What is Mathematics?
Major Unifying Themes in
Mind and Body Tools
Science of Teaching &
This section contains General References as well
as References on Specific Topics. When searching
for information, first scan the Specific Topics
A bibliography of important general resources
for leaders in the field of Brain Science and ICT
in mathematics education.
on Specific Topics
Items at the top of this list have not yet been merged into the main list, and they remain to be more carefully analyzed.
Brain images available for free use. See (accessed 7/14/04): http://www.brainconnection.com/topics/?main=gal/home.
The following issue of Brain Connection has several articles about math. See http://www.brainconnection.com/buzz/index.php3?main=buzz051904.
Abraham Flexner wrote a report published in 1910 that discussed the Medical Schools in the United States. (Flexner, Abraham. Medical Education in the United States and Canada: A Report to the Carnegie Foundation for the Advancement of Teaching, bull. 4. New York: The Carnegie Foundation; 1910, p. xii [Reprinted in Birmingham, AL: Classics of Medicine Library; 1990, p. xii]) There are lots of Web references to this report. For example, see www.aacte.org/Research/flexnerrpt.pdf. Quoting from this reference:
For nearly a year there has been discussion of a possible "Flexner" type study of teacher education. Indeed, a federally supported, comprehensive look at teacher education programs seems likely with the July 11, 2003 passage of H.R. 2660. This appropriations bill, which calls for the Institute of Education Sciences to look at the status of teacher preparation in the U.S., has been referred to the Senate.
This report appeared at a time when work was already going on to upgrade Medical Schools. The report was eventually backed up by a lot of money from the Carnegie Foundations and others. The overall result of the reform movement was a substantial improvement in Medical Schools in the US.
(Normal Curve Equivalent) Standard of measurement used to compare student outcomes in reading and math. Normal classroom progress typically results in a NCE gain score around zero. Students who experience NCE gains that are greater than zero are progressing at an accelerated rate from normal classroom expectation
* Normal Curve Equivalent Scores
Normal Curve Equivalent Scores Normalized standard score with a mean of 50 and a standard deviation of 21.06
Number Blindness: A Hidden
Challenge for Mathematics
by Ashish Ranpura
Quoting from the Website:
Educators often worry that some students just don't "get" math. In truth, some fundamental difficulties with math may be indicators of mild dyscalculia, or "number blindness."
A 1998 report published in the Journal of Pediatrics estimated that approximately five percent of the school age population has some degree of dyscalculia, a sort of "number blindness" that is an impairment of the ability to recognize or manipulate numbers. This [suggests] that for some children, creative teaching techniques and studious discipline are not enough for a productive math education. Mild dyscalculia may easily go unnoticed, leading some students into educational settings that can offer only frustration.
See also: http://www.brainconnection.com/topics/?main=sci-news/math-brain
Trying to Figure Out Why Math Is So Hard for Some. Theories Abound: Genetics, Gender, How It's Taught
By Valerie Strauss
Washington Post Staff Writer
Tuesday, December 2, 2003; Page A13
This article points to the work of Michelle Mazzocco, director of the Math Skills Development Project at Baltimore's Kennedy Krieger Institute, a clinical and research facility for pediatric developmental disabilities.
An interesting quote from the article:
"That's the question we are all asking and that is driving the research," said Michelle Mazzocco, director of the Math Skills Development Project at Baltimore's Kennedy Krieger Institute, a clinical and research facility for pediatric developmental disabilities.
"There could be so many different causes leading to what we call poor math achievement and math disability, which are not necessarily the same thing," she said. "It has taken researchers decades to understand the fundamental difficultie
s of reading, and we are now at the place with math research where reading researchers were 20, 30 years ago."
Date: Sat, 10 Aug 2002 02:22:48 -0400 (EDT)
From: NSF Custom News Service
To: CNS Subscribers <email@example.com>
Subject: [nsf02084] - Newsletters/Journals
The following document (nsf02084) is now available
the NSF Online Document System
Title: Foundations - Volume 3 - Professional Development
supports School Mathematics Reform
It may be found at:
Adding It Up: Helping Children Learn Mathematics (2001)
Full book is available free online at:
Summertime: Is the Living Too Easy?
By Jane Quinn
Many of the young people that youth agencies serve are about to lose a lot of what theyve spent the past nine months learning. Research indicates that all young people experience significant learning losses during the summer break from school, and that the magnitude of these declines varies by grade level, subject matter and family income.
For example, low-income children show greater academic declines than do their more affluent peers. Regardless of income level, students lose an average of 2.6 months of grade-level equivalency in mathematical computation over the summer.
Youth-serving agencies can help to reverse this troubling phenomenon.
Jane Quinn is assistant executive director for community schools at The Childrens Aid Society in New York City. Contact: firstname.lastname@example.org.
Comment: In the past few weeks I have seen several articles that make claims about student declines in reading and math over the summer. The level of the amount of loss claimed seems to vary, with the article given above claiming a larger math computation loss than in other articles. In both reading and math, there are lower-order and higher-order skills. If the lower-order skills have not yet been incorporated well into the procedural part of one's brain (learned to a high level of automaticity), then we can expect significant losses over a time of disuse.
Accessed 7/25/03: http://www.nytimes.com/2003/07/23/education/23SCHO.html
Basic Skills Forcing Cuts in Art Classes
By DAVID M. HERSZENHORN
nder pressure to find time for the extra English and math classes required by the Education Department's new standardized curriculum, the city's junior high schools are slashing art, music and other electives, an unintended cost in the push to help students master basic skills.
Some schools are also reducing foreign language, social studies and science instruction to accommodate the curriculum, which requires that 18 periods more than half of the 35 instructional periods in a typical week be dedicated to reading and math.
"The art, music and everything else are basically out the window," said Joseph D. Cantara, the principal of Intermediate School 237 in Flushing, Queens. "Something has to go. What went is all the art, the music and the foreign language."
My comment: My readings of Devilin's book "The Math Gene" is that he argues that the "math gene" and the "reading gene" are one and the same. In essence, reading, writing, and arithmetic are closely related. We understand the process of developing fluency in reading and writing, but we have not made similar progress in understanding fluency in math. Fluency in math is not mainly based on computational skills. Rather, it is the thinking, logical arguments, understanding, and so on that is the essence of understanding and doing math. More work on basic skills does not help much. What is needed is a significant change in how we teach math.
Hartshorn, Robert and Boren, Sue (1990). Experiential learning of mathematics: Using manipulatives. ERIC Digest. Accessed 11/4/03: http://www.ericfacility.net/databases/ERIC_Digests/ed321967.html. Quoting from the Website:
Experiential education is based on the idea that active involvement enhances students' learning. Applying this idea to mathematics is difficult, in part, because mathematics is so "abstract." One practical route for bringing experience to bear on students' mathematical understanding, however, is the use of manipulatives. Teachers in the primary grades have generally accepted the importance of manipulatives. Moreover, recent studies of students' learning of mathematical concepts and processes have created new interest in the use of manipulatives across all grades.
In this Digest "manipulatives" will be understood to refer to objects that can be touched and moved by students to introduce or reinforce a mathematical concept. The following discussion examines recent research about the use of manipulatives. It also speculates on some of the challenges that will affect their use in the future.
Calculators and arithmetical computatoinal skill: http://www.detnews.com/2004/schools/0405/26/d06-164049.htm
Quoting from the newspaper article:
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SAN DIEGO Allowing fourth-graders to use handheld calculators may be masking a serious deficiency in their basic computation skills, a new study suggests.
Teachers began allowing young children to use calculators in the mid-1970s. The notion was that they would learn addition, subtraction and other basic skills with or without calculators, and could move more quickly to complex problems and enjoy math more. Several studies supported this.
While average math scores on the National Assessment of Educational Progress, or NAEP, show that computation skills improved in the 1980s and 1990s, allowing students to use calculators for a few items made the difference between their mastering the material or not, says Brookings Institution scholar Tom Loveless.
Calculators change everything, says Loveless, who presents findings here Thursday at the annual conference of the American Educational Research Association. For a large number of 9-year-olds, when calculators ... are not available, they get wrong answers.
Authentic Assessment in Mathematics. The Geometry Forum Summer '94 Workshop at Swarthmore College, Swarthmore, PA. Accessed 10/27/03: http://mathforum.org/sum94/project2.html. Quoting from the Website:
The new mathematics curriculum calls for an instructional setting which is very different from the typical classroom settings of the past. This curriculum combines new as well as traditional topics; mathematics is
presented to students in the form of rich situational problems that actively engage the students. The situational lessons or real-life problems attempt to include four dimensions:
- thinking and reasoning--engaging in such activities as gathering data, exploring, investigating, interpreting, reasoning, modeling, designing, analyzing, formulating hypotheses, using trial and error, generalizing, and checking solutions
- settings--working individually or in small groups
- mathematical tools--using symbols, tables, graphs, drawings, calculators, computers, and manipulatives
- attitudes and dispositions--including persistence, self-regulation and reflection, participation, and enthusiasm.
In short, students work to construct new knowledge that is integrated with their prior knowledge. The role of the teacher is that of a facilitator. The learning helps students acquire mathematical power to cope with ambiguity, to perceive patterns, and to solve unconventional problems.
Battista, Michael T. (February 1999). The Mathematical
Miseducation of America's Youth [Online]. Phi Delta
Kappan. Accessed 4/23/02: http://www.pdkintl.org/kappan/kbat9902.htm.
Quoting from the Website:
To perform a reasonable analysis of the quality
of mathematics teaching requires an understanding not
only of the essence of mathematics but also of current
research about how students learn mathematical ideas, Mr.
Battista points out. Without extensive knowledge of both,
judgments made about what mathematics should be taught to
schoolchildren and how it should be taught are
necessarily naive and almost always wrong.
Bloom, B.S. (1984). The 2 Sigma problem: The search for
methods of group instruction as effective as one-to-one
tutoring. Educational Researcher. v13, n6, pp4-16.
This paper explores the learning gains (both in
speed of learning and in amount learned) that are
achieved through ono-on-one tutoring. It then explores
research on ways to achieve similar (or, perhaps less,
but still positive) gains through master learning, peer
tutoring, and so on.
Bransford, J.D.; A. L. Brown; & R.R. Cocking: editors
(1999). How people learn: Brain, mind, experience, and
school. Washington, D.C.: National Academy Press.
[Online]. Accessed (4/14/00)
Debaene, Stanislas (1997). The Number Sense: How the Mind
Created Mathematics. NY & Oxford: Oxford University
ERIC Clearinghouse for Science, Mathematics, and
Environmental Education [Online]. Accessed 3/17/02:
Quoting from the Website:
The Clearinghouse is a component of the
Educational Resources Information Center, sponsored by
the U.S. Department of Education. Our goal is to provide
access to the best information available for teaching and
learning about science, mathematics, and the environment.
Gardner, Howard (1987). The Mind's New Science: A History
of the Cognitive Revolution. Basic Books, Inc.: NY.
Genesee, Fred (2000). Brain Research: Implications for
Second Language Learning. ERIC Digest [Online].
Accessed 1/25/02: http://www.ed.gov/databases/ERIC_Digests/ed447727.html
This ERIC Digest article provides some insights
into Brain Science that seem relevant to learning
mathematics. Quoting from the Website:
There has been a long-standing interest among
second and foreign language educators in research on
language and the brain. Language learning is a natural
phenomenon; it occurs even without intervention. By
understanding how the brain learns naturally, language
teachers may be better able to enhance their
effectiveness in the classroom.
Gersten, Russell (February, 2002). Math Education and
Achievement: Scientifically Based Research [Online].
Accessed 3/3/02: http://www.ed.gov/nclb/research/gersten.html.
Gersten's paper is a transcript of a talk given
at a US Department of Education confernce on the science
of teaching and learning. Quoting from the first part of
This is actually an easy topic to be brief on
because there isn't a lot of scientific research in
math [education]. There's some. There's some
promising directions, but it is a somewhat depressing
There are two things going on. One, in elementary
education there is no question that [with]
most teachers, even most parents,--the reading is the
big emphasis there compared to math. But it's not that
simple. For other reasons, the math community of math
educators at least for forty-plus years has looked at
their role as reform, as change, as
Therefore, there hasn't been this steady tradition.
There are a few exceptions of really systematically
using the methods that Valerie and others talked about
earlier to build a knowledge base, but rather to study
using the more qualitative methods: teachers
understandings, kids understandings.
So, this is something that can change. There have
always been little glimmerings of change. There's a
slight increase in the amount, but overall the math
education community has been quite resistant to that,
where let's say in the reading field there have always
been at least two schools of thought, one in the
But rather than just dealing with how little we
know and getting us all depressed, I am going to give
some highlights of some work we recently did actually
for the state of Texas who was beginning a big
initiative in the area of math, getting kids ready for
algebra. So, it was basically, these kind of low
achieving kids who got to middle school and just were
weak in all areas of math. We tried to put together
the scientific research, using the procedures we've
heard about in terms of meta-analysis and all, in the
area of math for low achieving kids. I did this with
my colleagues Scott Baker and Dae Sik Lei.
I'm going to quickly go through the criteria, and
they resonate with what we've been hearing about
during the first session. We looked for studies that
used random assignment. We did include the
quasi-experiments, the ones that are kind of close,
but they only were included if they had measures
showed that the groups were comparable at the
beginning. So, if they just used the school down the
road, they were thrown out. They had to have at least
one math performance measure, which sounds weird. But
there were articles published in journals that either
had teachers grades or students attitudes or certain
interviews that we had no idea were they valid or
We found four categories. Notice the small number
of studies we found on this. Now, we limited ourselves
to low achieving students. These were students whose
documentation was well below grade level, at least
below the 35th percentile on some standardized
But some of the things that worked, and again we
don't have a lot of replications, but they were pretty
decent studies, is that when kids and/or their
teachers get ongoing information, every two weeks,
every four weeks, of where they are in math in terms
of either the state standards or some framework, it
invariably enhances performance.
This sounds kind of a little boring, it's not as
romantic, there's so much of romantic work done in
math. But the idea of having a system to know where
kids are and what they really know, rather than saying
this kid is struggling, this kid is struggling with
fractions, manipulating fractions, more than one, with
dividing fractions, with a sense of place value once
you get into the hundreds. That information can be
critical for low achieving kids, can be a life or
The second group we found, there was only six
studies, is peer assisted learning. It's usually
tutoring. This is something that could revolutionize
practice. Invariably, when kids are partnered up, and
it seems to be better if they're heterogeneous pairs,
there's one stronger student and one weaker student
and they switch off, achievement in math is always
So, peers can be excellent tutors. I'm not talking
here about cooperative groups of four, five, six kids.
It's two. And if you see the difference in classrooms
when there are two, it's very easy for the teacher to
quickly monitor and get a sense of what's going one.
Because kids are either working on stuff together,
giving each other feedback, taking turns, or they're
not. When it's a group of four or five, you're never
quite sure what's this group discussing, these two
kids look zoned out, but maybe they're finished.
Goleman, Daniel (1995). Emotional Intelligence: Why It
Can Matter More than IQ. Bantam Books: NY.
Hoff, David J. (February 19, 2003). Adding It All Up.
Accessed 2/20/03: http://www.edweek.com/ew/ewstory.cfm?
(free registration required).
This Education Week article summarizes the arguments for
and against moving our math education system in a direction
of decreasing computation and increasing an emphasis on
conceptual understanding. The article suggest that there are
some signs that research beginning to support the latter
Jasper Overview: What is the Jasper
Series? [Online]. Accessed 12/4/00: http://peabody.vanderbilt.edu/ctrs/
Quoting from the Website:
The Adventures of Jasper
Woodbury consists of 12 videodisc-based adventures (plus
video based analogs, extensions and teaching tips) that
focus on mathematical problem finding and problem
solving. Each adventure is designed from the perspective
of the standards recommended by the National Council of
Teachers of Mathematics (NCTM). In particular, each
adventure provides multiple opportunities for problem
solving, reasoning, communication and making connections
to other areas such as science, social studies,
literature and history (NCTM, 1989; 1991).
Jasper adventures are designed
for students in grades 5 and up. Each videodisc contains
a short (approximately 17 minute) video adventure that
ends in a complex challenge. The adventures are designed
like good detective novels where all the data necessary
to solve the adventure (plus additional data that are not
relevant to the solution) are embedded in the story.
Jasper adventures also contain "embedded teaching"
episodes that provide models of particular approaches to
solving problems. These episodes can be revisited on a
"just-in-time" basis as students need them to solve the
The developers of the Jasper
series have observed, as have other researchers in
education and psychology, that classroom learning is very
different from "natural" learning environments. Natural
learning environments, like those in which parents help
their children develop language, are often characterized
as "contextualized." Participants, in this case the
parent and the child, share a context, or a common frame
of reference, in which the learning takes place.
Additionally, in natural learning environments, the tasks
the teacher asks the learner to perform are authentic.
They arise naturally in the context, and the participants
care about the outcomes. Finally, the knowledge that is
being learned is often viewed as a tool to accomplish the
tasks, and the learner sees it as valuable knowledge that
can be used in new situations.
Jeremy; Swafford, Jane; and Findell, Bradford (Editors)
(2001). Adding It Up: Helping Children Learn Mathematics
This entire 480 page book produced by the
National Research Council is available free on the Web.
Quoting from the Website:
Adding it All Up explores how students in
pre-K through 8th grade learn mathematics and
recommends how teaching, curricula, and teacher
education should change to improve mathematics
learning during these critical years.
The committee identifies five interdependent
components of mathematical proficiency and describes
how students develop this proficiency. With examples
and illustrations, the book presents a portrait of
- Research findings on what children know about
numbers by the time they arrive in pre-K and the
implications for mathematics instruction.
- Details on the processes by which students
acquire mathematical proficiency with whole
numbers, rational numbers, and integers, as well as
beginning algebra, geometry, measurement, and
probability and statistics.
The committee discusses what is known from research
about teaching for mathematics proficiency, focusing
on the interactions between teachers and students
around educational materials and how teachers develop
proficiency in teaching mathematics.
Kulik, James A. (November 2002). School Mathematics and
Science Programs Benefit From Instructional Technology.
InfoBrief: Science Resources Statistics. Accessed 3/4/03:
Quoting from the Website:
Instructional developers have been working for
four decades to improve mathematics and science education
with computer technology, and they have made significant
contributions to student achievement during this time
according to a review of controlled evaluations of
instructional technology in elementary and secondary
schools. The review found that most evaluation studies
reported significant positive effects of instructional
technology on mathematics and science learning, but not
all technological approaches appeared to be equally
The forthcoming review, Effects of Using Instructional
Technology in Elementary and Secondary Schools: What
Controlled Evaluation Studies Say, includes discussion of
findings about mathematics and science in 36 controlled
evaluations published since 1990 and from earlier reviews
of controlled evaluations and less formal studies. The
review did not cover theoretical works, case studies,
policy or cost analyses, or other studies that
investigated learning processes or social dimensions of
technology without measuring learning outcomes.
The 36 evaluation studies examined four types of
computer applications in mathematics and science: (a)
integrated learning systems in mathematics; (b) computer
tutorials in science; (c) computer simulations in
science; and (d) microcomputer-based laboratories. The
findings for each are discussed below.
Kurzweil, Ray (1999). The age of spiritual machines: When
computers exceed human intelligence. NY: Viking. Quoting
How much do we humans enjoy our current status
as the most intelligent beings on earth? Enough to try to
stop our own inventions from surpassing us in smarts? If
so, we'd better pull the plug right now, because if Ray
Kurzweil is right we've only got until about 2020 before
computers outpace the human brain in computational power.
Kurzweil, artificial intelligence expert and author of
The Age of Intelligent Machines, shows that technological
evolution moves at an exponential pace. Further, he
asserts, in a sort of swirling postulate, time speeds up
as order increases, and vice versa. He calls this the
"Law of Time and Chaos," and it means that although
entropy is slowing the stream of time down for the
universe overall, and thus vastly increasing the amount
of time between major events, in the eddy of
technological evolution the exact opposite is happening,
and events will soon be coming faster and more furiously.
This means that we'd better figure out how to deal with
conscious machines as soon as possible--they'll soon not
only be able to beat us at chess, but also likely demand
civil rights, and might at last realize the very human
dream of immortality.
for Better Schools (RBS) has published a handbook on
lesson study, the form of professional development
credited by some researchers as key to Japan's steady
improvement in mathematics and science instruction. In
lesson study, teachers plan, observe, and refine research
lessons together. The handbook addresses the basic steps
of lesson study and provides guidance on pioneering the
method in your school. Also included are instructional
plans for mathematics, science and language arts. Lesson
Study: A Handbook of Teacher-Led Instructional Change is
available only from RBS. The introductory price is $19.95
($24.99 after December 31, 2002). Order information can
be found at http://www.rbs.org/catalog/pubs/pd55.shtml.
RBS, an educational research and development firm,
operates the Mid-Atlantic Eisenhower Consortium for
Mathematics and Science Education.
Logan, Robert K. (1999). The sixth language: Learning a
living in the computer age. Toronto, Canada: Stoddart
Publishing Company. Quoting from the Stoddard Publishing
The Internet is transforming learning and
commerce and accelerating the evolution of the
Information Age into the Knowledge Era. Web pages and
sites, intranets, extranets, and hypertext come together
in cyberspace to form one huge Global Network - the
realization of Marshall McLuhan's Global Village. How we
respond to the challenges and opportunities that are
presented by this burgeoning technology will have a huge
impact on whether we will be successful in our future
learning and work endeavours. In this provocative book,
Robert Logan submits that the Internet is more than just
a technological toy; rather, it constitutes a new link in
the linguistic chain, joining speech, writing,
mathematics, science, and computing. Characterized by
unique semantics and syntax, the Net forms another
distinct step in the evolution of human communication -
the sixth language. Incorporating the communications
theories of McLuhan and Harold Innis, Logan traces the
evolution of verbal languages to explain how we got here,
and engages in profound speculation on where we're going.
Middle School Math (From The Knowledge Loom).
[Online]. Accessed 4/2/02: http://knowledgeloom.org/practices
Quotng from the Website:
As part of its mission to improve K-12
mathematics and science teaching and learning, the
Eisenhower National Clearinghouse (ENC) collects
information and resources on best practices in Middle
School Mathematics. Drawing from standards as stated in
Principles and Standards for School Mathematics (PSSM,
2000) from the National Council of Teachers of
Mathematics (NCTM), ENC has identified the following
areas as essential to enhanced teaching and learning:
- Integrating Technology into Middle School
- Inquiry and Problem Solving in Middle School
- Effective Professional Development for Middle
- Assessment That Informs Practice in Middle School
Morovec, Hans (2000). Robot: Mere machine to transcendent
mind. Oxford University Press. Quoting from an Amazon.com
This is science fiction without the fiction--and
more mind-bending than anything you ever saw on Star
Trek. Moravec, a professor of robotics at Carnegie Mellon
University, envisions a not-too-distant future in which
robots of superhuman intelligence have picked up the
evolutionary baton from their human creators and headed
out into space to colonize the universe.
This isn't anything that a million sci-fi paperbacks
haven't already envisioned. The difference lies in
Moravec's practical-minded mapping of the technological,
economic, and social steps that could lead to that
vision. Starting with the modest accomplishments of
contemporary robotics research, he projects a likely
course for the next 40 years of robot development,
predicting the rise of superintelligent, creative,
emotionally complex cyberbeings and the end of human
labor by the middle of the next century.
Moursund, D. IT-Assisted Project-Based Learning
[Online]. Accessed 4/5/02: http://darkwing.uoregon.edu/~moursund/PBL/.
This is an extensive Website designed to support a short course, workshop, or self study on the topic of IT-Assisted Project-Based Learning. It contains an extensive annotated bibliography, with most of the items available on the web.
Moursund, D. (Accessed 7/14/01). http://otec.uoregon.edu/.
This is the Website of the Oregon Technology in
Education Council. It contains a huge amount of
information about the field of computers in education,
especially at the precollege level and in teacher
National Assessment of Educational Progress (NAEP).The nation's report card: 2003 mathematics and reading assessment report.. Accessed 11/18/03: http://nces.ed.gov/nationsreportcard/. Quoting from the report:
Average mathematics scores of both fourth- and eighth-grade students were higher in 2003 than in all the previous assessment years since 1990
Project-Based Instruction in Mathematics for the Liberal Arts. Accessed 9/14/04 : http://faculty.uscupstate.edu/mulmer/PBI_Index.shtml. Quoting from the Website:
The purpose of this web site is to provide projects and resources for instructors and students who wish to teach and learn college mathematics or post-algebra high school mathematics via project-based instruction. Check the site often for additions and improvements.
Project-Based Instruction in Mathematics for the Liberal Arts (PBI-MLA) was developed at the University of South Carolina Spartanburg. In six years, it developed a 30% higher success rate than traditional textbook-driven sections of College Mathematics..
Rudner, Lawrence M., Shafer, Mary Morello (1992).
Resampling: A Marriage of Computers and Statistics. ERIC
Digest. Quoting from the Website:
Resampling is simply a process for estimating
probabilities by conducting vast numbers of numerical
experiments. Today, resampling is done with the aid of
high speed computers.
In Science News, Peterson (1991) compares resampling
techniques to the trial-and-error way gamblers once used
to figure odds in card or dice games. Before the
invention of probability theory, gamblers would deal out
many hands of a card game to count the number of times a
particular hand occurred. Thus, by experimentation,
gamblers could figure the odds of getting a certain hand
in their game.
Probability theory freed researchers from the drudgery
of repeated experiments. With a few assumptions,
researchers could address a wide range of topics. While
the advances in statistics paved the way for elegant
analysis, the costs came high:
Salomon, G., & Perkins, D. (1988, September).
Teaching for transfer. Educational Leadership, 22-32.
Salomon and Perkins have developed the
high-road/low-road theory of transfer of learning. The
article listed here provides a good overview of the
domain of transfer of learning and how to teach transfer.
It also contains an extensive bibliography, so it is a
good starting point if you want to study the research on
transfer of learning.
Stigler, James and Hiebert,
James (1999), The Teaching Gap: Best Ideas from the World's
Teachers for Improving Education in the Classroom. NY: The
This excellent book is based on an analysis of
the video taping of eighth grade math classrooms in
Germany, Japan, and the United States that was done in
conjunction with the TIMSS.
Weisstein, Eric. World of Mathematics [Online].
Accessed 4/4/02: http://mathworld.wolfram.com/.
Quoting from the Website:
Eric Weisstein's World of Mathematics
(MathWorldTM) is the web's most complete mathematical
resource, assembled over more than a decade by internet
encyclopedist Eric W. Weisstein with assistance from the
mathematics and internet communities.
MathWorld is a comprehensive and interactive
mathematics encyclopedia intended for students,
educators, math enthusiasts, and researchers. Like the
vibrant and constantly evolving discipline of
mathematics, this site is continuously updated to include
new material and incorporate new discoveries.
Although it is often difficult to find explanations
for technical subjects that are both clear and
accessible, this website bridges the gap by placing an
interlinked framework of mathematical exposition and
illustrative examples at the fingertips of every internet
If you find MathWorld useful, you may also be
interested in the author's Treasure Troves of Science
site, which contains topically similar material about
astronomy, scientific biography, science books, physics,
and other areas of science.