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Problem Solving

‘‘Mathematics is something that you do,

not something that you understand.’’

Arthur B. Powell

Problem Solving

Problem Solving: As

Topic in Mathematics

or

Approach of Mathematics Teaching

Problem Solving

“We believe that the primary goals of mathematics learning are understandingand problem solving, and that these goals are inextricably related because learning mathematics with understanding is best supported by engaging in problem solving.”

Lester and Lambdin

Problem Solving

Problem Solving

WHAT IS PROBLEM SOLVING?

The meaning of mathematical

‘‘problem solving’’

is neither unique nor universal.

Definition:

Problem Solving

The National Council of Teachers of Mathematics (NCTM, 2000):

Problem solving is ‘engaging in a task for which the solution method is not known in advance’.

Definition:

Problem Solving

In order to find a solution, students must draw on their knowledge, and through this process, they will often develop new mathematical understandings. Solving problems is not only a goal of learning mathematics but also a major means of doing so.

Definition:

Problem Solving

Bell and Burkhardt:

‘Problem solving is the activity called into play when there is a demand to apply

knowledge, skill and experience to unfamiliar situations’.

Again we have the unfamiliar or unknown method of approaching the problem.

Definition:

Problem Solving

Mayer (1992):

Problem solving is cognitive processing directed at achieving a goal when no solution method is obvious to the problem solver. According to this definition, problem solving has four main characteristics.

Definition:

Problem Solving

Mayer (1992):

First, problem solving is cognitive—it

occurs within the problem solver’s cognitive system and can be inferred indirectly from

changes in the problem solver’s behavior.

Definition:

Problem Solving

Mayer (1992):

Second, problem solving is a process—it

involves representing and manipulating knowledge in the problem solver’s cognitive

system.

Definition:

Problem Solving

Mayer (1992):

Third, problem solving is directed—the problem solver’s thoughts are motivated

by goals.

Definition:

Problem Solving

Mayer (1992):

Fourth, problem solving is personal—the individual knowledge and skills of the problem solver help determine the difficulty or ease with which obstacles to solutions can be overcome.

Definition:

Problem Solving

Hayes (1989):

‘‘Whenever there is a gap between where you are now [an initial situation] and where you want to be [an adequate response], and you don’t know how to find a way [a sequence of actions] to cross that gap, you have a problem’’.

Definition:

Problem Solving

Mayer and Wittrock (1996):

‘‘A problem occurs when a problem solver wants to transform a problem situation from the given state into the goal state but lacks an obvious method for accomplishing the transformation’’.

Problem Solving

In the mathematics and mathematics education literature, no universally

accepted definition exists for the mathematical terms ‘‘task’’, ‘‘problem’’, or ‘‘exercise’’.

Problem Solving

Problem solving is an integral part of all mathematics learning, and so

it should not be an isolated part of the mathematics program.

Good problems will integrate multiple topics and will involve significant mathematics.

Standards: (NCTM)

Instructional programs from prekindergarten through grade 12 should enable all students to—

• build new mathematical knowledge through problem solving;

Problem Solving

Standards: (NCTM)

Instructional programs from prekindergarten through grade 12 should enable all students to—

• solve problems that arise in

mathematics and in other

contexts;

Problem Solving

Standards: (NCTM)

Instructional programs from prekindergarten through grade 12 should enable all students to—

• apply and adapt a variety of

appropriate strategies to

solve problems;

Problem Solving

Standards: (NCTM)

Instructional programs from prekindergarten through grade 12 should enable all students to—

• monitor and reflect on the

process of mathematical

problem solving.

Problem Solving

Problem Solving

How can problem solving help students learn mathematics?

Good problems give students the chance to solidify and extend what they know and, when well chosen, can stimulate mathematics learning.

Problem Solving

The teacher’s role in choosing worthwhile problems and mathematical tasks is crucial. By analyzing and adapting a problem, anticipating the mathematical ideas that can be brought out by working on the problem, and anticipating students’ questions, teachers can decide if particular problems will help to further their mathematical goals for the class.

Problem Solving

There are many, many problems that are interesting and fun but that may not lead to the development of the mathematical ideas that are important for a class at a particular time.

Problem Solving

Choosing problems wisely, and using and adapting problems from instructional materials, is a difficult

part of teaching mathematics.

Problem Solving

In the primary curriculum in England and Wales, problem solving comes under the strand of ‘Using and applying mathematics’. Guidance notes provided by the Department for Education and Skills (DfES, 2006a) identifies five themes within this strand:

Mathematics Across the Curriculum

Problem Solving

- Solving problems.
- Representing– analyse, record, do, check, confirm.
- Enquiring– plan, decide, organize, interpret, reason, justify.
- Reasoning– create, deduce, apply, explore, predict, hypothesize, test.
- Communicating– explain methods and solutions, choices, decisions, reasoning.

Problem Solving

Mayer (1985) identified the following

factors as contributing towards problem solving performance:

• Practice in recognizing problem types.

• Practice in representing problems – whether concretely, in pictures, in symbols, or in words.

• Practice in selecting relevant and irrelevant information in a problem.

Problem Solving

Orton and Frobisher (1996) identified the following processes within problem solving:

• Operational processes of collecting and ordering data.

• Mathematical process of searching for patterns.

• Reasoning processes of analysis and reflection.

• Communication processes in describing methods.

Problem Solving

The most famous ‘processes’ involved in problem solving by Polya(1957):

• Understanding the problem, e.g. looking for unknowns or data.

• Devising a plan, e.g. looking for related problems, restating the problem in a different way.

• Carrying out the plan, in particular checking each step.

• Looking back, e.g. checking the result, obtain the result differently, use the result for other problems.

Standard for Grades 9-12

Problem Solving

In high school, students’ repertoires of problem-solving strategies expand significantly because students are capable of employing more complex

methods and their abilities to reflect on their knowledge and act accordingly have grown. Thus, students should emerge from high school with the disposition, knowledge, and strategies to deal with the new challenges they will encounter.

Standard for Grades 9-12

Problem Solving

As in the earlier grades, problems and problem solving play an essential

role in students’ learning of mathematical content and in helping

students make connections across mathematical content areas.

?

Standard for Grades 9-12

Problem Solving

What should problem solving look like in grades 9 through 12?

Standard for Grades 9-12

Problem Solving

Problem solving plays a dual role in the high school curriculum. On the one hand, solving problems that have been strategically chosen and carefully sequenced is a fundamental vehicle for learning mathematical content.

Standard for Grades 9-12

Problem Solving

Most mathematical concepts or generalizations can be effectively introduced using a problem situation that helps students see important aspects of the idea to be generalized.

Standard for Grades 9-12

Problem Solving

On the other hand, a major goal of high school mathematics is to equip students with knowledge and tools that enable them to formulate, approach, and solve problems beyond those that they have studied.

Standard for Grades 9-12

Problem Solving

High school students should have significant opportunities to develop a broad repertoire of problem-solving (or heuristic) strategies. Theyshould have opportunities to formulate and refine problems because problems that occur in real settings do not often arrive neatly packaged. Students need experience in identifying problems and articulating them clearly enough to determine when they have arrived at solutions.

Problem Solving

The following problem serves multiple purposes. It gives students an opportunity to build their content knowledge during the problem solving process, to learn or practice some heuristic strategies, and to make connections among various ways of thinking about the same mathematical content.

Problem Solving

Problem:

How many rectangles are there on a standard 8 × 8 checkerboard?

Count only those rectangles (including squares) whose sides lie on grid lines.

For example, there are nine rectangles on a 2 × 2 board, as shown in next figure.

Problem Solving

Problem Solving

This problem provides students with an opportunity to review counting techniques and show their power, as well as to use their prior knowledge in other approaches to solving the problem.

Problem Solving

A teacher and a class with a problem-solving disposition will be quick to formulate interesting extensions such as, What would an analogous three-dimensional problem look like?

Problem Solving

What should be the teacher’s role in developing problem solving in grades 9 through 12?

Problem Solving

Successful problem solving requires knowledge of mathematical content, knowledge of problem-solving strategies, effective self-monitoring, and a productive disposition to pose and solve problems.

Problem Solving

A significant part of a teacher’s responsibility consists of planning problems that will give students the opportunity to learn important content through their explorations of the problems and to learn and practice a wide range of heuristic strategies.

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