Curriculum Standards

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NCTM Standards

Mathematics is a critically important discipline for students in our schools today. It is also generally understood that math is not, nor should it be, a "soft" discipline. Starting from the earliest primary grades, there are specific skills that students need to learn for a comprehensive understanding of math. In America, at least, these are expressed in the Standards of Learning from the National Council of Teachers of Mathematics (NCTM) and other standards boards, and followed by math teachers across the country.

It is NOT the intention of this "Puzzles in Education" project to supplant these standards, or in any way to "soften" the math standards that are currently in place.

However, what is recognized by enlightened educators is that along with specific skills, what is most important for young learners is the ability to understand how to evaluate unfamiliar problems and work through how to solve them. This is the skill of PROBLEM SOLVING.

Mechanical puzzles represent the essence of problem solving. When approaching a new puzzle, students can immediately understand what the purpose is, but the techniques that may be required to solve it may be a complete mystery. A well chosen collection of puzzles can offer a tremendously wide range of different types of problems for your students. Because they are fun as well as challenging, puzzles can teach your children to love and appreciate the problem solving process.

The importance of problem solving is well understood by National Council of Teachers of Mathematics. The following quotes are all excerpted by the NCTM Standards of Learning:

* Problem solving is the cornerstone of school mathematics. Without the ability to solve problems, the usefulness and power of mathematical ideas, knowledge, and skills are severely limited. Students who can efficiently and accurately multiply but who cannot identify situations that call for multiplication are not well prepared. Students who can both develop and carry out a plan to solve a mathematical problem are exhibiting knowledge that is much deeper and more useful than simply carrying out a computation. Unless students can solve problems, the facts, concepts, and procedures they know are of little use. The goal of school mathematics should be for all students to become increasingly able and willing to engage with and solve problems.

* The essence of problem solving is knowing what to do when confronted with unfamiliar problems. Teachers can help students become reflective problem solvers by frequently and openly discussing with them the critical aspects of the problem-solving process, such as understanding the problem and "looking back" to reflect on the solution and the process (Pólya, 1957). Through modeling, observing, and questioning, the teacher can help students become aware of their activity as they solve problems.

* Problem solving is also important because it can serve as a vehicle for learning new mathematical ideas and skills (Schroeder and Lester, 1989). A problem-centered approach to teaching mathematics uses interesting and well-selected problems to launch mathematical lessons and engage students. In this way, new ideas, techniques, and mathematical relationships emerge and become the focus of discussion. Good problems can inspire the exploration of important mathematical ideas, nurture persistence, and reinforce the need to understand and use various strategies, mathematical properties, and relationships.

* Most students enter grade 3 with enthusiasm for, and interest in, learning mathematics. In fact, nearly three-quarters of U.S. fourth graders report liking mathematics (Silver, Strutchens, and Zawojewski, 1997). They find it practical and believe that what they are learning is important. If the mathematics studied in grades 3-5 is interesting and understandable, the increasingly sophisticated mathematical ideas at this level can maintain students' engagement and enthusiasm. But if their learning becomes a process of simply mimicking and memorizing, they can soon begin to lose interest. Instruction at this level must be active and intellectually stimulating and must help students make sense of mathematics.

* Mathematical games can foster mathematical communication as students explain and justify their moves to one another. In addition, games can motivate students and engage them in thinking about and applying concepts and skills... Activities like this allow students to use communication as a tool to deepen their understanding of mathematics, as described in the "Communication Standard." The teacher (also can) reflect on her own mathematical learning that occurs as a result of using activities like games with her fifth-grade students.

* By reflecting on their solutions... students use a variety of mathematical skills, develop a deeper insight into the structure of mathematics, and gain a disposition toward generalizing. The teacher can ensure that classroom discussion continues until several solution paths have been considered, discussed, understood, and evaluated. It should become second nature for students to talk about connections among problems; to propose, critique, and value alternative approaches to solving. Although it is not the main focus of problem solving in the middle grades, learning about problem solving helps students become familiar with a number of problem-solving heuristics, such as looking for patterns, solving a simpler problem, making a table, and working backward. These general strategies are useful when no known approach to a problem is readily apparent. These processes may have been used in the elementary grades, but middle-grades students need additional experience and instruction in which they consider how to use these strategies appropriately and effectively.

* Students also should be encouraged to monitor and assess themselves. Good problem solvers realize what they know and don't know, what they are good at and not so good at; as a result they can use their time and energy wisely. They plan more carefully and more effectively and take time to check their progress periodically. These habits of mind are important not only in making students better problem solvers but also in helping students become better learners of mathematics.

* For several reasons, students should reflect on their problem solving and consider how it might be modified, elaborated, streamlined, or clarified. Through guided reflection, students can focus on the mathematics involved in solving a problem, thus solidifying their understanding of the concepts involved. They can learn how to generalize and extend problems, leading to an understanding of some of the structure underlying mathematics. Students should understand that the problem-solving process is not finished until they have looked back at their solution and reviewed their process.
Last Updated: June 5, 2007 top
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