what is problem solving as a teaching strategy

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Teaching problem solving.

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Tips and Techniques

Expert vs. novice problem solvers, communicate.

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

Encourage Independence

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

Be sensitive

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

Encourage Thoroughness and Patience

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

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

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

what is problem solving as a teaching strategy

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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.

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: Let students get ‘stuck’ and ‘unstuck’

Subscribe to the center for universal education bulletin, kate mills and km kate mills literacy interventionist - red bank primary school helyn kim helyn kim former brookings expert @helyn_kim.

October 31, 2017

This is the second in a six-part blog series on teaching 21st century skills , including problem solving , metacognition , critical thinking , and collaboration , in classrooms.

In the real world, students encounter problems that are complex, not well defined, and lack a clear solution and approach. They need to be able to identify and apply different strategies to solve these problems. However, problem solving skills do not necessarily develop naturally; they need to be explicitly taught in a way that can be transferred across multiple settings and contexts.

Here’s what Kate Mills, who taught 4 th grade for 10 years at Knollwood School in New Jersey and is now a Literacy Interventionist at Red Bank Primary School, has to say about creating a classroom culture of problem solvers:

Helping my students grow to be people who will be successful outside of the classroom is equally as important as teaching the curriculum. From the first day of school, I intentionally choose language and activities that help to create a classroom culture of problem solvers. I want to produce students who are able to think about achieving a particular goal and manage their mental processes . This is known as metacognition , and research shows that metacognitive skills help students become better problem solvers.

I begin by “normalizing trouble” in the classroom. Peter H. Johnston teaches the importance of normalizing struggle , of naming it, acknowledging it, and calling it what it is: a sign that we’re growing. The goal is for the students to accept challenge and failure as a chance to grow and do better.

I look for every chance to share problems and highlight how the students— not the teachers— worked through those problems. There is, of course, coaching along the way. For example, a science class that is arguing over whose turn it is to build a vehicle will most likely need a teacher to help them find a way to the balance the work in an equitable way. Afterwards, I make it a point to turn it back to the class and say, “Do you see how you …” By naming what it is they did to solve the problem , students can be more independent and productive as they apply and adapt their thinking when engaging in future complex tasks.

After a few weeks, most of the class understands that the teachers aren’t there to solve problems for the students, but to support them in solving the problems themselves. With that important part of our classroom culture established, we can move to focusing on the strategies that students might need.

Here’s one way I do this in the classroom:

I show the broken escalator video to the class. Since my students are fourth graders, they think it’s hilarious and immediately start exclaiming, “Just get off! Walk!”

When the video is over, I say, “Many of us, probably all of us, are like the man in the video yelling for help when we get stuck. When we get stuck, we stop and immediately say ‘Help!’ instead of embracing the challenge and trying new ways to work through it.” I often introduce this lesson during math class, but it can apply to any area of our lives, and I can refer to the experience and conversation we had during any part of our day.

Research shows that just because students know the strategies does not mean they will engage in the appropriate strategies. Therefore, I try to provide opportunities where students can explicitly practice learning how, when, and why to use which strategies effectively so that they can become self-directed learners.

For example, I give students a math problem that will make many of them feel “stuck”. I will say, “Your job is to get yourselves stuck—or to allow yourselves to get stuck on this problem—and then work through it, being mindful of how you’re getting yourselves unstuck.” As students work, I check-in to help them name their process: “How did you get yourself unstuck?” or “What was your first step? What are you doing now? What might you try next?” As students talk about their process, I’ll add to a list of strategies that students are using and, if they are struggling, help students name a specific process. For instance, if a student says he wrote the information from the math problem down and points to a chart, I will say: “Oh that’s interesting. You pulled the important information from the problem out and organized it into a chart.” In this way, I am giving him the language to match what he did, so that he now has a strategy he could use in other times of struggle.

The charts grow with us over time and are something that we refer to when students are stuck or struggling. They become a resource for students and a way for them to talk about their process when they are reflecting on and monitoring what did or did not work.

For me, as a teacher, it is important that I create a classroom environment in which students are problem solvers. This helps tie struggles to strategies so that the students will not only see value in working harder but in working smarter by trying new and different strategies and revising their process. In doing so, they will more successful the next time around.

Don’t Just Tell Students to Solve Problems. Teach Them How.

The positive impact of an innovative uc san diego problem-solving educational curriculum continues to grow.

Daniel Kane - [email protected]

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Not getting stuck

One of the overarching goals of this project is to teach both problem-identification and problem-solving skills that help students avoid getting stuck during the learning process. Stuck feelings lead to frustration – and when it’s a Science, Technology, Engineering and Math (STEM) project, that frustration can lead students to feel they don’t belong in a STEM major or a STEM career. Instead, the UC San Diego curriculum is designed to give students the tools that lead to reactions like “this class is hard, but I know I can do this!” – as Ogo, a celebrated high school biomedical sciences and technology teacher, put it.

Three years into the curriculum development effort, the light-hearted energy of the students combined with their intense focus points to success. On the last day of the class, Mourad Mjahed PhD, Director of the MESA Program at Southwestern College’s School of Mathematics, Science and Engineering came to UC San Diego to see the final project presentations made by his 22 MESA students.

“Industry is looking for students who have learned from their failures and who have worked outside of their comfort zones,” said Mjahed. The UC San Diego problem-solving curriculum, Mjahed noted, is an opportunity for students to build the skills and the confidence to learn from their failures and to work outside their comfort zone. “And from there, they see pathways to real careers,” he said.

What does it mean to explicitly teach problem solving?

This approach to teaching problem solving includes a significant focus on learning to identify the problem that actually needs to be solved, in order to avoid solving the wrong problem. The curriculum is organized so that each day is a complete experience. It begins with the teacher introducing the problem-identification or problem-solving strategy of the day. The teacher then presents case studies of that particular strategy in action. Next, the students get introduced to the day’s challenge project. Working in teams, the students compete to win the challenge while integrating the day’s technique. Finally, the class reconvenes to reflect. They discuss what worked and didn't work with their designs as well as how they could have used the day’s problem-identification or problem-solving technique more effectively.

The challenges are designed to be engaging – and over three years, they have been refined to be even more engaging. But the student engagement is about much more than being entertained. Many of the students recognize early on that the problem-identification and problem-solving skills they are learning can be applied not just in the classroom, but in other classes and in life in general.

Gabriel from Southwestern College is one of the students who saw benefits outside the classroom almost immediately. In addition to taking the UC San Diego problem-solving course, Gabriel was concurrently enrolled in an online computer science programming class. He said he immediately started applying the UC San Diego problem-identification and troubleshooting strategies to his coding assignments.

Gabriel noted that he was given a coding-specific troubleshooting strategy in the computer science course, but the more general problem-identification strategies from the UC San Diego class had been extremely helpful. It’s critical to “find the right problem so you can get the right solution. The strategies here,” he said, “they work everywhere.”

Phan echoed this sentiment. “We believe this curriculum can prepare students for the technical workforce. It can prepare students to be impactful for any career path.”

The goal is to be able to offer the course in community colleges for course credit that transfers to the UC, and to possibly offer a version of the course to incoming students at UC San Diego.

As the team continues to work towards integrating the curriculum in both standardized high school courses such as physics, and incorporating the content as a part of the general education curriculum at UC San Diego, the project is expected to impact thousands more students across San Diego annually.

Portrait of the Problem-Solving Curriculum

On a sunny Wednesday in July 2023, an experiential-learning classroom was full of San Diego community college students. They were about half-way through the three-week problem-solving course at UC San Diego, held in the campus’ EnVision Arts and Engineering Maker Studio. On this day, the students were challenged to build a contraption that would propel at least six ping pong balls along a kite string spanning the laboratory. The only propulsive force they could rely on was the air shooting out of a party balloon.

A team of three students from Southwestern College – Valeria, Melissa and Alondra – took an early lead in the classroom competition. They were the first to use a plastic bag instead of disposable cups to hold the ping pong balls. Using a bag, their design got more than half-way to the finish line – better than any other team at the time – but there was more work to do.

As the trio considered what design changes to make next, they returned to the problem-solving theme of the day: unintended consequences. Earlier in the day, all the students had been challenged to consider unintended consequences and ask questions like: When you design to reduce friction, what happens? Do new problems emerge? Did other things improve that you hadn’t anticipated?

Other groups soon followed Valeria, Melissa and Alondra’s lead and began iterating on their own plastic-bag solutions to the day’s challenge. New unintended consequences popped up everywhere. Switching from cups to a bag, for example, reduced friction but sometimes increased wind drag.

Over the course of several iterations, Valeria, Melissa and Alondra made their bag smaller, blew their balloon up bigger, and switched to a different kind of tape to get a better connection with the plastic straw that slid along the kite string, carrying the ping pong balls.

One of the groups on the other side of the room watched the emergence of the plastic-bag solution with great interest.

“We tried everything, then we saw a team using a bag,” said Alexander, a student from City College. His team adopted the plastic-bag strategy as well, and iterated on it like everyone else. They also chose to blow up their balloon with a hand pump after the balloon was already attached to the bag filled with ping pong balls – which was unique.

“I don’t want to be trying to put the balloon in place when it's about to explode,” Alexander explained.

Asked about whether the structured problem solving approaches were useful, Alexander’s teammate Brianna, who is a Southwestern College student, talked about how the problem-solving tools have helped her get over mental blocks. “Sometimes we make the most ridiculous things work,” she said. “It’s a pretty fun class for sure.”

Yoshadara, a City College student who is the third member of this team, described some of the problem solving techniques this way: “It’s about letting yourself be a little absurd.”

Alexander jumped back into the conversation. “The value is in the abstraction. As students, we learn to look at the problem solving that worked and then abstract out the problem solving strategy that can then be applied to other challenges. That’s what mathematicians do all the time,” he said, adding that he is already thinking about how he can apply the process of looking at unintended consequences to improve both how he plays chess and how he goes about solving math problems.

Looking ahead, the goal is to empower as many students as possible in the San Diego area and beyond to learn to problem solve more enjoyably. It’s a concrete way to give students tools that could encourage them to thrive in the growing number of technical careers that require sharp problem-solving skills, whether or not they require a four-year degree.

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Teaching Problem Solving

The other day, a physicist friend was working in the lab with her summer research students. They were talking about the work they’d been doing that summer and how there was no manual or instructions of any sort for any of it; no textbook, no lab procedure. It was as if they were making it up as they went along. Laughing about this, one of the students said, ‘You know what we need? We need an entire course with nothing but problems. Just give us one problem after another, and we figure out how to do them. Because that’s what real research is.’ The rest of the students laughed. And then all of them nodded. -Hanstedt, 2018, p. 41

Employers, college presidents, faculty, and students demonstrate remarkable consensus that problem solving is one of the most important outcomes of a college education (Bok, 2017; Hart Research Associates, 2015; Hora, Benbow, Oleson, 2016; Passow & Passow, 2017). At the time of this newsletter, there were 28 courses offered this year that included the words “problem*” and “solving” in Courses@Brown . Course descriptions ranged from focusing on how to apply techniques or skills, to solving problems, to tackling common problems encountered in the field, and concepts that included “problems” within their title. There are undoubtedly more courses that implicitly and explicitly focus on problem solving across campus. In light of this emphasis, it is important to ask, “What is a problem and what is problem solving?” and “How do I foster problem-solving skills in my course?” and eventually, "How will I be explicit about problem solving in my course and course description?" Although problem solving is often associated with STEM courses, this newsletter offers perspectives and teaching approaches from across the disciplines.

What is a “problem” and problem solving? Problems and problem solving may be context and discipline specific, but the concept and process have overarching components and similarities across contexts. Jonassen (2000, p. 65) defines a problem as an “unknown entity in some situation (the difference between a goal state and a current state)” such that “finding or solving for the unknown must have some social, cultural, or intellectual value.” Within courses, students may encounter a wide variety of current (e.g., a problem statement) and goal (e.g., a solution) states with different motivations for solving them. Students will be exposed to “well-structured” problems at one end of the spectrum, which have a typical solution path and solution, and “ill-structured” problems, which are highly context dependent and have no one solution path (Jonassen, 2000).

We bring in common case scenarios for students and try to develop the frameworks they need to approach a problem rather than just finding the answer. To help students think about the process, we scaffold scenarios over the years through self-study modules that students can complete on their own. The scenarios stay the same, but students can come back to them with new information and frameworks they have learned, a deeper toolbox to pull from in different clinical settings. This allows students to be lifelong learners and more flexible and adaptable in the future. -Dr. Steven Rougas, Director of the Doctoring Program, Alpert Medical School

Problem solving is a “goal-oriented” process that includes creating and manipulating problems as mental models (Jonassen, 2000). Brown faculty from a variety of disciplines were interviewed by Sheridan staff and asked, “What skills do students need to problem solve effectively?” They responded that students need to be able to do the following:

Reason, observe, and recognize patterns
Use current information to understand the past
Know how to break complex problems down into smaller, more manageable components
Make connections between concepts and disciplines
Creatively think of multiple solution paths

These skills, among others, target the following problem-solving steps (Pretz, Naples, & Sternbergy, 2003):

Recognize or identify a problem
Define and represent the problem mentally
Develop a solution strategy
Organize your knowledge about the problem
Allocate mental and physical resources for solving the problem
Monitor your progress toward the goal
Evaluate the solution for accuracy

Problem solving is an iterative process, and as such, these steps do not necessarily progress in a linear fashion. When creating homework assignments, projects, exams, etc., it is helpful to identify the specific skills you want students to practice, the strategies they should use, and how you will evaluate the solutions they produce.

How do I foster problem-solving skills in my course? Instructors can signpost the problem-solving skills students should develop in their courses by adapting existing problem sets to fit recommendations from the Transparency in Learning and Teaching Project (TILT). The process of increasing transparency in assignments includes communicating the assignment’s purpose, task, and criteria to students (Winkelmes et al., 2016):

The purpose usually links to one learning objective for the course, the skills students will develop as a result of completing the assignment, or a real-world application that students might experience outside of your classroom. In this way, the problem you have presented to the student becomes relevant because it has “some social, cultural, or intellectual value” (Jonassen, 2000, p. 65).
Next, the task states the strategy or strategies students should take to complete the assignment. This includes guiding students through organizing the information available to develop a strategy.
Finally, the criteria could be a rubric or annotated examples that are given to students before the assignment is due, so they are aware of the standards for the assignment.

In one study, researchers found that in courses where at least two assignments had features of transparent assignments, students self reported increases in their academic confidence, sense of belonging, and mastery of skills, such as problem solving (Winkelmes et al., 2016). Below are examples of different skills needed for problem solving with suggestions on how you can foster these skills through adapted or new assignments and in-class exercises.

Communication A key skill for problem solving is knowing how to define and represent the problem and its solutions. This is true for all students, regardless of discipline. For example, Berkenkotter (1982, p. 33) states, “A writer is a problem solver of a particular kind. Writers’ ‘solutions’ will be determined by how they frame their problems, the goals they set for themselves, and the means or plans they adopt for achieving those goals.” To help students understand and connect to research in their field, instructors can assign short articles and guide students through rhetorical practices to make expert thinking more explicit. Provide students multiple opportunities to refine their writing allows them to learn “how to frame their problems.”

The distant past can seem uncomfortably strange to modern observers. As we discuss our class readings, one thing I like to do with my students is to explore what seems weird or even offensive to them about our texts and the societies that produced them. Thinking about the disconnect between ancient and modern attitudes, outlooks, beliefs, and values can be an incredibly productive way to think about cultural difference over space and time. - Professor Jonathan Conant, History and Classics

Critical Thinking Critical thinking is the “ability to assess your assumptions, beliefs, and actions” (Merriam & Bierema, 2014, p. 222) with the intent to change your actions in the future and is necessary when solving problems. It is a skill required during all steps of the problem-solving process. Fostering critical thinking in your students is one way to create a more inclusive classroom because you are inherently asking students to challenge their assumptions and biases.

Instructors can use the following conditions to promote critical thinking in your classroom (Merriam & Bierema, 2014):

Foster critical reflection by examining assumptions (see Promoting Metacognition for specific reflective strategies), e.g., ask students to read a research article and identify possible assumptions that are made in the questions asked, methods used, or the interpretation of the results. For example, to foster critical reflection you could ask students to identify the sources of knowledge they value and use when completing homework and write a reflection on what assumptions they made about those sources. What are the identities of the people creating those sources of knowledge? What systems or people are gatekeepers of that knowledge?
Build a learning community where the expectation is that students can be wholly present, honest, ask questions, and productively fail (Kapur, 2016).
Practice dialogical conversation by teaching an awareness of power relations in the classroom such as microaggressions or micro-affirmations and how to use active listening (see Microaggressions and Micro-aggressions for examples and specific practices).
Provide students the opportunity to make connections between content and their experiences, e.g., by asking students on homework assignments how they apply concepts to a recent experience or asking students why they took your course and how it relates to their career goals.

Collaboration/Teamwork Instructors can develop aspects of problem solving by being intentional about team building, connecting students to alternative perspectives, and being explicit about the expectations of teamwork in the field (e.g., as a researcher, industry partner, consultant, etc.). You can create homework assignments using the TILT framework , which asks students to evaluate both their own and peers’ interactions in teams. There are several models or rubrics for how to assess teamwork, such as the AAC&U Teamwork Value Rubric , which focuses on students’ behaviors or the Comprehensive Assessment of Team Member Effectiveness (CATME) , which is a free packaged tool that gathers information from students and groups them into teams.

We use team-based learning exercises and collaborative problem solving. Students are assigned pre-reading to expand their knowledge so they are able to think through different aspects of a scenario before they come to class. In class, the discussion focuses on a team deciding and agreeing on what the next steps in a case will be. The problem-solving skills that this team discussion focuses on are interpersonal communication, being an active listener, and a collaborative team member. It is not high stakes, but together the team will succeed or fail. - Sarita Warrier, Assistant Dean for Medical Education, Alpert Medical School

A jigsaw is another collaborative approach to teach students how to break up a problem into smaller components. For example, in a class on Romanticism and Romantic philosophies, three groups of students would each be given the following questions about five poems: “How does the writer view nature?” (Group 1), “How does the writer view society?” (Group 2), “How does the writer view the purpose of poetry?” (Group 3). After discussion, three new groups, with representatives from each of these three clusters, might discuss a broader question, such as, “Using the information gathered in the first groups [...] what are Romanticism’s goals? What’s the agenda of the Romantic poets?” (Handstedt, 2018, pp. 121-122).

Reflection Activities or Assignments Expert researchers, practitioners, and educators incorporate reflection and iteration as part of their practice. Key steps of the problem-solving process include being reflective about the process and what is working or not working towards a goal. In a previous newsletter, Promoting Metacognition , the Sheridan Center provided a list of several activities and assignments you could use to help students be reflective in your course. These activities range from short minute papers , to semester-long reflective journals. Think-alouds, or having a student verbally solve a problem with another student, can also help students develop reflective problem-solving skills because it “provides a structure for students to observe both their own and another’s process of learning” (Barkley, 2010, p. 259).

For more strategies on how to engage students in these skills and topics, please see the Sheridan Center’s newsletter, Inclusive Teaching Through Active Learning . It is important to be explicit in how you approach problem solving and convey that information both through your course description, syllabi, and content.

Opportunities at Sheridan for Development of 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 analysis, oral communication, and problem solving. To help you think through how to integrate opportunities for students to problem solve effectively in your course, the Sheridan Center offers problem solving professional development opportunities for faculty and students in an effort to engage intergenerational, faculty-student teaching teams.

Problem-Solving Course Design Institute Increasing assignment transparency is at the core of Problem-Solving Course Design Institute (PSCDI). PSCDI is a two-day workshop for faculty, staff, postdocs, and graduate student teams to (re)design assignments that engage students in the problem-solving process. Upon successful completion, faculty participants will receive a $2,000 grant to implement their ideas. For more information on PSCDI and past recipents, please see this Sheridan web resource .

Problem-Solving Fellows Program Undergraduate students who are currently or plan to be peer educators (e.g., UTAs, lab TAs, peer mentors, etc.) are encouraged to take the course, UNIV 1110: The Theory and Teaching of Problem Solving. Within this course, we focus on developing effective problem solvers through students’ teaching practices. We discuss reflective practices necessary for teaching and problem solving; theoretical frames for effective learning; how culture, context, and identity impact problem solving and teaching; and the impact of the problem-solving cycle. For more information, please see this Sheridan web resource and contact Dr. Christina Smith, Sheridan Center (via [email protected] ).

Berkenkotter, C. (1982). Writing and problem solving. In T. Fulwiler & A. Young (Eds.), Language connections: Writing and reading across the curriculum (pp. 33-44). Urbana, Illinois: National Council of Teachers of English.

Barkley, E.F. (2010). Student engagement techniques: A handbook for college faculty . San Francisco, CA: Jossey-Bass.

Bok, D. (2017). The struggle to reform our colleges. Princeton, NJ: Princeton University Press.

Hanstedt, P. (2018). Creating wicked students: Designing courses for a complex world . Sterling, VA: Stylus.

Hart Research Associates. (2015). Falling short? College learning and career success . Survey carried out for AAC&U. Available: https://www.aacu.org/sites/default/files/files/LEAP/2015employerstudents…

Hora, M.T., Benbow, R. J., & Oleson, A. K.. (2016). Beyond the skills gap: Preparing college students for life and work . Cambridge, MA: Harvard University Press.

Jonassen, D. H. (2000). Toward a design theory of problem solving. Educational technology research and development , 48(4), 63-85.

Kapur, M. (2016). Examining productive failure, productive success, unproductive failure, and unproductive success in learning. Educational Psychologist , 51(2), 289-299.

Merriam, S. B., & Bierema, L. L. (2014). Adult learning: Linking theory and practice . John Wiley & Sons.

Passow, H.J., & Passow, C.H. (2017). What competencies should undergraduate engineering programs emphasize? A systematic review. Journal of Engineering Education , 106(3): 475-526.

Pretz, J.E., Naples, A. J., & Sternbergy, R. J. (2003). Recognizing, defining, and representing problems. In J. E. Davidson & R. J. Sternberg (Eds.), The psychology of problem solving (pp. 3-30). New York: Cambridge University Press.

Winkelmes, M.A., Bernacki, M., Butler, J., Zochowski, M., Golanics, J., & Weavil, K. H. (2016). A teaching intervention that increases underserved college students’ success. Peer Review , 18(1/2), 31–36.

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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

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

Why Use Problem-Based Learning?

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

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

Considerations for Using Problem-Based Learning

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

Students generally must:

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

Getting Started with Problem-Based Learning

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

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

Problem-Solving 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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Great Ideas

Exploring the Teacher’s Role in Problem-Solving

Developing problem-solving strategies this article is the second in a four-part series on problem-solving..

Problem-solving is what we do when we look at a task and don’t know what to do. This makes strategies very important – they are how we begin. When a child looks at a problem and says, “I don’t know,” our role as a teacher is to help them persevere – to stick with it and find a solution.

Strategies are the tools we use to get started when there is no obvious solution path. Look at these two problems below. Which one has the more straightforward solution path? Which is easier to start solving?

For many learners, the problem on the left, while not always easy to solve, is easier to start. There are some numbers given, along with clues about the fact that this involves addition and/or subtraction. The problem on the right has no numbers at all!

Here is a list of problem-solving strategies adapted from the book What’s Your Math Problem? by Linda Gojak. Which of these could you use to start working on the heartbeat problem?

Look for a pattern
Make a model
Solve a simpler problem
Work backward
Identify a sub-goal
Create a table
Create an organized list
Draw a picture or diagram
Account for all possibilities
Create a graph

Some of these strategies are intuitive for many students. How many of your students would act it out or draw a picture to help solve the bus problem? Other strategies require more teacher guidance and coaching to use effectively. When you create an organized list, how do you organize it? What items should be included in your list or table? When you solve a simpler problem, how do you decide how to simplify it?

Historically, problem-solving strategies have been developed chapter-by-chapter in traditional textbooks. Each chapter ends with a section on problem-solving that features a particular strategy. The given strategy is used to solve every problem in that section. This does not develop student thinking; it develops mimicry skills.

To develop these strategies thoughtfully, try this. Pose a problem like the heartbeat problem to your students. Give them time to think individually and to work with a partner or in a small group on the problem. Notice what strategies your students are using. Has one team thought about how many beats in a minute? Has one team drawn a calendar as an organizer for their information? Has one team recorded useful information like the number of days in a year and the number of hours in a day?

Take time to talk about the strategies students are using. For more fun, name the strategies for the students (D’wayne’s strategy or Anissa’s strategy). Allow students to share their work and build on it. If you’re going to list important facts like days in the year and hours in a day, how can you sequence those to be most helpful? That is organizing your list. If students are acting a problem out but spending a great deal of energy on costumes and scripts, encourage them to focus on the elements essential for math class.

Students benefit from seeing a variety of strategies used by their classmates. Some will be more effective or efficient than others. That’s okay. By making problem-solving a regular part of math class, students have opportunities to practice strategies and learn from seeing them used by others. Strategies are discussed as they are used, keeping the focus on how this strategy supports this problem.

In the next post for this series, we will focus on facilitating student discourse and questioning strategies. What are effective teacher moves that encourage students to talk about their thinking? The final post will address fostering perseverance with problem-solving.

If you would like to explore these ideas further, please watch this edWeb webinar about this topic. You’ll see example approaches to the problems included here as well as additional research and information about this topic.

Click HERE to download the resources for this article!

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Problem Solving in Science Learning

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Introduction: Problem Solving in the Science Classroom

Problem solving plays a central role in school science, serving both as a learning goal and as an instructional tool. As a learning goal, problem-solving expertise is considered as a means of promoting both proficiency in solving practice problems and competency in tackling novel problems, a hallmark of successful scientists and engineers. As an instructional tool, problem solving attempts to situate the learning of scientific ideas and practices in an applicative context, thus providing an opportunity to transform science learning into an active, relevant, and motivating experience. Problem solving is also frequently a central strategy in the assessment of students’ performance on various measures (e.g., mastery of procedural skills, conceptual understanding, as well as scientific and learning practices).

A problem is often defined as an unfamiliar task that requires one to make judicious decisions when searching through a problem...

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Yerushalmi, E., Eylon, BS. (2015). Problem Solving in Science Learning. In: Gunstone, R. (eds) Encyclopedia of Science Education. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2150-0_129

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Key Tips On Problem Solving Method Of Teaching

Problem-solving skills are necessary for all strata of life, and none can be better than classroom problem-solving activities. It can be an excellent way to introduce students to problem-solving skills, get them prepped and ready to solve real problems in real-life settings.

The ability to critically analyze a problem, map out all its elements and then prepare a solution that works is one of the most valuable skills; one must acquire in life. Educating your students about problem-solving techniques from an early age can be facilitated with in-class problem-solving activities. Such efforts encourage cognitive and social development and equip students with the tools they will need to tackle and resolve their lives.

So, what is a problem-solving method of teaching ?

Problem Solving is the act of defining a problem; determining the cause of the problem; identifying, prioritizing and selecting alternatives for a solution; and implementing a solution. In a problem-solving method, children learn by working on problems. This skill enables the students to learn new knowledge by facing the problems to be solved. It is expected of them to observe, understand, analyze, interpret, find solutions, and perform applications that lead to a holistic understanding of the concept. This method develops scientific process skills. This method helps in developing a brainstorming approach to learning concepts.

In simple words, problem-solving is an ongoing activity in which we take what we know to discover what we do not know. It involves overcoming obstacles by generating hypotheses, testing those predictions, and arriving at satisfactory solutions.

The problem-solving method involves three basic functions

Seeking information
Generating new knowledge
Making decisions

This post will include key strategies to help you inculcate problem-solving skills in your students.

First and foremostly, follow the 5-step model of problem-solving presented by Wood

Woods' problem-solving model

Identify the problem .

Allow your students to identify the system under study by interpreting the information provided in the problem statement. Then, prepare a list of what is known about the problem, and identify the knowledge needed to understand (and eventually) solve it. Once you have a list of known problems, identifying the unknown(s) becomes simpler. The unknown one is usually the answer to the problem; however, there may be other unknowns. Make sure that your students have a clear understanding of what they are expected to find.

While teaching problem solving, it is very important to have students know how to select, interpret, and use units and symbols. Emphasize the use of units and symbols whenever appropriate. Develop a habit of using appropriate units and symbols yourself at all times. Teach your students to look for the words only and neglect or assume to help identify the constraints.

Furthermore, help students consider from the beginning what a logical type of answer would be. What characteristics will it possess?

Think about it

Use the next stage to ponder the identified problem. Ideally, students will develop an imaginary image of the problem at hand during this stage. They need to determine the required background knowledge from illustrations, examples and problems covered in the course and collect pertinent information such as conversion factors, constants, and tables needed to solve the problem.

Plan a solution

Often, the type of problem will determine the type of solution. Some common problem-solving strategies are: compute; simplify; use an equation; make a model, diagram, table, or chart; or work backwards.

Help your students choose the best strategy by reminding them again what they must find or calculate.

Carry out the plan

Now that the major part of problem-solving has been done start executing the solution. There are possibilities that a plan may 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?

Other tips include

Ask open-ended questions.

When a student seeks help, you might be willing to give them the answer they are looking for so you can both move on. But what is recommend is that instead of giving answers promptly, try using open-ended questions and prompts. For example: ask What do you think will happen if..? Why do you think so? What would you do if you get into such situations? Etc.

Emphasize Process Over Product

For elementary students, reflecting on the process of solving a problem helps them develop a growth mindset. Getting an 'incorrect' response does not have to be a bad thing! What matters most is what they have done to achieve it and how they might change their approach next time. As a teacher, you can help students learn the process of reflection.

Model The Strategies

As children learn creative problem-solving techniques, there will probably be times when they will be frustrated or uncertain. Here are just a few simple ways to model what creative problem-solving looks like and sounds like.

Ask questions in case you don't understand anything.
Admit to not knowing the right answer.
Discuss the many possible outcomes of different situations.
Verbalize what you feel when you come across a problem.
Practising these strategies with your students will help create an environment where struggle, failure and growth are celebrated!

Encourage Grappling

Grappling is not confined to perseverance! This includes critical thinking, asking questions, observing evidence, asking more questions, formulating hypotheses and building a deep understanding of a problem.

There are numerous ways to provide opportunities for students to struggle. All that includes the engineering design process is right! Examples include:

Engineering or creative projects
Design-thinking challenges
Informatics projects
Science experiments

Make problem resolution relevant to the lives of your students

Limiting problem solving to class is a bad idea. This will affect students later in life because problem-solving is an essential part of human life, and we have had a chance to look at it from a mathematical perspective. Such problems are relevant to us, and they are not things that we are supposed to remember or learn but to put into practice in real life. These are things from which we can take very significant life lessons and apply them later in life.

What's your strategy? How do you teach Problem-Solving to your students? Do let us know in the comments.

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Think back to the first problem in this chapter, the ABC Problem . What did you do to solve it? Even if you did not figure it out completely by yourself, you probably worked towards a solution and figured out some things that did not work.

Unlike exercises, there is never a simple recipe for solving a problem. You can get better and better at solving problems, both by building up your background knowledge and by simply practicing. As you solve more problems (and learn how other people solved them), you learn strategies and techniques that can be useful. But no single strategy works every time.

Pólya’s How to Solve It

George Pólya was a great champion in the field of teaching effective problem solving skills. He was born in Hungary in 1887, received his Ph.D. at the University of Budapest, and was a professor at Stanford University (among other universities). He wrote many mathematical papers along with three books, most famously, “How to Solve it.” Pólya died at the age 98 in 1985. [1]

In 1945, Pólya published the short book How to Solve It , which gave a four-step method for solving mathematical problems:

Understand the problem.
Devise a plan.
Carry out the plan.
Looking back.

This is all well and good, but how do you actually do these steps?!?! Steps 1. and 2. are particularly mysterious! How do you “make a plan?” That is where you need some tools in your toolbox, and some experience to draw upon.

Much has been written since 1945 to explain these steps in more detail, but the truth is that they are more art than science. This is where math becomes a creative endeavor (and where it becomes so much fun). We will articulate some useful problem solving strategies, but no such list will ever be complete. This is really just a start to help you on your way. The best way to become a skilled problem solver is to learn the background material well, and then to solve a lot of problems!

We have already seen one problem solving strategy, which we call “Wishful Thinking.” Do not be afraid to change the problem! Ask yourself “what if” questions:

What if the picture was different?
What if the numbers were simpler?
What if I just made up some numbers?

You need to be sure to go back to the original problem at the end, but wishful thinking can be a powerful strategy for getting started.

This brings us to the most important problem solving strategy of all:

Problem Solving Strategy 2 (Try Something!). If you are really trying to solve a problem, the whole point is that you do not know what to do right out of the starting gate. You need to just try something! Put pencil to paper (or stylus to screen or chalk to board or whatever!) and try something. This is often an important step in understanding the problem; just mess around with it a bit to understand the situation and figure out what is going on.

And equally important: If what you tried first does not work, try something else! Play around with the problem until you have a feel for what is going on.

Problem 2 (Payback)

Last week, Alex borrowed money from several of his friends. He finally got paid at work, so he brought cash to school to pay back his debts. First he saw Brianna, and he gave her 1/4 of the money he had brought to school. Then Alex saw Chris and gave him 1/3 of what he had left after paying Brianna. Finally, Alex saw David and gave him 1/2 of what he had remaining. Who got the most money from Alex?

Think/Pair/Share

After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What did you figure out about the problem?

This problem lends itself to two particular strategies. Did you try either of these as you worked on the problem? If not, read about the strategy and then try it out before watching the solution.

Problem Solving Strategy 3 (Draw a Picture). Some problems are obviously about a geometric situation, and it is clear you want to draw a picture and mark down all of the given information before you try to solve it. But even for a problem that is not geometric, like this one, thinking visually can help! Can you represent something in the situation by a picture?

Draw a square to represent all of Alex’s money. Then shade 1/4 of the square — that’s what he gave away to Brianna. How can the picture help you finish the problem?

After you have worked on the problem yourself using this strategy (or if you are completely stuck), you can watch someone else’s solution.

Problem Solving Strategy 4 (Make Up Numbers). Part of what makes this problem difficult is that it is about money, but there are no numbers given. That means the numbers must not be important. So just make them up!

You can work forwards: Assume Alex had some specific amount of money when he showed up at school, say $100. Then figure out how much he gives to each person. Or you can work backwards: suppose he has some specific amount left at the end, like $10. Since he gave Chris half of what he had left, that means he had $20 before running into Chris. Now, work backwards and figure out how much each person got.

Watch the solution only after you tried this strategy for yourself.

If you use the “Make Up Numbers” strategy, it is really important to remember what the original problem was asking! You do not want to answer something like “Everyone got $10.” That is not true in the original problem; that is an artifact of the numbers you made up. So after you work everything out, be sure to re-read the problem and answer what was asked!

Problem 3 (Squares on a Chess Board)

How many squares, of any possible size, are on a 8 × 8 chess board? (The answer is not 64… It’s a lot bigger!)

Remember Pólya’s first step is to understand the problem. If you are not sure what is being asked, or why the answer is not just 64, be sure to ask someone!

Think / Pair / Share

After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What did you figure out about the problem, even if you have not solved it completely?

It is clear that you want to draw a picture for this problem, but even with the picture it can be hard to know if you have found the correct answer. The numbers get big, and it can be hard to keep track of your work. Your goal at the end is to be absolutely positive that you found the right answer. You should never ask the teacher, “Is this right?” Instead, you should declare, “Here’s my answer, and here is why I know it is correct!”

Problem Solving Strategy 5 (Try a Simpler Problem). Pólya suggested this strategy: “If you can’t solve a problem, then there is an easier problem you can solve: find it.” He also said: “If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?” In this case, an 8 × 8 chess board is pretty big. Can you solve the problem for smaller boards? Like 1 × 1? 2 × 2? 3 × 3?

Of course the ultimate goal is to solve the original problem. But working with smaller boards might give you some insight and help you devise your plan (that is Pólya’s step (2)).

Problem Solving Strategy 6 (Work Systematically). If you are working on simpler problems, it is useful to keep track of what you have figured out and what changes as the problem gets more complicated.

For example, in this problem you might keep track of how many 1 × 1 squares are on each board, how many 2 × 2 squares on are each board, how many 3 × 3 squares are on each board, and so on. You could keep track of the information in a table:


1	0	0	0
4	1	0	0
9	4	1	0

Problem Solving Strategy 7 (Use Manipulatives to Help You Investigate). Sometimes even drawing a picture may not be enough to help you investigate a problem. Having actual materials that you move around can sometimes help a lot!

For example, in this problem it can be difficult to keep track of which squares you have already counted. You might want to cut out 1 × 1 squares, 2 × 2 squares, 3 × 3 squares, and so on. You can actually move the smaller squares across the chess board in a systematic way, making sure that you count everything once and do not count anything twice.

Problem Solving Strategy 8 (Look for and Explain Patterns). Sometimes the numbers in a problem are so big, there is no way you will actually count everything up by hand. For example, if the problem in this section were about a 100 × 100 chess board, you would not want to go through counting all the squares by hand! It would be much more appealing to find a pattern in the smaller boards and then extend that pattern to solve the problem for a 100 × 100 chess board just with a calculation.

If you have not done so already, extend the table above all the way to an 8 × 8 chess board, filling in all the rows and columns. Use your table to find the total number of squares in an 8 × 8 chess board. Then:

Describe all of the patterns you see in the table.
Can you explain and justify any of the patterns you see? How can you be sure they will continue?
What calculation would you do to find the total number of squares on a 100 × 100 chess board?

(We will come back to this question soon. So if you are not sure right now how to explain and justify the patterns you found, that is OK.)

Problem 4 (Broken Clock)

This clock has been broken into three pieces. If you add the numbers in each piece, the sums are consecutive numbers. ( Consecutive numbers are whole numbers that appear one after the other, such as 1, 2, 3, 4 or 13, 14, 15.)

Can you break another clock into a different number of pieces so that the sums are consecutive numbers? Assume that each piece has at least two numbers and that no number is damaged (e.g. 12 isn’t split into two digits 1 and 2.)

Remember that your first step is to understand the problem. Work out what is going on here. What are the sums of the numbers on each piece? Are they consecutive?

After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What progress have you made?

Problem Solving Strategy 9 (Find the Math, Remove the Context). Sometimes the problem has a lot of details in it that are unimportant, or at least unimportant for getting started. The goal is to find the underlying math problem, then come back to the original question and see if you can solve it using the math.

In this case, worrying about the clock and exactly how the pieces break is less important than worrying about finding consecutive numbers that sum to the correct total. Ask yourself:

What is the sum of all the numbers on the clock’s face?
Can I find two consecutive numbers that give the correct sum? Or four consecutive numbers? Or some other amount?
How do I know when I am done? When should I stop looking?

Of course, solving the question about consecutive numbers is not the same as solving the original problem. You have to go back and see if the clock can actually break apart so that each piece gives you one of those consecutive numbers. Maybe you can solve the math problem, but it does not translate into solving the clock problem.

Problem Solving Strategy 10 (Check Your Assumptions). When solving problems, it is easy to limit your thinking by adding extra assumptions that are not in the problem. Be sure you ask yourself: Am I constraining my thinking too much?

In the clock problem, because the first solution has the clock broken radially (all three pieces meet at the center, so it looks like slicing a pie), many people assume that is how the clock must break. But the problem does not require the clock to break radially. It might break into pieces like this:

Were you assuming the clock would break in a specific way? Try to solve the problem now, if you have not already.

Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY 2.0 (http://creativecommons.org/licenses/by/2.0)], via Wikimedia Commons ↵

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Instructional Strategies for Teaching Problem Solving

Jesse holds two masters, a doctorate and has 15 years of academic experience in areas of education, linguistics, business and science across five continents.

Active vs. passive, ask-don't-tell, contextual inference, input hypothesis, scaffolding, lesson summary.

Entering his second year as a teacher, Mark wants to ensure that his instruction is more efficient and effective than his first year in the classroom. Mark reflects on how he interacted with his students; specifically, how he taught the material. Through reflection and reviewing last year's lesson plans, he realizes that he was mostly using traditional strategies, such as lecture, repetition, memorization, and other passive techniques. This year, Mark wants students to think actively rather than passively. His new plan is to integrate more strategies for teaching problem solving. Let's look at four field-tested strategies that Mark can implement.

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0:04 Active vs. Passive
0:42 Ask-Don't-Tell
2:06 Contextual Inference
3:02 Input Hypothesis
4:06 Scaffolding
5:59 Lesson Summary

Ask-don't-tell (ADT) is a teaching strategy that obligates the student to think actively rather than passively. Consider the following dialogue:

Student: ''What kind of writing does the author use in this article?''
Teacher: ''This article is an example of persuasive writing.''
Student: ''Ok, ok. Got it.''

In this scenario, it is clear that the teacher is simply providing information to the student. This is a passive process that is considered to be rather efficient, but not very effective. Now, let us consider a similar dialogue where the teacher uses ADT:

Teacher: ''What are the four types of writing we have studied?''
Student: ''We have studied expository , persuasive , descriptive, and narrative styles.''
Teacher: ''Good. Let's look at the article carefully. What characteristics best match one of the writing styles?''
Student: ''Oh, I see! The author uses arguments in an attempt to convince the reader to think in a certain way. This is definitely persuasive writing.''

In this second scenario, the teacher crafts strategic questions to redirect the student in a way that requires active thinking. Although perhaps more time consuming, this strategy of teaching problem solving tends to be much more effective.

Contextual inference (CI) is a strategy whereby the teacher provides sufficient context in order for the student to infer the missing information. Let's pose the following one-line incomplete sentence to the student:

Teacher: ''Birds can _____.''
Student: ''Sing? Chirp? Fly? Walk?''

In this scenario, many correct answers are possible, but it is unclear exactly what term the teacher is trying to elicit. This leads to confusion and frustration for everyone involved.

Now, let's consider the same sentence, but with more context:

Teacher: ''People can walk . Fish can swim . Birds can _____.''
Student: ''Oh, fly !''

Here, the student can identify a pattern: the concept of how different creatures move according to their nature. The anticipated term is immediately identified by the student due to the use of CI by the teacher.

The input hypothesis (or simply i+1 ) was first proposed by education researcher Stephen Krashen in the 1970s. Although i+1 is commonly associated with second language instruction, the hypothesis may be used throughout most academic content areas.

The idea is to strategically push students above their current level of comprehension (i) by increasing the difficulty level bit by bit (+1). The success in using this strategy depends on finding a balance between material that is too easy or too hard for the student; if the student is not challenged, he or she may lose interest. Conversely, if the task is too difficult, the student may become frustrated and give up.

An example in a science class would be to have the student not only describe how photosynthesis works, but to explain how it is important in ecology. In this sense, we are moving the learner from the cognitive activity of description (basic ability) to the cognitive activity of explaining (advanced ability), which directly helps to build critical thinking and problem solving skills.

Scaffolding in teaching is similar to scaffolding in construction: temporary supports are used to help a student practice and improve a target skill, at which time the scaffolding is either restructured for continual improvement, or removed completely. This allows the student to have guidance during the learning process, but also ensures that the student is able to eventually solve problems independently.

Suppose you are working with English language learners (or young children who are learning English as their native language). We know that most words have fixed syllabic stress patterns. Consider the following stress patterns:

Photograph: Ooo
Photographer: oOoo
Photographic: ooOo
Photographically: ooOooo

In this situation, the stressed syllable may come in the first, second, or even third position. The teacher establishes a system of communicating syllabic stress patterns without providing the solution (this is similar to, and may be used in conjunction with, ADT). Consider the following dialogue:

Student: ''How do I pronounce this word ?''
Teacher: ''Listen. PhotoGRAPHic .''
Teacher: ''How many parts?''
Student: ''Four.''
Teacher: ''Good. It looks like...?''
Student: ''Small, small, big, small?''
Teacher: ''Yes, well done! Now have a try.''
Student: ''Pho...to...GRAPH...ic. PhotoGRAPHic.''

In this way, students learn how to use the syllabic stress pattern system to independently figure out the structure and pronunciation of unknown words in the future as long as they have an auditory example.

This lesson introduced strategies and examples for teaching problem solving skills, including:

Ask, don't tell (ADT) : Helping students to think actively versus passively.
Contextual inference (CI) : Providing sufficient context to elicit target responses.
The input hypothesis (i+1) : Crafting challenging yet achievable tasks.
Scaffolding : Providing temporary supports to be removed or restructured as the learner progresses.

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Module 1: Problem Solving Strategies

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Pólya’s How to Solve It

George Pólya was a great champion in the field of teaching effective problem solving skills. He was born in Hungary in 1887, received his Ph.D. at the University of Budapest, and was a professor at Stanford University (among other universities). He wrote many mathematical papers along with three books, most famously, “How to Solve it.” Pólya died at the age 98 in 1985.1

1. Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY

$Screen Shot 2018-08-30 at 4.43.05 PM.png$

In 1945, Pólya published the short book How to Solve It , which gave a four-step method for solving mathematical problems:

First, you have to understand the problem.

After understanding, then make a plan.

Carry out the plan.

Look back on your work. How could it be better?

Problem Solving Strategy 1 (Guess and Test)

Make a guess and test to see if it satisfies the demands of the problem. If it doesn't, alter the guess appropriately and check again. Keep doing this until you find a solution.

Mr. Jones has a total of 25 chickens and cows on his farm. How many of each does he have if all together there are 76 feet?

Step 1: Understanding the problem

We are given in the problem that there are 25 chickens and cows.

All together there are 76 feet.

Chickens have 2 feet and cows have 4 feet.

We are trying to determine how many cows and how many chickens Mr. Jones has on his farm.

Step 2: Devise a plan

Going to use Guess and test along with making a tab

Many times the strategy below is used with guess and test.

Make a table and look for a pattern:

Procedure: Make a table reflecting the data in the problem. If done in an orderly way, such a table will often reveal patterns and relationships that suggest how the problem can be solved.

Step 3: Carry out the plan:

Chickens	Cows	Number of chicken feet	Number of cow feet	Total number of feet
20	5	40	20	60
21	4	42	16	58

Notice we are going in the wrong direction! The total number of feet is decreasing!

Better! The total number of feet are increasing!

15	10	30	40	70
12	13	24	52	76

Step 4: Looking back:

Check: 12 + 13 = 25 heads

24 + 52 = 76 feet.

We have found the solution to this problem. I could use this strategy when there are a limited number of possible answers and when two items are the same but they have one characteristic that is different.

Videos to watch:

1. Click on this link to see an example of “Guess and Test”

http://www.mathstories.com/strategies.htm

2. Click on this link to see another example of Guess and Test.

http://www.mathinaction.org/problem-solving-strategies.html

Check in question 1:

$clipboard_e6298bbd7c7f66d9eb9affcd33892ef0d.png$

Place the digits 8, 10, 11, 12, and 13 in the circles to make the sums across and vertically equal 31. (5 points)

Check in question 2:

Old McDonald has 250 chickens and goats in the barnyard. Altogether there are 760 feet . How many of each animal does he have? Make sure you use Polya’s 4 problem solving steps. (12 points)

Problem Solving Strategy 2 (Draw a Picture). Some problems are obviously about a geometric situation, and it is clear you want to draw a picture and mark down all of the given information before you try to solve it. But even for a problem that is not geometric thinking visually can help!

Videos to watch demonstrating how to use "Draw a Picture".

1. Click on this link to see an example of “Draw a Picture”

2. Click on this link to see another example of Draw a Picture.

Problem Solving Strategy 3 ( Using a variable to find the sum of a sequence.)

Gauss's strategy for sequences.

last term = fixed number ( n -1) + first term

The fix number is the the amount each term is increasing or decreasing by. "n" is the number of terms you have. You can use this formula to find the last term in the sequence or the number of terms you have in a sequence.

Ex: 2, 5, 8, ... Find the 200th term.

Last term = 3(200-1) +2

Last term is 599.

To find the sum of a sequence: sum = [(first term + last term) (number of terms)]/ 2

Sum = (2 + 599) (200) then divide by 2

Sum = 60,100

Check in question 3: (10 points)

Find the 320 th term of 7, 10, 13, 16 …

Then find the sum of the first 320 terms.

Problem Solving Strategy 4 (Working Backwards)

This is considered a strategy in many schools. If you are given an answer, and the steps that were taken to arrive at that answer, you should be able to determine the starting point.

Videos to watch demonstrating of “Working Backwards”

https://www.youtube.com/watch?v=5FFWTsMEeJw

Karen is thinking of a number. If you double it, and subtract 7, you obtain 11. What is Karen’s number?

1. We start with 11 and work backwards.

2. The opposite of subtraction is addition. We will add 7 to 11. We are now at 18.

3. The opposite of doubling something is dividing by 2. 18/2 = 9

4. This should be our answer. Looking back:

9 x 2 = 18 -7 = 11

5. We have the right answer.

Check in question 4:

Christina is thinking of a number.

If you multiply her number by 93, add 6, and divide by 3, you obtain 436. What is her number? Solve this problem by working backwards. (5 points)

Problem Solving Strategy 5 (Looking for a Pattern)

Definition: A sequence is a pattern involving an ordered arrangement of numbers.

We first need to find a pattern.

Ask yourself as you search for a pattern – are the numbers growing steadily larger? Steadily smaller? How is each number related?

Example 1: 1, 4, 7, 10, 13…

Find the next 2 numbers. The pattern is each number is increasing by 3. The next two numbers would be 16 and 19.

Example 2: 1, 4, 9, 16 … find the next 2 numbers. It looks like each successive number is increase by the next odd number. 1 + 3 = 4.

So the next number would be

25 + 11 = 36

Example 3: 10, 7, 4, 1, -2… find the next 2 numbers.

In this sequence, the numbers are decreasing by 3. So the next 2 numbers would be -2 -3 = -5

-5 – 3 = -8

Example 4: 1, 2, 4, 8 … find the next two numbers.

This example is a little bit harder. The numbers are increasing but not by a constant. Maybe a factor?

So each number is being multiplied by 2.

16 x 2 = 32

1. Click on this link to see an example of “Looking for a Pattern”

2. Click on this link to see another example of Looking for a Pattern.

Problem Solving Strategy 6 (Make a List)

Example 1 : Can perfect squares end in a 2 or a 3?

List all the squares of the numbers 1 to 20.

1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400.

Now look at the number in the ones digits. Notice they are 0, 1, 4, 5, 6, or 9. Notice none of the perfect squares end in 2, 3, 7, or 8. This list suggests that perfect squares cannot end in a 2, 3, 7 or 8.

How many different amounts of money can you have in your pocket if you have only three coins including only dimes and quarters?

Quarter’s dimes

0 3 30 cents

1 2 45 cents

2 1 60 cents

3 0 75 cents

Videos demonstrating "Make a List"

Check in question 5:

How many ways can you make change for 23 cents using only pennies, nickels, and dimes? (10 points)

Problem Solving Strategy 7 (Solve a Simpler Problem)

Geometric Sequences:

How would we find the nth term?

Solve a simpler problem:

1, 3, 9, 27.

1. To get from 1 to 3 what did we do?

2. To get from 3 to 9 what did we do?

Let’s set up a table:

Term Number what did we do

$what is problem solving as a teaching strategy$

Looking back: How would you find the nth term?

$what is problem solving as a teaching strategy$

Find the 10 th term of the above sequence.

Let L = the tenth term

$what is problem solving as a teaching strategy$

Problem Solving Strategy 8 (Process of Elimination)

This strategy can be used when there is only one possible solution.

I’m thinking of a number.

The number is odd.

It is more than 1 but less than 100.

It is greater than 20.

It is less than 5 times 7.

The sum of the digits is 7.

It is evenly divisible by 5.

a. We know it is an odd number between 1 and 100.

b. It is greater than 20 but less than 35

21, 23, 25, 27, 29, 31, 33, 35. These are the possibilities.

c. The sum of the digits is 7

21 (2+1=3) No 23 (2+3 = 5) No 25 (2 + 5= 7) Yes Using the same process we see there are no other numbers that meet this criteria. Also we notice 25 is divisible by 5. By using the strategy elimination, we have found our answer.

Check in question 6: (8 points)

Jose is thinking of a number.

The number is not odd.

The sum of the digits is divisible by 2.

The number is a multiple of 11.

It is greater than 5 times 4.

It is a multiple of 6

It is less than 7 times 8 +23

What is the number?

Click on this link for a quick review of the problem solving strategies.

https://garyhall.org.uk/maths-problem-solving-strategies.html

Teaching Methods and Strategies: The Complete Guide

You’ve completed your coursework. Student teaching has ended. You’ve donned the cap and gown, crossed the stage, smiled with your diploma and went home to fill out application after application.

Suddenly you are standing in what will be your classroom for the next year and after the excitement of decorating it wears off and you begin lesson planning, you start to notice all of your lessons are executed the same way, just with different material. But that is what you know and what you’ve been taught, so you go with it.

After a while, your students are bored, and so are you. There must be something wrong because this isn’t what you envisioned teaching to be like. There is.

Figuring out the best ways you can deliver information to students can sometimes be even harder than what students go through in discovering how they learn best. The reason is because every single teacher needs a variety of different teaching methods in their theoretical teaching bag to pull from depending on the lesson, the students, and things as seemingly minute as the time the class is and the subject.

Using these different teaching methods, which are rooted in theory of different teaching styles, will not only help teachers reach their full potential, but more importantly engage, motivate and reach the students in their classes, whether in person or online.

Teaching Methods

Teaching methods, or methodology, is a narrower topic because it’s founded in theories and educational psychology. If you have a degree in teaching, you most likely have heard of names like Skinner, Vygotsky , Gardner, Piaget , and Bloom . If their names don’t ring a bell, you should definitely recognize their theories that have become teaching methods. The following are the most common teaching theories.

Behaviorism

Behaviorism is the theory that every learner is essentially a “clean slate” to start off and shaped by emotions. People react to stimuli, reactions as well as positive and negative reinforcement, the site states.

Learning Theories names the most popular theorists who ascribed to this theory were Ivan Pavlov, who many people may know with his experiments with dogs. He performed an experiment with dogs that when he rang a bell, the dogs responded to the stimuli; then he applied the idea to humans.

Other popular educational theorists who were part of behaviorism was B.F. Skinner and Albert Bandura .

Social Cognitive Theory

Social Cognitive Theory is typically spoken about at the early childhood level because it has to do with critical thinking with the biggest concept being the idea of play, according to Edwin Peel writing for Encyclopedia Britannica . Though Bandura and Lev Vygotsky also contributed to cognitive theory, according to Dr. Norman Herr with California State University , the most popular and first theorist of cognitivism is Piaget.

There are four stages to Piaget’s Theory of Cognitive Development that he created in 1918. Each stage correlates with a child’s development from infancy to their teenage years.

The first stage is called the Sensorimotor Stage which occurs from birth to 18 months. The reason this is considered cognitive development is because the brain is literally growing through exploration, like squeaking horns, discovering themselves in mirrors or spinning things that click on their floor mats or walkers; creating habits like sleeping with a certain blanket; having reflexes like rubbing their eyes when tired or thumb sucking; and beginning to decipher vocal tones.

The second stage, or the Preoperational Stage, occurs from ages 2 to 7 when toddlers begin to understand and correlate symbols around them, ask a lot of questions, and start forming sentences and conversations, but they haven’t developed perspective yet so empathy does not quite exist yet, the website states. This is the stage when children tend to blurt out honest statements, usually embarrassing their parents, because they don’t understand censoring themselves either.

From ages 7 to 11, children are beginning to problem solve, can have conversations about things they are interested in, are more aware of logic and develop empathy during the Concrete Operational Stage.

The final stage, called the Formal Operational Stage, though by definition ends at age 16, can continue beyond. It involves deeper thinking and abstract thoughts as well as questioning not only what things are but why the way they are is popular, the site states. Many times people entering new stages of their lives like high school, college, or even marriage go through elements of Piaget’s theory, which is why the strategies that come from this method are applicable across all levels of education.

The Multiple Intelligences Theory

The Multiple Intelligences Theory states that people don’t need to be smart in every single discipline to be considered intelligent on paper tests, but that people excel in various disciplines, making them exceptional.

Created in 1983, the former principal in the Scranton School District in Scranton, PA, created eight different intelligences, though since then two others have been debated of whether to be added but have not yet officially, according to the site.

The original eight are musical, spatial, linguistic, mathematical, kinesthetic, interpersonal, intrapersonal and naturalistic and most people have a predominant intelligence followed by others. For those who are musically-inclined either via instruments, vocals, has perfect pitch, can read sheet music or can easily create music has Musical Intelligence.

Being able to see something and rearrange it or imagine it differently is Spatial Intelligence, while being talented with language, writing or avid readers have Linguistic Intelligence. Kinesthetic Intelligence refers to understanding how the body works either anatomically or athletically and Naturalistic Intelligence is having an understanding of nature and elements of the ecosystem.

The final intelligences have to do with personal interactions. Intrapersonal Intelligence is a matter of knowing oneself, one’s limits, and their inner selves while Interpersonal Intelligence is knowing how to handle a variety of other people without conflict or knowing how to resolve it, the site states. There is still an elementary school in Scranton, PA named after their once-principal.

Constructivism

Constructivism is another theory created by Piaget which is used as a foundation for many other educational theories and strategies because constructivism is focused on how people learn. Piaget states in this theory that people learn from their experiences. They learn best through active learning , connect it to their prior knowledge and then digest this information their own way. This theory has created the ideas of student-centered learning in education versus teacher-centered learning.

Universal Design for Learning

The final method is the Universal Design for Learning which has redefined the educational community since its inception in the mid-1980s by David H. Rose. This theory focuses on how teachers need to design their curriculum for their students. This theory really gained traction in the United States in 2004 when it was presented at an international conference and he explained that this theory is based on neuroscience and how the brain processes information, perform tasks and get excited about education.

The theory, known as UDL, advocates for presenting information in multiple ways to enable a variety of learners to understand the information; presenting multiple assessments for students to show what they have learned; and learn and utilize a student’s own interests to motivate them to learn, the site states. This theory also discussed incorporating technology in the classroom and ways to educate students in the digital age.

Teaching Styles

From each of the educational theories, teachers extract and develop a plethora of different teaching styles, or strategies. Instructors must have a large and varied arsenal of strategies to use weekly and even daily in order to build rapport, keep students engaged and even keep instructors from getting bored with their own material. These can be applicable to all teaching levels, but adaptations must be made based on the student’s age and level of development.

Differentiated instruction is one of the most popular teaching strategies, which means that teachers adjust the curriculum for a lesson, unit or even entire term in a way that engages all learners in various ways, according to Chapter 2 of the book Instructional Process and Concepts in Theory and Practice by Celal Akdeniz . This means changing one’s teaching styles constantly to fit not only the material but more importantly, the students based on their learning styles.

Learning styles are the ways in which students learn best. The most popular types are visual, audio, kinesthetic and read/write , though others include global as another type of learner, according to Akdeniz . For some, they may seem self-explanatory. Visual learners learn best by watching the instruction or a demonstration; audio learners need to hear a lesson; kinesthetic learners learn by doing, or are hands-on learners; read/write learners to best by reading textbooks and writing notes; and global learners need material to be applied to their real lives, according to The Library of Congress .

There are many activities available to instructors that enable their students to find out what kind of learner they are. Typically students have a main style with a close runner-up, which enables them to learn best a certain way but they can also learn material in an additional way.

When an instructor knows their students and what types of learners are in their classroom, instructors are able to then differentiate their instruction and assignments to those learning types, according to Akdeniz and The Library of Congress. Learn more about different learning styles.

When teaching new material to any type of learner, is it important to utilize a strategy called scaffolding . Scaffolding is based on a student’s prior knowledge and building a lesson, unit or course from the most foundational pieces and with each step make the information more complicated, according to an article by Jerry Webster .

To scaffold well, a teacher must take a personal interest in their students to learn not only what their prior knowledge is but their strengths as well. This will enable an instructor to base new information around their strengths and use positive reinforcement when mistakes are made with the new material.

There is an unfortunate concept in teaching called “teach to the middle” where instructors target their lessons to the average ability of the students in their classroom, leaving slower students frustrated and confused, and above average students frustrated and bored. This often results in the lower- and higher-level students scoring poorly and a teacher with no idea why.

The remedy for this is a strategy called blended learning where differentiated instruction is occurring simultaneously in the classroom to target all learners, according to author and educator Juliana Finegan . In order to be successful at blended learning, teachers once again need to know their students, how they learn and their strengths and weaknesses, according to Finegan.

Blended learning can include combining several learning styles into one lesson like lecturing from a PowerPoint – not reading the information on the slides — that includes cartoons and music associations while the students have the print-outs. The lecture can include real-life examples and stories of what the instructor encountered and what the students may encounter. That example incorporates four learning styles and misses kinesthetic, but the activity afterwards can be solely kinesthetic.

A huge component of blended learning is technology. Technology enables students to set their own pace and access the resources they want and need based on their level of understanding, according to The Library of Congress . It can be used three different ways in education which include face-to-face, synchronously or asynchronously . Technology used with the student in the classroom where the teacher can answer questions while being in the student’s physical presence is known as face-to-face.

Synchronous learning is when students are learning information online and have a teacher live with them online at the same time, but through a live chat or video conferencing program, like Skype, or Zoom, according to The Library of Congress.

Finally, asynchronous learning is when students take a course or element of a course online, like a test or assignment, as it fits into their own schedule, but a teacher is not online with them at the time they are completing or submitting the work. Teachers are still accessible through asynchronous learning but typically via email or a scheduled chat meeting, states the Library of Congress.

The final strategy to be discussed actually incorporates a few teaching strategies, so it’s almost like blended teaching. It starts with a concept that has numerous labels such as student-centered learning, learner-centered pedagogy, and teacher-as-tutor but all mean that an instructor revolves lessons around the students and ensures that students take a participatory role in the learning process, known as active learning, according to the Learning Portal .

In this model, a teacher is just a facilitator, meaning that they have created the lesson as well as the structure for learning, but the students themselves become the teachers or create their own knowledge, the Learning Portal says. As this is occurring, the instructor is circulating the room working as a one-on-one resource, tutor or guide, according to author Sara Sanchez Alonso from Yale’s Center for Teaching and Learning. For this to work well and instructors be successful one-on-one and planning these lessons, it’s essential that they have taken the time to know their students’ history and prior knowledge, otherwise it can end up to be an exercise in futility, Alonso said.

Some activities teachers can use are by putting students in groups and assigning each student a role within the group, creating reading buddies or literature circles, making games out of the material with individual white boards, create different stations within the classroom for different skill levels or interest in a lesson or find ways to get students to get up out of their seats and moving, offers Fortheteachers.org .

There are so many different methodologies and strategies that go into becoming an effective instructor. A consistent theme throughout all of these is for a teacher to take the time to know their students because they care, not because they have to. When an instructor knows the stories behind the students, they are able to design lessons that are more fun, more meaningful, and more effective because they were designed with the students’ best interests in mind.

There are plenty of pre-made lessons, activities and tests available online and from textbook publishers that any teacher could use. But you need to decide if you want to be the original teacher who makes a significant impact on your students, or a pre-made teacher a student needs to get through.

Read Also: – Blended Learning Guide – Collaborative Learning Guide – Flipped Classroom Guide – Game Based Learning Guide – Gamification in Education Guide – Holistic Education Guide – Maker Education Guide – Personalized Learning Guide – Place-Based Education Guide – Project-Based Learning Guide – Scaffolding in Education Guide – Social-Emotional Learning Guide

21 Essential Strategies in Teaching Math

Even veteran teachers need to read these.

$Examples of math strategies such as playing addition tic tac toe and emphasizing hands-on learning with manipulatives like dice, play money, dominoes and base ten blocks.$

We all want our kids to succeed in math. In most districts, standardized tests measure students’ understanding, yet nobody wants to teach to the test. Over-reliance on test prep materials and “drill and kill” worksheets steal instructional time while also harming learning and motivation. But sound instruction and good test scores aren’t mutually exclusive. Being intentional and using creative approaches to your instruction can get students excited about math. These essential strategies in teaching mathematics can make this your class’s best math year ever!

1. Raise the bar for all

WeAreTeachers

For math strategies to be effective, teachers must first get students to believe that they can be great mathematicians. Holding high expectations for all students encourages growth. As early as second grade, girls have internalized the idea that math is not for them . It can be a challenge to overcome the socially acceptable thought, I’m not good at math , says Sarah Bax, a math teacher at Hardy Middle School in Washington, D.C.

Rather than success being a function of how much math talent they’re born with, kids need to hear from teachers that anyone who works hard can succeed. “It’s about helping kids have a growth mindset ,” says Bax. “Practice and persistence make you good at math.” Build math equity and tell students about the power and importance of math with enthusiasm and high expectations.

(Psst … you can snag our growth mindset posters for your math classroom here. )

2. Don’t wait—act now!

Look ahead to the specific concepts students need to master for annual end-of-year tests, and pace instruction accordingly. Think about foundational skills they will need in the year ahead.

“You don’t want to be caught off guard come March thinking that students need to know X for the tests the next month,” says Skip Fennell, project director of Elementary Mathematics Specialists and Teacher Leaders Project and professor emeritus at McDaniel College in Westminster, Maryland. Know the specific standards and back-map your teaching from the fall so students are ready, and plan to use effective math strategies accordingly.

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3. Create a testing pathway

You may not even see the results of standardized tests until next school year, but you have to prepare students for it now. Use formative assessments to ensure that students understand the concepts. What you learn can guide your instruction and determine the next steps, says Fennell. “I changed the wording because I didn’t want to suggest that we are in favor of ‘teaching to the test.'”

Testing is not something separate from your instruction. It should be integrated into your planning. Instead of a quick exit question or card, give a five-minute quiz, an open-ended question, or a meaningful homework assignment to confirm students have mastered the math skill covered in the day’s lesson. Additionally, asking students to explain their thinking orally or in writing is a great way to determine their level of understanding. A capable digital resource, designed to monitor your students in real-time, can also be an invaluable tool, providing actionable data to inform your instruction along the way.

4. Observe, modify, and reevaluate

Sometimes we get stuck in a mindset of “a lesson a day” in order to get through the content. However, we should keep our pacing flexible, or kids can fall behind. Walk through your classroom as students work on problems and observe the dynamics. Talk with students individually and include “hinge questions” in your lesson plans to gauge understanding before continuing, suggests Fennell. In response, make decisions to go faster or slower or put students in groups.

5. Read, read, read!

Cover of Pitter Pattern and Equal Shmequal books for teaching 2nd grade as example of strategies in teaching mathematics

Although we don’t often think of reading as a math strategy, there’s almost nothing better to get students ready to learn a new concept than a great read-aloud. Kids love to be read to, and the more we show students how math is connected to the world around us, the more invested they become. Reading books with math connections helps children see how abstract concepts connect to their lives.

6. Personalize and offer choice

When students are given the opportunity to choose how they learn and demonstrate their understanding of a concept, their buy-in and motivation increase. It gives them the chance to understand their preferred learning style, provides agency over their own learning, and allows for the space to practice different strategies to solve math problems. Give students a variety of options, such as timed exercises, projects, or different materials , to show that they’ve mastered foundational skills. As students show what they’ve learned, teachers can track understanding, figure out where students need additional scaffolding or other assistance, and tailor lessons accordingly.

7. Plant the seeds!

Leave no child inside! A school garden is a great way to apply math concepts in a fun way while instilling a sense of purpose in your students. Measurement, geometry, and data analysis are obvious topics that can be addressed through garden activities, but also consider using the garden to teach operations, fractions, and decimals. Additionally, garden activities can help promote character education goals like cooperation, respect for the earth, and, if the crops are donated to organizations that serve those in need, the value of giving to others.

8. Add apps appropriately

The number of apps (interactive software used on touch-screen devices) available to support math instruction has increased rapidly in recent years. Kids who are reluctant to practice math facts with traditional pencil-and-paper resources will gladly do essentially the same work as long as it’s done on a touch screen. Many apps focus on practice via games, but there are some that encourage children to explore the content at a conceptual level.

9. Encourage math talk

$Lets Talk Math poster on wall next to backpack.$

Communicating about math helps students process new learning and build on their thinking. Engage students during conversations and have them describe why they solved a problem in a certain way. “My goal is to get information about what students are thinking and use that to guide my instruction, as opposed to just telling them information and asking them to parrot things back,” says Delise Andrews, who taught math (K–8) and is now a grade 3–5 math coordinator in the Lincoln Public Schools in Nebraska.

Instead of seeking a specific answer, Andrews wants to have deeper discussions to figure out what a student knows and understands. “True learning happens a lot around talking and doing math—not just drilling,” she says. Of course, this math strategy not only requires students to feel comfortable expressing their mathematical thinking, but also assumes that they have been trained to listen respectfully to the reasoning of their classmates.

Learn more: Free Let’s Talk Math Poster

10. The art of math

Almost all kids love art, and visual learners need a math strategy that works for them too, so consider integrating art and math instruction for one of the easiest strategies in teaching mathematics. Many concepts in geometry, such as shapes, symmetry, and transformations (slides, flips, and turns), can be applied in a fun art project. Also consider using art projects to teach concepts like measurement, ratios, and arrays (multiplication/division).

11. Seek to develop understanding

Meaningful math education goes beyond memorizing formulas and procedures. Memorization does not foster understanding. Set high goals, create space for exploration, and work with the students to develop a strong foundation. “Treat the kids like mathematicians,” says Andrews. Present a broad topic, review various strategies for solving a problem, and then elicit a formula or idea from the kids rather than starting with the formula. This creates a stronger conceptual understanding and mental connections with the material for the student.

12. Give students time to reflect

Sometimes teachers get so caught up in meeting the demands of the curriculum and the pressure to “get it all done” that they don’t give students the time to reflect on their learning. Students can be asked to reflect in writing at the end of an assignment or lesson, via class or small group discussion, or in interviews with the teacher. It’s important to give students the time to think about and articulate the meaning of what they’ve learned, what they still don’t understand, and what they want to learn more about. This provides useful information for the teacher and helps the student monitor their own progress and think strategically about how they approach mathematics.

13. Allow for productive struggle

When giving students an authentic problem, ask a big question and let them struggle to figure out several ways to solve it, suggests Andrews. “Your job as a teacher is to make it engaging by asking the right questions at the right time. So you don’t take away their thinking, but you help them move forward to a solution,” she says.

Provide as little information as possible but enough so students can be productive. Effective math teaching supports students as they grapple with mathematical ideas and relationships. Allow them to discover what works and experience setbacks along the way as they adopt a growth mindset about mathematics.

14. Emphasize hands-on learning

Different types of math manipulatives like blocks, play money, and dice.

WeAreTeachers; Teacher Created Resources

In math, there’s so much that’s abstract. Hands-on learning is a strategy that helps make the conceptual concrete. Consider incorporating math manipulatives whenever possible. For example, you can use LEGO bricks to teach a variety of math skills, including finding area and perimeter and understanding multiplication.

15. Build excitement by rewarding progress

Students—especially those who haven’t experienced success—can have negative attitudes about math. Consider having students earn points and receive certificates, stickers, badges, or trophies as they progress. Weekly announcements and assemblies that celebrate the top players and teams can be really inspiring for students. “Having that recognition and moment is powerful,” says Bax. “Through repeated practice, they get better, and they are motivated.” Through building excitement, this allows for one of the best strategies in teaching mathematics to come to fruition.

16. Choose meaningful tasks

Kids get excited about math when they have to solve real-life problems. For instance, when teaching sixth graders how to determine area, present tasks related to a house redesign, suggests Fennell. Provide them with the dimensions of the walls and the size of the windows and have them determine how much space is left for the wallpaper. Or ask them to consider how many tiles they would need to fill a deck. You can absolutely introduce problem-based learning, even in a virtual world.

17. Play math games

$Collage of First Grade Math Games, including Shape Guess Who? and Addition Tic-Tac-Toe$

Life Between Summers/Shape Guess Who via lifebetweensummers.com; 123 Homeschool 4 Me/Tic-Tac-Toe Math Game via 123homeschool4me.com; WeAreTeachers

Student engagement and participation can be a challenge, especially if you’re relying heavily on worksheets. Games, like these first grade math games , are an excellent way to make the learning more fun while simultaneously promoting strategic mathematical thinking, computational fluency , and understanding of operations. Games are especially good for kinesthetic learners and foster a home-school connection when they’re sent home for extra practice.

18. Set up effective math routines

Students generally feel confident and competent in the classroom when they know what to do and why they’re doing it. Establishing routines in your math class and training kids to use them can make math class efficient, effective, and fun! For example, consider starting your class with a number sense routine . Rich, productive small group math discussions don’t happen by themselves, so make sure your students know the “rules of the road” for contributing their ideas and respectfully critiquing the ideas of others.

19. Encourage teacher teamwork and reflection

You can’t teach in a vacuum. Collaborate with other teachers to improve your math instruction skills. Start by discussing the goal for the math lesson and what it will look like, and plan as a team to use the most effective math strategies. “Together, think through the tasks and possible student responses you might encounter,” says Andrews. Reflect on what did and didn’t work to improve your practice.

$Collage of Active Math Games as example of strategies in teaching mathematics$

Learn With Play at Home/Plastic Bottle Number Bowling via learnwithplayathome.com; Math Geek Mama/Skip-Counting Hopscotch via mathgeekmama.com; WeAreTeachers

Adding movement and physical activity to your instruction might seem counterintuitive as a math strategy, but asking kids to get out of their seats can increase their motivation and interest. For example, you could ask students to:

Make angles with their arms
Create a square dance that demonstrates different types of patterns
Complete a shape scavenger hunt in the classroom
Run or complete other exercises periodically and graph the results

The possibilities of these strategies in teaching mathematics are limited only by your imagination and the math concepts you need to cover. Check out these active math games .

21. Be a lifelong learner

Generally, students will become excited about a subject if their teacher is excited about it. However, it’s hard to be excited about teaching math if your understanding hasn’t changed since you learned it in elementary school. For example, if you teach how to divide fractions by fractions and your understanding is limited to following the “invert and multiply” rule, take the time to understand why the rule works and how it applies to the real world. When you have confidence in your own mathematical expertise, then you can teach math confidently and joyfully to best apply strategies in teaching mathematics.

What do you feel are the most important strategies in teaching mathematics? Share in the comments below.

Want more articles like this be sure to subscribe to our newsletters ., learn why it’s important to honor all math strategies in teaching math . plus, check out the best math websites for teachers ..

$We all want our students to be successful in math. These essential strategies in teaching mathematics can help.$

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Problem-Solving Strategies and Obstacles

Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Sean is a fact-checker and researcher with experience in sociology, field research, and data analytics.

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From deciding what to eat for dinner to considering whether it's the right time to buy a house, problem-solving is a large part of our daily lives. Learn some of the problem-solving strategies that exist and how to use them in real life, along with ways to overcome obstacles that are making it harder to resolve the issues you face.

What Is Problem-Solving?

In cognitive psychology , the term 'problem-solving' refers to the mental process that people go through to discover, analyze, and solve problems.

A problem exists when there is a goal that we want to achieve but the process by which we will achieve it is not obvious to us. Put another way, there is something that we want to occur in our life, yet we are not immediately certain how to make it happen.

Maybe you want a better relationship with your spouse or another family member but you're not sure how to improve it. Or you want to start a business but are unsure what steps to take. Problem-solving helps you figure out how to achieve these desires.

The problem-solving process involves:

Discovery of the problem
Deciding to tackle the issue
Seeking to understand the problem more fully
Researching available options or solutions
Taking action to resolve the issue

Before problem-solving can occur, it is important to first understand the exact nature of the problem itself. If your understanding of the issue is faulty, your attempts to resolve it will also be incorrect or flawed.

Problem-Solving Mental Processes

Several mental processes are at work during problem-solving. Among them are:

Perceptually recognizing the problem
Representing the problem in memory
Considering relevant information that applies to the problem
Identifying different aspects of the problem
Labeling and describing the problem

Problem-Solving Strategies

There are many ways to go about solving a problem. Some of these strategies might be used on their own, or you may decide to employ multiple approaches when working to figure out and fix a problem.

An algorithm is a step-by-step procedure that, by following certain "rules" produces a solution. Algorithms are commonly used in mathematics to solve division or multiplication problems. But they can be used in other fields as well.

In psychology, algorithms can be used to help identify individuals with a greater risk of mental health issues. For instance, research suggests that certain algorithms might help us recognize children with an elevated risk of suicide or self-harm.

One benefit of algorithms is that they guarantee an accurate answer. However, they aren't always the best approach to problem-solving, in part because detecting patterns can be incredibly time-consuming.

There are also concerns when machine learning is involved—also known as artificial intelligence (AI)—such as whether they can accurately predict human behaviors.

Heuristics are shortcut strategies that people can use to solve a problem at hand. These "rule of thumb" approaches allow you to simplify complex problems, reducing the total number of possible solutions to a more manageable set.

If you find yourself sitting in a traffic jam, for example, you may quickly consider other routes, taking one to get moving once again. When shopping for a new car, you might think back to a prior experience when negotiating got you a lower price, then employ the same tactics.

While heuristics may be helpful when facing smaller issues, major decisions shouldn't necessarily be made using a shortcut approach. Heuristics also don't guarantee an effective solution, such as when trying to drive around a traffic jam only to find yourself on an equally crowded route.

Trial and Error

A trial-and-error approach to problem-solving involves trying a number of potential solutions to a particular issue, then ruling out those that do not work. If you're not sure whether to buy a shirt in blue or green, for instance, you may try on each before deciding which one to purchase.

This can be a good strategy to use if you have a limited number of solutions available. But if there are many different choices available, narrowing down the possible options using another problem-solving technique can be helpful before attempting trial and error.

In some cases, the solution to a problem can appear as a sudden insight. You are facing an issue in a relationship or your career when, out of nowhere, the solution appears in your mind and you know exactly what to do.

Insight can occur when the problem in front of you is similar to an issue that you've dealt with in the past. Although, you may not recognize what is occurring since the underlying mental processes that lead to insight often happen outside of conscious awareness .

Research indicates that insight is most likely to occur during times when you are alone—such as when going on a walk by yourself, when you're in the shower, or when lying in bed after waking up.

How to Apply Problem-Solving Strategies in Real Life

If you're facing a problem, you can implement one or more of these strategies to find a potential solution. Here's how to use them in real life:

Create a flow chart . If you have time, you can take advantage of the algorithm approach to problem-solving by sitting down and making a flow chart of each potential solution, its consequences, and what happens next.
Recall your past experiences . When a problem needs to be solved fairly quickly, heuristics may be a better approach. Think back to when you faced a similar issue, then use your knowledge and experience to choose the best option possible.
Start trying potential solutions . If your options are limited, start trying them one by one to see which solution is best for achieving your desired goal. If a particular solution doesn't work, move on to the next.
Take some time alone . Since insight is often achieved when you're alone, carve out time to be by yourself for a while. The answer to your problem may come to you, seemingly out of the blue, if you spend some time away from others.

Obstacles to Problem-Solving

Problem-solving is not a flawless process as there are a number of obstacles that can interfere with our ability to solve a problem quickly and efficiently. These obstacles include:

Assumptions: When dealing with a problem, people can make assumptions about the constraints and obstacles that prevent certain solutions. Thus, they may not even try some potential options.
Functional fixedness : This term refers to the tendency to view problems only in their customary manner. Functional fixedness prevents people from fully seeing all of the different options that might be available to find a solution.
Irrelevant or misleading information: When trying to solve a problem, it's important to distinguish between information that is relevant to the issue and irrelevant data that can lead to faulty solutions. The more complex the problem, the easier it is to focus on misleading or irrelevant information.
Mental set: A mental set is a tendency to only use solutions that have worked in the past rather than looking for alternative ideas. A mental set can work as a heuristic, making it a useful problem-solving tool. However, mental sets can also lead to inflexibility, making it more difficult to find effective solutions.

How to Improve Your Problem-Solving Skills

In the end, if your goal is to become a better problem-solver, it's helpful to remember that this is a process. Thus, if you want to improve your problem-solving skills, following these steps can help lead you to your solution:

Recognize that a problem exists . If you are facing a problem, there are generally signs. For instance, if you have a mental illness , you may experience excessive fear or sadness, mood changes, and changes in sleeping or eating habits. Recognizing these signs can help you realize that an issue exists.
Decide to solve the problem . Make a conscious decision to solve the issue at hand. Commit to yourself that you will go through the steps necessary to find a solution.
Seek to fully understand the issue . Analyze the problem you face, looking at it from all sides. If your problem is relationship-related, for instance, ask yourself how the other person may be interpreting the issue. You might also consider how your actions might be contributing to the situation.
Research potential options . Using the problem-solving strategies mentioned, research potential solutions. Make a list of options, then consider each one individually. What are some pros and cons of taking the available routes? What would you need to do to make them happen?
Take action . Select the best solution possible and take action. Action is one of the steps required for change . So, go through the motions needed to resolve the issue.
Try another option, if needed . If the solution you chose didn't work, don't give up. Either go through the problem-solving process again or simply try another option.

You can find a way to solve your problems as long as you keep working toward this goal—even if the best solution is simply to let go because no other good solution exists.

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Dunbar K. Problem solving . A Companion to Cognitive Science . 2017. doi:10.1002/9781405164535.ch20

Stewart SL, Celebre A, Hirdes JP, Poss JW. Risk of suicide and self-harm in kids: The development of an algorithm to identify high-risk individuals within the children's mental health system . Child Psychiat Human Develop . 2020;51:913-924. doi:10.1007/s10578-020-00968-9

Rosenbusch H, Soldner F, Evans AM, Zeelenberg M. Supervised machine learning methods in psychology: A practical introduction with annotated R code . Soc Personal Psychol Compass . 2021;15(2):e12579. doi:10.1111/spc3.12579

Mishra S. Decision-making under risk: Integrating perspectives from biology, economics, and psychology . Personal Soc Psychol Rev . 2014;18(3):280-307. doi:10.1177/1088868314530517

Csikszentmihalyi M, Sawyer K. Creative insight: The social dimension of a solitary moment . In: The Systems Model of Creativity . 2015:73-98. doi:10.1007/978-94-017-9085-7_7

Chrysikou EG, Motyka K, Nigro C, Yang SI, Thompson-Schill SL. Functional fixedness in creative thinking tasks depends on stimulus modality . Psychol Aesthet Creat Arts . 2016;10(4):425‐435. doi:10.1037/aca0000050

Huang F, Tang S, Hu Z. Unconditional perseveration of the short-term mental set in chunk decomposition . Front Psychol . 2018;9:2568. doi:10.3389/fpsyg.2018.02568

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By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Ethical Culture’s Diversified Approach to Teaching Math

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June 7, 2024

By Emma Johnson, Communications Manager

At Ethical Culture Fieldston School, mathematics is a challenging journey of exploration, discovery, and empowerment. Our students engage in a learning experience that is both demanding and deeply rewarding, laying the foundation for a lifelong love of numbers and problem-solving.

This year, the Math Specialists at Ethical Culture — Jen Cooley, Becky Weintraub, Nina Liu, and Larry McMillan — challenged their students in ways that prioritize conceptual understanding and problem-solving over rote practices such as memorization and times tables. Students are encouraged to understand the “why” behind procedures, leading to a deeper comprehension of mathematical concepts.

One element of this approach is the 5th Grade “Problem of the Week.” McMillan gives his 5th Graders two challenging problems to solve at home each week. Students are tasked with solving one of the problems and exemplifying their strategies on a poster to share with their classmates. Differentiated learning caters to diverse styles, allowing students to choose the problem that resonates with them and fostering ownership and motivation. Believing that it’s important for students not to feel discouraged by difficulties, the math teachers design problems with multiple correct answers based on different approaches.

“It’s all about creating access and adjusting students’ mindset about math. We create multiple entry points to a problem,” says Cooley. “Students understand that there is more than one valid and efficient way to solve any problem.”

“Any sufficiently rich question can be extended,” adds McMillan. “A creative and thoughtful response to a rich question often leads students to a discovery about mathematics that begs for further investigation or suggests a way of confirming the solution.”

A recent Problem of the Week asked students the following question:

When considering the problem, students must comprehend it well enough to devise a strategy and own a solution well enough to explain it to others. When creating their posters, they must first annotate and paraphrase the question in their own words, explain their strategies for solving it, show their mathematical computations, and then identify where they had trouble, what they did correctly, and what they learned. Emphasizing the importance of differentiated learning, students can present their work on the posters in various ways — typed or handwritten, using tables or visuals, or choosing the poster’s orientation.

All students tackling this problem understood that they would eventually need to convert yards to inches. However, when and how they did so varied. Some students drew a football field, marking lines that halved the distance each time. Others created tables, first converting yards to inches and then dividing by two until their answer was less than one inch. Some worked in fractions, some in decimals, inviting comparisons of their advantages.

The variety in approaches is celebrated in the next step of the Problem of the Week process. Once their posters were complete, students participated in a “gallery walk,” silently reading each other’s posters and taking notes in their math journals to understand their peers’ work. They then gathered to share what they learned and discuss different strategies. This process allows students to take ownership of their work, practice explaining mathematical concepts, and appreciate multiple perspectives in problem-solving. This methodology extends beyond math, encouraging students to embrace creative solutions in other areas of their lives.

From Kindergarten onward, students engage in a journey that links abstract mathematical theories to real-world applications. Starting with basic concepts like shapes and numbers, they progressively build a strong foundation, recognizing math’s presence in everyday life and developing essential logical reasoning through activities focused on patterns. Math is seamlessly woven into their daily routines, whether in morning meetings, partner games, or guided explorations, ensuring that mathematical thinking becomes second nature.

As students progress into 2nd Grade, the emphasis shifts to building computational fluency while maintaining a focus on process over product. Small group settings encourage flexibility, accuracy, and efficiency in mathematical communication and problem-solving. In 3rd Grade, students may opt to join a Morning Math group that focuses on reasoning and problem-solving, exposing them to a broader range of mathematical challenges.

“From an early age, we deemphasize speed and instead try to develop fluency. Math is not an Olympic sport. Instead, we tell our students, ‘Your thinking and work matter and deserve recognition. Learning is messy, and it should be. Doing your best means embracing challenges and recognizing the value in where you get stuck,’” shares Weintraub.

This methodology continues as students advance through each grade and begin diving deeper into multiplication, division, factors, multiples, and properties of numbers. Real-world connections remain at the core of learning, with students applying mathematical concepts to analyze data, solve complex problems, and make informed decisions. Whether creating tables and graphs in tandem with Social Studies Workshop or exploring exponential growth and division strategies, students are constantly challenged to think critically and creatively.

This accessible and differentiated approach to teaching math also makes learning fun. Students are not deterred by difficulty and eagerly anticipate the next problem, looking for new ways to challenge themselves.

Students also have the opportunity to extend their math practice by joining the Math Club. McMillan invites any students looking for an added challenge to commit to Math Club, which invites them to explore mathematical concepts not typically covered in the regular curriculum. They investigate fascinating topics like fractal designs, Möbius strips, and probability. The latter is examined through the lens of fair and unfair games, culminating in designing games of chance, determining their probabilities, and presenting their games to the rest of the 5th Graders at the “Is it Fair? Fair!”. A club highlight is the rotational symmetry project, where students create and share intricate designs. This opportunity encourages students to commit to trying out these concepts in class, deepening their understanding and appreciation of math.

“5th Grade students have the skills and background knowledge to engage with a wider range of topics in mathematics than we can fit into math class,” says McMillan. “It is a lovely puzzle to determine if a particular group of students is better suited to investigate ciphers or applications of symmetry in art.”

At ECFS, we believe that every student is a mathematician in their own right. Through real-world investigations, collaborative problem-solving tasks, and a supportive learning environment, our teachers empower students to embrace challenges, think critically, and approach mathematics confidently and enthusiastically. Our math curriculum is not just about formulas and numbers; it’s about nurturing curious minds, fostering resilience, and preparing students for a future where mathematical literacy is key to success.

Five Questions for Marjorie Jean-Paul, Chief Advancement Officer at ECFS

Joyful learning extends to enriching after school program at ethical culture, ethical culture’s young playwrights invite you into the mind of a 4th grader, ethical culture renovates for the future.

Clinical Practice Guideline for the Treatment of Posttraumatic Stress Disorder (PTSD)

What is Cognitive Behavioral Therapy?

Cognitive behavioral therapy (CBT) is a form of psychological treatment that has been demonstrated to be effective for a range of problems including depression, anxiety disorders, alcohol and drug use problems, marital problems, eating disorders, and severe mental illness. Numerous research studies suggest that CBT leads to significant improvement in functioning and quality of life. In many studies, CBT has been demonstrated to be as effective as, or more effective than, other forms of psychological therapy or psychiatric medications.

It is important to emphasize that advances in CBT have been made on the basis of both research and clinical practice. Indeed, CBT is an approach for which there is ample scientific evidence that the methods that have been developed actually produce change. In this manner, CBT differs from many other forms of psychological treatment.

CBT is based on several core principles, including:

Psychological problems are based, in part, on faulty or unhelpful ways of thinking.
Psychological problems are based, in part, on learned patterns of unhelpful behavior.
People suffering from psychological problems can learn better ways of coping with them, thereby relieving their symptoms and becoming more effective in their lives.

CBT treatment usually involves efforts to change thinking patterns. These strategies might include:

Learning to recognize one’s distortions in thinking that are creating problems, and then to reevaluate them in light of reality.
Gaining a better understanding of the behavior and motivation of others.
Using problem-solving skills to cope with difficult situations.
Learning to develop a greater sense of confidence in one’s own abilities.

CBT treatment also usually involves efforts to change behavioral patterns. These strategies might include:

Facing one’s fears instead of avoiding them.
Using role playing to prepare for potentially problematic interactions with others.
Learning to calm one’s mind and relax one’s body.

Not all CBT will use all of these strategies. Rather, the psychologist and patient/client work together, in a collaborative fashion, to develop an understanding of the problem and to develop a treatment strategy.

CBT places an emphasis on helping individuals learn to be their own therapists. Through exercises in the session as well as “homework” exercises outside of sessions, patients/clients are helped to develop coping skills, whereby they can learn to change their own thinking, problematic emotions, and behavior.

CBT therapists emphasize what is going on in the person’s current life, rather than what has led up to their difficulties. A certain amount of information about one’s history is needed, but the focus is primarily on moving forward in time to develop more effective ways of coping with life.

Source: APA Div. 12 (Society of Clinical Psychology)

453: Fair Shake: Women and the Fight to Build a Just Economy The Strategy Skills Podcast: Strategy | Leadership | Critical Thinking | Problem-Solving

Welcome to Strategy Skills episode 453, featuring an interview with the authors of Fair Shake: Women and the Fight to Build a Just Economy, Naomi Cahn, June Carbone, and Nancy Levit. This book explains that the system that governs our economy—a winner-take-all economy—is the root cause of these myriad problems. The WTA economy self-selects for aggressive, cutthroat business tactics, which creates a feedback loop that sidelines women. The authors, three legal scholars, call this feedback loop “the triple bind”: if women don’t compete on the same terms as men, they lose; if women do compete on the same terms as men, they’re punished more harshly for their sharp elbows or actual misdeeds; and when women see that they can’t win on the same terms as men, they take themselves out of the game (if they haven’t been pushed out already). With odds like these stacked against them, it’s no wonder women feel like, no matter how hard they work, they can’t get ahead. Naomi Cahn is the Justice Anthony M. Kennedy Distinguished Professor of Law at the University of Virginia School of Law, as well as the Co-Director of the Family Law Center. Cahn is the author or editor of numerous books written for both academic and trade publishers, including Red Families v. Blue Families and Homeward Bound. In 2017, Cahn received the Harry Krause Lifetime Achievement in Family Law Award from the University of Illinois College of Law and in 2024 she was inducted into the Clayton Alumni Hall of Fame. June Carbone is the Robina Chair of Law, Science and Technology at the University of Minnesota Law School. Previously she has served as the Edward A. Smith/Missouri Chair of Law, the Constitution and Society at the University of Missouri at Kansas City; and as the Associate Dean for Professional Development and Presidential Professor of Ethics and the Common Good at Santa Clara University School of Law. She has written From Partners to Parents and co-written Red Families v. Blue Families; Marriage Markets; and Family Law. She is a co-editor of the International Survey of Family Law. Nancy Levit is the Associate Dean for Faculty and holds a Curator’s Professorship at the University of Missouri–Kansas City School of Law. Professor Levit has been voted Outstanding Professor of the Year five times by students and was profiled in Dean Michael Hunter Schwartz’s book, What the Best Law Teachers Do. She has received the N.T. Veatch Award for Distinguished Research and Creative Activity and the Missouri Governor’s Award for Teaching Excellence. She is the author of The Gender Line and co-author of Feminist Legal Theory; The Happy Lawyer; The Good Lawyer; and Jurisprudence—Classical and Contemporary. Get Fair Shake here: https://rb.gy/r2q7rw Here are some free gifts for you: Overall Approach Used in Well-Managed Strategy Studies free download: www.firmsconsulting.com/OverallApproach McKinsey & BCG winning resume free download: www.firmsconsulting.com/resumepdf Enjoying this episode? Get access to sample advanced training episodes here: www.firmsconsulting.com/promo

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‎The Strategy Skills Podcast: Strategy
Welcome to Strategy Skills episode 453, featuring an interview with the authors of Fair Shake: Women and the Fight to Build a Just Economy, Naomi Cahn, June Carbone, and Nancy Levit. This book explains that the system that governs our economy—a winner-take-all economy—is the root cause of these myri…

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