Problem Solving Skill Development
Routine to Non-Routine
Quantitative skills and methods (mathematics) represents a growing body of
rules and patterns that can be carefully, in other words intelligently,
used one at a time and one after another, alone and in sequence, to arrive at
repeatable, reproducible, observable and hence verifiable results.
To
develop problem solving skills, and avoid re-invention of the wheel, students
will be exposed to problems and situations in which the mathematical skills and
concepts they have met can be applied in routine or predictable manner. The
first aim of mathematics instruction is to give students those skills and
concepts - previously found or hard-won by previous generations - for solving
routine problems and puzzles in a straightforward or combinatorial or
opportunistic manner. For that, logic mastery would be useful for the
development of precision reading and writing skills. The well-practiced
ability to record problem solving steps and effort in a clear legible format
readable by peers, teachers and themselves would make aid and speed routine
problem solving.
Routine problem solving (challenging as it may be to some students) in which
mathematical skills and concepts are pieces of a jigsaw puzzle - one whose
solution is standard - even on display - represents a first step in developing
the critical thinking and problem solving skills of students. It provides
a standard for all further problem solving. Seeing what kinds of problems have
been met and/or solve before, and how, provides a model for further problem
solving. Greater knowledge of the kinds of problems met before and how
they have been solved provides a systematic base for further problem
solving.
Mathematics in the first instance, is an art form, a discipline, with simple
and then more complicated rules, patterns and methods to master. For many
routine problems or situations in daily life and in our cultures that students
need to learn to address and solve with ways that lead to repeatable and
reproducible results - reliable results. Once students have sufficient
drill and practice, sufficient exposure, the use of some skills and concepts
should become familiar, automatic, and their use no longer an adventure.
Problem solving from a state of ignorance is
over-rated. With a combinatorial or creative mind, standing on prior
knowledge of what has worked or not, is better. While creativity (the combination of previously mastered skills
and concepts, and the invention of new ones) is possible with any level of
knowledge, the ability to be creative and in that produce methods to solve
problems in a verifiable manner - a manner that peers can follow or reproduce - increases
with the level of knowledge and level of skill and competence. Students need to
learn when creativity is required and when previous methods give satisfactory
results. Problem solving situation with incomplete information of what
has been done - a partial state of ignorance - may be provided to show how a
greater knowledge of previous solution reduces problem solving
challenges.
Problem solving in an society where common problems repeat themselves and
thus become routine should be based routine solutions methods, methods
whose efficacy, suitability and limitations has been checked and understood by
the user. With practice, solving common problem should become
routine.
Empirical problem solving aims to find or apply methods with repeatable and
reproducible, and reliable results. That may turn open problems into routine
problems. Practice in solving problems which have become routine may
prepare students for open problems. Practice in solving routine problems and puzzles
in a straightforward or combinatorial or
opportunistic manner when solution methods are not given provides a model for
tackling non-routine problems, a model that stands on and then looks beyond
previous methods.
Remark: Routines and methods in society for
"solving" problems may lead to repeatable, reproducible and harmful
results. The ability to follow instructions carefully and precisely is a
plus for getting results but not a guarantee that the results will be ethnical
or that practices will be sustainable. So students should not be trained
to follow methods or instructions without reflection on the benefits and
limitations of the methods. Routine solution methods may be challenged
and should be for the everyone's sake. But those routine methods cannot
be challenge, cannot be considered and examine carefully if their study
is avoided.
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LAMP
(first
draft, June 2008) a program for adult
and teen mathematics education
Mathematics education standards implied by calculus should
be a factor, not the only one, yet not a forgotten nor hidden one in course design
Area Intro
Introduction
Arithmetic
Geometry
Algebra
Logic
Calculus
Musings - More Ideas
More About LAMP
Evaluation
Maths Cultural Origins
First Nation Education
Modern Mathematics
Before LAMP
Problem Solving Skills Routine to Non
Instructional Concepts
Student Cooperation
Maths Extrinsic Origins
Science Education
For further musings or thoughts see site books.
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