Math Framworks - Knightsen School District

7
• The concepts of proportional relationships underlie similarity.
• The level sets of functions of two variables are curves in the coordinate plane.
• Factoring a polynomial function into irreducible factors helps locate the
x-intercepts of its graph.
• Proofs are required to establish the truth of mathematical theorems.
Problem Solving
Problems occur in many forms. Some are simple and routine, providing
practice for skill development. Others are more complex and take a longer time
to complete. Whatever their nature, it is important that the kinds of problems
students are asked to solve balance situations in the real world with more abstract
situations. The process of solving problems generally has the following stages
(Geary 1994; Mayer 1985):
• Formulation, analysis, and translation
• Integration and representation
• Solutions and justifications
Formulation, analysis, and translation. Problems may be stated in an imprecise
form or in descriptions of puzzling or complex situations. The ability to recognize
potential mathematical relationships is an important problem-solving technique,
as is the identification of basic assumptions made directly or indirectly in the
description of the situation, including the identification of extraneous or missing
information. Important considerations in the formulation and analysis of any
problem situation include determining mathematical hypotheses, making conjectures,
recognizing existing patterns, searching for connections to known mathematical
structures, and translating the gist of the problem into mathematical
representations (e.g., equations).
Integration and representation. Important skills involved in the translation of a
mathematical problem into a solvable equation are problems of integration and
representation. Integration involves putting together different pieces of information
that are presented in complex problems, such as multistep problems. However
such problems are represented, a wide variety of basic and technical skills are
needed in solving problems; and, given this need, a mathematics program should
include a substantial number of ready-to-solve exercises that are designed specifically
to develop and reinforce such skills.
Solutions and justifications. Students should have a range of strategies to use in
solving problems and should be encouraged to think about all possible procedures
that might be used to aid in the solving of any particular problem, including but
not limited to the following:
• Referring to and developing graphs, tables, diagrams, and sketches
• Computing
• Finding a simpler related problem
• Looking for patterns
• Estimating, conjecturing, and verifying
• Working backwards
Chapter 1
Guiding Principles
and Key
Components
of an Effective
Mathematics
Program
The ability
to recognize
potential
mathematical
relationships is
an important
problem-solving
technique, as is the
identification of
basic assumptions.
California Department of Education Reposted 7-12-2007