## 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.

## 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.

## 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

## Sign Up For Our FREE Newsletter!

By signing up, you agree to receive useful information and to our privacy policy

## Search form

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

## IMAGES

## VIDEO

## COMMENTS

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

Problem solving places the focus on the student making sense of mathematical ideas. · Problem solving encourages students to believe in their ability to think

When you think about it, the whole aim of education is to equip children to solve problems, but not only in math. Aim of the new Mathematics Curriculum in

Math helps us have better problem-solving skills. Math helps us think analytically and have better reasoning abilities. Analytical thinking refers to the

know in mathematics, then problem-solving is the ... Problem solving is important because it ... of that individual's life as a constructive,.

Problem-solving skill is very crucial one whether in the field of teaching mathematics or in the daily life of any individual. This competency reflects two main

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

Problem solving is, however, more than a vehicle for teaching and reinforcing mathematical knowledge and helping to meet everyday challenges. It is also a skill

Problem solving is crucial in mathematics education because it transcends mathematics. By developing problem-solving skills, we learn not only how to tackle

Problem solving places the focus on the student making sense of mathematical ideas. · Problem solving encourages students to believe in their ability to think