The belief that "learning is limitless for learners who can solve problems" supports problem solving as a central focus for the curriculum. Problem solving plays an integral role in the mathematics program so that students are provided with some of the thinking and problem-solving skills necessary to help explain the world around them. Problem solving is the process of accepting a challenge and striving to resolve it. It is through problem solving that mathematics can be presented in a meaningful, applicable, and constructive manner for students.
In 1945, George Polya published the book How to Solve It in which he outlines a four-step model to use in the solution of problems. It involves understanding the problem, devising a plan, carrying out the plan, and looking back. Although problem solvers may approach a solution in the order outlined, they will often return to earlier stages because they have encountered an obstacle and it becomes obvious that another approach will work better. Students must analyze the problem and combine prior knowledge and experiences into a procedure that will arrive at a solution.
To establish a problem-solving environment in which students feel motivated and confident, a classroom teacher may consider a number of practices:
Students need to recognize the applicability of mathematics. Problem solving is the link between mathematics in the classroom and the real world. The problems should consist of a good mix of translation, process, and realistic problems.
Process problems are those that generally can not be solved using routine procedures; rather, their solution typically involves the application of some problem-solving strategy. Realistic problems are not well defined and require some further specification and refinement, often have multiple solutions, require the collection of information, involve collaboration with other people, cannot be solved in a few minutes, and involve some personal commitment on the part of the student. Translation problems involve translating written or verbal statements into mathematical expressions and then performing an algorithm.
Problem solving is a process that is learned by doing. Students will become better problem solvers if they think that the activity is important and relevant. This importance is enhanced by observing their teacher(s) solving problems and by their teacher(s) expecting them to do the same.
When mathematical concepts and operations are introduced, they should often follow rather than precede problem-solving opportunities. Before the formal terms and symbols are presented, students should learn to approach problems in a variety of ways. There are many strategies, processes, concepts, and skills that students should learn and be able to apply in order to become successful, life-long mathematical problem solvers. The Problem Solving Strand provides a number of problem-solving strategies that students should develop to complement and extend those learned in the elementary grades.
It is crucial that students develop a positive attitude towards mathematics. Without this positive attitude they will not realize their full potential. They must view mathematics as more than paper-and-pencil assignments with the primary goal of producing correct answers. They must see mathematics as a means to solving a variety of challenging problems. This involves manipulating concrete materials, using pictures and diagrams, collecting and analyzing data, and sharing their mathematical experiences.
Activity-based classrooms usually have:
When students learn mathematics in this type of environment, they come to view the acquisition of knowledge as an active process rather than a receptive one.
Many students learn mathematical concepts best through the manipulation of concrete materials because it assists them in building a mental representation of the concept. Manipulatives provide concrete introductions to abstract ideas. Each student should have an opportunity to have adequate "hands on" experiences with appropriate manipulatives before engaging in pencil-and-paper activities. All print resources, including textbooks, offer only the pictorial and symbolic representations of mathematical concepts. Therefore, it is highly recommended that every classroom have an assortment of manipulatives (purchased, constructed, or collected) that are accessible to students at all times.
Students need time for free exploration when each type of manipulative is first introduced. They must have the opportunity to play, experiment, and observe characteristics of the concrete materials. Students should talk to classmates and to their teacher (using appropriate mathematical terminology) about their experiences.
As objects are manipulated and new concepts introduced, teachers must help students make the connections between their actions and the concepts. A gradual transition to pictorial representations and, when appropriate, to symbolic representations is made. During this process, the appropriate verbal representation is incorporated. Students must eventually understand the relationship between the physical manipulation of materials and the concepts. The way students record their results of manipulative activities often affects their bridging of the gap between the concrete and the abstract.
Although not all objectives need to be introduced using manipulatives, most students will benefit enormously if given the opportunity to proceed from concrete to pictorial to abstract when learning new concepts. However, the manipulative is a tool for helping students to understand mathematical concepts, not an object of study itself.
It is extremely beneficial for students to use a variety of manipulatives when learning a major mathematical concept. This will help to assure that students do not develop a narrow view of the concept. It is important to ensure the comprehension of concepts rather than the memorization of rules and algorithms.
Many manipulatives can be collected (odds and ends, counters, dice, playing cards) and/or constructed (10-frames, geoboards, spinners) by teachers, students, and parents. Others, (linking cubes, mira, alge-tiles) are best purchased. The storage and distribution of manipulatives is an important consideration for the classroom teacher. Many excellent suggestions can be found in various teacher resources.
Mental calculation is a life skill that assists in solving many mathematically-related problems. It improves students' paper-and-pencil calculations and eliminates many common errors produced on electronic calculators. It can also improve the efficiency of pencil-and-paper calculations by reducing the number of steps needed to work out a written calculation. Mental calculation is also the cornerstone to all estimation. This mental ability develops the confidence that assures students that they have the skills to solve basic mathematical problems quickly.
The following instructional ideas are suggested to promote development of student mental calculation abilities:
Students involved in repetitive paper-and-pencil exercises often become accustomed to following, with minimal thought, pre-determined steps. Mental calculation forces students to think about numbers and number relationships.
Mathematics is a discipline that we often characterize by its precision in common usage. However, we do not always need, nor are we sometimes able, to attain a high degree of accuracy in our calculations. Whether to produce a rough estimate, a fine estimate, or an exact answer depends on the ultimate use of the estimate. Most often, practical situations involve estimations rather than exact numbers. Approximate numbers are often easier to comprehend and they can also help to develop consistency. When counting, measuring, or calculating, it is often advantageous to estimate prior to finding an exact solution. Development of the concept and skills of estimation helps students to adapt mathematics in a variety of situations.
The increased emphasis of estimation in the curriculum corresponds with the important role estimation assumes in daily life.
Course content should be presented within the context of its application in daily living, integrated within the various branches of mathematics, and related to other academic disciplines. Integration may include any one of the several forms.
Teachers need to be familiar with the mathematical competence required of students in the particular course of study, everyday life, and other academic disciplines. Cooperative planning and conferencing with other teachers is central to understanding differing contexts in which basic mathematical skills are used, and will assist teachers in providing practical learning experiences that encourage transfer of knowledge and skill.
Calculators should be an important contributing factor in students' number development. Quality designed calculator activities and problems enhance the growth and formation of students' understanding of mathematical patterns and relationships. The concepts of computation are usually understood in advance of the mastery of the algorithms. Therefore, with a calculator, more complex calculations can be performed and problems solved. This can eliminate the drudgery and frustration that may hinder rather than enhance mathematical thinking and creativity when students are expected to use and master algorithms they are not developmentally ready to comprehend.
All students should have regular access to calculators. The calculator should be used as a tool to:
Calculators do not replace the need for the acquisition of basic number facts. Because there is an increasing need to be able to calculate mentally, estimate, and determine the reasonableness of answers to computed problems, students need to learn strategies that assist them with number fact recall.
Students must use calculators regularly if they are to learn how, when, and why they are effective. If students use calculators during instruction, they should also be able to use them during assessment activities.
Computer hardware and software are becoming increasingly sophisticated in design in areas such as problem solving, manipulating, discovering, creative programming, games, tutorials, drill and practice, and managing. Interactive computer software holds great promise for application in the mathematics classroom. Its value in creating geometric displays, organizing and graphing data, simulating real-life situations, demonstrating mathematical relationships, and generating numerical sequences and patterns is evident. Word processing software, electronic spreadsheets, and database management software are all useful in Middle Level mathematics. Within these applications, it is crucial that software correlates with the curriculum and the ability levels of the students.
The microcomputer also has tremendous potential to help students with special needs. Students with learning difficulties, those who require enrichment activities, and students in multi-graded classrooms can all benefit from the individual assistance a computer can offer.
Parents/caregivers can be significant contributors to this learning process. Opportunities for parents/caregivers to be involved in the data collection and problem-solving processes allow them to display interest in the student's work. It also offers them the opportunity to become familiar with the student's program.