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 suggest

alternatives."

[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 98225

ronaward@henson.cc.wwu.edu