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What is Mathematics?

Major Unifying Themes in This Document


Foundational Information

Learning Theories

Mind and Body Tools

Science of Teaching & Learning

Project-Based Learning

Computational Mathematics

The Future




 Website Author
"Dr. Dave" Moursund


This section contains General References as well as References on Specific Topics. When searching for information, first scan the Specific Topics list.

General References

A bibliography of important general resources for leaders in the field of Brain Science and ICT in mathematics education.

References on Specific Topics

Arguments Against ICT in Education

Computer Algebra Systems

Developmental Theory

Handheld ICT Devices, Including Calculators

History of Calculators & Computers

Probabilistic Thinking

Project & Problem-Based Learning

Research Projects and Centers

Virtual Manipulatives

Top of Page

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):

The following issue of Brain Connection has several articles about math. See

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 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

Deborah Ball

Number Blindness: A Hidden
Challenge for Mathematics
May 2000
by Ashish Ranpura

Accessed 12/19/03:

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:

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 <>

Subject: [nsf02084] - Newsletters/Journals


The following document (nsf02084) is now available from

the NSF Online Document System


Title: Foundations - Volume 3 - Professional Development that

supports School Mathematics Reform

Type: Newsletters/Journals

Subtype: Education

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

Accessed 7/25/03:

Many of the young people that youth agencies serve are about to lose a lot of what they’ve 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 Children’s Aid Society in New York City. Contact:

 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:

Basic Skills Forcing Cuts in Art Classes

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: 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:

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.”

General References

Authentic Assessment in Mathematics. The Geometry Forum Summer '94 Workshop at Swarthmore College, Swarthmore, PA. Accessed 10/27/03: 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: 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 Press.

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:

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:

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 his talk:
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 topic.

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 re-conceptualizing.

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 experimental group.

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 reliable.

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 measure.

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 death issue.

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 improved.

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:
slug=23research.h22 (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 approach.

Jasper Overview: What is the Jasper Series? [Online]. Accessed 12/4/00:
ltc/Research/jasper_overview.html. 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 Jasper challenges.

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.

Kilpatrick, Jeremy; Swafford, Jane; and Findell, Bradford (Editors) (2001). Adding It Up: Helping Children Learn Mathematics [Online]:

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 mathematics learning:

  • 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 effective.

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 Reviews:

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.

Lesson Study

Research 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 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 Website:

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:
&spotlightid=1102&testflag=yes. 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 Mathematics
  • Inquiry and Problem Solving in Middle School Mathematics
  • Effective Professional Development for Middle School Mathematics
  • Assessment That Informs Practice in Middle School Mathematics

Morovec, Hans (2000). Robot: Mere machine to transcendent mind. Oxford University Press. Quoting from an book review:

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:

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).

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 education.

National Assessment of Educational Progress (NAEP).The nation's report card: 2003 mathematics and reading assessment report.. Accessed 11/18/03: 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 : 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 free Press.

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: 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 user.

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.

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