Bibliography Teaching for Depth

8/18/2019 Bibliography Teaching for Depth

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HeinemannA division of Reed Elsevier Inc.361 Hanover StreetPortsmouth, NH 03801–3912www.heinemann.com

Offices and agents throughout the world

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2002 by Heinemann

All rights reserved. No part of this book may be reproduced in any form or by any electronic ormechanical means, including information storage and retrieval systems, without permission in writingfrom the publisher, except by a reviewer, who may quote brief passages in a review.

The editor and publisher wish to thank those who have generously given permission toreprint borrowed material:

“The Human Face of Mathematics: Challenging Misconceptions” by Susan H. Picker and John S. Berryis used by permission of the authors.

“Math and the Westward Expansion: How an Interdisciplinary Project Changed My Thinking”by Peter Dubno, Jr., is used by permission of the author.

Figures in “The Math of Art” are reprinted by permission of artist Caissa Douwes.

“Getting Smarter: A Seventh-Grade Class Researches and Reflects on Its Discussion Habits” by MattWayne is used by permission of the author.

Student prompts in “From Windex to Wildstrom” are used by permission of Susan Wildstrom.

“If d y/d x = 4 x 3 + x 2 − 12/√

2 x 2 − 9, then” from Asylum by Amy Quan Barry. Copyright C 2001.Reprinted by permission of the University of Pittsburgh Press.

“Geometry” from Selected Poems by Rita Dove. Copyright C 1980 by Rita Dove. Published byVintage, 1993. Reprinted by permission of the author.

“Absolute Zero” by Elizabeth Fox is used by permission of the author.

Excerpt from curriculum letter to parents by Brooke Jackson is used by permission of the author.

“Mallarmé in Tournon” by Rodger Kamenetz is reprinted by permission of the author.

Excerpt of “Mr. Norton’s Wart Hog” from Very Much Like Desire by Diane Lefer. Copyright C 2000.Published by Carnegie Mellon University Press. Reprinted by permission of the author.“Record of Class Discussion” scoring form by Jody Madell is reprinted by permission of the author.

“Grace to Be Said at the Supermarket” from The Blue Swallows by Howard Nemerov. Published byUniversity of Chicago Press. Reprinted by permission of Margaret Nemerov.

“The Mark” from Works and Days, Volume XXVI by David Schubert, Quarterly Review of LiteraturePoetry Book Series. Reprinted by permission of the Quarterly Review of Literature.

Test reflection sheet by Cheryl Schafer is used by permission of the author.

“Palm Sunday” from Palm Sunday by Kurt Vonnegut. Copyright C 1981 by Kurt Vonnegut. Used bypermission of Dell Publishing, a division of Random House, Inc.

“What Did I Learn in School? (A Recitation)” from So Much to Do: Poems by Alan Ziegler by AlanZiegler. Copyright C

1981 by Release Press. Reprinted by permission of the author.

Library of Congress Cataloging-in-Publication DataTeaching for depth : where math meets the humanities / edited by Dale Worsley.

p. cm.Includes bibliographical references and index.ISBN 0-325-00245-2 (acid-free paper)1. Mathematics—Study and teaching (Middle school). 2. Mathematics—Study and teaching

(Secondary). 3. Literature in mathematics education. I. Worsley, Dale, 1948–.

QA11.2 .T43 2002510.712—dc21 2002009729

Consulting editor: Susan Ohanian Editor: Victoria MereckiProduction service: Lisa Garboski, bookworksProduction coordination: Vicki KasabianCover design: Joni DohertyTypesetter: TechBooks


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Teaching for Depth

Where Math Meets the Humanities

Edited by Dale Worsley

Annotated Bibliography and Websites

In the conclusion I referred to the world of possibilities that Teaching for Depth would

contain if it could. This annotated bibliography and list of websites are intended to direct

readers to at least the fringes of the world beyond the book's covers. The bibliography

comprises only the limited reading and research I was able to do, along with the

contributions of a few others (including Susan Schwartz Wildstrom and Matthew

Szenher, whose recommended reading list and website extensions and connections

appear at the end). I hope that readers will understand that it is merely a model of what

others may experience if they are willing to set out on similar journeys.

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Annotated Bibliography

Bartlett, F. C. 1932. Remembering: A Case Study in Experiential and Social Psychology.

London: Cambridge University Press. On Ian Hauser's reading list for those interested

in constructivist theory (see his chapter, "Every Class Is an English Class").

Barton, M. L., and C. Heidema. 2000. Teaching Reading in Mathematics. Aurora, CO:

Midcontinent Research for Education and Learning. Mary Lee Barton and Clare

Heidema have created a supplement to the comprehensive Teaching Reading in the

Content Areas, a collection of well-researched techniques to help teach the reading of

mathematics. They define reading as "decoding and comprehending what is read;

planning for and monitoring the effectiveness of one's reading; analyzing and

evaluating the content in light of one's prior knowledge, experiences and schemata;

and making inferences and generating conclusions based on the reader's unique

interpretation of what is read." They understand that "mathematics text demands that

readers use additional, content-specific skills." If math, in line with the themes of this

work, is going to become less isolated, then it must be understood. Teaching Reading

in Mathematics provides the basic literacy tools for accomplishing this. Tools include

concept definition mapping, semantic feature analysis, vocabulary elaboration

strategies, word sorts, anticipation/prediction guides, graphic organizers, and others.

Berdinasky, B., B. Cronnell, and J. A. Koehla. 1969. "Technical Report Number 15."

Austin, TX: Southwest Regional Laboratory for Educational Research and

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Development. On Ian Hauser's reading list for those interested in constructivist theory

(see his chapter, "Every Class Is an English Class").

Berger, R. 1996. "A Culture of Quality: A Reflection on Practice ." Occasional Paper

Series (1). Providence, RI: Annenberg Institute for School Reform. Ron Berger's

reflection on his exemplary project-based teaching includes a description of "The

Radon Project," in which mathematics played an integrated and meaningful part in

creating "the first-known comprehensive radon picture of any town in the state." The

way his students worked ("the room had the feel of a newspaper office") is how it

should look in every school.

Berlinski, D. 1995. A Tour of the Calculus. New York: Vintage. One of the books that

transformed the culture of Tammy Vu's classroom (see her chapter, "'I Would Have

Laughed. . .': A Math Classroom Transformed by Literature") and transformed my

understanding of higher order mathematics. Some math teachers I have spoken with

find his prose too flowery for their tastes but it gave me access to the wonder of the

concepts of calculus. I loved the dramatic scenarios and metaphoric visualizations.

The section quoted here mirrors in its language my own visualization of what Dewey

means by learners' "continuum of experience":

Continuity is an aspect of things as rooted in reality as the fact that material

objects occupy space; it is the contrast between the continuous and the discrete

that is the great generating engine by which the real numbers are constructed and

the calculus created. The concept of continuity is, like so many profound

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concepts, both simple and elusive, elementary and divinely enigmatic. A process

is continuous if it has no gaps, no place where the process itself falls into

abeyance. The flight of an eagle is an example. The great bird gathers its

shoulders, pushes off from a rotted tree stump, lifts into the wind, its wings

beating, soars upward on a thermal current, and then, its neck curved downward,

folds its wings together and dives toward the stream below. Although in the

course of flight the bird does different things, there is no moment when what it

does simply lapses so that it jumps from one part of its aerial repertoire to another.

(130)

Blattner, D. 1997. The Joy of π . New York: Walker and Company. This is the handy,

fact- filled, creatively designed book that gave me the sense of wonder at the number

π that I mentioned in the conclusion, in relation to the eighth-grade test. It is perhaps

worth telling an anecdote about the book. I had been reading it on the subway on my

way to a New York City Lab School graduation ceremony. I sat next to contributor

Peter Dubno in the auditorium. I was pretty much agog at the myriad wonders of the

number. (It is, for instance, associated with the relationship between circles and

squares. Whereas circles are "natural" and hard to measure, squares are more of a

human creation and much easier to measure, yet the relationship between them is

heavily tied up with the naturally occurring pi, which so mystically stretches out to

infinity. The number has many other odd attributes, such as its place in the measuring

of the heights of elephants. Apparently, the height of an elephant from foot to

shoulder = 2 x π x the diameter of its foot). After rattling on for a while about this to

Peter, he said, "It's not any more special than 5.0000 . . . , with zeros to infinity. It's

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just a number, in and of itself, like any other number." Somehow this didn't reduce

the wonder I felt at the properties of pi but made all numbers suddenly seem equally

wonderful to me.

Borowsky, E. J., and J. M. Borwein. 1991. The HarperCollins Dictionary of

Mathematics. New York: HarperCollins. I found this dictionary to be particularly

useful in defining concepts in mathematics and providing details on the lives of

mathematicians. Teachers and students could get an education in idle moments

thumbing through it, when not consulting it on specific words, ideas, and figures.

Brooks. J. G., and M. G. Brooks. 1993. In Search of Understanding: The Case for

Constructivist Classrooms. Alexandria, VA: Association for Supervision and

Curriculum Development. On Ian Hauser's reading list for those interested in

constructivist theory (see his chapter, "Every Class Is an English Class") and

recommended by Matthew Szenher in his chapter, "The Mathematician's Apprentice."

Brown, S. I. 1996. "Towards Humanistic Mathematics Education." First published in

First International Handbook in Mathematics Education, 1996, ed. Alan Bishop.

Now available at http://members.tripod.com/mumnet/sibrown/sib003.htm. Stephen I.

Brown's essay begins, "Humanistic mathematics education? What is it? Those who

have experienced mathematics as a depersonalized, uncontextualized, non-

controversial and asocial form of knowledge might very well consider the expression

humanistic mathematics education to be the epitome of an oxymoron (1)." Brown

proceeds to construct the philosophical and pedagogical frameworks that are critical

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to the humanizing of mathematics in the schools. "One needs to be aware of the ways

in which curriculum and texts influence each other; how teachers define themselves

professionally and personally; how students and their image of themselves are

affected by the culture of school and society; what concepts of application are both

used and ignored in an effort to 'apply' a discipline like mathematics to the real world;

what constitutes legitimate research in education; how teachers are 'trained'; how

authority in religion and other forms of dogma compare with authority of reason."

Brown, in this and other texts available on the tripod website, is the man to make us

so aware. (See Matthew Szenher's essay "Websites: Extensions and Connections" for

more on S. I. Brown.)

Bruner, J. S. 1971. Toward a Theory of Education. Cambridge: Harvard University Press.

On Ian Hauser's reading list for those interested in constructivist theory (see his

chapter, "Every Class Is an English Class").

Buhler, W. K. 1981. Gauss: A Biographical Study. New York: Springer-Verlag. Good for

a biographical study. Recommended in Mathematics Teacher.

Burghardt, M. D. 1995. Introduction to the Engineering Profession. New York:

HarperCollins. Recommended by David Hardy in his chapter, "Life at Imaginary

High."

Burns, M. 1987. A Collection of Math Lessons. White Plains, NY: Math Solutions

Publications. A pioneer in the field of constructivist mathematics, Marilyn Burns has

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created her own staff development and research organization, Math Solutions, which

produces high-quality resource materials, including the series A Collection of Math

Lessons. It includes volumes for grades 1–3, 3–6, and 6–8. If you want exemplars of

engaged, collaborative, inquiry-based math classrooms, investigate these books.

Calhoun, E. F. 1994. How To Use Action Research in the Self-Renewing School.

Alexandria, VA: Association for Supervision and Curriculum Development.

Recommended by Matt Wayne in his chapter, "Getting Smarter: A Seventh-Grade

Class Researches and Reflects on Its Discussion Habits."

Cambourne. B. 1988. The Whole Story: Natural Learning and the Acquisition of Literacy

in the Classroom. Auckland. N.Z.: Ashton Scholastic. On Ian Hauser's reading list for

those interested in constructivist theory (see his chapter, "Every Class Is an English

Class").

Clarke, D. 1997. Constructive Assessment in Mathematics: Practical Steps for Classroom

Teachers. Berkeley, CA: Key Curriculum. Math teachers who know the value of

portfolio assessment use this book.

Clay. M. M. 1979. Reading: The Patterning of Complex Behavior. Auckland. N.Z.:

Heinemann. This and other Clay references are on Ian Hauser's reading list for those

interested in constructivist theory (see his chapter, "Every Class Is an English Class").

———. 1984. The Early Detection of Reading Difficulties. Auckland N.Z.: Heinemann.

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———. 1993. An Observational Survey of Early Literacy Achievement. Portsmouth, NH:

Heinemann.

Cooney, T. J., S. I. Brown, J. A. Dossey, G. Schrage, and E.C. Wittmann. 1996.

Mathematics, Pedagogy, and Secondary Teacher Education. Portsmouth, NH:

Heinemann. Thomas J. Cooney and company (including Stephen I. Brown, a leader in

the field of humanistic math education, and listed previously) have prepared a

preservice and inservice resource book for math teachers that is organized as a kind

of novel. It is designed to address the NCTM's math "standards" while digging deep

into the realities encountered in the math teacher's mind, curriculum, and classroom.

COMAP, Inc. 1998. Mathematics: Modeling Our World. Cincinnati: South-Western

Educational Publishing. This standards-based textbook by The Consortium for

Mathematics and Its Applications is "dedicated to presenting mathematics through

contemporary applications." New York City Community School District Two has

adopted the textbook for the high schools in its district as part of the constructivist

ARISE curriculum. I participated in training sessions and became a math student

again, this time experiencing the pleasure and understanding I had failed to feel and

achieve in the teacher-centered math classrooms of my youth. The textbook delivers

handily on its promise to contextualize mathematics: "When the mathematics used to

solve problems gives inadequate solutions, people search for new ways to use

mathematics to achieve better solutions. . . . The search for new ways to use

mathematics to solve real problems is going on every day in the world around you."

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For anyone worried that the "basics" of math might be left out, the teacher's version

of the textbook begins each unit with a "Scope and Sequence Chart." The unit on

voting listed the following math skills and strategies: number sense, percentages,

preference diagram representation, graph theory, paradox, and matrices—and it was

only Unit One, the first of eight for the ninth grade. I loved the literature, multiple

applications of related concepts, and collaborative nature of the work in the

curriculum design. After trying it out for a few months, teachers celebrated their

students' abilities to create their own equations instead of just solving prefabricated

ones. They applauded the comfort levels already achieved by the students in

replacing variables with further variables. "They're doing fine with this," said Peter

Dubno (author of the chapter "Math and the Westward Expansion: How an

Interdisciplinary Project Changed My Thinking"). Another veteran teacher said,

"This curriculum is so much more than a math course. The nuance . . . there is always

so much more popping out. There is as much about language as about mathematics."

Connolly, P., and T. Vilardi. 1989. Writing to Learn Mathematics and Science. New

York: Teachers College Press. Generated by the progressive and innovative Institute

for Writing and Thinking at Bard College, this work collects the ideas of twenty-three

practitioners, academic researchers, and cross-disciplinary visitors to the field of

writing in math and science. It is replete with analysis, advice, case studies, and

informed perspective. I don't know of another book that so centrally collects the best

thinking on the subject of bringing "word people" and "number people" (as Reuben

Hersh phrases it in the final essay of the book) together for meaningful discourse. As

Reuben goes on to say, "the day will come, I believe, when the value of writing to

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learn will be universally acknowledged." Writing to Learn Mathematics and Science

brings brings that day much closer on our calendars.

Countryman, J. 1992. Writing to Learn Mathematics: Strategies that Work . Portsmouth,

NH: Heinemann. Joan Countryman's comprehensive, yet simple and instructive, book

on writing in the math classroom. It gives clear, cogent ideas, and samples of writing

in the math classroom in the following categories: writing to learn, autobiography,

journals, word problems, formal writing, evaluative writing, and reflective writing. I

give this book to math and humanities teachers alike to get the conversation started. It

closes the case on the value of writing.

Dauben, Joseph. 1979. Georg Cantor: His Mathematics and Philosophy of the Infinite.

New Jersey: Princeton University Press. Also good for a biographical study.

Dehaene, S. 1997. The Number Sense: How the Mind Creates Mathematics. New York:

Oxford University Press. Another of the "must-reads" for those interested in

plumbing the depths of this book's theme. The initial guiding questions are "Might

numbers be…almost as old as life itself? Might they be engraved in the very

architecture of our brains? Do we all possess a 'number sense,' a special intuition that

helps us make sense of numbers and mathematics?" The answers are thoroughly

elaborated, build on the history of neurological research into learning and

mathematics, and tell a good story. No argument for the function of intuition and the

emotions in understanding mathematics could be more persuasive or detailed. Its

significance to mathematics education is, or should be, inestimable.

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Dennis, D. 2000. "The Role of Historical Studies in Mathematics and Science

Educational Research." In Research Design in Mathematics and Science Education.

Mahwah, NJ: Lawrence Erlbaum. David Dennis answers the questions, What kinds of

historical investigations are desirable? Where and how should they be presented and

discussed? What sort of reforms of curricula can history inspire? What kind of

history, if any, should be presented directly to secondary students? or to teacher

candidates? What part should history play in educational philosophy and

epistemology? His essay has the potential to lend clarity to the reasoning in Teaching

for Depth as he develops his arguments along three lines: context (the use of history

to inspire students), content (the use of history to gain insights into concepts), and

critique (the use of history to see how certain views came to be valued over others).

Dewdney, A. K. 1999. A Mathematical Mystery Tour . New York: Wiley. Is mathematics

invented or discovered? This is the question that guides the dramatized scenes that

lead Dewdney to conclude that it is, essentially, discovered. The question is one that

students can grapple with energetically, especially as the arguments are subtle and

require much evidence to support them. Along the way, guided by Dewdney's

entertaining historical narrative, they will deepen their understanding of human

nature as well as of mathematics. "[T]he human mind has a certain ability to model

reality, an ability honed by millions of years of evolution (178)," claims one of

Dewdney's authoritative characters, who later also points out that "there is evidence

that we proceed in our researches at both a conscious and an unconscious level

(179)." The continuum of experience is illuminated even as math is seen as part of a

"holos" (as opposed to cosmos) beyond human experience. My experience has been

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that focusing on such questions as Dewdney's has the potential to engage even the

most alienated students because it gets to the true wonders of reality and of the

human mind.

Dewey, J. [1938] 1997. Experience and Education. Reprint, New York: Touchstone.

Discussed in the introduction, in reference to learners' "continuum of experience."

The summary on the cover states: "The great educational theorist's most concise

statement of his ideas about the needs, the problems, and the possibilities of

education—written after his experience with the progressive schools and in the light

of the criticisms his theories received."

Dunham, William. 1999. Euler, the Master of Us All . Washington, D.C.: Mathematical

Association of America. Good one for teachers or students doing biographical

research. Recommended in Mathematics Teacher.

Eglash, R. 1999. African Fractals: Modern Computing and Indigenous Design. New

Brunswick, NJ: Rutgers University Press. A fascinating ethnomathematical study of

fractals in the design work of African cultures, with gorgeous illustrations comparing

African art and computer design. Too academic for cover-to-cover reading in the high

school classroom, but a tantalizing and mind-blowing book to stimulate ideas and

change perceptions if presented in small chunks.

Emmer, M. 1990. "Mathematics and the Media." In The Popularization of Mathematics,

ed. A. G. Howson and J. P. Kahane. ICMI Study Series, 89–102. Cambridge:

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Cambridge University Press. Recommended by Susan H. Picker and John S. Berry to

elaborate on the themes they explore in their chapter, "The Human Face of

Mathematics: Challenging Misconceptions."

Fadiman, Clifton. 1962. The Mathematical Magpie: Being More Stories, Mainly

Transcendental, Plus Subsets of Essays, Rhymes, Music, Anecdotes, Epigrams, and

other Prime Oddments and Diversions, Rational or Irrational, All Derived from the

Infinite Domain of Mathematics. New York: Simon and Schuster. The title says it all.

William Wordsworth, Mark Twain, Samuel Beckett, Raymond Queneau, Saul

Steinberg—a prestigious and transcendentally intelligent and witty collection. Read

something from this book or its predecessor, Fantasia Mathematica, to a class every

day, and three quarters of the bridge to humanities will be crossed.

Fogarty, R. 1999. "Architects of the Intellect." In Educational Leadership. Alexandria,

VT: Association for Supervision and Curriculum Development. On Ian Hauser's

reading list for those interested in constructivist theory (see his chapter, "Every Class

Is an English Class").

Gardner, M. 1998. "A Quarter-Century of Recreational Mathematics." Scientific

American 279 (2): 68–76. Recommended by Sylvia Gross in her chapter, "From

Windex to Wildstrom: Conversations with My Teacher."

Gleick, J. Chaos. New York: Viking, 1987. James Gleick explores the gaps between

scientific and mathematical disciplines in this landmark report on the "new science"

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of chaos. To explain the growth of complexity in nature, one must peer into the dark

corners where transitions and anomalies take place. One will find there a deeper order

to the chaos that lurks behind the façade of order. Mathematics begins to look and

feel eerily poetic in the appearance of fractal patterns and strange attractors. I believe

that to be culturally, scientifically, and mathematically literate, one must read this

book. Students need their teachers to be literate at these levels. Elizabeth Fox's

chapter, "Encouraging Chaos," articulates the stakes and demonstrates how such

literacy might play out in the classroom.

Graves, D. H., and B. Sunstein. 1992. Portfolio Portraits. Portsmouth, NH: Heinemann.

Good case studies and strong inspiration for portfolio work in the classroom. Here's

an indicative statement from Sunstein, supporting the "calculus" of learning's smooth

continuity:

Portfolios mean more than evaluation or assessment. They are tied to our

definition of literacy. When we read and write constantly, when we reflect on who

we are and who we want to be, we cannot help but grow. Over time, portfolios

help us identify and organize the specifics of our reading and writing. They

catalogue our accomplishments and goals, from successes to instructive failures.

Portfolios ought to be documents of our personal literacy histories. Keeping a

portfolio is a long and disciplined process. We need to allow portfolios some

growing and breathing space before we freeze them into a definition or a

standardized mandate.

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Growney, J., ed. 2001. Numbers and Faces: A Collection of Poems with Mathematical

Imagery. Claremont, CA: Humanistic Mathematics Network. JoAnne Growney has

produced one of the finest, most focused collections of "math poems" I have seen.

Poems by our greatest poets, from W. H. Auden to Linda Pastan, from Pablo Neruda

to Sherman Alexie, are found here. It would be on my required reading list if I were a

math teacher. I will be using poems from the collection for some time to come in my

writing class. Note also that JoAnne Growney has produced two quality collections of

her own that offer poems on mathematical themes: My Dance is Mathematics and

Intersections.

———. 2001. My Dance is Mathematics. Bloomsberg, PA: Available from the author.

Hanley, S. 1994. "On Constructivism." Maryland Collaborative for Teacher Preparation.

Recommended by Matthew Szenher in his chapter, "The Mathematician's

Apprentice."

Heisenberg, W. 1971. Physics and Beyond: Encounters and Conversations. New York:

Harper & Row. Elizabeth Fox, author of the chapter "Encouraging Chaos,"

encourages teachers to familiarize themselves with this work that reads like a novel

as it describes the emergence of the uncertainty principle and quantum physics. The

cultural influence on mathematics, and vice versa, are clearly apparent in this story of

mysterious and liberating ideas.

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Hennings, D. G. 2000. "Contextually Relevant Word Study: Adolescent Vocabulary

Development across the Curriculum." Journal of Adolescent & Adult Literacy 44 (3):

270. This piece is the most comprehensive short treatment of the subject I have seen,

a perfect realization of the theme of Teaching for Depth at the linguistically atomic

level of the word.

Heschel, A. 1966. "Children and Youth." The Insecurity of Freedom: Essays on Human

Existence. Philadelphia: Jewish Publication Society of America. Recommended by

Sylvia Gross in her chapter, "From Windex to Wildstrom: Conversations with My

Teacher."

Hoffman, P. 1998. The Man Who Loved Only Numbers. New York: Hyperion. Another of

the books that transformed Tammy Vu's classroom (see her chapter, "'I Would Have

Laughed. . .': A Math Classroom Transformed by Literature"). Also recommended by

Susan Schwartz Wildstrom.

Howson, A. G. and J. P. Kahane. 1990. "A Study Overview." In The Popularization of

Mathematics, ed. A.G. Howson and J. P. Kahane. ICMI Study Series, 1–37.

Cambridge: Cambridge University Press. Recommended by Susan H. Picker and John

S. Berry to elaborate on the themes they explore in their chapter, "The Human Face

of Mathematics: Challenging Misconceptions."

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Hubbard, R., and B. Powers. 1993 The Art of Classroom Inquiry. Portsmouth, NH:

Heinemann. Recommended by Matt Wayne in his chapter, "Getting Smarter: A

Seventh-Grade Class Researches and Reflects on Its Discussion Habits."

Humanistic Mathematics Network. Journals, ed. Alvin White. Harvey Mudd College,

Claremont, CA. As I said in the conclusion, these journals are a steady source of

sophisticated thinking exactly along the lines of our theme. The poetry, stories,

essays, and pure freshness and spontaneity of thinking in these journals have the

potential to breathe life into any math or humanities classroom. (See Matthew

Szenher's essay, "Websites: Extensions and Connections" for more on the

Humanistics Mathematics Network.)

Institute for Learning. 1999. The Principles of Learning . Pittsburgh: Learning Research

and Development Center at the University of Pittsburgh. Recommended by Matt

Wayne in his chapter, "Getting Smarter: A Seventh-Grade Class Researches and

Reflects on Its Discussion Habits."

Juster, N. 1961. The Phantom Tollbooth. New York: Epstein & Carroll. The popular

children's book contains a wonderful emulation of Alice in Wonderland in the chapter

"Milo and the Mathemagician." Illustrated by Jules Feiffer.

Kline, M. 1953. Mathematics in Western Culture. New York: Oxford University Press. If

there were one book to recommend for perspective on this work's theme, it would be

Morris Kline's. In his first paragraphs he identifies the sources of prejudice against

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mathematics, quoting, amusingly (and frighteningly), St. Augustine: "The good

Christian should beware of mathematicians and all those who make empty

prophecies. The danger already exists that the mathematicians have made a covenant

with the devil to darken the spirit and to confine man in the bonds of Hell (vi)." I

quoted Kline on the nature and value of mathematics in my chapter with Kay

Rothman, "A Mathematical Correspondence Between Humanists." He promised in

that passage to demonstrate the full value of mathematics as a "major cultural force in

Western civilization," and he delivers. He teases apart the threads of every issue,

sketches every major personality, and inspires wonder at the content of mathematics.

No book that I have read is more historically comprehensive. It is accessible to

literate high school students. Passages can easily be excerpted to elucidate any

mathematical study in the classroom. It is a classic. Kline has written other works that

come recommended by math friends, including: Mathematics and the Physical

World; Calculus, an Intuitive and Physical Approach; Mathematics for Liberal Arts;

Mathematical Thought from Ancient to Modern Times, and Mathematics: The Loss of

Certainty.

Kuhs, T. 1997. Measure for Measure: Using Portfolios in K–8 Mathematics. Portsmouth,

NH: Heinemann. A practical book that demonstrates how to make portfolios work to

different purposes in the mathematics classroom and how to integrate them into the

culture of the school.

Krause, M. C. 2000. Multicultural Mathematics Materials. Reston, VA: National Council

of Teachers of Mathematics. A terrific collection of activities from global cultures

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rarely mentioned in math class. Applicable to grades 1–8, but the historical

background for each activity expands awareness for all ages of the depth of math

understanding inherent in the arts, crafts, and games of the peoples of the world.

Lakoff, G., and R. E. Núñez. 2000. Where Mathematics Comes From: How the Embodied

Mind Brings Mathematics into Being . New York: Basic Books. The authors attempt

to launch a discipline here, a cognitive science of mathematics. Working from the

premise that "[m]athematics is deep, fundamental, and essential to the human

experience," they consider that the intellectual content of mathematics is in its ideas,

not in its symbols, and that ideas are grounded in sensory–motor experience. Working

from the premises that "[m]ost of our thought and our systems of concepts are part of

the cognitive unconscious" and that "our ideas are shaped by our bodily

experiences—not in any simpleminded one-to-one way but indirectly, through the

grounding of our entire conceptual system in everyday life," they debunk romantic

ideas that math is "abstract and disembodied." On the contrary, it arises directly from

"the nature of our brains and our embodied experience." They have discovered that a

great many of the most fundamental mathematical ideas are inherently metaphorical

in nature, such as the number line, where numbers are conceptualized metaphorically

as points on a line. They emphatically defy the Platonists, on the basis that a

"disembodied mathematics transcending all bodies and minds and structuring the

universe" simply cannot be proved scientifically." Although it is not necessarily easy

to follow the rigorously argued—sometimes perhaps over -argued—development of

their argument, the implications for teaching are profound, as it ultimately puts a

"human face on mathematics." This shift in paradigm cannot help but make

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conversations with math students more humane. Coupled with Dewdney's A

Mathematical Mystery Tour , which basically agrees with the Platonists, the issue on

the origins of mathematics is joined.

Learning Media. 1996. Reading for Life: The Learner as a Reader . Wellington, N.Z.:

New Zealand Ministry of Education. On Ian Hauser's reading list for those interested

in constructivist theory (see his chapter, "Every Class Is an English Class").

Mathematics Teacher, Focus Issue: History 93 (8): entire volume. This issue of the

publication from the National Council of Teachers of Mathematics might well be

bound as a book and distributed as the single-most relevant publication for high

school teachers on the issue of history and mathematics. Any one of the articles

would make a terrific addition to this volume.

Mathews, H., and A. Brotchie, eds. 1998. Oulipo Compendium. London: Atlas. Elizabeth

Fox, in her chapter, "Encouraging Chaos," adapts an exercise from this weird and

inspiring book written by poets and artists who use mathematics as the basis for their

experiments. (Oulipo stands for Ouvroir de littérature potentielle, or "Workshop for

Potential Literature.") Raymond Queneau, one of the founders of the Oulipian

movement, humorously describes Oulipians as "rats who build the labyrinth from

which they plan to escape." The compendium is written in the form of a dictionary, in

which one finds dozens of invented forms of writing as well as a registry of

avant-gardists from Marcel Duchamp to John Ashbery.

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Maor, E. 1987. To Infinity and Beyond: A Cultural History of the Infinite. Princeton, NJ:

Princeton University Press. I am investigating Eli Maor's rich book as the deadline for

this work approaches. I am hungry to explore more deeply the layered history of the

idea of the infinite, which seems to have an infinite number of permutations, not to

mention an appealing appearance in its representations by the likes of M. C. Escher

(whose paintings are reproduced here in gorgeous color plates). "Such is the nature of

the infinite process that some series converge to their limits, while others which seem

to converge simply refuse to do so." The infinite, like life, seems to be a process, and

develops personality in its behaviors as it develops. The book promises to be very

pleasing to teachers and students willing to work through the ideas, which demand

concentration but not special knowledge. Matthew Szenher and Susan Wildstrom

recommend Maor's e: The Story of a Number, as well.

Morice, D. 2001. The Dictionary of Wordplay. New York: Teachers & Writers. Hundreds

of invented forms of poetry and wordplay are listed here via Dave Morice's obsessive

mind. It can't help but raise one's consciousness of the seemingly infinite variety of

mathematical patterns to be found in letters and words.

Moses, R. P., and C. E. Cobb. 2001. Radical Equations: Math, Literacy and Civil Rights.

Boston: Beacon. Moses and Cobb find literacy in mathematics to be a civil rights

issue. They analyze how community values and psychological health affect learning,

then devise constructivist methods that work within the context of African American

culture. It is a quietly revolutionary book that humanizes mathematics and gives us

useful tools for any classroom.

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Nelson, D., G. G. Joseph, and J. Williams. 1993. Multicultural Mathematics: Teaching

Mathematics from a Global Perspective. Oxford: Oxford University Press. Although

the authors worry that their volume, because of its title, will induce "visions of a

rhetorical, educationally unsound vehicle for antiracism," they are, in fact seeking to

show that "mathematics has a rich cultural heritage and that it can be taught from a

multicultural standpoint." They work from three premises: (1) that "[e]ducation must

(logically) incorporate material from several cultures," (2) that "[e]ducation must

incorporate material from several cultures to be effective," and (3) that "[e]ducation

ought (morally) to incorporate such material not primarily to enhance the self-image

of minority children but to help all children in the future to negotiate more effectively

in a multicultural environment." They realize the potential of these premises by

meticulously breaking down preconception and prejudice to deliver a far richer

understanding of the history of mathematics, its application in the classroom, and its

potential to give students not only greater math skills but deeper empathy for the

cultures around us. Aimed at elementary educators it is nevertheless so erudite and

the arguments are so original that it has the potential to change teaching in powerful

ways at the secondary level.

Newman, J. R. 1956. The World of Mathematics: A Small Library of the Literature of

Mathematics from A'h-Mosé the Scribe to Albert Einstein. Presented with

Commentaries and Notes by James R. Newman. New York: Simon and Schuster.

This compendium belongs in every high school math or humanities classroom to

demonstrate the long history of deep thought given to math and its relationship to

literature, history, art, and music. The greatest mathematicians, many scientists, and

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not a few literary figures, such as Aldous Huxley, are represented among its four

volumes. The essays are generally accessible at some level and, if for no other reason,

might be consulted to attach the big ideas that come up in the curriculum to the

human voices of their conceivers. Hardly any student's mathematical or philosophical

question cannot be addressed here. The words themselves are stirring, especially

considering their historical and intellectual significance. Galileo: "My purpose is to

set forth a very new science dealing with a very ancient subject. There is, in nature,

perhaps nothing older than motion. . . ." Student writing can benefit from a study of

the author's styles and be enriched by the vocabulary as well as the ideas. It also bears

mention that the chapters are introduced with apt epigrams of wit and beauty,

commentary in and of itself on the power of literature to excite and orient the

imagination. One of my favorites, by Eric Temple Bell, introduces Euclid: "The

cowboys have a way of trussing up a steer or a pugnacious bronco which fixes the

brute so that it can neither move nor think. This is the hog-tie, and it is what Euclid

did to geometry."

Orr, Eleanor Wilson. 1987. Twice as Less. New York: W. W. Norton & Co. Inc. Orr

opens her foreword, "I am a high school teacher—a teacher of mathematics and

science." She declares this with obvious pride, but she also declares it to lay the

foundation for a practical, and pivotal, study in how black english vernacular (BEV)

affects the learning of black students in math and science. "In this book I show how

the misunderstandings that had puzzled me relate to the students' nonstandard uses of

certain prepositions and conjunctions that in standard English distinguish certain

quantitative ideas, and I show where there is reason to believe that these nonstandard

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uses are rooted in the grammar of BEV. I emphasize, however, that it is the many

similarities between BEV and standard English that make the differences a

problem—more of a problem than they would be if the vocabularies and grammars of

the languages were totally distinct." Teasing out the differences between the

grammatical and syntactical structures of BEV and standard English in relation to

quantitative understanding required ten years of collecting data and consultation with

linguistic experts. In the end she boils the research down to illustrate just where the

traps are. A literacy program in any school with BEV speakers should include this

among its "must-read" professional books.

Padgett, R., ed. 1987. The Teachers & Writers Handbook of Poetic Forms. New York:

Teachers and Writers. The handbook was not written with mathematics in mind, but

many of the forms offer interesting mathematical patterns in their structure. Math

teachers who are interested in having students write poems will find good ideas here.

For instance, Tammy Vu, in her chapter, "'I Would Have Laughed. . .': A Math

Classroom Transformed by Literature," describes her use of the acrostic.

Paulos, J. A. 1995. A Mathematician Reads the Newspaper. New York: Doubleday. John

Allen Paulos' classic deconstruction of the way we use statistics. It has immediate

appeal to students who already feel a bit cynical and manipulated by the media and

helps to enlighten the rest. It is dedicated "To storytelling number-crunchers and

number-crunching storytellers."

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Picker, S. H., and J. S. Berry. 2000. " Investigating Pupils' Images of Mathematicians."

Educational Studies in Mathematics 43 (1): 65–94. A further elaboration of their

chapter in this work.

———. 2001. "Your Students' Images of Mathematicians and Mathematics."

Mathematics Teaching in the Middle School . 7 (4): 202–209. More elaboration of

their chapter in this work, with practical suggestions for solving mathematics'

ingrained image problems.

Ritchart, Ron, ed. 1997. Through Mathematical Eyes Exploring Relationships in Math

and Science. Portsmouth, NH: Heinemann. Part of the Moving Middle Schools series

of research papers, the book explains in the friendliest of terms how teaching for

understanding in a mindful classroom, using writing and projects, can eliminate the

need to even ask the question, Is this a humane way to teach mathematics?

Robson, E., and J. Wimp. 1979. Against Infinity: An Anthology of Contemporary

Mathematical Poetry. Parker Ford, PA: Primary Press. Ernest Robson and Jet Wimp

have undertaken a mission to bring about a reunion of the sciences and the arts that

would rival that of the metaphysical poets. "As co-editors of an anthology which

attempts once again to reconcile these willful disciplines, we feel we should declare

ourselves," they say in their introduction. They define mathematical poetry as "an

association of mathematical concepts, relationships, symbols or forms with

interesting verbalizations and/or graphic components." They present mathematical

poetry in three contemporary forms: sound poetry, visual (concrete) poetry, and

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conventional poetry (free or rhymed verse). Students will be very attracted to this

visually and intellectually playful anthology.

Rubenstein, R. N., and R. K. Schwartz. 2000. "Word Histories: Melding Mathematics

and Meanings." Mathematics Teacher 93 (8): 664–669. A wonderful article getting at

mathematics through the words it uses. Enlightening, practical, and entertaining, it

provides ideas for a level of understanding that should be a part of every

mathematical classroom.

Sears, P. 1986. Secret Writing: Keys to the Mysteries of Reading and Writing. New York:

Teachers & Writers. Peter Sears demonstrates how codes and ciphers work to help us

understand the conventions of English as well as other languages, give us access to

works of literature, understand the "language of numbers" more thoroughly, and

interpret the signs and behaviors around us. Here is a shift-coded message to readers

about our idealistic curricular ideas. It is a plus-four code using the English alphabet:

HVIEQ SR

Here is another coded message, for which the reader must find the key:

SREMAERD EHT OT SGNOLEB EFIL

Schifter, D. 1996. "A Constructivist Perspective on Teaching and Learning

Mathematics." In Constructivism: Theory, Perspectives and Practice, ed. C. T.

Fosnot. New York: Teachers College Press. On Ian Hauser's reading list for those

interested in constructivist theory (see his chapter, "Every Class Is an English Class").

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Schneider, M. S. 1994. A Beginner's Guide to Constructing the Universe: The

Mathematical Archetypes of Nature, Art and Science. New York: HarperCollins. To

quote the cover copy, Michael S. Schneider "leads us on a spectacular, lavishly

illustrated journey along the numbers one through ten to explore the mathematical

principles made visible in flowers, shells, crystals, plants, and the human body,

expressed in the symbolic language of folk sayings and fairy tales, myth and religion,

art and architecture. This is a new view of mathematics, not the one we learned at

school but a comprehensive guide to the patterns that recur through the universe and

underlie human affairs." It may not be the view of math we learned in school but it

should be.

Seidel, S., and J. Walters, E. Kirby, N. Olff, K. Powell, L. Scripp, I. S. Veenema. 1997.

Portfolio Practices: Thinking Through the Assessment of Children's Work .

Washington, DC: National Education Association. Steve Seidel and collaborators

have produced a thought-provoking manual that proves useful for integrating

portfolio practice within and across the disciplines. I have used it to help support

changes in schools seeking to become more reflective in their practice.

Singh, S. 1997. Fermat's Enigma: The Epic Quest to Solve the World's Greatest

Mathematical Problem. New York: Doubleday. Another of the books that

transformed the culture of Tammy Vu's classroom (see her chapter, "'I Would Have

Laughed. . .': A Classroom Transformed by Literature") as well as my own

experience of mathematics. It delivers deep pleasures in its evocation of the history of

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an idea, its descriptions of historical figures and eras, and its dramatic rendering of

the story of Andrew Wiles' ultimate solution to Fermat's last theorem. Part of the

magic of this book is in its subject; the theorem is so simple and its solution so

complex that it reads like a great mystery. It is in Singh's book that I discovered the

intriguing life of the young political martyr and great mathematician Évariste Galois,

who attempted, at the age of twenty, to write down all his mathematical ideas the

night before the duel that killed him. (Talk about your last-minute term paper!) Singh

quotes a tragically moving note he wrote to his friends:

I beg my patriots, my friends, not to reproach me for dying otherwise than for my

country. I died the victim of an infamous coquette and her two dupes. It is in a

miserable piece of slander that I end my life. Oh! Why die for something so little,

so contemptible? I call on heaven to witness that only under compulsion and force

have I yielded to a provocation which I have tried to avert by every means.

Surely such passions, when discovered by students, will help humanize the discipline.

Smith. F. 1975. Comprehension and Learning: A Conceptual Framework for Teachers.

New York: Holt, Rinehart and Winston. This and the following Smith work are on

Ian Hauser's reading list for those interested in constructivist theory (see his chapter,

"Every Class Is an English Class").

———. 1994. Understanding Reading. Hilldale, N.J.: Lawrence Erlbaum.

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Stewart, I. 1995. Nature's Numbers: The Unreal Reality of Mathematical Imagination.

New York: Basic Books. An uncomplicated explanation of how mathematics helps us

interpret the natural world. Demonstrates what mathematics has done for human

understanding and touches on the role of math in human culture. Great book for

independent reading and inquiry projects like those recommended by Susan

Wildstrom in Sylvia Gross' chapter, "From Windex to Wildstrom: Conversations with

My Teacher."

Stigler, J. W. and J. Hiebert. 1999. The Teaching Gap: Best Ideas from the World's

Teachers for Improving Education in the Classroom. New York: The Free Press.

Drawing on the conclusions of the Third International Mathematics and Science

Study (TIMSS) that reform is called for in mathematics education, James W. Stigler

and James Hiebert study teaching as it is actually practiced in math classrooms in

Germany, Japan, and the United States. They ultimately recommend a constructivist

approach based on the Japanese practice of "Lesson Study." Although they do not

look closely at the interdisciplinary themes of this work they nevertheless

demonstrate that students learn best when they are given the chance to undertake their

own explorations within a rigorous academic context. They also demonstrate the

value of a continuous process of school-based professional development. Anyone

interested in school reform needs to be familiar with this book.

Thompson, C. L., and J. S. Zeuli. 1997. The Frame and the Tapestry: Standards-Based

Reform and Professional Development. Ann Arbor: Michigan State University.

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Recommended by Matt Wayne in his chapter, "Getting Smarter: A Seventh-Grade

Class Researches and Reflects on Its Discussion Habits."

Tobias, S. 1993. Overcoming Math Anxiety (Revised and Expanded). New York: W. W.

Norton & Co., Inc. A seminal book, and origin of the household term. If teaching

math has become more humane, a good percentage of the responsibility can probably

be ascribed to this book. She asked the question, "Why do otherwise intelligent

adults, people who do well in subjects they like, have a specific disability in

mathematics?" She proved the answer to be not that they didn't have the necessary

"cognitive equipment" but that they didn't believe they had it. Ambiguity, intuition,

self-knowledge, gender bias, language, thought patterns . . . all play a role in the

learning, appreciation, and performance of mathematics, and Tobias shows just how.

She sees the different talents in different fields as "beacons in a continuum of human

curiosity in search of meaning." The opening paragraph of Chapter 4 in Overcoming

Math Anxiety reads: "Every concept in mathematics has two histories. The first is the

story of how, during the 10,000 or so years of human history, certain ideas slowly

emerged to make sense of the numerical relationships in the world around us. The

second is a personal history of how each of us, guided by all the discoveries that were

made before we were born, struggle individually to make sense of that world of

numbers for ourselves." Tobias is the cofounder of the innovative Math Anxiety

Clinic and author of an equally powerful sequel: Succeed with Math: Every Student's

Guide to Conquering Math Anxiety (which has an extraordinarily clear and useful

section on how to read mathematics texts using a "kit-building" metaphor).

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Treisman, U. 1992. "Studying Students Studying Calculus: A Look at the Lives of

Minority Math Students in College." College Mathematical Journal . 23(5), 362–372.

Recommended by Sylvia Gross in her chapter, "From Windex to Wildstrom:

Conversations with My Teacher."

Tsuruda, G. 1994. Putting It Together: Middle School Math in Transition. Gary Tsuruda

uses the panoply of ideas discussed (inadvertently but not coincidentally) in this book

to transform the culture of his middle school math classroom. Readers will get a good

idea of how it all comes together to make sense in his book.

Venables, D. R., and G. Peters. 1996. "Innovation and Practice: Changing Thinking and

Practice in Two Math Departments." Writing Within School Reform (6). Providence,

RI: The Annenberg Institute for School Reform. Two teachers talk persuasively of

integrating portfolios and accomplishing change in the cultures of their classrooms

and schools.

Von Glasersfeld, E. 1996. "Introduction: Aspects of Constructivism." In Constructivism:

Theory, Perpectives and Practice, ed. C. T. Fosnot. New York: Teachers College

Press. On Ian Hauser's reading list for those interested in constructivist theory (see his

chapter, "Every Class Is an English Class").

Vygotsky. L. 1978. Mind in Society: The Development of Higher Psychological

Processes. Cambridge: Harvard University Press. On Ian Hauser's reading list for

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those interested in constructivist theory (see his chapter, "Every Class Is an English

Class").

Wiggins, G., and J. McTighe. 1998. Understanding by Design. Alexandria, VA:

Association for Supervision and Curriculum Development. After a study of Grant

Wiggins and Jay McTighe's curricular ideas, a teacher I know and respect said, "You

don't need anything else." This is another "must read," or, perhaps more to the point,

"must understand," if one is to meaningfully integrate the disciplines in the schools.

The simple premise of the work is that educators need to figure out what they want

students to understand and how they will be assessed before planning activities. The

authors tease apart understanding from knowledge and skills, and demonstrate the

function of essential questions in getting to the understanding. As a staff developer I

have used the book's concepts extensively to help teachers plan for the deepest

possible engagement of their students.

Learning needs to be more experiential, more geared toward making students directly

confront the effects—and affect —of decisions, ideas, theories, and problems. The

absence of experience in learning may explain why so many important ideas are

misunderstood and learnings so fragile, as the misconception literature reveals. (57)

The authors are clearly aware of the significance of the "continuum of experience"

discussed by Dewey in Experience and Education, which is so much at the heart of

this book.

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Wildstrom, S. 1999. "Encouraging an Enjoyment of Mathematics through Reading and

Writing." Mathematical Association of America. Recommended by Sylvia Gross in

her chapter, "From Windex to Wildstrom: Conversations with My Teacher."

Worsley, D., and B. Mayer. 1989. The Art of Science Writing . New York: Teachers &

Writers. A compendium of ideas, samples, and bibliographic references for writing in

science and mathematics. A writing workshop plan is followed by specific ideas for

writing experiments. A FAQ section gets to the most common challenges of writing

in the math and science classroom. Samples of writing from scientists,

mathematicians, students, journalists, and poets provide models. An extensive

bibliography helps orient interested parties to the literature.

Zinn, H. 1995. A People's History of the United States: 1942–Present . New York:

HarperCollins. Howard Zinn keeps the mathematics of history in mind without fail

when he recounts historic events. How many people, of what description, were

involved in, or affected by, events? It is a soberingly democratic approach that takes

the gloss off idealization and puts backbone into the true potential of democracy to

account for and empower all citizens. The following short passage early on in the

book illustrates how he disabuses us of our myths and helps us get to the truths that

have the potential to liberate us politically and morally. Note that there is a kind of

"number sense" to historical thinking, without which history is but mythmaking.

When he arrived on Hispaniola in 1508 Las Casas says, "there were 60,000

people living on this island, including the Indians; so that from 1494 to 1508, over

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three million people had perished from war, slavery and the mines. Who in future

generations will believe this? I myself writing it as a knowledgable eyewitness

can hardly believe it. . . ."

Thus began the history, five hundred years ago, of European invasion of

the Indian settlements in the Americas. That beginning, when you read Las

Casas—even if his figures are exaggerations (were there 3 million Indians to

begin with, as he says, or less than a million, as some historians have calculated,

or 8 million as others now believe?)—is conquest, slavery, and death. When we

read the history books given to children in the United States, it all starts with

heroic adventure—there is no bloodshed—and Columbus Day is a celebration. (7)

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Websites

General

http://www.joma.org (Journal of Online Mathematics Applications—applets)

http://www.maa.org (Mathematical Association of America)

http://www.ams.org (American Mathematical Society)

http://www.awm-math.org (Association for Women in Mathematics)

http://www.enc.org/ (assorted mathematical sites and activities)

http://mathforum.org/~steve/ (Geometry Forum)

http://www.math.psu.edu/MathLists/Contents.html (Penn State)

http://euclid.math.fsu.edu/Science/math.html (Florida State)

http://www.geom.umn.edu/ (The Geometry Center)

http://camel.math.ca/Education/MallMath/ (Canadian site with math activities)

http://www.tc.cornell.edu/Services/Edu/MathSciGateway/math.asp (Theory Center)

http://www.npac.syr.edu/textbook/kidsweb/math.html (need to push the STOP key)

http://math.nist.gov/gams/

http://www.math.cudenver.edu/w4t/

http://galileo.imss.firenze.it/museo/4/index.html

Precalculus

http://www.math.toronto.edu/mathnet/answer.html (infinity)

http://members.tripod.com/~Paul_Kirby/vector/Vdotproduct.html

http://mecca.org/~halfacre/MATH/vector.htm

http://www.ti.com/calc/

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http://www.joma.org/http://www.maa.org/http://www.ams.org/http://www.awm-math.org/http://www.enc.org/http://mathforum.org/~steve/http://www.math.psu.edu/MathLists/Contents.htmlhttp://euclid.math.fsu.edu/Science/math.htmlhttp://www.geom.umn.edu/http://camel.math.ca/Education/MallMath/http://www.tc.cornell.edu/Services/Edu/MathSciGateway/math.asphttp://www.npac.syr.edu/textbook/kidsweb/math.htmlhttp://math.nist.gov/gams/http://www.math.cudenver.edu/w4t/http://galileo.imss.firenze.it/museo/4/index.htmlhttp://www.math.toronto.edu/mathnet/answer.htmlhttp://members.tripod.com/~Paul_Kirby/vector/Vdotproduct.htmlhttp://mecca.org/~halfacre/MATH/vector.htmhttp://www.ti.com/calc/http://www.ti.com/calc/http://mecca.org/~halfacre/MATH/vector.htmhttp://members.tripod.com/~Paul_Kirby/vector/Vdotproduct.htmlhttp://www.math.toronto.edu/mathnet/answer.htmlhttp://galileo.imss.firenze.it/museo/4/index.htmlhttp://www.math.cudenver.edu/w4t/http://math.nist.gov/gams/http://www.npac.syr.edu/textbook/kidsweb/math.htmlhttp://www.tc.cornell.edu/Services/Edu/MathSciGateway/math.asphttp://camel.math.ca/Education/MallMath/http://www.geom.umn.edu/http://euclid.math.fsu.edu/Science/math.htmlhttp://www.math.psu.edu/MathLists/Contents.htmlhttp://mathforum.org/~steve/http://www.enc.org/http://www.awm-math.org/http://www.ams.org/http://www.maa.org/http://www.joma.org/


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http://www.jalacy.com (conic sections)

http://mathworld.wolfram.com/topics/ConicSections.html

http://www.stg.brown.edu/projects/projects.old/classes/ma8/papers/dmargalit/

http://www.sisweb.com/math/algebra/conics.htm

http://www.mathforum.org/dr.math/

http://www.sosmath.com/calculus/geoser/geoser01.html

http://www.sparknotes.com/math/precalc/sequencesandseries/summary.html

http://www.schoolnet.ca/vp-pv/amof/index.html (combinations, permutations, etc.)

http://www.distancemath.com/preview/unit1/funct3.htm (functions)

http://dosxx.colorado.edu/~atlas/math/math81.html (logarithms)

http://engineering.uow.edu.au/Courses/Stats/File2419.html (combinations)

http://www.hoxie.org/math/algebra/sequen.htm

Calculus

http://www.hofstra.edu/~matscw/realworld.html (calculus "help" tutorials)

http://www.eyesoftime.com/teacher/math.htm (calculus)

http://www.seresc.k12.nh.us/www/alvirne.html (AP calculus site run by high school)

http://archives.math.utk.edu/visual.calculus/ (distance learning lessons in calculus)

http://www.rose-hulman.edu/Class/CalculusProbs (calculus problems at Rose-Hulman)

http://www.hsu.edu/faculty/lloydm/ti/prgmtabl.html

http://www.integrals.com (Mathematica website that will integrate)

http://www.uncwil.edu/courses/webcalc

http://www.wshs.fcps.k12.va.us/academic/math/staff/mdeegan/apcalc/notebook.htm

(summary sheet of ideas for AP exam)

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http://www.jalacy.com/http://www.stg.brown.edu/projects/projects.old/classes/ma8/papers/dmargalit/http://www.sisweb.com/math/algebra/conics.htmhttp://www.mathforum.org/dr.math/http://www.sosmath.com/calculus/geoser/geoser01.htmlhttp://www.sparknotes.com/math/precalc/sequencesandseries/summary.htmlhttp://www.schoolnet.ca/vp-pv/amof/index.htmlhttp://www.distancemath.com/preview/unit1/funct3.htmhttp://dosxx.colorado.edu/~atlas/math/math81.htmlhttp://engineering.uow.edu.au/Courses/Stats/File2419.htmlhttp://www.hoxie.org/math/algebra/sequen.htmhttp://www.hofstra.edu/~matscw/realworld.htmlhttp://www.eyesoftime.com/teacher/math.htmhttp://www.seresc.k12.nh.us/www/alvirne.htmlhttp://archives.math.utk.edu/visual.calculus/http://www.rose-hulman.edu/Class/CalculusProbshttp://www.hsu.edu/faculty/lloydm/ti/prgmtabl.htmlhttp://www.integrals.com/http://www.uncwil.edu/courses/webcalchttp://www.wshs.fcps.k12.va.us/academic/math/staff/mdeegan/apcalc/notebook.htmhttp://www.wshs.fcps.k12.va.us/academic/math/staff/mdeegan/apcalc/notebook.htmhttp://www.uncwil.edu/courses/webcalchttp://www.integrals.com/http://www.hsu.edu/faculty/lloydm/ti/prgmtabl.htmlhttp://www.rose-hulman.edu/Class/CalculusProbshttp://archives.math.utk.edu/visual.calculus/http://www.seresc.k12.nh.us/www/alvirne.htmlhttp://www.eyesoftime.com/teacher/math.htmhttp://www.hofstra.edu/~matscw/realworld.htmlhttp://www.hoxie.org/math/algebra/sequen.htmhttp://engineering.uow.edu.au/Courses/Stats/File2419.htmlhttp://dosxx.colorado.edu/~atlas/math/math81.htmlhttp://www.distancemath.com/preview/unit1/funct3.htmhttp://www.schoolnet.ca/vp-pv/amof/index.htmlhttp://www.sparknotes.com/math/precalc/sequencesandseries/summary.htmlhttp://www.sosmath.com/calculus/geoser/geoser01.htmlhttp://www.mathforum.org/dr.math/http://www.sisweb.com/math/algebra/conics.htmhttp://www.stg.brown.edu/projects/projects.old/classes/ma8/papers/dmargalit/http://www.jalacy.com/


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http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/

http://www.math.ucdavis.edu/~kouba/ProblemsList.html

http://www.math.psu.edu/dna/graphics.html

http://library.thinkquest.org/10030/calcucon.htm (general calculus materials)

http://archives.math.utk.edu/visual.calculus/3/mvt.3/index.html (mean value theorem)

http://www.ies.co.jp/math/java/calc/rolhei/rolhei.html (mean value theorem applet)

http://www.math.hmc.edu/calculus/tutorials/ (Harvey Mudd College tutorials)

Multivariable Calculus and Other Advanced Applications

http://www.ies.co.jp/math/java/calc/index.html (Java applets on assorted topics)

http://smard.cqu.edu.au/Database/Teaching (Java applets to support math education)

http://www.utc.edu/~cpmawata/

http://www.math.cudenver.edu/w4t/

http://www.mcasco.com/p1va.html (vectors)

http://mecca.org/~halfacre/MATH/vector.htm

http://www.sosmath.com/diffeq/diffeq.html (differential equations)

http://math.stcc.mass.edu/CalculusIII/157.html (tangent planes and extrema)

http://www.math.arizona.edu/~vector/Block2/pder/pder.html (partial derivatives)

http://www.math.gatech.edu/~carlen/2507/notes/lagMultipliers.html (Lagrange

multipliers)

http://web.mit.edu/wwmath/vectorc/index.html

http://www.math.ucla.edu/~ronmiech/Actuarial_Review/Multi_Var_Calc/Master/Master.

html (practice problems—multiple choice with explanations)

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http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/http://www.math.ucdavis.edu/~kouba/ProblemsList.htmlhttp://www.math.psu.edu/dna/graphics.htmlhttp://library.thinkquest.org/10030/calcucon.htmhttp://archives.math.utk.edu/visual.calculus/3/mvt.3/index.htmlhttp://www.ies.co.jp/math/java/calc/rolhei/rolhei.htmlhttp://www.math.hmc.edu/calculus/tutorials/http://www.ies.co.jp/math/java/calc/index.htmlhttp://smard.cqu.edu.au/Database/Teachinghttp://www.utc.edu/~cpmawata/http://www.math.cudenver.edu/w4t/http://www.mcasco.com/p1va.htmlhttp://mecca.org/~halfacre/MATH/vector.htmhttp://www.sosmath.com/diffeq/diffeq.htmlhttp://math.stcc.mass.edu/CalculusIII/157.htmlhttp://www.math.arizona.edu/~vector/Block2/pder/pder.htmlhttp://www.math.gatech.edu/~carlen/2507/notes/lagMultipliers.htmlhttp://web.mit.edu/wwmath/vectorc/index.htmlhttp://www.math.ucla.edu/~ronmiech/Actuarial_Review/Multi_Var_Calc/Master/Master.htmlhttp://www.math.ucla.edu/~ronmiech/Actuarial_Review/Multi_Var_Calc/Master/Master.htmlhttp://www.math.ucla.edu/~ronmiech/Actuarial_Review/Multi_Var_Calc/Master/Master.htmlhttp://www.math.ucla.edu/~ronmiech/Actuarial_Review/Multi_Var_Calc/Master/Master.htmlhttp://web.mit.edu/wwmath/vectorc/index.htmlhttp://www.math.gatech.edu/~carlen/2507/notes/lagMultipliers.htmlhttp://www.math.arizona.edu/~vector/Block2/pder/pder.htmlhttp://math.stcc.mass.edu/CalculusIII/157.htmlhttp://www.sosmath.com/diffeq/diffeq.htmlhttp://mecca.org/~halfacre/MATH/vector.htmhttp://www.mcasco.com/p1va.htmlhttp://www.math.cudenver.edu/w4t/http://www.utc.edu/~cpmawata/http://smard.cqu.edu.au/Database/Teachinghttp://www.ies.co.jp/math/java/calc/index.htmlhttp://www.math.hmc.edu/calculus/tutorials/http://www.ies.co.jp/math/java/calc/rolhei/rolhei.htmlhttp://archives.math.utk.edu/visual.calculus/3/mvt.3/index.htmlhttp://library.thinkquest.org/10030/calcucon.htmhttp://www.math.psu.edu/dna/graphics.htmlhttp://www.math.ucdavis.edu/~kouba/ProblemsList.htmlhttp://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/


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Puzzles and Problem Solving

http://www.geocities.com/CapeCanaveral/Lab/4661/index.html (Olympiad problems)

http://www.mandelbrot.org (Mandelbrot competition)

http://www.mathcounts.org (MathCounts competition)

http://www.math.smsu.edu/~les/POTW.html (Southern Missouri State math problem of

the week)

http://www.olemiss.edu/mathed/contest

http://www.math.hmc.edu/funfacts

http://www.kent.wednet.edu/pcpow (Kent County, WA, physics and math problem of the

week—prizes for correct answers!)

http://www.mathwright.com (source of computer notebooks and books for all)

http://www.mathnerds.com

http://www.ee.ryerson.ca/~elf/abacus/

http://members.aol.com/rmathmania/index.htm

http://www.ti.com/calc

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html (introduction to

Fibonacci numbers)

History

http://www.sphere.bc.ca/test/sruniverse.html (slide rules)

http://aleph0.clarku.edu/~djoyce/mathhist/mathhist.html (math history)

http://www.pbs.org/teachersource/math.htm (Public Broadcasting Service)

http://world.std.com/~reinhold/mathmovies.html (mathematics in movies)

http://www.mathnews.uwaterloo.ca/

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http://www.geocities.com/CapeCanaveral/Lab/4661/index.htmlhttp://www.mandelbrot.org/http://www.mathcounts.org/http://www.math.smsu.edu/~les/POTW.htmlhttp://www.olemiss.edu/mathed/contesthttp://www.math.hmc.edu/funfactshttp://www.kent.wednet.edu/pcpowhttp://www.mathwright.com/http://www.mathnerds.com/http://www.ee.ryerson.ca/~elf/abacus/http://members.aol.com/rmathmania/index.htmhttp://www.ti.com/calchttp://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.htmlhttp://www.sphere.bc.ca/test/sruniverse.htmlhttp://aleph0.clarku.edu/~djoyce/mathhist/mathhist.htmlhttp://www.pbs.org/teachersource/math.htmhttp://world.std.com/~reinhold/mathmovies.htmlhttp://www.mathnews.uwaterloo.ca/http://www.mathnews.uwaterloo.ca/http://world.std.com/~reinhold/mathmovies.htmlhttp://www.pbs.org/teachersource/math.htmhttp://aleph0.clarku.edu/~djoyce/mathhist/mathhist.htmlhttp://www.sphere.bc.ca/test/sruniverse.htmlhttp://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.htmlhttp://www.ti.com/calchttp://members.aol.com/rmathmania/index.htmhttp://www.ee.ryerson.ca/~elf/abacus/http://www.mathnerds.com/http://www.mathwright.com/http://www.kent.wednet.edu/pcpowhttp://www.math.hmc.edu/funfactshttp://www.olemiss.edu/mathed/contesthttp://www.math.smsu.edu/~les/POTW.htmlhttp://www.mathcounts.org/http://www.mandelbrot.org/http://www.geocities.com/CapeCanaveral/Lab/4661/index.html


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Susan Wildstrom's Recommended Reading List

Books

Abbott, Edwin: Flatland

Barrow, John D.: Pi in the Sky

Berlinski, David: A Tour of the Calculus

Conway and Guy: The Book of Numbers

Courant and Robbins: What is Mathematics?

Dudley, Underwood: The Trisectors, Mathematical Cranks

Dunham, William: Journey Through Genius, Euler The Master of Us All

Friedberg, Richard: An Adventurer ' s Guide to Number Theory

Gamov, George: One, Two Three, Infinity, Mr Tompkins…

Gardner, Martin: Diversions from Scientific American, Mathematical Circus,

Mathematical Carnival, Mathematical Magic Show, aha! Insight, ah! Gotcha, etc.

Gleick, James: Genius

Golomb, Solomon: Polyominoes

Hardy, G. H.: A Mathematician' s Apology

Hoffman, Paul: Archimedes' Revenge, The Man Who Loved Only Numbers

Hoffstadter, Douglas: Goedel, Escher, Bach; Metamagical Themas

Honsberger, Ross: Ingenuity in Mathematics and other books

Kanigel, Robert: The Man Who Knew Infinity

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Knuth, Donald: Surreal Numbers

Krause, Eugene: Taxicab Geometry

Larson, Loren: Problem Solving Through Problems

Logsdon, Mayme: A Mathematician Explains

Maor, Eli, e: The story of a number, To Infinity and Beyond, Trigonometric Delights

Nahim, Paul, −1 : An Imaginary Tale

Pappus, Theoni: The Joy of Mathematics, More Joy of Mathematics, Mathematical

Scandals

Paulos, John Allen: Innumeracy; Beyond Numeracy; A Mathematician Reads the

Newspaper; Once Upon a Number

Polya, George: How to Solve It

Rademacher, Hans, and Otto Toeplitz: The Enjoyment of Mathematics

Rusczyk, Richard, and Sandor Rehovsky: The Art of Problem Solving

Schechter, Bruce: My Brain is Open

Smullyan, Raymond: What is the Name of This Book, This Book Has No Name, Alice in

Puzzleland, The Lady or The Tiger, To Mock A Mockingbird , and other books.

Stewart, Ian: The Mathematical Tourist, Islands of Truth, and other books.

Sved, Martha: Journey into Geometries

The Trachtenberg System of Speed Mathematics

Wells, David: You Are a Mathematician

Wickelgren, Wayne: How to Solve Problems

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Journals and Magazines

College Mathematics Journal

Crux Mathematicorum

Discover

Math Horizons

Mathematics & Informatics Quarterly

Mathematics Magazine

Quantum

Science

Scientific American

The Arithmetic Teacher

The Mathematics Teacher

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Websites: Extensions and Connections

Matthew Szenher conducted a special investigation of certain websites for Teaching for

Depth. His ideas are reported here.

I took on the task of investigating Internet content for Teaching for Depth, understanding

the theme of the book to be this: If teachers help their students overcome the "otherness"

of mathematics and employ interdisciplinary practices, their students are more likely to

become curious and confident seekers of truth. In my investigation I tapped into

numerous sites and generally found them only superficially representative of this theme.

Take, for example, the page Mathematics in the Movies at

http://world.std.com/~reinhold/mathmovies.html. Initially I thought, "Wow, kids like

movies. This is a great interdisciplinary approach to inspire students to think about deep

questions." Unfortunately, the site offers no opinion on how to use movies to enrich

learning experiences. Nor do movies, as is clear from the site, ever delve into the use of

math in real and profound ways. Mathematical genius is usually just an easily dramatized

character trait. Most of the other sites that I found have the same problem: superficially

interesting content that on closer inspection offers little that is pedagogically useful.

On the other hand I found a few sites to be interesting because they led me to

people and organizations that truly seemed to embody, clarify, or extend the book's

themes. I've reviewed some of these sites below as a starting point for readers' ownInternet journeys. If I've learned one thing about the Internet in this experience, it is this:

one good site will almost invariably lead to more quality content. I caution that there is

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plenty of detritus on the Internet, but, occasionally, you find a cache of sites that will

change the way you teach.

The Writings of Stephen I. Brown at

http://members.tripod.com/mumnet/sibrown/sib001.html. Several of the eight essays at

this site overlap the themes of this book. Of these, the piece entitled "Toward Humanistic

Mathematics Education" is the most relevant (see the annotated bibliography). In it,

Brown attempts "to elaborate upon the concept of humanistic mathematics education"

(HME). He defines HME initially as a reaction to the traditional characterization of

mathematics as "a depersonalized, uncontextualized, noncontroversial and asocial form

of knowledge." Brown concedes that describing, placing, and defending HME is an

"enormous" task. This essay is indeed quite wordy and complex, requiring

uncompromised attention.

Ultimately Brown argues that HME would "inject a strong sense of personhood

into the doing of mathematics" and would encourage "creating or seeking heuristics that

enable one to face what is unknown (maybe unknowable)." HME also cultivates a

"sophisticated view of what might be involved in applying mathematics to the 'real

world.'" All this is in opposition to liberal mathematics education in which mathematics

is conveyed merely as exercises in deductive logic.

That mathematics is isolated and depersonalized, Brown argues, stems in fact

from the identification of deductive logic "as its distinguishing feature." The primacy of

logic, though, is "more fragile than we are led to believe" (though still a "critical

component of mathematical thought"). He asks, "If logic were all there were to

mathematics, couldn't a computer do research in the field just as effectively as a human?"

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Computer scientists have tried to create computerized mathematicians and failed because,

for one, computers lack the aesthetic sense to choose interesting problems to explore.

With this and other evidence, Brown calls the primacy of deductive logic in mathematics,

and in mathematics education, into question. Having demoted logic, Brown asks what

should supplement it.

Reviewing the history of mathematics education, with particular attention paid to

Dewey and related philosophers, Brown concludes that problem solving is an essential

part of mathematics. If taught constructively, an emphasis on problem solving leads to

partial fulfillment of Brown's ideal HME and provides a "view of mathematics that far

outstrips the notion of mathematics as deductive logic." After all, "there is a world of

difference between thinking of mathematics as either following or offering a logically

deductive pristine argument and creating or seeking heuristics that enable one to face

what is unknown (maybe unknowable)."

Still, "the concept [HME] is in need of major surgery." He argues that

mathematics education should be imbued "with a sense of purpose," though there are

difficulties in explaining purpose to a lay audience of students. Brown deftly describes

these problems.

This and some other of Brown's essays augment the material in this book, and

vice versa. Brown lacks detailed examples of HME, something we have in abundance.

Brown, on the other hand, presents a history of HME, whereas we do not. Brown's HME

and our concept of teaching for depth are surely fraternal, if not identical, twins. One

could certainly learn much about the one by studying the other.

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The Humanistic Mathematics Network

S. I. Brown, in "Toward Humanistic Mathematics Education," describes an association

called The Humanistic Mathematics Network who are (to quote Brown) "devoted to an

exploration of the relationship of mathematics to the humanities in a number of

philosophically interesting ways." (See the annotated bibliography for more on the

Humanistic Mathematics Network.)

The members of the Humanistic Mathematics Network desire their

students/classrooms to be/have:

• historical orientation; mathematics as a "human" endeavor;

• cooperative learning groups;

• a variety of assessment techniques, not just written tests;

• teacher/student and student/student interaction;

• surveys of student attitudes and opinions;

• interesting problems and open-ended questions, not just exercises;

• humor;

• attention to aesthetics;

• somewhat student driven;

• student ownership of mathematics, opportunity to create their own

meanings;

• students actively involved in learning, not just passive consumers;

• stimulating classroom environment;

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• access to and appropriate use of tools and technology;

• personal involvement and caring;

• less time restraints on examinations;

• time for reflection.

• (from http://www.cord.edu/faculty/haglund/hm.html)

Although there are numerous references to the HMN on the Internet, the network

itself doesn't seem to have an online presence. There is a collection of HMN essays and

contacts at http://mathforum.org/mathed/humanistic.math.html.

The Math Forum at http://mathforum.org/

My search for the Humanistic Mathematics Network online led me to the Swarthmore

College Online Math Forum, which seems to be the mathematics education portal on the

Internet. A subsection of the site is devoted to humanistic mathematics

(http://mathforum.org/mathed/humanistic.math.html), and other sections of interest to our

readers are:

• calculus reform (http://mathforum.org/mathed/calculus.reform.html)

• constructivism (http://mathforum.org/mathed/constructivism.html)

• interdisciplinary studies

(http://mathforum.org/mathed/interdisciplinary.math.html)

• math education reform

(http://mathforum.org/mathed/math.education.reform.html)

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http://www.cord.edu/faculty/haglund/hm.htmlhttp://mathforum.org/mathed/humanistic.math.htmlhttp://mathforum.org/http://mathforum.org/mathed/humanistic.math.htmlhttp://mathforum.org/mathed/calculus.reform.htmlhttp://mathforum.org/mathed/constructivism.htmlhttp://mathforum.org/mathed/interdisciplinary.math.htmlhttp://mathforum.org/mathed/math.education.reform.htmlhttp://mathforum.org/mathed/math.education.reform.htmlhttp://mathforum.org/mathed/interdisciplinary.math.htmlhttp://mathforum.org/mathed/constructivism.htmlhttp://mathforum.org/mathed/calculus.reform.htmlhttp://mathforum.org/mathed/humanistic.math.htmlhttp://mathforum.org/http://mathforum.org/mathed/humanistic.math.htmlhttp://www.cord.edu/faculty/haglund/hm.html


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These sections contain discussions of the history, goals, and efficacy of their

particular umbrella subject. Some, like the section on interdisciplinary studies, contain

lesson plans.

My experience with the forum has been frustrating. The forum like other online

resources contains copious information but appears to be unrefereed. There are many

oysters and few pearls, and the entries can be a

Bibliography Teaching for Depth

Documents

Transcript of Bibliography Teaching for Depth