math problem solving elementary

Problem Solving Activities: 7 Strategies

• Critical Thinking

Problem solving can be a daunting aspect of effective mathematics teaching, but it does not have to be! In this post, I share seven strategic ways to integrate problem solving into your everyday math program.

In the middle of our problem solving lesson, my district math coordinator stopped by for a surprise walkthrough. 

I was so excited!

We were in the middle of what I thought was the most brilliant math lesson– teaching my students how to solve problem solving tasks using specific problem solving strategies. 

It was a proud moment for me!

Each week, I presented a new problem solving strategy and the students completed problems that emphasized the strategy. 

Genius right? 

After observing my class, my district coordinator pulled me aside to chat. I was excited to talk to her about my brilliant plan, but she told me I should provide the tasks and let my students come up with ways to solve the problems. Then, as students shared their work, I could revoice the student’s strategies and give them an official name. 

What a crushing blow! Just when I thought I did something special, I find out I did it all wrong. 

I took some time to consider her advice. Once I acknowledged she was right, I was able to make BIG changes to the way I taught problem solving in the classroom. 

When I Finally Saw the Light

To give my students an opportunity to engage in more authentic problem solving which would lead them to use a larger variety of problem solving strategies, I decided to vary the activities and the way I approached problem solving with my students. 

Problem Solving Activities

Here are seven ways to strategically reinforce problem solving skills in your classroom. 

Seasonal Problem Solving

Many teachers use word problems as problem solving tasks. Instead, try engaging your students with non-routine tasks that look like word problems but require more than the use of addition, subtraction, multiplication, and division to complete. Seasonal problem solving tasks and daily challenges are a perfect way to celebrate the season and have a little fun too!

Cooperative Problem Solving Tasks

Go cooperative! If you’ve got a few extra minutes, have students work on problem solving tasks in small groups. After working through the task, students create a poster to help explain their solution process and then post their poster around the classroom. Students then complete a gallery walk of the posters in the classroom and provide feedback via sticky notes or during a math talk session.

Notice and Wonder

Before beginning a problem solving task, such as a seasonal problem solving task, conduct a Notice and Wonder session. To do this, ask students what they notice about the problem. Then, ask them what they wonder about the problem. This will give students an opportunity to highlight the unique characteristics and conditions of the problem as they try to make sense of it. 

Want a better experience? Remove the stimulus, or question, and allow students to wonder about the problem. Try it! You’ll gain some great insight into how your students think about a problem.

Math Starters

Start your math block with a math starter, critical thinking activities designed to get your students thinking about math and provide opportunities to “sneak” in grade-level content and skills in a fun and engaging way. These tasks are quick, designed to take no more than five minutes, and provide a great way to turn-on your students’ brains. Read more about math starters here ! 

Create your own puzzle box! The puzzle box is a set of puzzles and math challenges I use as fast finisher tasks for my students when they finish an assignment or need an extra challenge. The box can be a file box, file crate, or even a wall chart. It includes a variety of activities so all students can find a challenge that suits their interests and ability level.

Calculators

Use calculators! For some reason, this tool is not one many students get to use frequently; however, it’s important students have a chance to practice using it in the classroom. After all, almost everyone has access to a calculator on their cell phones. There are also some standardized tests that allow students to use them, so it’s important for us to practice using calculators in the classroom. Plus, calculators can be fun learning tools all by themselves!

Three-Act Math Tasks

Use a three-act math task to engage students with a content-focused, real-world problem! These math tasks were created with math modeling in mind– students are presented with a scenario and then given clues and hints to help them solve the problem. There are several sites where you can find these awesome math tasks, including Dan Meyer’s Three-Act Math Tasks and Graham Fletcher’s 3-Acts Lessons . 

Getting the Most from Each of the Problem Solving Activities

When students participate in problem solving activities, it is important to ask guiding, not leading, questions. This provides students with the support necessary to move forward in their thinking and it provides teachers with a more in-depth understanding of student thinking. Selecting an initial question and then analyzing a student’s response tells teachers where to go next. 

Ready to jump in? Grab a free set of problem solving challenges like the ones pictured using the form below. 

Which of the problem solving activities will you try first? Respond in the comments below.

Shametria Routt Banks

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2 Responses

This is a very cool site. I hope it takes off and is well received by teachers. I work in mathematical problem solving and help prepare pre-service teachers in mathematics.

Thank you, Scott! Best wishes to you and your pre-service teachers this year!

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6 Tips for Teaching Math Problem-Solving Skills

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

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

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

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

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

1. Link problem-solving to reading

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

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

2. Avoid boxing students into choosing a specific operation

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

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

3. Revisit ‘representation’

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

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

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

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

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

4. Give time to process

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

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

5. Ask questions that let Students do the thinking

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

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

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

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

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

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Usable Math

(formerly 4mality), a digital playground for math learning through problem solving and design.

Usable Math provides interactive problem solving practice for 3rd through 6th grade students learning mathematical reasoning and computation through creative writing, NoCode slideshow design, and human-AI collaboration.

• MATH MODULES

Math Friends

Featuring four coaches Estella Explainer, Chef Math Bear, How-to Hound, and Visual Vicuna who offer reading, computation, strategy, and visual strategies for solving math problems.

Estella Explainer

"I help children understand the language and meaning of questions using kid-friendly vocabulary."

Chef Math Bear

"I provide computational strategies (addition, subtraction, multiplication and division) for solving problems."

How-to-Hound

" I present strategic thinking clues (rounding, estimation, elimination of wrong answers). "

Visual Vicuna

" I offer ways to see problems and their solutions using animations, pictures, charts and graphs. "

The coaches annotate hints and provide feedback to help students with various levels of knowledge solve mathematical word problems using a wide range of strategies.

Math and ISTE Standards Based

Usable Math aims to teach mathematics concepts and problem solving skills based on the Massachusetts Mathematics Curriculum Framework and the Common Core State Standards for Mathematics. Usable Math supports ISTE Standards for Students : Empowered Learner (1.1), Knowledge Constructor (1.3), and Computational Thinker (1.5).

Open Education Resource

Usable Math is an open education resource project developed in the College of Education, University of Massachusetts Amherst. Usable Math received a 2023 classroom grant from MassCUE (Massachusetts Computer Using Educators) . An initial version called 4mality was developed with funding support from the Verizon Foundation and a grant from the US Department of Education, Institute of Education (IES).

BROWSE MATH MODULES

Storywriting, history, and science modules, a jenny-the-fisher math and citizen scientist adventure, math & science, a tai-the-math historian time travel adventure, math & history, ai-enhanced, a sofia-the-forester adventure, math & storywriting, math problem-solving and design modules, area and perimeter, total problems: 6.

Total problems: 8

Multiplication and Division

Algebraic Thinking

Total problems: 7

Measurement

Total problems: 10.

Geometry: Lines and Lines of Symmetry

Geometry: Maps + Grids + Ordered Pairs

Charts & Graphs

Geometry: Figures, Shapes and Angles

Total problems: 11

Add & Take Away

Place Value

Total problems: 14.

Total problems: 9

Total problems: 5

More coming soon, welcome to usable math. in this interactive website, you will find learning modules designed to develop mathematical problem solving skills among young learners in grades 3 to 6..

Our Modules explore standards-based math concepts including Fractions, Measurement, Geometry, Decimals, Money, and more. Usable Math is free to access using a computer, smartphone, or iPad.

What do we mean by Usable Math?

The word Usable can read as follows:

U Able meaning you can do math problem solving.

Us Able meaning together all of us can do math problem solving.

Usable meaning anyone is able to learn math problem solving - with practice, effort, and support.

What are the Usable Math Learning Modules?

Each learning module in Usable Math consists of a group of math word problems related to a specific mathematical concept. The problems are based on the Massachusetts Mathematics Curriculum Framework↗ as well as Common Core Standards↗ .

Each problem within a module consists of a question, three to four possible answer choices, and problem solving ideas and strategies provided by our four coaches: Estella Explainer, Chef Math Bear, How-to-Hound, and Visual Vicuna.

How are the Modules Displayed online?

Each module has been developed using Google Slides.

Click. Pause. Solve.

View each module in Slideshow.

How do teachers, students and families use each module?

We strive to make every module on Usable Math kid friendly . Clicking on a module from the selections on the Modules Homepage , each user controls what happens during the learning experience by clicking to open strategies and spending time thinking about them before answering the question. The goal is for students, by themselves, in small groups, or with a teacher, or a family member, to analyze and understand what the problem is asking them to solve before providing an answer.

A question appears without its answer choices or any problem solving strategies.

Click one time and Estella offers a problem solving strategy.

Click again and the Bear offers a different strategy.

Click again and the Hound presents a strategy.

Click again and the Vicuna has an additional strategy approach.

The next click gives the four answer choices, but not yet the correct answer.

The final click highlights the correct answer from among the answer choices.

Before going to the next problem, a motivational statement and gif appears offering encouragement to the users.

What is the purpose of the Motivational Statements between Problems?

Each motivational statement is intended to provide feedback and encouragement to students using the system. Following the insights of researchers into the use of praise and the development of growth mindsets in young learners, these motivational statements are designed to reward students’ effort, hard work, persistence, and belief in one’s self as a learner. We want youngsters to realize that they can learn anything with the right tools, the right beliefs, the right coaches, and their own work and practice.

Need more help? Or have a question?

Reach out to us and we will do our best to get back to you within 12 hours.

RESEARCH AND RESOURCES

We believe that every child deserves a strong foundation in mathematics. our platform is designed to provide engaging and effective math instruction to elementary school students, and we are proud to say that there is science behind the way we deliver this instruction..

UsableMath was formerly known as 4MALITY. As a result of our commitment to providing the best possible math instruction to elementary school students, we have rebranded our platform as UsableMath.com to better reflect our mission and approach to teaching mathematics.

Our platform is designed to provide engaging and effective math instruction to students in grades K-5, using a unique approach that emphasizes hands-on, problem-solving activities. We use interactive, multimedia elements such as videos, games, and simulations to help students understand key mathematical concepts and build a strong foundation of knowledge.

Math Coaches

The use of virtual coaches that provides students with personalized support and feedback, has become increasingly popular in the field of math education. Research has shown that learning companions can be effective in improving student engagement and motivation, as well as helping students to better understand mathematical concepts and build a stronger foundation of knowledge. UsableMath employs the concept of learning companions to help students succeed in mathematics. Our virtual math coaches serve as personal guides, providing students with individualized support and feedback as they work through mathematical concepts and problems. These coaches, or learning companions, are designed to be like friends or mentors, helping students to build their confidence, overcome challenges, and achieve their full potential.

How are we using Generative AI to enhance Usable Math Modules?

As developers of Usable Math, we are aware of both the educational potentials and complexities of Generative AI technologies. In our system, ChatGPT is used to support teachers and other adults to expand and enhance how math can be understood and taught in schools and homes. When you click on the AI icon, you are linked to a blog where we have recorded how AI proposes to solve selected math word problems found in Usable Math modules in a side-by-side view next to the hints we have authored from the perspectives of our four math coaches: Estella Explainer, Chef Math Bear, How-to-Hound, and Visual Vicuna. Our hope is that our strategies along with the AI-developed strategies will give adults more ways to inspire math learning among students.

Look for this icon for AI-enhanced guides.

Prompts for ChatGPT, BingAI and Other Generative AI Tools

Estella Explainer Prompt:

Take the personality of a math coach who provides strategies for understanding language and meaning of questions using kid-friendly vocabulary. The coach’s motto is "My job is to explain the math questions clearly so you know what you are supposed to do to solve the problem. Sometimes there are unfamiliar or confusing terms in the question. I will help you understand what they mean. The first math problem is {replace math word problem here}

Chef Math Bear Prompt:

Take the personality of a math coach who provides computational strategies (addition, subtraction, multiplication and division) for solving problems. The coach’s motto is “I am here to make sure that you know how to do the math needed to answer these questions. Sometimes you need to do addition, subtraction, multiplication or division. Some questions ask you to use fractions, decimals, large numbers, and probability. When you need ideas for what to do, I am ready.

How-to-Hound Prompt:

Take the personality of a math coach who uses strategic thinking clues (rounding, estimation, elimination of wrong answers) to solve math problems. The coach’s motto is “Answering math questions means you need a plan and my role is to help you figure out different strategies for solving problems. Sometimes you can get the correct answer by crossing out the wrong answers; other times you can round numbers up or down to make figuring a problem easier. I know other strategies as well.

Visual Vicuna Prompt:

Take the personality of a math coach who offers ways to see problems and their solutions using animations, pictures, charts and graphs. The coach’s motto is “I find math is a lot clearer when I take the numbers and words and put them into pictures and drawings or move objects around so I can see how to answer a question. When you find yourself unsure about a question, see if one of my ideas will explain what to do.

Growth Mindset Statements

As education researchers, we understand the important role that a positive attitude and motivation play in learner success. That's why we’ve integrated the use of growth mindset and motivational cues in Usable Math. After every math challenge, students receive messages that encourage them to adopt a growth mindset, reinforcing the idea that with effort and persistence, they can improve their math skills and achieve success.

A sample motivational cue from the Fractions module.

Collaborative Problem Solving

We believe in the power of collaboration and teamwork when it comes to learning mathematics. Our platform creates a learning climate that promotes collaborative problem solving, providing students with opportunities to work together and explore mathematical concepts in a supportive and inclusive environment. Whether you are a student, teacher, or parent, we invite you to explore our platform and experience the science behind the way we deliver math instruction to elementary school students. Read more about our work on the Journal of STEM Education↗

Papers, Presentations and Blogs

UsableMath GenAI Prompts: Learn Math with Our Tailor-Made Prompts for ChatGPT, Claude, and other GenAI tools. Usable Math Blog. https://blog.usablemath.org/usablemath-genai-prompts .

Maloy, R. W. & Gattupalli, S. (2024). Prompt Literacy. EdTechnica: The Open Encyclopedia of Educational Technology . https://edtechbooks.org/encyclopedia/prompt_literacy

Gattupalli, S., & Maloy, R. W. (2024). On Human-Centered AI in Education. https://doi.org/10.7275/KXAP-FN13

Gattupalli, S., Edwards, S.A, Maloy, R. W., & Rancourt, M. (2023, October). Designing for Learning: Key Decisions for an Open Online Math Tutor for Elementary Students. Digital Experiences in Mathematics Education . https://doi.org/10.1007/s40751-023-00128-3 .

Gattupalli, S., Maloy, R.W., Edwards, S.A. & Gearty, A. (2023, August 23). Prompt Literacy for STEM Educators: Enhance Your Teaching and Learning with Generative AI. Berkshire Resources for Learning and Innovation (BRLI) Teaching with Technology Conference, Pittsfield, MA. ScholarWorks@UMass.

Blending Gardens and Geometry: Socio-cultural Approaches in Math Ed. Usable Math Blog. https://blog.usablemath.org/blending-gardens-and-geometry-socio-cultural-approaches-in-math-education .

Maloy, R. W., Gattupalli, S., & Edwards, S. A. (2023). Developing Usable Math Online Tutor for Elementary Math Learners with NoCode Tools . Scholarworks@UMass.

Gattupalli, S., Maloy, R. W., & Edwards, S. A. (2023). Prompt Literacy: A Pivotal Educational Skill in the Age of AI . Scholarworks@UMass.

Gattupalli, S., Maloy, R. W., & Edwards, S. (2023). Comparing Teacher-Written and AI-Generated Math Problem Solving Strategies for Elementary School Students: Implications for Classroom Learning . https://doi.org/10.7275/8sgx-xj08

Making Math Usable for Young Learners . Guest post on Rachelle Dené Poth's EdTech blog Learning as I go: Experiences, Reflections, Lessons Learned . January, 2023.

Math Learning Digital Choice Board (2020) . ScholarWorks, University of Massachusetts Amherst.

Maloy, R.W., Razzaq, L., & Edwards, S.A. (2014). Learning by Choosing: Fourth Graders Use of an Online Multimedia Tutoring System for Math Problem Solving . Journal of Interactive Learning Research , 25(1), 51-64.

Razzaq, L., Maloy, R. W., Edwards, S. A., Arroyo, I., & Woolf, B.P. (2011). “4MALITY: Coaching Students with Different Problem Solving Strategies Using an Online Tutoring System” (p. 359-364). In J. A. Konstan, Ricardo Conejo, Jose L, Marzo & Nuria Oliver, User Modeling, Adaptation and Personalization: 19th International Conference, UMAP 2011, Girona, Spain, July 11-15 Proceedings . Berlin: Springer Verlag.

Maloy, R.W., Edwards,S. A. & Anderson G. (2010, January-June). “Teaching Math Problem Solving Using a Web-based Tutoring System, Learning Games, and Students’ Writing .” Journal of STEM Education: Innovations and Research, 11 (1&2).

Edwards, S. A., Maloy, R.W., & Anderson G. (2010, February). “Classroom Characters Coach Students to Success.” Teaching Children Mathematics, 16 (6), 342-349.

Edwards, S. A., Maloy, R. W., & Anderson G. (2009, Summer). “Reading Coaching of Math Word Problems.” Literacy Coaching Clearinghouse . http://www.literacycoachingonline.org/briefs.html .

MEET OUR TEAM

Sharon Edwards , Ph.D.

Teacher Education & Curriculum Studies

College of Education, University of Massachusetts Amherst

Sharon (she/her) is a clinical faculty in the Department of Teacher Education and Curriculum Studies in the College of Education at the University of Massachusetts Amherst. Sharon is the big brains behind the development of Usable Math online math tutor.

Email : sae at umass dot edu

Robert Maloy , Ph.D.

Elementary Math and History

Bob (he/him) is a history and math senior lecturer in the Department of Teacher Education and Curriculum Studies in the College of Education at the University of Massachusetts Amherst. Bob is the creative math content creator and storytelling artist behind Usable Math.

Email : rwm at umass dot edu

Sai Gattupalli

Math, Science & Learning Technologies (MSLT)

Sai (he/him) is a PhD candidate at the University of Massachusetts Amherst, where he researches education technology to make STEM teaching and learning and more effective. Sai is passionate about understanding learner culture to create effective learning experiences. Email : sgattupalli at umass dot edu Website : gattupalli.com

Marguerite Rancourt

Lead Teacher, Discovery School at Four Corners

Greenfield, Massachusetts

Marguerite (she/her) teaches fourth grade at the Discovery School in addition to serving as Lead Teacher for the school. She has created and taught professional development workshop for other elementary school teachers. In 2018, she received the Pioneer Valley Excellence in Teaching Award. Students in her class have been contributing to the design of system throughout the 2022-2023 school year.

Aubrey Coyne

Math Content Designer and Reviewer

College of Education, Commonwealth Honors College, University of Massachusetts Amherst.

Aubrey Coyne (she/her) is a sophomore at the University of Massachusetts Amherst. She is a math tutor and is studying to be an elementary teacher. Aubrey is passionate about finding ways to make learning accessible and enjoyable for all students.

Graduate Student, Math and Digital Media Research Assistant

Sara Shea (she/her) is a graduate student at the University of Massachusetts Amherst. She is currently part of the university’s Collaborative Teacher Education Pathway program, working towards earning her master’s degree in elementary education.

Katie Allan

Math and Digital Media Research Assistant

Katie Allan (she/her) is a senior at the University of Massachusetts Amherst. She is a math major with a concentration in education and passionate about math education.

SUGGESTIONS AND FEEDBACK

We welcome ideas from teachers, students, and families about the usable math system..

Please complete our UsableMath Module Review and Feedback↗ form.

Your responses will help us to improve how the system works instructionally and technically. Let us know any additional thoughts about the problems, characters, hints, gifs, mindset statements and more.

Your message has been received. We will get back to you shortly. The average response time is approximately 6 hours.

K-5 Math Centers

K-5 math ideas, 3rd grade math, need help organizing your k-5 math block, 5 ways to include math problem solving activities in your classroom.

Are you looking for math problem solving activities that are not too easy and not too hard, but juuust right? I’ve got something just for you and your students.

Solve and Explain Problem Solving Tasks are open-ended math tasks that provide just the right amount of challenge for your kids. Here’s a little more about them.

Open-ended math problem solving tasks:

• promote multiple solution paths and/or multiple solutions
• boost critical thinking and math reasoning skills
• increase opportunities for developing perseverance
• provide opportunities to justify answer choices
• strengthen kids written and oral communication skills

What Makes These So Great?

• All Common Core Standards are covered for your grade level
• 180+ Quality questions that are rigorous yet engaging
• They are SUPER easy to assemble
• Provide opportunities for meaningful math discussions
• Perfect for developing a growth mindset
• Easily identify student misconceptions so you can provide assistance
• Very versatile (check out the different ways to use them below)

You can find out more details for your grade level by clicking on the buttons below.

I’m sure you really want to know how can you use these with your kids. Check out the top 5 ideas on how to use Solve and Explain Problem Solving Tasks in your classroom.

How and When Can I Use Them?

Solve and Explain Tasks Cards are very versatile. You can use them for:

• Math Centers  – This is my favorite way to use these! Depending on your grade level, there are at least two (Kinder – 2nd) or three (3rd-5th) tasks types per Common Core standard. And each task type has 6 different questions. Print out each of the different tasks types on different color paper. Then, let students choose which one question from each task type they want to solve.

• Problem of the Day  – Use them as a daily math journal prompt. Print out the recording sheet and project one of the problems on your white board or wall.  Students solve the problem and then glue it in their spiral or composition notebooks.

• Early Finisher Activities  -No more wondering what to do next!Create an early finishers notebook where students can grab a task and a recording sheet. Place the cards in sheet protectors and make copies of the Early Finisher Activity Check-Off card for your kids to fill out BEFORE they pull a card out to work on. We want to make sure kids are not rushing through there first assignment before moving on to an early finisher activity.

• Weekly Math Challenges  – Kids LOVE challenges! Give students copies of one of the problems for homework. Then give them a week to complete it. Since many of the questions have multiple solutions and students have to explain how they got their answers, you can have a rich whole group discussion at the end of the week (even with your kindergarten and 1st grade students).

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• Formative Assessments  – Give your students a problem to solve. Then use the Teacher Scoring Rubric to see how your kids are doing with each standard. Since they have to explain their thinking, this is a great way to catch any misconceptions and give feedback to individual students.

So this wraps up the top 5 ways that you can use problem solving tasks in your classroom.  Click your grade level below to get Solve and Explain problem solving tasks for your classroom.

• Read more about: K-5 Math Ideas

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• Intro to division
• Understand fractions
• Equivalent fractions and comparing fractions
• More with multiplication and division
• Arithmetic patterns and problem solving
• Quadrilaterals
• Represent and interpret data
• Multiply by 1-digit numbers
• Multiply by 2-digit numbers
• Factors, multiples and patterns
• Add and subtract fractions
• Multiply fractions
• Understand decimals
• Plane figures
• Measuring angles
• Area and perimeter
• Units of measurement
• Decimal place value
• Add decimals
• Subtract decimals
• Multi-digit multiplication and division
• Divide fractions
• Multiply decimals
• Divide decimals
• Powers of ten
• Coordinate plane
• Algebraic thinking
• Converting units of measure
• Properties of shapes
• Ratios, rates, & percentages
• Arithmetic operations
• Negative numbers
• Properties of numbers
• Variables & expressions
• Equations & inequalities introduction
• Data and statistics
• Negative numbers: addition and subtraction
• Negative numbers: multiplication and division
• Fractions, decimals, & percentages
• Rates & proportional relationships
• Expressions, equations, & inequalities
• Numbers and operations
• Solving equations with one unknown
• Linear equations and functions
• Systems of equations
• Geometric transformations
• Data and modeling
• Volume and surface area
• Pythagorean theorem
• Transformations, congruence, and similarity
• Arithmetic properties
• Factors and multiples
• Reading and interpreting data
• Negative numbers and coordinate plane
• Ratios, rates, proportions
• Equations, expressions, and inequalities
• Exponents, radicals, and scientific notation
• Foundations
• Algebraic expressions
• Linear equations and inequalities
• Graphing lines and slope
• Expressions with exponents
• Quadratics and polynomials
• Equations and geometry
• Algebra foundations
• Solving equations & inequalities
• Working with units
• Linear equations & graphs
• Forms of linear equations
• Inequalities (systems & graphs)
• Absolute value & piecewise functions
• Exponents & radicals
• Exponential growth & decay
• Quadratics: Multiplying & factoring
• Quadratic functions & equations
• Irrational numbers
• Performing transformations
• Transformation properties and proofs
• Right triangles & trigonometry
• Non-right triangles & trigonometry (Advanced)
• Analytic geometry
• Conic sections
• Solid geometry
• Polynomial arithmetic
• Complex numbers
• Polynomial factorization
• Polynomial division
• Polynomial graphs
• Rational exponents and radicals
• Exponential models
• Transformations of functions
• Rational functions
• Trigonometric functions
• Non-right triangles & trigonometry
• Trigonometric equations and identities
• Analyzing categorical data
• Displaying and comparing quantitative data
• Summarizing quantitative data
• Modeling data distributions
• Exploring bivariate numerical data
• Study design
• Probability
• Counting, permutations, and combinations
• Random variables
• Sampling distributions
• Confidence intervals
• Significance tests (hypothesis testing)
• Two-sample inference for the difference between groups
• Inference for categorical data (chi-square tests)
• Advanced regression (inference and transforming)
• Analysis of variance (ANOVA)
• Scatterplots
• Data distributions
• Two-way tables
• Binomial probability
• Normal distributions
• Displaying and describing quantitative data
• Inference comparing two groups or populations
• Chi-square tests for categorical data
• More on regression
• Prepare for the 2020 AP®︎ Statistics Exam
• AP®︎ Statistics Standards mappings
• Polynomials
• Composite functions
• Probability and combinatorics
• Limits and continuity
• Derivatives: definition and basic rules
• Derivatives: chain rule and other advanced topics
• Applications of derivatives
• Analyzing functions
• Parametric equations, polar coordinates, and vector-valued functions
• Applications of integrals
• Differentiation: definition and basic derivative rules
• Differentiation: composite, implicit, and inverse functions
• Contextual applications of differentiation
• Applying derivatives to analyze functions
• Integration and accumulation of change
• Applications of integration
• AP Calculus AB solved free response questions from past exams
• AP®︎ Calculus AB Standards mappings
• Infinite sequences and series
• AP Calculus BC solved exams
• AP®︎ Calculus BC Standards mappings
• Integrals review
• Integration techniques
• Thinking about multivariable functions
• Derivatives of multivariable functions
• Applications of multivariable derivatives
• Integrating multivariable functions
• Green’s, Stokes’, and the divergence theorems
• First order differential equations
• Second order linear equations
• Laplace transform
• Vectors and spaces
• Matrix transformations
• Alternate coordinate systems (bases)

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How to Use Real-World Problems to Teach Elementary School Math: 6 Tips

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When you think back on elementary school math, do you have fond memories of the countless worksheets you completed on adding fractions or solving division problems? Probably not.

Researchers and educators have been pushing for years for schools to move away from teaching math through a set of equations with no context around them, and towards an approach that pushes kids to use numerical reasoning to solve real problems, mirroring the way that they’ll encounter the use of math as adults.

The strategy is largely about setting kids up for success in the professional world, and educators can lay the groundwork decades earlier, even in kindergarten .

Here are some tips for using a real world problem-solving approach to teaching math to elementary school students.

1. There’s more than one right answer and more than one right method

A “real world task” can be as simple as asking students to think of equations that will get them to a particular “target” number, say, 14. Students could say 7 plus 7 is 14 or they could say 25 minus 11 is 14. Neither answer is better than the other, and that lesson teaches kids that there are multiple ways to use math to solve problems.

2. Give kids a chance to explain their thinking

The process you use to solve a real world math problem can be just as important as arriving at the correct answer, said Robbi Berry, who teaches 5th grade in Las Cruces, N.M. Her students have learned not to ask her if a particular answer is correct, she said, because she’ll turn the question back on them, asking them to explain how they know that it is right. She also gives her students a chance to explain to one another how they arrived at a particular solution, “We always share our strategies so that the kids can see the different ways” to arrive at an answer, she said. Students get excited, she said, when one of their classmates comes up with an approach they never would have thought of. “Math is creative,” Berry said. “It’s not just learning and memorizing.”

3. Be willing to deal with some off-the-wall answers

Problem solving does not necessarily mean going to the word problems in your textbook, said Latrenda Knighten, a mathematics instructional coach in Baton Rouge, La. For little kids, it can be as simple as showing a group of geometric shapes and asking what they have in common. Students may go off track a bit by talking about things like color, she said, but teachers can steer them towards thinking about things like how a rectangle differs from a triangle.

4. Let your students push themselves

Tackling these richer, real-world problems can be tougher than solving equations on a worksheet. And that is a good thing, said Jo Boaler, a professor at Stanford University and an expert on math education. “It’s really good for your brain to struggle,” she said. “We don’t want kids getting right answers all the time because that’s not giving their brains a really good workout.” These types of problems require collaboration, a skill that many don’t associate with math, but that is key to how math reasoning works beyond the classroom. The complexity and difficulty of the tasks means that students “have to talk to each other and really figure out what to do, what’s a good method?”

5. Celebrate ‘favorite mistakes’ to encourage intellectual risk taking

Wrong answers should be viewed as learning opportunities, Berry said. When one of her students makes an error, she asks if she can share it with the class as a “favorite mistake.” Most of the time, students are comfortable with that, and the class will work together to figure how the misstep happened.

6. Remember there’s no such thing as a being born with a ‘math brain’

Some teachers believe that certain students are just naturally good at math, and others are not, Boaler said. But that’s not true. “Brains are constantly shaping, changing, developing, connecting, and there is no fixed anything,” said Boaler, who often works alongside neuroscientists. What’s more, many elementary school teachers lack confidence in their own math abilities, she said. “They think they can’t do [math],” Boaler said. “And they often pass those ideas on” to their students.

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Mathematics for Elementary Teachers

(18 reviews)

Michelle Manes, Honolulu, HI

Copyright Year: 2017

Publisher: University of Hawaii Manoa

Language: English

Formats Available

Conditions of use.

Learn more about reviews.

Reviewed by Kevin Voogt, Assistant Professor, Grace College on 4/20/23

There seem to be subjects missing that are typical of the common core mathematics for elementary teachers texts (e.g., Ratios/Proportions, clear Partitive/Measurement division ideas, percentages, certain ideas in Geometry, Measurement). read more

Comprehensiveness rating: 4 see less

There seem to be subjects missing that are typical of the common core mathematics for elementary teachers texts (e.g., Ratios/Proportions, clear Partitive/Measurement division ideas, percentages, certain ideas in Geometry, Measurement).

Content Accuracy rating: 5

I did not find mathematical errors in the text during my review.

Relevance/Longevity rating: 3

I think there is need for quite a few updates to the text in regards to what is covered in elementary mathematics through the common core. The topics listed in my review of the Comprehensiveness above are just a start. I also see a need to add more activities to each section where prospective elementary teachers could do more exploration of the mathematics rather than what seems to be a more traditional approach of having the text explain it followed by problem sets alone.

Clarity rating: 5

The wording was quite clear and had nice explanations throughout.

Consistency rating: 5

It seems consistent throughout - with recurrent use of the same technical terms as needed.

Modularity rating: 4

There were a few issues with being able to assign the texts at different points within the course, as is the case for many math texts, in that many of the sections rely heavily on prior knowledge. If reorganization were to occur, there would be some need to re-structure how certain sections are taught.

Organization/Structure/Flow rating: 4

The text lacks much of the wonderful mathematical connections that could be made between ideas. While some connections are made, they seem a little outdated at times. I also think it would make more sense to have the properties of operations within their corresponding sections on operations rather than after all 4 operations are introduced.

Interface rating: 5

I did not see any issues with the interface. It was pretty user-friendly.

Grammatical Errors rating: 5

I did not notice any errors during my review.

Cultural Relevance rating: 5

I did not see anything insensitive or offensive in the text.

The text is just a small sampling of the many methods that could be used in teaching these mathematical ideas. I would have liked to see more activities for elementary teachers built into the lessons in each chapter as a means for learning and exploring ideas to facilitate more discussion as this text is used. There also are so many more connections that could be made between mathematical ideas that were lost a bit, especially with the general organization. On the whole, it is a nice resource and I could see it as useful for students studying for their certification exams to get some perspectives on the mathematical ideas they might encounter.

Reviewed by Sandra Zirkes, Teaching Professor, Bowling Green State University on 4/14/23

The text covers whole numbers, fractions, decimals, and operations well, and it provides some topics in geometry and algebraic thinking. However, the topics of ratio, proportion, and percent, as well as a more thorough coverage of geometry and... read more

The text covers whole numbers, fractions, decimals, and operations well, and it provides some topics in geometry and algebraic thinking. However, the topics of ratio, proportion, and percent, as well as a more thorough coverage of geometry and measurement are missing.

All information in the text is mathematically accurate and the writing and diagrams are error-free.

Relevance/Longevity rating: 4

While all of the information in the text is accurate and thought-provoking, some specific approaches are outdated with respect to the current standards and pedagogy. Approaching the concept of place value through the "Dots and Boxes" method, without reference to base ten blocks that are overwhelmingly used in the elementary math classroom, limits the coverage of this important topic. Similarly, approaching fractions using the "pies and kids" scenario is not consistent with the standards which emphasize the understanding of all fractions as iterations of unit fractions.

The text is written using clear and understandable prose that is both mathematically accurate and accessible to college level pre-service teachers.

The text has a clear organization and focus and uses consistent approaches and terminology throughout.

Much of the text is easily divisible into smaller subsections for student use. With respect to reorganization and realignment for a particular course, while some topics are revisited at appropriate points in the text, if those original topics were not covered in the course, revisiting the topic may not provide enough basis for the new topic. For example, the understanding of decimals is highly reliant on a student's understanding of the Dots and Boxes approach to place value earlier in the text.

Organization/Structure/Flow rating: 5

The topics in the text are organized in a logical way that is consistent with the structure of a typical mathematics education course.

Interface rating: 4

Navigating the text itself was seamless and intuitive. However, the videos that I viewed had poor visual quality and there was no audio.

The text is well written with no grammatical errors.

There is no apparent cultural insensitivity in the text.

This text has a problem solving focus and emphasizes deep thinking and reasoning about mathematics. Its approaches are clear and understandable. While its approaches are mathematically correct and thought-provoking, it is missing some key topics such as ratio, proportion, percent, and a more thorough coverage of geometry and measurement, as well as some standards-based approaches such as base ten blocks and understanding fractions as iterations of unit fractions.

Reviewed by Fred Coon, Assistant Professor, Anderson University on 2/16/23

The text covers all major points to help develop future teachers. read more

Comprehensiveness rating: 5 see less

The text covers all major points to help develop future teachers.

Text appears to be accurate.

The content is consist with concepts that elementary teachers should know. The methods are small in diversity.

Topics where well explained.

Text appears to use understandable and consistent terms.

Modularity rating: 5

Units appear to be mostly independent and can be used as stand alone units.

The topics are presented in a manner that build on each other but can be rearrange if desired.

Interface was useful and aided in navigating text.

I found no errors.

The text has no culturally insensitive or offensive items that I noticed.

I would like to have seen more diversity in methods discussed.

Reviewed by Perpetual Opoku Agyemang, Professor of Mathematics, Holyoke Community College on 6/17/21

The content in this text is built to help its readers, especially, pre-service elementary education majors learn to think like a mathematician in some very specific ways. The content addresses the subject framework in a complete yet concise... read more

The content in this text is built to help its readers, especially, pre-service elementary education majors learn to think like a mathematician in some very specific ways. The content addresses the subject framework in a complete yet concise manner. Although it does not provide an effective index/or glossary, LCD was not extensively tackled using factor tree, multiples or tables to express it, I still give props to the author since there are a lot of pictorial examples and a question bank for most of the various concepts. Furthermore, Dots and Boxes game on chapter 1 was very engaging and fun.

This text is very accurate and informative using a variety of felicitous examples to suit a diverse student population.

Conventional concepts are presented in a current and applied manner which allows for easier association with similar organized and retained information. This text could use some updated fraction problems and examples involving mixed numbers. Some of the YouTube videos have no sound at all.

Clarity rating: 4

Content material was presented in an easy to understand prose. Introduction of concepts and new terms were usually done by association or relevant previous knowledge. Some of the concepts like Multiplying Fractions, have YouTube videos embedded in the introductions.

Terminologies and framework are consistent throughout the text. The use of different notations were consistent throughout the various chapters and subunits.

This text has easily divisible content as stand alone subunits. However, numbering these chapters and subunits would have gone a long way to help its readers.

The topics in this text are organized from basic to complex concepts in a logical, clear fashion.

This text has an awesome interface (Online, PDF and XML). Moreover, it is untainted by distractions that may confuse its reader. Hyperlinks should have been included in the content.

I did not spot any grammatical errors in this text.

This content material contains no recognizable cultural insensitivity. It could use more examples involving modern affairs that are inclusive of diverse backgrounds.

I truly love the concise format of this text and how many different examples it uses to explain the concepts. The Geometry of Arts and Science and Tangrams were so informative with fun activities. It's easy to tell when one example ends and another begins, although index/or glossary and a system of links from the table of contents would be greatly appreciated. I did not see Points on a Coordinate Plane. Additionally, the number of exercises per section is too small. Of course this can be remedied by adding more. As with any textbook, the reader will need to supplement certain sections and clarify particular terms and concepts to best fit their situation. Pre-service elementary education majors could transition to this book fairly easily and successfully teach K-6 students in the United States in alignment with current Common Core Math Standards.

Reviewed by April Slack, Math Instructor, Aiken Technical College on 5/13/21

This text covers elementary mathematics strands including place value, numbers and operations, fractions, patterns, algebraic thinking, decimals, and geometry. Measurement and Data and Statistics strands are not included in this particular text. ... read more

This text covers elementary mathematics strands including place value, numbers and operations, fractions, patterns, algebraic thinking, decimals, and geometry. Measurement and Data and Statistics strands are not included in this particular text. The last chapter supplies the audience with problem-based learning approaches that include some measurement, but not in the detail of previous chapters of the book. It does incorporate problem solving strategies and pedagogical techniques teachers may use in the classroom. Examples with solutions and clarifying notes are provided throughout the text. The text does address Common Core Standards as well as the eight mathematical process standards. The textbook also provide teachers with a conceptual understanding of elementary mathematics along with appropriate mathematical terminology. The text does not offer an index or glossary.

The mathematics content provided in this text is accurate and provides thorough examples of teaching elementary mathematics for pre-service teachers. I found the text to build conceptual understanding and procedural fluency rather than just focus on basic algorithms to solve math problems. This is especially important for pre-service teachers, as they need to truly understand the "why" behind the math tricks that are often taught in early grades. The embedded links throughout the text are all in working order, as well.

The problem-solving approach to mathematics is especially relevant for elementary pre-service teachers; the intended audience. The book does expand beyond elementary mathematics, however, this is deemed extremely useful for all levels of mathematics teachers. Knowing the mathematical concepts beyond elementary strands allows teachers to know where there students are going and the mathematical purpose of content standards at each grade level. Many of the pedagogical techniques presented in the text are aligned with current research and instructional strategies for the elementary classroom.

This text provides explanations and defines mathematical terminology and has accessible prose. Beginning with the problem solving chapter before the specific content strands allows teachers to apply and consider strategies throughout the text. Often times, textbooks save problem solving for the end, but this text addresses strategies upfront and spirals nicely throughout the text. Some of the examples and visual representations are intended for an audience with mathematical background knowledge and strengths. A pre-service teacher may need help with content review prior to understanding the selection of particular problems highlighted in the text.

The text is well-organized and consistent with terminology throughout. The text is also consistent with provided examples that are used by mathematics teachers in everyday classrooms. There are multiple examples throughout each of the content chapters for pre-service teachers to reference and use in their own experiences.

This book is an easy read and may be easily broken up for weekly reading assignments and reflections. It seems as if mathematics teachers had a hand in writing this book. Bulleted and numbered lists are used throughout the text. The text also presents examples in clear, colored blocks. Visual models are clear and concise.

The book is well-organized with headings, subheadings, and the use of italics and boldface make this book extremely student friendly. The topics and content presented in this text are clear and in a logical order. Bulleted and numbered lists are reader friendly and easily understood. I found having the problem solving chapter appear first in the text stresses the importance and relevance of helping students become natural problem solvers. Often times texts and even worksheets save problem solving until the end, which poses a problem with students in the classroom.

This book is very easily navigated. The contents tab and drop down menu allows for the reader to quickly navigate to particular chapters and specific content. The previous and next buttons located at the bottom of the text allows readers to toggle between chapters very quickly. All embedded links work as they should and visual models are clear and understandable. There are no distractors present when trying to navigate the text. There is no index / glossary offered with this text.

The text is free from grammatical errors.

Cultural Relevance rating: 4

This text is not culturally offensive in any way. The final chapter of the text is dedicated to problem based learning and is centered around Voyaging on Hōkūle`a. The text provides embedded links to culturally relevant videos and models that help illustrate the cultural practices of Polynesians.

This textbook has a solid foundation and is well-organized for it's intended audience, the elementary mathematics pre-service teacher. This text will help build conceptual understanding of mathematics that will lead to procedural fluency for teachers. The text also provides clear examples of instructional strategies to be used in today's classrooms. Methods courses for pre-service teachers will find this text extremely useful and easy to incorporate in elementary mathematics methods instruction.

Reviewed by Kane Jessen, Math Instructor, Community College of Aurora on 8/13/20

This textbook is intended to cover the mathematics topics necessary to prepare pre-service elementary education majors to successfully teach K-6 students in the United States in alignment with current Common Core Math Standards. The textbook is a... read more

This textbook is intended to cover the mathematics topics necessary to prepare pre-service elementary education majors to successfully teach K-6 students in the United States in alignment with current Common Core Math Standards. The textbook is a mostly comprehensive collection of K-6 Common Core elementary math topics ranging from non-numerical problem solving through summative PBL assessments incorporating algebra, geometry and authentic problem solving. However, several topics related to K-6 CCSS Standards are not covered or minimally covered. CCSS topics with minimal coverage include set theory, logic, integers, probability, graphing and data analysis. At the beginning of the book, there is an effective and accessible table of contents with links included. However, sections and subsections are labeled only with names and page numbers. The text does not contain an index, glossary or appendices. Chapter summaries and links to previous concepts/problems are not included but would support student learning if included. More visuals and historical explorations would increase comprehensiveness.

Content was found to be accurate, error-free and unbiased.

Relevance/Longevity rating: 5

The language and examples of this text are written with a constructivist and meta-pedagogical voice that is both academic and accessible. The author immediately addresses the importance of CCSS and consistently utilizes the “Exploding Dots” curriculum. The “Exploding Dots” curriculum is a brave and differentiated approach to holistically teaching multi-base mathematics to K-12 students. “Exploding Dots” has been a core focus of K-12 Global Math Project and was pioneered by James Tanton . As future teachers, students can expect to teach “Exploding Dots” or similar CCSS curriculum sometime during their teaching career.

The language of the text is well-written, accessible and clear. Some sections and examples could be expanded for clarity/depth. Prior definitions/review concepts are not consistently linked.

The text is internally consistent in terms of its own terminology, framework and graphics. The “Exploding Dots” infusion helps maintain continuity throughout the text but is not present in all modules.

This text follows the common sequence that many “Mathematics for Elementary Teachers” textbooks commonly follow. The text is organized into eight modules. The text initially builds upon itself without being overly self-referential. The text’s sections, subsections, definitions, axioms and problem banks are all well delineated but lack sections/subsection numbers/identifiers and links to previous concepts/definitions

This textbook has a solid flow and follows a common sequence shared by most for profit “Mathematics for Elementary Teachers” texts. The text is well organized and builds upon itself.

Minimal issues involving interface were observed. Observed interface issues include, one broken video link and unnumbered sections. Definitions and review topics are not linked or referenced with page numbers/sections, however, this creates minimal usability issues. The text contains adequate procedural visuals and also cultural and historical visuals that enhance the student learning experience.

This text is largely free from grammatical errors. Grammatical errors that were observed were minor and non-persistent.

The text is not culturally insensitive or offensive in any way. It consistently uses examples that are inclusive of a variety of races, ethnicities, and backgrounds. Textbook examples often include references to Hawaiin culture. These references are easily understandable and could be readily adapted for students in other places. In an effort to increase relevance, further additions to the text could be made to provide a more equitable and historical focus on women, minorities and problem based learning cross-sectional explorations similar to the Hōkūleʻa section.

This textbook has a solid structure and great flow, I thoroughly enjoyed reviewing this textbook. I am genuinely excited to incorporate Michelle Manes ‘Mathematics for Elementary Teachers’ into my upcoming semester’s curriculum. With subsequent editions and revisions, this textbook will become a wonderful text for students majoring in primary education, especially those who are either lacking in basic math skills or math confidence.

Reviewed by Reina Ojiri, Assistant Professor, Leeward Community College on 7/27/20

The book begins with a reference to the Common Core State Standards (CCSS) for Mathematics and the eight “Mathematical Practices". Though not all states have adopted and/or are currently using the Common Core Standards, with its incorporation at... read more

Comprehensiveness rating: 2 see less

The book begins with a reference to the Common Core State Standards (CCSS) for Mathematics and the eight “Mathematical Practices". Though not all states have adopted and/or are currently using the Common Core Standards, with its incorporation at the beginning of the text I initially thought that the Common Core standards would be revisited consistently throughout the text.

Though the "Think Pair Share" sections are great additions for discussion to the book they do not include common misconceptions or tips for instructors to use to help guide these discussion prompts. The focus on just one type of discussion "Think Pair Share" also does not give future teachers a broader experience with different cooperative learning strategies in the classroom. There are many strategies in addition to “Think Pair Share” that are also great and seeing the same strategy over and over did not provide variation or keep me engaged as I read through the text.

There are a few key concepts that are not included in the text including Measurement & Data and Statistics & Probability.

The text also does not include an effective index and/or glossary. I have found that students do use the index and/or glossary that is typically in the back of the book to help them find information in the text quickly.

Content Accuracy rating: 3

The content is error-free however some of the images included on the PDF version are blurry and hard to read. There does not seem to be consistency between the different readable versions of the text.

There also seems to be a bias for the dots and boxes strategy throughout the text and the content lacks current practices of teaching concepts.

Just like any text, this textbook needs to be updated to match current best practices and research in math education. Since this text is Attribution-ShareAlike which allows “others to remix, adapt, and build upon your work even for commercial purposes, as long as they credit you and license their new creations under the identical terms” it does seem that updates and instructor/course-specific content will be relatively easy to implement as needed.

Clarity rating: 3

This text is written in a way unique way that makes it easier for students to read through and follow. It is very student friendly however might not be as useful as an instructor text since the instructor needs to fill-in-the-blanks on their own.

Consistency rating: 3

The text is written with consistent terminology however the framework for each chapter is not consistent. Some chapters include Explorations and additional sections while others end consistently with a problem bank.

Modularity rating: 3

The text is divided into smaller reading sections however the titles of each section are not easily recognized by students. Though I imagine the titles were meant to be creative for each section, having something more straight forward to make it easier for students to navigate is more important than creativity especially for future teachers who might be teaching these concepts for the first time.

Organization/Structure/Flow rating: 3

It would be good to organize the material consistently throughout the text (e.g.each section should end with a problem bank). The variation in the different sections can be confusing to both the instructor and student when trying to find something in the text.

I also noticed that the online version does not include page numbers while the PDF version does. This is not helpful when referring students to particular sections of the book. The PDF version also has many completely blank pages. I am not sure if this was meant to be on purpose (for printing purposes) but these pages can be very distracting to the reader.

Interface rating: 2

Navigation throughout the text is fine however, there are noticeable differences between the online and PDF versions of the text. The images in the PDF versions are noticeably blurry and lower quality than those in the online version. In some instances, it seems as though images were screenshot and copied and pasted which could account for the image quality.

Some images, in particular, should not have been included at all and are unreadable, for example, the Hokulea on page 441.

I did not notice grammatical errors.

The connection to the Hawaiian culture was a nice touch.

I would use this text as a reference but would not adopt this book as the main text for my class.

Reviewed by Thomas Starmack, Professor, Bloomsburg University of Pennsylvania on 3/26/20

The book is somewhat dated and does not include current research based best practices like concrete, representational, then abstract. Like most authors, they make assumptions that students have the ability to understand abstract and start the... read more

The book is somewhat dated and does not include current research based best practices like concrete, representational, then abstract. Like most authors, they make assumptions that students have the ability to understand abstract and start the lesson there, which is contradictory to how the brain works and what current research says about effective math instruction and learning.

I agree the content is accurate, but in many areas the learner must have a very strong understanding of mathematical concepts, structures, and applications. There lacks current best practice and current NCTM recommendations to approaching the teaching of mathematical content.

Relevance/Longevity rating: 1

Although mathematical concepts at the elementary level remain the same, the approach to engaging students in learning and the methods of instruction have evolved greatly. The book lacks many of the newer approaches and is outdated. The arrangement of the concepts is okay. I would recommend that the big ideas of teaching math are in the beginning and providing an overview of what is mathematics and best approaches to teaching/learning mathematics. Then scaffold the specific concepts. Fractions is one of the most complex and abstract, and this book starts there as a first topic.

Once again, the book is okay in terms of math learning but dated on best practice approaches. The book does not use jargon per say, but does not provide the best approaches for students to learn how to effectively teach mathematics.

Consistency rating: 4

Yes the book is consistent throughout.

The text is divisible, just not relevant to today nor provides current approaches. The order of the content is not in line with a methods of teaching course I would follow.

Organization/Structure/Flow rating: 2

I think the topics are clear but dated and not in the order as described above.

The text provides a variety of interfaces, none of which are confusing for the student who has a very strong math background. The text does mislead students to think starting with abstract is how to instruct elementary students, which is contradictory to brain research and current best practices.

Grammatical Errors rating: 4

I did not notice any grammar errors.

Cultural Relevance rating: 3

I think the text is culturally appropriate. Not certain about the final chapter as it focuses on one population. Having a chapter or theme woven throughout the text that provides students with a stronger understanding that although mathematics is a universal language, there are cultural differences to teaching and learning as evidenced in the 1999 TIMSS report.

The text is outdated. The text is an okay resource but I would not be able to use as the main guide for learning in a college level methods of teaching elementary mathematics course.

Reviewed by Jamie Price, Assistant Professor, East Tennessee State University on 3/20/20

This book introduces the reader to the standards for mathematical practice (SMP) from the Common Core standards in the introduction. I appreciated this as these standards cover all grades and are a unifying theme of the Common Core standards, yet... read more

This book introduces the reader to the standards for mathematical practice (SMP) from the Common Core standards in the introduction. I appreciated this as these standards cover all grades and are a unifying theme of the Common Core standards, yet many times overlooked. In addition, many states, including mine, that are not following Common Core directly have adopted the SMPs. The book does not cover two of the mathematical strands, namely measurement and statistics/data. Among the strands that are covered, however, the author does a thorough job of explaining the content, using a unified theme throughout, such as dots and boxes introduced in place value that appear again in number operations. I particularly liked the final chapter of the book and its connection to Hawaiian culture. The author could easily incorporate ideas related to teaching and learning measurement into this chapter in order to make the book more comprehensive.

The content was very accurate. I did not come across any mathematical errors or biases. The author did a good job of incorporating "think, pair, share" elements throughout each chapter as a model for future teachers. To further guide future teachers, I would have liked to see the author include information in each chapter about common misconceptions students have when learning the related material and ideas on how to address those misconceptions. In my experience, I find that pre-service teachers are unaware of these misconceptions and it is helpful to make them aware of them so that they can anticipate them in their own classrooms.

The content presented in this book is up-to-date and will remain relevant for a long time. Due to the fact that this book focuses more on content rather than methods, I do not foresee a need for many updates moving forward.

The book is written in a very clear and concise way that is approachable to future and current elementary teachers. The author presents key words in bold throughout the book to draw attention to them. I liked the way that the author included videos as well as written explanations of ideas, such as in the Number and Operations chapter, section titled Addition: Dots and Boxes. The author explains, in words, how to use this method to add multi-digit numbers and follows the written example with a video explanation. This helps to reach a variety of learners and learning styles. The author also addresses common "jargon" associated with particular mathematical concepts, such as proper and improper fractions (section titled What is a Fraction?), and discusses how this jargon can be misleading for students.

Each chapter in the book includes an introduction, multiple opportunities for think-pair-share discussions, and several problem sets to practice. I appreciated the consistency in the Dots and Boxes method introduced in the Place Value chapter and then carried into the Number and Operations chapter.

The book uses a modular approach to present the material. Each module contains numerous sections that help to break up the content into smaller chunks so that the content does not seem overwhelming. The modules are set up in an order that makes sense for the mathematics, but a reader could begin reading at any module and still make sense of the content.

The organization of the topics makes sense according to the mathematics presented and is logical.

I did not find anything distracting or confusing in relation to the interface of the text. The book was easy to navigate, with a clearly defined table of contents. I was able to easily click through the various modules and sections within each module. The book uses figures well to provide engagement to the reader as well as to further clarify content. The use of videos embedded within the modules helps to strengthen understanding of the content. It did take me a minute to find the navigation link that allowed me to move to the next section in a module (right arrow at bottom right corner of the page), but once I found it I was able to navigate seamlessly to each subsequent section.

I did not find any grammatical errors in the text.

In my opinion, this was one of the biggest strengths of this text. The author did a nice job of incorporating Hawaiian culture into the text. For example, the author includes an image in the Place Value chapter (Number Systems section) that references the use of tally marks on a sign at Hanakapiai Beach. In addition, a full chapter was devoted to Voyaging on Hōkūle`a. I particularly liked how the author connected this idea to beginning teaching of elementary mathematics and encouraged future teachers to think about ways to see mathematics outside of traditional mathematical settings.

I am glad that I came across this resource. I primarily teach math methods courses for elementary pre-service teachers, but I found many aspects of this text that I can incorporate into my classes to help students think more deeply about the mathematics that they will teach. I appreciated the author's attempt to challenge students in their thinking about elementary mathematics. Initially, I was surprised to find that there was no "answer key" provided for the many problem sets that were included throughout the text. After reading the quote presented on the introductory page to the Problem Solving chapter, I realized that this may have been an intentional decision made by the author to encourage readers to go beyond "a trail someone else has laid." I find that many pre-service elementary teachers want to "just know the answer" when it comes to mathematics; a no answer key approach will encourage discussion and justification, two elements important to ensuring equity in the teaching and learning of mathematics.

Reviewed by Shay Kidd, Assistant Professor- Mathematics Education, University of Montana - Western on 12/30/19

The content that elementary teachers need to have that is not covered in this book is graphing, probability, statistics, exponents, visual displays of data. The coverage of operations is very specific in the examples and does not cover the wide... read more

The content that elementary teachers need to have that is not covered in this book is graphing, probability, statistics, exponents, visual displays of data. The coverage of operations is very specific in the examples and does not cover the wide range that should be presented in this type of text.

Content Accuracy rating: 4

While the core topics presented are correct, the number of problems that are provided without any solutions is alarming. The majority of problems that are provided are meant for the reader to perform but do not provide any type of answer key for checking the work. In this way, the book seems to assume the reader to have a solid knowledge of the topics already and this book discusses a few different approaches to these topics.

The specific content presented is up-to-date and usable.

The book's prose seem to be more of a teaching guide than a textbook. This is nice for the conversational aspect that a reader may want in their learning, but should be explained more or possibly a change of title for the book. Something more like "Exploring the concepts of Elementary Mathematics" would provide a more reading friendly approach the book offers.

The author has a consistent voice of teaching and presenting the material.

Modularity rating: 2

The break-up of the text with boxes is difficult to follow the purpose of each box. While some of the box styles are clear, such as the think, pair, share or problem boxes, others seem to break up the line of discussion. A problem box may be discussed more directly immediately following the box and the presentation of the problem. Most of the problem boxes are not discussed again in the main text. This cased issues for wanting to read with a specific purpose. When the reader wants to understand a problem more, there is generally not more discussion, but unclear about when that would be provided or not. Other times boxes were used without any "box type" provided and these were just to break up the flow of the text.

Place value was a major topic to start the book and had good coverage, then operations and fractions were discussed, then a return to place value with decimals. It would seem that a connection of place value and decimals would work better to follow the other place value discussion.

Interface rating: 3

There are several pages that have large blank parts or are totally blank. This may be due to the PDF version that I chose. When I did use the internet-connected version, there seems to be a dependence on youtube to help do some of the teaching.

There are a few minor issues that would be resolved with a good proofread.

The book does seem to be written with the Hawaiian culture in mind. This may be difficult for other cultures to connect to or understand but does not present any insensitivities.

The book's title suggests a full discussion of the topics that elementary education pre-service teachers would need to know and teach, but this book is very lacking in the topics required for this. I selected this book to review because I teach classes that would use the textbook, but I would not use this textbook as is. There are a few topics that I plan to add to my own instruction, but the book as a whole needs additional help to be able to stand alone. This really appears to be a teaching guide based on the constant think-pair-share setup. This also is a specific teaching and method that seems to require the students to already have much of the content mastered. It does not teach all the content that is required to the level of the discussion had.

Reviewed by Ryan Nivens, Associate Professor, East Tennessee State University on 10/25/19

The book covers all the expected mathematical strands except for measurement and data/stats. There are some obvious connections to the strands of mathematical practice from the Common Core standards. While the abstract specifically lists MP1, MP2,... read more

The book covers all the expected mathematical strands except for measurement and data/stats. There are some obvious connections to the strands of mathematical practice from the Common Core standards. While the abstract specifically lists MP1, MP2, and MP3, the introduction clearly lists all 8. The chapter "Voyaging on Hōkūle`a" contains activities that will require use of measurement and units, but there is no explanation on how measurement topics should be taught or approached. However, this chapter does provide a good project-based learning set of materials, and is an exceptional resource for navigation. The book also includes a chapter on Problem Solving, which is important for those students who must complete the EdTPA and address the 3rd subject specific emphasis area. All embedded links to Youtube videos or Vimeo videos are working and play within the textbook pages.

I find the mathematics to be entirely accurate. There are many teaching strategies, such as "think pair share" that are found throughout the chapters. This is particularly helpful for future teachers.

This book should last a very long time in terms of relevance.

This book is very clear, with mathematical words in bold and proper definitions provided. The text also addresses common math classroom jargon. For an excellent example of this, see the heading "What is a Fraction" in the chapter on Fractions. Toward the bottom is a sub-heading "Jargon: Improper Fractions" that has students consider the usefulness of proper and improper fractions.

This book is consistently laid out, with multiple examples, problems to try, and diagrams to support the transfer of information.

This book is entirely modular. You can pick it up, and easily start in any chapter and not be lost. The heading, subheading, use of italics and boldface make it easy to locate information. As a mathematics education book, this is quite nice.

A mathematician wrote this, the layout is logical without question.

The book is extremely easy to navigate, with a logical structure to the table of contents that you can easily click through. A drop down menu in the upper left corner allows you to view the outline of the book while still viewing a page, and you can collapse/expand chapters within the menu.

The many figures that are present throughout the textbook are perfectly displayed and fit the reading material.

There is nothing I find distracting in the layout and interface.

I could not find any errors.

An entire chapter is dedicated to Voyaging on Hōkūle`a, with exceptional videos and diagrams to illustrate the cultural practices of the early Polynesians.

I was excited to find this book in the Open Educational Resources library. As a professor who frequently teaches methods courses in mathematics for elementary teachers, I feel that this book may be a terrific book to use to replace previous texts that I've adopted. I would like to see a chapter on Measurement to make the Voyaging on Hōkūle`a chapter more useful. It is obvious from the first page you open to that this book was well planned and thought out. I'm impressed.

Reviewed by Monica Rose Gilmore, Graduate Student, CU Boulder on 7/1/19

This textbook goes into depth about different mathematical concepts that are important for elementary school teachers to understand in teaching mathematics. However, the text is missing a focus on statistics and probability, which are key areas of... read more

Comprehensiveness rating: 3 see less

This textbook goes into depth about different mathematical concepts that are important for elementary school teachers to understand in teaching mathematics. However, the text is missing a focus on statistics and probability, which are key areas of focus in elementary math classrooms. The text is also missing an index or glossary but does define new terms as they are introduced.

The content, mathematical diagrams and depictions are accurate and error-free. Each chapter also accurately shows various ways to understand mathematical concepts. However, the diagrams are geared towards an audience that already has some understanding of advanced mathematics.

The content is organized in a way that necessary updates would be straightforward to implement. More specifically, much of the content reflects current mathematical practices and activities endorsed by up-to-date research in mathematics education.

The text is written in accessible prose and provides context for jargon and technical terminology. Additionally, the text clearly separates different terms for different strategies and concepts. For example, in the Problem Solving Strategy section, the interface is divided into different strategies for the reader to explore. This is helpful in keeping new concepts and strategies organized for the reader.

The text is written with consistent terminology. More specifically, the text consistently gives examples of what concepts are called by mathematicians and teachers. This is helpful for pre-service teachers that might be teaching mathematical concepts and strategies for the first time.

The text is easily divided into smaller reading sections. These sections include not only explanations of mathematical concepts, but also theorems, activities and diagrams which can be referenced by the teacher at any point. Also, the text gives teachers ideas for activities and additional problems to try with students.

Though the topics in the texts are presented in a logical, clear fashion, it might be beneficial for pre-service or elementary teachers to see how to specifically scaffold the different concepts within those topics for elementary students at different grade levels. Additionally, the text could also demonstrate how students typically confuse topics so teachers and pre-service teachers are prepared to navigate new concepts for the class.

The interface is easy to navigate since the content clearly outlines chapters and the topics within them. Sections such as notation and vocabulary, think pair shares and theorems are clearly outlined, organized and conceptually scaffolded. However, it might be helpful to have an index so the reader does not have to click within each topic to find the concept they are exploring.

This text is free from grammatical errors.

This text is not culturally insensitive or offensive and includes examples from the Hawaiian culture. Though the text is mainly made up of mathematical explanations, there are a variety of people's names in different problems that could be attributed to a variety of cultures. Additionally, the text reflects Polya's advice (1945) to try adapt the problem until it makes sense. Though the text includes mainly mathematical explanations, it does call for adapting problems which could potentially be applied to a variety of students of different backgrounds.

Reviewed by Glenna Gustafson, Professor, Radford University on 5/22/19

This is book is fairly comprehensive and I feel could be used by most foundational courses in elementary mathematics. The structure and writing provide a good foundation for students learning the "why" behind the mathematics and becoming... read more

This is book is fairly comprehensive and I feel could be used by most foundational courses in elementary mathematics. The structure and writing provide a good foundation for students learning the "why" behind the mathematics and becoming mathematical thinkers. There were some areas that could possibly use more development. In geometry for example there was no discussion of perimeter, area, and volume. Estimation, measurement of weight, time, and probability also appears to be missing. The text is well organized and written so that the chapters do not have to be completed in the order in which they are presented. While there is not index or glossary, the author uses colored text boxes to explain specific content or terms.

The content of the text is accurate and represented in a variety of formats to support learning, Not only does it provide solutions to problems, but also the mathematical thinking behind those solutions.

The text is very relevant for K-6 elementary pre-service teachers. It would be beneficial to know the specific grade levels that the author considers as "elementary" since this does vary by location. The content is "standard" for most elementary math courses and would not need to be updated often and the consistent layout and formation would make changes easy to make.

The text is written in a conversational tone. The simplicity and straight-forwardness of the text should appeal to those students that have sometimes been overwhelmed by writing in more traditional math texts.

The text is organized consistently from chapter to chapter. The table of contents and chunking of content in the chapters is logical and clear, Each chapter includes graphics as well as sections for: Think-Pair-Share; Definitions; Theorems (when appropriate); and, Problems. This consistent structure makes navigation easy.

The table of contents and chunking of content in the chapters is logical and clear. This also makes it easy to not necessary to move sequentially through the text, but to have the option of reviewing or using only needed topics. Subtitles and graphic captioning are appropriate for the content.

The text is easy to read and the organization within each chapters makes navigation easy.

This text is easy to navigate. The inclusion of graphics, charts, photos, and videos support learning. There are several pages where graphics in the Geometry chapter are skewed in the PDF version, but this does not seem to be a problem in the online version, Not all of the video links work within the PDF version.

There were no obvious grammatical errors. Several of the errors that were found were typos and/or word omissions.

The text is culturally inclusive. One thing that should be noted is that it seems male names are over-represented in the Problem sections. A reference to Hawaiian culture and life is evident. The Hōkūle`a voyage found in the last chapter is a good example of problem based learning and the integration of math with other subject areas.

This would be a wonderful text to use as a supplement or compliment to an elementary math methods course. It is not as overwhelming as other math texts, and would provide pre-service teachers with a good foundational review of math concepts, including vocabulary and some pedagogy.

Reviewed by Karise Mace, Mathematics Instructor, Kuztown University on 5/16/19, updated 11/9/20

This book is fairly comprehensive for a one-semester course, although it does not include much detail about several topics. The section on number systems barely touches on Roman numerals and only mentions Mayan and Babylonian counting systems.... read more

This book is fairly comprehensive for a one-semester course, although it does not include much detail about several topics. The section on number systems barely touches on Roman numerals and only mentions Mayan and Babylonian counting systems. The sections on addition, subtraction, and division would be more robust if the author included other algorithms for these operations. The chapter on Geometry does not address perimeter, area, surface area and volume. The book does not include an index or glossary.

While the book is not error free, it is unbiased. Most of the errors seem to be typographical and/or related to web links or LaTeX. In the section on number systems, the author incorrectly explains how one million would be represented using Roman numerals and incorrectly claims that the Mayans did not use a symbol for zero. Further, the Mayan number system was not a true vigesimal system, as the text indicates.

This text uses a constructivist approach to help students build their understanding of the mathematics included in the book. It is well organized and written so that the chapters do not have to be completed in the order in which they are presented. Because of this, the text should be easy to update. When concepts that are presented earlier in the text are used in later chapters, the author includes a brief but thorough review that would allow students to understand the later chapter even if they had not read and completed the problems in the earlier chapter. The "dots and boxes" approach is timely, as it uses the idea of the "exploding dots" that are part of the Global Math Project (https://www.globalmathproject.org).

The textbook is clearly written and enjoyable to read...even for the math-phobic student. The tone is conversational and is even funny at times. The author defines important mathematical terminology in a way that is both mathematically accurate and accessible to students. The chapter on problem solving is fantastic and really gives students insight into how to think and problem solve like a mathematician. The pies per child model for fractions is not the most effective model for helping students understand fractions and this part of the text would be improved if the author replaced this type of modeling with pattern block modeling.

Overall, the text is consistent in its chapter structure and terminology use. However, there is inconsistent notation when using "dots and boxes."

The text is well-organized but can be reorganized in order to suit an instructor's preference. However, it would be best to complete the chapter on problem solving first, as it sets the stage for the rest of the book. Most of the chapters are structured more like an activity book with lots of great problems and thought provoking questions that will help students think deeply about the mathematical concepts being presented. With the exception of the chapter on problem solving, there is not a whole lot of text for students to read.

Although the topics presented could be reorganized to meet student needs, the order in which they are presented is logical and clear.

With only a few exceptions, the images it the text are clear. In the section titled "Careful Use of Language in Mathematics: =" some of the scale images need to be modified so that the items on the scale appear to actually sit on the pans. The same issue occurs in the section titled "Structural and Procedural Algebra." Some of the images in the sections titled "Platonic Solids" and "Symmetry" spill off of the page. The image that appears on page 89 and then again on page 144 would be more clear if a different font was used to label the line segments.

No grammatical errors were noted. However, there were a few typographical errors that could cause confusion for students as on page 219.

The text was culturally sensitive and nothing offensive was noted. As the focus of the text is purely mathematical, there are not many cultural references at all, unless they are references to historical cultures. The author does use names for hypothetical students that are diverse and represent a variety of ethnicities. The last chapter is an integrated unit that focuses on the Hawaiian culture. Unfortunately, the links and web addresses in this chapter do not work and/or are no longer active.

The book includes three sections at the end of the problem solving chapter in which the author articulately explains the language that mathematicians use to succinctly and precisely explain their problem solving and solutions. These sections will help students who may not think of themselves as mathematicians learn to think like mathematicians. So many mathematics textbooks are full of exercises but no true problems. On the other hand, this text is full of wonderful problem solving and critical thinking problems that are embedded in the sections as well as in the problem banks. The author also includes many "Think/Pair/Share" exercises and questions that will facilitate mathematical thinking and conversation among students. The constructivist approach used by the author will help students build deep understanding about the mathematics covered in the text. While there is some room for revision and improvement, this is a very good text to use with elementary education majors, and I definitely plan to use this book the next time I teach them.

Reviewed by Desley Plaisance, Associate Professor, Nicholls State University on 4/29/19

This textbook seems to be appropriate for the first course typically taught for elementary teachers which usually includes topics of problem solving, place value, number and operations. Most books are able to be used for a second course which... read more

This textbook seems to be appropriate for the first course typically taught for elementary teachers which usually includes topics of problem solving, place value, number and operations. Most books are able to be used for a second course which focuses on geometry. This book could not be used for the second course.

Content seems to be accurate.

Topics are somewhat static for a course like this, so the textbook will not become obsolete within a short period of time.

Appears to be clear.

The flow from topic to topic is consistent in presentation.

Divided into clear sections.

Topics are presented logically and in a similar order to most books of this type.

Easy to navigate with clear images and other items such as tables.

Book is written in simple language and appears to be free of grammatical errors.

Appears to be culturally diverse.

This book could definitely be used for a first course of elementary math for teachers with the teacher providing resources. As with many open books, the print and layout is very simple without cluttering pages with unnecessary items.

Reviewed by Lisa Cooper, Assistant Professor, LSUS on 4/26/19

This text covers many concepts appropriately; however, a few concepts are missing, such as; data analysis and statistics. For more than ten years, data-driven instruction has been a major focus in education along with many other uses. This text... read more

This text covers many concepts appropriately; however, a few concepts are missing, such as; data analysis and statistics. For more than ten years, data-driven instruction has been a major focus in education along with many other uses. This text has a table of contents but not an index and/or glossary; however, does define words in chapters when needed.

The content is well organized and accurate. Multiple representations and diverse examples are provided throughout the text which supports an unbiased approach to those entering elementary education.

The text is quite relevant to the classroom today, incorporating such resources as YouTube, varied strategies to promote differentiated instruction, scaffolding between concepts, and problem-solving opportunities. Some states may find issues with Common Core standards being addressed; however, mathematical practices could be interchanged with the "standards."

The text is written free from educational jargon; it is straightforward and easy to understand.

The text is consistent in its structure; color is not distracting, problems, strategies, diagrams, charts, and definitions are provided throughout.

The text is appealing with the page layout; it's not too busy or distracting. Colors are attractive and text is broken down into appropriate amounts.

The text has a well-organized flow with the layout of each topic/chapter.

The text has charts, pictures, diagrams, real-world examples throughout; several different versions of the text are offered too.

No grammatical errors were observed in my review of the text.

The text provides a variety of backgrounds, races, and ethnicities while providing learning experiences and pedagogical approaches to support student engagement and learning.

Reviewed by Demetrice Smith-Mutegi, Instructor/Coordinator, Marian University on 3/6/19

This text covers place value, numbers and operations, fractions, patterns, algebraic thinking, decimals, and geometry. However, elementary teachers are expected to also know and understand statistics and probability. This text does not address... read more

This text covers place value, numbers and operations, fractions, patterns, algebraic thinking, decimals, and geometry. However, elementary teachers are expected to also know and understand statistics and probability. This text does not address this mathematical concept.

The text makes non-traditional, yet, accurate representations of mathematical concepts. In some sections, different solutions are presented and explained. This eliminates bias and provides a diverse representation of ideas when solving math problems.

The text is representative of common core problem-solving standards, however, it does require mathematical knowledge beyond elementary school. The problem-solving nature of the text is very relevant to elementary pre-service and in-service teachers (the audience for the book).

The text language is clear and accessible. There is a section on terminology, which is very helpful. Additional diagrams would help to improve the clarity in some cases.

I was expecting to see videos embedded throughout, after seeing them in the first section. It would be great to have a consistent format throughout the text, however, I understand that it is not always feasible to do so. There were other obvious and clear patterns presented, color-coded sections (think/pair/share), problems, examples.

The chapters and subchapters can be easily accessed, breaking the material into smaller sections.

The topics were presented in a logical, clear fashion, however, not all of the chapters would end with a problem bank. In some cases, there were additional sections after the problem bank. It would be great if each section included key objectives or goals of the section.

The text comes in pdf, XML, and an online web version. The search feature on the online version was a valuable addition.

I did not observe any obvious grammatical errors.

Cultural awareness was very obvious in this text. While it was more relevant to Hawaiian culture, it also included cultural awareness of other cultures and backgrounds.

Overall, this text assumes that the student has successfully completed mathematics through basic calculus. There should be more support in this area, as some elementary math students are not prepared to complete problems with this focus.

This a great "discussion" text.

Reviewed by Kandy Noles Stevens, Assistant Professor of Education, Southwest Minnesota State University on 12/28/18

This text covers the areas applicable to elementary mathematics extremely well (with the exception of omitting probability and data analysis) and provides graphically visual boxes within the text to define terms and instructional strategies. ... read more

This text covers the areas applicable to elementary mathematics extremely well (with the exception of omitting probability and data analysis) and provides graphically visual boxes within the text to define terms and instructional strategies. Additionally, the text provides thinking routines that support understanding more than just the concept, but also, the how's and why's of conceptual understanding.

The content is accurate and organized in a way to supports student learning for those training to become elementary teachers of mathematics.

The content of the book is relevant to today's elementary classroom in that it provides future elementary educators with the content knowledge, but also pedagogical approaches that would support student learning. Additionally, the text is organized in a way that is consistent and provides scaffolding support for those who might struggle with any one of the concepts. For Minnesota standards the only item of note is that there is not a section devoted to probability or data analysis, but the latter is touched upon in other chapters. There are three mentions of the Common Core standards in the text. Minnesota is not an adopter of the CC mathematics standards, but the references to the CCSS are in regards to the practices of mathematics and not on standards specifically.

While a great text for training future math teachers, this book does not read as a "typical" mathematics textbook. Students who have struggled in the past with mathematics might find the authors' writing style to be approachable and accessible for all levels of mathematics competence and confidence.

The text is consistent in its terminology and the structure of the framework is uniform throughout, relying on supporting student learning through exercises, think-pair-share activities, and continuous dialog and reflection.

A majority of the chapters begin with a section that introduces the strand of elementary mathematics covered. Not all chapters have this introduction which may pose challenging to interrupt the mathematical progression of some established courses.

The text is very well organized and has an easy-to-read format and flow.

The text is graphically rich with succinct advanced organizers, diagrams, and photos to support learning.

The text is written with professional level writing and is free of grammatical errors.

A variety of races, ethnicities, and backgrounds are present in the exercises used to support student learning throughout. The end of the text involves a Hōkūle`a voyage as a part of a problem-based learning (integrated curriculum) experience. This was something that really made this text stand out in that it gave future elementary teachers an example of using mathematical concepts in authentic (and exciting) learning experiences. This Polynesian voyage would provide many students with an introduction to life culturally different from their own.

I have been a STEM educator for more than two decades and I come from a long line of mathematics educators. While wrapping up my reading of this text, I happened to have my father (a 46 year veteran mathematics educator) here visiting. I shared the text with him and several times I heard him utter, "I like the way this problem is set up". We both found the book to be very knowledgeable for mathematical conceptual understandings, but even more so for introducing ideas for instructional strategies and classroom discourse to help future teachers become equipped with speaking the "language of mathematics" to guide their future students.

Table of Contents

I. Problem Solving

• Introduction
• Problem or Exercise?
• Problem Solving Strategies
• Beware of Patterns!
• Problem Bank
• Careful Use of Language in Mathematics
• Explaning Your Work
• The Last Step

II. Place Value

• Dots and Boxes
• Other Rules
• Binary Numbers
• Other Bases
• Number Systems
• Even Numbers
• Exploration

III. Number and Operations

• Addition: Dots and Boxes
• Subtration: Dots and Boxes
• Multiplication: Dots and Boxes
• Division: Dots and Boxes
• Number Line Model
• Area Model for Multiplication
• Properties of Operations
• Division Explorations

IV. Fractions

• What is a Fraction?
• The Key Fraction Rule
• Adding and Subtracting Fractions
• What is a Fraction? Revisited
• Multiplying Fractions
• Dividing Fractions: Meaning
• Dividing Fractions: Invert and Multiply
• Dividing Fractions: Problems
• Fractions involving zero
• Egyptian Fractions
• Algebra Connections
• What is a Fraction? Part 3

V. Patterns and Algebraic Thinking

• Borders on a Square
• Careful Use of Language in Mathematics: =
• Growing Patterns
• Matching Game
• Structural and Procedural Algebra

VI. Place Value and Decimals

• Review of Dots & Boxes Model
• Division and Decimals
• More x -mals
• Terminating or Repeating?
• Operations on Decimals
• Orders of Magnitude

VII. Geometry

• Triangles and Quadrilaterals
• Platonic Solids
• Painted Cubes
• Geometry in Art and Science

VIII. Voyaging on Hokule?a

• Worldwide Voyage

Ancillary Material

About the book.

This book will help you to understand elementary mathematics more deeply, gain facility with creating and using mathematical notation, develop a habit of looking for reasons and creating mathematical explanations, and become more comfortable exploring unfamiliar mathematical situations. The primary goal of this book is to help you learn to think like a mathematician in some very specific ways. You will: • Make sense of problems and persevere in solving them. You will develop and demonstrate this skill by working on difficult problems, making incremental progress, and revising solutions to problems as you learn more. • Reason abstractly and quantitatively. You will demonstrate this skill by learning to represent situations using mathematical notation (abstraction) as well as creating and testing examples (making situations more concrete). • Construct viable arguments and critique the reasoning of others. You will be expected to create both written and verbal explanations for your solutions to problems. The most important questions in this class are “Why?” and “How do you know you're right?” Practice asking these questions of yourself, of your professor, and of your fellow students. Throughout the book, you will learn how to learn mathematics on you own by reading, working on problems, and making sense of new ideas on your own and in collaboration with other students in the class.

About the Contributors

Michelle Manes, Associate Professor, Department of Mathematics, University of Hawaii

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Unit 1: Algebra foundations

Unit 2: solving equations & inequalities, unit 3: working with units, unit 4: linear equations & graphs, unit 5: forms of linear equations, unit 6: systems of equations, unit 7: inequalities (systems & graphs), unit 8: functions, unit 9: sequences, unit 10: absolute value & piecewise functions, unit 11: exponents & radicals, unit 12: exponential growth & decay, unit 13: quadratics: multiplying & factoring, unit 14: quadratic functions & equations, unit 15: irrational numbers, unit 16: creativity in algebra.

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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120 Math Word Problems To Challenge Students Grades 1 to 8

Written by Marcus Guido

Hey teachers! 👋

Use Prodigy to spark a love for math in your students – including when solving word problems!

• Teaching Tools
• Subtraction
• Multiplication
• Mixed operations
• Ordering and number sense
• Comparing and sequencing
• Physical measurement
• Ratios and percentages
• Probability and data relationships

You sit at your desk, ready to put a math quiz, test or activity together. The questions flow onto the document until you hit a section for word problems.

A jolt of creativity would help. But it doesn’t come.

Whether you’re a 3rd grade teacher or an 8th grade teacher preparing students for high school, translating math concepts into real world examples can certainly be a challenge.

This resource is your jolt of creativity. It provides examples and templates of math word problems for 1st to 8th grade classes.

There are 120 examples in total.

The list of examples is supplemented by tips to create engaging and challenging math word problems.

120 Math word problems, categorized by skill

Addition word problems.

Best for: 1st grade, 2nd grade

1. Adding to 10: Ariel was playing basketball. 1 of her shots went in the hoop. 2 of her shots did not go in the hoop. How many shots were there in total?

2. Adding to 20: Adrianna has 10 pieces of gum to share with her friends. There wasn’t enough gum for all her friends, so she went to the store to get 3 more pieces of gum. How many pieces of gum does Adrianna have now?

3. Adding to 100: Adrianna has 10 pieces of gum to share with her friends. There wasn’t enough gum for all her friends, so she went to the store and got 70 pieces of strawberry gum and 10 pieces of bubble gum. How many pieces of gum does Adrianna have now?

4. Adding Slightly over 100: The restaurant has 175 normal chairs and 20 chairs for babies. How many chairs does the restaurant have in total?

5. Adding to 1,000: How many cookies did you sell if you sold 320 chocolate cookies and 270 vanilla cookies?

6. Adding to and over 10,000: The hobby store normally sells 10,576 trading cards per month. In June, the hobby store sold 15,498 more trading cards than normal. In total, how many trading cards did the hobby store sell in June?

7. Adding 3 Numbers: Billy had 2 books at home. He went to the library to take out 2 more books. He then bought 1 book. How many books does Billy have now?

8. Adding 3 Numbers to and over 100: Ashley bought a big bag of candy. The bag had 102 blue candies, 100 red candies and 94 green candies. How many candies were there in total?

Subtraction word problems

Best for: 1st grade, second grade

9. Subtracting to 10: There were 3 pizzas in total at the pizza shop. A customer bought 1 pizza. How many pizzas are left?

10. Subtracting to 20: Your friend said she had 11 stickers. When you helped her clean her desk, she only had a total of 10 stickers. How many stickers are missing?

11. Subtracting to 100: Adrianna has 100 pieces of gum to share with her friends. When she went to the park, she shared 10 pieces of strawberry gum. When she left the park, Adrianna shared another 10 pieces of bubble gum. How many pieces of gum does Adrianna have now?

Practice math word problems with Prodigy Math

Join millions of teachers using Prodigy to make learning fun and differentiate instruction as they answer in-game questions, including math word problems from 1st to 8th grade!

12. Subtracting Slightly over 100: Your team scored a total of 123 points. 67 points were scored in the first half. How many were scored in the second half?

13. Subtracting to 1,000: Nathan has a big ant farm. He decided to sell some of his ants. He started with 965 ants. He sold 213. How many ants does he have now?

14. Subtracting to and over 10,000: The hobby store normally sells 10,576 trading cards per month. In July, the hobby store sold a total of 20,777 trading cards. How many more trading cards did the hobby store sell in July compared with a normal month?

15. Subtracting 3 Numbers: Charlene had a pack of 35 pencil crayons. She gave 6 to her friend Theresa. She gave 3 to her friend Mandy. How many pencil crayons does Charlene have left?

16. Subtracting 3 Numbers to and over 100: Ashley bought a big bag of candy to share with her friends. In total, there were 296 candies. She gave 105 candies to Marissa. She also gave 86 candies to Kayla. How many candies were left?

Multiplication word problems

Best for: 2nd grade, 3rd grade

17. Multiplying 1-Digit Integers: Adrianna needs to cut a pan of brownies into pieces. She cuts 6 even columns and 3 even rows into the pan. How many brownies does she have?

18. Multiplying 2-Digit Integers: A movie theatre has 25 rows of seats with 20 seats in each row. How many seats are there in total?

19. Multiplying Integers Ending with 0: A clothing company has 4 different kinds of sweatshirts. Each year, the company makes 60,000 of each kind of sweatshirt. How many sweatshirts does the company make each year?

20. Multiplying 3 Integers: A bricklayer stacks bricks in 2 rows, with 10 bricks in each row. On top of each row, there is a stack of 6 bricks. How many bricks are there in total?

21. Multiplying 4 Integers: Cayley earns $5 an hour by delivering newspapers. She delivers newspapers 3 days each week, for 4 hours at a time. After delivering newspapers for 8 weeks, how much money will Cayley earn?

Division word problems

Best for: 3rd grade, 4th grade, 5th grade

22. Dividing 1-Digit Integers: If you have 4 pieces of candy split evenly into 2 bags, how many pieces of candy are in each bag?

23. Dividing 2-Digit Integers: If you have 80 tickets for the fair and each ride costs 5 tickets, how many rides can you go on?

24. Dividing Numbers Ending with 0: The school has $20,000 to buy new computer equipment. If each piece of equipment costs $50, how many pieces can the school buy in total?

25. Dividing 3 Integers: Melissa buys 2 packs of tennis balls for $12 in total. All together, there are 6 tennis balls. How much does 1 pack of tennis balls cost? How much does 1 tennis ball cost?

26. Interpreting Remainders: An Italian restaurant receives a shipment of 86 veal cutlets. If it takes 3 cutlets to make a dish, how many cutlets will the restaurant have left over after making as many dishes as possible?

Mixed operations word problems

27. Mixing Addition and Subtraction: There are 235 books in a library. On Monday, 123 books are taken out. On Tuesday, 56 books are brought back. How many books are there now?

28. Mixing Multiplication and Division: There is a group of 10 people who are ordering pizza. If each person gets 2 slices and each pizza has 4 slices, how many pizzas should they order?

29. Mixing Multiplication, Addition and Subtraction: Lana has 2 bags with 2 marbles in each bag. Markus has 2 bags with 3 marbles in each bag. How many more marbles does Markus have?

30. Mixing Division, Addition and Subtraction: Lana has 3 bags with the same amount of marbles in them, totaling 12 marbles. Markus has 3 bags with the same amount of marbles in them, totaling 18 marbles. How many more marbles does Markus have in each bag?

Ordering and number sense word problems

31. Counting to Preview Multiplication: There are 2 chalkboards in your classroom. If each chalkboard needs 2 pieces of chalk, how many pieces do you need in total?

32. Counting to Preview Division: There are 3 chalkboards in your classroom. Each chalkboard has 2 pieces of chalk. This means there are 6 pieces of chalk in total. If you take 1 piece of chalk away from each chalkboard, how many will there be in total?

33. Composing Numbers: What number is 6 tens and 10 ones?

34. Guessing Numbers: I have a 7 in the tens place. I have an even number in the ones place. I am lower than 74. What number am I?

35. Finding the Order: In the hockey game, Mitchell scored more points than William but fewer points than Auston. Who scored the most points? Who scored the fewest points?

Fractions word problems

Best for: 3rd grade, 4th grade, 5th grade, 6th grade

36. Finding Fractions of a Group: Julia went to 10 houses on her street for Halloween. 5 of the houses gave her a chocolate bar. What fraction of houses on Julia’s street gave her a chocolate bar?

37. Finding Unit Fractions: Heather is painting a portrait of her best friend, Lisa. To make it easier, she divides the portrait into 6 equal parts. What fraction represents each part of the portrait?

38. Adding Fractions with Like Denominators: Noah walks ⅓ of a kilometre to school each day. He also walks ⅓ of a kilometre to get home after school. How many kilometres does he walk in total?

39. Subtracting Fractions with Like Denominators: Last week, Whitney counted the number of juice boxes she had for school lunches. She had ⅗ of a case. This week, it’s down to ⅕ of a case. How much of the case did Whitney drink?

40. Adding Whole Numbers and Fractions with Like Denominators: At lunchtime, an ice cream parlor served 6 ¼ scoops of chocolate ice cream, 5 ¾ scoops of vanilla and 2 ¾ scoops of strawberry. How many scoops of ice cream did the parlor serve in total?

41. Subtracting Whole Numbers and Fractions with Like Denominators: For a party, Jaime had 5 ⅓ bottles of cola for her friends to drink. She drank ⅓ of a bottle herself. Her friends drank 3 ⅓. How many bottles of cola does Jaime have left?

42. Adding Fractions with Unlike Denominators: Kevin completed ½ of an assignment at school. When he was home that evening, he completed ⅚ of another assignment. How many assignments did Kevin complete?

43. Subtracting Fractions with Unlike Denominators: Packing school lunches for her kids, Patty used ⅞ of a package of ham. She also used ½ of a package of turkey. How much more ham than turkey did Patty use?

44. Multiplying Fractions: During gym class on Wednesday, the students ran for ¼ of a kilometre. On Thursday, they ran ½ as many kilometres as on Wednesday. How many kilometres did the students run on Thursday? Write your answer as a fraction.

45. Dividing Fractions: A clothing manufacturer uses ⅕ of a bottle of colour dye to make one pair of pants. The manufacturer used ⅘ of a bottle yesterday. How many pairs of pants did the manufacturer make?

46. Multiplying Fractions with Whole Numbers: Mark drank ⅚ of a carton of milk this week. Frank drank 7 times more milk than Mark. How many cartons of milk did Frank drink? Write your answer as a fraction, or as a whole or mixed number.

Decimals word problems

Best for: 4th grade, 5th grade

47. Adding Decimals: You have 2.6 grams of yogurt in your bowl and you add another spoonful of 1.3 grams. How much yogurt do you have in total?

48. Subtracting Decimals: Gemma had 25.75 grams of frosting to make a cake. She decided to use only 15.5 grams of the frosting. How much frosting does Gemma have left?

49. Multiplying Decimals with Whole Numbers: Marshall walks a total of 0.9 kilometres to and from school each day. After 4 days, how many kilometres will he have walked?

50. Dividing Decimals by Whole Numbers: To make the Leaning Tower of Pisa from spaghetti, Mrs. Robinson bought 2.5 kilograms of spaghetti. Her students were able to make 10 leaning towers in total. How many kilograms of spaghetti does it take to make 1 leaning tower?

51. Mixing Addition and Subtraction of Decimals: Rocco has 1.5 litres of orange soda and 2.25 litres of grape soda in his fridge. Antonio has 1.15 litres of orange soda and 0.62 litres of grape soda. How much more soda does Rocco have than Angelo?

52. Mixing Multiplication and Division of Decimals: 4 days a week, Laura practices martial arts for 1.5 hours. Considering a week is 7 days, what is her average practice time per day each week?

Comparing and sequencing word problems

Best for: Kindergarten, 1st grade, 2nd grade

53. Comparing 1-Digit Integers: You have 3 apples and your friend has 5 apples. Who has more?

54. Comparing 2-Digit Integers: You have 50 candies and your friend has 75 candies. Who has more?

55. Comparing Different Variables: There are 5 basketballs on the playground. There are 7 footballs on the playground. Are there more basketballs or footballs?

56. Sequencing 1-Digit Integers: Erik has 0 stickers. Every day he gets 1 more sticker. How many days until he gets 3 stickers?

57. Skip-Counting by Odd Numbers: Natalie began at 5. She skip-counted by fives. Could she have said the number 20?

58. Skip-Counting by Even Numbers: Natasha began at 0. She skip-counted by eights. Could she have said the number 36?

59. Sequencing 2-Digit Numbers: Each month, Jeremy adds the same number of cards to his baseball card collection. In January, he had 36. 48 in February. 60 in March. How many baseball cards will Jeremy have in April?

Time word problems

66. Converting Hours into Minutes: Jeremy helped his mom for 1 hour. For how many minutes was he helping her?

69. Adding Time: If you wake up at 7:00 a.m. and it takes you 1 hour and 30 minutes to get ready and walk to school, at what time will you get to school?

70. Subtracting Time: If a train departs at 2:00 p.m. and arrives at 4:00 p.m., how long were passengers on the train for?

71. Finding Start and End Times: Rebecca left her dad’s store to go home at twenty to seven in the evening. Forty minutes later, she was home. What time was it when she arrived home?

Money word problems

Best for: 1st grade, 2nd grade, 3rd grade, 4th grade, 5th grade

60. Adding Money: Thomas and Matthew are saving up money to buy a video game together. Thomas has saved $30. Matthew has saved $35. How much money have they saved up together in total?

61. Subtracting Money: Thomas has $80 saved up. He uses his money to buy a video game. The video game costs $67. How much money does he have left?

62. Multiplying Money: Tim gets $5 for delivering the paper. How much money will he have after delivering the paper 3 times?

63. Dividing Money: Robert spent $184.59 to buy 3 hockey sticks. If each hockey stick was the same price, how much did 1 cost?

64. Adding Money with Decimals: You went to the store and bought gum for $1.25 and a sucker for $0.50. How much was your total?

65. Subtracting Money with Decimals: You went to the store with $5.50. You bought gum for $1.25, a chocolate bar for $1.15 and a sucker for $0.50. How much money do you have left?

67. Applying Proportional Relationships to Money: Jakob wants to invite 20 friends to his birthday, which will cost his parents $250. If he decides to invite 15 friends instead, how much money will it cost his parents? Assume the relationship is directly proportional.

68. Applying Percentages to Money: Retta put $100.00 in a bank account that gains 20% interest annually. How much interest will be accumulated in 1 year? And if she makes no withdrawals, how much money will be in the account after 1 year?

Physical measurement word problems

Best for: 1st grade, 2nd grade, 3rd grade, 4th grade

72. Comparing Measurements: Cassandra’s ruler is 22 centimetres long. April’s ruler is 30 centimetres long. How many centimetres longer is April’s ruler?

73. Contextualizing Measurements: Picture a school bus. Which unit of measurement would best describe the length of the bus? Centimetres, metres or kilometres?

74. Adding Measurements: Micha’s dad wants to try to save money on gas, so he has been tracking how much he uses. Last year, Micha’s dad used 100 litres of gas. This year, her dad used 90 litres of gas. How much gas did he use in total for the two years?

75. Subtracting Measurements: Micha’s dad wants to try to save money on gas, so he has been tracking how much he uses. Over the past two years, Micha’s dad used 200 litres of gas. This year, he used 100 litres of gas. How much gas did he use last year?

76. Multiplying Volume and Mass: Kiera wants to make sure she has strong bones, so she drinks 2 litres of milk every week. After 3 weeks, how many litres of milk will Kiera drink?

77. Dividing Volume and Mass: Lillian is doing some gardening, so she bought 1 kilogram of soil. She wants to spread the soil evenly between her 2 plants. How much will each plant get?

78. Converting Mass: Inger goes to the grocery store and buys 3 squashes that each weigh 500 grams. How many kilograms of squash did Inger buy?

79. Converting Volume: Shad has a lemonade stand and sold 20 cups of lemonade. Each cup was 500 millilitres. How many litres did Shad sell in total?

80. Converting Length: Stacy and Milda are comparing their heights. Stacy is 1.5 meters tall. Milda is 10 centimetres taller than Stacy. What is Milda’s height in centimetres?

81. Understanding Distance and Direction: A bus leaves the school to take students on a field trip. The bus travels 10 kilometres south, 10 kilometres west, another 5 kilometres south and 15 kilometres north. To return to the school, in which direction does the bus have to travel? How many kilometres must it travel in that direction?

Ratios and percentages word problems

Best for: 4th grade, 5th grade, 6th grade

82. Finding a Missing Number: The ratio of Jenny’s trophies to Meredith’s trophies is 7:4. Jenny has 28 trophies. How many does Meredith have?

83. Finding Missing Numbers: The ratio of Jenny’s trophies to Meredith’s trophies is 7:4. The difference between the numbers is 12. What are the numbers?

84. Comparing Ratios: The school’s junior band has 10 saxophone players and 20 trumpet players. The school’s senior band has 18 saxophone players and 29 trumpet players. Which band has the higher ratio of trumpet to saxophone players?

85. Determining Percentages: Mary surveyed students in her school to find out what their favourite sports were. Out of 1,200 students, 455 said hockey was their favourite sport. What percentage of students said hockey was their favourite sport?

86. Determining Percent of Change: A decade ago, Oakville’s population was 67,624 people. Now, it is 190% larger. What is Oakville’s current population?

87. Determining Percents of Numbers: At the ice skate rental stand, 60% of 120 skates are for boys. If the rest of the skates are for girls, how many are there?

88. Calculating Averages: For 4 weeks, William volunteered as a helper for swimming classes. The first week, he volunteered for 8 hours. He volunteered for 12 hours in the second week, and another 12 hours in the third week. The fourth week, he volunteered for 9 hours. For how many hours did he volunteer per week, on average?

Probability and data relationships word problems

Best for: 4th grade, 5th grade, 6th grade, 7th grade

89. Understanding the Premise of Probability: John wants to know his class’s favourite TV show, so he surveys all of the boys. Will the sample be representative or biased?

90. Understanding Tangible Probability: The faces on a fair number die are labelled 1, 2, 3, 4, 5 and 6. You roll the die 12 times. How many times should you expect to roll a 1?

91. Exploring Complementary Events: The numbers 1 to 50 are in a hat. If the probability of drawing an even number is 25/50, what is the probability of NOT drawing an even number? Express this probability as a fraction.

92. Exploring Experimental Probability: A pizza shop has recently sold 15 pizzas. 5 of those pizzas were pepperoni. Answering with a fraction, what is the experimental probability that he next pizza will be pepperoni?

93. Introducing Data Relationships: Maurita and Felice each take 4 tests. Here are the results of Maurita’s 4 tests: 4, 4, 4, 4. Here are the results for 3 of Felice’s 4 tests: 3, 3, 3. If Maurita’s mean for the 4 tests is 1 point higher than Felice’s, what’s the score of Felice’s 4th test?

94. Introducing Proportional Relationships: Store A is selling 7 pounds of bananas for $7.00. Store B is selling 3 pounds of bananas for $6.00. Which store has the better deal?

95. Writing Equations for Proportional Relationships: Lionel loves soccer, but has trouble motivating himself to practice. So, he incentivizes himself through video games. There is a proportional relationship between the amount of drills Lionel completes, in x , and for how many hours he plays video games, in y . When Lionel completes 10 drills, he plays video games for 30 minutes. Write the equation for the relationship between x and y .

Geometry word problems

Best for: 4th grade, 5th grade, 6th grade, 7th grade, 8th grade

96. Introducing Perimeter:  The theatre has 4 chairs in a row. There are 5 rows. Using rows as your unit of measurement, what is the perimeter?

97. Introducing Area: The theatre has 4 chairs in a row. There are 5 rows. How many chairs are there in total?

98. Introducing Volume: Aaron wants to know how much candy his container can hold. The container is 20 centimetres tall, 10 centimetres long and 10 centimetres wide. What is the container’s volume?

99. Understanding 2D Shapes: Kevin draws a shape with 4 equal sides. What shape did he draw?

100. Finding the Perimeter of 2D Shapes: Mitchell wrote his homework questions on a piece of square paper. Each side of the paper is 8 centimetres. What is the perimeter?

101. Determining the Area of 2D Shapes: A single trading card is 9 centimetres long by 6 centimetres wide. What is its area?

102. Understanding 3D Shapes: Martha draws a shape that has 6 square faces. What shape did she draw?

103. Determining the Surface Area of 3D Shapes: What is the surface area of a cube that has a width of 2cm, height of 2 cm and length of 2 cm?

104. Determining the Volume of 3D Shapes: Aaron’s candy container is 20 centimetres tall, 10 centimetres long and 10 centimetres wide. Bruce’s container is 25 centimetres tall, 9 centimetres long and 9 centimetres wide. Find the volume of each container. Based on volume, whose container can hold more candy?

105. Identifying Right-Angled Triangles: A triangle has the following side lengths: 3 cm, 4 cm and 5 cm. Is this triangle a right-angled triangle?

106. Identifying Equilateral Triangles: A triangle has the following side lengths: 4 cm, 4 cm and 4 cm. What kind of triangle is it?

107. Identifying Isosceles Triangles: A triangle has the following side lengths: 4 cm, 5 cm and 5 cm. What kind of triangle is it?

108. Identifying Scalene Triangles: A triangle has the following side lengths: 4 cm, 5 cm and 6 cm. What kind of triangle is it?

109. Finding the Perimeter of Triangles: Luigi built a tent in the shape of an equilateral triangle. The perimeter is 21 metres. What is the length of each of the tent’s sides?

110. Determining the Area of Triangles: What is the area of a triangle with a base of 2 units and a height of 3 units?

111. Applying Pythagorean Theorem: A right triangle has one non-hypotenuse side length of 3 inches and the hypotenuse measures 5 inches. What is the length of the other non-hypotenuse side?

112. Finding a Circle’s Diameter: Jasmin bought a new round backpack. Its area is 370 square centimetres. What is the round backpack’s diameter?

113. Finding a Circle's Area: Captain America’s circular shield has a diameter of 76.2 centimetres. What is the area of his shield?

114. Finding a Circle’s Radius: Skylar lives on a farm, where his dad keeps a circular corn maze. The corn maze has a diameter of 2 kilometres. What is the maze’s radius?

Variables word problems

Best for: 6th grade, 7th grade, 8th grade

115. Identifying Independent and Dependent Variables: Victoria is baking muffins for her class. The number of muffins she makes is based on how many classmates she has. For this equation, m is the number of muffins and c is the number of classmates. Which variable is independent and which variable is dependent?

116. Writing Variable Expressions for Addition: Last soccer season, Trish scored g goals. Alexa scored 4 more goals than Trish. Write an expression that shows how many goals Alexa scored.

117. Writing Variable Expressions for Subtraction: Elizabeth eats a healthy, balanced breakfast b times a week. Madison sometimes skips breakfast. In total, Madison eats 3 fewer breakfasts a week than Elizabeth. Write an expression that shows how many times a week Madison eats breakfast.

118. Writing Variable Expressions for Multiplication: Last hockey season, Jack scored g goals. Patrik scored twice as many goals than Jack. Write an expression that shows how many goals Patrik scored.

119. Writing Variable Expressions for Division: Amanda has c chocolate bars. She wants to distribute the chocolate bars evenly among 3 friends. Write an expression that shows how many chocolate bars 1 of her friends will receive.

120. Solving Two-Variable Equations: This equation shows how the amount Lucas earns from his after-school job depends on how many hours he works: e = 12h . The variable h represents how many hours he works. The variable e represents how much money he earns. How much money will Lucas earn after working for 6 hours?

How to easily make your own math word problems & word problems worksheets

Armed with 120 examples to spark ideas, making your own math word problems can engage your students and ensure alignment with lessons. Do:

• Link to Student Interests:  By framing your word problems with student interests, you’ll likely grab attention. For example, if most of your class loves American football, a measurement problem could involve the throwing distance of a famous quarterback.
• Make Questions Topical:  Writing a word problem that reflects current events or issues can engage students by giving them a clear, tangible way to apply their knowledge.
• Include Student Names:  Naming a question’s characters after your students is an easy way make subject matter relatable, helping them work through the problem.
• Be Explicit:  Repeating keywords distills the question, helping students focus on the core problem.
• Test Reading Comprehension:  Flowery word choice and long sentences can hide a question’s key elements. Instead, use concise phrasing and grade-level vocabulary.
• Focus on Similar Interests:  Framing too many questions with related interests -- such as football and basketball -- can alienate or disengage some students.
• Feature Red Herrings:  Including unnecessary information introduces another problem-solving element, overwhelming many elementary students.

A key to differentiated instruction , word problems that students can relate to and contextualize will capture interest more than generic and abstract ones.

Final thoughts about math word problems

You’ll likely get the most out of this resource by using the problems as templates, slightly modifying them by applying the above tips. In doing so, they’ll be more relevant to -- and engaging for -- your students.

Regardless, having 120 curriculum-aligned math word problems at your fingertips should help you deliver skill-building challenges and thought-provoking assessments.

The result?

A greater understanding of how your students process content and demonstrate understanding, informing your ongoing teaching approach.

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25 Fun Math Problems For Elementary And Middle School (From Easy To Very Hard!)

Fun math problems and brainteasers are loved by mathematicians; they provide an opportunity to apply mathematical knowledge, logic, and problem-solving skills all at once.

In this article, we’ve compiled 25 fun math problems and brainteasers covering various topics and question types. They’re aimed at students in upper elementary (3rd-5th grade) and middle school (6th grade, 7th grade, and 8th grade). We’ve categorized them as:

How should teachers use these math problems?

Teachers could make use of these math problem solving questions in a number of ways. They can:

• incorporate the questions into a relevant math lesson.
• set tasks at the beginning of lessons.
• break up or extend a math worksheet.
• keep students thinking mathematically after the main lesson has finished.

Some are based on real life or historical math problems, and some include ‘bonus’ math questions to help extend the problem-solving fun! As you read through these problems, think about how you could adjust them to be relevant to your students and their grade level or to practice different math skills. 

These math problems can also be used as introductory puzzles for math games such as those introduced at the following links:

• Math games for grade 4
• Middle school math games

25 Math Problems

Want the fun and challenging questions from this blog wrapped up in a downloadable question and answer format? Then you're in luck!

Math word problems

1. home on time – easy .

Type: Elapsed time, Number, Addition

A movie theater screening starts at 2:35 pm. The movie lasts for 2 hours, 32 minutes after 23 minutes of previews. It takes 20 minutes to get home from the movie theater. What time should you tell your family that you’ll be home?

Answer: 5:50 pm

2. A nugget of truth – mixed

Type: Multiplication Facts, Multiplication, Multiples, Factors, Problem-Solving 

Chicken nuggets come in boxes of 6, 9 or 20, so you can’t order 7 chicken nuggets. How many other impossible quantities can you find (not including fractions or decimals)?

Answer: 1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 22, 23, 25, 28, 31, 34, 37, or 43

There is actually a theorem which can be used to prove that every integer quantity greater than 43 can be ordered.

3. A pet problem – mixed

Type: Number, Problem Solving, Forming and Solving Equations, Simultaneous Equations, Algebra

Eight of my pets aren’t dogs, five aren’t rabbits, and seven aren’t cats. How many pets do I have?

Answer: 10 pets (5 rabbits, 3 cats, 2 dogs)

Looking for more word problems, solutions and explanations? Read our article on word problems for elementary school.

4. the price of things – mixed.

Type: lateral thinking problem

A mouse costs $10, a bee costs $15, and a spider costs $20. Based on this, how much does a duck cost?

Answer: $5 ($2.50 per leg)

Math puzzles

5. a dicey math challenge – easy.

Type: Place value, number, addition, problem-solving

Roll three 6 sided dice to generate three place value digits. What’s the biggest number you can make out of these digits? What’s the smallest number you can make?

Add these two numbers together. What do you get?

Answer: If the digits are the same, the maximum is 666 and the minimum is 111. Then, if you add the numbers together, 666 + 111 = 777. If the digits are different, the maximum is 654 and the minimum is 456. Then, if you add the numbers together, 654 + 456 = 1,110.

Bonus: Who got a different result? Why?

6. PIN problem solving – mixed

Type: Logic, problem solving, reasoning

I’ve forgotten my PIN. Six incorrect attempts locks my account: I’ve used five! Two digits are displayed after each unsuccessful attempt: “2, 0” means 2 digits from that guess are in the PIN, but 0 are in the right place. No two digits in my PIN are the same.

What should my sixth attempt be?

Answer: 6347

7. So many birds – mixed

Type: Triangular Numbers, Sequences, Number, Problem Solving

On the first day of Christmas my true love gave me one gift. On the second day they gave me another pair of gifts plus a copy of what they gave me on day one. On day 3, they gave me three new gifts, plus another copy of everything they’d already given me. If they keep this up, how many gifts will I have after twelve days?

Answer: 364

Bonus: This could be calculated as 1 + (1 + 2) + (1 + 2 + 3) + … but is there an easier way? What percentage of my gifts do I receive on each day?

8. I 8 sum math questions – mixed

Type: Number, Place Value, Addition, Problem Solving, Reasoning

Using only addition and the digit 8, can you make 1,000? You can put 8s together to make 88, for example.

Answer: 888 + 88 + 8 + 8 + 8 = 1,000 Bonus: Which other digits allow you to get 1,000 in this way?

Fraction problems

4 friends entered a math quiz. One answered \frac {1}{5} of the math questions, one answered \frac {1}{10} , one answered \frac{1}{4} , and the other answered \frac{4}{25} . What percentage of the questions did they answer altogether?

Answer: 71%

10. Ancient problem solving – easy

Type: Fractions, Reasoning, Problem Solving

Ancient Egyptians only used unit fractions (like \frac {1}{2} , \frac{1}{3} or \frac{1}{4} . For \frac {2}{3} they’d write \frac{1}{3} + \frac{1}{3} . How might they write \frac{5}{8} ?

Answer: 

\frac {1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} is correct. So is \frac {1}{2} + \frac{1}{8} (They are both still unit fractions even though they have different denominators.)

Bonus: Which solution is better? Why? Can you find any more? What if subtractions are allowed?

Learn more about unit fractions here .

11. everybody wants a pizza the action – hard.

An infinite number of mathematicians buy pizza. The first wants \frac{1}{2} . The second wants \frac{1}{4} pizza. The third & fourth want \frac{1}{8} and \frac{1}{16} each, and so on. How many pizzas should they order?

Answer: 1 Each successive mathematician wants a slice that is exactly half of what is left:

12. Shade it black – hard

Type: Fractions, Reasoning, Problem Solving What fraction of this image is shaded black?

Answer:  

Look at the L-shaped part made up of two white and one black squares: 

\frac{1}{3} of this part is shaded. Zoom in on the top-right quarter of the image, which looks exactly the same as the whole image, and use the same reasoning to find what fraction of its L-shaped portion is shaded. Imagine zooming in to do the same thing again and again…

Multiplication and division problems

13. giving is receiving – easy.

Type: Number, Reasoning, Problem Solving

5 people give each other a present. How many presents are given altogether?

14. Sharing is caring – mixed

I have 20 candies. If I share them equally with my friends, there are 2 left over. If one more person joins us, there are 6 candies left. How many friends am I with?

Answer: 6 people altogether (so 5 friends!)

15. Multiplication facts secrets – mixed

Type: Area, 2D Shape, Rectangles

Here are 77 letters:

B Y H R C G N E O E A A H G C U R P U T S A S H H S B O R E O P E E M E E L A T P E F A D P H L TU T I E E O H L E N R Y T I I A G B M T N F C G E I I G

How many different rectangular grids could you arrange all 77 letters into?

Can you reveal the secret message?

Answer: Four: 1 × 77, 77 × 1, 11 × 7 and 7 × 11. If the letters are arranged into one of these, a message appears, reading down each column starting from the top left.

Bonus: Can you find any more integers with the same number of factors as 77? What do you notice about these factors (think about prime numbers)? Can you use this system to hide your own messages?

16. Laugh it up – hard

Type: Multiples, Least Common Multiple, Multiplication Facts, Division, Time

One friend jumps every \frac{1}{3} of a minute. Another jumps every 31 seconds. When will they jump together? Answer: After 620 seconds

Geometry problems

17. pictures of matchstick triangles – easy.

Type: 2D Shapes, Equilateral Triangles, Problem Solving, Reasoning

Look at the matchsticks arranged below. How many equilateral triangles are there?

Answer: 13 (9 small, 3 medium, 1 large)

Bonus: What if the biggest triangle only had two matchsticks on each side? What if it had four?

18. Dissecting squares – mixed

Type: Reasoning, Problem Solving

What’s the smallest number of straight lines you could draw on this grid such that each square has a line going through it?

19. Make it right – mixed

Type: Pythagorean theorem

This triangle does not agree with Pythagorean theorem. 

Adding, subtracting, multiplying or dividing each of the side lengths by the same whole number can fix it. What is the number?

Answer: 3 

The new side lengths are 3, 4 and 5 and  32 + 42 = 52.

20. A most regular math question – hard

Type: Polygons, 2D Shapes, tessellation, reasoning, problem-solving, patterns

What is the regular polygon with the largest number of sides that will self-tessellate?

Answer: Hexagon.

Regular polygons tessellate if one interior angle is a factor of 360 ° . The interior angle of a hexagon is 120°. This is the largest factor less than 180°.

Problem-solving questions

21. pleased to meet you – easy.

Type: Number Problem, Reasoning, Problem Solving

5 people meet; each shakes everyone else’s hand once. How many handshakes take place?

Person A shakes 4 people’s hands. Person B has already shaken Person A’s hand, so only needs to shake 3 more, and so on.

Bonus: How many handshakes would there be if you did this with your class?

22. All relative – easy

Type: Number, Reasoning, Problem-Solving

When I was twelve my brother was half my age. I’m 40 now, so how old is he?

23. It’s about time – mixed

Type: Time, Reasoning, Problem-Solving

When is “8 + 10 = 6” true?

Answer: When you’re telling the time (8am + 10 hours = 6pm)

24. More than a match – mixed

Type: Reasoning, Problem-Solving, Roman Numerals, Numerical Notation

Here are three matches:

How can you add two more matches, but get eight? Answer: Put the extra two matches in a V shape to make 8 in Roman Numerals:

25. Leonhard’s graph – hard

Type: Reasoning, Problem-Solving, Logic

Leonhard’s town has seven bridges as shown below. Can you find a route around the town that crosses every bridge exactly once?

Answer: No!

This is a classic real life historical math problem solved by mathematician Leonhard Euler (rhymes with “boiler”). The city was Konigsberg in Prussia (now Kaliningrad, Russia). Not being able to find a solution is different from proving that there aren’t any! Euler managed to do this in 1736, practically inventing graph theory in the process.

Math puzzles are everywhere!

Many of these 25 math problems are rooted in real life, from everyday occurrences to historical events. Others are just questions that might arise if you say “what if…?”. The point is that although there are many lists of such problem-solving math questions that you can make use of, with a little bit of experience and inspiration you could create your own on almost any topic – and so could your students. 

For a kick-starter on creating your own math problems, read our article on middle school math problem solving .

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

The content in this article was originally written by primary school teacher Tom Briggs and has since been revised and adapted for US schools by math curriculum specialist and former elementary math teacher Katie Keeton.

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1.3: Problem Solving Strategies

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• Page ID 9823

• Michelle Manes
• University of Hawaii

Think back to the first problem in this chapter, the ABC Problem. What did you do to solve it? Even if you did not figure it out completely by yourself, you probably worked towards a solution and figured out some things that did not work.

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

How to Solve It

George Pólya was a great champion in the field of teaching effective problem solving skills. He was born in Hungary in 1887, received his Ph.D. at the University of Budapest, and was a professor at Stanford University (among other universities). He wrote many mathematical papers along with three books, most famously, “How to Solve it.” Pólya died at the age 98 in 1985. [1]

George Pólya, circa 1973

• Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY 2.0 ( http://creativecommons.org/licenses/by/2.0 )], via Wikimedia Commons ↵

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

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

This is all well and good, but how do you actually do these steps?!?! Steps 1. and 2. are particularly mysterious! How do you “make a plan?” That is where you need some tools in your toolbox, and some experience to draw upon.

Much has been written since 1945 to explain these steps in more detail, but the truth is that they are more art than science. This is where math becomes a creative endeavor (and where it becomes so much fun). We will articulate some useful problem solving strategies, but no such list will ever be complete. This is really just a start to help you on your way. The best way to become a skilled problem solver is to learn the background material well, and then to solve a lot of problems!

We have already seen one problem solving strategy, which we call “Wishful Thinking.” Do not be afraid to change the problem! Ask yourself “what if” questions:

• What if the picture was different?
• What if the numbers were simpler?
• What if I just made up some numbers?

You need to be sure to go back to the original problem at the end, but wishful thinking can be a powerful strategy for getting started.

This brings us to the most important problem solving strategy of all:

Problem Solving Strategy 2 (Try Something!).

If you are really trying to solve a problem, the whole point is that you do not know what to do right out of the starting gate. You need to just try something! Put pencil to paper (or stylus to screen or chalk to board or whatever!) and try something. This is often an important step in understanding the problem; just mess around with it a bit to understand the situation and figure out what is going on.

And equally important: If what you tried first does not work, try something else! Play around with the problem until you have a feel for what is going on.

Last week, Alex borrowed money from several of his friends. He finally got paid at work, so he brought cash to school to pay back his debts. First he saw Brianna, and he gave her 1/4 of the money he had brought to school. Then Alex saw Chris and gave him 1/3 of what he had left after paying Brianna. Finally, Alex saw David and gave him 1/2 of what he had remaining. Who got the most money from Alex?

Think/Pair/Share

After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What did you figure out about the problem? This problem lends itself to two particular strategies. Did you try either of these as you worked on the problem? If not, read about the strategy and then try it out before watching the solution.

Problem Solving Strategy 3 (Draw a Picture).

Some problems are obviously about a geometric situation, and it is clear you want to draw a picture and mark down all of the given information before you try to solve it. But even for a problem that is not geometric, like this one, thinking visually can help! Can you represent something in the situation by a picture?

Draw a square to represent all of Alex’s money. Then shade 1/4 of the square — that’s what he gave away to Brianna. How can the picture help you finish the problem?

After you have worked on the problem yourself using this strategy (or if you are completely stuck), you can watch someone else’s solution.

Problem Solving Strategy 4 (Make Up Numbers).

Part of what makes this problem difficult is that it is about money, but there are no numbers given. That means the numbers must not be important. So just make them up!

You can work forwards: Assume Alex had some specific amount of money when he showed up at school, say $100. Then figure out how much he gives to each person. Or you can work backwards: suppose he has some specific amount left at the end, like $10. Since he gave Chris half of what he had left, that means he had $20 before running into Chris. Now, work backwards and figure out how much each person got.

Watch the solution only after you tried this strategy for yourself.

If you use the “Make Up Numbers” strategy, it is really important to remember what the original problem was asking! You do not want to answer something like “Everyone got $10.” That is not true in the original problem; that is an artifact of the numbers you made up. So after you work everything out, be sure to re-read the problem and answer what was asked!

(Squares on a Chess Board)

How many squares, of any possible size, are on a 8 × 8 chess board? (The answer is not 64... It’s a lot bigger!)

Remember Pólya’s first step is to understand the problem. If you are not sure what is being asked, or why the answer is not just 64, be sure to ask someone!

Think / Pair / Share

After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What did you figure out about the problem, even if you have not solved it completely?

It is clear that you want to draw a picture for this problem, but even with the picture it can be hard to know if you have found the correct answer. The numbers get big, and it can be hard to keep track of your work. Your goal at the end is to be absolutely positive that you found the right answer. You should never ask the teacher, “Is this right?” Instead, you should declare, “Here’s my answer, and here is why I know it is correct!”

Problem Solving Strategy 5 (Try a Simpler Problem).

Pólya suggested this strategy: “If you can’t solve a problem, then there is an easier problem you can solve: find it.” He also said: “If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?” In this case, an 8 × 8 chess board is pretty big. Can you solve the problem for smaller boards? Like 1 × 1? 2 × 2? 3 × 3?

Of course the ultimate goal is to solve the original problem. But working with smaller boards might give you some insight and help you devise your plan (that is Pólya’s step (2)).

Problem Solving Strategy 6 (Work Systematically).

If you are working on simpler problems, it is useful to keep track of what you have figured out and what changes as the problem gets more complicated.

For example, in this problem you might keep track of how many 1 × 1 squares are on each board, how many 2 × 2 squares on are each board, how many 3 × 3 squares are on each board, and so on. You could keep track of the information in a table:

Problem Solving Strategy 7 (Use Manipulatives to Help You Investigate).

Sometimes even drawing a picture may not be enough to help you investigate a problem. Having actual materials that you move around can sometimes help a lot!

For example, in this problem it can be difficult to keep track of which squares you have already counted. You might want to cut out 1 × 1 squares, 2 × 2 squares, 3 × 3 squares, and so on. You can actually move the smaller squares across the chess board in a systematic way, making sure that you count everything once and do not count anything twice.

Problem Solving Strategy 8 (Look for and Explain Patterns).

Sometimes the numbers in a problem are so big, there is no way you will actually count everything up by hand. For example, if the problem in this section were about a 100 × 100 chess board, you would not want to go through counting all the squares by hand! It would be much more appealing to find a pattern in the smaller boards and then extend that pattern to solve the problem for a 100 × 100 chess board just with a calculation.

If you have not done so already, extend the table above all the way to an 8 × 8 chess board, filling in all the rows and columns. Use your table to find the total number of squares in an 8 × 8 chess board. Then:

• Describe all of the patterns you see in the table.
• Can you explain and justify any of the patterns you see? How can you be sure they will continue?
• What calculation would you do to find the total number of squares on a 100 × 100 chess board?

(We will come back to this question soon. So if you are not sure right now how to explain and justify the patterns you found, that is OK.)

(Broken Clock)

This clock has been broken into three pieces. If you add the numbers in each piece, the sums are consecutive numbers. ( Consecutive numbers are whole numbers that appear one after the other, such as 1, 2, 3, 4 or 13, 14, 15.)

Can you break another clock into a different number of pieces so that the sums are consecutive numbers? Assume that each piece has at least two numbers and that no number is damaged (e.g. 12 isn’t split into two digits 1 and 2.)

Remember that your first step is to understand the problem. Work out what is going on here. What are the sums of the numbers on each piece? Are they consecutive?

After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What progress have you made?

Problem Solving Strategy 9 (Find the Math, Remove the Context).

Sometimes the problem has a lot of details in it that are unimportant, or at least unimportant for getting started. The goal is to find the underlying math problem, then come back to the original question and see if you can solve it using the math.

In this case, worrying about the clock and exactly how the pieces break is less important than worrying about finding consecutive numbers that sum to the correct total. Ask yourself:

• What is the sum of all the numbers on the clock’s face?
• Can I find two consecutive numbers that give the correct sum? Or four consecutive numbers? Or some other amount?
• How do I know when I am done? When should I stop looking?

Of course, solving the question about consecutive numbers is not the same as solving the original problem. You have to go back and see if the clock can actually break apart so that each piece gives you one of those consecutive numbers. Maybe you can solve the math problem, but it does not translate into solving the clock problem.

Problem Solving Strategy 10 (Check Your Assumptions).

When solving problems, it is easy to limit your thinking by adding extra assumptions that are not in the problem. Be sure you ask yourself: Am I constraining my thinking too much?

In the clock problem, because the first solution has the clock broken radially (all three pieces meet at the center, so it looks like slicing a pie), many people assume that is how the clock must break. But the problem does not require the clock to break radially. It might break into pieces like this:

Were you assuming the clock would break in a specific way? Try to solve the problem now, if you have not already.

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News & Views from edWeb

Solving Math Problems and Negative Feelings About Math

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As students progress through the upper elementary grades and into middle school, many are becoming caught in a negative feedback loop, with their limited understanding of math facts and underlying concepts resulting in poor grades and a lack of engagement, making successful math learning even less likely to occur.

Ways to teach math effectively and engage students in math learning were discussed during the edLeader Panel “ Differentiate Elementary Math Instruction to Increase Engagement and Flexibility: Build Your Toolbox .”

Alexa Poulin, Customer Marketing Manager at Unruly Studios and panel moderator, shared poll results about key challenges educators face when teaching math, and the data showed more than half of the educators found their students did not have the number sense and foundational knowledge needed for high-level thinking about math. Many educators also felt that math classes were not engaging and that many students were performing at levels that were multiple grades below their state standards.

Developing Math Knowledge and Positive Attitudes

The panelists agreed that students struggling with math need a fundamental understanding of how numbers work and relate to each other. Building a solid foundation of knowledge about base 10 and place value is crucial, as students can then apply that knowledge when solving math problems and thereby gain confidence and further engage in math learning.

Emma Simmons, Principal of Roxbury Prep Charter School (MA), discussed the importance of developing a culture of confidence and error acceptance in math classes so that a fear of failure does not lead to a lack of engagement and learning. This can be accomplished by encouraging discussions about how errors were found and what can be done to correct them. This way, errors are perceived as a routine part of the learning process, and any gaps in students’ understanding can be identified and remedied.

Another key part of the process is showing students that there is often more than one way to solve a math problem, and encouraging a willingness to try different approaches even if the students are not sure their answers will be correct. Students also need to be taught to “think holistically” about math problems, by considering what the problem is asking, what needs to be solved, and which steps need to be taken to arrive at a correct answer.

Strategies to Increase Learning and Engagement

To engage students in learning and flexible thinking, Elijah Ortiz, Fifth-Grade Educator and Teacher Leader at Concourse Village Elementary School (NY), explained three different strategies, all of which use a three-part approach. A “3-act task” can start with a brief video that students then discuss, after which additional information and a question are presented and prompt students to discuss how they should proceed and why. The third part of the process is revealing the answer and exploring different ways the answer can be found.

A “3-read protocol” is an approach to word problems that starts with reading a word problem like a story in order to determine what is happening and improve students’ comprehension of the problem. The second step is a choral reading that is followed by consideration of potential problem-solving strategies. The third step is another choral reading that includes the ending question to ensure understanding of the content.

The third approach is known as a “CRA frame,” in which students progress from a Concrete experience to a Representational one and then an Abstract one. This can start with the use of manipulatives, then proceed to students doing a drawing, and finally to the students performing a written math function or solving an equation.

Emily Semrad, Elementary Educator at Academy Adventures Midtown Charter School (AZ), pointed out that math vocabulary can be a crucial obstacle for students, creating confusion when the students encounter and try to differentiate between words such as “numerator” and “denominator.” To develop math vocabulary, she recommends starting with a brief video that students can just listen to and watch. Next, there can be a class discussion about what they saw and heard, including any familiar or unfamiliar terms. Students can then be encouraged to define key terms, write notes about them, and discuss them in relation to the math processes they are learning.

Semrad also has found that teamwork and competition can be effective ways to engage students in math learning and practice. One technique is a “relay race” in which students are divided into groups, and each member of a group is responsible for performing one part of a math function or providing one part of a solution to a word problem. There can also be a competition to see which group can come up with the most ways to solve a problem, thereby reinforcing the creative thinking that can lead to math solutions.

Through these and other techniques, educators can engage students in continuing to increase their knowledge and conceptual understanding of math, as well as becoming more engaged in the learning process.

Learn more about this edWeb broadcast, Differentiate Elementary Math Instruction to Increase Engagement and Flexibility: Build Your Toolbox , sponsored by  Unruly Studios

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Building Understanding in Mathematics is a free professional learning community that provides a platform, advice and support in helping educators learn methods that help students build understanding in mathematics.

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Article by Robert Low , based on this edLeader Panel

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Basic Elementary Math Problems with Solutions

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In early elementary school, students learn to add, subtract, multiply and divide using whole numbers. Read on for tips and sample problems to help your child, where he's just beginning to add or learning the basics of multiplication.

How to Help Your Child Solve Basic Elementary Math Problems

One of the most effective ways you can help your child with elementary math is to identify areas where he or she struggles and offer additional practice in those areas. You might begin by checking your child's addition and subtraction skills. In your child is in first grade, he or she should be able to add numbers up to 100. If he or she is a second-grader, your child should be able to add and subtract using more complicated, multi-digit numbers.

In third grade, most elementary students must memorize their multiplication facts. Repetition and consistency will help your child remember these facts. You can establish a practice schedule, and then go over the facts for a few minutes each night.

You also might find that having your child explain math concepts to you in his own words can help your child internalize the information. After he has explained a concept to you, completing a few problems together could help further his understanding.

Problems with Solutions by Grade Level

3. At a dog park, there are seven dogs playing fetch. Two more dogs arrive. How many dogs are there in all?

1. 171 + 201

2. 524 - 318

3. There are 178 guests at a party before 59 more guests arrive. How many guests are there in all?

2. 35 ÷ 7

3. Julianne wants to give five cookies to eight of her friends. How many cookies should she bake?

Other Articles You May Be Interested In

Before you can advance to more complex levels of math (like algebra) you have to master the order of operations. Read on to learn the simple steps involved in completing ordered operations math problems. This article provides tips for working your way through math problems with several steps.

Is your child struggling with math in elementary school? You may be able to get him or her excited about learning math by using an educational app on your iPad, iPhone, Android or other mobile device. Here are ten apps for elementary school children that may hold the key to math success.

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