```Date: Jun 26, 1995 2:14 AM
Author: Ronald A Ward
Subject: Focus--Thinking

On June 16, Michael South synthesized a lot of postings into one focus on "thinking."  After reading his post I was reminded of some things I'd read on "mathematical power," "mathematical modes of thought," "habits of mind," and "the new basics."  [I meant to post these sooner, but no time.]  Here they are:1.  From John Van De Walle's Elementary School Mathematics, page 2:	"In a world that is increasingly complex and dominated by quantitative information in every facet of its economy, MATHEMATICAL THINKING is not just more important but essential for even the most ordianry of jobs.   MATHEMATICAL THINKING is not at all the same as the computational skills of yesterday's school mathematics.  It involves the ability and the habits of reasoning and solving problems.  It includes having number sense--an intuition about numbers, their magnitudes, their effects in operations, and their relationships to real quantities and phenomena.  It implies the ability to meaningfully interpret charts and graphs and to understand basic concepts of probability and data interpretation.  It includes spatial sense--a familiarity with shapes and relationships among them.  These are the BASIC SKILLS OF TODAY'S SOCIETY.  Higher-order thinking skills remain entirely human.  These skills of the mind are expected of everyone in the modern workplace.		[CAPS ARE MINE]2.  From Everybody Counts, p. 31:  Mathematical MODES OF THOUGHT	Modeling--Representing worldly phenomena by mental constructs, often visual or symbolic, that capture important and useful features.	Optimization--Finding the best solution (least expensive or most efficient) by asking "what if" and exploring all possibilities.	Symbolism--Extending natural language to symbolic representation of abstract concepts in an economical form that makes possible both communication and computation.	Inference--Reasoning from data, from premises, from graphs, from incomplete and inconsistent sources.	Logical Analysis--Seeking implications of premises and searching for first principles to explain observed phenomena.	Abstraction--Singling out for special study certain properties common to many different phenomena.    [CAPS MINE]3A.  From the draft Assessment Standards:  "For all students to achieve MATHEMATICAL POWER, they need to become mathematical problem solvers, to value mathematics, to reason and communicate mathematically, and to be confident in using mathematics to make sense of real-world problem situations."		[CAPS MINE]3B.  From the Professional Standards: "MATHEMATICAL POWER includes the ability to explore, conjecture, and reason logically; to solve nonroutine problems;to communicate about and through mathematics;and to connect ideas within mathematics and between mathematics and other intellectual activity.  MATHEMATICAL POWER also involves the development of personal self-confidence and a disposition to seek, evaluate, and use quantitative and spatial information in solving problems and in making decisions.  Students' flexibility, perseverance, interest, curiosity, and inventiveness also affect the realization of mathematical power."  [CAPS MINE]3C.  From Everybody Counts: [MATHEMATICAL POWER is] "a capacity of mind of increasing value in a technological age that enables one to read critically,to identify fallacies, to detect bias, to assess risk, and to suggestalternatives."  [Also] "MATHEMATICAL POWER requires that students be able to discern relations, reason logically, and use a broad spectrum of mathematical methods to solve a wide variety of non-routine problems."  [CAPS MINE]4.  From Everybody Counts:  "The study of mathematics can help develop critical HABITS OF MIND--	to distinguish evidence from anecdote,	to recognize nonsense,	to understand chance, and	to value proof."   [CAPS AND PARAGRAPH RESTRUCTURING MINE]Ron Ward/Western Washington U/Bellingham, WA 98225ronaward@henson.cc.wwu.edu
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