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## 5 Teaching Mathematics Through Problem Solving

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.

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.

## Mathematics Tasks and Activities that Promote Teaching through Problem Solving

## Choosing the Right Task

- 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

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

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

- 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

- 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.

## Saying “This is Easy”

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:

## Using “Worksheets”

- 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 ”.

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

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

## Share This Book

## The Lesson Study Group

## Teaching Through Problem-solving

## What is Teaching Through Problem-Solving?

## Why Teaching Through Problem-Solving?

As students build their mathematical knowledge, they also:

- Learn to reason mathematically, using prior knowledge to build new ideas
- See the power of their explanations and carefully written work to spark insights for themselves and their classmates
- Expect mathematics to make sense
- Enjoy solving unfamiliar problems
- Experience mathematical discoveries that naturally deepen their perseverance

## Phases of a TTP Lesson

## WHAT STUDENTS DO

- Understand the problem and develop interest in solving it.
- Consider what they know that might help them solve the problem.

## WHAT TEACHERS DO

- Show several student journal reflections from the prior lesson.
- Pose a problem that students do not yet know how to solve.
- Interest students in the problem and in thinking about their own related knowledge.
- Independently try to solve the problem.
- Do not simply following the teacher’s solution example.
- Allow classmates to provide input after some independent thinking time.
- Circulate, using seating chart to note each student’s solution approach.
- Identify work to be presented and discussed at board.
- Ask individual questions to spark more thinking if some students finish quickly or don’t get started.
- Present and explain solution ideas at the board, are questioned by classmates and teacher. (2-3 students per lesson)
- Actively make sense of the presented work and draw out key mathematical points. (All students)
- Strategically select and sequence student presentations of work at the board, to build the new mathematics. (Incorrect approaches may be included.)
- Monitor student discussion: Are all students noticing the important mathematical ideas?
- Add teacher moves (questions, turn-and-talk, votes) as needed to build important mathematics.
- Consider what they learned and share their thoughts with class, to help formulate class summary of learning. Copy summary into journal.
- Write journal reflection on their own learning from the lesson.
- Write on the board a brief summary of what the class learned during the lesson, using student ideas and words where possible.
- Ask students to write in their journals about what they learned during the lesson.

## How Do Teachers Support Problem-solving?

## Additional Readings

## Can’t find a resource you need? Get in touch.

- What is Lesson Study?
- Why Lesson Study?
- Teacher Learning
- Content Resources
- Teaching Through Problem-solving (TTP)
- School-wide Lesson Study
- U.S. Networks
- International Networks

## METHODS OF TEACHING MATHEMATICS

Friday, may 20, 2011, module 9: problem solving method.

## 3. Formulating tentative hypothesis

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## PROBLEM SOLVING METHOD: METHODS OF TEACHING MATHEMATICS

Problem solving is a set of events in which human beings was rules to achieve some goals – Gagne

Problem solving involves concept formation and discovery learning – Ausube

Steps in Problem Solving / Procedure for Problem solving

## 3. Formulating tentative hypothesis

Define union of two sets. If A={2,3,5}. B={3,5,6} And C={4,6,8,9}.

Step 1: Identifying and Defining the Problem

- The union of two sets A and B is the set, which contains all the members of a set A and all the members of a set B.
- The union of two set A and B is express as ‘AUB ’
- The common elements are taken only once in the union of two sets

Step 3 : Formulating Tentative Hypothesis

Thus on the basis of given data, the child will be able to solve the problem in the following manner

In the example it is given that

Step 5 : Verifying of the result

After testing and verifying his hypothesis the child will be able to conclude that

Thus the child generalises the results and apply his knowledge in new situations.

- This method is psychological and scientific in nature
- It helps in developing good study habits and reasoning powers.
- It helps to improve and apply knowledge and experience.
- This method stimulates thinking of the child
- It helps to develop the power of expression of the child.
- The child learns how to act in new situation.
- It develops group feeling while working together.
- Teachers become familiar with his pupils.
- It develops analytical, critical and generalization abilities of the child.
- This method helps in maintaining discipline in the class.
- This is not suitable for lower classes
- There is lack of suitable books and references for children.
- It is not economical. It is wastage of time and energy.
- Teachers find it difficult to cover the prescribed syllabus.
- To follow this method talented teacher are required.
- There is always doubt of drawing wrong conclusions.
- Mental activities are more emphasized as compared to physical activities.

Source: The Teaching of mathematics by KULBIR SINGH SIDHU (Sterling Publisher Pvt Ltd)

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## Teaching Mathematics Through Problem Solving

By Tom McDougal and Akihiko Takahashi

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

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

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

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

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

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

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

## Teaching & Learning: Creating a Culture of Academic Integrity

## People also looked at

- 1 Department of Education, Uppsala University, Uppsala, Sweden
- 2 Department of Education, Culture and Communication, Malardalen University, Vasteras, Sweden
- 3 School of Natural Sciences, Technology and Environmental Studies, Sodertorn University, Huddinge, Sweden
- 4 Faculty of Education, Gothenburg University, Gothenburg, Sweden

## Introduction

## The Present Study

a) What is the effect of CL approach on students’ problem-solving in mathematics?

## Participants

FIGURE 1 . Flow chart for participants included in data collection and data analysis.

TABLE 1 . Background characteristics of classes and teachers in intervention and control groups.

## Intervention

## Implementation of the Intervention

## Control Group

## Tests of Mathematical Problem-Solving

## Measures of Peer Acceptance and Friendships

## Statistical Analyses

## What Is the Effect of the CL Approach on Students’ Problem-Solving in Mathematics?

## Is Social Acceptance and Friendships Associated With the Effect of CL on Students’ Problem-Solving in Mathematics?

## Limitations

## Implications

## Data Availability Statement

## Ethics Statement

## Author Contributions

The project was funded by the Swedish Research Council under Grant 2016-04,679.

## Conflict of Interest

## Publisher’s Note

## Acknowledgments

We would like to express our gratitude to teachers who participated in the project.

## Supplementary Material

CrossRef Full Text | Google Scholar

PubMed Abstract | CrossRef Full Text | Google Scholar

Received: 15 May 2021; Accepted: 09 August 2021; Published: 24 August 2021.

*Correspondence: Nina Klang, [email protected]

The home of mathematics education in New Zealand.

## Benefits of Problem Solving

- Problem solving places the focus on the student making sense of mathematical ideas. When solving problems students are exploring the mathematics within a problem context rather than as an abstract.
- Problem solving encourages students to believe in their ability to think mathematically. They will see that they can apply the maths that they are learning to find the solution to a problem.
- Problem solving provides ongoing assessment information that can help teachers make instructional decisions. The discussions and recording involved in problem solving provide a rich source of information about students' mathematical knowledge and understanding.
- Good problem solving activities provide an entry point that allows all students to be working on the same problem. The open-ended nature of problem solving allows high achieving students to extend the ideas involved to challenge their greater knowledge and understanding.
- Problem solving develops mathematical power. It gives students the tools to apply their mathematical knowledge to solve hypothetical and real world problems.
- Problem solving is enjoyable. It allows students to work at their own pace and make decisions about the way they explore the problem. Because the focus is not limited to a specific answer students at different ability levels can experience both challenges and successes on the same problem.
- Problem solving better represents the nature of mathematics. Research mathematicians apply this exact approach in their work on a daily basis.
- Once students understand a problem solving approach to maths, a single well framed mathematical problem provides the potential for an extended period of exploration.

## Mathematics as a Complex Problem-Solving Activity

By jacob klerlein and sheena hervey, generation ready.

“Problem-solving is not only a goal of learning mathematics, but also a major means of doing so.”

## Learning to problem solve

## Beliefs underpinning effective teaching of mathematics

- Every student’s identity, language, and culture need to be respected and valued.
- Every student has the right to access effective mathematics education.
- Every student can become a successful learner of mathematics.

## Why is problem-solving important?

- The ability to think creatively, critically, and logically
- The ability to structure and organize
- The ability to process information
- Enjoyment of an intellectual challenge
- The skills to solve problems that help them to investigate and understand the world

## Problems that are “Problematic”

- Are accessible and extendable
- Allow individuals to make decisions
- Promote discussion and communication
- Encourage originality and invention
- Encourage “what if?” and “what if not?” questions
- Contain an element of surprise (Adapted from Ahmed, 1987)

- Understand and explore the problem
- Find a strategy
- Use the strategy to solve the problem
- Look back and reflect on the solution

## Pólya’s Principals of Problem-Solving

Students move forward and backward as they move through the problem-solving process.

## Getting real

## Planning for talk

## Teaching Problem Solving in Math

## The Problem Solving Strategies

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

There’s also this visuals strategy wheel practice.

## The Problem Solving Steps

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

We did this over and over with example problems.

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

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

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

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

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

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

Act – Act on a strategy and try it out.

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

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

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