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## Problem Solving

## Transformations

Home > Fischler > Transformations > Vol. 1 (2016) > Iss. 1

## Article Title

The Problem-Solving Process in a Mathematics Classroom

Enrique Ortiz , University of Central Florida

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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

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## Theses & Dissertations

Basiliana Caroli Mrimi , Aga Khan University, Institute for Educational Development, Karachi

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Institute for Educational Development, Karachi

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Mathematics through problem solving.

What Is A 'Problem-Solving Approach'?

- interactions between students/students and teacher/students (Van Zoest et al., 1994)
- mathematical dialogue and consensus between students (Van Zoest et al., 1994)
- teachers providing just enough information to establish background/intent of the problem, and students clarifing, interpreting, and attempting to construct one or more solution processes (Cobb et al., 1991)
- teachers accepting right/wrong answers in a non-evaluative way (Cobb et al., 1991)
- teachers guiding, coaching, asking insightful questions and sharing in the process of solving problems (Lester et al., 1994)
- teachers knowing when it is appropriate to intervene, and when to step back and let the pupils make their own way (Lester et al., 1994)
- A further characteristic is that a problem-solving approach can be used to encourage students to make generalisations about rules and concepts, a process which is central to mathematics (Evan and Lappin, 1994).

- valuing the processes of mathematization and abstraction and having the predilection to apply them
- developing competence with the tools of the trade and using those tools in the service of the goal of understanding structure - mathematical sense-making (Schoenfeld, 1994, p.60).
- As Cobb et al. (1991) suggested, the purpose for engaging in problem solving is not just to solve specific problems, but to 'encourage the interiorization and reorganization of the involved schemes as a result of the activity' (p.187). Not only does this approach develop students' confidence in their own ability to think mathematically (Schifter and Fosnot, 1993), it is a vehicle for students to construct, evaluate and refine their own theories about mathematics and the theories of others (NCTM, 1989). Because it has become so predominant a requirement of teaching, it is important to consider the processes themselves in more detail.

The Role of Problem Solving in Teaching Mathematics as a Process

- developing skills and the ability to apply these skills to unfamiliar situations
- gathering, organising, interpreting and communicating information
- formulating key questions, analyzing and conceptualizing problems, defining problems and goals, discovering patterns and similarities, seeking out appropriate data, experimenting, transferring skills and strategies to new situations
- developing curiosity, confidence and open-mindedness (NCTM, 1980, pp.2-3).

Gardner, Howard (1985). Frames of Mind. N.Y: Basic Books.

Resnick, L. B. (1987). 'Learning in school and out', Educational Researcher, 16, 13-20..

Stacey, K. and Groves, S. (1985). Strategies for Problem Solving, Melbourne, Victoria: VICTRACC.

Related Article on Teaching Values | Other Articles

## Featured Sites:

Free math worksheets, charts and calculators

## 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

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## Making Math Meaningful for Young Children

## You are here

## Creating a math-rich classroom

## Encourage children to play mathematically

## Supporting Dual language learners

## Metacognitive Strategies in the Math Classroom

In other words, metacognition is the process of thinking about thinking.

## How do metacognitive strategies help students learn?

💡 Research Spotlight: In one study, students’ problem-solving processes were qualitatively shown to be supported by engaging in metacognitive regulation — the active monitoring and controlling of cognitive processes (Jin & Kim, 2018). Students were able to help monitor and adjust each other’s thinking through their conversations. As students said things like, “This makes no sense” or “I don’t understand this,” other students would respond with, “Let’s try to think of this another way.” Desoete and De Craene (2019) noted that metacognitive skills were associated with mathematical accuracy. Reflection is also linked to social-emotional learning, as students can benefit from reflecting on the thoughts, feelings, and emotional aspects of what they have learned.

## How can teachers encourage metacognitive strategies in the classroom?

Metacognitive strategies can be integrated into regular classroom instruction through:

- Collaborative activities, such as students working in groups while discussing solutions to a given problem (Jin and Kim 2018)
- Thorough explanation of a topic allows students to reflect on what they know about a topic and connect it to new information they learn (Denton, 2011)
- Formative assessments (Denton, 2011), such as verbal discussions or written evaluations in which students complete a chart to explain how they feel about their learning.

💡 Research Spotlight : In one study, students were guided to use metacognitive questioning as part of the process of solving math problems. These metacognitive questions included comprehension, connection, strategic and reflection questions (Mevarech & Kramarski, 2003).

## Metacognitive Strategies in Math

Why is metacognition important in the math classroom.

- When studying mathematics, metacognitive strategies can play an important role in knowledge acquisition, retention, and application . At the conceptual development stage, when students are first encountering new ideas and skills, thinking about the relationships between their prior knowledge and new knowledge tends to help students have better conceptual understanding (Mevarech & Kramarski, 2003).
- Giving students the time and space to reflect on their own thinking is critical for fostering student agency . Research shows that agency is time bound, where individuals draw on their patterns, habits, and identity to set goals or outcomes, creating plans or actions toward reaching that goal and evaluating how well the plan and actions are helping meet the goal in the current context or if a new plan is needed (Adie, Willis, & Van der Kleij, 2018; Poon, 2018; Klemencic, 2015). The ability to engage in metacognition allows students to recognize and reflect on how their own thinking helps them reach their goals.
- Metacognitive skill development is critical for all learners, including those with learning disabilities .

💡 Research Spotlight : Desoete and De Craene (2019) found that metacognitive activities can help students with learning disabilities build computational accuracy and mathematical reasoning.

## Helpful Metacognitive Strategies in Math

## About the Authors

Margaret Bowman is an Academic Designer in the Mathematics Department at McGraw Hill.

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

## When I Finally Saw the Light

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

## Seasonal Problem Solving

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## Getting the Most from Each of the Problem Solving Activities

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

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## The Role of the Teacher Changes in a Problem-Solving Classroom

## Problem-Based Classrooms Require Letting Go

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## Exploring the effects of role scripts and goal-orientation scripts in collaborative problem-solving learning

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## IMAGES

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## COMMENTS

Problem solving plays an important role in mathematics and should have a prominent role in the mathematics education of K-12 students. However, knowing how to incorporate problem solving meaningfully into the mathematics curriculum is not necessarily obvious to mathematics teachers.

Encouraging the development of key problem-solving skills should be a priority, along with giving learners opportunities to develop a productive disposition towards mathematics. Taking time to reflect upon our own behaviours as teachers is crucial as this may help us develop more independent learners who relish mathematical challenges.

Problem solving provides a working framework to apply mathematics, and well chosen mathematics problems provide students with the opportunity to solidify and extend what they know, and can stimulate students' mathematics learning (NCTM, 2001).

Learning Mathematics: A New Role for Problem Solving Mathematics makes sense to students and is easier to remember and apply when students understand the mathematics they are learning. Also, students who understand mathematical concepts and skills more readily learn new mathematical concepts and skills. students who learn

For further support on the development of key problem-solving skills, see our Problem-Solving Feature, which includes several articles and links to more activities which will give learners experience of specific problem-solving skills. Part 2: The Teacher's Role It is perhaps easy to underestimate the effect teacher behaviour can have on ...

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.

A true problem solving process will allow students to be flexible, intuitive, and creative. The students should be allowed to move from one step to another, and through many alternatives and...

4 Algebra Readiness, Cycle 1 The Effective Mathematics Classroom What are some best practices for mathematics instruction? In general, a best practice is a way of doing something that is shown to generate the desired results. In terms of mathematics instruction, we typically think of a best practice as a teaching strategy or lesson structure that promotes a deep student understanding of ...

The Role of Problem Solving in the Secondary School Mathemat ics Classroom Rachel Wing DiMatteo Frank K. Lester Jr. According to Principles and Standards for School Mathematics [7, p. 52]...

The efforts to improve the teaching and learning of mathematics involve varied approaches; the problem-solving approach is one of them. This research study sets out to understand how problem-solving tasks work in a mathematics classroom in relation to developing students' mathematical thinking.

The Role of Problem Solving in Teaching Mathematics as a Process Problem solving is an important component of mathematics education because it is the single vehicle which seems to be able to achieve at school level all three of the values of mathematics listed at the outset of this article: functional, logical and aesthetic.

The importance of problem-solving in learning mathematics comes from the belief that mathematics is primarily about reasoning, not memorization. Problem-solving allows students to develop understanding and explain the processes used to arrive at solutions, rather than remembering and applying a set of procedures.

2007; 37). Focusing on problem solving in lessons develops the students' high level thinking. For this reason, students perform selflearning in mathematisc lesso- ns with problem solving process. Problem solving plays an important role in mathematics education and most of learning is an occour as a result of problem solving process.

Problem solving in mathematics education has been a prominent research field that aims at understanding and relating the processes involved in solving problems to students' development of mathematical knowledge and problem solving competencies.

In a safe and supportive classroom they will feel comfortable taking risks and engaging in self-directed problem solving. Weaving math into all areas of the curriculum will heighten children's play experiences and allow all learners to experience success. Children will soon see themselves as capable mathematicians who apply their skills in a ...

When studying mathematics, metacognitive strategies can play an important role in knowledge acquisition, retention, and application. At the conceptual development stage, when students are first ...

Play moves math instruction beyond rote memorization to a more expansive understanding of mathematics. Encourage students to talk, think, reason, and wonder as they move through problems. Creating a sense of curiosity, even for simple concepts, engages students in a playful way.

Getting the Most from Each of the Problem Solving Activities. When students participate in problem solving activities, it is important to ask guiding, not leading, questions. This provides students with the support necessary to move forward in their thinking and it provides teachers with a more in-depth understanding of student thinking.

Flexibility and problem-solving are key skills. Problem- solving involves collaboration, communication, critical thinking, empathy, and integrity. If we listen to the business world, we...

Collaborative problem-solving (CPS) learning is increasingly valued for its role in promoting higher-order thinking of learners. Despite the widespread application of role scripts in CPS, little is known about the mechanisms by which roles influence learners' cognition and the impact of goal orientation on roles. In this study, we designed role scripts and goal-orientation scripts to ...

The mathematical thinking process is the explanation and collaboration of mathematics through problem-solving, reasoning and proof, communication, connections, and representation. In mathematics ...