what does problem solving look like in the classroom

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Teaching problem solving.

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Tips and Techniques

Expert vs. novice problem solvers, communicate.

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

Encourage Independence

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

Be sensitive

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

Encourage Thoroughness and Patience

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

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

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

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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.

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Problem Solving in the Classroom

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Success Story

Last week during our class meetings, I noticed a disturbing habit developing among my students. Sometimes they don't want to switch seats and move away from their best friends, and sometimes they want to be the last one standing (when we do an activity that has us sit down after our turn). Then we talked about how this might make everyone else feel and how it might affect our class community. We agreed that this was a problem because it did not make everyone feel welcome. Finally, I asked them for suggestions to solve the problem.

We have been working on problem solving all year. I started by teaching my students that solutions always need to be related, respectful, reasonable, and helpful. This is a challenge for students who often think of punishments before solutions. As we started talking about possible solutions to this problem, the first few solutions were not surprisingly more like punishments, such as, having the culprits sit out of future greetings and activities until they were being kind, or skipping offenders in the circle. However, the more we talked, the more they began to consider ways to prevent the problem from even occurring. Eventually we settled on two possible preventative solutions:

1) they could come to the circle separately and choose a place to sit away from close friends so they wouldn't be tempted to resist moving.

2) we could make assigned seats around the circle so that no one would feel uncomfortable about moving if necessary.

At this point, I told the class I would consider both solutions. It seems that I've taught them well about how to solve problems fairly because immediately one student suggested that I let the class vote. It was hard to argue with her logic and truthfully both solutions were acceptable. So this morning we had a vote. I had the kids close their eyes and raise their hands. They voted (20-3) to have assigned seats. When they opened their eyes and I announced the winning solution they started fist pumping with excitement.

I couldn't help but smile. I could never have imagined such a positive reaction to the idea of assigned seats for class activities. In fact, I suspect that had I forced the idea of assigned seats on them as a "punishment" or consequence, I would have heard lots of complaints and frustration. Yet when they could appreciate the problem and come to the solution on their own, they were more than willing to accept the idea. We immediately created a chart with assigned circle seats and by the afternoon they were already reminding each other where they needed to sit. Love it! Sarah Werstuik, Washington, D.C.

Teach Students the 4 Problem-Solving Steps

Another way to solve problems in the classroom is to teach students the 4 Problem-Solving Steps.

Post a copy of the 4 Problem-Solving Steps where students can refer to it (maybe next to a "peace table").

Problem-Solving Steps

• Do something else. (Find another game or activity.)
• Leave long enough for a cooling-off period, then follow-up with the next steps.
• Tell the other person how you feel. Let him or her know you don’t like what is happening.
• Listen to what the other person says about how he or she feels and what he or she doesn’t like.
• Share what you think you did to contribute to the problem.
• Tell the other person what you are willing to do differently.
• Work out a plan for sharing or taking turns.
• Put it on the class meeting agenda. (This can also be a first choice and is not meant as a last resort.)
• Talk it over with a parent, teacher, or friend.

After discussing these skills, have the children role-play the following hypothetical situations. Have them solve each of the situations four different ways (one for each of the steps).

• Fighting over whose turn it is to use the tetherball.
• Shoving in line.
• Calling people bad names.
• Fighting over whose turn it is to sit by the window in the car or bus.

Teachers can put the Four Problem-Solving Steps on a laminated poster for students to refer to. Some teachers require that children use these steps before they put a problem on the agenda. Other teachers prefer the class meeting process because it teaches other skills. Instead of making one better than the other (class meeting or one-on-one), let children choose which option they would prefer at the moment.

This tool and many others can be found in the Positive Discipline Teacher Tool Cards .

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Center for Teaching Innovation

Resource library.

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

Problem-Based Learning

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

Why Use Problem-Based Learning?

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

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

Considerations for Using Problem-Based Learning

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

Students generally must:

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

Getting Started with Problem-Based Learning

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

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

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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The Problem-solving Classroom

• Visualising
• Working backwards
• Reasoning logically
• Conjecturing
• Working systematically
• Looking for patterns
• Trial and improvement.

• stage of the lesson 
• level of thinking
• mathematical skill.
• The length of student response increases (300-700%)
• More responses are supported by logical argument.
• An increased number of speculative responses.
• The number of questions asked by students increases.
• Student - student exchanges increase (volleyball).
• Failures to respond decrease.
• 'Disciplinary moves' decrease.
• The variety of students participating increases.  As does the number of unsolicited, but appropriate contributions.
• Student confidence increases.
• conceptual understanding
• procedural fluency
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• adaptive reasoning
• productive disposition

Reimagining Assessment Assessing the Transfer of Critical Thinking and Problem Solving Skills

Jeff Heyck-Williams (He, His, Him) Director of the Two Rivers Learning Institute in Washington, DC

Educators are rethinking the purposes, forms, and nature of assessment. Beyond testing mastery of traditional content knowledge—an essential task, but not nearly sufficient—educators are designing assessment for learning as an integral part of the learning process.

Two Rivers embarked on a multi-year project to define and assess critical thinking and problem solving in project-based learning expeditions.

Two Rivers Public Charter School in Washington, D.C., is a network of EL Education schools serving over 700 students in preschool through 8th grade. Throughout our twelve-year history, we have continued to champion the importance of embracing a broader definition of student success than what has been handed to us by state and national policy. While we believe that it is essential for all students to be proficient in math, literacy, and the sciences, we believe that that is not enough. Students also need a rich set of social and cognitive skills that span beyond any given discipline.

Furthermore, we believe that we can best teach students these skills through hands-on interdisciplinary project-based learning. As EL Education schools, our projects are defined as expeditions lasting 10 to 12 weeks in which students tackle messy, real world problems that don’t have easy paths to solutions nor do they have one clear right answer. Through intentional design of these projects, teachers address the core content and basic skills defined by literacy and content standards; the social skills of collaboration and communication; the intrapersonal skills defined by character; and the broadly applicable cognitive skills of critical thinking and problem solving.

In the life of our schools, we have seen the powerful way that our students through project-based learning have embraced deeper learning outcomes, and exhibited the habits of effective critical thinking, collaboration, and personal character. However, our evidence that this is working is only found in anecdotes and in the quality of student work. We have been unable to demonstrate neither the degree to which students are developing these skills within projects nor their ability to transfer the skills beyond the context of the current project.

Focusing just on the dimensions of critical thinking and problem solving, our teachers expressed frustration at not knowing in concrete terms what those cognitive skills looked like when students exhibited them. Building on our understanding of the essential role that assessment for learning plays in the learning process and the very practical consideration of how we help teachers and students define and work towards developing these skills, we have embarked on a multi-year project to define and assess critical thinking and problem solving.

Critical thinking and problem solving, as we define it, are the set of non-discipline specific cognitive skills people use to analyze vast amounts of information and creatively solve problems. We have broken those skills down into these five core components:

• Schema Development: The ability to learn vast amounts of information and organize it in ways that are useful for understanding
• Metacognition and Evaluation: The ability to think critically about what one is doing and evaluate many potential choices
• Effective Reasoning: The ability to create claims and support them with logical evidence
• Problem Solving: The ability to identify the key questions in a problem, develop possible paths to a solution, and follow through with a solution
• Creativity and Innovation: The ability to formulate new ideas that are useful within a particular context

Our project is working to create learning progressions in each of these core components with accompanying rubrics. The progressions of learning and rubrics will both help define for students and teachers the skills that all students should be developing as well as function as evaluative tools to provide a picture where each student sits in the development of these skills and what are the next steps for further learning.

However, we believe it is not enough for students to be able to develop these skills within the highly scaffolded context of our expeditions. If they have truly learned the skills, they should have the ability to transfer them. With this in mind, we are working to create short content-neutral performance tasks that will give teachers and students valuable information about each of the five core components listed above. Our hypothesis is that through having students tackle short novel tasks, we will be able to draw clear conclusions about their learning of critical thinking and problem solving skills.

Through the course of this work, we hope that our process will be useful to other educators interested in achieving deeper learning outcomes for their students. We realize that deeper learning will not become a reality in most schools until teachers and leaders have a clear vision for what it looks like on a day-to-day basis and how we can clearly demonstrate student growth in these essential skills. We hope that our work will help to inform how to make deeper learning a concrete reality. It is a work in progress, and we invite you to share your thoughts and follow our progress at our website  https://learn.tworiverspcs.org .

Learn more about Two Rivers' Assessment for Learning Project on their grantee page .

Jeff Heyck-Williams (He, His, Him)

Director of the two rivers learning institute.

Jeff Heyck-Williams is the director of the Two Rivers Learning Institute and a founder of Two Rivers Public Charter School. He has led work around creating school-wide cultures of mathematics, developing assessments of critical thinking and problem-solving, and supporting project-based learning.

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The Role of the Teacher Changes in a Problem-Solving Classroom

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How can teachers help students develop problem-solving skills when they themselves, even though confronted with an array of problems every day, may need to become better problem solvers? Our experience leads us to conclude that there is an expertise in a certain kind of problem-solving that teachers possess but that broader problem-solving skills are sometimes wanting.There are a few reasons why this happens. One reason may be that teacher preparation programs remain focused on how to teach subjects and behavior management techniques. Another reason may be that professional development opportunities offered in schools are focused elsewhere. And, another reason could be that leaders still often fail to engage their faculties in solving substantive problems within the school community.

A recent issue of Education Leadership was dedicated to the topic, “Unleashing Problem Solvers”. One theme that ran through several of the articles was the changing role of the teacher. In a positive but traditional classroom, information is shared by the teacher and the students are asked to demonstrate application of that information. A problem-solving classroom is different. A problem-solving classroom requires extraordinary planning on the part of the teacher. For problems to have relevance, students are engaged in the identification of the problem. Teachers have to become experts at creating questions that require students to reach back to information and skills already attained, while figuring out what they need to learn next in order to solve the problem. Some of us are really good at asking these kinds of questions. Others are not.

Students have to become experts at reflecting on these questions as guides resulting in a gathering of new information and skills, and answers. Teachers have to be prepared to offer lessons that bridge the gaps between the skills and information already attained and those the performance of the students demonstrate remain needed. Often it involves teams of students and they are simultaneously learning collaboration and communication skills.

Problem-Based Classrooms Require Letting Go

Opportunities for teachers to work with each other, to learn from experts, to receive feedback from observers of their work, all allow for skill development. But at the same time, there is a more challenging effort required of the teacher. Problem-based classrooms require teachers to dare to let go of control of the learning and to take hold of the role of questioner, coach, supporter, and diagnostician. In addition to the lack of training teachers have in these skills, the leaders in charge of evaluating their work also have to know what problem-solving classrooms look like and how to capture that environment in an observation, how to give feedback on the teachers’ efforts. Of course, if problem- solving is a collaborative school community process, how does that change the leader’s role? Are leaders, themselves, ready to become facilitators of the process rather than the sole problem solver? Many talk about wanting that but most get rewarded for being the problem solver.

Questions are Essential

There is a place to begin and that place is the shared understanding of what problem-based learning actually is. Because teachers traditionally plan for a time for Q and A within classes, they and their leaders may think of questions as having a correct answer. In moving into a problem-based learning design, the questions also have to be more overarching, create cognitive dissonance, and provoke the learner to search for answers. Here is why it is important to come to an understanding about the types of questions to be asked and shifting the teaching and learning practices to be one of expecting more from the learner.

Students Need Problem-Solving Skills

Problem-based learning skills are skills that prepare for a changing environment in all fields. Current educators cannot imagine some of the careers our students will have over their lifetimes. We do know that change will be part of everyone’s work. Flexibility and problem-solving are key skills. Problem- solving involves collaboration, communication, critical thinking, empathy, and integrity. If we listen to the business world, we will hear that design thinking is the way of the future.

Tim Brown, CEO of IDEO says,

Design thinking is a human-centered approach to innovation that draws from the designer’s toolkit to integrate the needs of people, the possibilities of technology, and the requirements for business success.

The only way for educators to develop these skills in students is to build lessons and units that are interdisciplinary and demand these skills. If we begin from the earliest of grades and expect more as they ascend through the grades, students will have mastered not only their subjects, but the skills that will prepare them for the world of work. How do we best prepare our students? We think problem solving is key.

A nn Myers and Jill Berkowicz are the authors of The STEM Shift (2015, Corwin) a book about leading the shift into 21st century schools. Ann and Jill welcome connecting through Twitter & Email .

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What does problem-based math learning unlock for students? Part 1

Problem-based learning helps engage all students in math. This three-part series explains how—and how teachers can implement it in their classrooms.

Webinar series recap, part 1 of 3

Problem-based math learning helps teachers set the stage for memorable learning experiences and transfer the responsibility for the learning to students, which has been shown to help develop students’ problem-solving and math reasoning skills.

Our webinar series explores how this type of instruction engages all students in grade-level math every day, and how instructors can go about implementing problem-based learning in the classroom.  In part 1 of the webinar series, award-winning teacher Kristin Gray asks—and answers—the question: What does problem-based learning unlock for students?

Experience and explanation form a learning cycle

Imagine you’ve just gotten a new piece of technology: a phone, a TV, a computer. How do you learn to use it? Do you read the entire user guide first? Jump in and never touch the guide? Or turn it on and try some things, referencing the guide as needed? 

If the last option sounds like you, that’s very common—and it’s an example of learning through problem-solving. 

“It’s something we naturally do,” says Gray.  “ We’ve had a phone before so we would pick up this new phone and try doing things that we know worked on our last phone, and then we would experiment: Does it work the same on this phone? This bouncing between experience and explanation is really the foundation of how we learn through problem-solving.”

What learning through problem-solving looks like in the math classroom

If we think of instructional methods in the math classroom along a spectrum, on one end we might have a classroom where students are left to solve a problem and discover the relevant math on their own. On the other end, the instructional method might be to show students how to get the answer and then practice doing similar problems. 

The methods at both extremes are challenging, and it’s hard for instructors to go from one to the other, says Gray. “We need to install a soft landing space in the middle of these extremes—and we can think of that space as learning through problem-solving, or problem-based learning.” 

What does that look like in the math classroom? 

Students will tackle interesting problems, raise questions about the math required, receive an explanation, and apply it back to the problem—as with the example of learning new technology. 

“When we show students how to get the answer, we send the message that math is solely about answer-getting and learning processes. Answers are important, but we want to use problems to teach the math, not just teach students to get the answer,” says Gray. 

Practice is also key, she adds: “This place in the middle pulls the best from both extremes and puts them into a structure that supports teachers in teaching and students in learning.”

Why students should learn through problem-solving

Learning through problem-solving has the potential to engage all learners in math, says Gray. It influences the way teachers and students think of themselves as mathematicians and what it means to know and do math. 

In the 2000 NAEP survey, 70 percent of fourth and eighth graders reported that they enjoy activities that challenge their thinking, and enjoy thinking about problems in new ways. 

“Students are already naturally curious and like solving challenges and trying things in new ways, so that’s a great start,” says Gray. 

“No matter how kindly, clearly, patiently, or slowly teachers explain, they cannot make students understand,” says Gray. “Understanding takes place in the student’s mind as they connect new information with previously developed ideas. Teachers can help, but understanding is a by-product of solving problems.” 

Add understanding is motivating. It inspires perseverance and confidence. It supports making connections, not learning concepts in isolation. 

When students are given a new problem and are able to use prior knowledge to help solve it, that “promotes the development of autonomous learners,” says Gray. 

How Amplify Math supports problem-based learning

Amplify Math supports teachers in the planning and delivery of problem-based lessons. It also enables teachers to monitor student progress and differentiate instruction based on real-time data. 

Lessons start with warm-ups that tap into prior knowledge and move into problems that require collaboration to solve. Teachers monitor, engage, and ultimately synthesize student work into the main idea. There are also ample opportunities for practice and reflection. 

Learn more abou t Amplify Desmos Math .

Register to watch the rest of the series here . 

Visit Gray’s site, Math Minds, here .

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Nurturing Mistake Tolerance in the Classroom

Teachers can help students get over the fear of making a mistake by showing them that errors are just a part of the learning process.

Our students do not like making mistakes in front of their peers. As a matter of fact, I can say the same for many adults. This dislike leads to an avoidance of failure in the hopes of preserving notions of identity and self-efficacy. What I define as “fear branding” is the perception, often implicit in nature, of being outed as incompetent by one’s community.

This fear can lead students down trauma spirals of internalized inadequacies, classroom disruptions, and other barriers undermining their confidence as learners. The resulting outlook can last a lifetime, limiting individuals’ horizons and opportunities. These effects may be more pronounced among students who already face disadvantages in the classroom, including stereotypes about their educational capacity.

One of the most ambitious goals of educators is to teach students what to do when they don’t know what to do. Social scientists illustrate these fear-branded trauma spirals through theories such as Albert Bandura’s reciprocal determinism and Carol Dweck’s growth mindset ; practitioners, on the other hand, find tangible value in applying knowledge to execution. Teachers need practice-based instructional activities that engage students in problem-solving development to move beyond the barriers highlighted in erudition.

The challenge is that learning is messy. The nonlinearity of trial and error (e.g., starting and stopping, hesitating, confusion, drafting, amending) requires a mistake tolerance that must be nurtured in the classroom . As this tolerance builds, students not only openly embrace mistakes as part of the learning process but also acknowledge their intrinsic value as they mature into expert learners.

3 Ways to Grow Students’ Mistake Tolerance

1. Use jigsaw activities. These activities break up complex text into smaller, disjointed chunks that students work collaboratively to piece together, creating coherence and meaning. Arranging ideas and/or events chronologically, logically, or sequentially requires a grit for trial and error that expert learners sometimes take for granted. Jigsaw activities create opportunities for students to gain confidence through justifying their ordering by experimentation, negotiating text meaning, and identifying target language.

First, I introduce this activity to students using comic strips. Students practice shuffling comic strip panels in the correct order by identifying key story elements (e.g., plot, setting, characters, point of view, theme). I then increase the rigor by tasking students to correctly sequence reading passage sections, employing the same story-element identification.

This tactile approach to text engagement mirrors that of puzzling. Like puzzles, the comic strip panels or text sections within the jigsaw activity are in pieces that have to be assembled properly. Manipulating the puzzle components and figuring out where they fit within the larger picture requires problem-solving perseverance as students develop their abilities to plan and test ideas. From a social and emotional standpoint, completing jigsaw activities also helps students learn how to accept challenges, overcome problems, and deal with the frustrations of failure.

2. Guide students to get the GIST. These activities are summarizing exercises that help students focus on main ideas. Developed by James Cunningham, PhD, in 1982, GIST (Generating Interactions between Schemata and Texts) helps students improve reading comprehension and increase recall of complex texts. As the name suggests, GIST scaffolds the removal of extraneous detail as students evaluate and create information to convey the crux of what they read. It’s an adaptable strategy that can be used with a variety of informational and literary texts and is an effective tool to use in content areas.

Students read a text and respond to the six common journalists’ questions on the GIST template (who, what, when, where, why, and how). Using their responses, students identify the most important information by paring down the text into summaries of 20 words or less (the teacher predefines the GIST word count).

As students work to comply with the word constraint, you will notice the messiness of trial and error as students revise, alter, and refine their summaries to fit the predefined parameter. As a result, students build mistake tolerance in low-stakes routines using various combinations of elaborative rehearsal, reorganization, and contextualized language.

3. Actively model critical thinking. To create a classroom culture truly open to mistakes, teachers must not only embrace them among students, but actively model their own tolerance for mistakes. We should want our students to see that we, too, wrestle with getting ideas down on paper. Follow a plan to ensure that you’re modeling the type of thinking you intended. Stay in character as a learner, not a teacher.

For example, imagine that you’re working through a text or a task for the first time. Model the types of thinking you expect from the students. Like a good learner, ask yourself questions, and verbalize inner dialogue.

• What is the author trying to tell me? Is that a clue about what’s going to happen next?
• What happens next if I do this? Is this getting me closer to my goal?

Narrate actions you’re about to do, such as “I’d better write that down” or “That didn’t work. I’d better erase that step and start over.” Let students see and hear you struggle with your thinking. Students also need to see the strategies that good learners use to overcome challenges. It’s important for them to see that all learners encounter challenges and that it’s OK. So not only verbalize struggle but model the metacognitive and critical-thinking strategies that good learners use for overcoming challenges. Try modeling perseverance by building in some unsuccessful attempts and giving yourself a little pep talk after each one.

Learning is a continuous process that involves practice, adjustment, and refinement. When students are given the tools and information needed up front, trial and error can be a fun method of learning. Lived experience, which often comes with discovering the right way to do something while making mistakes along the way, is the basis for learning and one of the foundational characteristics related to the motivation of seasoned learners.

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Problem-solving skills are necessary in all areas of life, and classroom problem solving activities can be a great way to get students prepped and ready to solve real problems in real life scenarios. Whether in school, work or in their social relationships, the ability to critically analyze a problem, map out all its elements and then prepare a workable solution is one of the most valuable skills one can acquire in life.

Educating your students about problem solving skills from an early age in school can be facilitated through classroom problem solving activities. Such endeavors encourage cognitive as well as social development, and can equip students with the tools they’ll need to address and solve problems throughout the rest of their lives. Here are five classroom problem solving activities your students are sure to benefit from as well as enjoy doing:

1. Brainstorm bonanza

Having your students create lists related to whatever you are currently studying can be a great way to help them to enrich their understanding of a topic while learning to problem-solve. For example, if you are studying a historical, current or fictional event that did not turn out favorably, have your students brainstorm ways that the protagonist or participants could have created a different, more positive outcome. They can brainstorm on paper individually or on a chalkboard or white board in front of the class.

2. Problem-solving as a group

Have your students create and decorate a medium-sized box with a slot in the top. Label the box “The Problem-Solving Box.” Invite students to anonymously write down and submit any problem or issue they might be having at school or at home, ones that they can’t seem to figure out on their own. Once or twice a week, have a student draw one of the items from the box and read it aloud. Then have the class as a group figure out the ideal way the student can address the issue and hopefully solve it.

3. Clue me in

This fun detective game encourages problem-solving, critical thinking and cognitive development. Collect a number of items that are associated with a specific profession, social trend, place, public figure, historical event, animal, etc. Assemble actual items (or pictures of items) that are commonly associated with the target answer. Place them all in a bag (five-10 clues should be sufficient.) Then have a student reach into the bag and one by one pull out clues. Choose a minimum number of clues they must draw out before making their first guess (two- three). After this, the student must venture a guess after each clue pulled until they guess correctly. See how quickly the student is able to solve the riddle.

4. Survivor scenarios

Create a pretend scenario for students that requires them to think creatively to make it through. An example might be getting stranded on an island, knowing that help will not arrive for three days. The group has a limited amount of food and water and must create shelter from items around the island. Encourage working together as a group and hearing out every child that has an idea about how to make it through the three days as safely and comfortably as possible.

5. Moral dilemma

Create a number of possible moral dilemmas your students might encounter in life, write them down, and place each item folded up in a bowl or bag. Some of the items might include things like, “I saw a good friend of mine shoplifting. What should I do?” or “The cashier gave me an extra $1.50 in change after I bought candy at the store. What should I do?” Have each student draw an item from the bag one by one, read it aloud, then tell the class their answer on the spot as to how they would handle the situation.

Classroom problem solving activities need not be dull and routine. Ideally, the problem solving activities you give your students will engage their senses and be genuinely fun to do. The activities and lessons learned will leave an impression on each child, increasing the likelihood that they will take the lesson forward into their everyday lives.

You may also like to read

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• Activities for Teaching Tolerance in the Classroom
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How Montessori Promotes Problem Solving Skills for Kids

How does montessori promote problem solving skills for kids.

Problem solving is an essential skill that we all need for a successful life. We use problem solving daily at home and at work. Unfortunately, problem solving is something many kids (and adults!) struggle with. In this post, we’ll discuss how problem solving skills for kids are encouraged in a Montessori environment .

Allow the Child to Problem-Solve Independently

A common misconception is that it is the job of a good teacher or a good parent to solve problems for children. While well-intended, too much helping can actually be detrimental to a child’s development, particularly in the area of problem solving .

Dr. Montessori herself realized the dangers of making children too dependent on their caregivers:

“ The child who has never learned to work by himself, to set goals for his own acts, or to be the master of his own force of will is recognizable in the adult who lets others guide his will and feels a constant need for approval of others .” — Dr. Maria Montessori, Education and Peace

There is a subtle but important distinction between guiding a little one to the solution to a problem and solving the problem for the child.

Label the Problem

The first step in guiding a child to solve a problem independently is to help him or her identify exactly what the problem is. We do this by asking the child to name or label the problem. If the child is struggling to find the words, then the parent or teacher can provide those words, which empowers the child to properly identify the issue.

You can start by stating your observation of the problem, followed by a clarifying question. If a child is struggling with a puzzle, for instance, you could say, “You’re having trouble fitting that piece, is that right?”

This prompts the child to name the problem while still giving him or her the freedom to come to the conclusion independently.

Ask Questions

Mainstream models of teaching and parenting would have you believe that it’s the child’s role to ask the questions and a parent’s –or teacher’s— job to answer them. In Montessori, though, the opposite is more often the case.

When helping a child arrive at a solution to a problem, it’s usually helpful for the adult to prompt the child by asking some open-ended questions. This can assist the youngster in brainstorming possible strategies for resolving an issue successfully.

Going back to the puzzle example, you could prompt the child by asking questions such as, “Is there another way you could turn the piece?” or “What other pieces are there that match the colors in this piece?”.

This is much more effective that showing the child where the puzzle piece fits because it demonstrates the problem solving skills they’ll need for the next puzzle, activity , or problem.

Set the Child Up for Success

As with all things Montessori, encouraging effective problem solving for children involves preparing the environment. Here, it’s important to provide Montessori toys and activities that are developmentally appropriate but still provide a bit of a challenge.

The key is to present the youngster with a challenge that is just within reach. This “sweet spot,” if you will, encourages growth while not overly frustrating the child.

Of course, this will require a lot of observation on your part as the caregiver as you decide on the level of difficulty that is appropriate.

Wait, Then Wait Some More

Silence can be awkward, and as a society, we tend to detest waiting (I know I do!). If we are to impart problem solving skills to our children, though, we need to learn to be patient ourselves.

Next time you are tempted to “fix” something for your little one, make the conscious decision to stand back and wait instead. More often than not, your child will surprise you by finding a way to solve the problem, often an ingenious way you may have not even thought of yourself. This will be your reward for slowing down and practicing patience .

If you’re not already convinced of the power of waiting without interference, consider Montessori’s own words:

“ The fundamental help in development, especially with little children of 3 years of age, is not to interfere. Interference stops activity and stops concentration.” —Dr. Maria Montessori, The Child, Society and the World (Unpublished Speeches and Writing )

Benefits of Teaching Young Children Problem Solving Skills

Teaching problem-solving to little ones may sound straight-forward, but in practice, it requires a good deal of patience and perseverance on the part of the teacher or parent. That’s because it can be a lot easier to solve children’s problems for them in the moment, especially if you’re busy or stressed.

We can all relate!

When you’re feeling tempted to do things for your child rather than take the time to guide them to their own solution, gently remind yourself of the many benefits you’re providing your little one by prompting them to resolve their own problems.

Greater Self-Confidence

Each time a youngster is able to solve a new problem independently, they grow their self-confidence . Over time, this confidence provides them with the ability to tackle problems of ever-increasing difficulty with a sense of calm assurance.

Just think of what the children in your care will be able to accomplish once they know they can do hard things! There’s really no replacement for this sense of capability and confidence.

More Skills

Unfortunately, children who are deprived of opportunities to solve problems independently are robbed of important skills. Montessori herself described this scenario best:

“… the child’s individual liberty must be so guided that through his activity he may arrive at independence … the child who does not do, does not know how to do.” —Dr. Maria Montessori, The Montessori Method

On the other hand, when we afford children the opportunities necessary to learn problem solving skills, they learn these and much, much more. In essence, they become the highly capable children (and eventually adults!) that they were meant to be. Isn’t that what we all want for our kids?

Teaching problem solving skills to our little ones can be tough. The rewards are well worth it, though!

As your child grows in confidence and i ndependence , he or she will naturally exude more joy. You’ll still have moments of frustration, of course.

That’s just par for the course, but you’ll also have moments where you find yourself in absolute awe of the highly capable little person your child is becoming. That’s an accomplishment you can both be proud of!

What does problem-solving look like in your home or classroom?

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For everyone whose relationship with mathematics is distant or broken, Jo Boaler , a professor at Stanford Graduate School of Education (GSE), has ideas for repairing it. She particularly wants young people to feel comfortable with numbers from the start – to approach the subject with playfulness and curiosity, not anxiety or dread.

“Most people have only ever experienced what I call narrow mathematics – a set of procedures they need to follow, at speed,” Boaler says. “Mathematics should be flexible, conceptual, a place where we play with ideas and make connections. If we open it up and invite more creativity, more diverse thinking, we can completely transform the experience.”

Boaler, the Nomellini and Olivier Professor of Education at the GSE, is the co-founder and faculty director of Youcubed , a Stanford research center that provides resources for math learning that has reached more than 230 million students in over 140 countries. In 2013 Boaler, a former high school math teacher, produced How to Learn Math , the first massive open online course (MOOC) on mathematics education. She leads workshops and leadership summits for teachers and administrators, and her online courses have been taken by over a million users.

In her new book, Math-ish: Finding Creativity, Diversity, and Meaning in Mathematics , Boaler argues for a broad, inclusive approach to math education, offering strategies and activities for learners at any age. We spoke with her about why creativity is an important part of mathematics, the impact of representing numbers visually and physically, and how what she calls “ishing” a math problem can help students make better sense of the answer.

What do you mean by “math-ish” thinking?

It’s a way of thinking about numbers in the real world, which are usually imprecise estimates. If someone asks how old you are, how warm it is outside, how long it takes to drive to the airport – these are generally answered with what I call “ish” numbers, and that’s very different from the way we use and learn numbers in school.

In the book I share an example of a multiple-choice question from a nationwide exam where students are asked to estimate the sum of two fractions: 12/13 + 7/8. They’re given four choices for the closest answer: 1, 2, 19, or 21. Each of the fractions in the question is very close to 1, so the answer would be 2 – but the most common answer 13-year-olds gave was 19. The second most common was 21.

I’m not surprised, because when students learn fractions, they often don’t learn to think conceptually or to consider the relationship between the numerator or denominator. They learn rules about creating common denominators and adding or subtracting the numerators, without making sense of the fraction as a whole. But stepping back and judging whether a calculation is reasonable might be the most valuable mathematical skill a person can develop.

But don’t you also risk sending the message that mathematical precision isn’t important?

I’m not saying precision isn’t important. What I’m suggesting is that we ask students to estimate before they calculate, so when they come up with a precise answer, they’ll have a real sense for whether it makes sense. This also helps students learn how to move between big-picture and focused thinking, which are two different but equally important modes of reasoning.

Some people ask me, “Isn’t ‘ishing’ just estimating?” It is, but when we ask students to estimate, they often groan, thinking it’s yet another mathematical method. But when we ask them to “ish” a number, they're more willing to offer their thinking.

Ishing helps students develop a sense for numbers and shapes. It can help soften the sharp edges in mathematics, making it easier for kids to jump in and engage. It can buffer students against the dangers of perfectionism, which we know can be a damaging mindset. I think we all need a little more ish in our lives.

You also argue that mathematics should be taught in more visual ways. What do you mean by that?

For most people, mathematics is an almost entirely symbolic, numerical experience. Any visuals are usually sterile images in a textbook, showing bisecting angles, or circles divided into slices. But the way we function in life is by developing models of things in our minds. Take a stapler: Knowing what it looks like, what it feels and sounds like, how to interact with it, how it changes things – all of that contributes to our understanding of how it works.

There’s an activity we do with middle-school students where we show them an image of a 4 x 4 x 4 cm cube made up of smaller 1 cm cubes, like a Rubik’s Cube. The larger cube is dipped into a can of blue paint, and we ask the students, if they could take apart the little cubes, how many sides would be painted blue? Sometimes we give the students sugar cubes and have them physically build a larger 4 x 4 x 4 cube. This is an activity that leads into algebraic thinking.

Some years back we were interviewing students a year after they’d done that activity in our summer camp and asked what had stayed with them. One student said, “I’m in geometry class now, and I still remember that sugar cube, what it looked like and felt like.” His class had been asked to estimate the volume of their shoes, and he said he’d imagined his shoes filled with 1 cm sugar cubes in order to solve that question. He had built a mental model of a cube.

When we learn about cubes, most of us don’t get to see and manipulate them. When we learn about square roots, we don’t take squares and look at their diagonals. We just manipulate numbers.

I wonder if people consider the physical representations more appropriate for younger kids.

That’s the thing – elementary school teachers are amazing at giving kids those experiences, but it dies out in middle school, and by high school it’s all symbolic. There’s a myth that there’s a hierarchy of sophistication where you start out with visual and physical representations and then build up to the symbolic. But so much of high-level mathematical work now is visual. Here in Silicon Valley, if you look at Tesla engineers, they're drawing, they're sketching, they're building models, and nobody says that's elementary mathematics.

There’s an example in the book where you’ve asked students how they would calculate 38 x 5 in their heads, and they come up with several different ways of arriving at the same answer. The creativity is fascinating, but wouldn’t it be easier to teach students one standard method?

A depiction of various ways to calculate 38 x 5, numerically and visually. | Courtesy Jo Boaler

That narrow, rigid version of mathematics where there’s only one right approach is what most students experience, and it’s a big part of why people have such math trauma. It keeps them from realizing the full range and power of mathematics. When you only have students blindly memorizing math facts, they’re not developing number sense. They don’t learn how to use numbers flexibly in different situations. It also makes students who think differently believe there’s something wrong with them.

When we open mathematics to acknowledge the different ways a concept or problem can be viewed, we also open the subject to many more students. Mathematical diversity, to me, is a concept that includes both the value of diversity in people and the diverse ways we can see and learn mathematics. When we bring those forms of diversity together, it’s powerful. If we want to value different ways of thinking and problem-solving in the world, we need to embrace mathematical diversity.