## Teaching Problem Solving in Math

- Freebies , Math , Planning

Every year my students can be fantastic at math…until they start to see math with words. For some reason, once math gets translated into reading, even my best readers start to panic. There is just something about word problems, or problem-solving, that causes children to think they don’t know how to complete them.

Every year in math, I start off by teaching my students problem-solving skills and strategies. Every year they moan and groan that they know them. Every year – paragraph one above. It was a vicious cycle. I needed something new.

I put together a problem-solving unit that would focus a bit more on strategies and steps in hopes that that would create problem-solving stars.

## The Problem Solving Strategies

First, I wanted to make sure my students all learned the different strategies to solve problems, such as guess-and-check, using visuals (draw a picture, act it out, and modeling it), working backward, and organizational methods (tables, charts, and lists). In the past, I had used worksheet pages that would introduce one and provide the students with plenty of problems practicing that one strategy. I did like that because students could focus more on practicing the strategy itself, but I also wanted students to know when to use it, too, so I made sure they had both to practice.

I provided students with plenty of practice of the strategies, such as in this guess-and-check game.

There’s also this visuals strategy wheel practice.

I also provided them with paper dolls and a variety of clothing to create an organized list to determine just how many outfits their “friend” would have.

Then, as I said above, we practiced in a variety of ways to make sure we knew exactly when to use them. I really wanted to make sure they had this down!

Anyway, after I knew they had down the various strategies and when to use them, then we went into the actual problem-solving steps.

## The Problem Solving Steps

I wanted students to understand that when they see a story problem, it isn’t scary. Really, it’s just the equation written out in words in a real-life situation. Then, I provided them with the “keys to success.”

S tep 1 – Understand the Problem. To help students understand the problem, I provided them with sample problems, and together we did five important things:

- read the problem carefully
- restated the problem in our own words
- crossed out unimportant information
- circled any important information
- stated the goal or question to be solved

We did this over and over with example problems.

Once I felt the students had it down, we practiced it in a game of problem-solving relay. Students raced one another to see how quickly they could get down to the nitty-gritty of the word problems. We weren’t solving the problems – yet.

Then, we were on to Step 2 – Make a Plan . We talked about how this was where we were going to choose which strategy we were going to use. We also discussed how this was where we were going to figure out what operation to use. I taught the students Sheila Melton’s operation concept map.

We talked about how if you know the total and know if it is equal or not, that will determine what operation you are doing. So, we took an example problem, such as:

Sheldon wants to make a cupcake for each of his 28 classmates. He can make 7 cupcakes with one box of cupcake mix. How many boxes will he need to buy?

We started off by asking ourselves, “Do we know the total?” We know there are a total of 28 classmates. So, yes, we are separating. Then, we ask, “Is it equal?” Yes, he wants to make a cupcake for EACH of his classmates. So, we are dividing: 28 divided by 7 = 4. He will need to buy 4 boxes. (I actually went ahead and solved it here – which is the next step, too.)

Step 3 – Solving the problem . We talked about how solving the problem involves the following:

- taking our time
- working the problem out
- showing all our work
- estimating the answer
- using thinking strategies

We talked specifically about thinking strategies. Just like in reading, there are thinking strategies in math. I wanted students to be aware that sometimes when we are working on a problem, a particular strategy may not be working, and we may need to switch strategies. We also discussed that sometimes we may need to rethink the problem, to think of related content, or to even start over. We discussed these thinking strategies:

- switch strategies or try a different one
- rethink the problem
- think of related content
- decide if you need to make changes
- check your work
- but most important…don’t give up!

To make sure they were getting in practice utilizing these thinking strategies, I gave each group chart paper with a letter from a fellow “student” (not a real student), and they had to give advice on how to help them solve their problem using the thinking strategies above.

Finally, Step 4 – Check It. This is the step that students often miss. I wanted to emphasize just how important it is! I went over it with them, discussing that when they check their problems, they should always look for these things:

- compare your answer to your estimate
- check for reasonableness
- check your calculations
- add the units
- restate the question in the answer
- explain how you solved the problem

Then, I gave students practice cards. I provided them with example cards of “students” who had completed their assignments already, and I wanted them to be the teacher. They needed to check the work and make sure it was completed correctly. If it wasn’t, then they needed to tell what they missed and correct it.

To demonstrate their understanding of the entire unit, we completed an adorable lap book (my first time ever putting together one or even creating one – I was surprised how well it turned out, actually). It was a great way to put everything we discussed in there.

Once we were all done, students were officially Problem Solving S.T.A.R.S. I just reminded students frequently of this acronym.

Stop – Don’t rush with any solution; just take your time and look everything over.

Think – Take your time to think about the problem and solution.

Act – Act on a strategy and try it out.

Review – Look it over and see if you got all the parts.

Wow, you are a true trooper sticking it out in this lengthy post! To sum up the majority of what I have written here, I have some problem-solving bookmarks FREE to help you remember and to help your students!

You can grab these problem-solving bookmarks for FREE by clicking here .

You can do any of these ideas without having to purchase anything. However, if you are looking to save some time and energy, then they are all found in my Math Workshop Problem Solving Unit . The unit is for grade three, but it may work for other grade levels. The practice problems are all for the early third-grade level.

- freebie , Math Workshop , Problem Solving

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## 21 Essential Strategies in Teaching Math

Even veteran teachers need to read these.

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

## 1. Raise the bar for all

WeAreTeachers

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

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

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

## 2. Don’t wait—act now!

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

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

[contextly_auto_sidebar]

## 3. Create a testing pathway

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

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

## 4. Observe, modify, and reevaluate

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

## 5. Read, read, read!

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

## 6. Personalize and offer choice

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

## 7. Plant the seeds!

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

## 8. Add apps appropriately

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

## 9. Encourage math talk

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

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

Learn more: Free Let’s Talk Math Poster

## 10. The art of math

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

## 11. Seek to develop understanding

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

## 12. Give students time to reflect

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

## 13. Allow for productive struggle

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

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

## 14. Emphasize hands-on learning

WeAreTeachers; Teacher Created Resources

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

## 15. Build excitement by rewarding progress

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

## 16. Choose meaningful tasks

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

## 17. Play math games

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

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

## 18. Set up effective math routines

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

## 19. Encourage teacher teamwork and reflection

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

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

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

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

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

## 21. Be a lifelong learner

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

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

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

## You Might Also Like

## 24 Creative Ways to Use Math Manipulatives in Your Classroom

Students learn better when they’re engaged, and manipulatives in the classroom make it easy for kids to get excited. We Continue Reading

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## Center for Teaching

Teaching problem solving.

Print Version

## Tips and Techniques

Expert vs. novice problem solvers, communicate.

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

## Encourage Independence

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

## Be sensitive

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

## Encourage Thoroughness and Patience

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

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

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

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## Top 9 Math Strategies for Successful Learning (2021 and Beyond)

Written by Ashley Crowe

- Teaching Strategies

## Why are effective Math strategies so important for students?

Getting students excited about math problems, top 9 math strategies for engaging lessons.

- How teachers can refine math strategies

Math is an essential life skill. You use problem-solving every day. The math strategies you teach are needed, but many students have a difficult time making that connection between math and life.

Math isn’t just done with a pencil and paper. It’s not just solving word problems in a textbook. As an educator, you need fresh ways for math skills to stick while also keeping your students engaged.

In this article, we’re sharing 9 engaging math strategies to boost your students’ learning . Show your students how fun math can be, and let’s freshen up those lesson plans!

Unlike other subjects, math builds on itself. You can’t successfully move forward without a strong understanding of previous materials. And this makes math instruction difficult.

To succeed in math, students need to do more than memorize formulas or drill times tables. They need to develop a full understanding of what their math lessons mean , and how they translate into the real world. To reach that level of understanding, you need a variety of teaching strategies.

Conceptual understanding doesn’t just happen at the whiteboard. But it can be achieved by incorporating fun math activities into your lessons, including

- Hands-on practice
- Collaborative projects
- Gamified or game-based learning

Repetition and homework are important. But for these lessons to really stick, your students need to find the excitement and wonder in math.

Creating excitement around math can be an uphill battle. But it’s one you and your students can win!

Math is a challenging subject — both to teach and to learn. But it’s also one of the most rewarding. Finding the right mix of fun and learning can bring a lot of excitement to the classroom.

Think about what your students already love doing. Video games? Legos? Use these passions to create exciting math lesson plans your students can relate to.

Hands-on math practice can engage students that have disconnected from math. Putting away the pencils and textbooks and moving students out of their desks can re-energize your classroom.

If you’re teaching elementary or middle school math, find ways for your students to work together. Kids this age crave peer interaction. So don’t fight it — provide it!

Play a variety of math games or puzzles . Give them a chance to problem-solve together. Build real-world skills in the classroom while also boosting student confidence.

And be sure to celebrate all the wins! It is easy to get bogged down with instruction and testing. But even the smallest accomplishments are worth celebrating. And these rewarding moments will keep your students motivated and pushing forward.

Keep reading to uncover all of our top math strategies for keeping your students excited about math.

## 1. Explicit instruction

You can’t always jump straight into the fun. Explicit instruction still provides the best foundation for the activities to come.

Set up your lesson for the day at the whiteboard, along with materials to demonstrate the coming activities. Make sure to also focus on any new vocabulary and concepts.

Tip: don't stay here for too long. Once the lesson is introduced, move on to the next fun strategy for the day!

## 2. Conceptual understanding

Helping your students understand the concept behind the lesson is crucial, but not always easy. Even your highest performing students may only be following a pattern to solve problems, without grasping the “why.”

Visual aids and math manipulatives are some of your best tools to increase conceptual understanding. Math is not a two dimensional subject. Even the best drawing of a cone isn’t going to provide the same experience as holding one. Find ways to let your students examine math from all sides.

Math manipulatives don’t need to be anything fancy. Basic wooden blocks, magnets, molding clay and other toys can create great hands-on lessons. No need to invest in expensive or hard-to-find materials.

Math word problems are also a great time to break out a full-fledged demo. Hot Wheels cars can demonstrate velocity and acceleration. A tape measure is an interactive way to teach area and volume. These materials give your students a chance to bring math off the page and into real life.

## 3. Using concepts in Math vocabulary

There’s more than one way to say something. And the more ways you can describe a mathematical concept, the better. Subtraction can also be described as taking away or removing. Memorizing multiplication facts is useful, but seeing these numbers used to calculate area gives them new meaning.

Some math words are going to be unfamiliar. So to help students get comfortable with these concepts, demonstrate and label math ideas throughout your classroom . Understanding comes more easily when students are surrounded by new ideas.

For example, create a division corner in your station rotations , with blocks to demonstrate the concept of one number going into another. Use baskets and labels to have students separate the blocks into each part of the division problem: dividend, divisor, quotient and remainder.

Give students time to explore, and teach them big ideas with both academic and everyday terms. Demystify math and watch their confidence build!

## 4. Cooperative learning strategies

When students work together, it benefits everyone. More advanced students can lead, helping them solidify their knowledge. And they may have just the right words to describe an idea to others who are struggling.

It is rare in real-life situations for big problems to be solved alone. Cooperative learning allows students to view a problem from various angles. This can lead to more flexible, out-of-the-box thinking.

After reviewing a word problem together as a class, ask small student groups to create their own problems. What is something they care about that they can solve with these skills? Involve them as much as possible in both the planning and solving. Encourage each student to think about what they bring to the group. There’s no better preparation for the future than learning to work as a team.

## 5. Meaningful and frequent homework

When it comes to homework, it pays to think outside of textbooks and worksheets. Repetition is important, but how can you keep it fun?

Create more meaningful homework by including games in your curriculum plans. Encourage board game play or encourage families to play quiz-style games at home to improve critical thinking, problem solving and basic math skills.

Sometimes you need homework that doesn’t put extra work onto the parents. The end of the day is already full for many families. To encourage practice and give parents a break, assign game-based options like Prodigy Math Game for homework.

With Prodigy, students can enjoy a fun, video game experience that helps them stay excited and motivated to keep learning. They’ll practice math skills, while their parents have time to fix dinner. Plus, you’ll get progress reports that can help you plan future instruction . Win-win-win!

Set an Assessment through your Prodigy teacher account today to reinforce what you’re teaching in class and differentiate for student needs.

Ready to make homework fun?

## 6. Puzzle pieces math instruction

Some kids excel at math. But others pull back and may rarely participate. That lack of confidence is hard to break through. How can you get your reluctant students to join in?

Try giving each student a piece of the puzzle. When you’re presenting your class with a problem, this creates necessary collaboration to get to the solution.

Each student is given a piece of information needed to solve the problem. A number, a unit of measurement, or direction — break your problem into as many pieces as possible.

If you have a large class, break down three or more problems at a time. The first task: find the other students who are working on your problem (try color-coding or using symbols to distinguish each problem’s parts). Then watch the learning happen as everyone plays their own important role.

## 7. Verbalize math problems

There’s little time to slow down in the classroom. Instruction has to move fast to keep up with the expected standards. And students feel that, too.

When possible, try to set aside some time to ask about your students’ math struggles. Make sure they know that they can come to you when they get stuck. Keep the conversation open to their questions as much as possible.

One great way to encourage questions is to address common troubles students have encountered in the past. Where have your past classes struggled? Point these out during your explicit instruction, and let your students know this is a tricky area.

It’s always encouraging to know you’re not alone in finding something difficult. This also leaves the door open for questions, leading to more discovery and greater understanding.

## 8. Reflection time

Providing time to reflect gives the brain a chance to process the work completed. This can be done after both group and individual activities.

## Group Reflection

After a collaborative activity, save some time for the group to discuss the project . Encourage them to ask:

- What worked?
- What didn’t work?
- Did I learn a new approach?
- What could we have done differently?
- Did someone share something I had never thought of before?

These questions encourage critical thinking. They also show the value of working together with others to solve a problem. Everyone has different ways of approaching a problem, and they’re all valuable.

## Individual Reflection

One way to make math more approachable is to show how often math is used. Journaling math encounters can be a great way for students to see that math is all around.

Ask them to add a little bit to their journal every day, even just a line or two. Where did they encounter math outside of class? Or what have they learned in class that has helped them at home?

Math skills easily transfer outside of the classroom. Help them see how much they have grown, both in terms of academics and social emotional learning .

## 9. Making Math facts fun

As a teacher, you know math is anything but boring. But transferring that passion to your students is a tricky task. So how can you make learning math facts fun?

Play games! Math games are great classroom activities. Here are a few examples:

- Design and play a board game.
- Build structures and judge durability.
- Divide into groups for a quiz or game show.
- Get kids moving and measure speed or distance jumped.

Even repetitive tasks can be fun with the right tools. That’s why engaging games are a great way to help students build essential math skills. When students play Prodigy Math Game , for example, they learn curriculum-aligned math facts without things like worksheets or flashcards. This can help them become excited to play and learn!

## How teachers can refine Math strategies

Sometimes trying something new can make a huge difference for your students. But don’t stress and try to change too much at once.

You know your classroom and students best. Pick a couple of your favorite strategies above and try them out.

If you're looking to freshen up your math instruction, sign up for a free Prodigy teacher account. Your students can jump right into the magic of the Prodigy Math Game, and you’ll start seeing data on their progress right away!

## Classroom Q&A

With larry ferlazzo.

In this EdWeek blog, an experiment in knowledge-gathering, Ferlazzo will address readers’ questions on classroom management, ELL instruction, lesson planning, and other issues facing teachers. Send your questions to [email protected]. Read more from this blog.

## Four Teacher-Recommended Instructional Strategies for Math

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(This is the first post in a two-part series.)

The new question-of-the-week is:

What is the single most effective instructional strategy you have used to teach math?

This post is part of a longer series of questions and answers inviting educators from various disciplines to share their “single most effective instructional strategy.”

Two weeks ago, educators shared their recommendations when it came to teaching writing.

Last month , it was about teaching English-language learners.

There are many more to come!

Today, Cindy Garcia, Danielle Ngo, Patrick Brown, and Andrea Clark share their favorite math instructional strategies.

## ‘Concrete Representational Abstract’

Cindy Garcia has been a bilingual educator for 14 years and is currently a district instructional specialist for PK-6 bilingual/ESL mathematics. She is active on Twitter @CindyGarciaTX and on her blog:

The single most effective strategy that I have used to teach mathematics is the Concrete Representational Abstract (CRA) approach.

During the concrete step, students use physical materials (real-life objects or models) to explore a concept. Using physical materials allows the students to see and touch abstract concepts such as place value. Students are able to manipulate these materials and make sense of what works and what does not work. For example, students can represent 102, 120, and 201 with base 10 blocks and count each model to see the difference of the value of the digit 2 in each number.

During the representational step, students use pictures, images, or virtual manipulatives to represent concrete materials and complete math tasks. Students are making connections and gaining a deeper understanding of the concept by creating or drawing representations.

During the abstract step, students are now primarily using numbers and symbols. Students working at the abstract stage have a solid understanding of the concept.

The CRA approach is appropriate and applicable to all grade levels. It is not about the age of the student but rather the concept being taught. In 3rd grade, it is beneficial to students to have them use base 10 blocks to create an open-area model, then draw an open-area model, and finally use the multiplication algorithm. In algebra, it is STILL beneficial to practice using algebra tiles to multiply polynomials using an open-area model.

The CRA approach provides students P-12 to have multiple opportunities to explore concepts and make connections with prior concepts. Some teachers try to start teaching a concept at the abstract level, for example, the standard algorithm for multiplication. However, they soon find out that students have difficulty remembering the steps, don’t regroup, or don’t line up digits correctly. One of the main reasons is that students don’t understand this shortcut and they have not had the concrete & representational experiences to see how the shortcuts in the standard algorithm work.

## ‘Encouraging Discourse’

Danielle Ngo is a 3rd grade teacher and Lower School math coordinator at The Windward School . She has been a teacher for 10 years and works primarily with students who have language-based learning disabilities:

Growing up, so many of us were taught that there is one right answer to every math problem, and that there is one efficient way to arrive at that conclusion. The impetus to return to this framework when teaching math is a tempting one and one I’ve found myself having to fight actively against during my own classroom instruction. In my experience, the most effective way to counter this impulse is to mindfully increase the discourse present during my math lessons. Encouraging discourse benefits our students in several ways, all of which solidify crucial math concepts and sharpen higher-order thinking and reasoning skills:

Distributes math authority in the classroom: Allowing discourse between students—not just between the students and their teacher—establishes a classroom environment in which all contributions are respected and valued. Not only does this type of environment encourage students to advocate for themselves, to ask clarifying questions, and to assess their understanding of material, it also incentivizes students to actively engage in lessons by giving them agency and ownership over their knowledge. Learning becomes a collaborative effort, one in which each student can and should participate.

Promotes a deeper understanding of mathematical concepts: While the rote memorization of a process allows many students to pass their tests, this superficial grasp of math skills does not build a solid foundation for more complex concepts. Through the requisite explanation and justification of their thought processes, discourse pushes students to move beyond an understanding of math as a set of procedural tasks. Rather, rich classroom discussion gives students the freedom to explore the “why’s and how’s” of math—to engage with the concepts at hand, think critically about them, and connect new topics to previous knowledge. These connections allow students to develop a meaningful understanding of mathematical concepts and to use prior knowledge to solve unfamiliar problems.

Develops mathematical-language skills: Students internalize vocabulary words—both their definitions and correct usage—through repeated exposures to the words in meaningful contexts. Appropriately facilitated classroom discourse provides the perfect opportunity for students to practice using new vocabulary terms, as well as to restate definitions in their own words. Additionally, since many math concepts build on prior knowledge, classroom discussions allow students to revisit vocabulary words; use them in multiple, varied contexts; and thus keep the terms current.

## ‘Explore-Before-Explain’

Patrick Brown is the executive director of STEM and CTE for the Fort Zumwalt school district,in Missouri, an experienced educator, and a noted author :

The current COVID-19 pandemic is a sobering reminder that we are educating today’s students for a world that is increasingly complex and unpredictable. The sequence that we use in mathematics education can be pivotal in developing students’ understanding and ability to apply ideas to their lives.

An explore-before-explain mindset to mathematics teaching means situating learning in real-life situations and problems and using those circumstances as a context for learning. Explore-before-explain teaching is all about creating conceptual coherence for learners and students’ experiences must occur before explanations and practice-type activities.

Distance learning reaffirmed these ideas when I was faced with the challenge of teaching area and perimeter for the first-time to a 3 rd grade learner. I quickly realized that rather than viewing area and perimeter as topics to be explained and then practiced, situating learning in problem-solving scenarios and using household items as manipulatives can illustrate ideas and derive the mathematical formulas and relationships.

Using Lego bricks, we quickly transformed equations and word problems into problem-solving situations that could be built. Student Lego constructions were used as evidence for comparing and contrasting physically how area and perimeter are similar and different as well as mathematical ways to calculate these concepts (e.g., students quickly learned by using Legos that perimeter is the distance around a shape while area is the total shape of an object). Thus, situating learning and having students use data as evidence for mathematical understanding have been critical for motivating and engaging students in distance learning environments.

Using an explore-before-explain sequence of mathematics instruction helps transform traditional mathematics lessons into activities that promote the development of deeper conceptual understanding and transfer learning.

## A ‘Whiteboard Wall’

Andrea Clark is a grade 5-7 math and language arts teacher in Austin, Texas. She has a master’s in STEM education and has been teaching for over 10 years:

If you want to increase motivation, persistence, and participation in your math classroom, I recommend a whiteboard wall. Or some reusable dry erase flipcharts to hang on the wall. Or some dry erase paint. Anything to get your students standing up and working on math together on a nonpermanent surface.

The idea of using “vertical nonpermanent surfaces” in the math classroom comes from Peter Liljedahl’s work with the best conditions for encouraging and supporting problem-solving in the math classroom. He found that students who worked on whiteboards (nonpermanent surfaces) started writing much sooner than students who worked on paper. He also found that students who worked on whiteboards discussed more, participated more, and persisted for longer than students working on paper. Working on a vertical whiteboard (hung on the wall) increased all of these factors, even compared with working on horizontal whiteboards.

Adding additional whiteboard space for my students to write on the walls has changed my math classroom (I have a few moveable whiteboard walls covered in dry erase paint as well as one wall with large whiteboards from end to end). My students spent less time sitting down, more time collaborating, and more time doing high-quality math. They were more willing to take risks, even willing to erase everything they had done and start over if necessary. They were able to solve problems that were complex and challenging, covering the whiteboards with their thinking and drawing.

And my students loved it. They were excited to work together on the whiteboards. They were excited to come to math and work through difficult problems together. They moved around the room, talking to other groups and sharing ideas. The fact that the boards were on the wall meant that everyone could see what other groups were doing. I could see where every group was just by looking around the room. I could see who needed help and who needed more time to work through something. But my students could see everything, too. They could get ideas from classmates outside of their group, using others’ ideas to get them through a disagreement or a sticking point. It made formally presenting their ideas easier, too; everyone could just turn and look at the board of the students who were sharing.

I loved ending the math class with whiteboards covered in writing. It reminded me of all of the thinking and talking and collaborating that had just happened. And that was a good feeling at the end of the day. Use nonpermanent vertical surfaces and watch your math class come alive.

Thanks to Cindy, Danielle, Patrick, and Andrea for their contributions!

Please feel free to leave a comment with your reactions to the topic or directly to anything that has been said in this post.

Consider contributing a question to be answered in a future post. You can send one to me at [email protected] . When you send it in, let me know if I can use your real name if it’s selected or if you’d prefer remaining anonymous and have a pseudonym in mind.

You can also contact me on Twitter at @Larryferlazzo .

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## Module 1: Problem Solving Strategies

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

Pólya’s How to Solve It

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

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

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!

Problem Solving Strategy 1 (Guess and Test)

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

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

Step 1: Understanding the problem

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

All together there are 76 feet.

Chickens have 2 feet and cows have 4 feet.

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

Step 2: Devise a plan

Going to use Guess and test along with making a tab

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

Make a table and look for a pattern:

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

Step 3: Carry out the plan:

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

Better! The total number of feet are increasing!

Step 4: Looking back:

Check: 12 + 13 = 25 heads

24 + 52 = 76 feet.

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

Videos to watch:

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

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

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

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

Check in question 1:

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

Check in question 2:

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

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

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

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

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

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

Gauss's strategy for sequences.

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

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

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

Last term = 3(200-1) +2

Last term is 599.

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

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

Sum = 60,100

Check in question 3: (10 points)

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

Then find the sum of the first 320 terms.

Problem Solving Strategy 4 (Working Backwards)

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

Videos to watch demonstrating of “Working Backwards”

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

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

1. We start with 11 and work backwards.

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

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

4. This should be our answer. Looking back:

9 x 2 = 18 -7 = 11

5. We have the right answer.

Check in question 4:

Christina is thinking of a number.

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

Problem Solving Strategy 5 (Looking for a Pattern)

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

We first need to find a pattern.

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

Example 1: 1, 4, 7, 10, 13…

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

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

So the next number would be

25 + 11 = 36

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

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

-5 – 3 = -8

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

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

So each number is being multiplied by 2.

16 x 2 = 32

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

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

Problem Solving Strategy 6 (Make a List)

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

List all the squares of the numbers 1 to 20.

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

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

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

Quarter’s dimes

0 3 30 cents

1 2 45 cents

2 1 60 cents

3 0 75 cents

Videos demonstrating "Make a List"

Check in question 5:

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

Problem Solving Strategy 7 (Solve a Simpler Problem)

Geometric Sequences:

How would we find the nth term?

Solve a simpler problem:

1, 3, 9, 27.

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

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

Let’s set up a table:

Term Number what did we do

Looking back: How would you find the nth term?

Find the 10 th term of the above sequence.

Let L = the tenth term

Problem Solving Strategy 8 (Process of Elimination)

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

I’m thinking of a number.

The number is odd.

It is more than 1 but less than 100.

It is greater than 20.

It is less than 5 times 7.

The sum of the digits is 7.

It is evenly divisible by 5.

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

b. It is greater than 20 but less than 35

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

c. The sum of the digits is 7

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

Check in question 6: (8 points)

Jose is thinking of a number.

The number is not odd.

The sum of the digits is divisible by 2.

The number is a multiple of 11.

It is greater than 5 times 4.

It is a multiple of 6

It is less than 7 times 8 +23

What is the number?

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

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

Want to create or adapt books like this? Learn more about how Pressbooks supports open publishing practices.

## 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

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## Math Problem Solving Strategies That Make Students Say “I Get It!”

Even students who are quick with math facts can get stuck when it comes to problem solving.

As soon as a concept is translated to a word problem, or a simple mathematical sentence contains an unknown, they’re stumped.

That’s because problem solving requires us to consciously choose the strategies most appropriate for the problem at hand . And not all students have this metacognitive ability.

But you can teach these strategies for problem solving. You just need to know what they are.

We’ve compiled them here divided into four categories:

## Strategies for understanding a problem

Strategies for solving the problem, strategies for working out, strategies for checking the solution.

Get to know these strategies and then model them explicitly to your students. Next time they dive into a rich problem, they’ll be filling up their working out paper faster than ever!

Before students can solve a problem, they need to know what it’s asking them. This is often the first hurdle with word problems that don’t specify a particular mathematical operation.

Encourage your students to:

## Read and reread the question

They say they’ve read it, but have they really ? Sometimes students will skip ahead as soon as they’ve noticed one familiar piece of information or give up trying to understand it if the problem doesn’t make sense at first glance.

Teach students to interpret a question by using self-monitoring strategies such as:

- Rereading a question more slowly if it doesn’t make sense the first time
- Asking for help
- Highlighting or underlining important pieces of information.

## Identify important and extraneous information

John is collecting money for his friend Ari’s birthday. He starts with $5 of his own, then Marcus gives him another $5. How much does he have now?

As adults looking at the above problem, we can instantly look past the names and the birthday scenario to see a simple addition problem. Students, however, can struggle to determine what’s relevant in the information that’s been given to them.

Teach students to sort and sift the information in a problem to find what’s relevant. A good way to do this is to have them swap out pieces of information to see if the solution changes. If changing names, items or scenarios has no impact on the end result, they’ll realize that it doesn’t need to be a point of focus while solving the problem.

## Schema approach

This is a math intervention strategy that can make problem solving easier for all students, regardless of ability.

Compare different word problems of the same type and construct a formula, or mathematical sentence stem, that applies to them all. For example, a simple subtraction problems could be expressed as:

[Number/Quantity A] with [Number/Quantity B] removed becomes [end result].

This is the underlying procedure or schema students are being asked to use. Once they have a list of schema for different mathematical operations (addition, multiplication and so on), they can take turns to apply them to an unfamiliar word problem and see which one fits.

Struggling students often believe math is something you either do automatically or don’t do at all. But that’s not true. Help your students understand that they have a choice of problem-solving strategies to use, and if one doesn’t work, they can try another.

Here are four common strategies students can use for problem solving.

## Visualizing

Visualizing an abstract problem often makes it easier to solve. Students could draw a picture or simply draw tally marks on a piece of working out paper.

Encourage visualization by modeling it on the whiteboard and providing graphic organizers that have space for students to draw before they write down the final number.

## Guess and check

Show students how to make an educated guess and then plug this answer back into the original problem. If it doesn’t work, they can adjust their initial guess higher or lower accordingly.

## Find a pattern

To find patterns, show students how to extract and list all the relevant facts in a problem so they can be easily compared. If they find a pattern, they’ll be able to locate the missing piece of information.

## Work backward

Working backward is useful if students are tasked with finding an unknown number in a problem or mathematical sentence. For example, if the problem is 8 + x = 12, students can find x by:

- Starting with 12
- Taking the 8 from the 12
- Being left with 4
- Checking that 4 works when used instead of x

Now students have understood the problem and formulated a strategy, it’s time to put it into practice. But if they just launch in and do it, they might make it harder for themselves. Show them how to work through a problem effectively by:

## Documenting working out

Model the process of writing down every step you take to complete a math problem and provide working out paper when students are solving a problem. This will allow students to keep track of their thoughts and pick up errors before they reach a final solution.

## Check along the way

Checking work as you go is another crucial self-monitoring strategy for math learners. Model it to them with think aloud questions such as:

- Does that last step look right?
- Does this follow on from the step I took before?
- Have I done any ‘smaller’ sums within the bigger problem that need checking?

Students often make the mistake of thinking that speed is everything in math — so they’ll rush to get an answer down and move on without checking.

But checking is important too. It allows them to pinpoint areas of difficulty as they come up, and it enables them to tackle more complex problems that require multiple checks before arriving at a final answer.

Here are some checking strategies you can promote:

## Check with a partner

Comparing answers with a peer leads is a more reflective process than just receiving a tick from the teacher. If students have two different answers, encourage them to talk about how they arrived at them and compare working out methods. They’ll figure out exactly where they went wrong, and what they got right.

## Reread the problem with your solution

Most of the time, students will be able to tell whether or not their answer is correct by putting it back into the initial problem. If it doesn’t work or it just ‘looks wrong’, it’s time to go back and fix it up.

## Fixing mistakes

Show students how to backtrack through their working out to find the exact point where they made a mistake. Emphasize that they can’t do this if they haven’t written down everything in the first place — so a single answer with no working out isn’t as impressive as they might think!

## Need more help developing problem solving skills?

Read up on how to set a problem solving and reasoning activity or explore Mathseeds and Mathletics, our award winning online math programs. They’ve got over 900 teacher tested problem solving activities between them!

## Get access to 900+ unique problem solving activities

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## Teaching Problem Solving in Math: 5 Strategies For Becoming a Better Math Teacher

Effective strategies for teaching problem solving in math have evolved since I started teaching 20+ years ago – and, for the better! More and more teachers, just like you and I, are using problem solving to teach math concepts in a way that invites students to be active participants in the learning process.

If we want to teach kids math – and do it effectively, we need to explore how we can plan and structure our lessons so that children get the chance to explore math concepts and talk about their mathematical thinking. Here are 5 effective math teaching strategies you can use to better support your students in learning problem solving in math.

## 1. Know Your Curriculum! Get to know your math curriculum before teaching problem solving in math.

Before you can plan and deliver a math lesson you need a clear learning goal based on the expectations or learning standards laid out in your school’s math curriculum. Now if you want to deliver an effective math lesson, then you also need to know the standards that came before and after your learning goal.

Why you ask? Well, every math concept you teach is part of a larger math continuum or trajectory of learning. This trajectory is the developmental pathway that most children follow when they learn a math concept (or any concept for that matter). The Curriculum documents are set up to reflect this.

## When we see the bigger picture of what a math curriculum looks like across grade levels we can:

- review previous learning expectations before introducing new concepts
- make connections and build on previously learned math
- identify where students are in their trajectory of learning
- work on bridging gaps in learning for students that struggle
- see what we need to prepare students for in the years to come

Even after 20+ years of teaching I still use this strategy to better prepare myself for teaching math. Last year after spending the bulk of my career teaching grades 3 to 6 I found myself in kindergarten. Reviewing the curriculum to see what the kids would learn in grade 1 helped to inform my teaching. It helped me prioritize and make decisions around how in-depth I needed to go and how much time I needed to spend on different math expectations.

If you want to explore your math curriculum in greater depth check out your state’s math standards or (if you’re in Canada) your provincial expectations. Here are a few links you might find helpful:

Common Core: Mathematics Standards | Common Core State Standards Initiative (corestandards.org)

New York State Standards: Welcome to EngageNY | EngageNY

Ontario Math Curriculum: Mathematics (gov.on.ca)

## 2. Keep it relevant! Use real world math problems to teach math concepts.

If you teach children, then I’m sure you’ll agree that kids learn best when they can tap into learning experiences they can connect to. I mean, let’s be real here – we all learn better when we can relate to the material we are learning. Real-world math problems do just that by connecting math to everyday events relevant to your students’ lives, interests, and experiences.

## When we use real world math problems to teach mathematical concepts we create valuable opportunities for students to:

- see math in the everyday events of their life
- use real-world contexts to help them visualize the math needed to solve the problem
- form deeper mathematical connections beyond rote memorization of math facts, algorithms, and formulas

You can even take your use of real-world math problems a step further and connect them to experiences and events happening in your classroom, community, or around the world.

## 3. Solve the Problem! Solve the math problem multiple ways before giving it to your students.

So you’ve come up with the perfect problem and you’re ready to see what strategies your students will use to solve the math problem. To best support your students with this process, I highly recommend taking a few minutes to see how many ways you can solve it yourself before giving it to your students.

Generating multiple solutions helps you to think beyond the algorithms and consider some of the strategies and tools your students might use when they attempt the problem. It allows you to see the math through fresh eyes. And when you can see the math, you will become better at meeting the learning styles and needs of all your students.

## Knowing a variety of possible solutions before you teach your lesson sets the stage for you to:

- validate a variety of problem solving strategies used by students
- meet students where they are at in their trajectory of learning
- make connections between conceptual and abstract strategies more explicit for students
- analyze student errors and support students in seeing where they took a wrong turn

## 4. Know the Lingo! Be ready to use math vocabulary to talk about math with your students.

If our goal is to deepen our student’s understanding of the math they are learning through problem solving, we must encourage and support them as they learn to talk about math. One of the ways to do this is to brush up on our knowledge and use of math vocabulary.

You see, the more fluent we are in the language of mathematics, the better equipped we will be to model and guide our students in communicating about the math they are learning through problem solving.

## Having a solid math vocabulary is important because it allows students to:

- talk about math with their peers using a shared vocabulary
- communicate more clearly about the strategies they used to solve the math problem
- ask more specific questions when understanding breaks down

As you expand your math vocabulary to support student learning, please remember: You don’t need to know it all!

Your level of comfort with using math terminology will grow over time and with experience. Don’t be afraid to refer to your teacher’s manual, math textbook, Google, or favorite resource for teaching math. I still refer to my favourite, Big Ideas from Dr. Small, to confirm definitions and gain deeper insight into the math I am teaching regularly. When I taught grades 4 and 5, I would even look up definitions alongside my students. I strongly believe this sort of modeling teaches students strategies they can use to become more proficient in math.

If we want our students to be literate in the language of math, we must prepare and support them as they learn to identify, label, and talk about math in the context of problem-solving.

## 5. Use those manips! Encourage math exploration through the use of math manipulatives.

The use of math manipulatives in k to 6 classrooms is one of my favourite ways to support students in problem solving for a wide range of math concepts. When students use manipulatives to solve a math problem, they can focus on a highly engaging form of math ‘play’ that is purposeful and goal driven. If we rely too heavily on step-by-step processes and rote memorization to teach mathematical concepts, we miss vital opportunities for even our strongest math students to make sense of math on a deeper, conceptual level.

## The use of math manipulatives when learning problem solving in math provides:

- a fun and highly motivating way for kids to explore the math
- the freedom for kids to engage in trial and error
- the opportunity to explore the math by trying out multiple representations of the problem
- a concrete model of student thinking that can reinforce math concepts and confirm their ideas

When kids develop a strong conceptual foundation for math through the use of math manipulatives it empowers them to become more proficient and confident problem solvers

## Want more math ideas, tips, and strategies for teaching problem solving in math?

If you’re still reading this, can I just say THANK YOU for reading my very first blog post! I truly hope you feel inspired to expand your teaching toolbox and try even one of the strategies I suggested.

If you want to learn more about who I am check out my About Me page or join me over on Instagram where I am sharing more math goodness! So don’t be shy – Say hello and let me know…What questions do you have about teaching math? What strategies have helped you to become a better math teacher? Which strategies do you already use or would like to try from this list?

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## 20 Effective Math Teaching Strategies for Explicit Learning

Numerous math teaching strategies are there that can help teachers to connect students with mathematics and be effective problem-solving techniques for students also. The core of these strategies lies in only one thing, training students with practical problem-solving skills, even though the instructional strategies list is long and might use different and varied methods.

Indeed, using calculators can ease the process of solving mathematical problems; the students’ mental exercise and problem-solving practices can not be replaced, as quoted by NCTM.

Also, Mathematics Specialists for elementary kids can be more beneficial for effective math teaching for teachers and seamless learning for students.

## General Math Teaching Strategies to help teachers:

## 100+ Free Math Worksheets, Practice Tests & Quizzes

Frequent tests, preparing course material, class drills, memorizing formulas, instruction from worksheets, etc., are usually used to teach math. But teachers need to understand that remarkable test scores and quality & sound teaching are mutually exclusive. Quality and sound teaching requires effective teaching strategies, especially for math.

## 1. Higher expectations for all students

Teachers should keep equal higher expectations from all the students to encourage them for better growth.

For instance, female students from as early as the second standard tend to adopt the idea that subjects like mathematics are not for them. And it can be pretty challenging to downtrodden socially conventional thoughts like “math is not for me” or “I am not good at solving math,” etc.

Teachers need to explain to students that success in math depends on hard work rather than having a math talent. And Higher expectations idea is among the top excellent examples of teaching strategies encouraging students with growth mindsets.

## 2. Don’t leave essential concepts on schedule.

Teachers should focus on students’ foundational skills for teaching any concept instead of holding it for later. Teachers must teach foundational math concepts irrespective of scheduled-test when they need them.

Teachers only need to understand the specific standards and back-map their teachings from the beginning to prepare students for their end terms.

## 3. Follow a testing path.

Teachers might not even perceive the results of systematized tests until the following school year; however, they have to teach students now. Teachers can use one of the most praised approaches from the instructional strategies list. The approach is formative assessments, which help teachers ensure whether their students understand the concepts.

## The digital co-teacher made with ❤️ by teachers

ByteLearn saves you time and ensures every student gets the support they need

Additionally, teachers must integrate testing into their teaching plan, such as quick quiz contests in the classroom related to particular concepts or learnings of a specific day.

## 4. Observe students’ learning

Sometimes teachers get trapped in thinking of one lesson or concept to get through and cover the entire course. However, they must keep the teaching pace flexible; otherwise, the students can fall behind. Teachers should observe the students while they work on the math problems. They should also try talking individually with every student and ask “pivot questions” in their lesson plans to measure students’ understanding.

## 5. Link math with the outer world

Teachers must try this effective approach of showing their students how the entire world is connected to mathematics whenever they lose focus in math or get bored. This tactic will keep them engaged and more invested in learning math. Also, it will strengthen students’ belief in the contribution of mathematical thinking in other subjects.

## 6. Let students choose

One of the most powerful teaching strategies is allowing students to choose how they want to learn. It motivates students to participate more in math class and reflect their understanding of concepts. Additionally, it offers students to choose the best learning process and the power of their learning and gives them scope for various math problem-solving methods.

## 7. Talk about mathematics

Teachers should also prefer talking and discussing the math subject with the students. This strategy helps students process learning techniques. As teachers engage students in math talk and discuss some topics or why a particular problem is solved with that specific method, it will make them curious to know more about math which eventually captivates them to the subject.

## 8. Play math-related games

Only relying on books, worksheets, problems, and solutions will not keep students engaged in math. Instead, teachers need to make this learning and teaching process fun and engaging with some math-based fun classroom games. These games engage and promote strategic mathematical thinking, computational fluency, an understanding of mathematical operations, and much more.

## 9. Focus on Practical learning

Math teachers should always focus on practical learning above everything because it helps to develop a resilient conceptual foundation in students. Teachers must try to involve math manipulatives in the teaching plan as much as possible.

## 10. Develop understanding in students

None of the math teaching strategies can work better than developing an excellent understanding of mathematical concepts in students. Sound math teaching is beyond memorizing the formulas, tables, or procedures. Teachers need to focus on creating a better account of concepts and work closely with students to make a reinforced conceptual foundation for math.

## 11. Assign real-world and meaningful problems

One of the best points in the math instructional strategies list is assigning real-world and meaningful problems to students. This strategy excites students with math problems as they relate numbers, formulas, and mathematical concepts to real life and understand the benefits of learning math.

## 12. Encourage productive struggle

Math teachers’ priority and responsibility is to allow their students to struggle with authentic problems or big questions to find distinct methods to solve them. Teachers’ duty is limited to making math class exciting and engaging by asking appropriate questions. Teachers might help them find solutions, but taking away their thinking might be the biggest mistake.

## 13. Create excitement and offer rewards

When students do not experience any success in solving math problems, they become negative about math and lose interest. In such cases, teachers must use reward-based strategies to encourage students. Teachers should reward students with earning points, badges, certificates, etc. when they succeed in a given task or solve a math problem. Recognizing students’ efforts and making them understand that practice will make them better. It will motivate them in the best way.

## 14. Work on Mental math

Mental math is one of the most powerful teaching strategies for introducing math fluency to students . Solving mental math problems gives confidence to students that they can solve more complex issues. Additionally, it is the best way to recall math concepts and facts quickly.

## 15. Take help from math puzzles.

Teachers can use math puzzles to develop solid logical thinking in students. Additionally, math puzzles enhance combinative capabilities, reinforce the power of abstract thinking, operate with longitudinal images, impart critical thinking skills and develop mathematical memory. Like mental math problems, math puzzles also improve foundational math skills and enhance math fluency.

## 16. Go for teamwork

Teachers might not realize all the math teaching strategies independently so that they can team up with other teachers for improved teaching skills. Teachers can discuss the lesson outcomes, teaching plans, and lesson goals to implement any strategies effectively.

## Some effective Math Teaching Strategies for elementary kids:

Elementary-level children must learn mathematics and its basic concepts, but it is not a straightforward subject to teach. However, with some strategic approaches, teachers can easily teach math to kids .

## 1. Clear and Unambiguous Teaching

Kids require clear, unambiguous, direct, and structured instructions that tell them how to solve a problem or make them understand any concept. Teachers can break down things into small portions and teach students bit by bit instead of processing a big idea.

Teachers must keep an eye on whether students understand things; they must recap the things taught in the last class before starting the new class and summarize things at the end of class.

Examples of explicit teaching strategies include precise learning outcomes and teachers showing and modeling behaviors or thinking processes to students so they can also think out of the box.

Explicit teaching methods can be encapsulated in a six-step process. It begins with an orientation by asking pertinent questions to students, the teachers solving the sample problems in all the concepts, and letting students try and solve similar math questions by themselves, sometimes letting the students work alone, checking their answers, and re-teaching complex ideas and problems.

## 2. Cooperative teaching strategy

The cooperative teaching approach involves breaking the class strength into small groups and allowing students in all the groups to discuss things and work together to solve problems and learn concepts.

However, teachers should avoid this method in the first place, just after students enter the class because initially, students cannot easily focus on the problems.

Also, only teachers should form groups that have students with different abilities, such as a group should have a few intelligent students, some average and below average students.

These groups must be small, like having 5-6 students. Also, these groups must be changed sometimes so that students can learn to collaborate with all the other students in class.

Teachers can easily inculcate soft skills in students, such as everyone might not have the same opinion as they have, etc., with the help of a cooperative teaching strategy.

Moreover, teachers also need to observe the activities of students in each group to know how their students think and how they process the problems and solutions, whether they enjoy learning this way or not.

## 3. Flicked Classrooms

A flipped classroom approach is among the most effective teaching strategies that math teachers can use. Flipping the classroom means students doing things at home that they usually do in school or the classroom, like getting the facts about the math concept. On the other side, they would do things in the classroom or school, typically at home, like implementing the learning.

In simple words, flipping the classroom conveys the students’ idea, doing self-study for the next day’s work in classrooms. It could be anywhere from books or tutorials and the next day discussing the concepts of the topic, problem-solving, etc., in the class.

However, teachers must put some effort into providing students with sources to study at home. Also, teachers must be mindful about utilizing the class time for the same.

Most importantly, before using this strategy, teachers must teach students about self-study and how they can do it; students will soon give up when they fail to understand things.

The flipped classroom approach provides teachers and students with more productive hours in the classroom and flexible learning culture.

## 4. Visualization-based approach

It is one of the most efficient and powerful teaching strategies that seamlessly make math teaching and learning. The visualization-based approach includes images, figures, infographics, mind maps, and other visual objects that can be used in teaching math.

Often, students lose their focus on math, but these teaching strategies and examples will captivate them with the subject and concepts as they see new things beyond just listening to new concepts while learning in the classroom.

This strategy can help teachers retain information better because kids’ brains process images quicker than words.

This approach also creates a scope of learning by teaching as students share how and what they learned with such visuals.

Moreover, unlimited visual strategies are available for teachers, but it only takes the efforts of teachers to begin somewhere and try some things out using these visuals.

You may also like to read- Math word problems

## Final Thoughts On Powerful Math Teaching Strategies

Various math teaching strategies engage students with math without enforcing it. Though these strategies are powerful enough to make this change, using them in the right way and at the right time is more critical. Teachers first need to understand their students before implementing any strategy and observe how they respond to it.

Also, teachers should focus on teaching techniques and work in small portions, especially when the students already have a negative attitude toward mathematics.

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## How to Improve Math Skills of Your Students’

## 12 Unique Ways on How To Teach Math Better

## 10 Strategies for Motivating Students in Mathematics

## 130 Math Word Problems To Challenge Students Grades 1 to 8

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## Exploring the Power of the CUBES Math Strategy for Word Problems

Math problem-solving is one of the most challenging things we teach. The CUBES strategy is all about helping students tackle those tricky word problems with ease. Whether you're dealing with pesky volume or area questions, or trying to figure out how many more apples Sally has than Timmy, the CUBES math strategy can be a great way to help those learners who struggle with word problems develop a systematic method to approach these problems. By breaking down the problem into smaller, more manageable chunks, you can quickly solve even the most complicated math problems.

So if you've ever found yourself scratching your head in frustration over your struggling learners' battle with story problems, fear not! The CUBES math strategy is here to make your life a whole lot easier. Stick around to learn more about how this awesome strategy can help you help your struggling students become math problem-solving pros!

What you'll find on this page:

## What is the CUBES Math Strategy?

The CUBES math strategy is a tool designed to help give students a systematic approach to breaking down and solving math word problems. The acronym C.U.B.E.S stands for:

- C ircle key numbers & units
- U nderline the question
- B ox math action words
- E valuate the problem
- S olve the problem & check your work

By breaking down the problem into these steps, students can better understand the context of the problem and effectively solve it.

## Benefits of using the CUBES strategy in math word problems

While CUBES is not the ideal method for all math problem solving, especially as word problems become more complex, you can use the strategy as a starting point to guide struggling students in being more attentive and systematic when tackling word problems.

Many struggling learners struggle with executive functioning and need a clear-cut plan for tackling this next-level math skill, and incorporating a strategy like CUBES into your teaching can give them steps to approach word problems rather than leaving them overwhelmed and unsure where to begin. This can help students build confidence in their ability to successfully solve math story problems and prepare them to solve multi-step problems, ultimately enhancing their problem-solving skills.

## Step-by-Step Guide to Using the CUBES Strategy

Implementing the CUBES strategy means teaching students the key steps and working through a gradual release process until they can effectively do this themselves. This systematic approach helps students understand the problem and empowers them to tackle word problems with confidence. Here's a little more about each step your students will need to achieve:

## C- Circle the numbers & units

C stands for “circle the key information.” This includes the numbers, units, and core information needed to solve the problem. This includes identifying math vocabulary that represents a number, such as “several,” “half,” or “a dozen.” Some questions may not involve numbers at all, in which case you would circle each instance of the word “none.” It is also important to identify units (such as feet, miles, or kilograms) and whether there is a decimal point.

While we don't want students solely relying on keywords when faced with a word problem, it can be helpful for them to recognize which words are references to mathematical symbols. For example, “+” means addition, and “-” means subtraction.

## U- Underline the question

After students read the problem and circle numbers/units, they must underline the question. While this may seem silly, if students aren't attending to what is being asked, they won't get the right answer. Helping students stay on target is a key component of the CUBES strategy for solving math problems.

Once your student has underlined the question and knows exactly what he or she needs to solve, it's time to move on to step B: boxes and bullets.

## B- Box math action words

Notice this doesn't say keywords…Again, we don't want students focused solely on using keywords for math problem solving. Research has shown time and time again this is an ineffective strategy once problems become more complex.

That said, students need to look at word problems through the lens of critical readers. What in the problem gives them a clue as to what they need to do to solve it?

Just like the author of a story gives us details to help us infer and get to the story's resolution, the author of a word problem helps us find the path to the solution. We need to be critical readers to get there. This is where boxing key information can be helpful. Here are some common examples that are often viewed as keywords but are critical for students to attend to to solve problems accurately:

- Addition: add, added, both, sum, total
- Subtraction: difference between, less than
- Multiplication: times twice as many/much as of every
- Division: split equally among/between each share out of

## E- Evaluate or Equation

At this stage, it's time to implement your strategy to solve. For some students, this will be writing and solving the equation. Others may need to evaluate by drawing a picture or using manipulatives to model the problem.

Either way, by this stage your learners should have broken down the problem to the point that they feel confident implementing a method that will lead them to the final step – solving.

## S- Solve & Check

Once the strategy has been chosen, guide your students through the process of solving the problem. This may involve writing out the equation, solving for the unknown variable, and checking their work to ensure they have found the correct solution.

Encourage your students to show their work and explain their reasoning as they solve the problem. This will not only help them understand the process better but also allow you to provide feedback and support if needed.

It's important to emphasize the importance of checking their work to ensure they have found the correct solution. This may involve plugging the solution back into the original problem to verify it or checking their work for errors in calculations.

Once your students have successfully solved the problem, congratulate them on their hard work and encourage them to reflect on the process. Ask them questions such as what strategies worked well for them, what challenges they encountered, and how they can apply what they have learned to similar problems in the future.

By guiding your students through the process of problem-solving and encouraging them to reflect on their work, you are helping them develop essential critical thinking skills that will serve them well in all areas of their education and beyond.

## Tips for Implementing the CUBES Math Word Problem Strategy

Whenever you're preparing to implement a strategy with your struggling learners, it can be helpful to get some tips from teachers who have been there. In asking for advice from colleagues, here's what they had to say.

To teach the cubes strategy, you should:

- Teach the strategy as a whole. Because this is such a visual strategy, it’s useful to provide multiple examples of how to solve problems with CUBES on a poster or anchor chart. You can use the chart below as an example of what to include.
- Use a standard problem as an example. Before having students practice on their own, have them watch and listen as you model how to use the CUBES strategy on the board using a Problem of the Day or by writing in student journals. Visual learners will appreciate watching you write out each step and manipulate your complex number sentence cube.
- Use anchor charts you make together in class. Then transfer that knowledge into modeling one or two more examples with students using cubes they create out of construction paper, or if necessary, manipulatives like buttons or dry beans.
- Don't fall into the trap that the standard algorithm is the only way to solve once the strategy has been used. Let students draw pictures, use manipulatives, make number lines, or whatever other strategies you've taught. The CUBES math strategy is to help them break up the problem. It isn't the guiding principle of the math calculations.
- Students can use CUBES to filter out irrelevant details and focus on the essential details needed to solve the problem. By guiding students to evaluate the problem systematically, you can help students make informed decisions and tackle complex math challenges. It is great for learners who might get bogged down in all the details.

## Teaching Problem-Solving Skills

Many instructors design opportunities for students to solve “problems”. But are their students solving true problems or merely participating in practice exercises? The former stresses critical thinking and decision making skills whereas the latter requires only the application of previously learned procedures.

Problem solving is often broadly defined as "the ability to understand the environment, identify complex problems, review related information to develop, evaluate strategies and implement solutions to build the desired outcome" (Fissore, C. et al, 2021). True problem solving is the process of applying a method – not known in advance – to a problem that is subject to a specific set of conditions and that the problem solver has not seen before, in order to obtain a satisfactory solution.

Below you will find some basic principles for teaching problem solving and one model to implement in your classroom teaching.

## Principles for teaching problem solving

- Model a useful problem-solving method . Problem solving can be difficult and sometimes tedious. Show students how to be patient and persistent, and how to follow a structured method, such as Woods’ model described below. Articulate your method as you use it so students see the connections.
- Teach within a specific context . Teach problem-solving skills in the context in which they will be used by students (e.g., mole fraction calculations in a chemistry course). Use real-life problems in explanations, examples, and exams. Do not teach problem solving as an independent, abstract skill.
- Help students understand the problem . In order to solve problems, students need to define the end goal. This step is crucial to successful learning of problem-solving skills. If you succeed at helping students answer the questions “what?” and “why?”, finding the answer to “how?” will be easier.
- Take enough time . When planning a lecture/tutorial, budget enough time for: understanding the problem and defining the goal (both individually and as a class); dealing with questions from you and your students; making, finding, and fixing mistakes; and solving entire problems in a single session.
- Ask questions and make suggestions . Ask students to predict “what would happen if …” or explain why something happened. This will help them to develop analytical and deductive thinking skills. Also, ask questions and make suggestions about strategies to encourage students to reflect on the problem-solving strategies that they use.
- Link errors to misconceptions . Use errors as evidence of misconceptions, not carelessness or random guessing. Make an effort to isolate the misconception and correct it, then teach students to do this by themselves. We can all learn from mistakes.

## Woods’ problem-solving model

Define the problem.

- The system . Have students identify the system under study (e.g., a metal bridge subject to certain forces) by interpreting the information provided in the problem statement. Drawing a diagram is a great way to do this.
- Known(s) and concepts . List what is known about the problem, and identify the knowledge needed to understand (and eventually) solve it.
- Unknown(s) . Once you have a list of knowns, identifying the unknown(s) becomes simpler. One unknown is generally the answer to the problem, but there may be other unknowns. Be sure that students understand what they are expected to find.
- Units and symbols . One key aspect in problem solving is teaching students how to select, interpret, and use units and symbols. Emphasize the use of units whenever applicable. Develop a habit of using appropriate units and symbols yourself at all times.
- Constraints . All problems have some stated or implied constraints. Teach students to look for the words "only", "must", "neglect", or "assume" to help identify the constraints.
- Criteria for success . Help students consider, from the beginning, what a logical type of answer would be. What characteristics will it possess? For example, a quantitative problem will require an answer in some form of numerical units (e.g., $/kg product, square cm, etc.) while an optimization problem requires an answer in the form of either a numerical maximum or minimum.

## Think about it

- “Let it simmer”. Use this stage to ponder the problem. Ideally, students will develop a mental image of the problem at hand during this stage.
- Identify specific pieces of knowledge . Students need to determine by themselves the required background knowledge from illustrations, examples and problems covered in the course.
- Collect information . Encourage students to collect pertinent information such as conversion factors, constants, and tables needed to solve the problem.

## Plan a solution

- Consider possible strategies . Often, the type of solution will be determined by the type of problem. Some common problem-solving strategies are: compute; simplify; use an equation; make a model, diagram, table, or chart; or work backwards.
- Choose the best strategy . Help students to choose the best strategy by reminding them again what they are required to find or calculate.

## Carry out the plan

- Be patient . Most problems are not solved quickly or on the first attempt. In other cases, executing the solution may be the easiest step.
- Be persistent . If a plan does not work immediately, do not let students get discouraged. Encourage them to try a different strategy and keep trying.

Encourage students to reflect. Once a solution has been reached, students should ask themselves the following questions:

- Does the answer make sense?
- Does it fit with the criteria established in step 1?
- Did I answer the question(s)?
- What did I learn by doing this?
- Could I have done the problem another way?

If you would like support applying these tips to your own teaching, CTE staff members are here to help. View the CTE Support page to find the most relevant staff member to contact.

- Fissore, C., Marchisio, M., Roman, F., & Sacchet, M. (2021). Development of problem solving skills with Maple in higher education. In: Corless, R.M., Gerhard, J., Kotsireas, I.S. (eds) Maple in Mathematics Education and Research. MC 2020. Communications in Computer and Information Science, vol 1414. Springer, Cham. https://doi.org/10.1007/978-3-030-81698-8_15
- Foshay, R., & Kirkley, J. (1998). Principles for Teaching Problem Solving. TRO Learning Inc., Edina MN. (PDF) Principles for Teaching Problem Solving (researchgate.net)
- Hayes, J.R. (1989). The Complete Problem Solver. 2nd Edition. Hillsdale, NJ: Lawrence Erlbaum Associates.
- Woods, D.R., Wright, J.D., Hoffman, T.W., Swartman, R.K., Doig, I.D. (1975). Teaching Problem solving Skills.
- Engineering Education. Vol 1, No. 1. p. 238. Washington, DC: The American Society for Engineering Education.

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## Math Wheels for Note-taking?

## Teaching Math Problem Solving Strategies

Teaching math problem solving strategies in middle school.

There are many math problem solving strategies out there and some are very beneficial to upper elementary and middle school math students. Before we jump into them, I’ll share a little bit of my experiences in teaching math problem solving strategies over the past 25+ years.

During my second year of teaching ( in the early 90s) , I was teaching 5th grade, and our state math testing began to include a greater focus on problem solving and writing in math.

Over the next several years, the other math teachers and I used standard sentence starters to help math students practice explaining their problem solving process. These were starters like:

- “In this problem, I need to….”
- “From the problem, I know….”
- “I already know…”
- “To solve the problem, I will…”
- “I know my answer is correct because…”

## Benefits of Sentence Starters

By using these sentence starters, students ended up with several paragraphs (some short, some long) to explain how they approached and solved the math problem, AND how they knew they were correct .

Sometimes this process took quite a long time, but it was helpful, because:

- It made many students slow down and think a bit more about what they were doing mathematically.
- Students took a little more time to analyze the problem (rather than picking out the numbers and guessing at an operation!).

I was teaching 5th grade in elementary school at this time, and we had a full hour for math every day. So, fitting in problem solving practice a few times a week was pretty easy, after students understood the process.

I really liked spending the time on these types of math problems, because they often led to discussion of other math concepts, and they reinforced concepts already learned. I used math problems from a publication that focused on various strategies, like:

- Guess and Check
- Work Backwards
- Draw a Picture
- Use Logical Reasoning
- Create a Table
- Look for a Pattern
- Make an Organized List.

I LOVED these…I really did (do)! And the students I taught during those years became very good at reasoning and solving problems.

## Teaching Math Problem Solving in Middle School

When I moved to 6th grade math in middle school, I tried to keep teaching these strategies, but our math periods are only 44 minutes.

- I tried to use the problem solving as warm-ups some days, but it would often take 30 minutes or more, especially if we got into a good discussion, leaving little time for a lesson.
- I found that spending too many class periods using the problem solving ended up putting me too far behind in the curriculum (although I’d argue that my students became better thinkers:-), so I had to make some alterations.
- highlight/underline the question in the problem
- shorten up the writing to bullet points
- highlight/underline the important information in the problem

## Math Problem Solving Steps

Now, when I teach these problem solving strategies, our steps are: Find Out, Choose a Strategy, Solve, and Check Your Answer.

Find Out When they Find Out, students identify what they need to know to solve the problem.

- They underline the question the problem is asking them to answer and highlight the important information in the problem.
- They shouldn’t attempt to highlight anything until they’ve identified what question they are answering – only then can they decide what is important to that question.
- In this step, they also identify their own background knowledge about the concepts in that particular math problem.

Choose a Strategy This step requires students to think about what strategy will work well with the question they’ve been asked. Sometimes this is tough, so I give them some suggestions for when to use these particular strategies:

- Make an Organized List: when there are many possible answers/combinations; or when making a list may help identify a pattern.
- Guess and Check: when you can make an educated guess and then use an incorrect guess to help you decide if the next guess should be higher or lower. This is often used when you’re looking for 2 unknown numbers that meet certain requirements.
- Work Backwards: when you have the answer to a problem or situation, but the “starting” number is missing
- when data needs to be organized
- with ratios (ratio tables)
- when using the coordinate plane
- with directional questions
- with shape-related questions (area, perimeter, surface area, volume)
- or when it’s just hard to picture in your mind
- Find a Pattern: when numbers in a problem continue to increase, decrease or both
- when the missing number(s) can be expressed in terms of the same variable
- when the information can be used in a known formula (like area, perimeter, surface area, volume, percent)
- when a “yes” for one answer means “no” for another
- the process of elimination can be used

Solve Students use their chosen strategy to find the solution.

Check Your Answer I’ve found that many students think “check your answer” means to make sure they have an answer (especially when taking a test), so we practice several strategies for checking:

- Reread the question; make sure your solution answers the question.
- Redo the math problem and see if you get the same answer.
- Check with a different method, if possible.
- If you used an equation, substitute your answer into the equation.
- Ask – does your answer make sense/is it reasonable?

## Teaching the Math Problem Solving Strategies

- Students keep reference sheets in their binders, so they can quickly refer to the steps and strategies. A few newer reference math wheels can be found in this blog post .
- For example, I often find that a ‘Guess and Check’ problem can be solved algebraically, so we’ll do the guessing and checking together first, and then we’ll talk about an algebraic equation – some students can follow the line of thinking well, and will try it on their own the next time; for others, the examples are exposure, and they’ll need to see several more examples before they give it a try.

## Using Doodle Notes to Teach Problem Solving Strategies

This year, I’m trying something new – I created a set of Doodle Notes to use during our unit.

- The first page is a summary of the steps and possible strategies.
- There’s a separate page for each strategy, with a problem to work through AND an independent practice page for each

- There’s also a blank template, so I can create problem solving homework for students throughout the year, using the same format. I’m hoping that using the Doodle Notes format will make the problem solving strategies a little more fun, interesting, and easy to remember.

This was a long post about teaching math problem solving strategies! Thanks for sticking with me to the end!

## read next...

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## Conquering the Fraction Division Challenge

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Beyond Worksheets: Innovative Approaches to Teaching Math to Young Children

T his article explores creative and effective strategies beyond traditional worksheets to engage and inspire young learners in math.

Discover new ways to make math fun, exciting, and meaningful for your child’s early math education journey!

## Teaching Math to Young Children

When teaching math to young children, it is crucial to incorporate innovative approaches.

Here are the main points to get your inspiration going:

- Start by creating a lesson plan that includes age-appropriate activities, practice sheets, and real-world problem-solving.
- Incorporate technology when possible to engage students in their learning. For instance, introducing fun online math games can be a great way to make learning more interactive and enjoyable.
- Focus on building basic math skills while helping children understand why math is important.
- Encourage creative thinking and collaboration throughout the lesson and allow students to play a central role in their learning environment .
- Implement assessment strategies to measure student progress and make adjustments when necessary.

Now, let’s get into the details, shall we?

## Connect Math to Real World

Connect math to the real world to make it more relatable and engaging for students.

- For example, tutors should use everyday objects to demonstrate counting or incorporate shapes into art projects.
- Also, provide meaningful and contextual learning experiences that tap into children’s curiosity and creativity.

This helps learners develop their math, problem-solving, and critical thinking skills.

Teaching young children to connect math to the real world helps them see its relevance and use it as an important tool for problem-solving.

Help connect math to the real world by giving your students a better understanding of the world around them through available Sen Teacher Jobs Birmingham .

One way to make this connection for students is through practical math activities .

For example, building a simple structure from blocks of various shapes and sizes can help them understand basic geometry and measurements.

Teachers and parents should also explain how math is used in the real world .

Examples could include how architects use geometry when designing buildings or how engineers utilize mathematics when designing machines.

Exploring real-world applications of math can help students understand the importance and relevance of mathematics in our everyday lives.

Another method is to use visual aids, such as pictures and diagrams, to show how math can be used in different contexts.

- For example, showing a diagram of a pizza with the slices labeled by fractions can help students understand fractional parts and how they relate to one another.
- Likewise, showing them how their favorite foods are made or how a recipe uses measurements can help them understand basic math concepts.
- Once you show them real-world examples of how to use fractions, you can start teaching them more complicated formulas. For example, you can move from simple fractions to instructing them on how to compare fractions .

## Focus on Hands-on Activities and Games

When learning math, teachers should add fun activities and games.

- Students can do their math puzzles and challenge each other to solve them.
- Parents and teachers can also introduce card or board games related to the concepts they teach in class.
- Include group activities since they are an interactive way to teach math.
- Involve students in math competitions between teams, creating problem-solving challenges, or cooperative activities.

Since math can be intimidating for some students, teachers should provide support to make learning easier.

For example, they can use visual techniques and multiple examples or break down complex problems into smaller steps.

It helps students understand the concepts better and remember them for longer periods.

Both parents and teachers should encourage students to use online resources to practice and learn more.

They should also ensure their students have enough time to practice the concepts they learn in class. Repetition is important to master math concepts.

Don’t forget to celebrate each student’s achievements and encourage them to keep learning. Rewarding students when they accomplish a task or recognizing their efforts motivates them.

## Incorporate Technology

Technology creates a dynamic learning environment and makes learning math fun and engaging.

Parents play a crucial role here, as they can encourage kids to engage in interactive games or show videos that explain mathematical concepts as presented during maths online homeschool sessions.

Both parents and teachers can also use websites, apps, or virtual reality to help students explore and apply math concepts.

Technology can serve as a medium for assessment, allowing teachers to assess student performance and identify areas that need more attention.

Technology also enables children to work independently from home or in the classroom.

They can access learning materials and activities anytime and study math at their own pace.

In addition, teachers can assign online quizzes or exercises tailored to each student’s needs and provide real-time feedback.

Finally, encouraging kids to join math forums and share knowledge with others fosters creativity and collaboration.

## Make Flashcards

Make flashcards with a math problem on one side and the correct answer on the other. The students can quiz each other on math problems.

Flashcards can also act as assessment tools to identify areas where they need help.

Students can work through the cards at their own pace, which keeps them engaged and fosters learning in an active environment.

Tutors can adapt and reuse flashcards to ensure each student understands the material.

## Final Thoughts

In conclusion, incorporating innovative approaches to teaching math to young children can transform their learning experience, whether you’re a parent or a teacher.

By going beyond worksheets and embracing creative strategies, we can foster a love for math, build foundational skills, and empower our children for success in their math education and beyond.

So let’s work together to inspire young minds and make math a joyful and meaningful journey for every child!

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## 8 Math Division Tricks: Making Division Fun & Accessible

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## 1. The Halving Trick for Even Numbers

2. multiplication facts as division shortcuts, 3. the ’10s’, ‘100s’, and ‘1000s’ trick, 4. the ‘doubling and halving’ trick, 5. chunking method, 6. skip counting to find quotients, 7. reciprocals and inverse operations, 8. cross-division method.

Have you ever watched a child struggle with the concept of division , their frustration growing with each attempt? It’s a common scene in classrooms and homes around the world. Division, a fundamental pillar of mathematics, is not just a subject confined to the boundaries of school curriculums but also plays a crucial role in our daily lives. Yet, despite its importance, division can be a source of challenge for many students. This is where math division tricks come into play.

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These strategies are designed as shortcuts and tools to deepen understanding and improve problem-solving skills . By presenting division in a more accessible and less intimidating way, we aim to help students tackle division problems with confidence and ease.

Whether you’re a teacher seeking innovative teaching methods or a parent looking to support your child’s educational journey, this guide offers valuable insights and techniques to make division more manageable and enjoyable.

## 8 Math Division Tricks for Kids

One of the easiest ways to divide is using the halving trick for even numbers . This is especially handy when you’re dividing a number by 2. The concept is simple: if a number is even, you can split it into two equal parts.

Example: If you have 16 apples and want to share them equally between two people, you can use the halving trick. Since 16 is an even number, dividing it by 2 is as simple as finding half of 16, which is 8. So, each person gets 8 apples.

Understanding multiplication is a powerful tool for learning how to divide. This method relies on the idea that division is the inverse of multiplication. If you know your multiplication tables well, you can quickly reverse-engineer them to solve division problems.

Example: Suppose you’re faced with the division problem 63 ÷ 7. If you remember that 7 times 9 equals 63, then you can quickly figure out that 63 divided by 7 is 9. This shows how multiplication facts serve as math division tricks, making division for kids much simpler.

Here are some fun division facts worksheets :

Dividing large numbers by 10, 100, or 1000 can seem daunting, but there’s a straightforward trick: moving the decimal point . This is one of the most practical ways to divide large numbers without getting bogged down in complex calculations.

Example: If you need to divide 1,000 by 100, instead of working through the problem the long way, you can simply move the decimal point two places to the left (since you’re dividing by 100). So, 1,000 becomes 10.00, and when you remove the extra zeros , you’re left with 10. This trick demonstrates the easiest way to divide large numbers, making it a favorite among division math tricks .

This trick is an easy way to do division, especially when dealing with larger numbers that might initially seem intimidating. The idea is to simplify the division problem by doubling one number and halving the other. This works because the overall value of the problem doesn’t change, just the numbers you’re working with.

Example: Divide 80 by 4 using the ‘Doubling and Halving’ Trick

Apply the Trick:

Double the divisor: Doubling 4 gives you 8.

Halve the dividend: Halving 80 gives you 40.

Revised Problem: Now, instead of dividing 80 by 4, you divide 40 by 8.

Solution:

This method simplifies calculations, especially when dealing with more complex numbers. By doubling one number and halving the other, you can sometimes make the numbers more manageable without changing the result of the division.

The chunking method is an easy division method that breaks down dividing large numbers into more manageable steps. This method involves subtracting large chunks of multiples of the divisor from the dividend until you reach zero or a remainder smaller than the divisor.

Example: To divide 154 by 7 using the chunking method, start by seeing how many times 7 can fit into 154 in large chunks. You know that 7 times 10 is 70, so you can subtract 70 twice from 154 (140), leaving you with 14. Then, you see that 7 fits into 14 exactly 2 times. Add up all the times you subtracted 7, and you get 22. So, 154 divided by 7 is 22.

Skip counting is a visual and practical way to solve simple division problems, making it an excellent strategy for kids. It involves counting by the divisor until you reach the dividend. This method is particularly useful for visual learners and can be done with fingers, drawings, or counters.

Example: If you’re trying to divide 24 by 3, you start counting by 3s: 3, 6, 9, 12, 15, 18, 21, 24. Count how many steps you took to reach 24, which in this case is 8. Therefore, 24 divided by 3 equals 8.

Understanding reciprocals , and inverse operations are some of the math division tricks that can simplify division, especially as kids start dealing with fractions and more complex numbers. The reciprocal of a number is simply 1 divided by that number, and using it turns division into multiplication, which many students find easier.

Example: If you need to divide 1 by 2 (which is written as 1 ÷ 2), you can multiply 1 by the reciprocal of 2 instead. The reciprocal of 2 is 1/2, so you do 1 × 1/2, which equals 1/2. This approach makes it an easy way to divide, especially when working with fractions, by converting division problems into multiplication ones.

The cross-division method, also known as simplification before division, is a handy trick for reducing larger numbers before dividing. This method is particularly useful when the dividend and divisor share a common factor . By simplifying the numbers first, the division becomes much easier.

Example: Suppose you want to divide 48 by 6. You notice that both numbers are divisible by 6. You can simplify 48 divided by 6 to 8 divided by 1 (since 48 ÷ 6 = 8 and 6 ÷ 6 = 1). Your division problem is significantly simpler: 8 divided by 1, which obviously equals 8. This method is a great math division trick because it reduces the complexity of the problem before you even start the actual division.

## Common Challenges Kids Face When Learning Division

Learning division can be tricky for many kids. As they navigate the world of mathematics, they encounter several common challenges that can make understanding how to divide overwhelming. Here’s some of these hurdles:

- Grasping the basic idea behind division math problems can be tough. Kids often struggle with the concept that division essentially shares or groups numbers in equal parts.
- A strong foundation in multiplication is crucial to learn division effectively. Many children find it hard to recall multiplication facts, directly impacting their ability to solve division problems.
- When division for kids moves beyond exact answers, the concept of remainders can be confusing. Understanding what to do with the leftover numbers and how they fit into the context of a problem is a common challenge.
- Applying division to solve word problems adds another layer of difficulty. Kids must first decipher what the question is asking before they can even begin to apply their division skills.
- The process of long division , with its multiple steps, can be particularly daunting. Remembering the order of operations and where to write each number requires practice and patience.
- As division problems increase in complexity, so does the intimidation factor. Kids often feel overwhelmed when faced with dividing large numbers, unsure of how to start.

## 4 Tips for Practicing Division Tricks

Mastering math division tricks requires practice, patience, and strategies. Here are some tips on incorporating these tricks into daily math practice, making the learning process both effective and fun.

1. Interactive Online Division Games: Online interactive games are designed to make learning division fun and engaging. They often include a variety of math division tricks, allowing kids to practice and apply these strategies in a game-like setting. This approach reinforces learning and keeps children motivated.

Here are some fun division games that you can get started with:

2. Repetition and Consistency: Like any skill, becoming proficient with math division tricks requires regular practice. Dedicate a specific time each day for practicing division. Using worksheets can be particularly effective for this purpose. Worksheets allow for repetitive practice of the same type of problems, helping to cement the division tricks in memory. They can range from simple division problems to more complex ones, gradually increasing in difficulty as proficiency grows.

Get started with these printable division worksheets:

3. Daily Practice with Real-Life Examples: Incorporate math division tricks into everyday situations. Whether dividing snacks among friends or calculating the time each task takes during the day, using real-life examples helps reinforce these concepts in a practical, memorable way.

4. Encourage Self-Correction: Encourage kids to check their work after completing division exercises. This can be done by using multiplication to verify their division answers.

Mastering math division tricks can transform how children approach and solve division problems, making the process more enjoyable and less intimidating. By practicing these tricks regularly and incorporating them into daily learning, kids can build a strong foundation in division, setting them up for success in mathematics.

## Frequently Asked Questions (FAQs)

What is the 4 division trick.

The 4 division trick involves halving a number twice to quickly divide it by 4, making it a simple and effective strategy for division.

## How do you divide mentally fast?

To divide mentally fast, use known multiplication facts, round numbers to make calculations simpler, and apply division shortcuts like the halving trick.

## Which is harder multiplication or division?

The difficulty between multiplication and division varies by individual; however, many find division harder due to its more complex conceptual understanding and the steps involved in long division.

## How do you teach division to struggling students?

Teach division to struggling students by using visual aids, breaking down steps into manageable parts, and practicing with real-life examples to enhance understanding.

## What are the 3 rules of division?

The 3 rules of division are: dividing by 1 leaves the number unchanged, dividing a number by itself equals 1, and division by 0 is undefined.

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## 7 Real-World Math Strategies

We asked our audience how they’re using the real world to teach math and compiled their most intriguing responses.

Math used to be all rote memorization and pencil-to-paper equations disconnected from the real world, but more and more teachers are realizing the importance of making practical, relevant connections in math.

We asked our audience of educators: How do you use the real world to teach math? We’ve collected some of the most interesting answers, ways teachers are connecting math to the everyday lives of their students.

## The Real-World Math Wall

To get her fifth-grade students thinking about the math in their lives—and to head off the inevitable “we’ll never use this in the real world” complaint—Samantha Baumgardner, a teacher at Woodrow Petty Elementary School in Taft, Texas, has them bring in a common item and write three ways it relates to math on a notecard. These objects form the class’s real-world math wall.

An interesting rule: Once something is placed on the wall, the same item can’t be brought in again—pushing students to think outside the box to make real-world math connections. Students bring in objects like playing cards, cake pans, softball score sheets, and cookie recipes. One student brought in a medicine dropper and by way of explanation wrote, “The medicine dropper helps you measure water, and it also helps you with cooking steak.”

## A School Water Audit

At North Agincourt Jr. Public School in Scarborough, Canada, Krista Dunlop-Sheppard, a bilingual resource and home school teacher for grades 1–6, has her students conduct a water audit at home and at their school. Her students have a wide range of math ability: In a single class, she has students with learning disabilities, students who are gifted, and students who have no diagnoses but need extra support. Modifying a pilot project created by the Toronto Zoo Education Department , students add, subtract, find averages, and measure liquids—like the flow rate of all the water fountains, toilets, and urinals—to measure the amount of water their school uses in a day.

They also interview custodians to discover their daily water usage while mopping floors, and do online research to find out how much water the cafeteria dishwasher uses. When finished, students suggest ways for the school to conserve—like collecting rainwater in a barrel to water plants, and cleaning paint brushes in a bucket instead of using running water. In June, students repeat the water audit and see if the changes they implemented made a difference.

## Acting Out Restaurant Scenarios

There’s nothing wrong with using money to teach negative and positive numbers, or pizza to introduce fractions, but Justin Ouellette—a third- to fifth-grade International Baccalaureate educator at Suzhou Singapore International School in China—takes these exercises a memorable step further, bringing in menus so students can act out true-to-life restaurant scenarios. Going dutch on dinner and tipping reinforces addition, subtraction, decimals, and percents, Ouellette says.

If you need a good resource: Ouellette has used this free lesson plan about a fictional eatery called the Safari Restaurant.

## Integrating Math Into English and History

Making real-world math connections can happen outside of math class, too.

While reading Elie Wiesel’s Night , 12th-grade students at Kittatinny Regional High School in Hampton Township, New Jersey, calculate the volume of 11 million pennies to help them imagine the impact of the lives lost during the Holocaust. Ashley Swords, a resource center English teacher for grades nine to 12, uses pennies because they are small and plentiful and allow her to recontextualize a familiar, everyday object.

Students perform other calculations to amplify the impact of the lessons—at the school’s football field, for example, they determine the volume of Swords herself and then calculate how many football fields would be needed to bury the 11 million Holocaust victims if they were each Swords’s size and were buried in graves 10 feet deep. Guesses ranged from two to 20 football fields, with students finally concluding that it would take about 343 football fields.

Swords knew this lesson was a success when a group of six seniors got emotional after completing it and realizing the magnitude of the deaths in World War II.

## Math Recipes

Recipes were perhaps the most popular idea among the elementary teachers who responded to our request for real-world math examples. Fifth-grade teacher Gabi Sanfilippo of Meadow Ridge in Orland Park, Illinois, for example, asked her students to write down one to two ways they used math outside of the classroom during spring break—and more than half of her students wrote that they practiced using measurements and fractions while baking and cooking with their families.

In class, students practice halving, tripling, or quadrupling recipes based on how many people they’d cook or bake for. Most teachers don’t actually cook in class, but often students cook at home to practice their new skills.

Another educator, Elizabeth Eagan of Bastrop Independent School District in Texas, brings in a toaster oven to bake in class. She teaches the visually impaired, from newborns to 22-year-olds, using recipes in large print, braille, or audio to show the real-world application of adding and subtracting fractions.

Eagan prints out recipes at 129 percent for her low-vision students or converts the text to braille using braille transcription software , and then prints them using a braille printer . She purchased Stir It Up , a cookbook in both braille and print that makes it easier for families and teachers to help their students if they haven’t mastered the tactile language. Students may use electronic magnifying glasses like Pebble and Ruby. For audio, Eagan records herself reading the recipes, has a peer or parent help, or uses the app Seeing AI , which can scan and read recipes aloud.

## Grocery Store Field Trip

Many teachers make real-world math connections to grocery shopping. Leanna Agcaoili’s second-grade students at Joseph J. Fern Elementary School in Honolulu, Hawaii, are tasked with creating a healthy meal for their family on a $20 budget. On a grocery store field trip, students practice adding and subtracting one- and two-digit whole numbers—and learn about money and budgeting in the process.

Agcaoili says she’ll do a practice run in class next year, noting that the first time through some students had difficulty finding their ingredients.

## Graphing Favorite Halloween Candy Wrappers

After Halloween, Dottie Wright Berzins, a retired public school teacher, had her students bring in wrappers from their favorite Halloween candy. Depending on the age, the students then created graphs showing their favorite candies.

Younger ages built life-size candy bar graphs, marking the x- and y-axis with masking tape on the floor and using the wrappers to represent the bars. Older students constructed tally charts and paper graphs, and followed ads, tracked which brands advertised more, interviewed peers about their favorite candy, made predictions, and created what-if scenarios—like what if the price of chocolate increased—how would that variable impact the graph?

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February 28, 2024

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## Anyone can play Tetris, but architects, engineers and animators alike use the math concepts underlying the game

by Leah McCoy, The Conversation

With its bright colors, easy-to-learn rules and familiar music, the video game Tetris has endured as a pop culture icon over the last 40 years. Many people, like me, have been playing the game for decades, and it has evolved to adapt to new technologies like game systems, phones and tablets. But until January 2024, nobody had ever been able to beat it.

A teen from Oklahoma holds the Tetris title after he crashed the game on Level 157 and beat the game . Beating it means the player moved the tiles too fast for the game to keep up with the score, causing the game to crash. Artificial intelligence can suggest strategies that allow players to more effectively control the game tiles and slot them into place faster—these strategies helped crown the game's first winner.

But there's far more to Tetris than the elusive promise of winning. As a mathematician and mathematics educator , I recognize that the game is based on a fundamental element of geometry, called dynamic spatial reasoning. The player uses these geometric skills to manipulate the game pieces, and playing can both test and improve a player's dynamic spatial reasoning.

## Playing the game

A Russian computer scientist named Alexey Pajitnov invented Tetris in 1984. The game itself is very simple: The Tetris screen is composed of a rectangular game board with dropping geometric figures. These figures are called tetrominoes , made up of four squares connected on their sides in seven different configurations.

The game pieces drop from the top, one at a time, stacking up from the bottom. The player can manipulate each one as it falls by turning or sliding it and then dropping it to the bottom. When a row completely fills up, it disappears and the player earns points.

As the game progresses, the pieces appear at the top more quickly, and the game ends when the stack reaches the top of the board.

Dynamic spatial reasoning

Manipulating the game pieces gives the player an exercise in dynamic spatial reasoning. Spatial reasoning is the ability to visualize geometric figures and how they will move in space. So, dynamic spatial reasoning is the ability to visualize actively moving figures.

The Tetris player must quickly decide where the currently dropping game piece will best fit and then move it there. This movement involves both translation, or moving a shape right and left, and rotation, or twirling the shape in increments of 90 degrees on its axis.

Spatial visualization is partly inherent ability, but partly learned expertise. Some researchers identify spatial skill as necessary for successful problem solving, and it's often used alongside mathematics skills and verbal skills.

Spatial visualization is a key component of a mathematics discipline called transformational geometry, which is usually first taught in middle school. In a typical transformational geometry exercise, students might be asked to represent a figure by its x and y coordinates on a coordinate graph and then identify the transformations , like translation and rotation, necessary to move it from one position to another while keeping the piece the same shape and size.

Reflection and dilation are the two other basic mathematical transformations, though they're not used in Tetris. Reflection flips the image across any line while maintaining the same size and shape, and dilation changes the size of the shape, producing a similar figure.

For many students, these exercises are tedious, as they involve plotting many points on graphs to move a figure's position. But games like Tetris can help students grasp these concepts in a dynamic and engaging way.

Transformational geometry beyond Tetris

While it may seem simple, transformational geometry is the foundation for several advanced topics in mathematics. Architects and engineers both use transformations to draw up blueprints, which represent the real world in scale drawings .

Animators and computer graphic designers use concepts of transformations as well. Animation involves representing a figure's coordinates in a matrix array and then creating a sequence to change its position, which moves it across the screen. While animators today use computer programs that automatically move figures around, they are all based on translation.

Calculus and differential geometry also use transformation. The concept of optimization involves representing a situation as a function and then finding the maximum or minimum value of that function. Optimization problems often involve graphic representations where the student uses transformations to manipulate one or more of the variables.

Lots of real-world applications use optimization—for example, businesses might want to find out the minimum cost of distributing a product. Another example is figuring out the size of a theoretical box with the largest possible volume.

All of these advanced topics use the same concepts as the simple moves of Tetris.

Tetris is an engaging and entertaining video game, and players with transformational geometry skills might find success playing it. Research has found that manipulating rotations and translations within the game can provide a solid conceptual foundation for advanced mathematics in numerous science fields.

Playing Tetris may lead students to a future aptitude in business analytics, engineering or computer science—and it's fun. As a mathematics educator, I encourage students and friends to play on.

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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 ...

Then, I provided them with the "keys to success.". Step 1 - Understand the Problem. To help students understand the problem, I provided them with sample problems, and together we did five important things: read the problem carefully. restated the problem in our own words. crossed out unimportant information.

These essential strategies in teaching mathematics can make this your class's best math year ever! 1. Raise the bar for all. WeAreTeachers. For math strategies to be effective, teachers must first get students to believe that they can be great mathematicians. Holding high expectations for all students encourages growth.

A Math Word Problem Framework That Fosters Conceptual Thinking. This strategy for selecting and teaching word problems guides students to develop their understanding of math concepts. Word problems in mathematics are a powerful tool for helping students make sense of and reason with mathematical concepts. Many students, however, struggle with ...

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

Math is an essential life skill. You use problem-solving every day. The math strategies you teach are needed, but many students have a difficult time making that connection between math and life. Math isn't just done with a pencil and paper. It's not just solving word problems in a textbook.

The idea of using "vertical nonpermanent surfaces" in the math classroom comes from Peter Liljedahl's work with the best conditions for encouraging and supporting problem-solving in the math ...

Step 2: Devise a plan. Going to use Guess and test along with making a tab. Many times the strategy below is used with guess and test. Make a table and look for a pattern: Procedure: Make a table reflecting the data in the problem.

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 ...

A Powerful Rethinking of Your Math Classroom. We look at strategies you can reset this year—adjusting your testing regimen, tackling math anxiety, encouraging critical thinking, and fostering a mistake-friendly environment. The beginning of school is a great time for teachers—both veteran and early career—to consider ways they can improve ...

4. Make a List. This strategy is one of the most powerful ones. Students decide what information goes on the list from the word problem given. Organize the list by categories and make sure all the pieces of the problem are on the list. Lastly have students review the information that they organized on a list.

Math requires students to pay attention to details, plan, and self-monitor. Students also have to keep track of steps — and maybe even change direction while they work. Evidence-based math instruction helps these students because it breaks problems into multiple steps and reduces distractions.

Schema approach. This is a math intervention strategy that can make problem solving easier for all students, regardless of ability. Compare different word problems of the same type and construct a formula, or mathematical sentence stem, that applies to them all. For example, a simple subtraction problems could be expressed as:

I teach multiple strategies to solve math problems because of it: makes explicit what happens in our heads. helps students choose the most efficient strategy. provides scaffolding so that students can find a place to enter into the problem-solving process. motivates students to want to learn more.

Implement tasks that promote reasoning and problem solving. Effective teaching of mathematics engages students in solving and discussing tasks that promote mathematical reasoning and problem solving and allow multiple entry points and varied solution strategies. Use and connect mathematical representations. Effective teaching of mathematics ...

Effective strategies for teaching problem solving in math have evolved since I started teaching 20+ years ago - and, for the better! More and more teachers, just like you and I, are using problem solving to teach math concepts in a way that invites students to be active participants in the learning process.

Metacognitive Strategies. Metacognitive strategies enable students to become aware of how they think when solving problems in mathematics (Iris Center, 2019). One metacognitive strategy that can be used for problem-solving is UPS-Check. UPS-Check is a four-step process to assist students with understanding and solving problems in context ...

Numerous math teaching strategies are there that can help teachers to connect students with mathematics and be effective problem-solving techniques for students also. The core of these strategies lies in only one thing, training students with practical problem-solving skills, even though the instructional strategies list is long and might use different and varied methods.

Start your free trial now. Here are seven effective strategies for teaching elementary math: 1. Make it hands-on. Elementary math can be difficult because it involves learning new, abstract concepts that can be tricky for children to visualize. Try to imagine what it's like for a five-year-old to see an addition problem for the very first time.

The CUBES math strategy is a tool designed to help give students a systematic approach to breaking down and solving math word problems. The acronym C.U.B.E.S stands for: Circle key numbers & units; Underline the question ; Box math action words ; Evaluate the problem ; Solve the problem & check your work

Teach problem-solving skills in the context in which they will be used by students (e.g., mole fraction calculations in a chemistry course). Use real-life problems in explanations, examples, and exams. Do not teach problem solving as an independent, abstract skill. Help students understand the problem. In order to solve problems, students need ...

Math Problem Solving Steps. Now, when I teach these problem solving strategies, our steps are: Find Out, Choose a Strategy, Solve, and Check Your Answer. Find Out. When they Find Out, students identify what they need to know to solve the problem. They underline the question the problem is asking them to answer and highlight the important ...

Strategies for teaching problem solving include: 1. Teaching students an attack strategy to guide the process of problem solving 2. Teaching students to recognize and solve word problems according to the schema of the problem 3. Utilizing appropriate mathematical language to help students understand the meaning of each word in a problem.

Teaching young children to connect math to the real world helps them see its relevance and use it as an important tool for problem-solving. Help connect math to the real world by giving your ...

2. Repetition and Consistency: Like any skill, becoming proficient with math division tricks requires regular practice. Dedicate a specific time each day for practicing division. Using worksheets can be particularly effective for this purpose. Worksheets allow for repetitive practice of the same type of problems, helping to cement the division tricks in memory.

The Real-World Math Wall. To get her fifth-grade students thinking about the math in their lives—and to head off the inevitable "we'll never use this in the real world" complaint—Samantha Baumgardner, a teacher at Woodrow Petty Elementary School in Taft, Texas, has them bring in a common item and write three ways it relates to math on ...

Transformations may seem simple, but they underlie lots of more complex math concepts. Transformational geometry beyond Tetris While it may seem simple, transformational geometry is the foundation ...