problem solving elementary definition

Developing Problem-Solving Skills for Kids | Strategies & Tips

We've made teaching problem-solving skills for kids a whole lot easier! Keep reading and comment below with any other tips you have for your classroom!

Problem-Solving Skills for Kids: The Real Deal

Picture this: You've carefully created an assignment for your class. The step-by-step instructions are crystal clear. During class time, you walk through all the directions, and the response is awesome. Your students are ready! It's finally time for them to start working individually and then... 8 hands shoot up with questions. You hear one student mumble in the distance, "Wait, I don't get this" followed by the dreaded, "What are we supposed to be doing again?"

When I was a new computer science teacher, I would have this exact situation happen. As a result, I would end up scrambling to help each individual student with their problems until half the class period was eaten up. I assumed that in order for my students to learn best, I needed to be there to help answer questions immediately so they could move forward and complete the assignment.

Here's what I wish I had known when I started teaching coding to elementary students - the process of grappling with an assignment's content can be more important than completing the assignment's product. That said, not every student knows how to grapple, or struggle, in order to get to the "aha!" moment and solve a problem independently. The good news is, the ability to creatively solve problems is not a fixed skill. It can be learned by students, nurtured by teachers, and practiced by everyone!

Your students are absolutely capable of navigating and solving problems on their own. Here are some strategies, tips, and resources that can help:

Problem-Solving Skills for Kids: Student Strategies

These are strategies your students can use during independent work time to become creative problem solvers.

1. Go Step-By-Step Through The Problem-Solving Sequence 

Post problem-solving anchor charts and references on your classroom wall or pin them to your Google Classroom - anything to make them accessible to students. When they ask for help, invite them to reference the charts first.

2. Revisit Past Problems

If a student gets stuck, they should ask themself, "Have I ever seen a problem like this before? If so, how did I solve it?" Chances are, your students have tackled something similar already and can recycle the same strategies they used before to solve the problem this time around.

3. Document What Doesn’t Work

Sometimes finding the answer to a problem requires the process of elimination. Have your students attempt to solve a problem at least two different ways before reaching out to you for help. Even better, encourage them write down their "Not-The-Answers" so you can see their thought process when you do step in to support. Cool thing is, you likely won't need to! By attempting to solve a problem in multiple different ways, students will often come across the answer on their own.

4. "3 Before Me"

Let's say your students have gone through the Problem Solving Process, revisited past problems, and documented what doesn't work. Now, they know it's time to ask someone for help. Great! But before you jump into save the day, practice "3 Before Me". This means students need to ask 3 other classmates their question before asking the teacher. By doing this, students practice helpful 21st century skills like collaboration and communication, and can usually find the info they're looking for on the way.

Problem-Solving Skills for Kids: Teacher Tips

These are tips that you, the teacher, can use to support students in developing creative problem-solving skills for kids.

1. Ask Open Ended Questions

When a student asks for help, it can be tempting to give them the answer they're looking for so you can both move on. But what this actually does is prevent the student from developing the skills needed to solve the problem on their own. Instead of giving answers, try using open-ended questions and prompts. Here are some examples:

2. Encourage Grappling

Grappling  is everything a student might do when faced with a problem that does not have a clear solution. As explained in this article from Edutopia , this doesn't just mean perseverance! Grappling is more than that - it includes critical thinking, asking questions, observing evidence, asking more questions, forming hypotheses, and constructing a deep understanding of an issue.

There are lots of ways to provide opportunities for grappling. Anything that includes the Engineering Design Process is a good one! Examples include:

• Engineering or Art Projects
• Design-thinking challenges
• Computer science projects
• Science experiments

3. Emphasize Process Over Product

For elementary students, reflecting on the process of solving a problem helps them develop a growth mindset . Getting an answer "wrong" doesn't need to be a bad thing! What matters most are the steps they took to get there and how they might change their approach next time. As a teacher, you can support students in learning this reflection process.

4. Model The Strategies Yourself! 

As creative problem-solving skills for kids are being learned, there will likely be moments where they are frustrated or unsure. Here are some easy ways you can model what creative problem-solving looks and sounds like.

• Ask clarifying questions if you don't understand something
• Admit when don't know the correct answer
• Talk through multiple possible outcomes for different situations 
• Verbalize how you’re feeling when you find a problem

Practicing these strategies with your students will help create a learning environment where grappling, failing, and growing is celebrated!

Problem-Solving Skill for Kids

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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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Problem-Solving

Jabberwocky

Problem-solving is the ability to identify and solve problems by applying appropriate skills systematically.

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

Problem-solving involves three basic functions:

Seeking information

Generating new knowledge

Making decisions

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

Here is a five-stage model that most students can easily memorize and put into action and which has direct applications to many areas of the curriculum as well as everyday life:

Expert Opinion

Here are some techniques that will help students understand the nature of a problem and the conditions that surround it:

• List all related relevant facts.
• Make a list of all the given information.
• Restate the problem in their own words.
• List the conditions that surround a problem.
• Describe related known problems.

It's Elementary

For younger students, illustrations are helpful in organizing data, manipulating information, and outlining the limits of a problem and its possible solution(s). Students can use drawings to help them look at a problem from many different perspectives.

Understand the problem. It's important that students understand the nature of a problem and its related goals. Encourage students to frame a problem in their own words.

Describe any barriers. Students need to be aware of any barriers or constraints that may be preventing them from achieving their goal. In short, what is creating the problem? Encouraging students to verbalize these impediments is always an important step.

Identify various solutions. After the nature and parameters of a problem are understood, students will need to select one or more appropriate strategies to help resolve the problem. Students need to understand that they have many strategies available to them and that no single strategy will work for all problems. Here are some problem-solving possibilities:

Create visual images. Many problem-solvers find it useful to create “mind pictures” of a problem and its potential solutions prior to working on the problem. Mental imaging allows the problem-solvers to map out many dimensions of a problem and “see” it clearly.

Guesstimate. Give students opportunities to engage in some trial-and-error approaches to problem-solving. It should be understood, however, that this is not a singular approach to problem-solving but rather an attempt to gather some preliminary data.

Create a table. A table is an orderly arrangement of data. When students have opportunities to design and create tables of information, they begin to understand that they can group and organize most data relative to a problem.

Use manipulatives. By moving objects around on a table or desk, students can develop patterns and organize elements of a problem into recognizable and visually satisfying components.

Work backward. It's frequently helpful for students to take the data presented at the end of a problem and use a series of computations to arrive at the data presented at the beginning of the problem.

Look for a pattern. Looking for patterns is an important problem-solving strategy because many problems are similar and fall into predictable patterns. A pattern, by definition, is a regular, systematic repetition and may be numerical, visual, or behavioral.

Create a systematic list. Recording information in list form is a process used quite frequently to map out a plan of attack for defining and solving problems. Encourage students to record their ideas in lists to determine regularities, patterns, or similarities between problem elements.

Try out a solution. When working through a strategy or combination of strategies, it will be important for students to …

Keep accurate and up-to-date records of their thoughts, proceedings, and procedures. Recording the data collected, the predictions made, and the strategies used is an important part of the problem solving process.

Try to work through a selected strategy or combination of strategies until it becomes evident that it's not working, it needs to be modified, or it is yielding inappropriate data. As students become more proficient problem-solvers, they should feel comfortable rejecting potential strategies at any time during their quest for solutions.

Monitor with great care the steps undertaken as part of a solution. Although it might be a natural tendency for students to “rush” through a strategy to arrive at a quick answer, encourage them to carefully assess and monitor their progress.

Feel comfortable putting a problem aside for a period of time and tackling it at a later time. For example, scientists rarely come up with a solution the first time they approach a problem. Students should also feel comfortable letting a problem rest for a while and returning to it later.

Evaluate the results. It's vitally important that students have multiple opportunities to assess their own problem-solving skills and the solutions they generate from using those skills. Frequently, students are overly dependent upon teachers to evaluate their performance in the classroom. The process of self-assessment is not easy, however. It involves risk-taking, self-assurance, and a certain level of independence. But it can be effectively promoted by asking students questions such as “How do you feel about your progress so far?” “Are you satisfied with the results you obtained?” and “Why do you believe this is an appropriate response to the problem?”

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Mathematical Reasoning & Problem Solving

In this lesson, we’ll discuss mathematical reasoning and methods of problem solving with an eye toward helping your students make the best use of their reasoning skills when it comes to tackling complex problems.

Previously Covered:

• Over the course of the previous lesson, we reviewed some basics about chance and probability, as well as some basics about sampling, surveys, etc. We also covered some ideas about data sets, how they’re represented, and how to interpret the results.

Approaches to Problem Solving

When solving a mathematical problem, it is very common for a student to feel overwhelmed by the information or lack a clear idea about how to get started.

To help the students with their problem-solving “problem,” let’s look at some examples of mathematical problems and some general methods for solving problems:

Identify the following four-digit number when presented with the following information:

• One of the four digits is a 1.
• The digit in the hundreds place is three times the digit in the thousands place.
• The digit in the ones place is four times the digit in the ten’s place.
• The sum of all four digits is 13.
• The digit 2 is in the thousands place.

Help your students identify and prioritize the information presented.

In this particular example, we want to look for concrete information. Clue #1 tells us that one digit is a 1, but we’re not sure of its location, so we see if we can find a clue with more concrete information.

We can see that clue #5 gives us that kind of information and is the only clue that does, so we start from there.

Because this clue tells us that the thousands place digit is 2, we search for clues relevant to this clue. Clue #2 tells us that the digit in the hundreds place is three times that of the thousands place digit, so it is 6.

So now we need to find the tens and ones place digits, and see that clue #3 tells us that the digit in the ones place is four times the digit in the tens place. But we remember that clue #1 tells us that there’s a one somewhere, and since one is not four times any digit, we see that the one must be in the tens place, which leads us to the conclusion that the digit in the ones place is four. So then we conclude that our number is:

If you were following closely, you would notice that clue #4 was never used. It is a nice way to check our answer, since the digits of 2614 do indeed add up to be thirteen, but we did not need this clue to solve the problem.

Recall that the clues’ relevance were identified and prioritized as follows:

• clue #3 and clue #1

By identifying and prioritizing information, we were able to make the information given in the problem seem less overwhelming. We ordered the clues by relevance, with the most relevant clue providing us with a starting point to solve the problem. This method also utilized the more general method of breaking a problem into smaller and simpler parts to make it easier to solve.

Now let’s look at another mathematical problem and another general problem-solving method to help us solve it:

Two trees with heights of 20 m and 30 m respectively have ropes running from the top of each tree to the bottom of the other tree. The trees are 40 meters apart. We’ll assume that the ropes are pulled tight enough that we can ignore any bending or drooping. How high above the ground do the ropes intersect?

Let’s solve this problem by representing it in a visual way , in this case, a diagram:

You can see that we have a much simpler problem on our hands after drawing the diagram. A, B, C, D, E, and F are vertices of the triangles in the diagram. Now also notice that:

b = the base of triangle EFA

h = the height of triangle EFA and the height above the ground at which the ropes intersect

If we had not drawn this diagram, it would have been very hard to solve this problem, since we need the triangles and their properties to solve for h. Also, this diagram allows us to see that triangle BCA is similar to triangle EFC, and triangle DCA is similar to triangle EFA. Solving for h shows that the ropes intersect twelve meters above the ground.

Students frequently complain that mathematics is too difficult for them, because it is too abstract and unapproachable. Explaining mathematical reasoning and problem solving by using a variety of methods , such as words, numbers, symbols, charts, graphs, tables, diagrams, and concrete models can help students understand the problem better by making it more concrete and approachable.

Let’s try another one.

Given a pickle jar filled with marbles, about how many marbles does the jar contain?

Problems like this one require the student to make and use estimations . In this case, an estimation is all that is required, although, in more complex problems, estimates may help the student arrive at the final answer.

How would a student do this? A good estimation can be found by counting how many marbles are on the base of the jar and multiplying that by the number of marbles that make up the height of the marbles in the jar.

Now to make sure that we understand when and how to use these methods, let’s solve a problem on our own:

How many more faces does a cube have than a square pyramid?

Reveal Answer

The answer is B. To see how many more faces a cube has than a square pyramid, it is best to draw a diagram of a square pyramid and a cube:

From the diagrams above, we can see that the square pyramid has five faces and the cube has six. Therefore, the cube has one more face, so the answer is B.

Before we start having the same problem our model student in the beginning did—that is, being overwhelmed with too much information—let’s have a quick review of all the problem-solving methods we’ve discussed so far:

• Sort and prioritize relevant and irrelevant information.
• Represent a problem in different ways, such as words, symbols, concrete models, and diagrams.
• Generate and use estimations to find solutions to mathematical problems.

Mathematical Mistakes

Along with learning methods and tools for solving mathematical problems, it is important to recognize and avoid ways to make mathematical errors. This section will review some common errors.

Circular Arguments

These involve drawing a conclusion from a premise that is itself dependent on the conclusion. In other words, you are not actually proving anything. Circular reasoning often looks like deductive reasoning, but a quick examination will reveal that it’s far from it. Consider the following argument:

• Premise: Only an untrustworthy man would become an insurance salesman; the fact that insurance salesmen cannot be trusted is proof of this.
• Conclusion: Therefore, insurance salesmen cannot be trusted.

While this may be a simplistic example, you can see that there’s no logical procession in a circular argument.

Assuming the Truth of the Converse

Simply put: The fact that A implies B doesn’t not necessarily mean that B implies A. For example, “All dogs are mammals; therefore, all mammals are dogs.”

Assuming the Truth of the Inverse

Watch out for this one. You cannot automatically assume the inverse of a given statement is true. Consider the following true statement:

If you grew up in Minnesota , you’ve seen snow.

Now, notice that the inverse of this statement is not necessarily true:

If you didn’t grow up in Minnesota , you’ve never seen snow.

Faulty Generalizations

This mistake (also known as inductive fallacy) can take many forms, the most common being assuming a general rule based on a specific instance: (“Bridge is a hard game; therefore, all card games are difficult.”) Be aware of more subtle forms of faulty generalizations.

Faulty Analogies

It’s a mistake to assume that because two things are alike in one respect that they are necessarily alike in other ways too. Consider the faulty analogy below:

People who absolutely have to have a cup of coffee in the morning to get going are as bad as alcoholics who can’t cope without drinking.

False (or tenuous) analogies are often used in persuasive arguments.

Now that we’ve gone over some common mathematical mistakes, let’s look at some correct and effective ways to use mathematical reasoning.

Let’s look at basic logic, its operations, some fundamental laws, and the rules of logic that help us prove statements and deduce the truth. First off, there are two different styles of proofs: direct and indirect .

Whether it’s a direct or indirect proof, the engine that drives the proof is the if-then structure of a logical statement. In formal logic, you’ll see the format using the letters p and q, representing statements, as in:

If p, then q

An arrow is used to indicate that q is derived from p, like this:

This would be the general form of many types of logical statements that would be similar to: “if Joe has 5 cents, then Joe has a nickel or Joe has 5 pennies “. Basically, a proof is a flow of implications starting with the statement p and ending with the statement q. The stepping stones we use to link these statements in a logical proof on the way are called axioms or postulates , which are accepted logical tools.

A direct proof will attempt to lay out the shortest number of steps between p and q.

The goal of an indirect proof is exactly the same—it wants to show that q follows from p; however, it goes about it in a different manner. An indirect proof also goes by the names “proof by contradiction” or reductio ad absurdum . This type of proof assumes that the opposite of what you want to prove is true, and then shows that this is untenable or absurd, so, in fact, your original statement must be true.

Let’s see how this works using the isosceles triangle below. The indirect proof assumption is in bold.

Given: Triangle ABC is isosceles with B marking the vertex

Prove: Angles A and C are congruent.

Now, let’s work through this, matching our statements with our reasons.

• Triangle ABC is isosceles . . . . . . . . . . . . Given
• Angle A is the vertex . . . . . . . . . . . . . . . . Given
• Angles A and C are not congruent . . Indirect proof assumption
• Line AB is equal to line BC . . . . . . . . . . . Legs of an isosceles triangle are congruent
• Angles A and C are congruent . . . . . . . . The angles opposite congruent sides of a triangle are congruent
• Contradiction . . . . . . . . . . . . . . . . . . . . . . Angles can’t be congruent and incongruent
• Angles A and C are indeed congruent . . . The indirect proof assumption (step 3) is wrong
• Therefore, if angles A and C are not incongruent, they are congruent.

“Always, Sometimes, and Never”

Some math problems work on the mechanics that statements are “always”, “sometimes” and “never” true.

Example: x < x 2 for all real numbers x

We may be tempted to say that this statement is “always” true, because by choosing different values of x, like -2 and 3, we see that:

Example: For all primes x ≥ 3, x is odd.

This statement is “always” true. The only prime that is not odd is two. If we had a prime x ≥ 3 that is not odd, it would be divisible by two, which would make x not prime.

• Know and be able to identify common mathematical errors, such as circular arguments, assuming the truth of the converse, assuming the truth of the inverse, making faulty generalizations, and faulty use of analogical reasoning.
• Be familiar with direct proofs and indirect proofs (proof by contradiction).
• Be able to work with problems to identify “always,” “sometimes,” and “never” statements.

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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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K-5 Math Centers

K-5 math ideas, 3rd grade math, need help organizing your k-5 math block, math problem solving 101.

Have you ever given your students a money word problem where someone buys an item from a store, but your students come up with an answer where the person that bought the item ends up with more money than he or she came in with?

Word problem solving is one of those things that many of our children struggle with.  When used effectively, questioning and dramatization can be powerful tools for our students to use when solving these types of problems.

I came up with this approach after co-teaching a lesson with a 3rd grade teachers.  Her kids were having extreme difficulty comprehending a word problem she presented.  So we devised a lesson that would help students better understand problem solving.

The approach we took included the use of several literacy skills, like reading comprehension and writing. First, we started the lesson with a “think aloud” modeled by the teacher.We read and displayed the problem below but excluded ALL of the numbers. See the images below:

The purpose of reading the problem without the numbers is to get the students to understand what is actually happening in the problem.  Typically some students focus solely on keywords when solving word problems, but I do not advise using this approach exclusively.  With math problems, the context of the problem and actions in the problem determine how the child should go about solving it.

Read the Problem Without Numbers & Ask Questions:

After reading the problem (without numbers) to the students, I asked the following questions:

• Can you describe what is happening in your own words?
• What is the main idea of the problem?
• How could you act this out?

Make a Plan & Ask Questions:

After the students articulated what was happening in the problem, we made a plan to solve the problem.  I used the following guiding questions:

• Sample Answers include- We know that Kai has some goldfish. Kai donated or gave away some of the goldfish.
• Sample Answers include – We need to know how many goldfish Kai has.  We also need to know how many he gave anyway.  We also need to know how many bowls there are.
• Sample Answers include-   We need to find out how many fish belong in each bowl.

The class discussed the answers to the questions above. As we discussed the questions above the responses were written out on a problem solving template.

As part of this process, we clarified student understanding of the problem and determined what we needed to find and do to solve the problem.  Next, we walked the students through the process of showing their work using pictures.  Lastly, we checked our answers by writing an equation that matched the pictures to finally solve the problem.

Team Work Counts

After going through the process with the class, we decided to split the students into small groups of 3 and 4 to solve a math problem together.  The groups were expected to use the same process that we used to solve the problem.  It took a while but check out one of the final products below.

Benefits to Using this Process:

• Students understood what the problem is asking them to do
• Students are required to think and communicate as a team
• Students avoid making errors that can come with only using keywords
• Students are required to record their math reasoning using the problem solving template
• After using this process a couple of times, students get used to explaining and justifying their answers
• You become the facilitator of the learning by asking more questions, thereby making students independent thinkers

Things to Consider Include:

• This process in NOT quick.  It requires TIME.  You should not rush the process and expect to have it completed in 20 – 30 minutes in one day.
• This process is not a one time lesson.  Students may not get it the first time.  It should be seen a routine that can be used when solving word problems.

Be sure to let me know how this process works in your classroom in the comments below.

• Read more about: K-5 Math Ideas

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SEL Problem Solving: How to Teach Students to be Problem Solvers in 2024

If you are an elementary teacher looking to learn how to help your students solve problems independently, then you found the right place! Problem solving skills prepare kids to face dilemmas and obstacles with confidence. Students who have problem solving skills are more independent than students who do not. In this post, we’ll go into detail about what problem solving skills are and why they are important. In addition, we’ll share tips and ideas for how to teach problem solving skills in an elementary classroom setting. Read all about helping students solve problems in and out of the classroom below!

What Does Solving Problems Mean?

Solving problems means brainstorming solutions to the problem after identifying and analyzing the problem and why it occurred. It is important to brainstorm different solutions by looking at all angles of the problem and creating a list of possible solutions. Then you can pick the solution that fits the best.

Why is it Important for Kids to Solve Problems?

It is important for kids to solve problems by brainstorming different solutions so that they can pick the best solution. This teaches them that there can be many different solutions to a problem and they vary in effectiveness. Teaching kids to solve problems helps them be independent in making choices. 

How Do I Know If I Need to Teach Problem Solving in My Classroom?

The students in your 1st, 2nd, 3rd, 4th or 5th grade classroom would benefit from problem solving lessons and activities if any of these statements are true:

• Student confidence is lacking.
• Students are getting into conflicts with each other.
• They come to you to solve problems they could have solved on their own.
• Students are becoming easily frustrated.
• Recess is a hard time for your class.

5 Reasons To Promote Problem Solving In Your Elementary Classroom

Below are 5 reasons to promote problem solving in your elementary classroom.

1. Problem solving builds confidence 

Students’ confidence will grow as they learn problem solving skills because they will believe in their own abilities to solve problems. The more experience they have using their problem solving skills, the more confident they will become. Instead of going to others to solve problems for them, they will look inside themselves at their own abilities. 

2. Problem solving creates stronger friendships

Students who can problem solve create stronger friendships because they won’t let arguments or running into issues stop them from being friends with a person. Instead they work with their friend to get through their problem together and get through the bump in the road, instead of giving up on the friendship. 

3. Problem solving skills increase emotional intelligence 

Having emotional intelligence is incredibly helpful when solving problems. As students learn problem solving skills, they will use emotional intelligence to think about the feelings of others involved in the conflict. They will also think about how the problem is affecting others. 

4. Problem solving skills create more independent kids

Students who can problem solve become more independent than kids who cannot because they will try to solve their problems first instead of going to an adult. They won’t look at adults as being the only people who can solve their problems. They will be equipped with the skill set to tackle the problems they are experiencing by themselves or with peers. However, it is important to make the distinction with kids between problems they can solve on their own and problems they need an adult for. 

5. Teaching problem solving skills causes students to be more reflective 

Reflecting is part of the problem solving process. Students need to reflect on the problem and what caused it when deciding how to solve the problem. Once students choose the best solution to their problem, they need to reflect on whether or not the solution was effective. 

5 Tips and Ideas for Teaching Problem Solving Skills 

Below are tips and ideas for teaching problem solving.

1. Read Aloud Picture Books about Problem Solving Skills 

Picture books are a great way to introduce and teach an SEL topic. It gets students thinking about the topic and activating their background knowledge. Check out this list of picture books for teaching problem solving skills !

2. Watch Videos about Problem Solving Skills 

There are tons of free online videos out there that promote social emotional learning. It’s a fun and engaging way to teach SEL skills that your students will enjoy. Check out these videos for teaching problem solving skills !

3. Explicitly Teach Vocabulary Related to Problem Solving Skills 

Vocabulary words can help students develop understanding of problem solving and create connections through related words. Our problem solving SEL unit includes ten vocabulary cards with words related to the SEL topic. It is important for students to be able to see, hear, and use relevant vocabulary while learning. One idea for how to use them is to create an SEL word wall as students learn the words.

4. Provide Practice Opportunities

When learning any skill, students need time to practice. Social emotional learning skills are no different! Our problem solving SEL unit includes scenario cards, discussion cards, choice boards, games, and much more. These provide students with opportunities to practice the skills independently, with partners or small groups, or as a whole class.

5. Integrate Other Content Areas

Integrating other content areas with this topic is a great way to approach this SEL topic. Our problem solving SEL unit includes reading, writing, and art activities.

Skills Related to Problem Solving

Problem-solving, in the context of social emotional learning (SEL) or character education, refers to the process of identifying, analyzing, and resolving challenges or obstacles in a thoughtful and effective manner. While “problem-solving” is the commonly used term, there are other words and phrases that can convey a similar meaning. These alternative words highlight different aspects of finding solutions, critical thinking, and decision-making. Here are some other words used in the context of problem-solving:

• Troubleshooting: Identifying and resolving problems or difficulties by analyzing their root causes.
• Critical thinking: Applying logical and analytical reasoning to evaluate and solve problems.
• Decision-making: Considering options and making choices to address and solve problems effectively.
• Analytical problem-solving: Using data, evidence, and systematic thinking to address challenges and find solutions.
• Creative problem-solving: Generating innovative ideas and approaches to overcome obstacles and find solutions.
• Resourcefulness: Finding effective solutions using available resources and thinking outside the box.
• Solution-oriented: Focusing on identifying and implementing solutions rather than dwelling on problems.
• Adaptability: Adjusting strategies and approaches to fit changing circumstances and overcome challenges.
• Strategic thinking: Planning and organizing actions to achieve desired outcomes and resolve problems.
• Systems thinking: Considering the interconnectedness and relationships between different elements when solving problems.

These terms encompass the concept of problem-solving and reflect the qualities of critical thinking, decision-making, and finding effective solutions within the context of social emotional learning (SEL) or character education.

Download the SEL Activities

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In closing, we hope you found this information about teaching problem solving skills helpful! If you did, then you may also be interested in these posts.

• SEL Best Practices for Elementary Teachers
• Social Emotional Learning Activities
• 75+ SEL Videos for Elementary Teachers
• Teaching SEL Skills with Picture Books
• How to Create a Social Emotional Learning Environment
• Read more about: ELEMENTARY TEACHING , SOCIAL EMOTIONAL LEARNING IN THE CLASSROOM

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Guiding Students to Ask Questions and Define Problems in Science

Teachers can use these strategies to help students in grades 6 to 12 develop skills that are crucial to scientific study and exploration.

Without clear questions and well-defined problems, scientific investigations lack direction and focus, leading to inconclusive or irrelevant results. That’s why the first science and engineering practice is asking questions and defining problems . This is the foundation upon which scientific inquiry and problem-solving are built.

More important, asking questions and defining problems are essential skills in life that enable individuals to think critically, solve problems, and make informed decisions.

As STEAM professional development specialists, we’re fortunate to work with hundreds of teachers and students in the collective subjects of STEAM. We’ve identified several reasons why students have a hard time asking questions and defining problems.

We’ve witnessed a lack of confidence, limited prior exposure, fixed mindsets, overemphasis on answers, fear of judgment, and students not seeing the relevance to their lives. 

supporting students in Developing These Skills

Teachers of grades six through 12 play a crucial role in helping students develop their ability to ask questions and define problems in science. To begin, we employ the 5E Model of Instruction (engage, explore, explain, elaborate, evaluate) with our teachers.

In this student-centered model, instruction begins with students asking questions and exploring phenomena rather than teachers delivering content or information. Even if teachers don’t employ the 5E Model, we work with them on practical applications to have students ask and explore before the teachers explain . 

We’ve identified four overarching ideas that teachers can start working on tomorrow with their students.

• Encourage curiosity: Create an environment that encourages curiosity by simply presenting phenomena connected to the curriculum and standards and then allowing students to ask questions related to the phenomena. The Wonder of Science has science phenomena aligned to all grades and disciplinary core ideas .
• Offer opportunities for inquiry: Provide opportunities for students to engage in inquiry-based learning , which encourages them to ask their own questions, investigate topics of interest, and define their own problems. This can help students take ownership of their learning and develop skills in problem-solving, research, and analysis.
• Provide scaffolding: Offer guidance and support as students engage in questioning and problem-definition activities. Model effective questioning techniques, provide resources for research, and offer feedback on student-generated questions and problem statements. This helps create a safe and inclusive classroom environment where students feel comfortable and are capable of expressing their ideas and opinions.
• Use real-world examples: Help students connect their questions and problem statements to real-world contexts that are relevant and meaningful to their lives. Teachers can use case studies and encourage students to identify issues in their local community, current events, or global challenges so that they see the relevance and applicability of their learning. This can help students develop a broader perspective and understand how their learning is relevant and applicable in the real world, preparing them for future challenges and opportunities.

In addition to the four overarching ideas, we’ve put together four practical techniques and routines to support our colleagues and students. The links below can show how to specifically employ the techniques and routines in the classroom. Although our perspective is STEAM, the routines and techniques can be used in a variety of settings.

• Question Formulation Technique : QFT is a structured approach to asking questions. It improves communication skills, critical thinking, and problem-solving skills as well as metacognition. It also increases student autonomy and ownership of learning.
• Need to Know Questions : These questions are used to guide students’ inquiry and drive their learning. “Need to know” questions help students engage in critical thinking and inquiry, while also providing a framework for their learning and guiding their research and investigation. Students can take ownership of their learning and develop a deeper understanding of the topic or issue at hand.
• See, Think, Wonder : This routine is designed to encourage students to engage with visual stimuli. By using “see, think, wonder,” students develop their observation and interpretation skills, ask refined questions, and learn how to approach visual stimuli with a critical and curious mindset. Ultimately,  it promotes inquiry, creativity, and critical thinking.
• Design Thinking Bootleg : This is a set of tools to support the design thinking process. The approach promotes asking questions and defining problems through its iterative and human-centered process, which involves several stages that encourage inquiry and problem definition. By using design thinking, students develop critical thinking skills, empathize and connect to the perspectives of others, enhance their communication skills, and reflect and employ metacognition. 

The ability to ask questions and define problems is a critical skill in the field of science. Whether working in a laboratory, conducting field research, or simply trying to understand the world around us, asking questions and defining problems are the first steps toward knowledge and discovery, and they’re essential skills in many other aspects of life as well.

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1.2: Problem or Exercise?

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The main activity of mathematics is solving problems. However, what most people experience in most mathematics classrooms is practice exercises. An exercise is different from a problem.

In a problem , you probably don’t know at first how to approach solving it. You don’t know what mathematical ideas might be used in the solution. Part of solving a problem is understanding what is being asked, and knowing what a solution should look like. Problems often involve false starts, making mistakes, and lots of scratch paper!

In an exercise , you are often practicing a skill. You may have seen a teacher demonstrate a technique, or you may have read a worked example in the book. You then practice on very similar assignments, with the goal of mastering that skill.

What is a problem for some people may be an exercise for other people who have more background knowledge! For a young student just learning addition, this might be a problem:

\[\textit{Fill in the blank to make a true statement} \: \_\_\_ + 4 = 7 \ldotp \nonumber \]

But for you, that is an exercise!

Both problems and exercises are important in mathematics learning. But we should never forget that the ultimate goal is to develop more and better skills (through exercises) so that we can solve harder and more interesting problems.

Learning math is a bit like learning to play a sport. You can practice a lot of skills:

• hitting hundreds of forehands in tennis so that you can place them in a particular spot in the court,
• breaking down strokes into the component pieces in swimming so that each part of the stroke is more efficient,
• keeping control of the ball while making quick turns in soccer,
• shooting free throws in basketball,
• catching high fly balls in baseball,

But the point of the sport is to play the game. You practice the skills so that you are better at playing the game. In mathematics, solving problems is playing the game!

On Your Own

For each question below, decide if it is a problem or an exercise . (You do not need to solve the problems! Just decide which category it fits for you.) After you have labeled each one, compare your answers with a partner.

• This clock has been broken into three pieces. If you add the numbers in each piece, the sums are consecutive numbers.(Note: 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).

• A soccer coach began the year with a $500 budget. By the end of December, the coach spent $450. How much money in the budget was not spent?
• What is the product of 4,500 and 27?
• Arrange the digits 1–6 into a “difference triangle” where each number in the row below is the difference of the two numbers above it.
• Simplify the following expression: $$\frac{2 + 2(5^{3} - 4^{2})^{5} - 2^{2}}{2(5^{3} - 4^{2})} \ldotp$$
• What is the sum of \(\frac{5}{2}\) and \(\frac{3}{13}\)?
• You have eight coins and a balance scale. The coins look alike, but one of them is a counterfeit. The counterfeit coin is lighter than the others. You may only use the balance scale two times. How can you find the counterfeit coin?

• How many squares, of any possible size, are on a standard 8 × 8 chess board?
• What number is 3 more than half of 20?
• Find the largest eight-digit number made up of the digits 1, 1, 2, 2, 3, 3, 4, and 4 such that the 1’s are separated by one digit, the 2’s are separated by two digits, the 3’s by three digits, and the 4’s by four digits.

Why Every Educator Needs to Teach Problem-Solving Skills

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

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

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

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

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

Problem-Solving Skills Help Students…

   develop resilience.

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

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

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

    Set and Achieve Goals

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

  Resolve Conflicts

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

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

    Achieve Success

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

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

Problem-Solving Skills Can Be Measured and Taught

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

What Are Performance-Based Assessments?

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

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

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

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

Preview a Performance-Based Assessment

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

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

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

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

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

Explore CAE’s Problem-Solving Assessments

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

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

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

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

• Measuring students’ problem-solving skills through a performance-based assessment    
• Using the problem-solving assessment data to inform instruction and tailor interventions
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Problem solving facts for kids

Problem solving is a mental activity related to intelligence and thinking . It consists of finding solutions to problems. A problem is a situation that needs to be changed. It suggests that the solution is not totally obvious, for then it would not be a problem. A great deal of human life is spent solving problems. Social life is based on the notion that together we might solve problems which we could not as individuals.

The word "problem" comes from a Greek word meaning an "obstacle" (something that is in your way). If someone has a problem, they have to find a way of solving the problem. The way to solve it is called a solution. Some problem-solving techniques have been developed and used in artificial intelligence , computer science , engineering , and mathematics . Some are related to mental problem-solving techniques studied in gestalt psychology , cognitive psychology . and chess .

Problems can be classified as ill-defined or well-defined. Ill-defined problems are those that do not have clear goals, solution paths, or expected solution. An example is how to face threats which might perhaps be made in the future. Well-defined problems have specific goals, clearly defined solution paths, and clear expected solutions. These problems also allow for more initial planning than ill-defined problems.

Being able to solve problems involves the ability to understand what the goal of the problem is and what rules could be applied to solving the problem. Sometimes the problem requires abstract thinking and coming up with a creative solution.

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Problem Solving

Problem or Exercise?

The main activity of mathematics is solving problems. However, what most people experience in most mathematics classrooms is practice exercises. An exercise is different from a problem.

In a problem , you probably don’t know at first how to approach solving it. You don’t know what mathematical ideas might be used in the solution. Part of solving a problem is understanding what is being asked, and knowing what a solution should look like. Problems often involve false starts, making mistakes, and lots of scratch paper!

In an exercise , you are often practicing a skill.  You may have seen a teacher demonstrate a technique, or you may have read a worked example in the book.  You then practice on very similar assignments, with the goal of mastering that skill.

Note: What is a problem for some people may be an exercise for other people who have more background knowledge! For a young student just learning addition, this might be a problem:

But for you, that is an exercise!

Both problems and exercises are important in mathematics learning. But we should never forget that the ultimate goal is to develop more and better skills (through exercises) so that we can solve harder and more interesting problems.

Learning math is a bit like learning to play a sport. You can practice a lot of skills:

• hitting hundreds of forehands in tennis so that you can place them in a particular spot in the court,
• breaking down strokes into the component pieces in swimming so that each part of the stroke is more efficient,
• keeping control of the ball while making quick turns in soccer,
• shooting free throws in basketball,
• catching high fly balls in baseball,

But the point of the sport is to play the game. You practice the skills so that you are better at playing the game. In mathematics, solving problems is playing the game!

On Your Own

For each question below, decide if it is a problem or an exercise . (You do not need to solve the problems! Just decide which category it fits for you.) After you have labeled each one, compare your answers with a partner.

1. This clock has been broken into three pieces. If you add the numbers in each piece, the sums are consecutive numbers.(Note: 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).

2. A soccer coach began the year with a $500 budget.  By the end of December, the coach spent $450.  How much money in the budget was not spent?

3. What is the product of 4,500 and 27?

4. Arrange the digits 1–6 into a “difference triangle” where each number in the row below is the difference of the two numbers above it.

5. Simplify the following expression:

7. You have eight coins and a balance scale. The coins look alike, but one of them is a counterfeit. The counterfeit coin is lighter than the others. You may only use the balance scale two times. How can you find the counterfeit coin?

8. How many squares, of any possible size, are on a standard 8 × 8 chess board?

9. What number is 3 more than half of 20?

10. Find the largest eight-digit number made up of the digits 1, 1, 2, 2, 3, 3, 4, and 4 such that the 1’s are separated by one digit, the 2’s are separated by two digits, the 3’s by three digits, and the 4’s by four digits.

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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Word Problems Explained For Elementary School Teachers & Parents

Sophie bartlett.

Solving word problems in elementary school is an essential part of the math curriculum. Here are over 30 math word problems to practice with children, plus expert guidance on how to solve them.

This blog is part of our series of blogs designed for teachers, schools, and parents supporting home learning .

What is a word problem?

• Isn’t brilliant arithmetic enough?

Mastery helps children to explore math in greater depth

How to teach children to solve word problems , math word problems for kindergarten to grade 5, topic based word problems.

A word problem in math is a math question written as one sentence or more that requires children to apply their math knowledge to a ‘real-life’ scenario. 

This means that children must be familiar with the vocabulary associated with the mathematical symbols they are used to, in order to make sense of the word problem. 

For example:

Word Problems Grade 4 Number and Base 10

11 grade 4 number and base 10 questions to develop your students' reasoning and problem solving skills.

Isn’t brilliant arithmetic enough?

In short, no. Students need to build good reading comprehension, even in math. Overtime math problems become increasingly complex and require students to possess deep conceptual understanding and the ability to recall and apply knowledge rapidly and accurately. 

As students progress through their mathematical education, they will need to be able to apply mathematical reasoning and develop mathematical arguments and proofs using math language. They will also need to be dynamic, applying their math knowledge to a variety of increasingly sophisticated problems. 

To support this schools are adopting a ‘mastery’ approach to math

“Teaching for mastery”, is defined with these components: 

• Math teaching for mastery rejects the idea that a large proportion of people ‘just can’t do math’.  
• All students are encouraged by the belief that by working hard at math they can succeed. 
• Procedural fluency and conceptual understanding are developed in tandem because each supports the development of the other. 
• Significant time is spent developing deep knowledge of the key ideas that are needed to support future learning. The structure and connections within the mathematics are emphasized, so that students develop deep learning that can be sustained.

(The Essence of Maths Teaching for Mastery, 2016)

Fluency in arithmetic is important; however, with this often lies the common misconception that once a child has learned the number skills appropriate to their grade level/age, they should be progressed to the next grade level/age of number skills. 

The mastery approach encourages exploring the breadth and depth of these math concepts (once fluency is secure) through reasoning and problem solving. 

Here are two simple strategies that can be applied to many word problems before solving them.

• What do you already know?
• How can this problem be drawn/represented pictorially?

Let’s see how this can be applied to word problems to help achieve the answer.

Solving a simple word problem

There are 28 students in a class.

The teacher has 8 liters of orange juice.

She pours 225 milliliters of orange juice for every student.

How much orange juice is left over?

1. What do you already know?

• There are 1,000ml in 1 liter
• Pours = liquid leaving the bottle = subtraction
• For every = multiply
• Left over = requires subtraction at some point

2. How can this problem be drawn/represented pictorially?

The bar model , also known as strip diagram , is always a great way of representing problems. However, if you are not familiar with this, there are always other ways of drawing it out. 

Read more: What is a bar model

For example, for this question, you could draw 28 students (or stick man x 28) with ‘225 ml’ above each one and then a half-empty bottle with ‘8 liters’ marked at the top.

Now to put the math to work. This is a 5th grade multi-step problem, so we need to use what we already know and what we’ve drawn to break down the steps.

Solving a more complex, mixed word problem

Mara is in a bookshop.

She buys one book for $6.99 and another that costs $3.40 more than the first book.

She pays using a $20 bill.

What change does Mara get? (What is the remainder?)

• More than = add
• Using decimals means I will have to line up the decimal points correctly in calculations
• Change from money = subtract

See this example of bar modelling for this question:

Now to put the math to work using what we already know and what we’ve drawn to break down the steps.

Mara is in a bookshop. 

She buys one book for $6.99 and another that costs $3.40 more than the first book. 1) $6.99 + ($6.99 + $3.40) = $17.38

What change does Mara get? 2) $20 – $17.38 = $2.62

The more children learn about math as they go through elementary school, the trickier the word problems they face will become.

Below you will find some information about the types of word problems your child will be coming up against on a year by year basis, and how word problems apply to each elementary grade. 

Word problems in kindergarten

Throughout kindergarten a child is likely to be introduced to word problems with the help of concrete resources (manipulatives, such as pieces of physical apparatus like coins, cards, counters or number lines) to help them understand the problem.

An example of a word problem for kindergarten would be

Chris has 3 red bounce balls and 2 green bounce balls. How many bounce balls does Chris have in all?

Word problems in 1st grade

First grade is a continuation of kindergarten when it comes to word problems, with children still using concrete resources to help them understand and visualize the problems they are working on

An example of a word problem for first grade would be:

A class of 10 children each have 5 pencils in their pencil cases. How many pencils are there in total?

Word problems in 2nd grade

In second grade, children will move away from using concrete resources when solving word problems, and move towards using written methods. Teachers will begin to demonstrate the adding and subtracting within 100, adding up to 4- two-digit numbers at a time.

This is also the year in which 2-step word problems will be introduced. This is a problem which requires two individual calculations to be completed.

2nd grade word problem: geometry properties of shape

Shaun is making shapes out of plastic straws.

At the vertices where the straws meet, he uses blobs of modeling clay to fix them together

Here are some of the shapes he makes:

One of Sean’s shapes is a triangle. Which is it? Explain your answer.

Answer: shape B as a triangle has 3 sides (straws) and 3 vertices, or angles (clay)

2nd grade word problem: statistics

2nd grade is collecting pebbles. This pictogram shows the different numbers of pebbles each group finds.

Word problems in 3rd grade

At this stage of their elementary school career, children should feel confident using the written method for addition and subtraction. They will begin multiplying and dividing within 100. 

This year children will be presented with a variety of problems, including 2-step problems and be expected to work out the appropriate method required to solve each one. 

3rd grade word problem: number and place value

My number has four digits and has a 7 in the hundreds place.

The digit which has the highest value in my number is 2.

The digit which has the lowest value in my number is 6.

My number has 3 fewer tens than hundreds.

What is my number?

Answer: 2,746

Word problems in 4th grade

One and two-step word problems continue in fourth grade, but this is also the year that children will be introduced to word problems containing decimals.

4th grade word problem: fractions and decimals

Stan, Frank and John are washing their cars outside their houses.

Stan has washed 0.5 of his car.

Frank has washed 1/5 of his car.

Norm has washed 2/5 of his car.

Who has washed the most?

Explain your answer.

Answer: Stan (he has washed 0.5 whereas Frank has only washed 0.2 and Norm 0.4)

Word problems in 5th grade

In fifth grade children move on from 2-step word problems to multi-step word problems . These will include fractions and decimals.

Here are some examples of the types of math word problems in fifth grade will have to solve.

5th grade word problem: ratio and proportion

The Angel of the North is a large statue in England. It is 20 meters tall and 54 meters wide. 

Ally makes a scale model of the Angel of the North. Her model is 40 centimeters tall. How wide is her model?

Answer: 108cm

5th grade word problem: algebra

Amina is making designs with two different shapes.

She gives each shape a value.

Calculate the value of each shape.

Answer: 36 (hexagon) and 25. 

5th grade word problem: measurement

Answer: 1.7 liters or 1,700ml

The following examples give you an idea of the kinds of math word problems your child will encounter in elementary school

4th grade word problem: place value

This machine subtracts one hundredth each time the button is pressed. The starting number is 8.43. What number will the machine show if the button is pressed six times? Answer: 8.37

Download free number and place value word problems for grades 2, 3, 4 and 5

2nd grade word problem: addition and subtraction

Sam has 64 sweets. He gets given 12 more. He then gives 22 away. How many sweets is he left with? Answer: 54

Download free addition and subtraction word problems for for grades 2, 3, 4 and 5

2nd grade word problem: addition

Sammy thinks of a number. He subtracts 70. His new number is 12. What was the number Sammy thought of? Answer: 82

5th grade word problem: subtraction

The temperature at 7pm was 4oC. By midnight, it had dropped by 9 degrees. What was the temperature at midnight? Answer: -5oC

3rd grade word problem: multiplication

Eggs are sold in boxes of 12. The egg boxes are delivered to stores in crates. Each crate holds 9 boxes. How many eggs are in a crate? Answer: 108

Download free multiplication word problems for grades 2, 3, 4 and 5. 

5th grade word problem: division

A factory produces 3,572 paint brushes every day. They are packaged into boxes of 19. How many boxes does the factory produce every day? Answer: 188

Download free division word problems for grades 2, 3, 4 and 5. 

Free resource: Use these four operations word problems to practice addition, subtraction, multiplication and division all together.

4th grade word problem: fractions

At the end of every day, a chocolate factory has 1 and 2/6 boxes of chocolates left over. How many boxes of chocolates are left over by the end of a week? Answer: 9 and 2/6 or 9 and 1/3

Download free fractions and decimals word problems for grades 2, 3, 4 and 5. 

2nd grade word problem: money

Lucy and Noor found some money on the playground at recess. Lucy found 2 dimes and 1 penny, and Noor found 2 quarters and a dime. How many cents did Lucy and Noor find? Answer: Lucy = $0.21, Noor = $0.60; $0.21 + $0.61 = $0.81

3rd grade word problem: area

A rectangle measures 6cm by 5cm.

What is its area? Answer: 30cm2

3rd grade word problem: perimeter

The swimming pool at the Sunshine Inn hotel is 20m long and 7m wide. Mary swims around the edge of the pool twice. How many meters has she swum? Answer: 108m

5th grade word problem: ratio (crossover with measurement)

A local council has spent the day painting double yellow lines. They use 1 pot of yellow paint for every 100m of road they paint. How many pots of paint will they need to paint a 2km stretch of road? Answer: 20 pots

5th grade word problem: PEMDAS

Draw a pair of parentheses in one of these calculations so that they make two different answers. What are the answers?

50 – 10 × 5 =

5th grade word problem: volume

This large cuboid has been made by stacking shipping containers on a boat. Each individual shipping container has a length of 6m, a width of 4m and a height of 3m. What is the volume of the large cuboid? Answer: 864m3

Remember: The word problems can change but the math won’t 

It can be easy for children to get overwhelmed when they first come across word problems, but it is important that you remind them that while the context of the problem may be presented in a different way, the math behind it remains the same. 

Word problems are a good way to bring math into the real world and make math more relevant for your child. So help them practice, or even ask them to turn the tables and make up some word problems for you to solve. 

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problem-solving

Definition of problem-solving

Examples of problem-solving in a sentence.

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“Problem-solving.” Merriam-Webster.com Dictionary , Merriam-Webster, https://www.merriam-webster.com/dictionary/problem-solving. Accessed 12 Apr. 2024.

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