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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.
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| 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 |
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| 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:
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| ¡á 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. |
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| ¡áThe 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
(Polya, 1957). Through modeling, observing, and
questioning, the teacher can help students become
aware of their activity as they solve problems.
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| ¡áProblem 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. |
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| ¡áMost 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.
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| ¡á 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.
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| ¡á 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. |
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| ¡á 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. |
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¡á 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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