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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. |
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Last Updated:
June 5, 2007 |
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