advantages and disadvantages of problem solving method of teaching mathematics

5 Advantages and Disadvantages of Problem-Based Learning [+ Activity Design Steps]

Written by Marcus Guido

• Teaching Strategies

• Advantages of Problem-Based Learning
• Disadvantages of Problem-Based Learning
• Steps to Designing Problem-Based Learning Activities

Used since the 1960s, many teachers express concerns about the effectiveness of problem-based learning (PBL) in certain classroom settings.

Whether you introduce the student-centred pedagogy as a one-time activity or mainstay exercise, grouping students together to solve open-ended problems can present pros and cons.

Below are five advantages and disadvantages of problem-based learning to help you determine if it can work in your classroom.

If you decide to introduce an activity, there are also design creation steps and a downloadable guide to keep at your desk for easy reference.

1. Development of Long-Term Knowledge Retention

Students who participate in problem-based learning activities can improve their abilities to retain and recall information, according to a literature review of studies about the pedagogy .

The literature review states “elaboration of knowledge at the time of learning” -- by sharing facts and ideas through discussion and answering questions -- “enhances subsequent retrieval.” This form of elaborating reinforces understanding of subject matter , making it easier to remember.

Small-group discussion can be especially beneficial -- ideally, each student will get chances to participate.

But regardless of group size, problem-based learning promotes long-term knowledge retention by encouraging students to discuss -- and answer questions about -- new concepts as they’re learning them.

2. Use of Diverse Instruction Types

You can use problem-based learning activities to the meet the diverse learning needs and styles of your students, effectively engaging a diverse classroom in the process. In general, grouping students together for problem-based learning will allow them to:

• Address real-life issues that require real-life solutions, appealing to students who struggle to grasp abstract concepts
• Participate in small-group and large-group learning, helping students who don’t excel during solo work grasp new material
• Talk about their ideas and challenge each other in a constructive manner, giving participatory learners an avenue to excel
• Tackle a problem using a range of content you provide -- such as videos, audio recordings, news articles and other applicable material -- allowing the lesson to appeal to distinct learning styles

Since running a problem-based learning scenario will give you a way to use these differentiated instruction approaches , it can be especially worthwhile if your students don’t have similar learning preferences.

3. Continuous Engagement

Providing a problem-based learning challenge can engage students by acting as a break from normal lessons and common exercises.

It’s not hard to see the potential for engagement, as kids collaborate to solve real-world problems that directly affect or heavily interest them.

Although conducted with post-secondary students, a study published by the Association for the Study of Medical Education reported increased student attendance to -- and better attitudes towards -- courses that feature problem-based learning.

These activities may lose some inherent engagement if you repeat them too often, but can certainly inject excitement into class.

4. Development of Transferable Skills

Problem-based learning can help students develop skills they can transfer to real-world scenarios, according to a 2015 book that outlines theories and characteristics of the pedagogy .

The tangible contexts and consequences presented in a problem-based learning activity “allow learning to become more profound and durable.” As you present lessons through these real-life scenarios, students should be able to apply learnings if they eventually face similar issues.

For example, if they work together to address a dispute within the school, they may develop lifelong skills related to negotiation and communicating their thoughts with others.

As long as the problem’s context applies to out-of-class scenarios, students should be able to build skills they can use again.

5. Improvement of Teamwork and Interpersonal Skills

Successful completion of a problem-based learning challenge hinges on interaction and communication, meaning students should also build transferable skills based on teamwork and collaboration . Instead of memorizing facts, they get chances to present their ideas to a group, defending and revising them when needed.

What’s more, this should help them understand a group dynamic. Depending on a given student, this can involve developing listening skills and a sense of responsibility when completing one’s tasks. Such skills and knowledge should serve your students well when they enter higher education levels and, eventually, the working world.

1. Potentially Poorer Performance on Tests

Devoting too much time to problem-based learning can cause issues when students take standardized tests, as they may not have the breadth of knowledge needed to achieve high scores. Whereas problem-based learners develop skills related to collaboration and justifying their reasoning, many tests reward fact-based learning with multiple choice and short answer questions. Despite offering many advantages, you could spot this problem develop if you run problem-based learning activities too regularly.

2. Student Unpreparedness

Problem-based learning exercises can engage many of your kids, but others may feel disengaged as a result of not being ready to handle this type of exercise for a number of reasons. On a class-by-class and activity-by-activity basis, participation may be hindered due to:

• Immaturity  -- Some students may not display enough maturity to effectively work in a group, not fulfilling expectations and distracting other students.
• Unfamiliarity  -- Some kids may struggle to grasp the concept of an open problem, since they can’t rely on you for answers.
• Lack of Prerequisite Knowledge  -- Although the activity should address a relevant and tangible problem, students may require new or abstract information to create an effective solution.

You can partially mitigate these issues by actively monitoring the classroom and distributing helpful resources, such as guiding questions and articles to read. This should keep students focused and help them overcome knowledge gaps. But if you foresee facing these challenges too frequently, you may decide to avoid or seldom introduce problem-based learning exercises.

3. Teacher Unpreparedness

If supervising a problem-based learning activity is a new experience, you may have to prepare to adjust some teaching habits . For example, overtly correcting students who make flawed assumptions or statements can prevent them from thinking through difficult concepts and questions. Similarly, you shouldn’t teach to promote the fast recall of facts. Instead, you should concentrate on:

• Giving hints to help fix improper reasoning
• Questioning student logic and ideas in a constructive manner
• Distributing content for research and to reinforce new concepts
• Asking targeted questions to a group or the class, focusing their attention on a specific aspect of the problem

Depending on your teaching style, it may take time to prepare yourself to successfully run a problem-based learning lesson.

4. Time-Consuming Assessment

If you choose to give marks, assessing a student’s performance throughout a problem-based learning exercise demands constant monitoring and note-taking. You must take factors into account such as:

• Completed tasks
• The quality of those tasks
• The group’s overall work and solution
• Communication among team members
• Anything you outlined on the activity’s rubric

Monitoring these criteria is required for each student, making it time-consuming to give and justify a mark for everyone.

5. Varying Degrees of Relevancy and Applicability

It can be difficult to identify a tangible problem that students can solve with content they’re studying and skills they’re mastering. This introduces two clear issues. First, if it is easy for students to divert from the challenge’s objectives, they may miss pertinent information. Second, you could veer off the problem’s focus and purpose as students run into unanticipated obstacles. Overcoming obstacles has benefits, but may compromise the planning you did. It can also make it hard to get back on track once the activity is complete. Because of the difficulty associated with keeping activities relevant and applicable, you may see problem-based learning as too taxing.

If the advantages outweigh the disadvantages -- or you just want to give problem-based learning a shot -- follow these steps:

1. Identify an Applicable Real-Life Problem

Find a tangible problem that’s relevant to your students, allowing them to easily contextualize it and hopefully apply it to future challenges. To identify an appropriate real-world problem, look at issues related to your:

• Students’ shared interests

You must also ensure that students understand the problem and the information around it. So, not all problems are appropriate for all grade levels.

2. Determine the Overarching Purpose of the Activity

Depending on the problem you choose, determine what you want to accomplish by running the challenge. For example, you may intend to help your students improve skills related to:

• Collaboration
• Problem-solving
• Curriculum-aligned topics
• Processing diverse content

A more precise example, you may prioritize collaboration skills by assigning specific tasks to pairs of students within each team. In doing so, students will continuously develop communication and collaboration abilities by working as a couple and part of a small group. By defining a clear purpose, you’ll also have an easier time following the next step.

3. Create and Distribute Helpful Material

Handouts and other content not only act as a set of resources, but help students stay focused on the activity and its purpose. For example, if you want them to improve a certain math skill , you should make material that highlights the mathematical aspects of the problem. You may decide to provide items such as:

• Data that helps quantify and add context to the problem
• Videos, presentations and other audio-visual material
• A list of preliminary questions to investigate

Providing a range of resources can be especially important for elementary students and struggling students in higher grades, who may not have self-direction skills to work without them.

4. Set Goals and Expectations for Your Students

Along with the aforementioned materials, give students a guide or rubric that details goals and expectations. It will allow you to further highlight the purpose of the problem-based learning exercise, as you can explain what you’re looking for in terms of collaboration, the final product and anything else. It should also help students stay on track by acting as a reference throughout the activity.

5. Participate

Although explicitly correcting students may be discouraged, you can still help them and ask questions to dig into their thought processes. When you see an opportunity, consider if it’s worthwhile to:

• Fill gaps in knowledge
• Provide hints, not answers
• Question a student’s conclusion or logic regarding a certain point, helping them think through tough spots

By participating in these ways, you can provide insight when students need it most, encouraging them to effectively analyze the problem.

6. Have Students Present Ideas and Findings

If you divided them into small groups, requiring students to present their thoughts and results in front the class adds a large-group learning component to the lesson. Encourage other students to ask questions, allowing the presenting group to elaborate and provide evidence for their thoughts. This wraps up the activity and gives your class a final chance to find solutions to the problem.

Wrapping Up

The effectiveness of problem-based learning may differ between classrooms and individual students, depending on how significant specific advantages and disadvantages are to you. Evaluative research consistently shows value in giving students a question and letting them take control of their learning. But the extent of this value can depend on the difficulties you face.It may be wise to try a problem-based learning activity, and go forward based on results.

Create or log into your teacher account on Prodigy -- an adaptive math game that adjusts content to accommodate player trouble spots and learning speeds. Aligned to US and Canadian curricula, it’s used by more than 350,000 teachers and 10 million students. It may be wise to try a problem-based learning activity, and go forward based on results.

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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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Teaching Mathematics Through Problem Solving

By Tom McDougal, Akihiko Takahashi

What do your students do when faced with a math problem they don't know how to solve? Most students give up pretty quickly. At best, they seek help from another student or the teacher. At worst, they shut down, seeing their failure as more evidence that they just aren't good at math. Neither of these behaviors will serve students in the long run. Inevitably, someday, every one of your students will encounter problems that they will not have explicitly studied in school and their ability to find a solution will have important consequences for them.

In the Common Core State Standards for Mathematics, the very first Standard for Mathematical Practice is that students should “understand problems and persevere in solving them.”1 Whether you are beholden to the Common Core or not, this is certainly something you would wish for your students. Indeed, the National Council of Teachers of Mathematics (NCTM) has been advocating for a central role for problem solving at least since the release of Agenda for Action in 1980, which said, “Problem solving [must] be the focus of school mathematics… .”2 

The common instructional model of “I do, we do, you do,” increases student dependence on the teacher and decreases students’ inclination to persevere. How, then, can teachers develop perseverance in problem solving in their students?

First we should clarify what we mean by “problem solving.” According to NCTM, “Problem solving means engaging in a task for which the solution is not known in advance.”3 A task does not have to be a word problem to qualify as a problem — it could be an equation or calculation that students have not previously learned to solve. Also, the same task can be a problem or not, depending on when it is given. Early in the year, before students learn a particular skill, the task could be a problem; later, it becomes an exercise, because now they know how to solve it.

In Japan, math educators have been thinking about how to develop problem solving for several decades. They studied George Polya's How to Solve It ,4 NCTM's Agenda for Action , and other documents, and together, using a process called lesson study , they began exploring what it would mean to make problem solving “the focus of school mathematics.” And they succeeded. Today, most elementary mathematics lessons in Japan are organized around the solving of one or a very few problems, using an approach known as “teaching through problem solving.”

“Teaching through problem solving” needs to be clearly distinguished from “teaching problem solving.” The latter, which is not uncommon in the United States, focuses on teaching certain strategies — guess-and-check, working backwards, drawing a diagram, and others. In a lesson about problem solving, students might work on a problem and then share with the class how using one of these strategies helped them solve the problem. Other students applaud, the students sit down, and the lesson ends. These lessons are usually outside the main flow of the curriculum; indeed, they are purposely independent of any curriculum.

In “teaching through problem solving,” on the other hand, the goal is for students to learn precisely that mathematical idea that the curriculum calls for them to learn next.

A “teaching through problem solving” lesson would begin with the teacher setting up the context and introducing the problem. Students then work on the problem for about 10 minutes while the teacher monitors their progress and notes which students are using which approaches. Then the teacher begins a whole-class discussion. Similar to a “teaching problem solving” lesson, the teacher may call on students to share their ideas, but, instead of ending the lesson there, the teacher will ask students to think about and compare the different ideas — which ideas are incorrect and why, which ideas are correct, which ones are similar to each other, which ones are more efficient or more elegant. Through this discussion, the lesson enables students to learn new mathematical ideas or procedures. This approach is represented in Figure 1.

Let's illustrate this with an example from a hypothetical fifth-grade lesson based on the most popular elementary mathematics textbook in Japan. (This textbook has been translated into English as Mathematics International and is available at http://GlobalEdResources.com . 5) During most Japanese lessons, the textbook is closed, but the textbook shows how the authors think the lesson might play out.

When the lesson begins, the blackboard is completely empty. The teacher starts by displaying, either with a poster or using a projector, the picture from the textbook of four different rabbit cages, shown in Figure 2 (it is not uncommon for Japanese elementary students to care for rabbits in several rabbit hutches, so this is a familiar context).

Figure 2 (Mathematics International, Grade 5, p. A93)

“What do you notice about the cages?” the teacher asks. Some students notice that some of the cages are different sizes. The teacher then asks, “Should each cage have the same number of rabbits?” No, say the students, smaller cages should have fewer rabbits, so the rabbits aren't too crowded.

The teacher then displays the pictures in Figure 3. “What do you think?” the teacher asks, as he puts them up one at a time for dramatic effect. “Are these equally crowded, or do you think some cages are more crowded than others?” There is some discussion about the rabbits in cage B, and students decide that just because they are bunched together right now, they probably won't stay that way. Students recognize that cages A and B are the same size, and since cage A has more rabbits (9 vs. 8), it is more crowded. The teacher writes that observation on the board: “When two cages are the same size, the one with more rabbits is more crowded.”

Figure 3 (Mathematics International, Grade 5, p. A93)

“What about the others?” he asks. “How can we decide which are more crowded?” This last question becomes the key mathematical question of the lesson, and the teacher writes it on the board: “Let's think about how to compare crowdedness.” Students copy this problem in their notebooks while he writes.

The teacher gives students a piece of paper with the pictures from Figure 3 to glue in their notebooks and gives them 5 minutes to think about the problem. Several students take a ruler and begin measuring. “Why are you doing that?” the teacher quietly asks one of them. “I want to figure out the area,” the student says. “Oh! You think the area might be important. Write that idea in your notebook.” Other students count the rabbits and decide that B and C are equally crowded because they look like they are the same size, but they are unsure about D.

The teacher stops the students and asks for ideas. He first calls on a student who thinks that B and C are the same size. He records her idea on the board: “Arthi says B and C look like they are the same size and have the same number of rabbits, so they are equally crowded.” A student who found the areas says that they are not. The teacher records this idea on the board: “Karen thinks you need to know the area.” He turns to the first student. “Arthi, what do you think?” he asks. She and other students agree. The teacher posts a table with the areas of the four cages (Figure 4). “Let's copy this table into our notebooks, and think about the problem some more.”

Figure 4 (Mathematics International, Grade 5, p. A94)

Students work independently for another 5 minutes while the teacher monitors their progress, encourages them to keep thinking, and reminds them to record their ideas in their notebook. He anticipates the following five ideas and notes which students are using them:

Idea 1: B and C have the same number of rabbits, but C has a smaller area, so C is more crowded. Unsure about A vs. C. 

Idea 2: If you make 5 copies of A and 6 copies of C, they would have the same area (30 m2). A would then have 45 rabbits while C would have 48 rabbits, so C is more crowded. 

Idea 3: If you make 8 copies of A and 9 copies of C, they would have the same number of rabbits (72). A would have an area of 48 m2 while C would have an area of 45 m2, so B is more crowded. 

Idea 4: Divide: (area) ÷ (# of rabbits) = amount of area per rabbit 

Idea 5: Divide: (# of rabbits) ÷ (area) = number of rabbits per unit area

The teacher invites students to explain their ideas to the class, selecting students based on the order above, while he records each idea on the blackboard. He asks students to compare Idea 1 to the thinking used to compare A and B. He writes on the board: “If either the area or the number of rabbits is the same, it's easy to compare.” The student with Idea 2 says, “I found a way to make the area the same,” and explains. This prompts the student with Idea 3 to say, “I used kind of the same approach to make the number of rabbits the same.”

When a student with Idea 4 comes up, she begins, “I decided to divide the area by the number of rabbits.” The teacher stops her. He writes: “(area) ÷ (# of rabbits).” Then he asks the class, “Why is she doing this? Who can explain her thinking?” Another student says, “That gives the amount of area for each rabbit.” He lets the student finish her idea:

A: 9÷6 = 1.5 C: 8÷5 = 1.6

The teacher asks the class to clarify what the 1.5 and 1.6 mean (m2 per rabbit) and what that says about the crowdedness of each cage.

He then invites a student to explain Idea 5: “I divided the other way…”

A: 6÷9 = 0.66… C: 5÷8 = 0.625

“Why is he doing this?” the teacher asks the class. “What does this 0.66… mean? What does 0.625 mean?” (“Rabbits per square meter,” the students answer.)

The teacher then asks the class to look for similarities across the five ideas, which are all visible on the blackboard. Some students note that Ideas 2 and 3 use multiplication while Ideas 4 and 5 use division, a superficial similarity. But some students notice the more significant connection that 2 and 5 are both about making the area the same, while 3 and 4 are both about making the number of rabbits the same.

“We haven't talked about cage D yet,” the teacher points out. “How shall we compare A, C, and D? Please try using one of these ideas.”

Students work in their notebooks for a few minutes. Students who try using multiplication (Idea 2 or 3) discover that the method is cumbersome. The teacher invites students who used Ideas 4 and 5 to share their calculations, adding them to the lists from before: Idea 4:

A: 9÷6 = 1.5 C: 8÷5 = 1.6 D: 15÷9 = 1.66… (m2/rabbit) Idea 5: A: 6÷9 = 0.66… C: 5÷8 = 0.625 D: 9÷15 = 0.6 (rabbits/m2)

“What do you think about these ideas?” asks the teacher, and students respond, “They are easy!” So the teacher writes a summary on the board, “Using division, it is easy to compare crowdedness.” He asks the students to write a reflection in their notebooks. One student who used multiplication writes, “I tried using multiplication, but dividing is easier. Next time I want to try that.” And the lesson ends.

In the students’ previous experience with comparing quantities, a single quantity was important, such as the number of apples or kilograms or square meters. Their prior experience with division was about finding a missing multiplier or multiplicand, which was itself a single quantity. This problem presented students for the first time with a situation in which two numbers needed to be considered. So by working on a problem about rabbits and cages, students learn that division can be used to compute a new type of quantity, a per unit quantity, that expresses the relationship between rabbits and area and can be used to compare crowdedness. In subsequent lessons, students will see how division can be used to compute other types of per unit quantities, such as the productivity of two farms in crops grown per acre of land or the cost per pencil.

What was the teacher's role in helping students learn this new mathematical idea? He never explained anything to the students, but the task had to be carefully constructed, and the teacher had to be very deliberate in how he directed the lesson, or the lesson wouldn't have worked.

The task was accessible to all students in the beginning by the fact that two cages had the same area (A and B) and two cages had the same number of rabbits (B and C), but since it wasn't clear whether B and C were the same size, students were pushed to think formally about area. And, while using multiplication was feasible for comparing cages A and C, the area of cage D was such that multiplication was cumbersome for comparing all three cages. Students who might have been happy with using multiplication and uncomfortable with the decimal values that result from division were pushed by cage D to appreciate the efficiency of using division.

The teacher's role in the lesson can be compared to the role of a film director, who carefully stages each scene and makes cuts between cameras to create the desired effect. Early in the lesson, the teacher highlighted the idea, raised by students, that equal areas or equal numbers of rabbits made comparisons easier. This was the foundation for the idea of dividing to find a “per unit quantity,” square meters per one rabbit or rabbits per one square meter. By starting with a discussion of incorrect or partially correct ideas and writing them on the board, the teacher valued those ideas. This encourages students to try: Even if they can’t solve the whole problem, they might come up with something to contribute. When a student first suggested the idea of dividing, the teacher asked other students to explain the thinking behind it. This enabled students who did not themselves think of dividing to make the idea their own. And by carefully organizing student ideas on the board (Figure 5), the teacher made it easier for students to compare those ideas with each other and to follow the flow of learning in the lesson.

Figure 5 (includes items from Mathematics International, Grade 5, pp. A93-94)

Although the lesson vignette above is fictional, videos of lessons like it can be found at http://tinyurl.com/kuwb4bg . The grade 3 lesson “Multiplication Algorithm” and the grade 5 lesson “Do I Have a Window Seat or an Aisle Seat?” are particularly good, both for the quality of the lessons and for the quality of the videos themselves.

Japanese educators believe that regular lessons that teach through problem solving, interspersed with occasional practice days, help their students learn mathematics more thoroughly than didactic instruction coupled with a greater amount of practice. Certainly Japanese students have performed very well on the TIMSS and PISA international studies of mathematics achievement. But perhaps more important, teaching through problem solving habituates students to being confronted with unfamiliar problems, to struggling at length with those problems, and to learning from those problems. This is a way to cultivate perseverance in problem solving.

Reading this article and watching videos, however, will not equip most teachers to incorporate teaching through problem solving into their practice. The teacher who wishes to do so is faced with several challenges. The first challenge is that few curricula are designed to support such lessons; most are designed to support fairly direct instruction by the teacher. The second problem is that students are not used to learning this way and may resist. And the third problem is that teaching this way is hard. It requires ways of thinking about a lesson that are unfamiliar to almost all U.S. teachers. One needs to be absolutely clear about what the mathematical goal of the lesson is; that goal is never for students to simply solve a problem. One needs to anticipate the various solutions, correct and incorrect, that are likely to come from students, as well as the ways students will get stuck. One needs to plan how the discussion around the various student ideas will address misconceptions and build toward the mathematical goal of the lesson. One needs to think about how the ideas will be organized on the board so that students can easily compare them.

Japanese teachers certainly did not learn to teach this way by reading articles or watching videos. They learned it — and continue to learn it — by trying it, together, one lesson at a time through a process called lesson study .6,7 A full treatment of lesson study would be another article in itself, but U.S. teachers who are interested in learning to teach through problem solving can find more information about lesson study at http://LessonStudyGroup.net  and at http://LSAlliance.org . Lesson Study Alliance organizes the annual Chicago Lesson Study Conference, which features live lessons by teachers who are working to incorporate teaching through problem solving into their practice.

1. National Governors Association Center for Best Practices, Council of Chief State School Officers, Common Core State Standards for Mathematics (Washington, DC: National Governors Association Center for Best Practices, Council of Chief State School Officers, 2010); online at www.corestandards.org/math/ . 2. National Council of Teachers of Mathematics, An Agenda for Action: Recommendations for School Mathematics of the 1980s (Washington, DC: NCTM, 1980); online at www.nctm.org/standards/content.aspx?id=17278 .  3. National Council of Teachers of Mathematics, Principles and Standards for School Mathematics (Washington, DC: NCTM, 2000); online at http://www.nctm.org/standards/content.aspx?id=16909 . 4. George Polya, How to Solve It: A New Aspect of Mathematical Method (Princeton, NJ: Princeton University Press, 1945).  5. T. Fujii and S. Iitaka, Mathematics International , Grades 1-6 (Tokyo: Tokyo Shoseki Co., Ltd., 2012).  6. Akihiko Takahashi, “Implementing Lesson Study in North American Schools and School Districts” (no date); online at http://hrd.apec.org/images/a/ae/51.2.pdf .  7. Akihiko Takahashi and Makoto Yoshida, “Ideas for Establishing Lesson-Study Communities.” Teaching Children Mathematics , May 2004.

Tom McDougal is executive director of Lesson Study Alliance in Chicago, a nonprofit organization that promotes and supports Lesson Study. He taught middle and high school mathematics and was an elementary math specialist.

Akihiko Takahashi is associate professor of mathematics education at DePaul University in Chicago. He taught students in grades 1-6 for 19 years in Japan, where he helped lead the national shift to teaching mathematics through problem solving.

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Learning to teach mathematics through problem posing: teachers’ beliefs and performance on problem posing

• Published: 07 September 2020
• Volume 105 , pages 325–347, ( 2020 )

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• Xinlian Li 1 ,
• Naiqing Song 1 ,
• Stephen Hwang 2 &
• Jinfa Cai 2  

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Any effort to integrate problem-posing instruction in school mathematics must attend to teachers’ beliefs about the advantages of teaching through problem posing and especially their beliefs about the challenges of teaching in this way. This study investigated teachers who were learning how to teach mathematics through problem posing. The primary foci were the teachers’ beliefs about teaching through problem posing and their problem-posing performance. The findings demonstrate that the participating teachers were able to engage successfully in problem-posing tasks and that they held a number of different beliefs about the advantages and challenges of teaching through problem posing. After participating in three workshops, a sample of teachers showed gains in problem-posing performance and in the scope and specificity of their beliefs about teaching through problem posing.

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Li, X., Song, N., Hwang, S. et al. Learning to teach mathematics through problem posing: teachers’ beliefs and performance on problem posing. Educ Stud Math 105 , 325–347 (2020). https://doi.org/10.1007/s10649-020-09981-0

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