problem solving in the mathematics classroom

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5 Teaching Mathematics Through Problem Solving

Janet Stramel

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

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

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

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

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

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

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

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

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

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

Key features of a good mathematics problem includes:

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

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

But look at the b ack.

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

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

Mathematics Tasks and Activities that Promote Teaching through Problem Solving

Choosing the Right Task

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

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

Low Floor High Ceiling Tasks

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

The strengths of using Low Floor High Ceiling Tasks:

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

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

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

Math in 3-Acts

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

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

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

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

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

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

Number Talks

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

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

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

• Inside Mathematics Number Talks
• Number Talks Build Numerical Reasoning

Saying “This is Easy”

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

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

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

Instead, you and your students could say the following:

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

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

Using “Worksheets”

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

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

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

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

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

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

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

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

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

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

when we teach students how to problem solve

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

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

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

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

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20 Effective Math Strategies To Approach Problem-Solving 

Katie Keeton

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

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

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

What are problem-solving strategies?

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

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

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

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

The ultimate guide to problem solving techniques

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

20 Math Strategies For Problem-Solving

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

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

Strategies to understand the problem

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

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

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

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

1. Read the problem aloud

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

2. Highlight keywords 

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

3. Summarize the information

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

4. Determine the unknown

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

5. Make a plan

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

Strategies for solving the problem 

1. draw a model or diagram.

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

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

2. Act it out

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

3. Work backwards

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

For example,

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

4. Write a number sentence

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

5. Use a formula

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

Strategies for checking the solution 

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

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

Here are five strategies to help students check their solutions. 

1. Use the Inverse Operation

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

2. Estimate to check for reasonableness

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

3. Plug-In Method

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

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

4. Peer Review

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

5. Use a Calculator

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

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

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

Polya’s 4 steps include:

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

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

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

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

How Third Space Learning improves problem-solving 

Resources .

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

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

One-on-one tutoring 

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

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

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

Problem-solving

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

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

READ MORE : 8 Common Core math examples

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

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

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

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

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6 tips for supporting   problem-based learning in your math classroom

Most math teachers have been here before. We’ve all transposed the names of our own students for those in a word problem. We’ve all used our school community goings-on as fodder for plot lines: “If 89 middle schoolers are traveling on buses to Outdoor School, and each bus can transport 35 students and four chaperones.…” But how “real world” are we really getting?

We—Kailey Rhodes and Kristen Tsutsui, math teachers and authors of this blog post—wanted to know teachers’ experience with real-world problem-solving, what’s going well, and what’s in their way. So we surveyed some. In this post, we’ll walk you through what we asked, learned, and think as we move forward—and we’ll also share the resources our teachers shared with us. But first, let us introduce you to an official definition of real-world context in the math classroom.

The PISA Mathematics Framework

PISA is an international assessment administered to 15-year-olds globally. In their 2022 Mathematics Framework , PISA explains the underpinnings of their assessment as it relates to math literacy, reasoning, and problem-solving. It also stipulates that to uplift these underpinnings, mathematics problems should be presented in real-world contexts: personal, occupational, societal, and scientific.

Educators, take a moment to read the descriptions of these contexts . As you read, ask yourself the question we asked in our survey: “Which do you naturally find yourself gravitating toward in your classroom?”

• Personal: “Problems classified in the personal context category focus on activities of one’s self, one’s family, or one’s peer group. Personal contexts include (but are not limited to) those involving food preparation, shopping, games, personal health, personal transportation, sports, travel, personal scheduling, and personal finance.”
• Occupational: “Problems classified in the occupational context category are centered on the world of work. Items categorized as occupational may involve (but are not limited to) such things as measuring, costing, and ordering materials for building, payroll/accounting, quality control, scheduling/inventory, design/architecture, and job-related decision-making. Occupational contexts may relate to any level of the workforce, from unskilled work to the highest levels of professional work, although items in the PISA survey must be accessible to 15-year-old students.”
• Societal: “Problems classified in the societal context category focus on one’s community (whether local, national, or global). They may involve (but are not limited to) such things as voting systems, public transport, government, public policies, demographics, advertising, national statistics, and economics. Although individuals are involved in all of these things in a personal way, in the societal context category, the focus of problems is on the community perspective.”
• Scientific: “Problems classified in the scientific category relate to the application of mathematics to the natural world and issues and topics related to science and technology. Particular contexts might include (but are not limited to) such areas as weather or climate, ecology, medicine, space science, genetics, measurement, and the world of mathematics itself. Items that are intra-mathematical, where all the elements involved belong in the world of mathematics, fall within the scientific context.”

Which context did you most resonate with? If you were to focus on one context per quarter, how would you order them? What is your biggest struggle with real-world context and problem-based learning? How do you bring the “real world” inside your classroom?

We surveyed a focus group of math teachers, from kindergarten to IB, to see what they thought.

Survey says!

When we asked our teachers, “Which do you naturally find yourself gravitating toward in your classroom?” personal and scientific contexts were the winners, with societal coming in fourth. Our guess for this is that societal is not only ever-changing but is also often politically adjacent, something educators can, understandably, be wary of approaching.

Naturally, personal real-world math context offers an entry point into students’ interests, which is paramount in the math classroom. One teacher said, “Good projects that connect to the curriculum and also interest the students are worth their weight in gold. I really wish publishers did a better job of planning good projects; in most books I’ve used, the projects, if they exist at all, are an afterthought and poorly done.”

When it comes to other contexts, like societal, the bullseye of curriculum connection, student interest level, and math teacher time is a hard one to hit. As one teacher said, “With more time I would like to start new math topics with ‘real world problems’ and have the students brainstorm what knowledge would be useful to solve them, building resilience in the face of complex problems. However, these kinds of freeform explorations take time that I often feel like I don’t have in my class.”

This theme of “not enough time” appeared often, with teachers expressing a desire for more real-world presence, including cross-collaboration with other colleagues. A teacher said, “I have tried to coordinate with science classes to talk about the mathematical aspects of science concepts they are learning, but it can be hard to coordinate and map those kinds of things onto my own curriculum.”

This was echoed in the teachers’ comments, along with many wishful statements about what they “would do…if.”

With all the time and resources, what would teachers do?

We asked teachers to describe their dream scenario: what they would do if they had more resources, time, and permission. In most responses, what stood out was teachers’ love for math’s interconnectedness and innate curiosity. Some dreamed big:

• “An interdisciplinary project across all subjects that would allow students to see how math applies to all facets of the world”
• “Something like ‘a history of mathematical thought,’ bringing history, culture, psychology, ecology, science, and engineering into the curriculum”
• “Students being given the opportunity to explore a problem they are passionate about and explore ways that mathematics can be used to help solve it”
• “Students working alongside professionals in various math-related fields so they can see, firsthand, how the math they are learning is currently being used”

It’s clear that teachers want to provide classroom experiences that both underscore math’s omnipresence in the world around us and ignite students’ interests. It’s also clear that when your survey takers have to type qualifiers like, “But this would take a lot of time” and “We don’t have the resources for this,” the “real world” is actually what’s in the way. So, what can be done?

From ideal to real: Helpful tips & a relaxing thought on problem-based learning

No one knows better than math teachers that you can’t add time to your school day.  But, through our interactions with teachers, we walked away with some resources and tips to share with you. Here are the gems:

• Split it into quadrants. Most teachers surveyed would order their quarters and contexts like this: first quarter, personal; second quarter, societal; third quarter, scientific; fourth quarter, occupational.
• Switch with science. One teacher said this: “Actually switch classes with a science teacher to reiterate how what they are learning in science is related to what we do in math. And then they can come to my class to do the same so they can really understand the connection.”
• Draw a parallel. “Have a project that seems like it might belong in an arts classroom, like creating a piece of clothing,” another teacher suggested. “They have to use multiple modes of mathematical knowledge to do this, like unit conversions (centimeters to inches to yards), spatial constraints (if fabric is a certain size, can I fit my pattern in it? How big do the pieces have to be to fit on the body?), and area (how much fabric do I need?).”
• Estimate. “I use Estimation 180 , but I gamify it to address average, mean, median, and mode. I have students secretly record their estimations, and I write them all on the board. We discuss the average classroom guess and how close our range is. Estimation is everywhere.”
• Level up. “I’ve been using Skew the Script to heighten engagement for all my students and teach them about larger global issues affecting society. Their After The AP Data Science Challenge aims to solve the real (unsolved) problem of finding a model to predict the best and worst colleges for conquering student debt.”
• Build the roster: “Try to arrange a handful of yearly speakers and field trips you can count on to bring the math to life. My husband is a civil engineer and uses the Pythagorean theorem. You’d be surprised how many folks would love to come in and talk to a math class—with your teacher guidance and enthusiasm, of course.”

And now for the relaxing thought: Start small. As with any endeavor in the classroom, no matter the discipline, the key is to work incrementally. We already know how important increments are, right, math teachers?

What is “real,” anyway?

Let’s return to the age-old question of, “When will we ever use this?” It plagues math teachers perhaps more than any other discipline, and the task of “real-world” problem-solving often feels like a “gotcha” moment. The fact is, sometimes, we math teachers don’t have a great real-world example of dividing negative fractions in our back pocket. But does that mean that it isn’t worth learning? Must every concept in math be nested perfectly within the “real” world?

In one particular mic-drop moment, one of our teachers shared, “I think the label ‘real-world’ has been used extensively in math education, and I’ve found it a bit frustrating. In many regards, math is the least ‘real’ discipline we teach children. It is, largely, the art of abstraction! That it is ‘real’ is in many ways the least interesting and important thing about it as a system of thinking, and this is often lost when continually looked at with the question ‘When will we ever use this?’”

Maybe when we’ll “use this” is when we’re thinking through a problem with many facets and variables. Math is, after all, less of a “what” and more of a “how.” How should we go about solving this? How do we use computational thinking to solve problems? How do we spiral what we already know with what we don’t?

We’re teaching kids how to think —and isn’t that the most real-world skill we can possibly teach?

To learn more about how NWEA can support you in teaching math, visit our website .

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How to Turn Your Math Classroom Into a ‘Thinking Classroom’

The researcher Peter Liljedahl evangelizes for practices that prioritize and stimulate more hard thinking in classrooms.

In traditional math classrooms—and in classrooms where challenging, unfamiliar work is often assigned, more generally—the progression " I do, we do, you do” often becomes the default approach to learning, according to researcher and professor of mathematics at Simon Fraser University, Peter Liljedahl.

It makes sense in many cases, particularly when difficult concepts need to be addressed in time windows that are compressed by bell schedules, holiday breaks, and summer vacation. But used too frequently, this approach, Liljedahl recently told the Cult of Pedagogy , inhibits higher order thinking and results in students who “mimic” teachers. Students who engage in too much rote work miss out on some of the challenging, sometimes confusing work that builds up their self esteem to face difficult thinking tasks in the future.

“By and large students spend most of their class time not thinking, at least not in ways we know they need to think in order to be successful in mathematics.” Liljedahl explains. “If they’re not thinking, they’re not learning.”

Liljedahl, the author of Building Thinking Classrooms in Mathematics, Grades K-12: 14 Teaching Practices for Enhancing Learning advocates for “Thinking Classrooms”, which offers a different take on how classroom work is organized, how tasks are assigned, and how students learn and work together. His conclusions are based on more than a decade of research, experimentation, and collaboration with over 400 K-12 teachers.

In a 2017 article for Edutopia , Liljedahl clarified that a “non-thinking” classroom is “predicated on an assumption that the students either could not or would not think.” When presented with difficult problems, students in these classrooms have a hard time pushing through to find their own solutions, he argued, and often wait for teachers to step in to do the heavy lifting for them. 

To stimulate independent thinking, Liljedahl says, reorganize some of your classroom approaches: Start with hard puzzles and problems that push kids to their limits; confront the fundamental passivity of classroom seating; and use highly structured group activities to promote discussion, peer review, and iterative thinking. 

Starting With ‘Thinking Tasks’

Instead of starting a lesson with direct instruction, give students novel “thinking tasks” they can work on, ideally in groups. Liljedahl describes these tasks as problem solving activities and mental puzzles that, early on in the school year, should be “highly engaging, non-curricular tasks” to motivate students and get them in the mindset of challenging themselves. As the school year progresses and students become more accustomed to this mode of working and thinking, the activities and challenges can be replaced with tasks directly related to the curriculum. 

Liljedahl points out that the tasks should be carefully sequenced so they get incrementally more challenging. “The goal of thinking classrooms is not to get students to think about engaging with non-curricular tasks day in and day out—that turns out to be rather easy,” he told Cult of Pedagogy . “Rather, the goal is to get more of your students thinking, and thinking for longer periods of time, within the context of curriculum, which leads to longer and deeper learning.” 

Liljedahl has a long list of sample “thinking tasks” to look through. They include challenging dice-related problems, such as: Imagine a typical 6-sided die, and notice that the sum of opposite faces is always seven. The one is across from the six, the two is across from the five, and so on. Now imagine that you were making your own six-sided die that did not have this restriction. How many different dice could be made?

Use Standing, Randomized Group Work  

Central to Liljedahl’s approach is student collaboration and group work. Instead of grouping students by ability, or allowing them to choose their own groups to work in, his research has shown him students work more effectively—and are more likely to contribute—in randomized groups. According to Liljedahl , interviews with students show that randomized groups “break down social barriers within the room, increase knowledge mobility, reduce stress, and increase enthusiasm for mathematics.” 

At Design39Campus , a K-8 school in San Diego, eighth grade math teacher Kyle Asmus puts Liljedahl’s approach into practice by using a random group generator (such as this one from Classtools or this one from Keamk ) to ensure that different kids work together across multiple exercises. 

To ensure that all students feel included, Asmus assigns group roles: The scribe writes down possible solutions; the speaker communicates the group’s thinking to the broader classroom; the inquirer asks the teacher questions; and the manager makes sure the rest of the group stays on track. 

Having students stand while they engage in this collaborative, messy thinking is yet another way to engage them, according to Liljedahl: It makes them much less likely to withdraw from the work, or assume that others will handle it. “It turns out that when students are sitting, they feel anonymous,” Liljedahl told Cult of Pedagogy . “And when students feel anonymous, they disengage.” 

Work on Non-Permanent, Vertical Surfaces  

In a thinking classroom, students put notebooks away and participate in group work while standing at vertical non-permanent surfaces such as whiteboards, blackboards, or windows—surfaces that Liljedahl believes promote more risk-taking. 

According to Liljedahl, his experimentation with students shows that when comparing a group working on a whiteboard versus a group working on flip chart paper, the group working on whiteboards start working within 20 seconds. 

“They’ll start making notations on the board. They’ll try anything and everything because they feel like they can just erase it if it’s wrong,” he told Cult of Pedagogy . Meanwhile, students working on chart paper take upwards of three minutes to make a single notation, because, Liljedahl said, they often wait until what they write is perfect—“and that hesitation leads to a lower form of thinking.”

Design39 students also practice Liljedahl’s “vertical surfaces” learning technique, which Asmus says promotes “high quality collaboration” and facilitates “questions and rich conversations” among students. According to Asmus, when students are presented with challenging tasks and work vertically, the time for students to get to a task is not only quicker, but the time they spend on a task is longer. 

The large surfaces spread out around the room also allow students to see the work their peers are doing in other groups, and build off of each other’s understanding. “It makes it easy for the whole class to see what everybody else is doing so we can be inspired by each other’s ideas,” said Iniyaa, a student in Asmus’ class. 

Answer the Right Questions

As students work vertically in groups, teachers can easily see how they’re progressing and bounce around the room. Questions will undoubtedly arise, but Liljedahl says teachers should avoid answering questions asked for the purpose of reducing student effort and getting to an answer more quickly—such as “is this right?” Instead, they should prioritize addressing student questions that will lead to further independent thinking. 

In her elementary school classroom in Brooklyn, teacher Tori Filler says that instead of rushing to provide hints and solutions to the hard parts of a lesson, she often asks them to evaluate what’s tough about it, and encourages them to sort it out on their own before stepping in to help. 

If most of the class is struggling, ask more questions and turn the struggle into a productive discussion for everyone. Asking questions like “What makes this hard?” or “What have we tried?” gets students to think metacognitively and develop the skills to push through challenging work on their own. 

Evaluate What You Value

To succeed in a thinking classroom, students need to develop skills like perseverance, academic courage, collaboration, and curiosity, among others. But according to Liljedahl, if we want students to develop these competencies, then we should find ways to evaluate them on it. 

“What we choose to evaluate tells students what we value, and, in turn, students begin to value it as well,” he writes. 

He argues for a mix of both formative and summative assessment in math classrooms that focus less on end products and student ranking, and more on the work leading up to those end products and collaboration between groups to get there. 

The educators at Bite Sized Learning , for example, suggest evaluating students on the work they produce, but also on how well they persevere and make an effort in response to challenges, how well they set individual goals and monitor progress toward achieving them, and how well they share information and resources with group members to solve problems and make decisions. 

Liljedahl’s own formative assessments focus on informing students “about where they are and where they’re going in their learning.” This can take the form of observations, check-for-understanding questions, or even unmarked quizzes. Summative assessments, meanwhile, “should focus more on the processes of learning than on the products, and should include the evaluation of both group and individual work,” Liljedahl writes. 

• Inside Mathematics
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• standard 1 make sense problems persevere solving them

Standard 1: Make Sense of Problems & Persevere in Solving Them

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Classroom Observations

Teachers who are developing students’ capacity to “make sense of problems and persevere in solving them” develop ways of framing mathematical challenges that are clear and explicit, and then check-in repeatedly with students to help them clarify their thinking and their process. An early childhood teacher might ask her students to work in pairs to evaluate their approach to a problem, telling a partner to describe their process, saying “what [they] did, and what [they] might do next time.” A middle childhood teacher might post a set of different approaches to a solution, asking students to identify “what this mathematician was thinking or trying out” and evaluating the success of the strategy. An early adolescence teacher might have students articulate a specific way of laying out the terrain of a problem and evaluating different starting points for solving. A teacher of adolescents and young adults might frame the task as a real-world design conundrum, inviting students to engage in a “tinkering” process of working toward mathematical proof, changing course as necessary as they develop their thinking. Visit the video excerpts below to view multiple examples of teachers engaging students in sense-making and mathematical perseverance.

The Standard

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

Practice Standards

• Make sense of problems & persevere in solving them
• Reason abstractly & quantitatively
• Construct viable arguments & critique the reasoning of others
• Model with mathematics
• Use appropriate tools strategically
• Attend to precision
• Look for & make use of structure
• Look for & express regularity in repeated reasoning

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Connections to Classroom Practices

Connections to Classroom Practices (29)

Center for Teaching

Teaching problem solving.

Print Version

Tips and Techniques

Expert vs. novice problem solvers, communicate.

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

Encourage Independence

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

Be sensitive

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

Encourage Thoroughness and Patience

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

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

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

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Math and Special Education Blog

8 problem solving strategies for the math classroom.

Posted by Colleen Uscianowski · February 25, 2014

Would you draw a picture, make a list  possible number pairs that have the ratio 5:3, or guess and check? 

Explicit strategy instruction should be an integral part of your math classroom, whether you're teaching kindergarten or 12th grade.

Teach students that they can choose from a list of strategies to solve a problem, and often there isn't one correct way of finding a solution.

Demonstrate how you solve a word problem by thinking aloud as you choose and execute a strategy.

Ask students if they would solve the problem differently and praise students for coming up with unique ways of arriving at an answer.

Here are some problem-solving strategies I've taught my students:

Below is a helpful chart to remind students of the many problem-solving strategies they can use when solving word problems. This useful handout is a great addition to students' strategy binders, math notebooks, or math journals.  

How do you teach problem-solving in your classroom? Feel free to share advice and tips below!    

Sign up to receive a FREE copy of our problem-solving poster.

Problem Solving Within the Mathematics Classroom: Challenges and Recommendations

• October 2021

• The University of the West Indies at Mona
• This person is not on ResearchGate, or hasn't claimed this research yet.

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Students’ Collaborative Problem Solving in Mathematics Classrooms

An Empirical Study

• Open Access
• © 2024

You have full access to this open access Book

• Yiming Cao 0

School of Mathematical Sciences, Beijing Normal University, Beijing, China

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• Is open access, which means that you have free and unlimited access
• Explores students’ group collaboration process from the perspective of cognitive and social interaction
• Investigates teachers’ intervention behavior and the cognitive process behind it
• Offers evidence about the validity of prior theories to different situation

Part of the book series: Perspectives on Rethinking and Reforming Education (PRRE)

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About this book

This open access book provides key insights into the social fundamentals of learning and indications of social interactive modes conducive and restrictive of that learning in China. Combining theoretical and technical advances in an innovative research design, this book focuses on collaborative problem solving in mathematics to increase the visibility of social interactions in teachers’ designing, students’ learning and teachers’ instructional intervention. It also explores students’ cognitive and social interaction as well as teacher intervention in students’ group collaboration.

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The social essentials of learning: an experimental investigation of collaborative problem solving and knowledge construction in mathematics classrooms in australia and china.

Utilizing Multiple Methods in Mathematics Problem Solving: Contrasts and Commonalities Between Two Canadian and Chinese Elementary Schools

Collaborative Learning to Improve Problem-Solving Skills: A Relation Affecting Through Attitude Toward Mathematics

• Collaborative Problem Solving
• Group Collaboration
• Problem Solving
• Collaborative Learning
• Social Interaction
• Cognitive Interaction
• Student Participation
• Student Interaction
• Teacher Guidance
• Teacher Intervention

Table of contents (11 chapters)

Front matter, research on collaborative problem solving teaching in a secondary school mathematics classroom, examining junior high school students’ collaborative knowledge building: based on the comparison of high- & low-performance groups’ mathematical problem-solving, how did students solve mathematics tasks collaboratively an investigation of chinese students’ participation in groups.

• Shu Zhang, Yiming Cao

Research on Individual Authority and Group Authority Relations in Collaborative Problem Solving in Middle School Mathematics

The development and use of opportunity to learn (otl) in the collaborative problem solving: evidence from chinese secondary mathematics  classroom.

• Yinan Sun, Boran Yu

The Characteristics of Mathematical Communication in Secondary School Students’ Collaborative Problem Solving

A study of conflict discourse in mathematical collaborative problem solving.

• Jingbo Zhao

Research on Student Interaction in Peer Collaborative Problem Solving in Mathematics

• Zhengyi Zhang

Differences Between Experienced and Preservice Teachers in Noticing Students’ Collaborative Problem-Solving Processes

Teacher intervention in collaborative mathematics problem solving in secondary school.

• Yixuan Liu, Hang Wei

Research on the Evaluation of Students’ Collaborative Problem-Solving

• Bingxuan Du

Back Matter

Editors and affiliations, about the editor, bibliographic information.

Book Title : Students’ Collaborative Problem Solving in Mathematics Classrooms

Book Subtitle : An Empirical Study

Editors : Yiming Cao

Series Title : Perspectives on Rethinking and Reforming Education

DOI : https://doi.org/10.1007/978-981-99-7386-6

Publisher : Springer Singapore

eBook Packages : Education , Education (R0)

Copyright Information : The Editor(s) (if applicable) and The Author(s) 2024

Hardcover ISBN : 978-981-99-7385-9 Published: 04 January 2024

Softcover ISBN : 978-981-99-7388-0 Published: 04 January 2024

eBook ISBN : 978-981-99-7386-6 Published: 03 January 2024

Series ISSN : 2366-1658

Series E-ISSN : 2366-1666

Edition Number : 1

Number of Pages : XII, 297

Number of Illustrations : 31 b/w illustrations, 46 illustrations in colour

Topics : Mathematics Education , Education, general , Education, general

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