discuss broadly problem solving strategy in teaching elementary pupils

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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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:

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 ↵

Mathematics for Elementary Teachers Copyright © 2018 by Michelle Manes is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License , except where otherwise noted.

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The Problem-solving Classroom

• Visualising
• Working backwards
• Reasoning logically
• Conjecturing
• Working systematically
• Looking for patterns
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• stage of the lesson 
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• 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
• strategic competence
• adaptive reasoning
• productive disposition

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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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4 Tips on Teaching Problem Solving (From a Student)

A student shares her insights into the most important skill you can teach. (Hint: It’s not perseverance.)

Education is one of the most important things in the world, but at most schools, students are told to memorize facts, formulas, and functions without any applicability to the real challenges we will face later. Instead, give us challenges; give us problems that focus on real-world scenarios; give us a chance to understand the world we’re entering and to be prepared for it before we’re thrown in headfirst.

At Two Rivers Public Charter School, they taught us how to problem solve, and they made it relevant. Here are four tips that engaged me in my learning that you can adapt in your classroom:

1. Give Your Students Hard Problems

In the real world, we’re not going to have nice problems that will be easy to understand. We are going to have complex problems that require a lot more preparation than most math, science, or English classes will give us. The challenges in the real world won’t be simple, and the problems that are supposed to prepare us for that world shouldn't be either.

2. Make Problem Solving Relevant to Your Students’ Lives

In the seventh grade, we looked at statistics concerning racial murders and the jury system. That’s something that is going to affect students later in life, and we got a chance to look at it from a mathematical perspective. Problems like that are actually relevant to us, and they’re not things we’re supposed to just memorize or learn. They are things from which we can take very important life lessons, and then actually apply them later on in life.

Related Article: Solving Real World Issues Through Problem-Based Learning

In the eighth grade, we wrote policy briefs in relation to gene editing and presented them to the National Academies of Sciences, Engineering, and Medicine. We talked to researchers who worked with CRISPR-Cas9 (a gene editing tool used to modify specific genes in organisms), and we studied how gene editing is evolving and how we can use this modern technology for science applications. At the same time, in English, we read The Giver by Lois Lowry and analyzed whether the society in the book was ethical to gain an understanding of what ethical means and how it’s applicable in real situations, like gene editing.

This wasn’t something where we were being told, “Somebody’s going to buy 60 watermelons at a store.” This was actually happening in real life, and the only people really discussing this were people whom it wasn't even going to affect. This science won’t come into widespread use until much later, and we’re going to be the first ones who are actually in danger from the possible consequences of it. By presenting our policy briefs, we had a chance to make an impact and get our voice out there at only 14.

3. Teach Your Students How to Grapple (It’s More Powerful Than Perseverance)

Grappling is like perseverance, but it goes beyond that. Perseverance means trying again and again, even after you’ve failed. Grappling implies trying even before you fail the first time. It’s thinking, “First, I’ll work with it independently. Okay, I’m really not understanding it. Let me go back to my notes. Okay, I have solved for the first part of it. Now I have the second part of it. Okay, I got the question wrong; let me try again. Maybe I can ask my peer now.” Grappling is working hard to make sure you understand the problem fully, and then using every resource at your fingertips to solve it.

4. Put More Importance on Student Understanding Than on Getting the Right Answer

I am graduating from Two Rivers with a practical view of the world. I don’t think that many students come out of middle school saying, “It was great.” And I don’t think many students have had this introduction to our society and its benefits and drawbacks. I’m also coming out of here with incredible problem-solving skills and the ability to look at any problem and have 10,000 ways to solve it in my mind already—because we don’t just memorize functions or the periodic table. We understand why, and we work to understand how to solve a problem instead of just getting the answer.

As students preparing for the real world, it is so much more impactful for us if our learning is relevant and challenging than if it is just about memorizing the right answers.

Two Rivers Public Charter School

Per pupil expenditures, free / reduced lunch, demographics:.

This blog post is part of our Schools That Work series, which features key practices from Two Rivers Public Charter School .

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5 Teaching Mathematics Through Problem Solving

Janet Stramel

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

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

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

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

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

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

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

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

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

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

Key features of a good mathematics problem includes:

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

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

But look at the b ack.

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

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

Mathematics Tasks and Activities that Promote Teaching through Problem Solving

Choosing the Right Task

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

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

Low Floor High Ceiling Tasks

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

The strengths of using Low Floor High Ceiling Tasks:

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

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

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

Math in 3-Acts

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

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

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

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

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

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

Number Talks

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

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

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

• Inside Mathematics Number Talks
• Number Talks Build Numerical Reasoning

Saying “This is Easy”

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

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

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

Instead, you and your students could say the following:

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

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

Using “Worksheets”

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

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

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

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

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

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

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

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

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

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

when we teach students how to problem solve

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

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

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

Mathematics Methods for Early Childhood Copyright © 2021 by Janet Stramel is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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9 Strategies We Can Teach Students to Problem Solve

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Back to the main Bully Prevention Guide .

In the school setting and in life in general, a parent or teacher cannot be with a student 24 hours a day for 365 days a year. We must empower our students to deal with conflict and problem solve. We know that students can solve many problems at the lowest level; we also know there are going to be problems that they cannot solve which require adult support. If we can equip students with several tools and strategies, they can become problem solvers of many situations. Without the skills, many of these skills can lead to bully behaviors.

Here are nine easy strategies we can teach at school and at home that will support students "going to solution."

PU #22 - Student Strategy: Tell Them To Stop

This strategy may sound obvious, but when I talk with students many of them start a conversation such as this, "Jimmy keeps calling me frog face." I then ask Billy, "Did you tell Jimmy to stop." As student such as Billy, routinely tells me, "No!" This is the first place to start. I give them the words to tell Jimmy. They practice with me, and then they can go and use the strategy. This strategy stops quite a few problems. If Jimmy persists, the student may come back to me and say, "Mr. Konen, I used the words we practiced to tell Jimmy to stop, but he did not." Then an adult will need to support. My conversations with Jimmy start by asking, "Tell me what's going on between you and Billy." I do this to see if Jimmy is going to own any of his behavior, or if he is going to blame Billy for some other situation. Sometimes this leads me to do some re-teaching with Billy. Many times I continue my verbiage, "Billy came to me and said you were calling him frog face. Billy and I also worked on words to say to make sure it stopped, but is sounds like it is still going on. Can you tell me more about this?" At this point, a student has two routes to decide, "own their behavior," or "deny it." Many times a student will own their behavior, come up with a plan with me to fix it with Billy, and vow to stop. If it continues, we can then take it to the next level much faster. I write down these interactions in my notebook; keeping track of behavioral patterns, and what students were involved is important to getting the bully behavior stopped. I may even write a Level 1 discipline referral if I deem it more important to track. Then, sometimes a student choses to "deny it," and I then spend more time with Billy finding out if there were any witnesses. I then investigate further with these witnesses. My whole goal is to get enough information to make my case to Jimmy and for him to own his behavior.

This is strategy is sometimes hard for a student, especially a student who avoids conflict. If we can teach our students to stick up for themselves, and say "No," or "Stop," I believe they will be able to do it in much more serious situations!

PU #23 - Student Strategy: Go To Another Activity

A student, "Jimmy," sometimes is so involved in a game and the competitive nature of the game becomes confrontational. Students want to play and make sure it is fair for everyone who is involved. Asking Jimmy to go to another activity can be difficult as an adult; we know we may in for an argument. To Jimmy, it seems unfair that he should go to another activity when Billy isn't being fair in the game.

The goal of this strategy is for a student to self-reflect, and chose another game. When the student can chose another game, they are able to see that that their current participation is leading to bigger problems. Whether the problems self-generated from or derived from other students, they know if they keep playing, they are not going to be able to regulate themselves.

Adults must reinforce this as a viable option to problem solving, and when a student choses to use it, we must praise their smart choice.

PU #24 - Student Strategy: Rock, Paper, Scissors, Go!

Fairness and justice are qualities we all want in life. Unfortunately, we know that life is not always fair. When students have a disagreement or believe there are two separate viable solutions, a simple game of "Rock, Paper, Scissors" may be an easy tool for them to use.

Students must agree to the outcome of the game before they play. If they cannot accept the outcome, then this strategy is not going to work. Seems simple, but sometimes it is not. Sometimes a student will risk the outcome of the problem to this 50/50 odds game and they lose. Negative behaviors from losing may come to the forefront, and another problem-solving strategy may need to be employed.

The idea behind the game is to solve a problem in a game format, requiring less stress then arguing. This strategy is great for games outside, as well as making decisions in the classroom where both solutions can be used.

PU #25 - Student Strategy: Use "I messages…"

The ability for a student to use their words to solve a problem is vital to them becoming a successful problem solver. Unfortunately, some students use physical aggression or hurtful language to solve problems. This leads to bully behavior if it forms a pattern. We must teach our students to use words that are productive, proactive, and let others know you how you feel.

The use of "I messages," is a good strategy for students to learn when they are solving problems and dealing with conflict. An example of an "I message" states, "Jimmy, I feel sad when Billy is made fun of." Starting with the word, "I," lets the other student know that what Jimmy said affects how you feel. It is not accusatory and it is much harder for Jimmy to attack a feeling then accusing him of making fun of Billy. When a Jimmy is accused, he mostly likely will be defensive and wanting to fight back.

A teacher can practice "I" with students before they use them. This practice and modeling will help them go to solution much faster when conflict arises. This is a strategy that many adults have difficulties using as shifting blame tends to be easier. Trying to go solution with the other person tends take more time and work. Many people don't like using this strategy as they may they are being judged as weak due to the fact of using "I messages" that discuss their personal emotions. If we can teach this skill, it can be a useful tool for the rest of a student's life.

PU #26 - Student Strategy: Apologize

It seems in this day-in-age, an apology can be difficult. Many people do not want to be viewed as being weak, or admit they are guilty, or even think that an apology can be a practicable solution. Teaching students that an apology is part of the culture of our school when we make a mistake or hurt someone. We teach the words on how to give an apology. We do not give out apologies that are forced. Apologies must be sincere, tell the person specifically what they are apologizing for, as well as letting them know it is not going to occur again.

Teaching a student when to use an apology can be tricky. Deciding when an apology can be a useful strategy is powerful self-reflection tool. A student must step outside of the situation, assess that what they were doing or saying garners an apology. For a student to do this on their must be practiced, modeled, and praised.

We also teach students how to accept an apology. We never say after an apology, "It's okay." It is not "okay," but we can teach students to accept the apology. We also go further and tell them to make sure the behavior does not happen again.

PU #27 - Student Strategy: Talk It Out

Two students in a disagreement sometimes needs more time to "talk to it." Students, like many adults, want to avoid conflict and want a problem to be solved immediately. If the problem is not solved immediately, they tend to hold grudges, possibly spread rumors, and cause more drama.

Many times this strategy needs adult support. I like to bring students together that have previously been friends, but are not talking. One student, and many times both students, want to get the problem fixed so they can rekindle their friendship. I spend time prepping both students on what it will look like in my office, the words they can use, and discuss the perception from the other student. This gives them those tools, some power, and ultimately, some choice in how they will handle the situation.

For students who have matured enough to be able to spend time talking out a problem, having them do it on their own is a tool they will use the rest of the lives. Parents and teachers modeling language, actions, and problem solving skills are creating solution-orientated students who can be successful with the "talk it out" strategy.

PU #28 - Student Strategy: Ignore It

The ability for a student to ignore a problem can be a successful tool. The saying goes, "What we pay attention to is what we get more of..." If a student puts a lot of time and effort into combatting negative attention directed towards them, it might continue to occur. The ability for a student to ignore another person's bully behavior takes courage.

When I discuss this strategy with a student, I talk with them about power. I let them know there is a lot of power in the words we chose to use and just as much in the words we do not use. I ask them who retains the power in their situation.

If a student who is exhibiting bully behavior continues to use the language or action because they know they are getting to that student, that student loses his or her power. A student retains their power by not letting the other student know how they feel. If they can ignore the problem, the other student may just stop because they are not getting the attention they are trying to ascertain.

If ignore strategy is not working and the bully behavior persists, adult intervention and support may be needed.

PU #29 - Student Strategy: Wait and Cool Off

In the heat of a battle or argument, taking a break to cool off can be strategy that students can use to solve a problem. Self-reflection by the student is required. A student must understand his or her own emotions and know that stepping away from a problem or conflict can be successful.

Sometimes students get into a conflict and they attack each other because they do not have the words in the heat of the moment to solve the problem. Giving wait time to answer, gather yourself, and possibly do some breathing to calm down can be a great tool to problem solve.

PU #30 - Student Strategy: Walk Away and Let It Go.

Similar to the "Ignore it" strategy, "walk away and let it go" is another strategy that students can use to go to solution when they encounter conflict and bully behavior. When a student decides to walk away from a problem, it takes courage not to engage with the person exhibiting the bully behavior.

When a student walks away, they are physically putting distance between the bully behavior and themselves. In addition, they are letting the problem go. This strategy also does not give attention to the person bullying, one more possible reason for the bullying to stop.

Of course if this problem persists, adult support and intervention may be needed. Letting an adult know of the situation is important.

MORE BULLYING PREVENTION GUIDE RELATED READINGS :

• 3 Types of Bullying in School + 1 Immense Social Challenge
• 5 Reasons Why Schools Have a Difficult Time Stopping Bully Behaviors
• 5 Reasons Why We Need to Define Bully Behavior and Stop Generalizing Events as "Bullying"
• 6+ Steps to Addressing Bullying When It Occurs
• 9 Strategies We Can Teach Students to Problem Solve (Currently here)
• 7 Ways How to Raise a Defender of Bully Prevention
• Provocative Victims and 7+ Practices for Victory
• One School Wide Philosophy: High-Trusting Relationships
• 13 Ideas to Combat Bullying at the Community Level
• 6 of the Most Hideous Cyber Bullying Tactics Used By Students
• 11 Communication Strategies to Combat Bullying
• 20 Ideas To Successfully Use Bully Data
• 5 Reasons Why Strong Instruction Affects Bully Prevention
• #1 Instructional Lesson for All Students on Bully Prevention
• Power of Buddy Classrooms: 19 Ideas
• 8 Ways to Teach Empathy
• 10 Ways to Empower Defenders
• 9 Reasons Culture Trumps Strategy
• One School Wide Strategy: Kindness Campaign
• Recent Posts

• Learning How to Say No and Set Boundaries with Parents - November 21, 2022
• If You Had Only One Behavior Strategy to Use in Your Classroom, What Would It Be? - September 26, 2022
• Live Your Code: 7 Strategies That Will Help You Be the Most Effective Educator You Can Be - August 15, 2022

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20 Effective Math Strategies To Approach Problem-Solving 

Katie Keeton

Math strategies for problem-solving help students use a range of approaches to solve many different types of problems. It involves identifying the problem and carrying out a plan of action to find the answer to mathematical problems.  

Problem-solving skills are essential to math in the general classroom and real-life. They require logical reasoning and critical thinking skills. Students must be equipped with strategies to help them find solutions to problems.

This article explores mathematical problem solving strategies, logical reasoning and critical thinking skills to help learners with solving math word problems independently in real-life situations. 

What are problem-solving strategies?

Problem-solving strategies in math are methods students can use to figure out solutions to math problems. Some problem-solving strategies: 

• Draw a model
• Use different approaches
• Check the inverse to make sure the answer is correct

Students need to have a toolkit of math problem-solving strategies at their disposal to provide different ways to approach math problems. This makes it easier to find solutions and understand math better. 

Strategies can help guide students to the solution when it is difficult ot know when to start.

The ultimate guide to problem solving techniques

Download these ready-to-go problem solving techniques that every student should know. Includes printable tasks for students including challenges, short explanations for teachers with questioning prompts.

20 Math Strategies For Problem-Solving

Different problem-solving math strategies are required for different parts of the problem. It is unlikely that students will use the same strategy to understand and solve the problem. 

Here are 20 strategies to help students develop their problem-solving skills. 

Strategies to understand the problem

Strategies that help students understand the problem before solving it helps ensure they understand: 

• The context
• What the key information is
• How to form a plan to solve it

Following these steps leads students to the correct solution and makes the math word problem easier .

Here are five strategies to help students understand the content of the problem and identify key information. 

1. Read the problem aloud

Read a word problem aloud to help understand it. Hearing the words engages auditory processing. This can make it easier to process and comprehend the context of the situation.

2. Highlight keywords 

When keywords are highlighted in a word problem, it helps the student focus on the essential information needed to solve it. Some important keywords help determine which operation is needed.  For example, if the word problem asks how many are left, the problem likely requires subtraction.  Ensure students highlight the keywords carefully and do not highlight every number or keyword. There is likely irrelevant information in the word problem.

3. Summarize the information

Read the problem aloud, highlight the key information and then summarize the information. Students can do this in their heads or write down a quick summary.  Summaries should include only the important information and be in simple terms that help contextualize the problem.

4. Determine the unknown

A common problem that students have when solving a word problem is misunderstanding what they are solving. Determine what the unknown information is before finding the answer.  Often, a word problem contains a question where you can find the unknown information you need to solve. For example, in the question ‘How many apples are left?’ students need to find the number of apples left over.

5. Make a plan

Once students understand the context of the word problem, have dentified the important information and determined the unknown, they can make a plan to solve it.  The plan will depend on the type of problem. Some problems involve more than one step to solve them as some require more than one answer.  Encourage students to make a list of each step they need to take to solve the problem before getting started.

Strategies for solving the problem 

1. draw a model or diagram.

Students may find it useful to draw a model, picture, diagram, or other visual aid to help with the problem solving process.  It can help to visualize the problem to understand the relationships between the numbers in the problem. In turn, this helps students see the solution.

Similarly, you could draw a model to represent the objects in the problem:

2. Act it out

This particular strategy is applicable at any grade level but is especially helpful in math investigation in elementary school . It involves a physical demonstration or students acting out the problem using movements, concrete resources and math manipulatives .  When students act out a problem, they can visualize and contectualize the word problem in another way and secure an understanding of the math concepts.  The examples below show how 1st-grade students could “act out” an addition and subtraction problem:

3. Work backwards

Working backwards is a popular problem-solving strategy. It involves starting with a possible solution and deciding what steps to take to arrive at that solution.  This strategy can be particularly helpful when students solve math word problems involving multiple steps. They can start at the end and think carefully about each step taken as opposed to jumping to the end of the problem and missing steps in between.

For example,

To solve this problem working backwards, start with the final condition, which is Sam’s grandmother’s age (71) and work backwards to find Sam’s age. Subtract 20 from the grandmother’s age, which is 71.  Then, divide the result by 3 to get Sam’s age. 71 – 20 = 51 51 ÷ 3 = 17 Sam is 17 years old.

4. Write a number sentence

When faced with a word problem, encourage students to write a number sentence based on the information. This helps translate the information in the word problem into a math equation or expression, which is more easily solved.  It is important to fully understand the context of the word problem and what students need to solve before writing an equation to represent it.

5. Use a formula

Specific formulas help solve many math problems. For example, if a problem asks students to find the area of a rug, they would use the area formula (area = length × width) to solve.   Make sure students know the important mathematical formulas they will need in tests and real-life. It can help to display these around the classroom or, for those who need more support, on students’ desks.

Strategies for checking the solution 

Once the problem is solved using an appropriate strategy, it is equally important to check the solution to ensure it is correct and makes sense. 

There are many strategies to check the solution. The strategy for a specific problem is dependent on the problem type and math content involved.

Here are five strategies to help students check their solutions. 

1. Use the Inverse Operation

For simpler problems, a quick and easy problem solving strategy is to use the inverse operation. For example, if the operation to solve a word problem is 56 ÷ 8 = 7 students can check the answer is correct by multiplying 8 × 7. As good practice, encourage students to use the inverse operation routinely to check their work. 

2. Estimate to check for reasonableness

Once students reach an answer, they can use estimation or rounding to see if the answer is reasonable.  Round each number in the equation to a number that’s close and easy to work with, usually a multiple of ten.  For example, if the question was 216 ÷ 18 and the quotient was 12, students might round 216 to 200 and round 18 to 20. Then use mental math to solve 200 ÷ 20, which is 10.  When the estimate is clear the two numbers are close. This means your answer is reasonable. 

3. Plug-In Method

This method is particularly useful for algebraic equations. Specifically when working with variables.  To use the plug-in method, students solve the problem as asked and arrive at an answer. They can then plug the answer into the original equation to see if it works. If it does, the answer is correct.

If students use the equation 20m+80=300 to solve this problem and find that m = 11, they can plug that value back into the equation to see if it is correct. 20m + 80 = 300 20 (11) + 80 = 300 220 + 80 = 300 300 = 300 ✓

4. Peer Review

Peer review is a great tool to use at any grade level as it promotes critical thinking and collaboration between students. The reviewers can look at the problem from a different view as they check to see if the problem was solved correctly.   Problem solvers receive immediate feedback and the opportunity to discuss their thinking with their peers. This strategy is effective with mixed-ability partners or similar-ability partners. In mixed-ability groups, the partner with stronger skills provides guidance and support to the partner with weaker skills, while reinforcing their own understanding of the content and communication skills.  If partners have comparable ability levels and problem-solving skills, they may find that they approach problems differently or have unique insights to offer each other about the problem-solving process.

5. Use a Calculator

A calculator can be introduced at any grade level but may be best for older students who already have a foundational understanding of basic math operations. Provide students with a calculator to allow them to check their solutions independently, accurately, and quickly. Since calculators are so readily available on smartphones and tablets, they allow students to develop practical skills that apply to real-world situations.  

Step-by-step problem-solving processes for your classroom

In his book, How to Solve It , published in 1945, mathematician George Polya introduced a 4-step process to solve problems. 

Polya’s 4 steps include:

• Understand the problem
• Devise a plan
• Carry out the plan

Today, in the style of George Polya, many problem-solving strategies use various acronyms and steps to help students recall. 

Many teachers create posters and anchor charts of their chosen process to display in their classrooms. They can be implemented in any elementary, middle school or high school classroom. 

Here are 5 problem-solving strategies to introduce to students and use in the classroom.

How Third Space Learning improves problem-solving 

Resources .

Third Space Learning offers a free resource library is filled with hundreds of high-quality resources. A team of experienced math experts carefully created each resource to develop students mental arithmetic, problem solving and critical thinking. 

Explore the range of problem solving resources for 2nd to 8th grade students. 

One-on-one tutoring 

Third Space Learning offers one-on-one math tutoring to help students improve their math skills. Highly qualified tutors deliver high-quality lessons aligned to state standards. 

Former teachers and math experts write all of Third Space Learning’s tutoring lessons. Expertly designed lessons follow a “my turn, follow me, your turn” pedagogy to help students move from guided instruction and problem-solving to independent practice. 

Throughout each lesson, tutors ask higher-level thinking questions to promote critical thinking and ensure students are developing a deep understanding of the content and problem-solving skills.

Problem-solving

Educators can use many different strategies to teach problem-solving and help students develop and carry out a plan when solving math problems. Incorporate these math strategies into any math program and use them with a variety of math concepts, from whole numbers and fractions to algebra. 

Teaching students how to choose and implement problem-solving strategies helps them develop mathematical reasoning skills and critical thinking they can apply to real-life problem-solving.

READ MORE : 8 Common Core math examples

There are many different strategies for problem-solving; Here are 5 problem-solving strategies: • draw a model  • act it out  • work backwards  • write a number sentence • use a formula

Here are 10 strategies of problem-solving: • Read the problem aloud • Highlight keywords • Summarize the information • Determine the unknown • Make a plan • Draw a model  • Act it out  • Work backwards  • Write a number sentence • Use a formula

1. Understand the problem 2. Devise a plan 3. Carry out the plan 4. Look back

Some strategies you can use to solve challenging math problems are: breaking the problem into smaller parts, using diagrams or models, applying logical reasoning, and trying different approaches.

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13 Effective Learning Strategies: A Guide to Using them in your Math Classroom

Differentiated Instruction: 9 Differentiated Curriculum And Instruction Strategies For Teachers 

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Home » Blog » General » Teaching Problem-Solving Skills to Elementary Students: Activities & Discussions

Teaching Problem-Solving Skills to Elementary Students: Activities & Discussions

Introduction

Problem-solving is an essential life skill that helps students navigate the challenges they face in their daily lives. By teaching children how to identify, analyze, and resolve problems, educators can empower them to develop resilience and independence. In this blog post, we will explore an easy-to-implement no-prep activity designed to teach problem-solving skills to elementary students. We will also provide discussion questions to stimulate further exploration of the topic, as well as mention related skills and resources.

No-Prep Activity: The Problem-Solving Chain

This simple activity encourages students to work together to solve a problem by following a step-by-step process. Here’s how it works:

• Divide the students into pairs or small groups.
• Present a common problem scenario, such as the one involving Serena and Kate in the prompt.
• Identify the problem.
• Decide if it’s a big or small problem.
• If it’s a small problem, brainstorm ways to solve the problem themselves.
• Choose the best solution and try it out.
• Encourage the students to discuss their solutions and the reasoning behind their choices.
• Repeat the activity with different problem scenarios to reinforce the problem-solving process.

The Problem-Solving Chain activity helps students practice their problem-solving skills in a collaborative and structured environment, which can boost their confidence in tackling real-life challenges.

Discussion Questions

After completing the no-prep activity, engage your students in a conversation about problem-solving with the following discussion questions:

• Why is it important to identify whether a problem is big or small? How can this help us in solving the problem?
• Can you think of a time when you faced a problem and solved it on your own? How did you feel afterward?
• What are some strategies we can use when we’re feeling overwhelmed by a problem?
• How can working together with others help us solve problems more effectively?
• Why is it important to learn problem-solving skills at a young age?

Related Skills

Problem-solving is just one aspect of social-emotional learning (SEL). To help students develop a well-rounded set of SEL skills, consider teaching them about:

• Effective communication: Listening to others, expressing thoughts and feelings clearly, and resolving conflicts peacefully.
• Empathy: Understanding and sharing the feelings of others, which can lead to better cooperation and problem-solving.
• Resilience: Bouncing back from setbacks and learning from mistakes.
• Teamwork: Collaborating with others to achieve common goals and solve problems.
• Decision-making: Evaluating the pros and cons of different options and making informed choices.

If you found this blog post helpful and would like to explore more activities and resources for teaching problem-solving skills and other SEL topics, we invite you to sign up for free sample materials at Everyday Speech. Our comprehensive library offers a wide range of engaging materials designed to help educators teach essential life skills to students of all ages.

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Elementary teachers’ experience of engaging with Teaching Through Problem Solving using Lesson Study

• Original Article
• Published: 13 May 2022
• Volume 35 , pages 901–927, ( 2023 )

Cite this article

• Mairéad Hourigan   ORCID: orcid.org/0000-0002-6895-1895 1 &
• Aisling M. Leavy   ORCID: orcid.org/0000-0002-1816-0091 1  

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For many decades, problem solving has been a focus of elementary mathematics education reforms. Despite this, in many education systems, the prevalent approach to mathematics problem solving treats it as an isolated activity instead of an integral part of teaching and learning. In this study, two mathematics teacher educators introduced 19 Irish elementary teachers to an alternative problem solving approach, namely Teaching Through Problem Solving (TTP), using Lesson Study (LS) as the professional development model. The findings suggest that the opportunity to experience TTP first-hand within their schools supported teachers in appreciating the affordances of various TTP practices. In particular, teachers reported changes in their beliefs regarding problem solving practice alongside developing problem posing knowledge. Of particular note was teachers’ contention that engaging with TTP practices through LS facilitated them to appreciate their students’ problem solving potential to the fullest extent. However, the planning implications of the TTP approach presented as a persistent barrier.

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Introduction

A fundamental goal of mathematics education is to develop students’ ability to engage in mathematical problem solving. Despite curricular emphasis internationally on problem solving, many teachers are uncertain how to harness students’ problem solving potential (Cheeseman, 2018 ). While many problem solving programmes focus on providing students with step-by-step supports through modelling, heuristics, and other structures (Polya, 1957 ), Goldenberg et al. ( 2001 ) suggest that the most effective approach to developing students’ problem solving ability is by providing them with frequent opportunities over a prolonged period to solve worthwhile open-ended problems that are challenging yet accessible to all. This viewpoint is in close alignment with reform mathematics perspectives that promote conceptual understanding, where students actively construct their knowledge and relate new ideas to prior knowledge, creating a web of connected knowledge (Hiebert, 2003 ; Lester, 2013 ; Takahashi, 2006 ; Watanabe, 2001 ).

There is consensus in the mathematics education community that problem solving should not be taught as an isolated topic focused solely on developing problem solving skills and strategies or presented as an end-of-chapter activity (Takahashi, 2006 , 2016 ; Takahashi et al., 2013 ). Instead, problem solving should be integrated across the curriculum as a fundamental part of mathematics teaching and learning (Cai & Lester, 2010 ; Takahashi, 2016 ).

A ‘Teaching Through Problem Solving’ (TTP) approach, a problem solving style of instruction that originated in elementary education in Japan, meets these criteria treating problem solving as a core practice rather than an ‘add-on’ to mathematics instruction.

Teaching Through Problem Solving (TTP)

Teaching Through Problem Solving (TTP) is considered a powerful means of promoting mathematical understanding as a by-product of solving problems, where the teacher presents students with a specially designed problem that targets certain mathematics content (Stacey, 2018 ; Takahashi et al., 2013 ). The lesson implementation starts with the teacher presenting a problem and ensuring that students understand what is required. Students then solve the problem either individually or in groups, inventing their approaches. At this stage, the teacher does not model or suggest a solution procedure. Instead, they take on the role of facilitator, providing support to students only at the right time (Hiebert, 2003 ; Lester, 2013 ; Takahashi, 2006 ). As students solve the problem, the teacher circulates, observes the range of student strategies, and identifies work that illustrates desired features. However, the problem solving lesson does not end when the students find a solution. The subsequent sharing phase, called Neriage (polishing ideas), is considered by Japanese teachers to be the heart of the lesson rather than its culmination. During Neriage, the teacher purposefully selects students to share their strategies, compares various approaches, and introduces increasingly sophisticated solution methods. Effective questioning is central to this process, alongside careful recording of the multiple solutions on the board. The teacher concludes the lesson by formalising and consolidating the lesson’s main points. This process promotes learning for all students (Hiebert, 2003 ; Stacey, 2018 ; Takahashi, 2016 ; Takahashi et al., 2013 ; Watanabe, 2001 ).

The TTP approach assumes that students develop, extend, and enrich their understandings as they confront problematic situations using existing knowledge. Therefore, TTP fosters the symbiotic relationship between conceptual understanding and problem solving, as conceptual understanding is required to solve challenging problems and make sense of new ideas by connecting them with existing knowledge. Equally, problem solving promotes conceptual understanding through the active construction of knowledge (Hiebert, 2003 ; Lambdin, 2003 ; Takahashi, 2006 ). Consequently, students simultaneously develop more profound understandings of the mathematics content while cultivating problem solving skills (Kapur, 2010 ; Stacey, 2018 ).

Relevant research affirms that teachers acknowledge the merits of this approach (Sullivan et al., 2014 ) and most students report positive experiences (Russo & Minas, 2020 ). The process is considered to make students’ thinking and learning visible (Ingram et al., 2020 ). Engagement in TTP has resulted in teachers becoming more aware of and confident in their students’ problem solving abilities and subsequently expecting more from them (Crespo & Featherstone, 2006 ; Sakshaug & Wohlhuter, 2010 ).

Demands of TTP

Adopting a TTP approach challenges pre-existing beliefs and poses additional knowledge demands for elementary teachers, both content and pedagogical (Takahashi, 2008 ).

Research has consistently reported a relationship between teacher beliefs and the instructional techniques used, with evidence of more rule-based, teacher-directed strategies used by teachers with traditional mathematics beliefs (Stipek et al., 2001 ; Swan, 2006 ; Thompson, 1985 ). These teachers tend to address problem solving separately from concept and skill development and possess a simplistic view of problem solving as translating a problem into abstract mathematical terms to solve it. Consequently, such teachers ‘are very concerned about developing skilfulness in translating (so-called) real-world problems into mathematical representations and vice versa’ (Lester, 2013 , p. 254). Early studies of problem solving practice reported direct instructional techniques where the teacher would model how to solve the problem followed by students practicing similar problems (Chapman, 2015 ; Hiebert, 2003 ; Lester, 2013 ). This naïve conception of problem solving is reflected in many textbook problems that simply require students to apply previously learned routine procedures to solve problems that are merely thinly disguised number operations (Lester, 2013 ; Singer & Voica, 2013 ). Hence, the TTP approach requires a significant shift for teachers who previously considered problem solving as an extra activity conducted after the new mathematics concepts are introduced (Lester, 2013 ; Takahashi et al., 2013 ) or whose personal experience of problem solving was confined to applying routine procedures to word problems (Sakshaug &Wohlhuter, 2010 ).

Alongside beliefs, teachers’ knowledge influences their problem solving practices. Teachers require a deep understanding of the nature of problem solving, in particular viewing problem solving as a process (Chapman, 2015 ). To be able to understand the stages problem solvers go through and appreciate what successful problem solving involves, teachers benefit from experiencing solving problems from the problem solver’s perspective (Chapman, 2015 ; Lester, 2013 ).

It is also essential that teachers understand what constitutes a worthwhile problem when selecting or posing problems (Cai, 2003 ; Chapman, 2015 ; Lester, 2013 ; O’Shea & Leavy, 2013 ). This requires an understanding that problems are ‘mathematical tasks for which the student does not have an obvious way to solve it’ (Chapman, 2015 , p. 22). Teachers need to appreciate the variety of problem characteristics that contribute to the richness of a problem, e.g. problem structures and cognitive demand (Klein & Leiken, 2020 ; O’Shea & Leavy, 2013 ). Such understandings are extensive, and rather than invest heavily in the time taken to construct their mathematics problems, teachers use pre-made textbook problems or make cosmetic changes to make cosmetic changes to these (Koichu et al., 2013 ). In TTP, due consideration must also be given to the problem characteristics that best support students in strengthening existing understandings and experiencing new learning of the target concept, process, or skill (Cai, 2003 ; Takahashi, 2008 ). Specialised content knowledge is also crucial for teachers to accurately predict and interpret various solution strategies and misconceptions/errors, to determine the validity of alternative approaches and the source of errors, to sequence student approaches, and to synthesise approaches and new learning during the TTP lesson (Ball et al., 2008 ; Cai, 2003 ; Leavy & Hourigan, 2018 ).

Teachers should also be knowledgeable regarding appropriate problem solving instruction. It is common for teachers to teach for problem solving (i.e., focusing on developing students’ problem solving skills and strategies). Teachers adopting a TTP approach engage in reform classroom practices that reflect a constructivist-oriented approach to problem solving instruction where the teacher guides students to work collaboratively to construct meaning, deciding when and how to support students without removing their autonomy (Chapman, 2015 ; Hiebert, 2003 ; Lester, 2013 ). Teachers ought to be aware of the various relevant models of problem solving, including Polya’s ( 1957 ) model that supports teaching for problem solving (Understand the problem-Devise a plan-Carry out the plan-Look back) alongside models that support TTP (e.g., Launch-Explore-Summarise) (Lester, 2013 ; Sullivan et al., 2021 ). While knowledge of heuristics and strategies may support teachers’ problem solving practices, there is consensus that teaching heuristics and strategies or teaching about problem solving does not significantly improve students’ problem solving ability. Teachers require a thorough knowledge of their students as problem solvers, for example, being aware of their abilities and factors that hinder their success, including language (Chapman, 2015 ). Knowledge of content and student, alongside content and teaching (Ball et al., 2008 ), is essential during TTP planning when predicting student approaches and errors. Such knowledge is also crucial during TTP implementation when determining the validity of alternative approaches, identifying the source of errors (Explore phase), sequencing student approaches, and synthesising the range of approaches and new learning effectively (Summarise phase) (Cai, 2003 ; Leavy & Hourigan, 2018 ).

Supports for teachers

Given the extensive demands of TTP, adopting this approach is arduous in terms of the planning time required to problem pose, predict approaches, and design questions and resources (Lester, 2013 ; Sullivan et al., 2010 ; Takahashi,  2008 ). Consequently, it is necessary to support teachers who adopt a TTP approach (Hiebert, 2003 ). Professional development must facilitate them to experience the approach themselves as learners and then provide classroom implementation opportunities that incorporate collaborative planning and reflection when trialling the approach (Watanabe, 2001 ). In Japan, a common form of professional development to promote, develop, and refine TTP implementation among teachers and test potential problems for TTP is Japanese Lesson Study (LS) (Stacey, 2018 ; Takahashi et al., 2013 ). Another valuable support is access to a repository of worthwhile problems. In Japan, government-authorised textbooks and teacher manuals provide a sequence of lessons with rich well-tested problems to introduce new concepts. They also detail alternative strategies used by students and highlight the key mathematical aspects of these strategies (Takahashi, 2016 ; Takahashi et al., 2013 ).

Teachers’ reservations about TTP

Despite the acknowledged benefits of TTP for students, some teachers report reluctance to employ TTP, identifying a range of obstacles. These include limited mathematics content knowledge or pedagogical content knowledge (Charalambous, 2008 ; Sakshaug & Wohlhuter, 2010 ) and a lack of access to resources or time to develop or modify appropriate resources (Ingram et al., 2020 ; Russo & Hopkins, 2019 ; Sullivan et al., 2015 ). Other barriers for teachers with limited experience of TTP include giving up control, struggling to support students without directing them, and a tendency to demonstrate how to solve the problem (Cheeseman, 2018 ; Crespo & Featherstone, 2006 ; Klein & Leiken, 2020 ; Takahashi et al., 2013 ). Resistance to TTP is also associated with some teachers’ perception that this approach would lead to student disengagement and hence be unsuitable for lower-performing students (Ingram et al., 2020 ; Russo & Hopkins, 2019 ; Sullivan et al., 2010 ).

Problem solving practices in Irish elementary mathematics education

Within the Irish context, problem solving is a central tenet of elementary mathematics curriculum documents (Department of Education and Science (DES), 1999 ) with recommendations that problem solving should be integral to students’ mathematical learning. However, research reveals a mismatch between intended and implemented problem solving practices (Dooley et al., 2014 ; Dunphy et al., 2014 ), where classroom practices reflect a narrow approach limited to problem solving as an ‘add on’, only applied after mathematical procedures had been learned and where problems are predominantly sourced from dedicated sections of textbooks (Department of Education and Skills (DES), 2011 ; Dooley et al., 2014 ; National Council for Curriculum and Assessment (NCCA), 2016 ; O’Shea & Leavy, 2013 ). Regarding the attained curriculum, Irish students have underperformed in mathematical problem solving, relative to other skills, in national and international assessments (NCCA, 2016 ; Shiel et al., 2014 ). Consensus exists that there is scope for improvement of problem solving practices, with ongoing calls for Irish primary teachers to receive support through school-based professional development models alongside creating a repository of quality problems (DES, 2011 ; Dooley et al., 2014 ; NCCA, 2016 ).

Lesson Study (LS) as a professional development model

Reform mathematics practices, such as TTP, challenge many elementary teachers’ beliefs, knowledge, practices, and cultural norms, particularly if they have not experienced the approach themselves as learners. To support teachers in enacting reform approaches, they require opportunities to engage in extended and targeted professional development involving collaborative and practice-centred experiences (Dudley et al., 2019 ; Murata et al., 2012 ; Takahashi et al., 2013 ). Lesson Study (LS) possesses the characteristics of effective professional development as it embeds ‘…teachers’ learning in their everyday work…increasing the likelihood that their learning will be meaningful’ (Fernandez et al., 2003 , p. 171).

In Japan, LS was developed in the 1980s to support teachers to use more student-centred practices. LS is a school-based, collaborative, reflective, iterative, and research-based form of professional development (Dudley et al., 2019 ; Murata et al., 2012 ). In Japan, LS is an integral part of teaching and is typically conducted as part of a school-wide project focused on addressing an identified teaching–learning challenge (Takahashi & McDougal, 2016 ). It involves a group of qualified teachers, generally within a single school, working together as part of a LS group to examine and better understand effective teaching practices. Within the four phases of the LS cycle, the LS group works collaboratively to study and plan a research lesson that addresses a pre-established goal before implementing (teach) and reflecting (observe, analyse and revise) on the impact of the lesson activities on students’ learning.

LS has become an increasingly popular professional development model outside of Japan in the last two decades. In these educational contexts, it is necessary to find a balance between fidelity to LS as originally envisaged and developing a LS approach that fits the cultural context of a country’s education system (Takahashi & McDougal, 2016 ).

Relevant research examining the impact of LS on qualified primary mathematics teachers reports many benefits. Several studies reveal that teachers demonstrated transformed beliefs regarding effective pedagogy and increased self-efficacy in their use due to engaging in LS (Cajkler et al., 2015 ; Dudley et al., 2019 ; Fernandez, 2005 ; Gutierez, 2016 ). Enhancements in participating teachers’ knowledge have also been reported (Cajkler et al., 2015 ; Dudley et al., 2019 ; Fernandez, 2005 ; Gutierez, 2016 ; Murata et al., 2012 ). Other gains recounted include improvements in practice with a greater focus on students (Cajkler et al., 2015 ; Dudley et al., 2019 ; Flanagan, 2021 ).

Context of this study

A cluster of urban schools, coordinated by their local Education Centre, engaged in an initiative to enhance teachers’ mathematics problem solving practices. The co-ordinator of the initiative approached the researchers, both mathematics teacher educators (MTEs), seeking a relevant professional development opportunity. Aware of the challenges of problem solving practice within the Irish context, the MTEs proposed an alternative perspective on problem solving: the Teaching Through Problem Solving (TTP) approach. Given Cai’s ( 2003 ) recommendation that teachers can best learn to teach through problem solving by teaching and reflecting as opposed to taking more courses, the MTEs identified LS as the best fit in terms of a supportive professional development model, as it is collaborative, experiential, and school-based (Dudley et al., 2019 ; Murata et al., 2012 ; Takahashi et al., 2013 ). Consequently, LS would promote teachers to work collaboratively to understand the TTP approach, plan TTP practices for their educational context, observe what it looks like in practice, and assess the impact on their students’ thinking (Takahashi et al., 2013 ). In particular, the MTEs believed that the LS phases and practices would naturally support TTP structures, emphasizing task selection and anticipating students’ solutions. Given Lester’s ( 2013 ) assertion that each problem solving experience a teacher engages in can potentially alter their knowledge for teaching problem solving, the MTEs sought to explore teachers’ perceptions of the impact of engaging with TTP through LS on their beliefs regarding problem solving and their knowledge for teaching problem solving.

Research questions

This paper examines two research questions:

Research question 1: What are elementary teachers’ reported problem solving practices prior to engaging in LS?

Research question 2: What are elementary teachers’ perceptions of what they learned from engaging with TTP through LS?

Methodology

Participants.

The MTEs worked with 19 elementary teachers (16 female, three male) from eight urban schools. Schools were paired to create four LS groups on the basis of the grade taught by participating class teachers, e.g. Grade 3 teacher from school 1 paired with Grade 4 teacher from school 2. Each LS group generally consisted of 4–5 teachers, with a minimum of two teachers from each school, along with the two MTEs. For most teachers, LS and TTP were new practices being implemented concurrently. However, given the acknowledged overlap between the features of the TTP and LS approaches, for example, the focus on problem posing and predicting student strategies, the researchers were confident that the content and structure were compatible. Also, in Japan, LS is commonly used to promote TTP implementation among teachers (Stacey, 2018 ; Takahashi et al., 2013 ).

All ethical obligations were adhered to throughout the research process, and the study received ethical approval from the researchers’ institutional board. Of the 19 participating LS teachers invited to partake in the research study, 16 provided informed consent to use their data for research purposes.

Over eight weeks, the MTEs worked with teachers, guiding each LS group through the four LS phases involving study, design, implementation, and reflection of a research lesson that focused on TTP while assuming the role of ‘knowledgeable others’ (Dudley et al., 2019 ; Hourigan & Leavy, 2021 ; Takahashi & McDougal, 2016 ). An overview of the timeline and summary of each LS phase is presented in Table 1 .

LS phase 1: Study

This initial study phase involved a one-day workshop. The process and benefits of LS as a school-based form of professional development were discussed in the morning session and the afternoon component was spent focusing on the characteristics of TTP. Teachers experienced the TPP approach first-hand by engaging in the various lesson stages. For example, they solved a problem (growing pattern problem) themselves in pairs and shared their strategies. They also predicted children’s approaches to the problem and possible misconceptions and watched the video cases of TTP classroom practice for this problem. Particular focus was placed on the importance of problem selection and prediction of student strategies before the lesson implementation and the Neriage stage of the lesson. Teachers also discussed readings related to LS practices (e.g. Lewis & Tsuchida, 1998 ) and TTP (e.g., Takahashi, 2008 ). At the end of the workshop, members of each LS group were asked to communicate among themselves and the MTEs, before the planning phase, to decide the specific mathematics focus of their LS group’s TTP lesson (Table 1 ).

LS phase 2: Planning

The planning phase was four weeks in duration and included two 1½ hour face-to-face planning sessions (i.e. planning meetings 1 and 2) between the MTEs and each LS group (Table 1 ). Meetings took place in one of the LS group’s schools. At the start of the first planning meeting, time was dedicated to Takahashi’s ( 2008 ) work focusing on the importance of problem selection and prediction of student strategies to plan the Neriage stage of the TTP lesson. The research lesson plan structure was also introduced. Ertle et al.’s ( 2001 ) four column lesson plan template was used. It was considered particularly compatible with the TTP approach, given the explicit attention to expected student response and the teacher’s response to student activity/response.

The planning then moved onto the content focus of each LS group’s TTP research lesson. LS groups selected TTP research lessons focusing on number (group A), growing patterns (group B), money (group C), and 3D shapes (group D). Across the planning phase, teachers invested substantial time extensively discussing the TTP lesson goals in terms of target mathematics content, developing or modifying a problem to address these goals, and exploring considerations for the various lesson stages. Drawing on Takahashi’s ( 2008 ) article, it was re-emphasised that no strategies would be explicitly taught before students engaged with the problem. While one LS group modified an existing problem (group B) (Hourigan & Leavy, 2015 ), the other three LS groups posed an original problem. To promote optimum teacher readiness to lead the Neriage stage, each LS group was encouraged to solve the problem themselves in various ways considering possible student strategies and their level of mathematical complexity, thus identifying the most appropriate sequence of sharing solutions.

LS phase 3: Implementation

The implementation phase involved one teacher in each LS group teaching the research lesson (teach 1) in their school. The remaining group members and MTEs observed and recorded students’ responses. Each LS group and the MTEs met immediately for a post-lesson discussion to evaluate the research lesson. The MTEs presented teachers with a series of focus questions: What were your observations of student learning? Were the goals of the lesson achieved? Did the problem support students in developing the appropriate understandings? Were there any strategies/errors that we had not predicted? How did the Neriage stage work? What aspects of the lesson plan should be reconsidered based on this evidence? Where appropriate, the MTEs drew teachers’ attention to particular lesson aspects they had not noticed. Subsequently, each LS group revised their research lesson in response to the observations, reflections, and discussion. The revised lesson was retaught 7–10 days later by a second group member from the paired LS group school (teach 2) (Table 1 ). The post-lesson discussion for teach 2 focused mainly on the impact of changes made after the first implementation on student learning, differences between the two classes, and further changes to the lesson.

LS phase 4: Reflection

While reflection occurred after both lesson implementations, the final reflection involved all teachers from the eight schools coming together for a half-day meeting in the local Education Centre to share their research lessons, experiences, and learning (Table 1 ). Each LS group made a presentation, identifying their research lesson’s content focus and sequence of activity. Artefacts (research lesson plan, materials, student work samples, photos) were used to support observations, reflections, and lesson modifications. During this meeting, teachers also reflected privately and in groups on their initial thoughts and experience of both LS and TTP, the benefits of participation, the challenges they faced, and they provided suggestions for future practice.

Data collection

The study was a collective case study (Stake, 1995 ). Each LS group constituted a case; thus, the analysis was structured around four cases. Data collection was closely aligned with and ran concurrent to the LS process. Table 2 details the links between the LS phases and the data collection process.

The principal data sources (Table 2 ) included both MTEs’ fieldnotes (phase (P) 1–4), and reflections (P1–4), alongside email correspondence (P1–4), individual teacher reflections (P1, 2, 4) (see reflection tasks in Table 3 ), and LS documentation including various drafts of lesson plans (P2–4) and group presentations (P4). Fieldnotes refer to all notes taken by MTEs when working with the LS groups, for example, during the study session, planning meetings, lesson implementations, post-lesson discussions, and the final reflection session.

The researchers were aware of the limitations of self-report data and the potential mismatch between one’s perceptions and reality. Furthermore, data in the form of opinions, attitudes, and beliefs may contain a certain degree of bias. However, this paper intentionally focuses solely on the teachers’ perceived learning in order to represent their ‘lived experience’ of TTP. Despite this, measures were taken to assure the trustworthiness and rigour of this qualitative study. The researchers engaged with the study over a prolonged period and collected data for each case (LS group) at every LS phase (Table 2 ). All transcripts reflected verbatim accounts of participants’ opinions and reflections. At regular intervals during the study, research meetings interrogated the researchers’ understandings, comparing participating teachers’ observations and reflections to promote meaning-making (Creswell, 2009 ; Suter, 2012 ).

Data analysis

The MTEs’ role as participant researchers was considered a strength of the research given that they possessed unique insights into the research context. A grounded theory approach was adopted, where the theory emerges from the data analysis process rather than starting with a theory to be confirmed or refuted (Glaser, 1978 ; Strauss & Corbin, 1998 ). Data were examined focusing on evidence of participants’ problem solving practices prior to LS and their perceptions of their learning as a result of engaging with TTP through LS. A systematic process of data analysis was adopted. Initially, raw data were organised into natural units of related data under various codes, e.g. resistance, traditional approach, ignorance, language, planning, fear of student response, relevance, and underestimation. Through successive examinations of the relationship between existing units, codes were amalgamated (Creswell, 2009 ). Progressive drafts resulted in the firming up of several themes. Triangulation was used to establish consistency across multiple data sources. While the first theme, Vast divide between prevalent problem solving practices and TTP , addresses research question 1, it is considered an overarching theme, given the impact of teachers’ established problem solving understandings and practices on their receptiveness to and experience of TTP. The remaining five themes ( Seeing is believing : the value of practice centred experiences ; A gained appreciation of the relevance and value of TTP practices ; Enhanced problem posing understandings ; Awakening to students’ problem solving potential ; and Reservations regarding TTP) represent a generalised model of teachers’ perceived learning due to engaging with TTP through LS, thus addressing research question 2. Although one of the researchers was responsible for the initial coding, both researchers met regularly during the analysis to discuss and interrogate the established codes and to agree on themes. This process served to counteract personal bias (Suter, 2012 ).

As teacher reflections were anonymised, it was not possible to track teachers across LS phases. Consequently, teacher reflection data are labelled as phase and instrument only. For example, ‘P2, teacher reflection’ communicates that the data were collected during LS phase 2 through teacher reflection. However, the remaining data are labelled according to phase, instrument, and source, e.g. ‘P3, fieldnotes: group B’. While phase 4 data reflect teachers’ perceptions after engaging fully with the TTP approach, data from the earlier phases reflect teachers’ evolving perceptions at a particular point in their unfolding TTP experience.

Discussion of findings

The findings draw on the analysis of the data collected across the LS phases and address the research questions. Within the confines of this paper, illustrative quotes are presented to provide insights into each theme. An additional layer of analysis was completed to ensure a balanced representation of teachers’ views in reporting findings. This process confirmed that the findings represent the views of teachers across LS groups, for example, within the first theme presented ( Vast divide ), the eight quotes used came from eight different teacher reflections. Equally, the six fieldnote excerpts selected represent six different teachers’ views across the four LS groups. Furthermore, in the second theme ( Seeing is believing ), the five quotes presented were sourced from five different participating teachers’ reflections and the six fieldnote excerpts included are from six different teachers across the four LS groups. Subsequent examination of the perceptions of those teachers not included in the reporting of findings confirmed that their perspectives were represented within the quotes used. Hence, the researchers are confident that the findings represent the views of teachers across all LS groups. For each theme, sources of evidence that informed the presented conclusions will be outlined.

Vast divide between prevalent problem solving practices and TTP

This overarching theme addresses the research question ‘What were elementary teachers’ reported problem solving practices prior to engaging in LS?’.

At the start of the initiative, within the study session (fieldnotes), all teachers identified mathematics problem solving as a problem of practice. The desire to develop problem solving practices was also apparent in some teachers’ reflections (phase 1 (P1), N  = 8):

I am anxious about it. Problem solving is an area of great difficulty throughout our school (P1, teacher reflection).

During both study and planning phase discussions, across all LS groups, teachers’ reports suggested the almost exclusive use of a teaching for problem solving approach, with no awareness of the Teaching Through Problem Solving (TTP) approach; a finding also evidenced in both teacher reflections (P1, N  = 7) and email correspondence:

Unfamiliar, not what I am used to. I have no experience of this kind of problem solving. This new approach is the reverse way to what I have used for problem solving (P1, teacher reflection) Being introduced to new methods of teaching problem solving and trying different approaches is both exciting and challenging (P1: email correspondence)

Teachers’ descriptions of their problem solving classroom practices in both teacher reflections (P1, N  = 8) and study session discussions (fieldnotes) suggested a naïve conception of problem solving, using heuristics such as the ‘RUDE (read, underline, draw a picture, estimate) strategy’ (P1, fieldnotes) to support students in decoding and solving the problem:

In general, the problem solving approach described by teachers is textbook-led, where concepts are taught context free first and the problems at the end of the chapter are completed afterwards (P1, reflection: MTE2)

This approach was confirmed as widespread across all LS groups within the planning meetings (fieldnotes).

In terms of problem solving instruction, a teacher-directed approach was reported by some teachers within teacher reflections (P1, N  = 5), where the teacher focused on a particular strategy and modelled its use by solving the problem:

I tend to introduce the problem, ensure everyone understands the language and what is being asked. I discuss the various strategies that children could use to solve the problem. Sometimes I demonstrate the approach. Then children practice similar problems … (P1: Teacher reflection)

However, it was evident within the planning meetings, that this traditional approach to problem solving was prevalent among the teachers in all LS groups. During the study session (field notes and teacher reflections (P1, N  = 7)), there was a sense that problem solving was an add-on as opposed to an integral part of mathematics teaching and learning. Again, within the planning meetings, discussions across all four LS groups verified this:

Challenge: Time to focus on problems not just computation (P1: Teacher reflection). From our discussions with the various LS groups’ first planning meeting, text-based teaching seems to be resulting in many teachers teaching concepts context-free initially and then matching the concept with the relevant problems afterwards (P2, reflection: MTE1)

However, while phase 1 teacher reflections suggested that a small number of participating teachers ( N  = 4) possessed broader problem solving understandings, subsequently during the planning meetings, there was ample evidence (field notes) of problem-posing knowledge and the use of constructivist-oriented approaches that would support the TTP approach among some participating teachers in each of the LS groups:

Challenge: Spend more time on meaningful problems and give them opportunities and time to engage in activities, rather than go too soon into tricks, rhymes etc (P1, Teacher reflection). The class are already used to sharing strategies and explaining where they went wrong (P2, fieldnotes, Group B) Teacher: The problem needs to have multiple entry points (P2, fieldnotes: Group C)

While a few teachers reported problem posing practices, in most cases, this consisted of cosmetic adjustments to textbook problems. Overall, despite evidence of some promising practices, the data evidenced predominantly traditional problem solving views and practices among participating teachers, with potential for further broadening of various aspects of their knowledge for teaching problem solving including what constitutes a worthwhile problem, the role of problem posing within problem solving, and problem solving instruction. Within phase 1 teacher reflections, when reporting ‘challenges’ to problem solving practices (Table 3 ), a small number of responses ( N  = 3) supported these conclusions:

Differences in teachers’ knowledge (P1: Teacher reflection). Need to challenge current classroom practices (P1: Teacher reflection).

However, from the outset, all participating teachers consistently demonstrated robust knowledge of their students as problem solvers, evidenced in phase 1 teacher reflections ( N  = 10) and planning meeting discussions (P2, fieldnotes). However, in these early phases, teachers generally portrayed a deficit view, focusing almost exclusively on the various challenges impacting their students’ problem solving abilities. While all teachers agreed that the language of problems was inhibiting student engagement, other common barriers reported included student motivation and perseverance:

They often have difficulties accessing the problem – they don’t know what it is asking them (P2, fieldnotes: Group C) Sourcing problems that are relevant to their lives. I need to change every problem to reference soccer so the children are interested (P1: teacher reflection) Our children deal poorly with struggle and are slow to consider alternative strategies (P2, fieldnotes: Group D)

Despite showcasing a strong awareness of their students’ problem solving difficulties, teachers initially demonstrated a lack of appreciation of the benefits accrued from predicting students’ approaches and misconceptions relating to problem solving. While it came to the researchers’ attention during the study phase, its prevalence became apparent during the initial planning meeting, as its necessity and purpose was raised in three of the LS groups:

What are the benefits of predicting the children’s responses? (P1, fieldnotes). I don’t think we can predict- we will have to wait and see (P2, fieldnotes: Group A).

This finding evidences teachers’ relatively limited knowledge for teaching problem solving, given that this practice is fundamental to TTP and constructivist-oriented approaches to problem solving instruction.

Perceived impacts of engaging with TTP through LS

In response to the research question ‘What are elementary teachers’ perceptions of what they learned from engaging with TTP through LS?’, thematic data analysis identified 5 predominant themes, namely, Seeing is believing : the value of practice centred experiences ; A gained appreciation of the relevance and value of TTP practices ; Enhanced problem posing understandings ; Awakening to students’ problem solving potential ; and Reservations regarding TTP.

Seeing is believing: the value of practice centred experiences

Teachers engaged with TTP during the study phase as both learners and teachers when solving the problem. They were also involved in predicting and analysing student responses when viewing the video cases, and engaged in extensive reading, discussion, and planning for their selected TTP problem within the planning phase. Nevertheless, teachers reported reservations about the relevance of TTP for their context within both phase 2 teacher reflections ( N  = 5) as well as within the planning meeting discourse of all LS groups. Teachers’ keen awareness of their students’ problem solving challenges, coupled with the vast divide between the nature of their prior problem solving practices and the TTP approach, resulted in teachers communicating concern regarding students’ possible reaction during the planning phase:

I am worried about the problem. I am concerned that if the problem is too complex the children won’t respond to it (P2, fieldnotes: Group B) The fear that the children will not understand the lesson objective. Will they engage? (P2, Teacher reflection)

Acknowledging their apprehension regarding students’ reactions to TTP, from the outset, all participating teachers communicated a willingness to trial TTP practices:

Exciting to be part of. Eager to see how it will pan out and the learning that will be taken from it (P1, teacher reflection) They should be ‘let off’ (P2, fieldnotes: Group A).

It was only within the implementation phase, when teachers received the opportunity to meaningfully observe the TTP approach in their everyday work context, with their students, that they explicitly demonstrated an appreciation for the value of TTP practices. It was evident from teacher commentary across all LS groups’ post-lesson discussions (fieldnotes) as well as in teacher reflections (P4, N  = 10) that observing first-hand the high levels of student engagement alongside students’ capacity to engage in desirable problem solving strategies and demonstrate sought-after dispositions had affected this change:

Class teacher: They engaged the whole time because it was interesting to them. The problem is core in terms of motivation. It determines their willingness to persevere. Otherwise, it won’t work whether they have the skills or not (P3, fieldnotes: Group C) LS group member: The problem context worked really well. The children were all eager and persevered. It facilitated all to enter at their own level, coming up with ideas and using their prior knowledge to solve the problem. Working in pairs and the concrete materials were very supportive. It’s something I’d never have done before (P3, fieldnotes: Group A)

Although all teachers showcased robust knowledge of their students’ problem solving abilities prior to engaging in TTP, albeit with a tendency to focus on their difficulties and factors that inhibited them, teachers’ contributions during post-lesson discussions (fieldnotes) alongside teacher reflections (P4, N  = 9) indicate that observing TTP in action supported them in developing an appreciation of value of the respective TTP practices, particularly the role of prediction and observation of students’ strategies/misconceptions in making the students’ thinking more visible:

You see the students through the process (P3, fieldnotes: Group C) It’s rare we have time to think, to break the problem down, to watch and understand children’s ways of thinking/solving. It’s really beneficial to get a chance to re-evaluate the teaching methods, to edit the lesson, to re-teach (P4, teacher reflection)

Analysis of the range of data sources across the phases suggests that it was the opportunity to experience TTP in practice in their classrooms that provided the ‘proof of concept’:

I thought it wasn’t realistic but bringing it down to your own classroom it is relevant (P4, teacher reflection).

Hence from the teachers’ perspective, they witnessed the affordances of TTP practices in the implementation phase of the LS process.

A gained appreciation of the relevance and value of TTP practices

While during the early LS phases, teachers’ reporting suggested a view of problem solving as teaching to problem solve, data from both fieldnotes (phases 3 and 4) and teacher reflections (phase 4) demonstrate that all teachers broadened their understanding of problem solving as a result of engaging with TTP:

Interesting to turn lessons on their head and give students the chance to think, plan and come up with possible strategies and solutions (P4, Teacher reflection)

On witnessing the affordances of TTP first-hand in their own classrooms, within both teacher reflections (P4, N  = 12) and LS group presentations, the teachers consistently reported valuing these new practices:

I just thought the whole way of teaching was a good way, an effective way of teaching. Sharing and exploring more than one way of solving is vital (P4, teacher reflection) There is a place for it in the classroom. I will use aspects of it going forward (P4, fieldnotes: Group C)

In fact, teachers’ support for this problem solving approach was apparent in phase 3 during the initial post-lesson discussions. It was particularly notable when a visitor outside of the LS group who observed teach 1 challenged the approach, recommending the explicit teaching of strategies prior to engagement. A LS group member’s reply evidenced the group’s belief that TTP naturally exposes students to the relevant learning: ‘Sharing and questioning will allow students to learn more efficient strategies [other LS group members nodding in agreement]’ (P3, fieldnotes; Group A).

In turn, within phase 4 teacher reflections, teachers consistently acknowledged that engaging with TTP through LS had challenged their understandings about what constitutes effective problem solving instruction ( N  = 12). In both teacher reflections (P4, N  = 14) and all LS group presentations, teachers reported an increased appreciation of the benefits of adopting a constructivist-oriented approach to problem solving instruction. Equally for some, this was accompanied by an acknowledgement of a heightened awareness of the limitations of their previous practice :

Really made me re-think problem solving lesson structures. I tend to spoon-feed them …over-scaffold, a lot of teacher talk. … I need to find a balance… (P4, teacher reflection) Less is more, one problem can be the basis for an entire lesson (P4, teacher reflection)

What was unexpected, was that some teachers (P4, N  = 8) reported that engaging with TTP through LS resulted in them developing an increased appreciation of the value of problem solving and the need for more regular opportunities for students to engage in problem solving:

I’ve come to realise that problem solving is critical and it should be focused on more often. I feel that with regular exposure to problems they’ll come to love being problem solvers (P4, teacher reflection)

Enhanced problem posing understandings

In the early phases of LS, few teachers demonstrated familiarity with problem characteristics (P2 teacher reflection, N  = 5). However, there was growth in teachers’ understandings of what constitutes a worthwhile problem and its role within TTP within all LS groups’ post-lesson discussions and presentations (fieldnotes) and teacher reflections (P4, N  = 10):

I have a deepened understanding of how to evaluate a problem (P2, teacher reflection) It’s essential to find or create a good problem with multiple strategies and/or solutions as a springboard for a topic. It has to be relevant and interesting for the kids (P4, teacher reflection)

As early as the planning phase, a small group of teachers’ reflections ( N  = 2) suggested an understanding that problem posing is an important aspect of problem solving that merits significant attention:

It was extremely helpful to problem solve the problem (P2, teacher reflection)

However, during subsequent phases, this realisation became more mainstream, evident within all LS groups’ post-lesson discussions and presentations (fieldnotes) and teacher reflections (P4, N  = 12):

During the first planning meeting, I was surprised and a bit anxious that we would never get to having created a problem. In hindsight, this was time well spent as the problem was crucial (P4, teacher reflection) I learned the problem is key. We don’t spend enough time picking the problem (P4, fieldnotes: Group C).

Alongside this, in all LS groups’ dialogues during the post-lesson discussion and presentations (fieldnotes) and teacher reflections (P4, N  = 15), teachers consistently demonstrated an enhanced awareness of the interdependence between the quality of the problem and students’ problem solving behaviours:

Better perseverance if the problem is of interest to them (P4, teacher reflection) It was an eye-opener to me, relevance is crucial, when the problem context is relevant to them, they are motivated to engage and can solve problems at an appropriate level…They all wanted to present (P3, fieldnotes: Group C)

The findings suggest that engaging with TTP through LS facilitated participating teachers to develop an enhanced understanding of the importance of problem posing and in identifying the features of a good mathematics problem, thus developing their future problem posing capacity. In essence, the opportunity to observe the TTP practices in their classrooms stimulated an enhanced appreciation for the value of meticulous attention to detail in TTP planning.

Awakening to students’ problem solving potential

In the final LS phases, teachers consistently reported that engaging with TTP through LS provided the opportunity to see the students through the process , thus supporting them in examining their students’ capabilities more closely. Across post-lesson discussions and presentations (fieldnotes) and teacher reflections (P4, N  = 14), teachers acknowledged that engagement in core TTP practices, including problem posing, prediction of students’ strategies during planning, and careful observation of approaches during the implementation phase, facilitated them to uncover the true extent of their students’ problem solving abilities, heightening their awareness of students’ proficiency in using a range of approaches:

Class teacher: While they took a while to warm up, I am most happy that they failed, tried again and succeeded. They all participated. Some found a pattern, others used trial and error. Others worked backward- opening the cube in different ways. They said afterward ‘That was the best maths class ever’ (P3, fieldnotes: Group D) I was surprised with what they could do. I have learned the importance of not teaching strategies first. I need to pull back and let the children solve the problems their own way and leave discussing strategies to the end (P4, teacher reflection)

In three LS groups, class teachers acknowledged in the post-lesson discussion (fieldnotes) that engaging with TTP had resulted in them realising their previous underestimation of [some or all] of their students’ problem solving abilities . Teacher reflections (P4, N  = 8) and LS group presentations (fieldnotes) also acknowledged this reality:

I underestimated my kids, which is awful. The children surprised me with the way they approached the problem. In the future I need to focus on what they can do as much as what might hinder them…they are more able than we may think (P4, reflection)

In all LS groups, teachers reported that their heightened appreciation of students’ problem solving capacities promoted them to use a more constructivist-orientated approach in the future:

I learned to trust the students to problem solve, less scaffolding. Children can be let off to explore without so much teacher intervention (P3, fieldnotes: Group D)

Some teachers ( N  = 3) also acknowledged the affective benefits of TTP on students:

I know the students enjoyed sharing their different strategies…it was great for their confidence (P4, teacher reflection)

Interestingly, in contrast with teachers’ initial reservations, their experiential and school-based participation in TTP through LS resulted in a lessening of concern regarding the suitability of TTP practices for their students. Hence, this practice-based model supported teachers in appreciating the full extent of their students’ capacities as problem solvers.

Reservations regarding TTP

When introduced to the concept of TTP in the study session, one teacher quickly addressed the time implications:

It is unrealistic in the everyday classroom environment. Time is the issue. We don’t have 2 hours to prep a problem geared at the various needs (P1, fieldnotes)

Subsequently, across the initiative, during both planning meetings, the reflection session and individual reflections (P4, N  = 14), acknowledgements of the affordances of TTP practices were accompanied by questioning of its sustainability due to the excessive planning commitment involved:

It would be hard to maintain this level of planning in advance of the lesson required to ensure a successful outcome (P4, teacher reflection)

Given the extensive time dedicated to problem posing, solving, prediction, and design of questions as well as selection or creation of materials both during and between planning meetings, there was agreement in the reflection session (fieldnotes) and in teacher reflections (P4, N  = 10) that while TTP practices were valuable, in the absence of suitable support materials for teachers, adjustments were essential to promote implementation:

There is definitely a role for TTP in the classroom, however the level of planning involved would have to be reduced to make it feasible (P4, teacher reflection) The TTP approach is very effective but the level of planning involved is unrealistic with an already overcrowded curriculum. However, elements of it can be used within the classroom (P4, teacher reflection)

A few teachers ( N  = 3) had hesitations beyond the time demands, believing the success of TTP is contingent on ‘a number of criteria…’ (P4, teacher reflection):

A whole-school approach is needed, it should be taught from junior infants (P4, teacher reflection) I still have worries about TTP. We found it difficult to decide a topic initially. It lends itself to certain areas. It worked well for shape and space (P4, teacher reflection)

Conclusions

The reported problem solving practice reflects those portrayed in the literature (NCCA, 2016 ; O’Shea & Leavy, 2013 ) and could be aptly described as ‘pendulum swings between emphases on basic skills and problem solving’ (Lesh & Zawojewski, 2007 in Takahashi et al., 2013 , p. 239). Teachers’ accounts depicted problem solving as an ‘add on’ occurring on an ad hoc basis after concepts were taught (Dooley et al., 2014 ; Takahashi et al., 2013 ), suggesting a simplistic view of problem solving (Singer & Voica, 2013 ; Swan, 2006 ). Hence, in reality there was a vast divide between teachers’ problem solving practices and TTP. Alongside traditional beliefs and problem solving practices (Stipek et al., 2001 ; Swan, 2006 ; Thompson, 1985 ), many teachers demonstrated limited insight regarding what constitutes a worthwhile problem (Klein & Lieken, 2020 ) or the critical role of problem posing in problem solving (Cai, 2003 ; Takahashi, 2008 ; Watson & Ohtani, 2015 ). Teachers’ reports suggested most were not actively problem posing, with reported practices limited to cosmetic changes to the problem context (Koichu et al., 2013 ). Equally, teachers demonstrated a lack of awareness of alternative approaches to teaching for problem solving (Chapman, 2015 ) alongside limited appreciation among most of the affordances of a more child-centred approach to problem solving instruction (Hiebert, 2003 ; Lester, 2013 ; Swan, 2006 ). Conversely, there was evidence that some teachers held relevant problem posing knowledge and utilised practices compatible with the TTP approach.

All teachers displayed relatively strong understandings of their students as problem solvers from the outset; however, they initially focused almost exclusively on factors impacting students’ limited problem solving capacity (Chapman, 2015 ). Teachers’ perceptions of their students’ problem solving abilities alongside the vast divide between teachers’ problem solving practice and the proposed TTP approach resulted in teachers being initially concerned regarding students’ response to TTP. This finding supports studies that reported resistance by teachers to the use of challenging tasks due to fears that students would not be able to manage (Ingram et al., 2020 ; Russo & Hopkins, 2019 ; Sullivan et al., 2010 ). Equally, teachers communicated disquiet from the study phase regarding the time investment required to adopt the TTP approach, a finding common in similar studies (Ingram et al., 2020 ; Russo & Hopkins, 2019 ; Sullivan et al., 2015 ). Hence, the transition to TTP was uneasy for most teachers, given the significant shift it represented in terms of moving beyond a teaching to problem solve approach alongside the range of teacher demands (Takahashi et al., 2013 ).

Nevertheless, despite initial reservations, all teachers reported that engagement with TTP through LS affected their problem solving beliefs and understandings. What was particularly notable was that they reported an awakening to students’ problem solving potential . During LS’s implementation and reflection stages, all teachers acknowledged that seeing was believing concerning the benefits of TTP for their students (Kapur, 2010 ; Stacey, 2018 ). In particular, they recognised students’ positive response (Russo & Minas, 2020 ) enacted in high levels of engagement, perseverance in finding a solution, and the utilisation of a range of different strategies. These behaviours were in stark contrast to teachers’ reports in the study phase. Teachers acknowledged that students had more potential to solve problems autonomously than they initially envisaged. This finding supports previous studies where teachers reported that allowing students to engage with challenging tasks independently made students’ thinking more visible (Crespo & Featherstone, 2006 ; Ingram et al., 2020 ; Sakshand & Wohluter, 2010 ). It also reflects Sakshaug and Wohlhuter’s ( 2010 ) findings of teachers’ tendency to underestimate students’ potential to solve problems. Interestingly, at the end of LS, concern regarding the appropriateness of the TTP approach for students was no longer cited by teachers. This finding contrasts with previous studies that report teacher resistance due to fears that students will become disengaged due to the unsuitability of the approach (challenging tasks) for lower-performing students (Ingram et al., 2020 ; Russo & Hopkins, 2019 ; Sullivan et al., 2015 ). Hence, engaging with TTP through LS supported teachers in developing an appreciation of their students’ potential as problem solvers.

Teachers reported enhanced problem posing understandings, consisting of newfound awareness of the connections between the quality of the problem, the approach to problem solving instruction, and student response (Chapman, 2015 ; Cai, 2003 ; Sullivan et al., 2015 ; Takahashi, 2008 ). They acknowledged that they had learned the importance of the problem in determining the quality of learning and affecting student engagement, motivation and perseverance, and willingness to share strategies (Cai, 2003 ; Watson & Oktani, 2015 ). These findings reflect previous research reporting that engagement in LS facilitated teachers to enhance their teacher knowledge (Cajkler et al., 2015 ; Dudley et al., 2019 ; Gutierez, 2016 ).

While all teachers acknowledged the benefits of the TTP approach for students (Cai & Lester, 2010 ; Sullivan et al., 2014 ; Takahashi, 2016 ), the majority confirmed their perception of the relevance and value of various TTP practices (Hiebert, 2003 ; Lambdin, 2003 ; Takahashi, 2006 ). They referenced the benefits of giving more attention to the problem, allowing students the opportunity to independently solve, and promoting the sharing of strategies and pledged to incorporate these in their problem solving practices going forward. Many verified that the experience had triggered them to question their previous problem solving beliefs and practices (Chapman, 2015 ; Lester, 2013 ; Takahashi et al., 2013 ). This study supports previous research reporting that LS challenged teachers’ beliefs regarding the characteristics of effective pedagogy (Cajkler et al., 2015 ; Dudley et al., 2019 ; Fernandez, 2005 ; Gutierez, 2016 ). However, teachers communicated reservations regarding TTP , refraining from committing to TTP in its entirety, highlighting that the time commitment required for successful implementation on an ongoing basis was unrealistic. Therefore, teachers’ issues with what they perceived to be the excessive resource implications of TTP practices remained constant across the initiative. This finding supports previous studies that report teachers were resistant to engaging their students with ‘challenging tasks’ provided by researchers due to the time commitment required to plan adequately (Ingram et al., 2020 ; Russo & Hopkins, 2019 ; Sullivan et al., 2015 ).

Unlike previous studies, teachers in this study did not perceive weak mathematics content or pedagogical content knowledge as a barrier to implementing TTP (Charalambous, 2008 ; Sakshaug & Wohlhuter, 2010 ). However, it should be noted that the collaborative nature of LS may have hidden the knowledge demands for an individual teacher working alone when engaging in the ‘Anticipate’ element of TTP particularly in the absence of appropriate supports such as a bank of suitable problems.

The findings suggest that LS played a crucial role in promoting reported changes, serving both as a supportive professional development model (Stacey, 2018 ; Takahashi et al., 2013 ) and as a catalyst, providing teachers with the opportunity to engage in a collaborative, practice-centred experience over an extended period (Dudley et al., 2019 ; Watanabe, 2001 ). The various features of the LS process provided teachers with opportunities to engage with, interrogate, and reflect upon key TTP practices. Reported developments in understandings and beliefs were closely tied to meaningful opportunities to witness first-hand the affordances of the TTP approach in their classrooms with their students (Dudley et al., 2019 ; Fernandez et al., 2003 ; Takahashi et al., 2013 ). We suggest that the use of traditional ‘one-off’ professional development models to introduce TTP, combined with the lack of support during the implementation phase, would most likely result in teachers maintaining their initial views about the unsuitability of TTP practices for their students.

In terms of study limitations, given that all data were collected during the LS phases, the findings do not reflect the impact on teachers’ problem solving classroom practice in the medium to long term. Equally, while acknowledging the limitations of self-report data, there was no sense that the teachers were trying to please the MTEs, as they were forthright when invited to identify issues. Also, all data collected through teacher reflection was anonymous. The relatively small number of participating teachers means that the findings are not generalisable. However, they do add weight to the body of relevant research. This study also contributes to the field as it documents potential challenges associated with implementing TTP for the first time. It also suggests that despite TTP being at odds with their problem solving practice and arduous, the opportunity to experience the impact of the TTP approach with students through LS positively affected teachers’ problem solving understandings and beliefs and their commitment to incorporating TTP practices in their future practice. Hence, this study showcases the potential role of collaborative, school-based professional development in supporting the implementation of upcoming reform proposals (Dooley et al., 2014 ; NCCA, 2016 , 2017 , 2020 ), in challenging existing beliefs and practices and fostering opportunities for teachers to work collaboratively to trial reform teaching practices over an extended period (Cajkler et al., 2015 ; Dudley et al., 2019 ). Equally, this study confirms and extends previous studies that identify time as an immense barrier to TTP. Given teachers’ positivity regarding the impact of the TTP approach, their consistent acknowledgement of the unsustainability of the unreasonable planning demands associated with TTP strengthens previous calls for the development of quality support materials in order to avoid resistance to TTP (Clarke et al., 2014 ; Takahashi, 2016 ).

The researchers are aware that while the reported changes in teachers’ problem solving beliefs and understandings are a necessary first step, for significant and lasting change to occur, classroom practice must change (Sakshaug & Wohlhuter, 2010 ). While it was intended that the MTEs would work alongside interested teachers and schools to engage further in TTP in the school term immediately following this research and initial contact had been made, plans had to be postponed due to the commencement of the COVID 19 pandemic. The MTEs are hopeful that it will be possible to pick up momentum again and move this initiative to its natural next stage. Future research will examine these teachers’ perceptions of TTP after further engagement and evaluate the effects of more regular opportunities to engage in TTP on teachers’ problem solving practices. Another possible focus is teachers’ receptiveness to TTP when quality support materials are available.

In practical terms, in order for teachers to fully embrace TTP practices, thus facilitating their students to avail of the many benefits accrued from engagement, teachers require access to professional development (such as LS) that incorporates collaboration and classroom implementation at a local level. However, quality school-based professional development alone is not enough. In reality, a TTP approach cannot be sustained unless teachers receive access to quality TTP resources alongside formal collaboration time.

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Acknowledgements

The authors acknowledge the participating teachers’ time and contribution to this research study.

This work was supported by the Supporting Social Inclusion and Regeneration in Limerick’s Programme Innovation and Development Fund.

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Hourigan, M., Leavy, A.M. Elementary teachers’ experience of engaging with Teaching Through Problem Solving using Lesson Study. Math Ed Res J 35 , 901–927 (2023). https://doi.org/10.1007/s13394-022-00418-w

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DOI : https://doi.org/10.1007/s13394-022-00418-w

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[2023] 50 Instructional Strategies Examples for Every Elementary Classroom

• October 15, 2023
• Instructional Coaching

Quick Answer: Instructional strategies are techniques and methods used by teachers to engage students in the learning process and promote deeper understanding. Here are 50 examples of instructional strategies that can be used in elementary classrooms:

• Problem-Solving
• Didactic Questioning
• Demonstration
• Storytelling
• Drill & Practice
• Spaced Repetition
• Project-Based Learning
• Concept Mapping
• Case Studies
• Reading for Meaning
• Science Experiments
• Field Trips
• Simulations
• Service Learning
• Peer Instruction
• Brainstorming
• Role-Playing
• Think-Pair-Share
• Learning Centers
• Computer-Based Instruction
• Research Projects
• Graphic Organizers
• Cooperative Learning
• Visual Aids
• Manipulatives
• Socratic Seminars
• Think Alouds
• Interactive Whiteboards
• Flipped Classroom
• Differentiated Instruction
• Guided Reading
• Think-Tac-Toe
• Problem-Based Learning
• Concept Attainment
• Direct Instruction
• Reciprocal Teaching
• Inquiry-Based Learning
• Self-Assessment
• Exit Tickets
• Peer Feedback
• Metacognition
• Reflective Journals

Quick Tips and Facts:

• Different instructional strategies cater to different learning styles and promote active engagement.
• It is important to vary instructional strategies to keep students engaged and meet their diverse needs.
• Effective instructional strategies promote critical thinking, problem-solving, and collaboration skills.
• It is essential to provide clear instructions and scaffold learning when using instructional strategies.
• Regularly assess student understanding and adjust instructional strategies accordingly.

Watch the video on YouTube .

Instructional strategies are the techniques and methods that teachers use to facilitate learning in the classroom. These strategies aim to engage students, promote deeper understanding, and cater to different learning styles. By employing a variety of instructional strategies, teachers can create a dynamic and inclusive learning environment that meets the diverse needs of their students.

50 Instructional Strategies Examples for Every Elementary Classroom

1. Problem-Solving

Problem-solving activities encourage critical thinking and help students develop problem-solving skills. Teachers can present real-world problems or scenarios for students to analyze and solve collaboratively.

Lectures involve the teacher presenting information to the whole class. This strategy is useful for introducing new concepts or providing background knowledge. However, it is important to make lectures interactive and engaging to maintain student interest.

3. Didactic Questioning

Didactic questioning involves the teacher asking questions to guide student thinking and promote deeper understanding. By asking open-ended questions, teachers can encourage students to think critically and articulate their thoughts.

4. Demonstration

Demonstrations involve the teacher showing students how to perform a task or complete a process. This strategy is particularly effective for teaching practical skills or complex procedures.

5. Storytelling

Storytelling is a powerful instructional strategy that engages students and helps them make connections to the content. Teachers can use stories to introduce new concepts, illustrate real-life examples, or spark student interest.

6. Drill & Practice

Drill and practice activities involve repetitive practice of skills or concepts. These activities help reinforce learning and improve retention. However, it is important to balance drill and practice with other instructional strategies to maintain student engagement.

7. Spaced Repetition

Spaced repetition involves revisiting previously learned material at spaced intervals. This strategy helps reinforce learning and improve long-term retention. Teachers can incorporate spaced repetition through regular review activities or quizzes.

8. Project-Based Learning

Project-based learning involves students working on a project or task that requires them to apply knowledge and skills to real-world situations. This strategy promotes critical thinking, problem-solving, collaboration, and creativity.

9. Concept Mapping

Concept mapping is a visual representation of relationships between concepts. This strategy helps students organize and connect information, promoting deeper understanding and critical thinking.

10. Case Studies

Case studies involve analyzing real or hypothetical scenarios to apply knowledge and problem-solving skills. This strategy encourages critical thinking, decision-making, and collaboration.

11. Reading for Meaning

Reading for meaning involves actively engaging with texts to understand and analyze the content. Teachers can use strategies such as close reading, annotation, and discussion to promote comprehension and critical thinking.

12. Science Experiments

Science experiments provide hands-on opportunities for students to explore scientific concepts and develop inquiry skills. These activities promote critical thinking, problem-solving, and collaboration.

13. Field Trips

Field trips offer students the opportunity to learn outside the classroom and make real-world connections to the curriculum. Teachers can plan field trips to museums, historical sites, or nature reserves to enhance learning experiences.

Games are an engaging way to reinforce learning and promote active participation. Teachers can incorporate educational games that align with the curriculum to make learning fun and interactive.

15. Simulations

Simulations allow students to experience real-world scenarios in a controlled environment. This strategy promotes problem-solving, decision-making, and critical thinking skills.

16. Service Learning

Service learning involves students engaging in community service projects that connect to the curriculum. This strategy promotes civic engagement, empathy, and critical thinking.

17. Peer Instruction

Peer instruction involves students teaching and learning from each other. This strategy promotes collaboration, communication, and critical thinking skills.

Debates provide opportunities for students to research, analyze, and present arguments on a given topic. This strategy promotes critical thinking, communication, and persuasive skills.

19. Fishbowl

Fishbowl discussions involve a small group of students having a discussion while the rest of the class observes. This strategy promotes active listening, critical thinking, and respectful communication.

20. Brainstorming

Brainstorming encourages students to generate ideas and solutions to a problem or question. This strategy promotes creativity, critical thinking, and collaboration.

21. Role-Playing

Role-playing involves students taking on different roles or perspectives to understand a concept or scenario. This strategy promotes empathy, critical thinking, and communication skills.

22. Think-Pair-Share

Think-pair-share involves students individually thinking about a question or prompt, discussing their thoughts with a partner, and then sharing with the whole class. This strategy promotes critical thinking, collaboration, and communication.

23. Learning Centers

Learning centers are designated areas in the classroom where students can engage in independent or small group activities related to the curriculum. This strategy promotes independent learning, collaboration, and hands-on exploration.

24. Computer-Based Instruction

Computer-based instruction involves using technology to deliver instruction and engage students in interactive learning activities. This strategy promotes digital literacy, critical thinking, and problem-solving skills.

Essays require students to express their thoughts and ideas in a structured written format. This strategy promotes critical thinking, communication, and writing skills.

26. Research Projects

Research projects involve students conducting independent research on a topic of interest. This strategy promotes critical thinking, information literacy, and presentation skills.

27. Journaling

Journaling involves students reflecting on their learning experiences and expressing their thoughts and feelings in writing. This strategy promotes self-reflection, critical thinking, and self-expression.

28. Graphic Organizers

Graphic organizers are visual tools that help students organize and connect information. This strategy promotes critical thinking, comprehension, and organization skills.

Jigsaw is a cooperative learning strategy where students work in small groups to become experts on a specific topic and then teach their findings to the rest of the class. This strategy promotes collaboration, critical thinking, and communication skills.

30. Cooperative Learning

Cooperative learning involves students working together in small groups to achieve a common goal. This strategy promotes collaboration, communication, and critical thinking skills.

31. Visual Aids

Visual aids such as charts, diagrams, and images help students visualize and understand complex concepts. This strategy promotes comprehension, critical thinking, and engagement.

32. Manipulatives

Manipulatives are hands-on materials that students can use to explore and understand abstract concepts. This strategy promotes kinesthetic learning, critical thinking, and problem-solving skills.

33. Socratic Seminars

Socratic seminars involve students engaging in a structured discussion around a text or topic. This strategy promotes critical thinking, communication, and respectful dialogue.

34. Think Alouds

Think alouds involve the teacher verbalizing their thought process while solving a problem or completing a task. This strategy helps students develop metacognitive skills and understand problem-solving strategies.

35. Interactive Whiteboards

Interactive whiteboards allow teachers and students to interact with digital content and engage in collaborative activities. This strategy promotes active learning, critical thinking, and technology literacy.

36. Flipped Classroom

The flipped classroom model involves students learning new content at home through videos or online resources and using class time for hands-on activities and discussions. This strategy promotes self-directed learning, critical thinking, and collaboration.

37. Differentiated Instruction

Differentiated instruction involves tailoring instruction to meet the diverse needs of students. This strategy promotes individualized learning, critical thinking, and engagement.

38. Guided Reading

Guided reading involves small group instruction where students read and discuss texts at their instructional level. This strategy promotes reading comprehension, critical thinking, and collaboration.

39. Think-Tac-Toe

Think-tac-toe is a choice-based activity where students select tasks from a grid to demonstrate their understanding of a topic. This strategy promotes critical thinking, creativity, and self-directed learning.

40. Problem-Based Learning

Problem-based learning involves students working on authentic, real-world problems that require critical thinking and problem-solving skills. This strategy promotes collaboration, creativity, and inquiry.

41. Concept Attainment

Concept attainment involves presenting students with examples and non-examples of a concept to help them understand its defining characteristics. This strategy promotes critical thinking, analysis, and classification skills.

42. Direct Instruction

Direct instruction involves the teacher providing explicit instruction and modeling of skills or concepts. This strategy promotes clarity, comprehension, and skill development.

43. Reciprocal Teaching

Reciprocal teaching involves students taking turns leading discussions and summarizing key points while reading a text. This strategy promotes comprehension, critical thinking, and collaboration.

44. Inquiry-Based Learning

Inquiry-based learning involves students exploring questions, problems, or phenomena through investigation and discovery. This strategy promotes curiosity, critical thinking, and problem-solving skills.

45. Self-Assessment

Self-assessment involves students reflecting on their own learning and progress. This strategy promotes metacognition, self-reflection, and goal-setting.

46. Exit Tickets

Exit tickets are brief assessments or reflections that students complete at the end of a lesson or class. This strategy provides feedback to the teacher and promotes self-reflection and metacognition.

47. Peer Feedback

Peer feedback involves students providing constructive feedback to their peers on their work or ideas. This strategy promotes collaboration, communication, and critical thinking.

48. Metacognition

Metacognition involves students thinking about their own thinking and learning processes. This strategy promotes self-awareness, self-regulation, and critical thinking.

49. Mnemonics

Mnemonics are memory aids or techniques that help students remember information. This strategy promotes memorization, critical thinking, and recall.

50. Reflective Journals

Reflective journals involve students writing about their learning experiences, thoughts, and feelings. This strategy promotes self-reflection, critical thinking, and self-expression.

Incorporating a variety of instructional strategies in the elementary classroom is essential for engaging students, promoting deeper understanding, and meeting their diverse needs. By using strategies such as problem-solving, project-based learning, simulations, and cooperative learning, teachers can create a dynamic and inclusive learning environment. It is important to regularly assess student understanding and adjust instructional strategies accordingly to ensure effective learning outcomes.

Recommended Links:

• Lesson Planning
• Classroom Management
• 2023 List of Instructional Strategies for Teaching

Reference Links:

• 30 Instructional Strategies Examples for Every Kind of …

Marti is a seasoned educator and strategist with a passion for fostering inclusive learning environments and empowering students through tailored educational experiences. With her roots as a university tutor—a position she landed during her undergraduate years—Marti has always been driven by the joy of facilitating others' learning journeys.

Holding a Bachelor's degree in Communication alongside a degree in Social Work, she has mastered the art of empathetic communication, enabling her to connect with students on a profound level. Marti’s unique educational background allows her to incorporate holistic approaches into her teaching, addressing not just the academic, but also the emotional and social needs of her students.

Throughout her career, Marti has developed and implemented innovative teaching strategies that cater to diverse learning styles, believing firmly that education should be accessible and engaging for all. Her work on the Teacher Strategies site encapsulates her extensive experience and dedication to education, offering readers insights into effective teaching methods, classroom management techniques, and strategies for fostering inclusive and supportive learning environments.

As an advocate for lifelong learning, Marti continuously seeks to expand her knowledge and skills, ensuring her teaching methods are both evidence-based and cutting edge. Whether through her blog articles on Teacher Strategies or her direct engagement with students, Marti remains committed to enhancing educational outcomes and inspiring the next generation of learners and educators alike.

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10 Ways to Teach Your Children to Be Problem Solvers

Problem-solving is vital in navigating the complexities of life and is best nurtured from a young age. Let’s explore a variety of approaches, each contributing to the development of a child’s ability to think critically and resolve challenges effectively.

Strategy 1: Modeling Problem-Solving Behavior

Parents are the first role models children observe and learn from. Demonstrating problem-solving skills in everyday life plays a crucial role in teaching children how to handle challenges.

Impact of Demonstrating Problem-Solving

• Observational Learning: Children learn by observing their parents. When a parent faces a challenge and vocalizes their thought process, it provides a practical, real-world example of problem-solving.
• Developing Cognitive Skills: As parents articulate their problem-solving steps, children learn to think critically and analytically. This process helps in developing their cognitive skills.

How to Model Problem-Solving

• Think Out Loud: Parents should verbalize their thoughts when encountering a problem. For instance, if deciding between buying different products, explain the pros and cons of each option out loud.
• Show Emotion Management: It’s beneficial to express how certain problems make you feel and how you manage these emotions. This teaches emotional regulation alongside problem-solving.
• Involve Children in Solutions: For age-appropriate problems, involve children in the decision-making process. Ask for their opinions and discuss the potential outcomes.
• Boosts Confidence: When children see their parents tackling problems effectively, it boosts their confidence in handling their issues.
• Enhances Critical Thinking: This method promotes critical thinking and decision-making skills in children.
• Prepares for Real-life Situations: Children get better prepared for real-life situations, understanding that problems are a normal part of life and can be approached logically and calmly.

Strategy 2: Encouraging Creative Play

Creative play and DIY projects are not just forms of entertainment for children; they are essential tools for developing problem-solving skills.

How Creative Play Fosters Problem-Solving

• Stimulates Imagination: Engaging in activities like building forts, crafting, or imaginative play scenarios encourages children to think outside the box, an essential aspect of problem-solving.
• Encourages Experimentation: Creative play often involves trial and error, teaching children that it’s okay to fail and try again, a key component of solving problems.
• Develops Cognitive Flexibility: When children create and explore in an unstructured environment, they learn to adapt and change their approaches, which is vital in problem-solving.

DIY Projects as Learning Tools

• Hands-On Experience: DIY projects provide hands-on opportunities for children to encounter and solve real-world problems. They learn to follow steps, use tools, and understand the process of creating something from start to finish.
• Collaborative Problem-Solving: Working on projects with others, including parents or siblings, enhances their ability to work as a team and solve problems together.
• Boosts Self-Efficacy: Completing a project successfully instills a sense of accomplishment and confidence in their problem-solving abilities.
• Enhances Critical Thinking: Children learn to think critically about how to use materials and what steps to take to achieve their desired outcome.
• Promotes Persistence: Creative play teaches persistence as children learn that not every attempt leads to immediate success.
• Encourages Independent Thinking: These activities allow children to make decisions, fostering independent thought and decision-making skills.

Strategy 3: Systematic Problem-Solving Approach

A systematic method for problem-solving helps children approach challenges in a more organized and effective manner.

Step-by-Step Problem-Solving Method

Identify emotions:.

Begin by helping children recognize and name their emotions related to the problem (e.g., frustration, confusion). This step is crucial for emotional regulation and clear thinking.

Define the Problem:

Guide children to articulate the problem clearly. Encourage them to state the issue in their own words, which helps in understanding the challenge more deeply.

Brainstorm Solutions:

Encourage children to think of as many solutions as possible, without initially judging the ideas. This brainstorming phase fosters creativity and open-mindedness.

Evaluate Solutions:

Guide children to consider the pros and cons of each solution. Ask questions like, “What could happen if you try this?” to help them think through the outcomes.

Choose a Solution:

Encourage children to select a solution based on their evaluation. This step empowers them to make decisions and take ownership of the problem-solving process.

Implement the Solution:

Guide them in putting their chosen solution into action. This step translates their theoretical understanding into practical application.

Reflect on the Outcome:

After the solution has been implemented, discuss with children what worked well and what could be improved. This reflection helps in learning from the experience.

• Develops Critical Thinking: This approach enhances critical thinking skills by requiring children to analyze problems and consider various solutions.
• Encourages Independence: By following these steps, children learn to rely on their own abilities to solve problems.
• Builds Resilience: Children learn that not every problem is solved on the first try, which builds resilience and persistence.

Strategy 4: Reading and Discussing Problem-Solving Stories

Stories and books are powerful tools for teaching problem-solving. They offer relatable scenarios where characters face and overcome challenges, providing real-life lessons in a fictional setting.

Using Stories to Teach Problem-Solving

Selecting appropriate books:.

Choose stories that focus on characters solving problems. Books like “Ladybug Girl and Bumblebee Boy” by Jacky Davis and “The Curious George Series” by Margaret and H.E. Rey are great examples where characters face and resolve dilemmas.

Discussion During Reading:

Engage children in discussions about the story. Ask questions like, “What problem is the character facing?” and “How did they solve it?” This helps children understand the problem-solving process.

Relating to Personal Experiences:

Encourage children to connect the story’s events to their own lives. Discuss how they might handle similar situations, fostering empathy and personal connection.

Encouraging Active Participation:

Have children predict outcomes or suggest alternative solutions for the characters. This engages their critical thinking and imagination.

Role-Playing:

Involve children in role-playing exercises based on the stories. Acting out different scenarios helps solidify the problem-solving methods demonstrated by the characters.

• Enhances Comprehension: Discussing the story’s problems and solutions improves children’s comprehension and analytical skills.
• Builds Empathy: Identifying with characters and their challenges helps develop empathy and emotional intelligence.
• Encourages Creative Thinking: By exploring different solutions within a safe, fictional context, children can expand their creative problem-solving abilities.

Strategy 5: Promoting Autonomy and Learning from Failure

Fostering autonomy in children is a critical aspect of their development. It involves allowing them to make decisions, take risks, and, most importantly, learn from their mistakes.

Allowing Mistakes and Failures

• Avoiding Helicopter Parenting: Overprotective or “helicopter” parenting can hinder a child’s ability to develop problem-solving skills. Allowing children to face challenges and sometimes fail teaches them resilience and self-reliance.
• Learning Opportunities : Mistakes and failures are valuable learning opportunities. They teach children that not every attempt will be successful and that persistence is key.
• Encouraging Risk-Taking: Encourage children to take calculated risks. This helps them learn to weigh options and make decisions based on their judgments.

Guiding Through Failures

• Supportive Environment: Create a supportive environment where children feel safe to fail. Encourage them to try again and guide them through the process of analyzing what went wrong.
• Constructive Feedback: Provide constructive feedback that focuses on the effort and strategy rather than the outcome. This approach helps children understand that failure is a part of the learning process.
• Builds Problem-Solving Skills: Experiencing failure and learning to overcome it is an integral part of developing problem-solving skills.
• Promotes Growth Mindset: It encourages a growth mindset where children understand that abilities can be developed through dedication and hard work.
• Enhances Emotional Intelligence: Learning from failures helps children manage their emotions and cope with setbacks in a healthy manner.

Strategy 6: Utilizing Open-Ended Questions

Open-ended questions are a powerful tool in encouraging critical thinking and problem-solving in children. These questions do not have a predetermined answer, allowing children to explore their thoughts and ideas freely.

Implementing Open-Ended Questions:

• Types of Questions: Ask questions that cannot be answered with a simple ‘yes’ or ‘no’. Examples include, “How could we solve this problem together?” or “What do you think would happen if…?”
• Encouraging Explanation: Prompt children to explain their reasoning with questions like, “How did you come to that conclusion?” or “Can you tell me more about your thought process?”
• Fostering Imagination: Use questions that encourage children to use their imagination, such as “What would you do if you were in this situation?” or “How would you handle this differently?”

Benefits of Open-Ended Questions:

• Develops Problem-Solving Skills: These questions make children contemplate different aspects of a problem and potential solutions, enhancing their problem-solving abilities.
• Enhances Communication Skills: Open-ended questions require children to articulate their thoughts clearly, improving their communication skills.
• Builds Confidence: As children express their ideas and are heard, it boosts their self-esteem and confidence in their abilities.

Creating a Supportive Environment:

• Active Listening: Actively listen to the child’s responses without interrupting. This shows that their thoughts and opinions are valued.
• Non-Judgmental Responses: Respond to their answers in a non-judgmental way, encouraging them to share more freely.
• Encourage Exploration: Encourage children to explore different answers and viewpoints, reinforcing that there are often multiple ways to approach a problem.

Strategy 7: Fostering Open-Mindedness

Teaching children to be open-minded is crucial for developing effective problem-solving skills. It involves considering various perspectives and integrating different views into solutions.

Encouraging Multiple Perspectives:

• Understanding Different Viewpoints: Encourage children to think about how others might view a situation. Ask questions like, “What do you think someone else would do in this case?” or “Can you think of a different way to look at this problem?”
• Empathy in Problem-Solving: Teach children to consider the feelings and perspectives of others involved in a problem. This not only helps in finding more compassionate solutions but also in building strong interpersonal skills.

Integrating Diverse Solutions:

• Combining Ideas: Encourage children to combine different ideas to find a novel solution. This could involve brainstorming sessions where multiple solutions are discussed and combined.
• Learning from Different Cultures: Expose children to problem-solving methods from different cultures and backgrounds. This broadens their understanding and appreciation of diverse approaches.
• Enhances Creativity: Open-mindedness in problem-solving fosters creativity, as children learn to think outside their usual boundaries.
• Builds Critical Thinking: Considering multiple perspectives requires children to critically evaluate each viewpoint, enhancing their critical thinking skills.
• Promotes Tolerance and Understanding: Fostering open-mindedness helps children develop tolerance and understanding towards different ideas and cultures.

Strategy 8: Incorporating Problem-Solving into Family Culture

Integrating problem-solving into family culture involves turning everyday challenges into learning opportunities and making this practice an enjoyable part of family life.

Practical Ways to Integrate Problem-Solving:

• Family Meetings: Regular family meetings can be an effective way to discuss and solve family issues together. It encourages collaboration and collective decision-making.
• Shared Challenges: Involve the entire family in solving practical problems, such as planning a family vacation or budgeting for a big purchase. This teaches children the value of planning and compromise.
• Fun Problem-Solving Activities: Incorporate games and activities that involve problem-solving skills, like puzzles, strategy games, or scavenger hunts. This makes the process fun and engaging.

Encouraging a Positive Attitude Towards Challenges:

• Modeling Positivity: Show a positive attitude when facing challenges, demonstrating that problems are opportunities for growth and learning.
• Celebrating Solutions: Whenever a problem is solved, whether it’s big or small, celebrate the achievement. This reinforces problem-solving as a positive and rewarding experience.
• Fosters Teamwork: Engaging in family problem-solving activities helps in building teamwork and cooperation skills.
• Develops Practical Life Skills: Children learn practical life skills that are essential for their future, like financial planning, time management, and organization.
• Strengthens Family Bonds: Working together on problems strengthens family relationships and fosters a sense of unity and support.

Strategy 9: Engaging in Role-Playing Activities

Role-playing is an effective educational tool that allows children to simulate real-life situations. It provides a safe environment to practice problem-solving skills by acting out various scenarios.

Implementing Role-Playing in Problem-Solving:

• Creating Scenarios: Develop scenarios that children are likely to encounter, such as resolving a disagreement with a friend or handling a difficult situation at school. These should be age-appropriate and relevant to their experiences.
• Encouraging Different Perspectives: In role-playing, children can take on different roles, allowing them to see a problem from various viewpoints. This helps them understand the importance of empathy and considering multiple perspectives in problem-solving.
• Guided Discussion: After the role-play, have a discussion about the experience. Ask questions like, “How did you feel in that role?” or “What could have been done differently to solve the problem?”
• Enhances Communication Skills: Role-playing requires children to articulate their thoughts and feelings, improving their communication skills.
• Builds Emotional Intelligence: By putting themselves in someone else’s shoes, children develop empathy and emotional understanding.
• Practical Application of Skills: It allows children to apply problem-solving strategies in a controlled, low-stakes environment, helping them internalize these skills.

Variations of Role-Playing:

• Use of Props and Costumes: Incorporating props and costumes can make the activity more engaging and realistic.
• Incorporating Real-life Situations: Use real-life events as a basis for role-playing scenarios. This makes the exercise more relevant and practical.

Strategy 10: Encouraging Reflective Thinking

Reflective thinking is a critical component of the learning process. It involves looking back at the steps taken during problem-solving, analyzing the effectiveness of different strategies, and considering what could be improved.

Process of Reflective Thinking:

• After-Action Review: After a problem has been addressed, encourage children to reflect on the process. Ask questions like, “What part of our solution worked well?” or “What challenges did we face, and how did we overcome them?”
• Encouraging Honesty and Openness: Create an environment where children feel comfortable discussing both successes and failures openly. This honesty is crucial for genuine reflection and growth.
• Focus on Learning, Not Just Outcome: Emphasize the importance of the learning process over the outcome. This approach helps children understand that the value lies not only in solving the problem but also in the lessons learned along the way.
• Improves Problem-Solving Skills: Reflective thinking helps children understand what strategies are effective and which are not, refining their problem-solving skills over time.
• Fosters a Growth Mindset: It promotes the idea that skills and intelligence can be developed through dedication and hard work.
• Builds Self-Awareness: Reflecting on one’s own thought processes and decisions enhances self-awareness and personal development.

Guiding Children in Reflective Thinking:

• Modeling Reflection: Demonstrate reflective thinking yourself. After solving a problem, talk about what you learned from the experience and what you might do differently next time.
• Writing Journals: Encourage children to keep a journal where they can write down their thoughts about different problems they encounter and how they solved them. This can be a powerful tool for reflection.

Empowering the Next Generation: Fostering Critical Thinking and Problem-Solving at Las Vegas Day School

As we navigate a world that is increasingly complex and interconnected, equipping our children with the ability to think critically and solve problems is more important than ever. By implementing these strategies, parents and educators can provide children with the tools they need to face challenges confidently and effectively.

For families looking to further support their children’s educational journey, Las Vegas Day School (LVDS) offers an encouraging environment where these skills can be honed and developed. LVDS emphasizes a well-rounded approach to learning, where problem-solving is integrated into the curriculum, preparing students not just for academic success but for life-long resilience and adaptability. Visit LVDS to learn more about their programs and how they can support your child’s growth into a confident problem-solver and independent thinker.

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