. Once a solution has been reached, students should ask themselves the following questions:

Does the answer make sense?
Does it fit with the criteria established in step 1?
Did I answer the question(s)?
What did I learn by doing this?
Could I have done the problem another way?

If you would like support applying these tips to your own teaching, CTE staff members are here to help. View the CTE Support page to find the most relevant staff member to contact.

Fissore, C., Marchisio, M., Roman, F., & Sacchet, M. (2021). Development of problem solving skills with Maple in higher education. In: Corless, R.M., Gerhard, J., Kotsireas, I.S. (eds) Maple in Mathematics Education and Research. MC 2020. Communications in Computer and Information Science, vol 1414. Springer, Cham. https://doi.org/10.1007/978-3-030-81698-8_15
Foshay, R., & Kirkley, J. (1998). Principles for Teaching Problem Solving. TRO Learning Inc., Edina MN. (PDF) Principles for Teaching Problem Solving (researchgate.net)
Hayes, J.R. (1989). The Complete Problem Solver. 2nd Edition. Hillsdale, NJ: Lawrence Erlbaum Associates.
Woods, D.R., Wright, J.D., Hoffman, T.W., Swartman, R.K., Doig, I.D. (1975). Teaching Problem solving Skills.
Engineering Education. Vol 1, No. 1. p. 238. Washington, DC: The American Society for Engineering Education.

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Teaching problem solving

Strategies for teaching problem solving apply across disciplines and instructional contexts. First, introduce the problem and explain how people in your discipline generally make sense of the given information. Then, explain how to apply these approaches to solve the problem.

Introducing the problem

Explaining how people in your discipline understand and interpret these types of problems can help students develop the skills they need to understand the problem (and find a solution). After introducing how you would go about solving a problem, you could then ask students to:

frame the problem in their own words
define key terms and concepts
determine statements that accurately represent the givens of a problem
identify analogous problems
determine what information is needed to solve the problem

Working on solutions

In the solution phase, one develops and then implements a coherent plan for solving the problem. As you help students with this phase, you might ask them to:

identify the general model or procedure they have in mind for solving the problem
set sub-goals for solving the problem
identify necessary operations and steps
draw conclusions
carry out necessary operations

You can help students tackle a problem effectively by asking them to:

systematically explain each step and its rationale
explain how they would approach solving the problem
help you solve the problem by posing questions at key points in the process
work together in small groups (3 to 5 students) to solve the problem and then have the solution presented to the rest of the class (either by you or by a student in the group)

In all cases, the more you get the students to articulate their own understandings of the problem and potential solutions, the more you can help them develop their expertise in approaching problems in your discipline.

Why Every Educator Needs to Teach Problem-Solving Skills

Strong problem-solving skills will help students be more resilient and will increase their academic and career success .

Want to learn more about how to measure and teach students’ higher-order skills, including problem solving, critical thinking, and written communication?

Problem-solving skills are essential in school, careers, and life.

Problem-solving skills are important for every student to master. They help individuals navigate everyday life and find solutions to complex issues and challenges. These skills are especially valuable in the workplace, where employees are often required to solve problems and make decisions quickly and effectively.

Problem-solving skills are also needed for students’ personal growth and development because they help individuals overcome obstacles and achieve their goals. By developing strong problem-solving skills, students can improve their overall quality of life and become more successful in their personal and professional endeavors.

Problem-Solving Skills Help Students…

develop resilience.

Problem-solving skills are an integral part of resilience and the ability to persevere through challenges and adversity. To effectively work through and solve a problem, students must be able to think critically and creatively. Critical and creative thinking help students approach a problem objectively, analyze its components, and determine different ways to go about finding a solution.

This process in turn helps students build self-efficacy . When students are able to analyze and solve a problem, this increases their confidence, and they begin to realize the power they have to advocate for themselves and make meaningful change.

When students gain confidence in their ability to work through problems and attain their goals, they also begin to build a growth mindset . According to leading resilience researcher, Carol Dweck, “in a growth mindset, people believe that their most basic abilities can be developed through dedication and hard work—brains and talent are just the starting point. This view creates a love of learning and a resilience that is essential for great accomplishment.”

Set and Achieve Goals

Students who possess strong problem-solving skills are better equipped to set and achieve their goals. By learning how to identify problems, think critically, and develop solutions, students can become more self-sufficient and confident in their ability to achieve their goals. Additionally, problem-solving skills are used in virtually all fields, disciplines, and career paths, which makes them important for everyone. Building strong problem-solving skills will help students enhance their academic and career performance and become more competitive as they begin to seek full-time employment after graduation or pursue additional education and training.

Resolve Conflicts

In addition to increased social and emotional skills like self-efficacy and goal-setting, problem-solving skills teach students how to cooperate with others and work through disagreements and conflicts. Problem-solving promotes “thinking outside the box” and approaching a conflict by searching for different solutions. This is a very different (and more effective!) method than a more stagnant approach that focuses on placing blame or getting stuck on elements of a situation that can’t be changed.

While it’s natural to get frustrated or feel stuck when working through a conflict, students with strong problem-solving skills will be able to work through these obstacles, think more rationally, and address the situation with a more solution-oriented approach. These skills will be valuable for students in school, their careers, and throughout their lives.

Achieve Success

We are all faced with problems every day. Problems arise in our personal lives, in school and in our jobs, and in our interactions with others. Employers especially are looking for candidates with strong problem-solving skills. In today’s job market, most jobs require the ability to analyze and effectively resolve complex issues. Students with strong problem-solving skills will stand out from other applicants and will have a more desirable skill set.

In a recent opinion piece published by The Hechinger Report , Virgel Hammonds, Chief Learning Officer at KnowledgeWorks, stated “Our world presents increasingly complex challenges. Education must adapt so that it nurtures problem solvers and critical thinkers.” Yet, the “traditional K–12 education system leaves little room for students to engage in real-world problem-solving scenarios.” This is the reason that a growing number of K–12 school districts and higher education institutions are transforming their instructional approach to personalized and competency-based learning, which encourage students to make decisions, problem solve and think critically as they take ownership of and direct their educational journey.

Problem-Solving Skills Can Be Measured and Taught

Research shows that problem-solving skills can be measured and taught. One effective method is through performance-based assessments which require students to demonstrate or apply their knowledge and higher-order skills to create a response or product or do a task.

What Are Performance-Based Assessments?

With the No Child Left Behind Act (2002), the use of standardized testing became the primary way to measure student learning in the U.S. The legislative requirements of this act shifted the emphasis to standardized testing, and this led to a decline in nontraditional testing methods .

But many educators, policy makers, and parents have concerns with standardized tests. Some of the top issues include that they don’t provide feedback on how students can perform better, they don’t value creativity, they are not representative of diverse populations, and they can be disadvantageous to lower-income students.

While standardized tests are still the norm, U.S. Secretary of Education Miguel Cardona is encouraging states and districts to move away from traditional multiple choice and short response tests and instead use performance-based assessment, competency-based assessments, and other more authentic methods of measuring students abilities and skills rather than rote learning.

Performance-based assessments measure whether students can apply the skills and knowledge learned from a unit of study. Typically, a performance task challenges students to use their higher-order skills to complete a project or process. Tasks can range from an essay to a complex proposal or design.

Preview a Performance-Based Assessment

Want a closer look at how performance-based assessments work? Preview CAE’s K–12 and Higher Education assessments and see how CAE’s tools help students develop critical thinking, problem-solving, and written communication skills.

Performance-Based Assessments Help Students Build and Practice Problem-Solving Skills

In addition to effectively measuring students’ higher-order skills, including their problem-solving skills, performance-based assessments can help students practice and build these skills. Through the assessment process, students are given opportunities to practically apply their knowledge in real-world situations. By demonstrating their understanding of a topic, students are required to put what they’ve learned into practice through activities such as presentations, experiments, and simulations.

This type of problem-solving assessment tool requires students to analyze information and choose how to approach the presented problems. This process enhances their critical thinking skills and creativity, as well as their problem-solving skills. Unlike traditional assessments based on memorization or reciting facts, performance-based assessments focus on the students’ decisions and solutions, and through these tasks students learn to bridge the gap between theory and practice.

Performance-based assessments like CAE’s College and Career Readiness Assessment (CRA+) and Collegiate Learning Assessment (CLA+) provide students with in-depth reports that show them which higher-order skills they are strongest in and which they should continue to develop. This feedback helps students and their teachers plan instruction and supports to deepen their learning and improve their mastery of critical skills.

Explore CAE’s Problem-Solving Assessments

CAE offers performance-based assessments that measure student proficiency in higher-order skills including problem solving, critical thinking, and written communication.

College and Career Readiness Assessment (CCRA+) for secondary education and
Collegiate Learning Assessment (CLA+) for higher education.

Our solution also includes instructional materials, practice models, and professional development.

We can help you create a program to build students’ problem-solving skills that includes:

Measuring students’ problem-solving skills through a performance-based assessment
Using the problem-solving assessment data to inform instruction and tailor interventions
Teaching students problem-solving skills and providing practice opportunities in real-life scenarios
Supporting educators with quality professional development

Get started with our problem-solving assessment tools to measure and build students’ problem-solving skills today! These skills will be invaluable to students now and in the future.

Ready to Get Started?

Learn more about cae’s suite of products and let’s get started measuring and teaching students important higher-order skills like problem solving..

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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Case-based Teaching and Problem-based Learning

Case-based teaching.

With case-based teaching, students develop skills in analytical thinking and reflective judgment by reading and discussing complex, real-life scenarios. The articles in this section explain how to use cases in teaching and provide case studies for the natural sciences, social sciences, and other disciplines.

Teaching with Case Studies (Stanford University)

This article from the Stanford Center for Teaching and Learning describes the rationale for using case studies, the process for choosing appropriate cases, and tips for how to implement them in college courses.

The Case Method (University of Illinois)

Tips for teachers on how to be successful using the Case Method in the college/university classroom. Includes information about the Case Method values, uses, and additional resource links.

National Center for Case Study Teaching in Science (National Science Teaching Association)

This site offers resources and examples specific to teaching in the sciences. This includes the “UB Case Study Collection,” an extensive list of ready-to-use cases in a variety of science disciplines. Each case features a PDF handout describing the case, as well as teaching notes.

The Michigan Sustainability Cases Initiative (CRLT Occasional Paper)

This paper describes the Michigan Sustainability Cases Initiative, including links to the full library of cases, and it offers advice both for writing cases and facilitating case discussions effectively.

The Case Method and the Interactive Classroom (Foran, 2001, NEA Higher Education Journal)

First-person account of how a sociology faculty member at University of California, Santa Barbara began using case studies in his teaching and how his methods have evolved over time as a professor.

Problem-based Learning

Problem-based learning (PBL) is both a teaching method and an approach to the curriculum. It consists of carefully designed problems that challenge students to use problem solving techniques, self-directed learning strategies, team participation skills, and disciplinary knowledge. The articles and links in this section describe the characteristics and objectives of PBL and the process for using PBL. There is also a list of printed and web resources.

Problem-Based Learning Network (Illinois Mathematics and Science Academy)

Site includes an interactive PBL Model, Professional Development links, and video vignettes to illustrate how to effectively use problem-based learning in the classroom. The goals of IMSA's PBLNetwork are to mentor educators in all disciplines, to explore problem-based learning strategies, and to connect PBL educators to one another.

Problem-Based Learning: An Introduction (Rhem, 1998, National Teaching and Learning Forum)

This piece summarizes the benefits of using problem-based learning, its historical origins, and the faculty/student roles in PBL. Overall, this is an easy to read introduction to problem-based learning.

Problem-Based Learning (Stanford University, 2001)

This issue of Speaking of Teaching identifies the central features of PBL, provides some guidelines for planning a PBL course, and discusses the impact of PBL on student learning and motivation.

Problem-Based Learning Clearinghouse (University of Delaware)

Collection of peer reviewed problems and articles to assist educators in using problem-based learning. Teaching notes and supplemental materials accompany each problem, providing insights and strategies that are innovative and classroom-tested. Free registration is required to view and download the Clearinghouse’s resources.

See also: The International Journal of Problem-Based Learning

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5 Teaching Mathematics Through Problem Solving

Janet Stramel

In his book “How to Solve It,” George Pólya (1945) said, “One of the most important tasks of the teacher is to help his students. This task is not quite easy; it demands time, practice, devotion, and sound principles. The student should acquire as much experience of independent work as possible. But if he is left alone with his problem without any help, he may make no progress at all. If the teacher helps too much, nothing is left to the student. The teacher should help, but not too much and not too little, so that the student shall have a reasonable share of the work.” (page 1)

What is a problem in mathematics? A problem is “any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method” (Hiebert, et. al., 1997). Problem solving in mathematics is one of the most important topics to teach; learning to problem solve helps students develop a sense of solving real-life problems and apply mathematics to real world situations. It is also used for a deeper understanding of mathematical concepts. Learning “math facts” is not enough; students must also learn how to use these facts to develop their thinking skills.

According to NCTM (2010), the term “problem solving” refers to mathematical tasks that have the potential to provide intellectual challenges for enhancing students’ mathematical understanding and development. When you first hear “problem solving,” what do you think about? Story problems or word problems? Story problems may be limited to and not “problematic” enough. For example, you may ask students to find the area of a rectangle, given the length and width. This type of problem is an exercise in computation and can be completed mindlessly without understanding the concept of area. Worthwhile problems includes problems that are truly problematic and have the potential to provide contexts for students’ mathematical development.

There are three ways to solve problems: teaching for problem solving, teaching about problem solving, and teaching through problem solving.

Teaching for problem solving begins with learning a skill. For example, students are learning how to multiply a two-digit number by a one-digit number, and the story problems you select are multiplication problems. Be sure when you are teaching for problem solving, you select or develop tasks that can promote the development of mathematical understanding.

Teaching about problem solving begins with suggested strategies to solve a problem. For example, “draw a picture,” “make a table,” etc. You may see posters in teachers’ classrooms of the “Problem Solving Method” such as: 1) Read the problem, 2) Devise a plan, 3) Solve the problem, and 4) Check your work. There is little or no evidence that students’ problem-solving abilities are improved when teaching about problem solving. Students will see a word problem as a separate endeavor and focus on the steps to follow rather than the mathematics. In addition, students will tend to use trial and error instead of focusing on sense making.

Teaching through problem solving focuses students’ attention on ideas and sense making and develops mathematical practices. Teaching through problem solving also develops a student’s confidence and builds on their strengths. It allows for collaboration among students and engages students in their own learning.

Consider the following worthwhile-problem criteria developed by Lappan and Phillips (1998):

The problem has important, useful mathematics embedded in it.
The problem requires high-level thinking and problem solving.
The problem contributes to the conceptual development of students.
The problem creates an opportunity for the teacher to assess what his or her students are learning and where they are experiencing difficulty.
The problem can be approached by students in multiple ways using different solution strategies.
The problem has various solutions or allows different decisions or positions to be taken and defended.
The problem encourages student engagement and discourse.
The problem connects to other important mathematical ideas.
The problem promotes the skillful use of mathematics.
The problem provides an opportunity to practice important skills.

Of course, not every problem will include all of the above. Sometimes, you will choose a problem because your students need an opportunity to practice a certain skill.

Key features of a good mathematics problem includes:

It must begin where the students are mathematically.
The feature of the problem must be the mathematics that students are to learn.
It must require justifications and explanations for both answers and methods of solving.

Problem solving is not a neat and orderly process. Think about needlework. On the front side, it is neat and perfect and pretty.

But look at the b ack.

It is messy and full of knots and loops. Problem solving in mathematics is also like this and we need to help our students be “messy” with problem solving; they need to go through those knots and loops and learn how to solve problems with the teacher’s guidance.

When you teach through problem solving , your students are focused on ideas and sense-making and they develop confidence in mathematics!

Mathematics Tasks and Activities that Promote Teaching through Problem Solving

Choosing the Right Task

Selecting activities and/or tasks is the most significant decision teachers make that will affect students’ learning. Consider the following questions:

Teachers must do the activity first. What is problematic about the activity? What will you need to do BEFORE the activity and AFTER the activity? Additionally, think how your students would do the activity.
What mathematical ideas will the activity develop? Are there connections to other related mathematics topics, or other content areas?
Can the activity accomplish your learning objective/goals?

Low Floor High Ceiling Tasks

By definition, a “ low floor/high ceiling task ” is a mathematical activity where everyone in the group can begin and then work on at their own level of engagement. Low Floor High Ceiling Tasks are activities that everyone can begin and work on based on their own level, and have many possibilities for students to do more challenging mathematics. One gauge of knowing whether an activity is a Low Floor High Ceiling Task is when the work on the problems becomes more important than the answer itself, and leads to rich mathematical discourse [Hover: ways of representing, thinking, talking, agreeing, and disagreeing; the way ideas are exchanged and what the ideas entail; and as being shaped by the tasks in which students engage as well as by the nature of the learning environment].

The strengths of using Low Floor High Ceiling Tasks:

Allows students to show what they can do, not what they can’t.
Provides differentiation to all students.
Promotes a positive classroom environment.
Advances a growth mindset in students
Aligns with the Standards for Mathematical Practice

Examples of some Low Floor High Ceiling Tasks can be found at the following sites:

YouCubed – under grades choose Low Floor High Ceiling
NRICH Creating a Low Threshold High Ceiling Classroom
Inside Mathematics Problems of the Month

Math in 3-Acts

Math in 3-Acts was developed by Dan Meyer to spark an interest in and engage students in thought-provoking mathematical inquiry. Math in 3-Acts is a whole-group mathematics task consisting of three distinct parts:

Act One is about noticing and wondering. The teacher shares with students an image, video, or other situation that is engaging and perplexing. Students then generate questions about the situation.

In Act Two , the teacher offers some information for the students to use as they find the solutions to the problem.

Act Three is the “reveal.” Students share their thinking as well as their solutions.

“Math in 3 Acts” is a fun way to engage your students, there is a low entry point that gives students confidence, there are multiple paths to a solution, and it encourages students to work in groups to solve the problem. Some examples of Math in 3-Acts can be found at the following websites:

Dan Meyer’s Three-Act Math Tasks
Graham Fletcher3-Act Tasks ]
Math in 3-Acts: Real World Math Problems to Make Math Contextual, Visual and Concrete

Number Talks

Number talks are brief, 5-15 minute discussions that focus on student solutions for a mental math computation problem. Students share their different mental math processes aloud while the teacher records their thinking visually on a chart or board. In addition, students learn from each other’s strategies as they question, critique, or build on the strategies that are shared.. To use a “number talk,” you would include the following steps:

The teacher presents a problem for students to solve mentally.
Provide adequate “ wait time .”
The teacher calls on a students and asks, “What were you thinking?” and “Explain your thinking.”
For each student who volunteers to share their strategy, write their thinking on the board. Make sure to accurately record their thinking; do not correct their responses.
Invite students to question each other about their strategies, compare and contrast the strategies, and ask for clarification about strategies that are confusing.

“Number Talks” can be used as an introduction, a warm up to a lesson, or an extension. Some examples of Number Talks can be found at the following websites:

Inside Mathematics Number Talks
Number Talks Build Numerical Reasoning

Saying “This is Easy”

“This is easy.” Three little words that can have a big impact on students. What may be “easy” for one person, may be more “difficult” for someone else. And saying “this is easy” defeats the purpose of a growth mindset classroom, where students are comfortable making mistakes.

When the teacher says, “this is easy,” students may think,

“Everyone else understands and I don’t. I can’t do this!”
Students may just give up and surrender the mathematics to their classmates.
Students may shut down.

Instead, you and your students could say the following:

“I think I can do this.”
“I have an idea I want to try.”
“I’ve seen this kind of problem before.”

Tracy Zager wrote a short article, “This is easy”: The Little Phrase That Causes Big Problems” that can give you more information. Read Tracy Zager’s article here.

Using “Worksheets”

Do you want your students to memorize concepts, or do you want them to understand and apply the mathematics for different situations?

What is a “worksheet” in mathematics? It is a paper and pencil assignment when no other materials are used. A worksheet does not allow your students to use hands-on materials/manipulatives [Hover: physical objects that are used as teaching tools to engage students in the hands-on learning of mathematics]; and worksheets are many times “naked number” with no context. And a worksheet should not be used to enhance a hands-on activity.

Students need time to explore and manipulate materials in order to learn the mathematics concept. Worksheets are just a test of rote memory. Students need to develop those higher-order thinking skills, and worksheets will not allow them to do that.

One productive belief from the NCTM publication, Principles to Action (2014), states, “Students at all grade levels can benefit from the use of physical and virtual manipulative materials to provide visual models of a range of mathematical ideas.”

You may need an “activity sheet,” a “graphic organizer,” etc. as you plan your mathematics activities/lessons, but be sure to include hands-on manipulatives. Using manipulatives can

Provide your students a bridge between the concrete and abstract
Serve as models that support students’ thinking
Provide another representation
Support student engagement
Give students ownership of their own learning.

Adapted from “ The Top 5 Reasons for Using Manipulatives in the Classroom ”.

any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method

should be intriguing and contain a level of challenge that invites speculation and hard work, and directs students to investigate important mathematical ideas and ways of thinking toward the learning

involves teaching a skill so that a student can later solve a story problem

when we teach students how to problem solve

teaching mathematics content through real contexts, problems, situations, and models

a mathematical activity where everyone in the group can begin and then work on at their own level of engagement

20 seconds to 2 minutes for students to make sense of questions

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5 Advantages and Disadvantages of Problem-Based Learning [+ Activity Design Steps]

Written by Marcus Guido

Easily differentiate learning and engage your students with Prodigy Math.

Teaching Strategies

Advantages of Problem-Based Learning

Disadvantages of problem-based learning, steps to designing problem-based learning activities.

Used since the 1960s, many teachers express concerns about the effectiveness of problem-based learning (PBL) in certain classroom settings.

Whether you introduce the student-centred pedagogy as a one-time activity or mainstay exercise, grouping students together to solve open-ended problems can present pros and cons.

Below are five advantages and disadvantages of problem-based learning to help you determine if it can work in your classroom.

If you decide to introduce an activity, there are also design creation steps and a downloadable guide to keep at your desk for easy reference.

1. Development of Long-Term Knowledge Retention

Students who participate in problem-based learning activities can improve their abilities to retain and recall information, according to a literature review of studies about the pedagogy .

The literature review states “elaboration of knowledge at the time of learning” -- by sharing facts and ideas through discussion and answering questions -- “enhances subsequent retrieval.” This form of elaborating reinforces understanding of subject matter , making it easier to remember.

Small-group discussion can be especially beneficial -- ideally, each student will get chances to participate.

But regardless of group size, problem-based learning promotes long-term knowledge retention by encouraging students to discuss -- and answer questions about -- new concepts as they’re learning them.

2. Use of Diverse Instruction Types

You can use problem-based learning activities to the meet the diverse learning needs and styles of your students, effectively engaging a diverse classroom in the process. In general, grouping students together for problem-based learning will allow them to:

Address real-life issues that require real-life solutions, appealing to students who struggle to grasp abstract concepts
Participate in small-group and large-group learning, helping students who don’t excel during solo work grasp new material
Talk about their ideas and challenge each other in a constructive manner, giving participatory learners an avenue to excel
Tackle a problem using a range of content you provide -- such as videos, audio recordings, news articles and other applicable material -- allowing the lesson to appeal to distinct learning styles

Since running a problem-based learning scenario will give you a way to use these differentiated instruction approaches , it can be especially worthwhile if your students don’t have similar learning preferences.

3. Continuous Engagement

Providing a problem-based learning challenge can engage students by acting as a break from normal lessons and common exercises.

It’s not hard to see the potential for engagement, as kids collaborate to solve real-world problems that directly affect or heavily interest them.

Although conducted with post-secondary students, a study published by the Association for the Study of Medical Education reported increased student attendance to -- and better attitudes towards -- courses that feature problem-based learning.

These activities may lose some inherent engagement if you repeat them too often, but can certainly inject excitement into class.

4. Development of Transferable Skills

Problem-based learning can help students develop skills they can transfer to real-world scenarios, according to a 2015 book that outlines theories and characteristics of the pedagogy .

The tangible contexts and consequences presented in a problem-based learning activity “allow learning to become more profound and durable.” As you present lessons through these real-life scenarios, students should be able to apply learnings if they eventually face similar issues.

For example, if they work together to address a dispute within the school, they may develop lifelong skills related to negotiation and communicating their thoughts with others.

As long as the problem’s context applies to out-of-class scenarios, students should be able to build skills they can use again.

5. Improvement of Teamwork and Interpersonal Skills

Successful completion of a problem-based learning challenge hinges on interaction and communication, meaning students should also build transferable skills based on teamwork and collaboration . Instead of memorizing facts, they get chances to present their ideas to a group, defending and revising them when needed.

What’s more, this should help them understand a group dynamic. Depending on a given student, this can involve developing listening skills and a sense of responsibility when completing one’s tasks. Such skills and knowledge should serve your students well when they enter higher education levels and, eventually, the working world.

1. Potentially Poorer Performance on Tests

Devoting too much time to problem-based learning can cause issues when students take standardized tests, as they may not have the breadth of knowledge needed to achieve high scores. Whereas problem-based learners develop skills related to collaboration and justifying their reasoning, many tests reward fact-based learning with multiple choice and short answer questions. Despite offering many advantages, you could spot this problem develop if you run problem-based learning activities too regularly.

2. Student Unpreparedness

Problem-based learning exercises can engage many of your kids, but others may feel disengaged as a result of not being ready to handle this type of exercise for a number of reasons. On a class-by-class and activity-by-activity basis, participation may be hindered due to:

Immaturity -- Some students may not display enough maturity to effectively work in a group, not fulfilling expectations and distracting other students.
Unfamiliarity -- Some kids may struggle to grasp the concept of an open problem, since they can’t rely on you for answers.
Lack of Prerequisite Knowledge -- Although the activity should address a relevant and tangible problem, students may require new or abstract information to create an effective solution.

You can partially mitigate these issues by actively monitoring the classroom and distributing helpful resources, such as guiding questions and articles to read. This should keep students focused and help them overcome knowledge gaps. But if you foresee facing these challenges too frequently, you may decide to avoid or seldom introduce problem-based learning exercises.

3. Teacher Unpreparedness

If supervising a problem-based learning activity is a new experience, you may have to prepare to adjust some teaching habits . For example, overtly correcting students who make flawed assumptions or statements can prevent them from thinking through difficult concepts and questions. Similarly, you shouldn’t teach to promote the fast recall of facts. Instead, you should concentrate on:

Giving hints to help fix improper reasoning
Questioning student logic and ideas in a constructive manner
Distributing content for research and to reinforce new concepts
Asking targeted questions to a group or the class, focusing their attention on a specific aspect of the problem

Depending on your teaching style, it may take time to prepare yourself to successfully run a problem-based learning lesson.

4. Time-Consuming Assessment

If you choose to give marks, assessing a student’s performance throughout a problem-based learning exercise demands constant monitoring and note-taking. You must take factors into account such as:

Completed tasks
The quality of those tasks
The group’s overall work and solution
Communication among team members
Anything you outlined on the activity’s rubric

Monitoring these criteria is required for each student, making it time-consuming to give and justify a mark for everyone.

5. Varying Degrees of Relevancy and Applicability

It can be difficult to identify a tangible problem that students can solve with content they’re studying and skills they’re mastering. This introduces two clear issues. First, if it is easy for students to divert from the challenge’s objectives, they may miss pertinent information. Second, you could veer off the problem’s focus and purpose as students run into unanticipated obstacles. Overcoming obstacles has benefits, but may compromise the planning you did. It can also make it hard to get back on track once the activity is complete. Because of the difficulty associated with keeping activities relevant and applicable, you may see problem-based learning as too taxing.

If the advantages outweigh the disadvantages -- or you just want to give problem-based learning a shot -- follow these steps:

1. Identify an Applicable Real-Life Problem

Find a tangible problem that’s relevant to your students, allowing them to easily contextualize it and hopefully apply it to future challenges. To identify an appropriate real-world problem, look at issues related to your:

Students’ shared interests

You must also ensure that students understand the problem and the information around it. So, not all problems are appropriate for all grade levels.

2. Determine the Overarching Purpose of the Activity

Depending on the problem you choose, determine what you want to accomplish by running the challenge. For example, you may intend to help your students improve skills related to:

Collaboration
Problem-solving
Curriculum-aligned topics
Processing diverse content

A more precise example, you may prioritize collaboration skills by assigning specific tasks to pairs of students within each team. In doing so, students will continuously develop communication and collaboration abilities by working as a couple and part of a small group. By defining a clear purpose, you’ll also have an easier time following the next step.

3. Create and Distribute Helpful Material

Handouts and other content not only act as a set of resources, but help students stay focused on the activity and its purpose. For example, if you want them to improve a certain math skill , you should make material that highlights the mathematical aspects of the problem. You may decide to provide items such as:

Data that helps quantify and add context to the problem
Videos, presentations and other audio-visual material
A list of preliminary questions to investigate

Providing a range of resources can be especially important for elementary students and struggling students in higher grades, who may not have self-direction skills to work without them.

4. Set Goals and Expectations for Your Students

Along with the aforementioned materials, give students a guide or rubric that details goals and expectations. It will allow you to further highlight the purpose of the problem-based learning exercise, as you can explain what you’re looking for in terms of collaboration, the final product and anything else. It should also help students stay on track by acting as a reference throughout the activity.

5. Participate

Although explicitly correcting students may be discouraged, you can still help them and ask questions to dig into their thought processes. When you see an opportunity, consider if it’s worthwhile to:

Fill gaps in knowledge
Provide hints, not answers
Question a student’s conclusion or logic regarding a certain point, helping them think through tough spots

By participating in these ways, you can provide insight when students need it most, encouraging them to effectively analyze the problem.

6. Have Students Present Ideas and Findings

If you divided them into small groups, requiring students to present their thoughts and results in front the class adds a large-group learning component to the lesson. Encourage other students to ask questions, allowing the presenting group to elaborate and provide evidence for their thoughts. This wraps up the activity and gives your class a final chance to find solutions to the problem.

Wrapping Up

The effectiveness of problem-based learning may differ between classrooms and individual students, depending on how significant specific advantages and disadvantages are to you. Evaluative research consistently shows value in giving students a question and letting them take control of their learning. But the extent of this value can depend on the difficulties you face.It may be wise to try a problem-based learning activity, and go forward based on results.

Create or log into your teacher account on Prodigy -- an adaptive math game that adjusts content to accommodate player trouble spots and learning speeds. Aligned to US and Canadian curricula, it’s used by more than 350,000 teachers and 10 million students. It may be wise to try a problem-based learning activity, and go forward based on results.

Problem Solving Method Of Teaching

The problem-solving method of teaching is the learning method that allows children to learn by doing. This is because they are given examples and real-world situations so that the theory behind it can be understood better, as well as practice with each new concept or skill taught on top of what was previously learned in class before moving onto another topic at hand.

What is your preferred problem-solving technique?

Answers : - I like to brainstorm and see what works for me - I enjoy the trial and error method - I am a linear thinker

Share it with me by commenting.

For example, while solving a problem, the child may encounter terms he has not studied yet. These will further help him understand their use in context while developing his vocabulary. At the same time, being able to practice math concepts by tapping into daily activities helps an individual retain these skills better.

One way this type of teaching is applied for younger students particularly is through games played during lessons. By allowing them to become comfortable with the concepts taught through these games, they can put their knowledge into use later on. This is done by developing thinking processes that precede an action or behavior. These games can be used by teachers for different subjects including science and language.

For younger students still, the method of teaching using real-life examples helps them understand better. Through this, it becomes easier for them to relate what they learned in school with terms used outside of school settings so that the information sticks better than if all they were given were theoretical definitions. For instance, instead of just studying photosynthesis as part of biology lessons, children are asked to imagine plants growing inside a dark room because there is no sunlight present. When questioned about the plants, children will be able to recall photosynthesis more easily because they were able to see its importance in real life.

Despite being given specific examples, the act of solving problems helps students think for themselves. They learn how to approach situations and predict outcomes based on what they already know about concepts or ideas taught in class including the use of various skills they have acquired over time. These include problem-solving strategies like using drawings when describing a solution or asking advice if they are stuck to unlock solutions that would otherwise go beyond their reach.

Teachers need to point out in advance which method will be used for any particular lesson before having children engage with it. By doing this, individuals can prepare themselves mentally for what is to come. This is especially true for students who have difficulty with a particular subject. In these cases, the teacher can help them get started by providing a worked example for reference or breaking the problem down into manageable chunks that are easier to digest.

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Ultimately, the goal of teaching using a problem-solving method is to give children the opportunity to think for themselves and to be able to do so in different contexts. Doing this helps foster independent learners who can utilize the skills they acquired in school for future endeavors.

The problem-solving method of teaching allows children to learn by doing. This is because they are given examples and real-world situations so that the theory behind it can be understood better, as practice with each new concept or skill taught on top of what was previously learned in class before moving onto another topic at hand.

One way this type of teaching is applied for younger students particularly is through games played during lessons. By allowing them to become comfortable with the concepts taught through these games, they are able to put their knowledge into use later on. This is done by developing thinking processes that precede an action or behavior. These games can be used by teachers for different subjects including science and language.

For instance, a teacher may ask students to imagine they are plants in a dark room because there is no sunlight present. When questioned about the plants, children will be able to recall photosynthesis more easily because they were able to see its importance in real life.

It is important for teachers to point out in advance which method will be used for any particular lesson before having children engage with it. By doing this, individuals can prepare themselves mentally for what is to come. This is especially true for students who have difficulty with a particular subject. In these cases, the teacher can help them get started by providing a worked example for reference or breaking the problem down into manageable chunks that are easier to digest.

lesson before having children engage with it. By doing this, individuals can prepare themselves mentally for what is to come. This is especially true for students who have difficulty with a particular subject. In these cases, the teacher can help them get started by providing a worked example for reference or breaking the problem down into manageable chunks that are easier to digest.

The teacher should have a few different ways to solve the problem.

For example, the teacher can provide a worked example for reference or break down the problem into chunks that are easier to digest.

The goal of teaching using a problem-solving method is to give children the opportunity to think for themselves and to be able to do so in different contexts. Successful problem solving allows children to become comfortable with concepts taught through games that develop thinking processes that precede an action or behavior.

Introduce the problem

The problem solving method of teaching is a popular approach to learning that allows students to understand new concepts by doing. This approach provides students with examples and real-world situations, so they can see how the theory behind a concept or skill works in practice. In addition, students are given practice with each new concept or skill taught, before moving on to the next topic. This helps them learn and retain the information better.

Explain why the problem solving method of teaching is effective.

The problem solving method of teaching is effective because it allows students to learn by doing. This means they can see how the theory behind a concept or skill works in practice, which helps them understand and remember the information better. This would not be possible if they are only told about the new concept or skill, or read a textbook to learn on their own. Since students can see how the theory works in practice through examples and real-world situations, the information is easier for them to understand.

List some advantages of using the problem solving method of teaching.

Some advantages of using the problem solving method of teaching are that it helps students retain information better since they are able to practice with each new concept or skill taught until they master it before moving on to another topic. This also allows them to learn by doing so they will have hands-on experience with facts which helps them remember important facts faster rather than just hearing about it or reading about it on their own. Furthermore, this teaching method is beneficial for students of all ages and can be adapted to different subjects making it an approach that is versatile and easily used in a classroom setting. Lastly, the problem solving method of teaching presents new information in a way that is easy to understand so students are not overwhelmed with complex material.

The problem solving method of teaching is an effective way for students to learn new concepts and skills. By providing them with examples and real-world situations, they can see how the theory behind a concept or skill works in practice. In addition, students are given practice with each new concept or skill taught, before moving on to the next topic. This them learn and retain the information better.

What has been your experience with adopting a problem-solving teaching method?

How do you feel the usefulness of your lesson plans changed since adopting this method?

What was one of your most successful attempts in using this technique to teach students, and why do you believe it was so successful?

Were there any obstacles when trying to incorporate this technique into your class?

Did it take a while for all students to get used to the new type of teaching style before they felt comfortable enough to participate in discussions and ask questions about their newly acquired knowledge?

What are your thoughts on this method?

“I have had the opportunity to work in several districts, including one where they used problem solving for all subjects. I never looked back after that experience--it was exciting and motivating for students and teachers alike."

"The problem solving method of teaching is great because it makes my subject matter more interesting with hands-on activities."

Active Learning, Teaching through problem-solving allows for active learning, Children understand the theory better by getting involved in real-world situations, Practice, Continuous practice is integral to problem-solving teaching, Each new skill or concept is practiced after being learned in class, Relevance, Problem-solving techniques make learning more relevant, Real-world examples related to the topic are presented, Incremental Learning, Each new topic builds on previous lessons, Relating new problems to ones solved in previous sessions, Overcome Challenges, Enhances ability to overcome real-world situations, Children understand the application of skills learned, Variety, Problem-solving allows flexibility in teaching methods, Problems can be practical, conceptual, or theoretical, Critical Thinking, Improves children's critical thinking skills, Adding alternative paths to a solution, Confidence, Boosts children's confidence in handling problems, Children feel empowered after successfully solving a problem, Adaptability, Increases adaptability to new learning situations, Children can apply learned strategies to new problems, Engagement, Problem-solving increases engagement and interest, Children find solving real-world examples interesting

What is the role of educators in facilitating problem-solving method of teaching?

Role of Educators in Facilitating Problem-Solving Understanding the Problem-Solving Method The problem-solving method of teaching encourages students to actively engage their critical thinking skills to analyze and seek solutions to real-world problems. As such, educators play a crucial part in facilitating this learning style to ensure the effective attainment of desired skills. Encouraging Collaboration and Communication One of the ways educators can facilitate problem-solving is by promoting collaboration and communication among students. Working as a team allows students to share diverse perspectives while considering multiple solutions, thereby fostering an open-minded and inclusive environment that is crucial for effective problem-solving. Creating a Safe Space for Failure Educators must recognize that failure is an integral component of the learning process in a problem-solving method. By establishing a safe environment that allows students to fail without facing judgment or embarrassment, teachers enable students to develop perseverance, resilience, and an enhanced ability to learn from mistakes. Designing Relevant and Engaging Problems The selection and design of appropriate problems contribute significantly to the success of the problem-solving method of teaching. Educators should focus on presenting issues that are relevant, engaging, and age-appropriate, thereby sparking curiosity and interest amongst students, which further improves their problem-solving abilities. Scaffolding Learning Scaffolding is essential in the problem-solving method for providing adequate support when required. Teachers need to break down complex problems into smaller, manageable steps, and gradually remove support as students develop the necessary skills, thus promoting their self-reliance and independent thinking. Providing Constructive Feedback Constructive feedback from educators is invaluable in facilitating the problem-solving method of teaching, as it enables students to reflect on their progress, recognize areas for improvement, and actively develop their critical thinking and problem-solving abilities. In conclusion, the role of educators in facilitating the problem-solving method of teaching comprises promoting collaboration, creating a safe space for failure, designing relevant problems, scaffolding learning, and providing constructive feedback. By integrating these elements, educators can help students develop essential life-long skills and effectively navigate the complex world they will experience.

The problem-solving method of teaching is a dynamic and interactive instructional strategy that engages students directly with challenges that resemble those they might encounter outside of the classroom. Within this framework, educators are not just conveyors of knowledge, but rather facilitators of learning who empower their students to think critically and deeply. Below, we look into the nuanced role educators play in making the problem-solving method impactful.Firstly, educators must curate an atmosphere that is conducive to inquiry and exploration. They set the tone by modeling an inquisitive mindset, posing thought-provoking questions, and encouraging students to ask why, how, and what if without hesitation. This intellectual curiosity promotes the kind of deep thinking that underpins successful problem-solving.Another key responsibility is to scaffold the complexity of problems. Educators do so by assessing the readiness of their students and designing tasks that are at the appropriate level of difficulty. They must ensure challenges are neither too easy – risking boredom and disengagement – nor too difficult – potentially causing frustration and disheartenment. By striking this balance, educators help students to experience incremental success and build their problem-solving capacities over time.Educators must also provide students with relevant tools and methodologies. This might involve teaching specific problem-solving strategies such as the scientific method, design thinking, or computational thinking. Educators help students to become conversant in these approaches, allowing them to tackle problems methodically and effectively.Assessment is another pivotal area where educators play a vital role in the problem-solving method. The traditional means of assessment may not always capture the depth of understanding and learning that occurs in problem-solving scenarios. Therefore, educators develop alternative forms of assessment, such as reflective journals, portfolios, and presentations, to better gauge student learning and thinking processes.Finally, educators must be adept at facilitating group dynamics. Collaborative problem-solving can be powerful, but it also invites a range of interpersonal challenges. Thus, educators need to guide students in conflict resolution, equitable participation, and recognizing the contribution of each member to the collective effort.Educators facilitate the problem-solving method by fostering inquiry, balancing problem difficulty, equipping students with methodologies, rethinking assessment, and nurturing group cooperation. In doing so, they are not simply providing students with content knowledge but are equipping them with crucial life skills that transcend educational settings and prepare them for real-world challenges.

Can interdisciplinary approaches be incorporated into problem-solving teaching methods, and if so, how?

Interdisciplinary Approaches in Problem-Solving Teaching Methods Integration of Interdisciplinary Approaches Incorporating interdisciplinary approaches into problem-solving teaching methods can be achieved by integrating various subject areas when presenting complex problems that require students to draw from different fields of knowledge. By doing so, learners will develop a deeper understanding of the interconnectedness of various disciplines and improve their problem-solving skills. Project-Based Learning Activities Implementing project-based learning activities in the classroom allows students to work collaboratively on real-world problems. By involving learners in tasks that necessitate the integration of diverse subjects, they develop the ability to transfer skills acquired in one context to novel situations, thereby expanding their problem-solving abilities. Role of Teachers in Interdisciplinary Teaching Teachers play a crucial role in the successful incorporation of interdisciplinary methods in problem-solving teaching. They must be prepared to facilitate student-centered learning and engage in ongoing professional development tailored towards interdisciplinary education. In doing so, educators can create inclusive learning environments that encourage individualized discovery and the application of diverse perspectives to solve complex problems. Benefits of Interdisciplinary Teaching Methods Adopting interdisciplinary teaching methods in problem-solving education not only enhances students' problem-solving abilities but also fosters the development of critical thinking, creativity, and collaboration. These essential skills enable learners to navigate and adapt to an increasingly interconnected world and have been shown to contribute to students' academic and professional success. In conclusion, incorporating interdisciplinary approaches into problem-solving teaching methods can be achieved through the integration of various subject areas, implementing project-based learning activities, and the active role of teachers in interdisciplinary education. These methods benefit students by developing problem-solving skills, critical thinking, creativity, and collaboration, preparing them for future success in an interconnected world.

Interdisciplinary approaches in problem-solving teaching methods present a contemporary framework for preparing students to tackle the complexities of real-world issues. This approach can bridge the gap between various academic disciplines, offering students a more holistic and connected way of thinking.**Embracing Complexity through Interdisciplinary Problem-Solving**Problem-solving in education is no longer confined to single-subject exercises. Interdisciplinary problem-solving recognizes the multifaceted nature of real issues and encourages students to tackle them by drawing from multiple disciplines. For instance, when examining the impacts of urbanization, students might incorporate knowledge from sociology, economics, environmental science, and urban planning.**Strategies for Implementing an Interdisciplinary Approach**Various strategies can be employed to incorporate interdisciplinary methods effectively:1. **Cross-Curricular Projects**: These require students to apply knowledge and skills across different subject areas, fostering an understanding of each discipline’s unique contribution to the whole problem.2. **Thematic Units**: By designing units around broad themes, educators can seamlessly weave multiple subjects into the exploration of a single topic, prompting students to see connections between different areas of study.3. **Collaborative Teaching**: When educators from different disciplines co-teach, they can provide a combined perspective that enriches the learning experience and demonstrates the value of integrating knowledge.4. **Inquiry-Based Learning**: Encourages students to ask questions and conduct research across multiple disciplines, leading to comprehensive investigations and solutions.**Outcome-Benefits of Interdisciplinary Teaching**The merits of an interdisciplinary approach within problem-solving teaching methods are manifold:1. **Complex Problem Understanding**: It can elevate a student’s ability to deconstruct complicated issues by understanding various factors and viewpoints.2. **Adaptability**: Students learn to apply knowledge pragmatically, enabling them to adapt to new and unforeseen problems.3. **Enhanced Cognitive Abilities**: The process can promote cognitive growth, supporting the development of higher-order thinking skills like analysis and synthesis.4. **Real-World Relevance**: Students find meaning and motivation in their work when they see its relevance outside the classroom walls.In summary, integrating interdisciplinary approaches into problem-solving methods is a highly effective way to provide students with robust and adaptable skills for the future. By engaging in project-based learning activities, enjoying the support of proactive educators, and seeing the interconnectivity across subjects, students can foster critical thinking, creativity, and collaborative abilities that transcend traditional learning boundaries. As we navigate a rapidly evolving and interrelated global landscape, such approaches to education become not just advantageous but essential.

In what ways can technology be integrated into the problem-solving method of instruction?

**Role of Technology in Problem-Solving Instruction** Technology can be integrated into the problem-solving method of instruction by enhancing student engagement, promoting collaboration, and supporting personalized learning. **Enhancing Student Engagement** One way technology supports the problem-solving method is by increasing students' interest through interactive and dynamic tools. For instance, digital simulations and educational games can help students develop critical thinking and problem-solving skills in a fun, engaging manner. These tools provide real-world contexts and immediate feedback, allowing students to experiment, take risks, and learn from their mistakes. **Promoting Collaboration** Technology also promotes collaboration among students, as online platforms facilitate communication and cooperation. Utilizing tools like video conferencing and shared workspaces, students can collaborate on group projects, discuss ideas, and solve problems together. This collaborative approach fosters a sense of community, mutual support, and collective problem-solving. Moreover, it helps students develop essential interpersonal skills, such as teamwork and communication, which are crucial in today's workplaces. **Supporting Personalized Learning** Finally, technology can be used to provide personalized learning experiences tailored to individual learners' needs, interests, and abilities. With access to adaptive learning platforms or online resources, students can progress at their own pace, focus on areas where they need improvement, and explore topics that interest them. This kind of personalized approach allows instructors to identify areas where students struggle and offer targeted support, enhancing the problem-solving learning experience. In conclusion, integrating technology into the problem-solving method of instruction can improve the learning process in various ways. By fostering student engagement, promoting collaboration, and facilitating personalized learning experiences, technology can be employed as a valuable resource to develop students' problem-solving skills effectively.

The integration of technology into the problem-solving method of instruction can significantly enhance the educational process, as it offers diverse opportunities for students to engage with challenging concepts and develop practical skills. The deliberate use of technology can stimulate student interaction with course material and encourage a more dynamic approach to learning.**Interactive Problem-Solving Scenarios**Technology can simulate complex scenarios requiring students to apply their knowledge creatively to solve problems. Through interactive case studies and gamified learning environments, students can engage with these scenarios in a manner that is both compelling and educative. Such simulations often incorporate branching choices, offering an exploration of consequences which creates a deeper understanding of the material.**Data Analysis Tools**Incorporating data analysis tools into problem-solving instruction can offer students hands-on experience with real-world data sets. By learning to manipulate and analyze data through software, students can identify patterns, test hypotheses, and make evidence-based conclusions. These skills are particularly valuable in STEM fields, economics, and social sciences.**Global Connectivity & Resources**Through global connectivity, technology enables access to a vast array of resources that can be utilized to enrich problem-solving tasks. Platforms such as IIENSTITU offer courses that are designed to incorporate technology into pedagogical strategies effectively. Moreover, access to international databases, research materials, and expert lectures from around the world ensures that students are exposed to diverse perspectives and approaches to problem-solving.**Interactive Whiteboards and Projection**Interactive whiteboards and projection technology make it possible to visualize complex problems and work though them interactively in the classroom. This technology allows for collaborative diagramming and mapping of ideas, which can aid in visual learning and the synthesis of information in group settings.**Adaptive Learning Software**Educational technology that adapts to individual student performance and preferences enables personalized instruction. Adaptive learning software assesses students' skills and tailors the difficulty of problems accordingly, ensuring that each student is engaged at the appropriate level of challenge.**Formative Assessment through Technology**Technology-enabled formative assessments give teachers and students real-time feedback on understanding and performance. These tools can help identify areas of difficulty, track progress, and adjust teaching strategies to help students develop their problem-solving abilities more effectively.**Facilitating Research and Inquiry**The ability to conduct research and inquiry is central to problem solving. When students are provided with the tools to explore, research, and verify information on the internet securely, they are empowered to seek out answers to their questions and develop solutions based on evidence.**Closing Thoughts**In integrating technology into problem-solving instruction, it's important to ensure that the use of any tool or platform is pedagogically sound, enhances the learning objectives, and actually serves to improve students' problem-solving capabilities. As education evolves with the digital age, so too does the art and science of teaching problem solving, where technology becomes an indispensable ally in preparing students for the challenges of the future.

I graduated from the Family and Consumption Sciences Department at Hacettepe University. I hold certificates in blogging and personnel management. I have a Master's degree in English and have lived in the US for three years.

A rectangular puzzle piece with a light green background and a blue geometric pattern sits in the center of the image. The puzzle piece has a curved edge along the top, and straight edges along the bottom and sides. The pattern on the piece consists of a thin green line that wraps around the outside edge and a thick blue line that follows the contours of the shape. The inside of the piece is filled with various shapes of the same color, including circles, triangles, and squares. The overall effect of the piece is calming and serene. It could be part of a larger puzzle that has yet to be solved.

What are Problem Solving Skills?

A woman in a white shirt is looking down and holding her head in her hands. She has long blonde hair and blue eyes. Her lips are slightly pursed, and her eyebrows are slightly furrowed. She looks sad and contemplative, as if she is lost in thought. Her arms are crossed in front of her chest, and her head is slightly tilted to the side. Her expression is thoughtful and her posture is relaxed. She is standing in front of a plain white wall, and the light casts shadows on her face. She appears to be alone in the room, and her posture conveys a sense of loneliness and introspection.

How To Solve The Problems? Practical Problem Solving Skills

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A Problem Solving Method: Brainstorming

A close-up of a group of people holding puzzle pieces in their hands. A man is looking at the piece he is holding, while two other people are carefully looking at the pieces they are holding in their hands. The pieces have a wooden texture, and each one is a different color. One person is holding a light blue piece, while another person is holding a red piece. All the pieces are shaped differently, and some are curved while others are straight. The pieces all fit together to form a larger puzzle.

How To Develop Problem Solving Skills?

3 Ways to Strengthen Math Instruction

Students’ math scores have plummeted, national assessments show , and educators are working hard to turn math outcomes around.

But it’s a challenge, made harder by factors like math anxiety , students’ feelings of deep ambivalence about how math is taught, and learning gaps that were exacerbated by the pandemic’s disruption of schools.

This week, three educators offered solutions on how districts can turn around poor math scores in a conversation moderated by Peter DeWitt, an opinion blogger for Education Week.

Here are three takeaways from the discussion. For more, watch the recording on demand .

1. Intervention is key

Research shows that early math skills are a key predictor of later academic success.

“Children who know more do better, and math is cumulative—so if you don’t grasp some of the earlier concepts, math gets increasingly harder,” said Nancy Jordan, a professor of education at the University of Delaware.

For example, many students struggle with the concept of fractions, she said. Her research has found that by 6th grade, some students still don’t really understand what a fraction is, which makes it harder for them to master more advanced concepts, like adding or subtracting fractions with unlike denominators.

At that point, though, teachers don’t always have the time in class to re-teach those basic or fundamental concepts, she said, which is why targeted intervention is so important.

Conceptual photo of of a young boy studying mathematics using fingers in primary school.

Still, Jordan’s research revealed that in some middle schools, intervention time is not a priority: “If there’s an assembly, or if there is a special event or whatever, it takes place during intervention time,” she said. “Or ... the children might sit on computers, and they’re not getting any really explicit instruction.”

2. ‘Gamify’ math class

Students today need new modes of instruction that meet them where they are, said Gerilyn Williams, a math teacher at Pinelands Regional Junior High School in Little Egg Harbor Township, N.J.

“Most of them learn through things like TikTok or YouTube videos,” she said. “They like to play games, they like to interact. So how can I bring those same attributes into my lesson?”

Part of her solution is gamifying instruction. Williams avoids worksheets. Instead, she provides opportunities for students to practice skills that incorporate elements of game design.

That includes digital tools, which provide students with the instant feedback they crave, she said.

But not all the games are digital. Williams’ students sometimes play “trashketball,” a game in which they work in teams to answer math questions. If they get the question right, they can crumble the piece of paper and throw it into a trash can from across the room.

“The kids love this,” she said.

Gerilyn Williams, a middle school math teacher in New Jersey, stands in her classroom.

Williams also incorporates game-based vocabulary into her instruction, drawing on terms from video games.

For example, “instead of calling them quizzes and tests, I call them boss battles,” she said. “It’s less frightening. It reduces that math anxiety, and it makes them more engaging.

“We normalize things like failure, because when they play video games, think about what they’re doing,” Williams continued. “They fail—they try again and again and again and again until they achieve success.”

3. Strengthen teacher expertise

To turn around math outcomes, districts need to invest in teacher professional development and curriculum support, said Chaunté Garrett, the CEO of ELLE Education, which partners with schools and districts to support student learning.

“You’re not going to be able to replace the value of a well-supported and well-equipped mathematics teacher,” she said. “We also want to make sure that that teacher has a math curriculum that’s grounded in the standards and conceptually based.”

Students will develop more critical thinking skills and better understand math concepts if teachers are able to relate instruction to real life, Garrett said—so that “kids have relationships that they can pull on, and math has some type of meaning and context to them outside of just numbers and procedures.”

Tonya Clarke, coordinator of K–12 mathematics in the division of school leadership and improvement for Clayton County Public Schools in Jonesboro, Ga., in the hallway at Adamson Middle School.

It’s important for math curriculum to be both culturally responsive and relevant, she added. And teachers might need training on how to offer opportunities for students to analyze and solve real-world problems.

“So often, [in math problems], we want to go back to soccer and basketball and all of those things that we lived through, and it’s not that [current students] don’t enjoy those, but our students live social media—they literally live it,” Garrett said. “Those are the things that have to live out in classrooms right now, and if we’re not doing those things, we are doing a disservice.”

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ORIGINAL RESEARCH article

This article is part of the research topic.

Invention Education and STEM: Perspectives and Possibilities

Systematic Review of Invention Education Research Landscape: State of the Discipline and Future Directions Provisionally Accepted

1 Saline High School, United States
2 Eastern Michigan University, United States

The final, formatted version of the article will be published soon.

Invention and innovation education and its associated practices (e.g., problem-finding, problem-defining, learning from failure, iterative problem-solving, innovation-focused curricula, collaboration, and maker spaces) are moving from the periphery to the center of education at an ever-increasing pace. Although the research and literature on invention and innovation education, collectively termed as Invention Education (IvE) in this research, is on the rise, to our knowledge no attempt has been made to systematically review the literature available on the topic. To address this gap, we identify, collect, and systematically review scientific literature on IvE. We conduct Bibliometrix-based and targeted analysis to identify the topics, sources, authors, and articles most cited, as well as prominent countries publishing IvE literature. Another objective of this research is to uncover the intellectual, conceptual, and social structures of IvE. A third objective is to identify the progress made and the challenges being faced in furthering IvE and propose future directions. Our review shows that the field has seen substantial growth, especially in recent years particularly in the USA. Research shows IvE’s importance in nurturing a well-rounded, innovative, and skilled future workforce, emphasizing creativity, critical thinking, collaboration, adaptability, and problem-solving skills. Although with a plethora of curricula and K-20 programs in USA, followed by South Korea, and China, IvE lacks unifying conceptualization, definitions and frameworks. The lack of commonly accepted terms and theoretical bases, and difficulties integrating invention into STEM coursework, are compounded by barriers like resource limitations, curriculum constraints, and the need for teacher training and support. The review underscores the need for IvE to address and dismantle inventor stereotypes and cultivate a diverse and inclusive generation of innovators. It points to the impact of gender and stereotypes on participation in IvE programs and the importance of promoting equity and access to IvE opportunities for all students. The article concludes with a discussion of challenges and future research directions to address them.

Keywords: Invention education, Innovation education, Systematic Literature Review, Bibliometrix, problem-solving

Received: 28 Aug 2023; Accepted: 01 May 2024.

Copyright: © 2024 Dalela and Ahmed. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY) . The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

* Correspondence: Miss. Suhani L. Dalela, Saline High School, Saline, 48176, Michigan, United States

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Teaching Methods and Learning Activities

Teaching strategies and activities that accommodate a range of learning styles, interests, and skill levels must be used to engage and motivate students in the learning process. I use the following four powerful teaching techniques to get students interested and motivated in my class:

1. Project-Based Learning (PBL) Project-Based Learning is an engaging teaching method where students actively investigate real-world issues and challenges. By doing so, they gain a deeper understanding of the subject matter. It promotes teamwork, critical thinking, and original problem-solving among pupils.

2. Gamification Gamification is the process of applying aspects of game design to non-gaming contexts, such as education, in order to increase involvement, commitment, and participation. A few examples of these could be point scoring, multiplayer competition, game regulations, and prizes.

3. Inquiry-Based Learning

Inquiry-Based Learning begins with asking questions, creating issues, or presenting scenarios instead than just giving predetermined answers or a clear cut route to knowledge. Through inquiry and probing, the method inspires children to look into, investigate, and find answers.

4 Implementation and impact

Because students have different requirements and experiences, each of these methods and activities requires careful planning and flexibility in implementation. Teachers may build a dynamic and inclusive learning environment that not only engages and encourages students but also cultivates a lifetime love of learning by utilizing a combination of these tactics.

Reason for Selecting PBL (Project-Based Learning)

PBL was selected because it provides an all-encompassing and student-centered method of instruction. Constructivist theory, which holds that students actively create their own knowledge via interactions and experiences, is in line with PBL. PBL encourages critical thinking, teamwork, and problem-solving abilities in students by exposing them to real-world issues or difficulties. By making learning relevant and meaningful to students’ daily lives, this approach encourages pupils to learn on an intrinsic level.

Why Gamification Was Selected:

Gamification was selected in order to take use of the game design’s motivational elements in an educational setting. Gamification, in accordance with behaviorist theory, uses positive reinforcement mechanisms—like points, awards, and achievements—to encourage and reinforce desired actions. According to the theory that motivated students are more likely to participate actively in the learning process, this approach turns learning into an interesting and dynamic experience.

Motivation from Within via Rewards:

Positive reinforcement bolsters behavior, according to operant conditioning. Associating learning with favorable results, gamification employs incentives such as points, badges, or leaderboard positions to promote good learning practices and so stimulate intrinsic motivation.

Feeling of Achievement:

Completing tasks or levels in video games frequently leaves players with a strong sense of accomplishment. The idea that pupils become motivated when they feel like they have accomplished something is supported by the achievement motivation hypothesis. The integration of these components through gamification fosters an enjoyable educational process.

Inquiry-Based Learning

Investigative-Based The purpose of education was to foster curiosity, critical thinking, and self-sufficient problem-solving abilities. Lee et al. (2004) defined inquiry-based learning as an “array of classroom practices that promote student learning through guided and, increasingly, independent investigation of complex questions and problems, often for which there is no single answer” (p. 9). This approach is consistent with the cognitive constructivist paradigm, which emphasizes that learning is an active mental process in which people build their own knowledge via introspection and investigation.

Application and Impact:

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Nunez Community College’s STEAM Division will offer seven single- and multi-day educational camps for grades K-12 during June and July 2024. All camps will take place on the Nunez campus in Chalmette.

Nunez Community College’s STEAM Division has a full schedule of short camps on tap for the summer to keep kids’ minds sharp and their hands busy. Science, Technology, Engineering, Arts and Mathematics (STEAM) at Nunez will cover everything from robotics to artificial intelligence to space physics over the course of eight consecutive camps. Students in every grade from kindergarten through 12th will get an opportunity to learn about cutting edge technology delivered via age-appropriate lessons and hands-on activities in the safe environment of the Nunez campus in Chalmette.

“Our camps are designed to engage students with hands-on activities that promote critical thinking and problem solving in a fun and inviting environment,” said Nunez STEAM Director Dr. Julie Rexford.

Sign your child up for a STEAM camp at Nunez.edu/STEAM . Keep up with the program and find camp flyers with more information on the STEAM at Nunez Facebook page.

The full schedule of STEAM Summer Camps includes:

Robotics Workshop – Grades 4-8, June 6th, $60 Students in grades 4-8 will participate in various missions and challenges as they learn to build and program the Lego Spike Prime robot. This camp will teach the fundamentals of fabrication and coding while building the student's problem solving and critical thinking skills. This is a one-day camp. Students will bring their own lunch.

CEEF (Consumer Energy Education Foundation) AI Energy – Grades 4-8, June 10-14, $300 Learn about Artificial Intelligence and how it relates to the Energy sector through hands on activities.

Careers in STEM – Grades 4-8, June 17-21, $300 This camp is designed for students to explore various fields of STEM through hands on activities.

NASA Astro Camp – Grades 3-6, July 8-12, $300 Students will participate in various hands-on activities related to space and space exploration. This is a full-week camp for students in grades 3-6. Students will bring their own lunch.

Cybersecurity Camp – Grades 9-12, July 15-19, $300 Students will be introduced to the various aspects of Cybersecurity through hands on activities. This camp is designed for students in high school who are interested in cybersecurity or information technology. Students will bring their own lunch.

Robotics Camp – Grades 3-7, July 22-26, $300 Students in Grades 3-7 will learn to fabricate and program a Lego Spike Prime robot. Students will then modify their designs to accomplish various challenges. This is a one-week workshop - students will provide their own lunch and water bottle.

Robotics Workshop – Grades K-2, July 30, $60 Students will be introduced to the world of robotics including fabrication and programming.

About Nunez Community College Nunez Community College is a student-centered institution that delivers relevant and innovative curriculum integrating the arts, sciences, and humanities leading to academic credentials and workforce opportunities. Nunez serves a vital role in the community by engaging with partners to support student success and personal growth. Nunez Community College is accredited by the Southern Association of Colleges and Schools Commission on Colleges (SACSCOC) to award associate degrees, technical diplomas, and certificates. Degree-granting institutions also may offer credentials such as certificates and diplomas at approved degree levels. Questions about the accreditation of Nunez Community College may be directed in writing to the Southern Association of Colleges and Schools Commission on Colleges at 1866 Southern Lane, Decatur, GA 30033-4097, by calling (404) 679-4500, or by using information available on SACSCOC’s website ( www.sacscoc.org ). Nunez is located at 3710 Paris Road, Chalmette, LA. For more information, visit www.nunez.edu .

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Solution of engineering design and truss topology problems with improved forensic-based investigation algorithm based on dynamic oppositional based learning

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Published: 02 May 2024

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Funda Kutlu Onay ORCID: orcid.org/0000-0002-8531-4054 1

The forensic-based investigation (FBI) is a metaheuristic algorithm inspired by the criminal investigation process. The collaborative efforts of the investigation and pursuit teams demonstrate the FBI’s involvement during the exploitation and exploration phases. When choosing the promising population, the FBI algorithm’s population selection technique focuses on the same region. This research aims to propose a dynamic population selection method for the original FBI and thereby enhance its convergence performance. To achieve this objective, the FBI may employ dynamic oppositional learning (DOL), a dynamic version of the oppositional learning methodology, to dynamically navigate to local minima in various locations. Therefore, the proposed advanced method is named DOLFBI. The performance of DOLFBI on the CEC2019 and CEC2022 benchmark functions is evaluated by comparing it with several other popular metaheuristics in the literature. As a result, DOLFBI yielded the lowest fitness value in 18 of 22 benchmark problems. Furthermore, DOLFBI has shown promising results in solving real-world engineering problems. It can be argued that DOLFBI exhibits the best convergence performance in cantilever beam design, speed reducer, and tension/compression problems. DOLFBI is often utilized in truss engineering difficulties to determine the minimal weight. Its success is comparable to other competitive MAs in the literature. The Wilcoxon signed-rank and Friedman rank tests further confirmed the study’s stability. Convergence and trajectory analyses validate the superior convergence concept of the proposed method. When the proposed study is compared to essential and enhanced MAs, the results show that DOLFBI has a competitive framework for addressing complex optimization problems due to its robust convergence ability compared to other optimization techniques. As a result, DOLFBI is expected to achieve significant success in various optimization challenges, feature selection, and other complex engineering or real-world problems.

Avoid common mistakes on your manuscript.

1 Introduction

Converging to optimal results for practical and complex optimization problems is a common challenge in real-world problems. In particular, complex and difficult-to-converge engineering and mathematical problems have led to the development of various optimization techniques. Various derivative-based methods are used in the optimization of mathematical equations. Among these methods are Newton-based approach [ 1 ], Broyden–Fletcher–Goldfarb–Shanno Algorithm [ 2 ], Adadelta [ 3 ], AdaGrad [ 4 ], etc. On the other hand, derivative-independent optimization techniques such as population-based, sequential model-based, local optimization hill climbing, and global optimization [ 5 ]. Metaheuristic algorithms (MAs), typically designed and implemented based on population dynamics, yield effective solutions for mathematical and real-world problems. MAs have a rapid and effective convergence strategy due to their ability to handle non-convex problems and their structure, which does not rely on derivatives.

The inspiration of MAs from natural and mathematical phenomena leads to increased cases due to randomness. However, metaheuristic techniques illuminate many issues related to effective search and convergence methods. MAs progress through two stages: exploitation and exploration. The balance of these two phases is required for a metaheuristic technique. If this balance is not maintained, either the local search becomes excessively practical and misses the global optimum point, or the global search becomes excessively practical. Even if the global optimal zone is found, it may not be converged to the local optimum point [ 6 ].

There are four types of MAs: evolution-based, swarm-based, physics/mathematics-based, and human-based. Evolutionary algorithms simulate natural selection and genetic crossover processes. This category includes algorithms such as genetic algorithms (GA) [ 7 ], evolution strategies (ES) [ 8 ], and genetic programming (GP) [ 9 ]. Swarm-based algorithms solve optimization problems by mimicking the behavior of living organisms that naturally move in groups. Algorithms such as particle swarm optimization (PSO) [ 10 , 11 , 12 ], ant colony optimization (ACO) [ 13 ], and artificial bee colony (ABC) [ 14 ] are evaluated in this group. Physics and mathematics-based algorithms are designed to solve problems that require optimization using principles from physics and mathematics. Examples of some physics and mathematics-based metaheuristic algorithms include simulated annealing (SA) [ 15 ], gravitational search (GSA) [ 16 ], and black hole algorithm (BH) [ 17 ]. Human-based metaheuristic algorithms are algorithms inspired by human behavior or interactions. Each human-based algorithm utilizes human social skills to explore and enhance solutions, mimicking various human interactions and sources of information. Each human-based algorithm uses human social skills to examine and improve solutions while mimicking different human interactions or sources of information. Tabu Search (TS) [ 18 ], teaching learning-based optimization (TLB) [ 19 ], and forensic-based investigation algorithm (FBI) [ 20 ] improved in this study are also considered under this category.

MAs may differ in their application areas, performance, advantages, and disadvantages. At this point, MA development, enhancement, and hybridization have recently gained significant attention in the literature as crucial subjects for achieving more effective and efficient optimization solutions. New methods can be proposed by integrating techniques that involve parameter adjustments, changing operators, selection strategies, elitism, and local search, and by enhancing population distribution within the framework of metaheuristic algorithms. Oppositional based learning (OBL) [ 21 ] is a learning paradigm and one of the approaches utilized for improvement. OBL can be used in artificial intelligence to address problems like data mining, classification, prediction, and pattern recognition. OBL improves learning by utilizing the conflicting characteristics of two opposed notions. This situation develops as a means for the metaheuristic algorithm to improve efficiency and performance in reaching the ideal result across multiple solution spaces by picking diverse populations. Dynamic oppositional based learning (DOL) [ 22 ] is a dynamically adapted version of the OBL. In contrast to OBL, DOL seeks to improve results by making the process more flexible and adaptive. With DOL, reacting to changing conditions more efficiently and getting the best results thus far may be feasible. DOL can update features dynamically to meet shifting data distributions and evolving features over time. When new data is added, or old data is discarded, it can recalculate and optimize conflicting notions, emphasizing DOL’s adaptable nature.

The following are some of the uses of the OBL and its derivatives in metaheuristic and usage areas: Balande and Shrimankar created the OBL learning paradigm in collaboration with TLB, a human-based metaheuristic for optimizing the permutation flow-shop scheduling problem [ 23 ]. Izci et al. enhanced the arithmetic optimization algorithm with modified OBL (mOBL-AOA) [ 24 ] and applied it to benchmarks. Elaziz et al. used DOL with atomic orbit search (AOSD) for the feature selection problems [ 25 ]. Sharma et al. introduced DOL-based bald eagle search for global optimization issues, naming it self-adaptive bald eagle search (SABES). Shahrouzi et al. proposed static and dynamic OBL with colliding bodies optimization, a robust optimization technique tested using global optimization benchmarks in numerous engineering applications [ 26 ]. Khaire et al. integrated the OBL and sailfish optimization algorithm to identify the prominent features from a high-dimensional dataset [ 27 ]. Wang et al. have proposed hybrid aquila optimizer and artificial rabbits optimization algorithms with dynamic chaotic OBL (CHAOARO) for some engineering problems [ 28 ]. Yildiz et al. utilized a hybrid flow direction optimizer-dynamic OBL for constrained mechanical design problems [ 29 ].

Although the algorithms stated above are widely employed in numerous sectors in the literature, new methods for MAs are continually being presented. This is because not all optimization issues can be solved by a metaheuristic method. The no free lunch (NFL) theorem provides additional evidence for this [ 30 ]. As a result, when a novel approach is proposed, it is validated against real-world issues and mathematical benchmark functions. This study proposes using the DOL paradigm to create the FBI algorithm. The FBI algorithm includes stages for investigation and pursuit. Here, two distinct demographic groupings are involved in two stages. The goal of the investigation phase is to locate where suspicion is most likely to exist. The uncertainty probability is computed to ascertain this. The location indexes calculated in the original FBI are chosen randomly. In DOLFBI, however, the DOL updated the location and relayed to the chase team. Similar activities are carried out during the pursuit stage, and the location information collected by DOL is transmitted back to the investigation stage. This method is repeated until the optimal location is determined using the probability values.

The primary motivations and contributions of this paper are given as follows:

The present work suggests an enhanced algorithm for forensic-based learning through the use of oppositional based learning (DOLFBI).

The DOL paradigm has been added to the FBI for opposite population selection. The study proposes improving the FBI algorithm’s random population selection process by applying opposite integers because opposed numbers narrow the search space, allowing for more effective scanning and faster convergence.

Benchmark suites (CEC2019 and CEC2022), engineering, and truss topology problems are all being used to evaluate convergence capabilities. In particular, truss topology problems are significant and complex optimization problems specific to civil engineering. In other words, the goal of the truss topology problem is to minimize the weight of distinct structural components produced from different node numbers to determine the optimal sections.

The proposed method has been tested with metaheuristics commonly used in the literature, and its convergence ability has been studied using average and best findings. Furthermore, the Wilcoxon sign and Friedman rank tests were used to validate the results.

The article’s organization is as follows: Sect. 2.1 and their subsections have the working principle, algorithmic structure, and mathematical model of the FBI algorithm. Section 2.2 discusses the DOL paradigm and its impact and added value. Section 3 consists of the working flow of the FBI algorithm developed with DOL and the pseudo-codes of the proposed method DOLFBI. Section 4 includes parameter settings of the proposed and compared algorithms, properties, and experimental results of the CEC2019 and CEC2022 benchmarks, well-known engineering problems, and truss topology optimization problems, supporting the results with statistical tests. Sect. 5 includes an overview of the study and future objectives.

2 Materials and methods

The basic FBI algorithm and the DOL principle, which form the basis of the proposed DOLFBI algorithm, will be explored in detail under subheadings in this section.

2.1 Forensic-based investigation algorithm

This section describes the forensic investigation procedure, the details of the suggested algorithm and its mathematical model. A visual of the investigation process and a flow diagram of the FBI algorithm’s operation are presented for further clarity.

2.1.1 Steps of the forensic investigation process

Forensic investigation is one of the most risky tasks in which law enforcement is frequently involved for any country. For each research, a different path may be required. In some cases, the incident immediately enters the suspect management phase, and in others, the criminal’s name may be disclosed due to multiple investigations. However, research activities tend to be comparable [ 20 ].

Salet [ 31 ] stated that a large-scale forensic investigation by police officers consists of five steps, and Fig. 1 illustrates these five steps. Steps 2, 3 and 4 are defined as a cyclical process.

The phases of the investigation process

These phases, along with their explanations, are outlined as follows:

Open a case: The information discovered by the first police officers on the scene launches the investigation. Team members look into the crime scene, the victim, potential suspects, and background information. They also locate and question potential witnesses.

Interpretation of findings: Team members attempt to gain an overview of all accessible information. The team attempts to connect this knowledge to their present situation perception.

Direction of inquiry: This is the stage at which team members construct distinct hypotheses based on their Interpretation of the findings. Based on these findings, the team approves, modifies, or terminates a new direction or current research recommendations.

Actions: In light of the lines of research and the determined priorities, the team makes decisions on other actions. Priorities are vital at this step, and the most promising research direction is explored first. As a result, new information may be presented, and the team assesses its meaning or implications based on the information available. Changes in research and action may be required to interpret new findings.

Prosecution: This process is repeated until a clear and unambiguous picture of the incident is obtained. It ends when a serious suspect is identified before determining whether or not to prosecute them.

There are no hard and fast rules governing the number of police officers involved in an investigation, and this number is frequently tied to the gravity, difficulty, and complexity of the case.

2.1.2 Mathematical model of the algorithm

The FBI algorithm is a human-based metaheuristic algorithm inspired by police officers’ forensic investigation processes. An investigation can be launched after receiving notification of criminal activity. The investigation involves identifying physical evidence, acquiring information, collecting and preserving evidence, and questioning and interrogating witnesses and suspects [ 32 ]. Based on the information gathered and witness declarations, all probable suspects within the search area are identified, and likely locations are determined. The FBI makes two assumptions: there is a single most sought suspect in an incident, and that individual remains in hiding throughout the investigation. The process results in the capture and arrest of the suspect.

An investigation team is organized to investigate "suspicious places," prospective hiding places for the suspect. After the investigative team has determined the most likely location, a search area is established, and a pursuit team is assembled. All tracking team members travel to the indicated site with team members capable of apprehending the suspect.

The pursuit team proceeds toward the suspected location following the head office’s orders and reports all information regarding the suspected location. The investigation and pursuit teams collaborate closely throughout the investigate-find-approach process. The investigation team directs the tracking team to approach the spots. By periodically reporting the findings of their searches, the pursuing team hopes to update the information and maximize the accuracy of future evaluations.

The algorithm includes two major stages: the investigation stage (Phase A) and the pursuit stage (Phase B). The investigator team runs Phase A and Phase B by the police team. In Phase A, ${X_{{A_i}}}$ shows the i th suspected location, $i=1,2,...,N_A$ . $N_A$ indicates the number of the suspected locations to be investigated. In Phase B, ${X_{{B_i}}}$ shows the i th suspected location, $i=1,2,...,N_B$ . $N_B$ indicates the number of the suspected locations to be investigated. Here, $N_A$ and $N_B$ equal N , such that N shows the population size. Since the forensic investigation is a cyclical process, the process ends when the current iteration count ( t ) reaches the maximum iteration count ( t Max).

Stage A1 The interpretation of the findings portion of the forensic investigation process corresponds to Stage A1. The team analyzes the data and pinpoints any suspect spots. Every conceivable suspect location is investigated in light of other discoveries. First, a new suspicious location named ${X_{{A1_i}}}$ is extracted from ${X_{{A_i}}}$ based on information about ${X_{{A_i}}}$ and other suspicious locations. The general formula of the movement for this study, in which each individual is assumed to act under the influence of other individuals, is as Eq. ( 1 ):

where Dim is the problem size (dimension), R corresponds to a random number in the range [0,1], ${a_1}$ shows the number of individuals which affect the movement of ${X_{{A_{ij}}}}$ and ${a_1}$ $\in $ $\{ 1,2,...,n - 1\}$ . Here, as a result of trial and error tests, the best and shortest convergence is observed if the value of ${a_1}$ is 2. Accordingly, the new suspect location ${X_{{A_{i}}}}$ is revised as in Eq. ( 2 ).

where k , h , and i indexes correspond to three suspected locations and $\{k,h,i\}$ $\in $ $\{ 1,2,...,N\}$ . k and h are randomly chosen numbers. N shows the population size and also the number of suspected locations, where Dim is the problem size (dimension), and $R_1$ is a random number in the range [0,1]. Therefore, the expressions $({R_1} - 0.5) * 2) $ and $({R_1} - 0.5) * 2)$ represent the range [−1,1].

Stage A2 corresponds to the direction of the inquiry phase. Investigators compare the probability of each suspicious location with that of the others to determine the most likely suspicious location. The probability of each location is estimated using $P({X_{{A_i}}})$ , Eq. ( 3 ) and a high $P({X_{{A_i}}})$ value means a high probability for the location.

where pW is worst (lowest) possibility and pB is the best (highest) possibility. ${p_{{A_i}}}$ indicates the possibility of i th location.

Updating a search location is influenced by the directions of other suspected locations. Instead of updating all directions, randomly selected directions in the updated location are changed. In this stage, the movement of ${X_{{A_i}}}$ depends on the best individual and other random individuals. Like Stage A1, the general formula for motion is in Eq. ( 4 ).

Here, ${X_\textrm{best}}$ represents the best location; ${a_2}$ are number of individuals which affect ${X_{A{2_i}}}$ and ${a_2}$ $\in $ $\{ 1,2,...,n - 1\}$ ; c is the effectiveness coefficient of the remaining individuals and c $\in $ $\left[ { - 1,1} \right] $ . ${a_c}=3$ has been taken in the experiments. Thus, Eq. ( 5 ) obtains the new suspect position.

where $R_5$ is the random number in the range [0,1]; and p,q,r, and i are four suspected locations selected 1,..,N. p, q, and r are randomly chosen, and $j=1,2,...,$ Dim.

Stage B1 can be expressed as the "action" phase. Once the best location information has been received from the investigative team, all agents in the pursuit team must approach the target in a coordinated manner to arrest the suspect. Each agent ( $B_i$ ) approaches the position with the best probability according to Eq. ( 6 ). An update is made if the newly approached site generates a higher probability than the previous location.

${R_6}$ and ${R_7}$ are the random numbers in the range [0,1].

Stage B2 is the stage in which the process of "actions" is expanded. Locations are updated according to the probabilities of new locations reported to the headquarters by the police agents in case of any movement. The headquarters commands the tracking team to approach this location. In the process, agent $B_i$ moves toward the best position, and agent $B_i$ is influenced by another team member ( $B_R$ ). Agent $B_i$ ’s new position is calculated as in Eq. ( 7 ). If the probability of $B_R$ is better than the probability of $B_i$ ; otherwise, it is formalized as Eq. ( 8 ). The new-found location is updated if it is more probable than the old one.

Here, ${R_8}$ , ${R_9}$ , ${R_10}$ and ${R_11}$ are the random numbers; R and i represent the two police agents, and they are selected from 1, ..., N . R is selected randomly in this group.

The optimum location for the suspect will be advised to the investigation team by the pursuit team. They perform this to help them increase the accuracy of their analysis and evaluation. Forensic investigative procedures might repeat themselves. The operating steps of the FBI algorithm are summarized in Fig. 2 .

The flowchart of FBI algorithm

2.2 Dynamic oppositional based learning

Xu et al. [ 33 ] have proposed a method that overcomes the difficulties of opposition-based learning (OBL), quasi-reflection-based learning methods (QRBL), and quasi-opposite-based learning methods (QOBL) and named as dynamic-opposite learning (DOL). QOBL proposed by Rahnamayan et al. [ 34 ], one of the variants of oppositional based learning (OBL), aims to increase the chance of approaching the solution by using quasi-opposite numbers instead of opposite numbers. According to the probability theorem, randomly initialized candidate solutions are further away from the global solution than the opposite prediction. Therefore, opposite numbers can effectively reduce the search space area and increase the convergence speed. The quasi-opposite number is formed from the interval between the median and the opposite number of the current population [ 35 ]. (QRBL) is proposed by Ergezer et al. [ 36 ] to extend the search space between current and central locations. However, these OBL approaches will miss the local optimal point if there is one between the current and opposite values. Thus, a system that dynamically broadens the search space should be considered in this situation. In this instance, DOL prevails. First and foremost, DOL needs to define the opposing point and opposite number. The mathematical expression of DOL is as in Eqs. ( 9 ), ( 10 ).

where lb and ub represent the lower and upper bounds, rand is a random number in the range of [0,1], $X_i$ refers to a real number used as agent positions in between [ lb , ub ], NP denotes the population size, and i is the current agent selected from [1, NP ]. ${OP_i}$ is obtained based on OBL and ${DO_i}$ corresponds to dynamic opposition number.

3 The proposed DOLFBI algorithm

It has been mentioned in Sect. 2.1.2 that the original FBI algorithm consisted of two main phases: the investigation phase (Phase A) and the pursuit phase (Phase B). In Stage A1, the investigation process is followed by detecting suspicious locations. In Stage A2, on the other hand, the location with the highest probability and the current best location is determined by calculating the suspicion probability of each location. By applying DOL over the best available position ${X{A_2}}$ (from Eq. ( 5 )), the position update for Phase A is performed in the next iteration.

Similar situations exist for Stage B. In Stage B1, also called the action phase, the best location information from the exploration team is received, and the tracking team approaches this location in a coordinated manner. Then, in Stage B2, where the action has been expanded, in case of any movement, the locations are updated according to the probabilities of the new locations reported to the headquarters by the police teams, and headquarters orders the monitoring team to approach this location. If the probability is higher than the old location, the current location becomes the new location.

The pursuit team updates the location with DOL and sends it back to Stage A before informing the investigative team about the suspect’s best location. The process continues in this way until it produces the best result. Since the algorithm has two different population sets (A and B), DOL is applied to both population groups. The algorithm of the developed DOLFBI is included in Algorithms 1 , 2 , and 3 .

Pseudocode of Investigation Team Process

Pseudocode of Pursuit Team Process

Pseudocode of dynamic oppositional based learning

Pseudocode of DOLFBI

4 Simulation results

The flow in this section is given as follows: First, the values of the specific parameters used in the proposed and compared traditional and advanced methods are expressed. The algorithms are applied to the CEC2019 and CEC2022 test data within these parameters. As a result, the best and average outcomes are tabulated. The performance of the proposed method for known engineering challenges is then compared. Similarly, the proposed strategy is studied for 20, 24, and 72-truss optimization issues. The acquired results are validated using the Wilcoxon sign and Friedman rank tests.

4.1 Parameter settings

The population number and maximum iteration parameters, determined to be 30 and 5000, respectively, have been crucial parameters impacting the original FBI. A DOL strategy is being used to improve the procedure. The jumping rate (Jr) parameter, which is set to 0.25, affects DOL. These criteria serve as the foundation for all comparisons done within the framework of experimental studies. Table 1 shows the parameter settings for the traditional and enhanced methods utilized in the comparison.

4.2 Benchmark test suites

CEC2019 and CEC2022 were employed as benchmarks in this study. The CEC2019 test functions comprise ten multimodal functions listed in Table 22 . The first three CEC2019 functions have 9, 16, and 18 dimensions. Other CEC2019 functions have a dimension of 10. All of the global minimum values converge to 1. The benchmark functions in CEC2022 are unimodal, basic, hybrid, and composition. These are all minimization problems. Table 23 gives their comprehensive descriptions and specifications. The first five functions are shifted and rotated functions [ 37 ]. F11 is unimodal, which means it has a single minimum point. F12–F15 are multimodal, with multiple local minimum points. F16–F18 are hybrid functions developed by combining distinct functions. F16, for example, is derived from the functions of Bent Cigar, HGBat, and Rastrigin. F19 through F22 are composition functions. All functions are tested in the [ $-100$ ,100] range, each with a global minimum value.

First, Tables 2 and 3 show the comparison results for the CEC2019 benchmark set. As a result, the optimum convergence for the F2, F3, F4, F5, F6, F7, and F8 problems is found using DOLFBI using both the best value and the mean value. Most of the compared methods for the F1 problem, including DOLFBI, converge to the global optimum value of 1, and this convergence value is reached on average; however, when analyzed in terms of standard deviation, SMA, HGS, and AVOA provide full convergence for all runs.

Tables 4 and 5 show the findings achieved compared to the improved approaches. Although the analyses produced similar results for DOLFBI, DOLFBI for F10 converged better than the advanced approaches this time. Except for F9 and F10, DOLFBI is a success for the CEC2019 benchmark set based on the ten challenges (Tables 6 and 7 ).

The CEC2022 set contains 12 problems. Traditional and upgraded approaches used in 2019 are being investigated for 2022. The convergence of DOLFBI is thriving, according to the findings of F11–F20 (for the first 11 problems) in Tables 8 and 9 . Although MFO appears to get the best convergence for F22, it attained this convergence value in DOLFBI but fell below MFO in standard deviation.

Convergence behaviors for both benchmark sets are plotted in Figs. 7 and 8 using the average convergence curve from 30 runs. From this, it is concluded that the drawings agree with the result tables. Here is the DOLFBI curve plotted in red and dotted; only the relationship to conventional methods is considered. The first 1000 iterations are plotted to show the convergence behavior clearly. If it is generally interpreted, it can be deduced that many methods converge to the optimum value for F1. It is possible to observe this similar behavior for F2, F12, F18, F20, and F22. The problems where early convergence is most prominent are F3, F4, F6, F8, F10, and F17. Although the global optimum value of the F10 benchmark function is 1, it usually converges to about 20. The successful convergence in the F10 function determines how many times it converges to 1 in 30 different convergence performances. Thus, it can be seen that DOLFBI often converges to 1 and, therefore, performs better on average than other methods. Considering Figs. 7 and 8 , CEC2019 problems are more challenging than CEC2022.

DOLFBI’s trajectory analysis is in Figs. 3 and 4 . This analysis is performed in 2000 iterations and 100 populations. The first column shows the positions of all individuals in the population at the end of 2000 iterations for dimensions x1 and x2 only. The red dot indicates the global minimum point, while the black dots represent the candidate solutions. It can be seen that at the end of the iteration, the candidate solutions are concentrated around the global optimum. The second column denotes the trajectory for the first dimension. Although the algorithm oscillates sharply at the beginning of the iterations, it converges to the optimum position at the end. The third column represents the average fitness value over 30 different runs, while the fourth column shows the convergence curve. According to the detailed trajectory analysis, it can be said that DOLFBI exhibits a consistent and robust convergence throughout the iterations.

Trajectory analysis of the DOLFBI

Trajectory analysis of the DOLFBI (Cont)

Box plots visualize statistical measures such as standard deviation, mean, minimum maximum, and quartile. Therefore, box-plot analysis is performed between DOLFBI and the compared algorithms. Box-plot analyses are visualized in Figs. 5 and 6 . Generally speaking, it can be seen that DOLFBI converges better with lower standard deviations except for F6, F8, and F18. Some algorithms produce extreme values in the F3, F9, F15, F18, F19, F20, and F21 benchmark functions in 30 runs. DOLFBI achieves an efficient convergence in these functions with a low standard deviation.

Box-plot analysis of the compared algorithms

Box-plot analysis of the compared algorithms (Cont)

The proposed DOLFBI scans the search space utilizing a dynamic oppositional learning strategy, which makes it superior to other compared approaches. This method uses opposite numbers. According to this strategy, opposite numbers play a major role in the generation-to-generation transfer of promising populations. The layout of the FBI algorithm considers two significant population groups (investigation and pursuit teams). Consequently, the DOL’s impact on the FBI became more noticeable Figs. 7 and 8 .

Convergence analysis of the compared algorithms (Cont)

4.3 Engineering problems

This section presents the engineering design problems to which the proposed method is applied and their comparative results. Each problem has its parameters and constraints. Accordingly, the most optimal values and best cost values of these parameters are emphasized for each problem.

ENG1: This problem involves finding the optimal parameters of a cantilever beam. It has one constraint and five variables. Five variables represent five block lengths. Figure 9 a shows the cantilever beam structure. x 1, x 2, x 3, x 4, and x 5 are the height and width values of the square structure.

ENG2: This problem deals with minimizing the pressure vessel design. The problem has four constraints and four variables. Figure 9 b shows the pressure vessel structure. $T_s$ (thickness of the shell), $T_h$ (the thickness of the head), R (inner radius), and L (length of the cylindrical section) are the variables of the problem.

ENG3: Tension/compression aims to minimize the weight of a tension/compression spring while adhering to constraints on shear stress, surge frequency, and minimum deflection [ 38 ]. In Fig. 9 c, d represents the wire diameter, D mean coil diameter, and P number of active coils.

ENG4: This engineering problem involves improving the performance and efficiency of a speed reducer or gearbox [ 39 ]. The structure of the speed reducer is given in Fig. 9 d. The variables that need to be optimized are face width $(x_1)$ , module of gear $(x_2)$ , count of gear in the pinion $(x_3)$ , first shaft’s length between bearings $(x_4)$ , second shaft length between bearings $(x_5)$ , first shaft’s diameter $(x_6)$ , and second shaft’s diameter $(x_7)$ .

ENG5: The crashworthiness design problem aims to minimize the vehicle’s weight to enhance its ability to protect occupants during a collision [ 40 ]. The crashworthiness structure is given in Fig. 9 e. The variables of this design are expressed as thicknesses of B-Pillar inner, B-Pillar reinforcement, floor side inner, cross-members, door beam, door beltline reinforcement, and roof rail (x1-x7), materials of B-Pillar inner and floor side inner (x8 and x9), and barrier height and hitting position (x10 and x11).

ENG6: The beam will be optimized to achieve minimum cost by varying the weld and member dimensions. The problem’s constraints include limits on shear stress, bending stress, buckling load, and end deflection. [ 41 ]. Welded beam design consists of four variables: the thickness of weld ( h ), the length of the clamped bar ( l ), the height of the bar ( t ), and the thickness of the bar ( b ). The structure of the welded beam is shown in Fig. 9 f.

Engineering problems

Here, the convergence performance of DOLFBI for the six engineering problems mentioned above is investigated. It is also interpreted by making comparisons with other pioneering and new metaheuristics. First, the ENG1 (cantilever beam design) results are in Table 10 . Accordingly, it can be interpreted that the DOL approach improves FBI, and DOLFBI converges better than other metaheuristics. DOLFBI converges to the smallest (good) value 1.301205963 with [5.951125854, 4.874066596, 4.464381903, 3.478196725, 2.138494389] ideal parameters [ 42 , 43 , 44 ]. Second, the ENG2 (pressure vessel design) real-world problem is reported in Table 11 . Here it can be seen that DOLFBI lags behind FBI by one thousandth, but comes significantly closer compared to other methods. The best cost value of DOLFBI is obtained as 5885.333014, while the ideal parameters calculated are [0.7782, 0.3846, 40.3196, 199.9999] [ 45 ]. This method is followed most closely by GWO with 5887.323 value. The third problem, ENG3 (tension/compression spring design), is detailed in Table 12 . DOLFBI is the method that gives the best convergence value 0.012665965, followed by HS after FBI. The best cost value of DOLFBI is obtained with the ideal parameters [0.051764678, 0.358539609, 11.18294909]. The fourth problem, ENG4 (speed reducer) result, is reported in Table 13 . Here, DOLFBI and FBI have the same convergence performance with the best cost value of 2993.761765. The GOA follows this result with 2994.4245. The fifth is ENG5 (crashworthiness problem), which is more challenging than the others because it is a problem with many parameters and constraints. According to the results in Table 14 , it can be interpreted that DOLFBI outperformed FBI, and that the DOL approach improved and improved the FBI significantly but still lagged behind SMO and LIACOR by one thousandth. The best cost value is 22.84298 with SMO, while it is 22.84300988 when DOLFBI is used. Finally, for ENG6 (welded beam design), DOLFBI has lagged behind RSA alone. However, it is seen that it gives more effective results than other compared methods (Table 15 ).

When interpreted in general, it would not be wrong to comment that the DOL approach improves the FBI, although DOLFBI falls in the second and third places in the literature for some challenging problems.

4.4 Truss topology optimization

The arrangement and layout of beam members in a structural system are addressed by structural truss topology. A truss is a frame of triangularly interconnected pieces (such as beams, bars, or rods). The cage’s triangular design provides stability, strength, and stiffness. Truss elements can be assembled in various configurations to meet specific design and engineering needs. The optimization of 20, 24, and 72-bar truss systems is examined in this work. The subheadings provide details on each topology (Table 16 ).

20-Bar Truss Problem : This structural problem has nine nodes leading to 14 degrees of freedom and is shown in Fig. 10 . It is given as a benchmark problem by Kaveh and Zolghadr [ 46 ] and Tejani et al. [ 47 ]. The design parameters and constraints of the issues are given in Table 16 .

Ground topology structure of 20-truss problem

24-Bar Truss Problem : The second structural problem is 24-bar truss and shown in Fig. 11 and the constraints are given in Table 16 .

Ground topology structure of 24-truss problem

72-Bar Truss Problem : The last truss problem is 72-bar truss has been previously used by Mohan et al. [ 48 ]. It has been split into and the constraint size is 198 and these are given in Table 16 .

In the 72-bar problem, the elements are clustered in 16 groups. These are C1 (A1−A4), C2 (A5−A12), C3 (A13−A16), C4 (A17−A18), C5 (A19−A22), C6 (A23−A30), C7 (A31−A34), C8 (A35−A36), C9 (A37−A40), C10 (A41−A48), C11 (A49−A52), C12 (A53−A54), C13 (A55−A58), C14 (A59−A66), C15 (A67−A70), and C16 (A71−A72). As it can be seen in Table 12 , some clusters (C3 and C16) have been removed for all algorithms and also removed from the Fig. 12 .

Ground topology structure of 72-truss problem

Table 17 compares DOLFBI with the methods in the literature for 24-bar truss topology optimization. Elements less than zero are not taken into account. In addition, the extracted element in all compared methods is not included in the table, for example, 4, 5, 11, 17, 18. Considering the efficient metaheuristic methods in the literature, DOLFBI ranks third. ITLBO takes first place with a minimum weight of 120.0798. However, it can be said that DOLFBI has a competitive convergence behavior. 20-bar truss topology optimization results are reported in Table 18 . In this problem, DOLFBI and ITLBO converge to the same and lowest weight values as 154.7988. In this problem, where there are 20 total sections, eight are selected as 1, 5, 8, 11, 13, 15, 18, and 20. Table 19 gives the optimum sections and weights of the 72-bar truss structure. 72-bar truss has a structure that can converge to the optimum weight with few elements. After IOWA, DOLFBI is in second place with 450.388 weight value. The optimum topology of 20 and 24-bar truss problems are given in Fig. 13 .

20 and 24-truss topology optimization with DOLFBI

4.5 Statistical tests

In this study, Wilcoxon sign rank (WSR) and Friedman rank statistical tests are used to show the effectiveness and difference of the proposed method. These two tests are nonparametric statistical tests. The Wilcoxon paired-pairs test is a nonparametric hypothesis test that compares the median of two paired groups and determines whether they are the same distributed [ 49 ]. Thus, WSR shows whether there is a significant difference between any two metaheuristic algorithms. In this study, a $5\%$ significance level research is carried out. Friedman rank test determines a rank value for the proposed algorithm [ 50 ] (Tables 17 , 18 , 19 ).

Table 20 tabulates the WSR results. The p value between DOLFBI and alternative metaheuristic algorithms is displayed in this table. The h value between DOLFBI and methods is 0 for other functions except for FBI in F8 and HBA in F3. This means that DOLFBI differs significantly from the methods compared in the literature. Table 21 reports the Friedman results. According to the Friedman rank results, HGS in F1, HGS, HBA, and AVOA in F19 have the first rank. Considering the function-based rank average of all algorithms, DOLFBI is in the first rank, and then the FBI is in the second rank. In the last ranks, there are SCA and SMA. In CEC2019 and CEC2022 functions, it is clear that DOLFBI precedes other algorithms and exhibits competitive convergence.

4.6 Discussion

The study integrated dynamic oppositional based learning (DOL), an effective population determination method, into one of the newly proposed algorithms, the forensic-based investigation algorithm (FBI). In this case, a more successful convergence was sought by employing DOL, which picks individuals using the opposite approach rather than the original FBI’s random population selection algorithm. With the proposed study, it appears conceivable to improve the convergence outcomes of metaheuristic algorithms, which are regarded to be weak, particularly during the population selection stage.

The paper’s proposed method is compared to basic and enhanced metaheuristic methods. Basic approaches lack the search space’s contrasting strategy. Selecting opposite solution possibilities from the search space reduces the probability of becoming stuck in the local optimum. The enhanced approaches, on the other hand, were likewise developed using the combined opposite selection strategy. However, the FBI exhibits an excellent convergence performance due to the exploitation and exploration capabilities of the Investigation and Pursuit stages.

The impact and contribution of the DOL are evident numerically in the results, especially in the FBI, because two critical population groups are obtained. Additionally, DOLFBI, an enhanced version of FBI, is expected to yield promising results in numerous engineering or challenging real-world scenarios, feature selection with its binary form, and various optimization problems.

5 Conclusions

The DOLFBI presented in this study was created by combining the DOL paradigm with the FBI algorithm. The DOL paradigm, a variant of OBL, promotes efficiency by generating opposing populations and finding the global optimum as quickly and correctly as possible. DOL adaptively and dynamically determines randomly selected populations in traditional FBI for DOLFBI. Unlike primary and complex metaheuristic algorithms, DOLFBI outperforms or lags behind FBI and other algorithms. Twenty-two challenging benchmark problems from CEC2019 and CEC2022 are compared using traditional and sophisticated methodologies.

Compared to standard algorithms, DOLFBI converges to a lower fitness value than other approaches in 18 of 22 benchmark problems (excluding F1, F9, F10, and F22). According to the findings of a comparison with other OBL-based enhanced MAs in the literature, DOLFBI achieved the best convergence for 19 problems except F1, F9, and F22. The Wilcoxon sign and Friedman rank tests validate the results and statistical tests for DOLFBI. Only FBI in the F8 function and $h=0$ in the F3 function in HBA are generated in WSR; in all other circumstances, $h=1$ is generated. According to the Friedman test, DOLFBI ranks top in the CEC2019 and CEC2022 comparison problems. As a result, F4, F10, F14, and F17 demonstrate early convergence behavior. This study also includes a trajectory and qualitative analysis for DOLFBI. The trajectory analysis shows that the oscillation of the first dimension toward the last iteration is fixed and approaches an optimum.

DOLFBI has the best convergence in cantilever beam design, speed reducer, and tension/compression problems in terms of engineering challenges. It is ranked second for welded beam design, third for pressure vessel design, and fourth for crashworthiness.

DOLFBI performs successfully not only in mathematical problems but also in real-world problems. Based on an analysis of the tabulated findings, trajectory analyses, convergence curves, and box-whisker plots, it is evident that DOL yields encouraging outcomes for several FBI improvement issues.

Finally, DOLFBI is employed in the design of 20-, 24-, and 72-bar truss topology optimization challenges. As a result, it is compared to other methods in the literature. The results show that it comes in second position behind TLBO, with an optimal weight of 122.057 at 24-bar. They are on par with ITLBO and in first place with a weight value of 154.7988 at 20 bar. Finally, it ranks second after IWOA for the 72-truss issue, with an optimum value of 450.388.

A binary version of the proposed method can be created and used as a feature selection method in future investigations. Furthermore, the suggested method is adaptable to neural networks, extreme learning machines, and deep learning architectures.

Although dynamic OBL enhances convergence performance, the running time for exploring opposing regions can be computed as O(dim*N). In addition, although this change in running time causes DOLFBI to run slower, this does not cause an extra burden in algorithm complexity. DOLFBI, on the other hand, calls the objective function one last time to maintain the existing optimum value. As a result, the overall number of called functions grows.

Data availability

Source codes used in analyzing the datasets are available from the corresponding author upon reasonable request.

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Kutlu Onay, F. Solution of engineering design and truss topology problems with improved forensic-based investigation algorithm based on dynamic oppositional based learning. Neural Comput & Applic (2024). https://doi.org/10.1007/s00521-024-09737-4

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Received : 06 October 2023

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DOI : https://doi.org/10.1007/s00521-024-09737-4

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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.
Teaching Problem Solving
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.
Problem-Based Learning (PBL)
Problem-Based Learning (PBL) is a teaching method in which complex real-world problems are used as the vehicle to promote student learning of concepts and principles as opposed to direct presentation of facts and concepts. In addition to course content, PBL can promote the development of critical thinking skills, problem-solving abilities, and ...
Teaching Problem Solving
Problem solving is a necessary skill in all disciplines and one that the Sheridan Center is focusing on as part of the Brown Learning Collaborative, which provides students the opportunity to achieve new levels of excellence in six key skills traditionally honed in a liberal arts education - critical reading, writing, research, data ...
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 ...
Ch. 5 Problem Based Learning
There is no single method for designing problem-based inquiry learning environments. Various techniques have been used to generate the problem and stimulate learning. Promoting student-ownership, using a particular medium to focus attention, telling stories, simulating and recreating events, and utilizing resources and data on the Internet are ...
Teaching Problem-Solving Skills
Some common problem-solving strategies are: compute; simplify; use an equation; make a model, diagram, table, or chart; or work backwards. Choose the best strategy. Help students to choose the best strategy by reminding them again what they are required to find or calculate. Be patient.
Teaching problem solving
Working on solutions. In the solution phase, one develops and then implements a coherent plan for solving the problem. As you help students with this phase, you might ask them to: identify the general model or procedure they have in mind for solving the problem. set sub-goals for solving the problem. identify necessary operations and steps.
Why Every Educator Needs to Teach Problem-Solving Skills
Resolve Conflicts. In addition to increased social and emotional skills like self-efficacy and goal-setting, problem-solving skills teach students how to cooperate with others and work through disagreements and conflicts. Problem-solving promotes "thinking outside the box" and approaching a conflict by searching for different solutions.
Problem-Based Learning: What and How Do Students Learn?
Problem-based approaches to learning have a long history of advocating experience-based education. Psychological research and theory suggests that by having students learn through the experience of solving problems, they can learn both content and thinking strategies. Problem-based learning (PBL) is an instructional method in which students learn through facilitated problem solving. In PBL ...
Teaching and Learning as Problem Solving
An approach to teaching and learning ically emphasized general problem-solving strate- consistent with the problem-solving metaphor would gies, since it was felt that these strategies would ensure that this type of information is an integral be universally effective. part of what students are taught in school.
PDF Problem Based Learning: A Student-Centered Approach
Problem-based learning is a teaching method in which students' learn through the complex and open ended problems. These problems are real world problems and are used to encourage students' learning through ... Problem solving can incites for learning. 7). Throughout the learning process, critical reflection happens
The process of implementing problem-based learning in a teacher
This is an example of students learning by solving problems using PBL and reflecting on their experiences (Barrows ... B. Y. (2005). A study on the teaching experiment of problem-based learning on materials and methods of teaching mandarin at elementary school teacher education. Journal of Research on Elementary Education, 1, 163-192. ...
(PDF) Principles for Teaching Problem Solving
structured problem solving. 7) Use inductive teaching strategies to encourage synthesis of mental models and for. moderately and ill-structured problem solving. 8) Within a problem exercise, help ...
Problem-Solving Method In Teaching
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 ...
Frontiers
Objectives: The purpose of this study is to compare the effects of the problem-solving method vs. the traditional method on motivation and learning during physical education courses. Methods: Fifty-three students ( M age 15 ± 0.1 years), in their 1st year of the Tunisian secondary education system, voluntarily participated in this study, and ...
(Pdf) Learning and Problem Solving: the Use of Problem Solving Method
Abstract. Problem-based learning is a recognized teaching method in which complex real-world problems are used as the vehicle to promote student learning of concepts and principles as opposed to ...
The effectiveness of collaborative problem solving in promoting
Collaborative problem-solving is the organic integration of collaborative learning and problem-based learning, which takes learners as the center of the learning process and uses problems with ...
Case-based Teaching and Problem-based Learning
Problem-based Learning. Problem-based learning (PBL) is both a teaching method and an approach to the curriculum. It consists of carefully designed problems that challenge students to use problem solving techniques, self-directed learning strategies, team participation skills, and disciplinary knowledge. The articles and links in this section ...
Teaching Mathematics Through Problem Solving
Teaching about problem solving begins with suggested strategies to solve a problem. For example, "draw a picture," "make a table," etc. You may see posters in teachers' classrooms of the "Problem Solving Method" such as: 1) Read the problem, 2) Devise a plan, 3) Solve the problem, and 4) Check your work. There is little or no ...
The problem-solving method: Efficacy for learning and motivation in the
Methods. Fifty-three students (M age 15 ± 0.1 years), in their 1st year of the Tunisian secondary education system, voluntarily participated in this study, and randomly assigned to a control or experimental group.Participants in the control group were taught using the traditional methods, whereas participants in the experimental group were taught using the problem-solving method.
5 Advantages and Disadvantages of Problem-Based Learning [+ Activity
Used since the 1960s, many teachers express concerns about the effectiveness of problem-based learning (PBL) in certain classroom settings. Whether you introduce the student-centred pedagogy as a one-time activity or mainstay exercise, grouping students together to solve open-ended problems can present pros and cons.. Below are five advantages and disadvantages of problem-based learning to ...
Problem Solving Method Of Teaching
The problem-solving method of teaching is the learning method that allows children to learn by doing. This is because they are given examples and real-world situations so that the theory behind it can be understood better, as well as practice with each new concept or skill taught on top of what was previously learned in class before moving onto another topic at hand.
3 Ways to Strengthen Math Instruction
Here are three takeaways from the discussion. For more, watch the recording on demand. 1. Intervention is key. Research shows that early math skills are a key predictor of later academic success.
Frontiers
Invention and innovation education and its associated practices (e.g., problem-finding, problem-defining, learning from failure, iterative problem-solving, innovation-focused curricula, collaboration, and maker spaces) are moving from the periphery to the center of education at an ever-increasing pace. Although the research and literature on invention and innovation education, collectively ...
Teaching Methods and Learning Activities
Project-Based Learning is an engaging teaching method where students actively investigate real-world issues and challenges. By doing so, they gain a deeper understanding of the subject matter. It promotes teamwork, critical thinking, and original problem-solving among pupils. 2. Gamification
Problem solving in mathematics education: tracing its ...
Mathematical problem solving has been a prominent theme and research area in the mathematics education agenda during the last four decades. Problem-solving perspectives have influenced and shaped mathematics curriculum proposals and ways to support learning environments worldwide (Törner et al., 2007; Toh et al., 2023).Various disciplinary communities have identified and contributed to ...
Cureus
Traditional teaching methods may lack the immersive approach needed for effective learning. The incorporation of escape rooms serves as a novel and experiential way to bridge this gap, providing students with a hands-on, scenario-based learning.</p> <p>Instructional Methods and Materials Used: The escape room experience is designed to mirror ...
Nunez STEAM Division Combines Fun and Learning for Summer 2024
This camp will teach the fundamentals of fabrication and coding while building the student's problem solving and critical thinking skills. This is a one-day camp. Students will bring their own lunch. CEEF (Consumer Energy Education Foundation) AI Energy - Grades 4-8, June 10-14, $300
Solution of engineering design and truss topology problems ...
The forensic-based investigation (FBI) is a metaheuristic algorithm inspired by the criminal investigation process. The collaborative efforts of the investigation and pursuit teams demonstrate the FBI's involvement during the exploitation and exploration phases. When choosing the promising population, the FBI algorithm's population selection technique focuses on the same region. This ...

Center for Teaching Innovation

Problem-Based Learning

Why Use Problem-Based Learning?

Considerations for Using Problem-Based Learning

Getting Started with Problem-Based Learning

Center for Teaching

Tips and Techniques

Encourage Independence

Be sensitive

Encourage Thoroughness and Patience

Teaching Guides

Quick Links

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Problem-Based Learning (PBL)

By working with PBL, students will:

How to Begin PBL

Designing Classroom Instruction

How to Orchestrate a PBL Activity

Teacher’s Role in PBL

The Role of the Students

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Teaching Problem-Solving Skills

Principles for teaching problem solving

Woods’ problem-solving model

Think about it

Plan a solution

Carry out the plan

Catalog search

Teaching problem solving

Introducing the problem

Working on solutions

Why Every Educator Needs to Teach Problem-Solving Skills

Want to learn more about how to measure and teach students’ higher-order skills, including problem solving, critical thinking, and written communication?

Problem-Solving Skills Help Students…

Set and Achieve Goals

Resolve Conflicts

Achieve Success

Problem-Solving Skills Can Be Measured and Taught

What Are Performance-Based Assessments?

Preview a Performance-Based Assessment

Performance-Based Assessments Help Students Build and Practice Problem-Solving Skills

Explore CAE’s Problem-Solving Assessments

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Problem-Solving Method in Teaching

Table of Contents

Meaning of Problem-Solving Method

Benefits of Problem-Solving Method

Steps in Problem-Solving Method

Verification of the concluded solution or Hypothesis

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5 Teaching Mathematics Through Problem Solving

Mathematics Tasks and Activities that Promote Teaching through Problem Solving

Choosing the Right Task

Low Floor High Ceiling Tasks

Math in 3-Acts

Number Talks

Saying “This is Easy”

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5 Advantages and Disadvantages of Problem-Based Learning [+ Activity Design Steps]

Advantages of Problem-Based Learning

1. Development of Long-Term Knowledge Retention