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.

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

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 suggest
[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

Ron Ward/Western Washington U/Bellingham, WA 98225