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## Problem Solving

## 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

The home of mathematics education in New Zealand.

## What is Problem Solving?

## Introduction

## Four Stages of Problem Solving

34 → 17 → 52 → 26 → 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1

## Scientific Approach

## Problem Solving in Mathematics

## Use Established Procedures

## Look for Clue Words

Common clue words for addition problems:

Common clue words for subtraction problems:

Common clue words for multiplication problems:

Common clue words for division problems:

## Read the Problem Carefully

- Ask yourself if you've seen a problem similar to this one. If so, what is similar about it?
- What did you need to do in that instance?
- What facts are you given about this problem?
- What facts do you still need to find out about this problem?

## Develop a Plan and Review Your Work

- Define your problem-solving strategy or strategies. This might mean identifying patterns, using known formulas, using sketches, and even guessing and checking.
- If your strategy doesn't work, it may lead you to an ah-ha moment and to a strategy that does work.

If it seems like you’ve solved the problem, ask yourself the following:

- Does your solution seem probable?
- Does it answer the initial question?
- Did you answer using the language in the question?
- Did you answer using the same units?

If you feel confident that the answer is “yes” to all questions, consider your problem solved.

## Tips and Hints

Some key questions to consider as you approach the problem may be:

- What are the keywords in the problem?
- Do I need a data visual, such as a diagram, list, table, chart, or graph?
- Is there a formula or equation that I'll need? If so, which one?
- Will I need to use a calculator? Is there a pattern I can use or follow?

## What is Mathematical Problem Solving

## Learn More About Mathematical Problem Solving in These Related Titles

## Problem Solving

Definition of problem solving.

The procedure used to solve a problem is called Problem Solving.

## More About Problem Solving

## Video Examples: 7 Step Problem Solving

## Example of Problem Solving

## Solved Example on Problem Solving

## Related Worksheet

- Locating-Integers-on-a-Number-Line-Gr-6
- Subtracting-Decimals-to-Hundredths-Gr-5
- Adding-Whole-Numbers-without-Regrouping---3-+-3-Gr-3
- Extending-and-Recognizing-Geometric-Patterns-by-its-Rules-Gr-4
- Likelihood-of-an-Event-Gr-7

## HighSchool Math

## 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

## 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]

## IMAGES

## VIDEO

## COMMENTS

(The term "problem solving" refers to mathematical tasks that have the potential to provide intellectual challenges for enhancing students' mathematical

Problem-Solving involves using the knowledge gained in Understanding, along with the intelligent practice and experience gained from Fluency, in order to inform

in mathematics a problem is a question which needs a mathematical solution. • problems may be written in words or using numbers and variables. • problem solving

The importance of problem-solving in learning mathematics comes from the belief that mathematics is primarily about reasoning, not memorization.

On the other hand, the processes of mathematics are the ways of using the skills creatively in new situations. Mathematical processes include problem solving

A multistep math problem-solving plan involves looking for clues, developing a game plan, solving the problem, and carefully reflecting on

What is Mathematical Problem Solving? Definition of Mathematical Problem Solving: The actions an individual takes to establish the means of achieving a

Definition Of Problem Solving. The procedure used to solve a problem is called Problem Solving. More About Problem Solving. Some of the problems solving

According to NCTM (2010), the term “problem solving” refers to mathematical tasks that have the potential to provide intellectual challenges for enhancing

In fact, problem-solving instruction creates opportunities for students to apply their knowledge of mathematical concepts, integrate and connect