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## 6 Tips for Teaching Math Problem-Solving Skills

Solving word problems is tougher than computing with numbers, but elementary teachers can guide students to do the deep thinking involved.

A growing concern with students is the ability to problem-solve, especially with complex, multistep problems. Data shows that students struggle more when solving word problems than they do with computation , and so problem-solving should be considered separately from computation. Why?

Consider this. When we’re on the way to a new destination and we plug in our location to a map on our phone, it tells us what lane to be in and takes us around any detours or collisions, sometimes even buzzing our watch to remind us to turn. When I experience this as a driver, I don’t have to do the thinking. I can think about what I’m going to cook for dinner, not paying much attention to my surroundings other than to follow those directions. If I were to be asked to go there again, I wouldn’t be able to remember, and I would again seek help.

If we can switch to giving students strategies that require them to think instead of giving them too much support throughout the journey to the answer, we may be able to give them the ability to learn the skills to read a map and have several ways to get there.

Here are six ways we can start letting students do this thinking so that they can go through rigorous problem-solving again and again, paving their own way to the solution.

## 1. Link problem-solving to reading

When we can remind students that they already have many comprehension skills and strategies they can easily use in math problem-solving, it can ease the anxiety surrounding the math problem. For example, providing them with strategies to practice, such as visualizing, acting out the problem with math tools like counters or base 10 blocks, drawing a quick sketch of the problem, retelling the story in their own words, etc., can really help them to utilize the skills they already have to make the task less daunting.

We can break these skills into specific short lessons so students have a bank of strategies to try on their own. Here's an example of an anchor chart that they can use for visualizing . Breaking up comprehension into specific skills can increase student independence and help teachers to be much more targeted in their problem-solving instruction. This allows students to build confidence and break down the barriers between reading and math to see they already have so many strengths that are transferable to all problems.

## 2. Avoid boxing students into choosing a specific operation

It can be so tempting to tell students to look for certain words that might mean a certain operation. This might even be thoroughly successful in kindergarten and first grade, but just like when our map tells us where to go, that limits students from becoming deep thinkers. It also expires once they get into the upper grades, where those words could be in a problem multiple times, creating more confusion when students are trying to follow a rule that may not exist in every problem.

We can encourage a variety of ways to solve problems instead of choosing the operation first. In first grade, a problem might say, “Joceline has 13 stuffed animals and Jordan has 17. How many more does Jordan have?” Some students might choose to subtract, but a lot of students might just count to find the amount in between. If we tell them that “how many more” means to subtract, we’re taking the thinking out of the problem altogether, allowing them to go on autopilot without truly solving the problem or using their comprehension skills to visualize it.

## 3. Revisit ‘representation’

The word “representation” can be misleading. It seems like something to do after the process of solving. When students think they have to go straight to solving, they may not realize that they need a step in between to be able to support their understanding of what’s actually happening in the problem first.

Using an anchor chart like one of these ( lower grade , upper grade ) can help students to choose a representation that most closely matches what they’re visualizing in their mind. Once they sketch it out, it can give them a clearer picture of different ways they could solve the problem.

Think about this problem: “Varush went on a trip with his family to his grandmother’s house. It was 710 miles away. On the way there, three people took turns driving. His mom drove 214 miles. His dad drove 358 miles. His older sister drove the rest. How many miles did his sister drive?”

If we were to show this student the anchor chart, they would probably choose a number line or a strip diagram to help them understand what’s happening.

If we tell students they must always draw base 10 blocks in a place value chart, that doesn’t necessarily match the concept of this problem. When we ask students to match our way of thinking, we rob them of critical thinking practice and sometimes confuse them in the process.

## 4. Give time to process

Sometimes as educators, we can feel rushed to get to everyone and everything that’s required. When solving a complex problem, students need time to just sit with a problem and wrestle with it, maybe even leaving it and coming back to it after a period of time.

This might mean we need to give them fewer problems but go deeper with those problems we give them. We can also speed up processing time when we allow for collaboration and talk time with peers on problem-solving tasks.

## 5. Ask questions that let Students do the thinking

Questions or prompts during problem-solving should be very open-ended to promote thinking. Telling a student to reread the problem or to think about what tools or resources would help them solve it is a way to get them to try something new but not take over their thinking.

These skills are also transferable across content, and students will be reminded, “Good readers and mathematicians reread.”

## 6. Spiral concepts so students frequently use problem-solving skills

When students don’t have to switch gears in between concepts, they’re not truly using deep problem-solving skills. They already kind of know what operation it might be or that it’s something they have at the forefront of their mind from recent learning. Being intentional within their learning stations and assessments about having a variety of rigorous problem-solving skills will refine their critical thinking abilities while building more and more resilience throughout the school year as they retain content learning in the process.

Problem-solving skills are so abstract, and it can be tough to pinpoint exactly what students need. Sometimes we have to go slow to go fast. Slowing down and helping students have tools when they get stuck and enabling them to be critical thinkers will prepare them for life and allow them multiple ways to get to their own destination.

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## Problem Solving in Mathematics

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The main reason for learning about math is to become a better problem solver in all aspects of life. Many problems are multistep and require some type of systematic approach. There are a couple of things you need to do when solving problems. Ask yourself exactly what type of information is being asked for: Is it one of addition, subtraction, multiplication , or division? Then determine all the information that is being given to you in the question.

Mathematician George Pólya’s book, “ How to Solve It: A New Aspect of Mathematical Method ,” written in 1957, is a great guide to have on hand. The ideas below, which provide you with general steps or strategies to solve math problems, are similar to those expressed in Pólya’s book and should help you untangle even the most complicated math problem.

## Use Established Procedures

Learning how to solve problems in mathematics is knowing what to look for. Math problems often require established procedures and knowing what procedure to apply. To create procedures, you have to be familiar with the problem situation and be able to collect the appropriate information, identify a strategy or strategies, and use the strategy appropriately.

Problem-solving requires practice. When deciding on methods or procedures to use to solve problems, the first thing you will do is look for clues, which is one of the most important skills in solving problems in mathematics. If you begin to solve problems by looking for clue words, you will find that these words often indicate an operation.

## Look for Clue Words

Think of yourself as a math detective. The first thing to do when you encounter a math problem is to look for clue words. This is one of the most important skills you can develop. If you begin to solve problems by looking for clue words, you will find that those words often indicate an operation.

Common clue words for addition problems:

Common clue words for subtraction problems:

- How much more

Common clue words for multiplication problems:

Common clue words for division problems:

Although clue words will vary a bit from problem to problem, you'll soon learn to recognize which words mean what in order to perform the correct operation.

## Read the Problem Carefully

This, of course, means looking for clue words as outlined in the previous section. Once you’ve identified your clue words, highlight or underline them. This will let you know what kind of problem you’re dealing with. Then do the following:

- Ask yourself if you've seen a problem similar to this one. If so, what is similar about it?
- What did you need to do in that instance?
- What facts are you given about this problem?
- What facts do you still need to find out about this problem?

## Develop a Plan and Review Your Work

Based on what you discovered by reading the problem carefully and identifying similar problems you’ve encountered before, you can then:

- Define your problem-solving strategy or strategies. This might mean identifying patterns, using known formulas, using sketches, and even guessing and checking.
- If your strategy doesn't work, it may lead you to an ah-ha moment and to a strategy that does work.

If it seems like you’ve solved the problem, ask yourself the following:

- Does your solution seem probable?
- Does it answer the initial question?
- Did you answer using the language in the question?
- Did you answer using the same units?

If you feel confident that the answer is “yes” to all questions, consider your problem solved.

## Tips and Hints

Some key questions to consider as you approach the problem may be:

- What are the keywords in the problem?
- Do I need a data visual, such as a diagram, list, table, chart, or graph?
- Is there a formula or equation that I'll need? If so, which one?
- Will I need to use a calculator? Is there a pattern I can use or follow?

Read the problem carefully, and decide on a method to solve the problem. Once you've finished working the problem, check your work and ensure that your answer makes sense and that you've used the same terms and or units in your answer.

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## 1.3: Problem Solving Strategies

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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.

## 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]

George Pólya, circa 1973

- 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 ↵

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

- First, you have to understand the problem.
- After understanding, then make a plan.
- Carry out the plan.
- Look back on your work. How could it be better?

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.

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!

## (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.)

## (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.

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## What can QuickMath do?

QuickMath will automatically answer the most common problems in algebra, equations and calculus faced by high-school and college students.

- The algebra section allows you to expand, factor or simplify virtually any expression you choose. It also has commands for splitting fractions into partial fractions, combining several fractions into one and cancelling common factors within a fraction.
- The equations section lets you solve an equation or system of equations. You can usually find the exact answer or, if necessary, a numerical answer to almost any accuracy you require.
- The inequalities section lets you solve an inequality or a system of inequalities for a single variable. You can also plot inequalities in two variables.
- The calculus section will carry out differentiation as well as definite and indefinite integration.
- The matrices section contains commands for the arithmetic manipulation of matrices.
- The graphs section contains commands for plotting equations and inequalities.
- The numbers section has a percentages command for explaining the most common types of percentage problems and a section for dealing with scientific notation.

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## 10 Strategies for Problem Solving in Math

Created: December 25, 2023

Last updated: January 6, 2024

When faced with problem-solving, children often get stuck. Word puzzles and math questions with an unknown variable, like x, usually confuse them. Therefore, this article discusses math strategies and how your students may use them since instructors often have to lead students through this problem-solving maze.

## What Are Problem Solving Strategies in Math?

If you want to fix a problem, you need a solid plan. Math strategies for problem solving are ways of tackling math in a way that guarantees better outcomes. These strategies simplify math for kids so that less time is spent figuring out the problem. Both those new to mathematics and those more knowledgeable about the subject may benefit from these methods.

There are several methods to apply problem-solving procedures in math, and each strategy is different. While none of these methods failsafe, they may help your student become a better problem solver, particularly when paired with practice and examples. The more math problems kids tackle, the more math problem solving skills they acquire, and practice is the key.

## Strategies for Problem-solving in Math

Even if a student is not a math wiz, a suitable solution to mathematical problems in math may help them discover answers. There is no one best method for helping students solve arithmetic problems, but the following ten approaches have shown to be very effective.

## Understand the Problem

Understanding the nature of math problems is a prerequisite to solving them. They need to specify what kind of issue it is ( fraction problem , word problem, quadratic equation, etc.). Searching for keywords in the math problem, revisiting similar questions, or consulting the internet are all great ways to strengthen their grasp of the material. This step keeps the pupil on track.

Math for Kids

## Guess and Check

One of the time-intensive strategies for resolving mathematical problems is the guess and check method. In this approach, students keep guessing until they get the answer right.

After assuming how to solve a math issue, students should reintroduce that assumption to check for correctness. While the approach may appear cumbersome, it is typically successful in revealing patterns in a child’s thought process.

## Work It Out

Encourage pupils to record their thinking process as they go through a math problem. Since this technique requires an initial comprehension of the topic, it serves as a self-monitoring method for mathematics students. If they immediately start solving the problem, they risk making mistakes.

Students may keep track of their ideas and fix their math problems as they go along using this method. A youngster may still need you to explain their methods of solving the arithmetic questions on the extra page. This confirmation stage etches the steps they took to solve the problem in their minds.

## Work Backwards

In mathematics, a fresh perspective is sometimes the key to a successful solution. Young people need to know that the ability to recreate math problems is valuable in many professional fields, including project management and engineering.

Students may better prepare for difficulties in real-world circumstances by using the “Work Backwards” technique. The end product may be used as a start-off point to identify the underlying issue.

In most cases, a visual representation of a math problem may help youngsters understand it better. Some of the most helpful math tactics for kids include having them play out the issue and picture how to solve it.

One way to visualize a workout is to use a blank piece of paper to draw a picture or make tally marks. Students might also use a marker and a whiteboard to draw as they demonstrate the technique before writing it down.

## Find a Pattern

Kids who use pattern recognition techniques can better grasp math concepts and retain formulae. The most remarkable technique for problem solving in mathematics is to help students see patterns in math problems by instructing them how to extract and list relevant details. This method may be used by students when learning shapes and other topics that need repetition.

Students may use this strategy to spot patterns and fill in the blanks. Over time, this strategy will help kids answer math problems quickly.

When faced with a math word problem, it might be helpful to ask, “What are some possible solutions to this issue?” It encourages you to give the problem more thought, develop creative solutions, and prevent you from being stuck in a rut. So, tell the pupils to think about the math problems and not just go with the first solution that comes to mind.

## Draw a Picture or Diagram

Drawing a picture of a math problem can help kids understand how to solve it, just like picturing it can help them see it. Shapes or numbers could be used to show the forms to keep things easy. Kids might learn how to use dots or letters to show the parts of a pattern or graph if you teach them.

Charts and graphs can be useful even when math isn’t involved. Kids can draw pictures of the ideas they read about to help them remember them after they’ve learned them. The plan for how to solve the mathematical problem will help kids understand what the problem is and how to solve it.

## Trial and Error Method

The trial and error method may be one of the most common problem solving strategies for kids to figure out how to solve problems. But how well this strategy is used will determine how well it works. Students have a hard time figuring out math questions if they don’t have clear formulas or instructions.

They have a better chance of getting the correct answer, though, if they first make a list of possible answers based on rules they already know and then try each one. Don’t be too quick to tell kids they shouldn’t learn by making mistakes.

## Review Answers with Peers

It’s fun to work on your math skills with friends by reviewing the answers to math questions together. If different students have different ideas about how to solve the same problem, get them to share their thoughts with the class.

During class time, kids’ ways of working might be compared. Then, students can make their points stronger by fixing these problems.

Check out the Printable Math Worksheets for Your Kids!

There are different ways to solve problems that can affect how fast and well students do on math tests. That’s why they need to learn the best ways to do things. If students follow the steps in this piece, they will have better experiences with solving math questions.

Jessica is a a seasoned math tutor with over a decade of experience in the field. With a BSc and Master’s degree in Mathematics, she enjoys nurturing math geniuses, regardless of their age, grade, and skills. Apart from tutoring, Jessica blogs at Brighterly. She also has experience in child psychology, homeschooling and curriculum consultation for schools and EdTech websites.

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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.

## Share This Book

## How to Improve Problem-Solving Skills: Mathematics and Critical Thinking

In today’s rapidly changing world, problem-solving has become a quintessential skill. When we discuss the topic, it’s natural to ask, “What is problem-solving?” and “How can we enhance this skill, particularly in children?” The discipline of mathematics offers a rich platform to explore these questions. Through math, not only do we delve into numbers and equations, but we also explore how to improve problem-solving skills and how to develop critical thinking skills in math. Let’s embark on this enlightening journey together.

## What is Problem-Solving?

At its core, problem-solving involves identifying a challenge and finding a solution. But it’s not always as straightforward as it sounds. So, what is problem-solving? True problem-solving requires a combination of creative thinking and logical reasoning. Mathematics, in many ways, embodies this blend. When a student approaches a math problem, they must discern the issue at hand, consider various methods to tackle it, and then systematically execute their chosen strategy.

But what is problem-solving in a broader context? It’s a life skill. Whether we’re deciding the best route to a destination, determining how to save for a big purchase, or even figuring out how to fix a broken appliance, we’re using problem-solving.

## How to Develop Critical Thinking Skills in Math

Critical thinking goes hand in hand with problem-solving. But exactly how to develop critical thinking skills in math might not be immediately obvious. Here are a few strategies:

- Contextual Learning: Teaching math within a story or real-life scenario makes it relevant. When students see math as a tool to navigate the world around them, they naturally begin to think critically about solutions.
- Open-ended Questions: Instead of merely seeking the “right” answer, encourage students to explain their thought processes. This nudges them to think deeply about their approach.
- Group Discussions: Collaborative learning can foster different perspectives, prompting students to consider multiple ways to solve a problem.
- Challenging Problems: Occasionally introducing problems that are a bit beyond a student’s current skill level can stimulate critical thinking. They will have to stretch their understanding and think outside the box.

## What are the Six Basic Steps of the Problem-Solving Process?

Understanding how to improve problem-solving skills often comes down to familiarizing oneself with the systematic approach to challenges. So, what are the six basic steps of the problem-solving process?

- Identification: Recognize and define the problem.
- Analysis: Understand the problem’s intricacies and nuances.
- Generation of Alternatives: Think of different ways to approach the challenge.
- Decision Making: Choose the most suitable method to address the problem.
- Implementation: Put the chosen solution into action.
- Evaluation: Reflect on the solution’s effectiveness and learn from the outcome.

By embedding these steps into mathematical education, we provide students with a structured framework. When they wonder about how to improve problem-solving skills or how to develop critical thinking skills in math, they can revert to this process, refining their approach with each new challenge.

## Making Math Fun and Relevant

At Wonder Math, we believe that the key to developing robust problem-solving skills lies in making math enjoyable and pertinent. When students see math not just as numbers on a page but as a captivating story or a real-world problem to be solved, their engagement skyrockets. And with heightened engagement comes enhanced understanding.

As educators and parents, it’s crucial to continuously ask ourselves: how can we demonstrate to our children what problem-solving is? How can we best teach them how to develop critical thinking skills in math? And how can we instill in them an understanding of the six basic steps of the problem-solving process?

The answer, we believe, lies in active learning, contextual teaching, and a genuine passion for the beauty of mathematics.

## The Underlying Beauty of Mathematics

Often, people perceive mathematics as a rigid discipline confined to numbers and formulas. However, this is a limited view. Math, in essence, is a language that describes patterns, relationships, and structures. It’s a medium through which we can communicate complex ideas, describe our universe, and solve intricate problems. Understanding this deeper beauty of math can further emphasize how to develop critical thinking skills in math.

## Why Mathematics is the Ideal Playground for Problem-Solving

Math provides endless opportunities for problem-solving. From basic arithmetic puzzles to advanced calculus challenges, every math problem offers a chance to hone our problem-solving skills. But why is mathematics so effective in this regard?

- Structured Challenges: Mathematics presents problems in a structured manner, allowing learners to systematically break them down. This format mimics real-world scenarios where understanding the structure of a challenge can be half the battle.
- Multiple Approaches: Most math problems can be approached in various ways . This teaches learners flexibility in thinking and the ability to view a single issue from multiple angles.
- Immediate Feedback: Unlike many real-world problems where solutions might take time to show results, in math, students often get immediate feedback. They can quickly gauge if their approach works or if they need to rethink their strategy.

## Enhancing the Learning Environment

To genuinely harness the power of mathematics in developing problem-solving skills, the learning environment plays a crucial role. A student who is afraid of making mistakes will hesitate to try out different approaches, stunting their critical thinking growth.

However, in a nurturing, supportive environment where mistakes are seen as learning opportunities, students thrive. They become more willing to take risks, try unconventional solutions, and learn from missteps. This mindset, where failure is not feared but embraced as a part of the learning journey, is pivotal for developing robust problem-solving skills.

## Incorporating Technology

In our digital age, technology offers innovative ways to explore math. Interactive apps and online platforms can provide dynamic problem-solving scenarios, making the process even more engaging. These tools can simulate real-world challenges, allowing students to apply their math skills in diverse contexts, further answering the question of how to improve problem-solving skills.

## More than Numbers

In summary, mathematics is more than just numbers and formulas—it’s a world filled with challenges, patterns, and beauty. By understanding its depth and leveraging its structured nature, we can provide learners with the perfect platform to develop critical thinking and problem-solving skills. The key lies in blending traditional techniques with modern tools, creating a holistic learning environment that fosters growth, curiosity, and a lifelong love for learning.

Join us on this transformative journey at Wonder Math. Let’s make math an adventure, teaching our children not just numbers and equations, but also how to improve problem-solving skills and navigate the world with confidence. Enroll your child today and witness the magic of mathematics unfold before your eyes!

## FAQ: Mathematics and Critical Thinking

1. what is problem-solving in the context of mathematics.

Problem-solving in mathematics refers to the process of identifying a mathematical challenge and systematically working through methods and strategies to find a solution.

## 2. Why is math considered a good avenue for developing problem-solving skills?

Mathematics provides structured challenges and allows for multiple approaches to find solutions. This promotes flexibility in thinking and encourages learners to view problems from various angles.

## 3. How does contextual learning enhance problem-solving abilities?

By teaching math within a story or real-life scenario, it becomes more relevant for the learner. This helps them see math as a tool to navigate real-world challenges , thereby promoting critical thinking.

## 4. What are the six basic steps of the problem-solving process in math?

The six steps are: Identification, Analysis, Generation of Alternatives, Decision Making, Implementation, and Evaluation.

## 5. How can parents support their children in developing mathematical problem-solving skills?

Parents can provide real-life contexts for math problems , encourage open discussions about different methods, and ensure a supportive environment where mistakes are seen as learning opportunities.

## 6. Are there any tools or apps that can help in enhancing problem-solving skills in math?

Yes, there are various interactive apps and online platforms designed specifically for math learning. These tools provide dynamic problem-solving scenarios and simulate real-world challenges, making the learning process engaging.

## 7. How does group discussion foster critical thinking in math?

Group discussions allow students to hear different perspectives and approaches to a problem. This can challenge their own understanding and push them to think about alternative methods.

## 8. Is it necessary to always follow the six steps of the problem-solving process sequentially?

While the six steps provide a structured approach, real-life problem-solving can sometimes be more fluid. It’s beneficial to know the steps, but adaptability and responsiveness to the situation are also crucial.

## 9. How does Wonder Math incorporate active learning in teaching mathematics?

Wonder Math integrates mathematics within engaging stories and real-world scenarios, making it fun and relevant. This active learning approach ensures that students are not just passive recipients but active participants in the learning process.

## 10. What if my child finds a math problem too challenging and becomes demotivated?

It’s essential to create a supportive environment where challenges are seen as growth opportunities. Remind them that every problem is a chance to learn, and it’s okay to seek help or approach it differently.

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Ratio word problems Six carefully selected ratio word problems with solutions to help you master ratios. Proportion word problems Four carefully selected proportion word problems with solutions.

Convert square feet to acres See how you can use proportion to convert square feet to acres

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Average word problems.

Average word problems Word problems about finding the average from easy to challenging

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## Common Core Math Explained: 8 Common Core Math Examples To Use In The Classroom

Samantha dock.

Navigating the world of Common Core math can be tricky for teachers who are trying to meet the individual needs of their students. It can be helpful to have a bank of Common Core math examples to hand when planning your lessons.

Embraced by the majority of states in the U.S., the Common Core math standards help to develop students’ conceptual understanding, problem-solving skills, and real-world applications. In this article, we explore what Common Core math is, 8 Common Core math examples and top tips for educators teaching Common Core math.

## What is Common Core math?

How is common core math different from traditional math, 8 common core math examples, tips for teaching common core math, common core math and the wider world.

Common Core math is a set of educational standards for mathematics adopted by forty states in the United States. Each standard outlines the math knowledge students should know and be able to do at each grade level, from kindergarten through to high school.

These standards aim to provide a more focused and coherent set of learning goals for students, with an emphasis on conceptual understanding, problem-solving, and critical thinking skills.

Often, Common Core math involves multiple strategies and approaches to solving problems. In turn, this encourages students to understand the underlying concepts rather than simply memorizing algorithms.

One of the aims of the Common Core State Standards is to move away from traditional memorization of procedures and algorithms towards a deeper understanding of connections between mathematical concepts.

Common Core math standards are organized by grade level and cover a wide range of mathematical topics, including:

- Arithmetic
- Probability

Each standard is divided into domains, which represent broad categories of mathematical content such as:

- Counting and cardinality
- Operations and algebraic thinking
- Numbers and operations in base ten
- Measurement and data

Common Core math standards have been controversial in some areas due to concerns about curriculum changes, standardized testing, and complexity. But some argue that these standards provide a more coherent approach to mathematics education and better prepare students for higher education than traditional math.

Read more: Why is Math Important?

Common Core math and traditional math represent two different approaches to teaching mathematics.

Traditional math typically refers to methods of teaching mathematics that were used before the adoption of the Common Core standards. These methods often focused on rote memorization of formulas and procedures, with less emphasis on understanding the concepts or on real-life application of mathematical skills.

Here are some key differences between the two:

- Focus on Conceptual Understanding vs. Memorization A strong emphasis is placed on developing students’ conceptual understanding of math concepts under the common core. It aims to help learners understand the “why” behind mathematical procedures rather than just memorizing algorithms. Traditional math often focuses more on rote memorization of formulas and procedures without necessarily understanding the underlying concepts needed to approach math questions.
- Problem-Solving and Critical Thinking vs. Rote Practice Common Core math problems encourage critical thinking skills. They promote multiple approaches to solving new math problems and require students to justify their reasoning. Often, traditional math involves repetitive practice of standard procedures with less emphasis on problem-solving and critical thinking.
- Real-World Applications vs. Abstract Exercises Connections between mathematical concepts to real-world situations is valued under the common core. This helps students see the relevance of the math skills they are learning. Tasks and problems require the application of mathematical skills in practical contexts. Traditional math lessons focus more on abstract exercises and textbook problems that may not always have clear real-world connections.
- Depth of Understanding vs. Breadth of Coverage Rather than covering a wide range of topics, Common Core math aims for depth of understanding and maths mastery. Fewer topics at each grade level allow for deeper exploration and mastery of key concepts. In contrast, traditional math tends to cover a broader range of topics in less depth.
- Flexibility and Multiple Strategies vs. One Correct Method Students are encouraged to use multiple strategies and approaches to solve problems through the Common Core math standards. Flexibility and creativity are valued when approaching problem-solving. Emphasis on a single “correct” method or algorithm for solving problems is the general approach in traditional math. Overall, Common Core State Standards aim to develop students’ mathematical proficiency in alignment with the demands of the modern world. This includes the need for critical thinking, problem-solving, and application of mathematical concepts to real-world situations

Overall, Common Core State Standards aim to develop students’ mathematical proficiency in alignment with the demands of the modern world. This includes the need for critical thinking, problem-solving, and application of mathematical concepts to real-world situations.

Third Space Learning provides one-on-one math instruction for students who need it most. Personalized one-on-one math lessons are designed by math experts and aligned to your state’s math standards — including the Common Core State Standards.

## 3rd to 6th Grade Common Core Math test

Help your students prepare for their state math test with these 3rd - 6th grade Common Core practice math tests.

## Common Core math example 1: Make sense of problems and persevere in solving them

Students should not only be able to understand problems and make sense of them, but persevere in finding solutions.

Finding solutions may involve math skills such as:

- Analyzing problems
- Making conjectures
- Planning approaches to solving math problems

Example: A student is faced with a word problem about finding the area of a garden. They must take the time to carefully read and understand the problem before attempting to solve it.

This problem may require several approaches to answer the math question. Small group work and discussion can encourage students to persevere through the challenge and try different strategies until they find a solution.

## Common Core math example 2: Reason abstractly and quantitatively

In order to reason abstractly, students need to be able to make sense of quantities and their relationships in mathematical situations.

This will be easier for students if they can take abstract information from context and quantify information. Being able to decontextualize and contextualize mathematical ideas will benefit students.

Example: A graph shows the relationship between the number of hours worked and the amount earned. Students can analyze the graph to determine patterns and make predictions about future earnings based on proportional relationships between hours worked and money earned.

For example, if John worked for 13 hours, how much money would he earn?

## Common Core math example 3: Construct viable arguments and critique the reasoning of others

Introducing math vocabulary in the classroom helps students construct viable arguments and critique the mathematical reasoning of others. Exposure to mathematics language and sentence stems will help students to reason mathematically, construct arguments, and justify their thinking, without creating cognitive overload.

Example: During a class discussion about strategies for solving a particular math problem, you might ask students to present their solutions — justifying and explaining their reasoning.

They can also be encouraged to critique each other’s approaches, identify strengths and weaknesses in their arguments and offer alternative methods.

## Common Core math example 4: Model mathematics

Math lessons should prepare students to use math to solve real-world problems. It may help students to do this if you represent mathematical concepts with visual models and math manipulatives .

Example: Subtraction of fractions is a skill that many students struggle with. Using a visual model to describe and analyze the word problem can release cognitive load for students.

For example, Paul had 11 ⅔ yards of twine. He used 6 ½ yards to make macrame wall hangings, how many yards of twine does Paul have left?

## Common Core math example 5: Use appropriate tools strategically

To solve math problems effectively and efficiently, students must be able to select and use appropriate tools. This includes recognizing when and how to use tools, as well as evaluating effectiveness and efficiency.

Example: When solving a complex geometry problem, students should recognise the effectiveness of using a protractor and ruler to accurately measure angles and lengths.

For example, Given an angle ABC where point B is the vertex of the angle, construct an angle bisector of angle ABC using a ruler and a protractor. Then, using the angle bisector you have constructed, draw a line segment from point B to the bisected angle’s line that is exactly 5 cm long. Measure and report the angle sizes of the two new angles created by the angle bisector.

## Common Core math example 6: Attend to precision

Calculations need to be carried out precisely. To do this, students need to be aware of key mathematical terminology for the Common Core Standards they are studying. This involves using appropriate units and labels and stating mathematical results clearly.

Example: A student ensures that their work is clear and organized. They pay attention to detail, avoiding errors and inaccuracies in their calculations. Below is a worked example of a student showing how to solve a word problem involving multiple percentages.

## Common Core math example 7: Look for and make use of structure

Solving math problems accurately means students need to recognize and use mathematical patterns and structure. They should be able to identify relationships between mathematical ideas and make connections between different mathematical representations.

Example: When solving a multiplication of decimals problem, a student recognizes that breaking down the whole numbers and decimal parts into their factors makes the problem easier to solve. They identify the underlying structure of the problem and use it to their advantage.

## Common Core math example 8: Look for and express regularity in repeated reasoning

Identifying and generalizing patterns and regularities in mathematical situations is key to proficiency in problem soving and reasoning . Students should be able to notice repeated reasoning and use it to solve math problems efficiently.

Example: A student identifies similarities between a problem they’re working on and a previous math problem. They utilize the patterns in the prior example to complete the new problem. This also helps them to solve similar problems in the future.

Teachers need to understand Common Core math standards to recognize the appropriate instructional strategies and promote a growth mindset in the classroom .

Here are 8 tips for maximising student progress when teaching the Common Core State Standards:

## 1. Understand the Common Core State Standards

Familiarize yourself with the math Common Core State Standards for your specific grade level. Take the time to understand the mathematical practices, domains and teaching strategies required for your grade.

## 2. Focus on conceptual understanding

Prioritize conceptual understanding over rote memorization. You can achieve this by helping students understand the “why” behind math concepts and skills. Always encourage them to explain and justify their reasoning.

## 3. Promote multiple approaches

Offer your students a range of math strategies and approaches to problem-solving. The more methods in their math bank, the better equipped they are to find a solution. Asking students to share their thinking process helps those who are not grasping the content from the math instruction.

## 4. Real-world connections

Connecting mathematical concepts to real-world situations makes learning more meaningful and relevant. You can do this is by implementing math problems where students work collaboratively to solve complex, open-ended word problems with real-world relevance.

## 5. Use visual representations

Diagrams, models, and manipulatives support students’ understanding of mathematical concepts by making abstract concepts more concrete and accessible. For example, you could use algebra tiles when students are first learning how to solve algebraic equations and inequalities to help them contexutalize the abstract nature of algebra.

## 6. Encourage discourse and collaboration

Promote a classroom environment where students feel comfortable sharing their ideas, asking questions and engaging in mathematical discourse.

Encourage discourse by using techniques such as turn and talk, or the 3 reads method for word problems.

## 7. Assess progress

Use formative and summative assessments to monitor students’ progress and understanding of mathematical concepts and adjust instruction accordingly based on assessment data.

Some examples of a formative assessment are:

- Exit tickets
- Rating scales
- Thumbs up or thumbs down

Summative assessments include:

- Check for understanding quizzes
- End-of-topic quizzes

Assessment resources:

- Practice state assessments

## 8. Professional development

Continuously seek professional development opportunities to deepen your understanding of Common Core math and improve your teaching practices. Collaborate with colleagues and participate in workshops, conferences, and online courses.

Embracing Common Core principles can help equip students for future challenges

Educators’ commitment to teaching Common Core math goes beyond math instruction. It’s about nurturing critical thinking and problem solving, ensuring students are prepared for the wider world.

Math lessons are no longer simply giving students math worksheets and grading them on the correct answer. The American education system has developed a math curriculum that anchors mathematical concepts in real-world relevance, promotes diverse problem-solving strategies, and encourages a collaborative learning environment.

Educators have a responsibility to ensure students have the tools and mathematical literacy they need to succeed.

## Common Core math examples FAQ

1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning

1. Focus: Emphasizes focusing deeply on a smaller number of key topics at each grade level. This is done to ensure students develop a deep understanding of foundational mathematical ideas. 2. Coherence: Emphasizes the importance of coherence in mathematical instruction. This is done to support students in making meaningful connections between different mathematical ideas, helping them see how concepts are related and reinforcing their understanding over time 3. Rigor: Focuses on increasing the rigor of mathematical instruction by demanding that students engage in conceptual understanding, procedural fluency, and application of mathematical concepts in real-world contexts. In this context, rigor means ensuring that students develop a deep understanding of mathematical concepts, are able to apply their knowledge in various contexts, and can solve complex problems through reasoning and critical thinking.

Forty states have fully adopted Common Core math, while Minnesota partially embraces it. South Carolina, Oklahoma, Indiana, Florida, and Arizona initially adopted but later repealed Common Core. Alaska, Nebraska, Texas, and Virginia never adopted it.

Do you have students who need extra support in math? Give your students more opportunities to consolidate learning and practice skills through personalized math tutoring with their own dedicated online math tutor. Each student receives differentiated instruction designed to close their individual learning gaps, and scaffolded learning ensures every student learns at the right pace. Lessons are aligned with your state’s standards and assessments, plus you’ll receive regular reports every step of the way. Personalized one-on-one math tutoring programs are available for: – 2nd grade tutoring – 3rd grade tutoring – 4th grade tutoring – 5th grade tutoring – 6th grade tutoring – 7th grade tutoring – 8th grade tutoring Why not learn more about how it works ?

## Ultimate Guide to Problem Solving Techniques [FREE]

Are you trying to build problem solving and reasoning skills in the classroom?

Here are 9 ready-to-go printable problem solving techniques that all your students should know, including challenges, short explanations and questioning prompts.

## Privacy Overview

## Teaching and Learning

Elevating math education through problem-based learning, by lisa matthews feb 14, 2024.

## Image Credit: rudall30 / Shutterstock

Imagine you are a mountaineer. Nothing excites you more than testing your skill, strength and resilience against some of the most extreme environments on the planet, and now you've decided to take on the greatest challenge of all: Everest, the tallest mountain in the world. You’ll be training for at least a year, slowly building up your endurance. Climbing Everest involves hiking for many hours per day, every day, for several weeks. How do you prepare for that?

The answer, as in many situations, lies in math. Climbers maximize their training by measuring their heart rate. When they train, they aim for a heart rate between 60 and 80 percent of their maximum. More than that, and they risk burning out. A heart rate below 60 percent means the training is too easy — they’ve got to push themselves harder. By combining this strategy with other types of training, overall fitness will increase over time, and eventually, climbers will be ready, in theory, for Everest.

## Knowledge Through Experience

The influence of constructivist theories has been instrumental in shaping PBL, from Jean Piaget's theory of cognitive development, which argues that knowledge is constructed through experiences and interactions , to Leslie P. Steffe’s work on the importance of students constructing their own mathematical understanding rather than passively receiving information .

You don't become a skilled mountain climber by just reading or watching others climb. You become proficient by hitting the mountains, climbing, facing challenges and getting right back up when you stumble. And that's how people learn math.

So what makes PBL different? The key to making it work is introducing the right level of problem. Remember Vygotsky’s Zone of Proximal Development? It is essentially the space where learning and development occur most effectively – where the task is not so easy that it is boring but not so hard that it is discouraging. As with a mountaineer in training, that zone where the level of challenge is just right is where engagement really happens.

I’ve seen PBL build the confidence of students who thought they weren’t math people. It makes them feel capable and that their insights are valuable. They develop the most creative strategies; kids have said things that just blow my mind. All of a sudden, they are math people.

## Skills and Understanding

Despite the challenges, the trend toward PBL in math education has been growing , driven by evidence of its benefits in developing critical thinking, problem-solving skills and a deeper understanding of mathematical concepts, as well as building more positive math identities. The incorporation of PBL aligns well with the contemporary broader shift toward more student-centered, interactive and meaningful learning experiences. It has become an increasingly important component of effective math education, equipping students with the skills and understanding necessary for success in the 21st century.

## At the heart of Imagine IM lies a commitment to providing students with opportunities for deep, active mathematics practice through problem-based learning. Imagine IM builds upon the problem-based pedagogy and instructional design of the renowned Illustrative Mathematics curriculum, adding a number of exclusive videos, digital interactives, design-enhanced print and hands-on tools.

The value of imagine im's enhancements is evident in the beautifully produced inspire math videos, from which the mountaineer scenario stems. inspire math videos showcase the math for each imagine im unit in a relevant and often unexpected real-world context to help spark curiosity. the videos use contexts from all around the world to make cross-curricular connections and increase engagement..

This article was sponsored by Imagine Learning and produced by the Solutions Studio team.

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## 10 Helpful Worksheet Ideas for Primary School Math Lessons

M athematics is a fundamental subject that shapes the way children think and analyze the world. At the primary school level, laying a strong foundation is crucial. While hands-on activities, digital tools, and interactive discussions play significant roles in learning, worksheets remain an essential tool for reinforcing concepts, practicing skills, and assessing understanding. Here’s a look at some helpful worksheets for primary school math lessons.

## Comparison Chart Worksheets

Comparison charts provide a visual means for primary school students to grasp relationships between numbers or concepts. They are easy to make at www.storyboardthat.com/create/comparison-chart-template , and here is how they can be used:

- Quantity Comparison: Charts might display two sets, like apples vs. bananas, prompting students to determine which set is larger.
- Attribute Comparison: These compare attributes, such as different shapes detailing their number of sides and characteristics.
- Number Line Comparisons: These help students understand number magnitude by placing numbers on a line to visualize their relative sizes.
- Venn Diagrams: Introduced in later primary grades, these diagrams help students compare and contrast two sets of items or concepts.
- Weather Charts: By comparing weather on different days, students can learn about temperature fluctuations and patterns.

## Number Recognition and Counting Worksheets

For young learners, recognizing numbers and counting is the first step into the world of mathematics. Worksheets can offer:

- Number Tracing: Allows students to familiarize themselves with how each number is formed.
- Count and Circle: Images are presented, and students have to count and circle the correct number.
- Missing Numbers: Sequences with missing numbers that students must fill in to practice counting forward and backward.

## Basic Arithmetic Worksheets

Once students are familiar with numbers, they can start simple arithmetic.

- Addition and Subtraction within 10 or 20: Using visual aids like number lines, counters, or pictures can be beneficial.
- Word Problems: Simple real-life scenarios can help students relate math to their daily lives.
- Skip Counting: Worksheets focused on counting by 2s, 5s, or 10s.

## Geometry and Shape Worksheets

Geometry offers a wonderful opportunity to relate math to the tangible world.

- Shape Identification: Recognizing and naming basic shapes such as squares, circles, triangles, etc.
- Comparing Shapes: Worksheets that help students identify differences and similarities between shapes.
- Pattern Recognition: Repeating shapes in patterns and asking students to determine the next shape in the sequence.

## Measurement Worksheets

Measurement is another area where real-life application and math converge.

- Length and Height: Comparing two or more objects and determining which is longer or shorter.
- Weight: Lighter vs. heavier worksheets using balancing scales as visuals.
- Time: Reading clocks, days of the week, and understanding the calendar.

## Data Handling Worksheets

Even at a primary level, students can start to understand basic data representation.

- Tally Marks: Using tally marks to represent data and counting them.
- Simple Bar Graphs: Interpreting and drawing bar graphs based on given data.
- Pictographs: Using pictures to represent data, which can be both fun and informative.

## Place Value Worksheets

Understanding the value of each digit in a number is fundamental in primary math.

- Identifying Place Values: Recognizing units, tens, hundreds, etc., in a given number.
- Expanding Numbers: Breaking down numbers into their place value components, such as understanding 243 as 200 + 40 + 3.
- Comparing Numbers: Using greater than, less than, or equal to symbols to compare two numbers based on their place values.

## Fraction Worksheets

Simple fraction concepts can be introduced at the primary level.

- Identifying Fractions: Recognizing half, quarter, third, etc., of shapes or sets.
- Comparing Fractions: Using visual aids like pie charts or shaded drawings to compare fractions.
- Simple Fraction Addition: Adding fractions with the same denominator using visual aids.

## Money and Real-Life Application Worksheets

Understanding money is both practical and a great way to apply arithmetic.

- Identifying Coins and Notes: Recognizing different denominations.
- Simple Transactions: Calculating change, adding up costs, or determining if there’s enough money to buy certain items.
- Word Problems with Money: Real-life scenarios involving buying, selling, and saving.

## Logic and Problem-Solving Worksheets

Even young students can hone their problem-solving skills with appropriate challenges.

- Sequences and Patterns: Predicting the next item in a sequence or recognizing a pattern.
- Logical Reasoning: Simple puzzles or riddles that require students to think critically.
- Story Problems: Reading a short story and solving a math-related problem based on the context.

Worksheets allow students to practice at their own pace, offer teachers a tool for assessment, and provide parents with a glimpse into their child’s learning progression. While digital tools and interactive activities are gaining prominence in education, the significance of worksheets remains undiminished. They are versatile and accessible and, when designed creatively, can make math engaging and fun for young learners.

The post 10 Helpful Worksheet Ideas for Primary School Math Lessons appeared first on Mom and More .

## Math Solver - Nerd AI 4+

Powered by chatgpt api, codeway dijital hizmetler anonim sirketi, designed for iphone.

- #16 in Education
- 4.7 • 17.9K Ratings
- Offers In-App Purchases

## iPhone Screenshots

Description.

Powered by the latest AI technology, Nerd AI is here to help you breeze through your studies. With a range of powerful features, Nerd AI transforms the way you approach writing, problem-solving, language learning, summarizing, coding, and expanding your knowledge on any topic. SCAN AND SOLVE Snap a photo of your question and get answers in a second with step-by-step explanations. WRITE EFFORTLESSLY Say goodbye to the writing struggles and hello to a world of effortless creativity. With our cutting-edge features, you can generate compelling essays, captivating blogs, and polished presentation scripts with utmost ease. Customize the word count and writing tone to match your requirements or cater to your specific audience perfectly. IMPROVE LANGUAGE SKILLS Ready to level up your language skills? Nerd AI has got your back! Our intelligent grammar hints and text translation tools will help you become a linguistic maestro. REWRITE Shape your words like a pro with Nerd AI's rewriting capabilities. Whether you need to paraphrase, simplify, continue texts, or fine-tune your writing drafts, we've got you covered. Make your writing shine and convey your thoughts with precision. LEARN TO CODE Do you want to learn to code but have no idea where to start? No problem! Nerd AI provides comprehensive assistance with any programming language. Understand the logic behind any code as Nerd AI explains it and take your coding skills to new heights. Input your code, and Nerd AI will check and optimize it for you, paving the way for coding success! TRIVIA QUEST: Knowledge is power, and Nerd AI loves to challenge your intellect! Engage in stimulating trivia quests covering a wide range of topics. Feel the power of Nerd AI as it transforms the way you write, learn languages, summarize texts, rewrite content, and delve into the world of coding. Download now and experience the joy of seamless learning and creativity! Privacy Policy: https://static.nerdai.app/privacy-en.html Terms of Use: https://static.nerdai.app/terms-conditions-en.html

Version 2.6.0

Hey there! The Nerd AI team sends big thanks! We've fixed some bugs based on your feedback. Your experience matters to us. Please leave us a review on the AppStore!

## Ratings and Reviews

17.9K Ratings

## App Needs a Little Bit Of Work, Even Though Amazing!

Great app nothing wrong with it but when I do need some help with like math, story problems, or maybe a different subject. There does need to be some more options for homework help. The only down side is that when it comes to story problems, or graphing, or multiple choice sometimes it lacks on it. For example the A.I understand what i’m trying to input from the picture. Yes maybe if I write it down it can understand my problem better but there is only a 1000 word cap and somethings I can’t input on my keyboard. lastly when it comes to taking the picture it will pick up on small details like 19. which is the problem i am on or the fill in the ____. Also the question or the date on the bottom possibly they can add feature to take those things off so the A.I doesn’t get confused by it. Thank you for reading have a Bless day!

## Actually explain.

My friend used this app and i caught a glance at it and downloaded it because at first i couldn’t believe an app that give you accurate answers. This app is pretty impressive. It actually tells you what to do instead of going straight into the answer, then demanding that you buy premium for questions thousands of people struggle on. The explanations are actually pretty simple to understand and as someone who is trying to get better grades in math this is amazing. I recommend you do actually get this app if you want to have step-by-step explanations which can get you ahead in your class.

## 😎This is the most AMAZING THING I’ve ever been to

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## App Privacy

The developer, Codeway Dijital Hizmetler Anonim Sirketi , indicated that the app’s privacy practices may include handling of data as described below. For more information, see the developer’s privacy policy .

## Data Used to Track You

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## Data Linked to You

The following data may be collected and linked to your identity:

## Data Not Linked to You

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- User Content
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## Information

- Nerd AI Premium W7 Free $6.99
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Free Math Worksheets — Over 100k free practice problems on Khan Academy Looking for free math worksheets? You've found something even better! That's because Khan Academy has over 100,000 free practice questions. And they're even better than traditional math worksheets - more instantaneous, more interactive, and more fun!

6 Tips for Teaching Math Problem-Solving Skills Solving word problems is tougher than computing with numbers, but elementary teachers can guide students to do the deep thinking involved. By Dani Fry February 6, 2023 JulPo / iStock A growing concern with students is the ability to problem-solve, especially with complex, multistep problems.

Step 1: Understanding the problem. We are given in the problem that there are 25 chickens and cows. All together there are 76 feet. Chickens have 2 feet and cows have 4 feet. We are trying to determine how many cows and how many chickens Mr. Jones has on his farm.

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt.

Get ready for 3rd grade math! Learn the skills that will set you up for success in addition and subtraction; multiplication and division; fractions; patterns and problem solving; area and perimeter; telling time; and data.

Take a guided, problem-solving based approach to learning Everyday Math. These compilations provide unique perspectives and applications you won't find anywhere else. Mathematical Fundamentals

Problem Solving in Mathematics Hero Images/Getty Images By Deb Russell Updated on July 09, 2019 The main reason for learning about math is to become a better problem solver in all aspects of life. Many problems are multistep and require some type of systematic approach. There are a couple of things you need to do when solving problems.

Get math help in your language. Works in Spanish, Hindi, German, and more. Online math solver with free step by step solutions to algebra, calculus, and other math problems. Get help on the web or with our math app.

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.

Get math help in your language. Works in Spanish, Hindi, German, and more. Online math solver with free step by step solutions to algebra, calculus, and other math problems. Get help on the web or with our math app.

Algebra (all content) 20 units · 412 skills. Unit 1 Introduction to algebra. Unit 2 Solving basic equations & inequalities (one variable, linear) Unit 3 Linear equations, functions, & graphs. Unit 4 Sequences. Unit 5 System of equations. Unit 6 Two-variable inequalities. Unit 7 Functions. Unit 8 Absolute value equations, functions, & inequalities.

Example: 2x-1=y,2y+3=x What can QuickMath do? QuickMath will automatically answer the most common problems in algebra, equations and calculus faced by high-school and college students. The algebra section allows you to expand, factor or simplify virtually any expression you choose.

10 Strategies for Problem Solving in Math Jessica Kaminski December 25, 2023 January 6, 2024 Teaching When faced with problem-solving, children often get stuck. Word puzzles and math questions with an unknown variable, like x, usually confuse them.

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.

At its core, problem-solving involves identifying a challenge and finding a solution. But it's not always as straightforward as it sounds. So, what is problem-solving? True problem-solving requires a combination of creative thinking and logical reasoning. Mathematics, in many ways, embodies this blend.

Four carefully selected proportion word problems with solutions. Convert square feet to acres See how you can use proportion to convert square feet to acres Percentage word problems Percentage word problems Some very good word problems about percentage. Percent error word problems

Maths problem solving is when a mathematical task challenges pupils to apply their knowledge, logic and reasoning in unfamiliar contexts. Problem solving questions often combine several elements of maths. We know from talking to the hundreds of school leaders and maths teachers that we work with as one to one online maths tutoring providers ...

In turn, this kind of work relating new concepts to real-life contexts enhances conceptual and problem-solving skills. For example, a student may already know that 6 x 2 = 12.

Free math problem solver answers your algebra homework questions with step-by-step explanations.

Common Core math example 1: Make sense of problems and persevere in solving them. Students should not only be able to understand problems and make sense of them, but persevere in finding solutions. Finding solutions may involve math skills such as: Analyzing problems. Making conjectures.

The Art of Problem Solving, Volume 1, is the classic problem solving textbook used by many successful MATHCOUNTS programs, and has been an important building block for students who, like the authors, performed well enough on the American Mathematics Contest series to qualify for the Math Olympiad Summer Program which trains students for the United States International Math Olympiad team.

Math problem-solving skills go beyond the classroom, preparing children for real-world challenges. Parents play a crucial role in making math engaging and practical. With the rise of online math tutors and innovative platforms like Byju's Math Companion, math learning has evolved. The classroom is just the beginning, and the world becomes a ...

Skills and Understanding. Despite the challenges, the trend toward PBL in math education has been growing, driven by evidence of its benefits in developing critical thinking, problem-solving skills and a deeper understanding of mathematical concepts, as well as building more positive math identities.The incorporation of PBL aligns well with the contemporary broader shift toward more student ...

Word Problems: Simple real-life scenarios can help students relate math to their daily lives. Skip Counting: Worksheets focused on counting by 2s, 5s, or 10s. Geometry and Shape Worksheets

It Struggled With Basic Math. Khanmigo, a ChatGPT-powered bot, made frequent calculation errors during a Journal reporter's test. By . Matt Barnum. Updated Feb. 16, 2024 9:50 am ET. Share.

With a range of powerful features, Nerd AI transforms the way you approach writing, problem-solving, language learning, summarizing, coding, and expanding your knowledge on any topic. Snap a photo of your question and get answers in a second with step-by-step explanations. Say goodbye to the writing struggles and hello to a world of effortless ...