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Mathematics Reform The Implications of Problem Solving in Middle School Mathematics

The Significance of Mathematics Problem Solving Creative and critical thinking skills are mandatory abilities for functioning and succeeding in the competitive world of the twenty-first century. Problem solving is a skill that allows children to investigate and discover mathematics for themselves through everyday life. Solving problems is the purpose of mathematics in the first place.

A “Problem” Defined: The National Council of Teachers Mathematics defines problem solving as “engaging in a task for which the solution method is not known in advance.” Mathematics problem solving, as defined by Schoenfeld is a process wherein students encounter a problem for which they have no immediate resolution, nor an algorithm that they can directly apply to reach a solution. True mathematical problems are truly problematic and involve significant mathematics to provide the intellectual contexts necessary for students’ mathematical development.

Cognitive Processes Involved in Solving a Problem: The solver must:  combine, decompose, and consider relationships among numbers and information in the problem.  actively search for patterns, experiment with their own conjectures, describe and evaluate quantitative and spatial situations, and make decisions under uncertainty.  decide which mathematical operations to use, what numbers and information to include and exclude, and whether their answers make sense.  investigate their own mathematical understanding in order to devise some sort of plan or strategy that will lead to a solution by drawing on prior knowledge. Create Evaluate Analyze Apply and Understand Knowledge— Remember

Common Core State Standards for Mathematical Practice The Common Core State Standards for Mathematics (CCSSM) was adopted in Arkansas in 2010.  The recent shift in the CCSSM curriculum regards problem solving, not as simply a branch of mathematics to be mastered, but as a means of teaching and learning mathematics. Standards for Mathematical Practice: 1) Make sense of problems and persevere in solving them. 2) Reason Abstractly and quantitatively. 3) Construct viable arguments; critique the reasoning of others. 4) Model with mathematics. 5) Use appropriate tools strategically. 6) Attend to precision. 7) Look for and make use of structure. 8) Look for and make use of regularity in repeated reasoning.

National Council of Teachers Mathematics: Process Standards As stated in the NCTM’s Standards of School Mathematics, problem solving instruction in school systems should enable students to:  construct new mathematical knowledge through problem solving,  solve problems that occur in mathematics and in other backgrounds,  apply and adapt an assortment of appropriate strategies to solve problems, and  monitor and reflect on the process of mathematical problem solving.

Connections between NCTM’s Process Standards (2000) and CCSSM Practices: NCTM 1) Problem Solving 2) Reasoning and Proof 3) Communication 4) Connections 5) Representations CSSM 1) Make sense of problems and persevere in solving them. 5) Use appropriate tools strategically. 2) Reason Abstractly and quantitatively 3) Critique the reasoning of others. 8) Look for and make use of regularity in repeated reasoning. 3) Construct viable arguments 6) Attend to precision. 7) Look for and make use of structure. 4) Model with Mathematics

What is a Problem-Solving Approach to Mathematics? A problem-solving approach to mathematics can be characterized as: a shift away from a skill-oriented, teacher directed approach to learning and teaching problem-solving skills in which students first learn some basic math ideas and skills, then some general problem- solving strategies, followed by application and problems.  Such an approach to mathematics leads simply to the acquisition of unstable, disconnected concepts, procedures, and strategies. a shift away from an emphasis on problem-solving strategies as an end to be reached, toward problem-solving strategies as the process to an end.

What is a Problem-Solving Approach to Mathematics? Role of the Student:Role of the Teacher Students play a very active role in their learning. The teacher acts as a facilitator and resource while students construct their own understandings from their own experiences Students explore problem situations with teacher guidance and invent their own solution strategies. Mathematics teachers should guide students to develop efficient strategies. Students have more opportunity to express their ideas and justify their answers verbally. As an observer, the teacher is constantly assessing Students have opportunities to engage in cognitively demanding questions posed in problems as well as by the teacher and fellow peers. The teacher must select appropriate tasks and organizing classroom discourse.

Selecting Appropriate Problems Worthwhile problems should govern students’ attention to particular aspects of mathematical content. Worthwhile tasks should be intriguing, with a level of challenge that invites exploration, speculation, and hard work.  The most important criterion of a worthwhile mathematical problem is that the problem should serve as a means for students to learn important mathematics.  Such a problem does not have to be complicated or fancy, nor does it have to trick or stump the student.

Classroom Discourse Selecting worthwhile problems is not sufficient in itself; worthwhile problems must also be implemented and applied in ways that foster meaningful connections. The nature of the classroom discourse, involving both students and teacher, is a very important consideration.  The amount of time allocated to solving problems and discussing solution efforts is an important factor.  “Teachers must also decide what aspects of a task to highlight, how to organize and orchestrate the work of the students, what questions to ask to challenge those with varied levels of expertise, and how to support students without taking over the process of thinking for them and thus eliminating the challenge” (NCTM, 2000).

Problem-Solving Environment A problem-solving environment is one in which students experience problems that have similar characteristics to those they will encounter in the outside world. Finding a useful way to think about and understand the problem is far more important than simply arriving at an answer quickly. Problem-solving is not only the goal of learning mathematics, but also a major means of doing so. Teaching through problem solving involves fewer problems, with more time spent on each problem due to the discussion of mathematical elements and alternative solutions.

Problem-Solving Environment (Cont’d.) The focus is on conceptual understanding, rather than on procedural understanding, because it is expected that students will learn algorithms and master basic skills as they engage in explorations of worthwhile problems. The learning environment of teaching via problem solving provides a natural setting for students to present various solutions to their group or class and learn mathematics through social interactions, meaningful negotiation, and reaching shared understanding.  Students’ learning and understanding of mathematics can be enhanced through considering one another’s ideas and debating the validity of alternative approaches.

References: Post, T. R. (1992). Teaching Mathematics in Grades K-8: Research-Based Methods. (2nd ed). Boston: Allyn and Bacon. Schoenfeld, A.H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D.A. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 334-370). New York: Macmillan.

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