Math Education Digital Filing Cabinet

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Contents



Background for Digital Filing Cabinets

On December 10, 1948 the General Assembly of the United Nations adopted and proclaimed the Universal Declaration of Human Rights. The following is quoted from Article 26:

  1. Everyone has the right to education. Education shall be free, at least in the elementary and fundamental stages. Elementary education shall be compulsory. Technical and professional education shall be made generally available and higher education shall be equally accessible to all on the basis of merit.
  2. Education shall be directed to the full development of the human personality and to the strengthening of respect for human rights and fundamental freedoms. It shall promote understanding, tolerance and friendship among all nations, racial or religious groups, and shall further the activities of the United Nations for the maintenance of peace.

The world is making progress toward achieving free, universal elementary school education. This, of course, is merely a step toward providing free PreK-12 or PreK-16 or lifelong education for all people of all ages throughout the world. Information and Communication Technology (ICT) is making a steadily growing contribution toward eventual achievement of these visionary goals.

The Information Age Education Digital Filing Cabinet projects are are designed to aid in providing universal free education to people of all ages throughout the world.

The following IAE-pedia pages are designed to be part of every Digital Filing Cabinet. They are intended for teachers at all levels and in all disciplines.

  • Open Source Textbooks. This Web Page explores the idea of providing free, open source textbooks and instructional materials to students took keep, edit, add marginal notes and comments, and so on.

Introduction to Math Education DFC

Each academic discipline has its own discipline-specific educational goals and ways of achieving these goals. In our current school curriculum, it is useful to think of how to improve education in specific disciplines. This Web Page focuses specifically on the idea of a Math Education Digital Filing Cabinet.

It is important to remember, however, that most problems people encounter are interdisciplinary. Math is an important aid to representing and attempting to solve problems in every academic discipline. Thus, math needs to be taught in a manner that facilitates transfer of learning to other disciplines, and other disciplines need to be taught in a manner that helps students learn to make effective use of math in the disciplines.

The Math Education Digital Filing Cabinet project is based on three assumptions:

  1. That all people of the world are entitled to a free, good quality education. Good quality is to be determined by contemporary standards; however, it should prepare students to become and remain responsible citizens and lifelong learners who can adjust to life in a changing world.
  2. This education should be designed to empower learners by helping them gain levels of expertise in diverse areas that meet their own specific needs and interests, the needs and interests of their community, and the needs and interests of the world.
  3. Knowledge and skills in math and in using math to help represent and solve problems are an important outcome of a good education.

The Math Digital Filing Cabinet project is very large and is just in its infancy. This IAE-pedia page is being used to explore various aspects of the project. Eventually there will be separate Digital Filing Cabinet drawers for various groups of math teachers. For example, the needs of an elementary school teacher are quite different than the needs of a College of Education Math Methods teacher or a university Department of Mathematics faculty member who provides math content instruction to preservice elementary and secondary school teachers.

Math Education for Teachers of Math

Math is a broad, deep discipline with a long history. A person can spend a lifetime studying and doing research on math content and still know only a small fraction of the totality of collected math knowledge. Similar statements hold for a person exploring the history of math, the teaching of math, and the applications of math in various non-math disciplines.

A person who is teaching math or teaching teachers to teach math needs to be knowledgeable in three overlapping areas of mathematics:

  • Math content knowledge.
  • Math pedagogical knowledge.
  • Math pedagogical content knowledge (PCK).

The diagram given below is applicable in every academic discipline.

PCK.jpeg


Math Content Knowledge

There is a huge and steadily growing accumulation of math content knowledge. On a worldwide basis, many thousands of math researchers are contributing to this accumulation.

The challenge of this huge and steadily growing accumulation of math content knowledge can be examined from how it affects elementary teachers, secondary school math teachers, and higher education math teachers. The situation is roughly as follows:

  1. A typical elementary school teacher has studied math up through the 11th or 12th grade, has taken a Math For Elementary Teachers course or sequence of courses in college,and has taken a Math Methods course. This persons "peak" math content is the material covered in Math for Elementary Teachers, which may have College Algebra as a prerequisite.
  2. A typical secondary school math teacher has a math content preparation that lies in the range of two years of college math to a bachelor's degree in math. In a number of states, there is a strong emphasis on high school math teachers having a bachelor's degree in math.
  3. A typical teacher of math in a higher education institution has math content preparation that lies someplace in the range of a bachelors degree in math to a Doctorate in math.

For most teachers of math, there is a considerable difference between their highest level of math content course work and their current level of math content knowledge and skill. On the one hand, we know that people tend to forget the details of coursework that they are not using on a regular basis.Thus, for example, a typical fourth grade teacher will gradually forget most of the details of math content learned in high school and above.

On the other hand, the research-oriented math faculty in a college of university university will be routinely actively engaged in maintaining and expanding their math content knowledge. Thus, especially in their areas of research, their content knowledge will be well above the level achieved while in school.

Math Pedagogical Knowledge

A relatively strong rule of thumb is that teachers teach the way they were taught. The pedagogical knowledge gained by years and years of observing teachers (being taught by teachers) create a powerful mind set on how teaching is done.

Think about your experiences as a math student in elementary school, in secondary school, and in college. During these years of your schooling, you learned how elementary teachers typically teach math, how secondary school math teachers typically teach math, and how college math teachers typically teach math.

In secondary school, for example, the math class might begin with students handing in an assignment that they started working on during the previous math class. This is followed by a discussion of assignment problems, presentation of some new material, a new assignment, and seat work for the remainder of the period. The class may well include students doing some work at the chalk board, and the teacher will likely use an overhead projector, computer projector, chalkboard, or white board in the presentation. The amount of interaction between the teacher and students may vary considerably depending on the students and the teacher. Some teachers may have students interact in small groups to explore a problem of mathematical task.

In college math courses, demonstration and lecture tend to dominate. The teacher demonstrates and explains how to solve various problems that were in the homework assignment. The teacher lectures and demonstrates on the new material to be presented. Students take notes, and they ask questions about parts of the demonstration and lecture that they do not understand. From time to time the teacher asks a question and accepts ananswer from some volunteer in the class.

A student in a preservice teacher eduction program has repeatedly seen examples of math teaching and has gained quite a bit of math pedagogical knowledge. Research indicates that teachers tend to teach the way they were taught. It is hard for a teacher at any level to break the math pedagogical knowledge patterns they grew up with.

This creates an interesting and large challenge to the math education community as research suggests new and possible better ways to facilitate student learning.

Here is an example. Consider the idea of student-centered teaching, small group discussions, and team projects in teaching and learning math. How does a preservice teacher who has seldom or never participated in such teaching/learning environments learn to make effective use of these teaching techniques?

For another example, how does one make effective use of a computer hooked to a projection system and to the Internet while teaching a math unit of study? At the current time, relatively few students are seeing good examples of this in their elementary school secondary school, and college math courses. The idea of virtual manipulatives is related to this. Many elementary school math teachers are comfortable with students using physical manipulatives. What are advantages disadvantages of using computer-based manipulatives?

Math Pedagogical Content Knowledge

The idea of pedagogical content knowledge (PCK) has received a lot of attention and has been the focus of quite a bit of research and teacher education since it was first proposed by Lee Shulman in the mid 1980s. Quoting from the [http://tpck.pbwiki.com/Pedagogical%20Content%20Knowledge%20(PCK) Technology Pedagogical Content Knowledge Website:

This knowledge includes knowing what teaching approaches fit the content, and likewise, knowing how elements of the content can be arranged for better teaching. This knowledge is different from the knowledge of a disciplinary expert and also from the general pedagogical knowledge shared by teachers across disciplines. PCK is concerned with the representation and formulation of concepts, pedagogical techniques, knowledge of what makes concepts difficult or easy to learn, knowledge of students’ prior knowledge and theories of epistemology. It also involves knowledge of teaching strategies that incorporate appropriate conceptual representations, to address learner difficulties and misconceptions and foster meaningful understanding. It also includes knowledge of what the students bring to the learning situation, knowledge that might be either facilitative or dysfunctional for the particular learning task at hand. This knowledge of students includes their strategies, prior conceptions (both “naïve” and instructionally produced); misconceptions students are likely to have about a particular domain and potential misapplications of prior knowledge.

Liping Ma is well known for her 1999 book Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the U.S. Her book provides good examples of math PCK needed by elementary school teachers. She argues that even though elementary school math teachers in China have had quite a bit less formal math instruction than similar teachers in the United States, the Chine teachers have better PCK because their elementary and secondary school teachers had better PCK.

Information and Communication Technology

Information and Communication Technology (ICT) is affecting the content, pedagogy, and PCK of every academic discipline. Here is a very brief summary of the current and future situation:

  1. Computers can solve or significantly help to solve some of the problems in each academic discipline. The ICT content in any particular discipline varies with the discipline. However, ICT is now important enough in each discipline so that the content being taught to students needs to reflect capabilities and limitations of ICT in the discipline.
  2. ICT provides a variety of pedagogical aids. For example, computer-assisted learning and distance learning are of growing importance in each academic discipline.
  3. Teachers in each discipline are faced by the challenge of how to help students learn to use ICT as an aid to knowing and using the discipline being taught. ICE brings some general aids to teaching and learning, such as computer-assisted learning, distance learning, and use of multimedia in classroom instructional. Each specific discipline has its own ways of dong this and its own discipline-specific materials that are relevant to such tasks.

For math education, all of these ideas are illustrated in:

In Brief Summary

Our math education system can be substantially improved. Each teacher needs to think carefully about his or her current level of expertise in:

  • math content relevant to themselves and to their students
  • math pedagogy
  • math PCK
  • information and communication technology (ICT) in math content, math pedagogy, and math PCK

Thus, a Math Education Digital Filing Cabinet needs to provide preservice teachers, inservice teachers, and teachers of teachers with materials to help in this teacher education endeavor.

Physical and Digital Math Filing Cabinet

Nowadays, a math teacher needs both a physical filing cabinet (more generally, storage space for physical materials) and a DFC.

Physical Math Filing Cabinet

Here are a few examples of possible content math teacher's physical filing cabinets and storage areas.

  • Permission slips, hall passes, attendance forms, and so on.
  • Bulletin board materials.
  • Chalk and/or white board markers of various colors, calculators, pencils, paper, scissors, rulers, protractors, compasses, various kinds of tape and glue, and etc.
  • Math games and puzzles, along with some information about suitable uses to improve the quality of math education that students using the games and puzzles are apt to gain.
  • Archival copies of grade books from past years.
  • A library of books, magazines, journals, and articles to support personal needs and needs of one's students.
  • Overhead projector foils. There are lots of different possibilities. For example, many teachers find it useful to have various sizes of black line masters of graph paper. Of course, blank foils and appropriate marking pens are a needed part of this collection.
  • Physical manipulatives. Examples include blocks, spinners, geoboards, dice, and so on. Teachers making use of class sets of such manipulatives need a lot of drawer and/or shelf space.
  • Video tapes, audio tapes, CDs, and DVDs. In each case, a teacher may have both prerecorded and blank recordable media.
  • Storage boxes and containers.
  • Papers and tests received from and/or ready to be handed back to students.
  • Etc. etc. etc.

Some items might be stored in either a physical filing cabinet or a DFC, or both. Some possible examples include grade books, quizzes and tests, lesson plans, handouts for use by students.

A physical filing cabinet might consist of some materials stored at home, some stored in one's classroom, and some stored elsewhere, such as a storage room in one's school. ideally, all of the contents would be easily accessible when you want to access them. This is an obvious problem with the storage and retrieval of physical materials. As an example, usually one wants to avoid the cost of having duplicate copies of reference books that one might want to refer to at home and at school.

With physical materials, there is an ongoing problem of guarding against possible disasters. What happens if there is a fire or flood in the space where your materials are stored? What happens if materials are stolen or maliciously destroyed? If a teacher moves to a new teaching job, who gets to keep the physical materials?

Here is an important question. As the collection grows, how does one organize it so that needed materials are quickly retrieved? Here are two interesting aspects of an answer.

First, one only tends to collect and store materials that are specifically relevant to the job. One has a good working knowledge of how to use the materials on the job. Thus, one has a type of personal "ownership" of the materials.

Second, the materials are often stored in a manner so that one's kinesthetic sense, visual memory, and a quick glance tend to help in quick retrieval. It is an interesting exercise to compare and contrast this with retrieval of information stored in a computer.

Math DFC: Where to Put It?

Anything that can be stored in a computer can be part of your DFC. A major question is, where do you want to store your Math DFC contents. Two general choices are:

  • On your own personal computer or computers.
  • On a server. This might be a local server, such as in your school building or district. It might be a server located thousands of miles away. (Indeed, you may have no idea where your materials are being stored.)

Nowadays, you are apt to have part of your DFC on your own personal computer and part on servers. In either case, you need to be concerned with having the material regularly backed up in a "off site" location and having your materials protected from physical and electronic threats. You also need to decide what parts of your DFC you want to make available to other people.

Suppose, for example, that all of the contents of your DFC are stored on a laptop computer that you regularly carry between home and school. You might have a home desktop microcomputer where you keep a backup copy of your DFC. If your storage and backup are just on these two computers, have the risk of both computers being damaged at the same time by fire, storm, or flood. Thus, you still need to have some form of off site backup storage.

When you want to share a document with a particular person you can send it as an email attachment, copy it onto a CD or DVD and hand the person the physical medium, copy the material onto a thumb drive and watch as the person copies it into his or her computer, and so on. In all of these cases you have considerable control over who receives the material. Of course, you don't know who might get copies from this person.

The Information Age Education (IAE) Websites provide an example of storage on a server. The company running the server provides automatic backup. In addition, Information Age Education staff periodically make an off site backup.

IAE is currently running two different Websites. The http://i-a-e.org/ Website uses Joomla! and the http://I-A-E-pedia.org/ Website uses MediaWiki.

The Web Page you are currently reading is stored on a server, along with the MediaWiki software that is used in creating, editing, storing, and retrieving the material. All of the content can be accessed from any place in the world by anyone who can access the Internet. Most of the pages in the http://I-A-E-pedia.org/ Website are open to online editing by readers. That is, people from around the world can log onto the site, add and/or edit pages, add comments, and so on.

The content stored on the http://i-a-e.org/ Website (along with Joomla! software) is open to reading and download by people throughout the world. However, the content posted on this Website cannot be changed by the readers.

A third alternative available when storing on a server is to have the contents password protected. Then, only people who know the password can log on and access the content.

In summary, storage on a server run by a reputable and responsible organization provides for off site backup and for ease of access by people throughout the world. Access to the content may be restricted through use of some sort of password protection system.

Math DFC: Making It Yours

It does little good to collect lots of stuff into a DFC and have no idea what is there or how and why you might want to use it some day. You need to have personal ownership and understanding of the content of your DFC.

Today's search engines make it relatively easy to retrieve thousands (indeed, millions) of articles on various topics that are relevant to teaching and learning math. Each time you find an article of personal use to you, think about adding it to your DFC. One way to do this is via an annotated bibliography. Put an entry into the "references" section of DFC that contains a proper citation to the article, including a link to its website if the article is on the web. Then write a brief paragraph (in essence, a note to yourself) explaining what this article means to you and why or how you might want to use it in the future.

Teachers are used to the idea of developing lesson plans, using the lesson plans, and writing comments to themselves about what they will do the same and what they will do different the next time they use the lesson plans. This is an excellent example of steadily increasing the value of material in one's filing cabinets. At the end of a teaching day, spend just a few minutes writing notes in each of the lesson plans you have used that day. Think of these as notes to your future self—things that you want your future self to know about the next time the lesson plan is used.

Another aspect of personalization is organizing the material in a form that helps you to quickly find and retrieve a particular item you are interested in. If your personal DFC is on your own computer, you may want to use a structure of file folders. An elementary school teacher, for example, might want to have a separate file folder for each of the subject areas she or he teaches.

A DFC can be searched electronically, and it can have an indexing system that is specifically designed to fit the needs of its owner. This is a key idea. Think in terms of deciding upon a list of Categories (using the term the way a Wiki does), being able to easily add or delete categories, and being able to index the various entries in one's DFC. The combination of indexing using Categories and use of a search engine is a powerful aid to finding what is in your DFC.

A Sample Collection of Relevant Materials

Math is considered to be one of the basics of education. We want all students to gain contemporary levels of knowledge and skills in reading, writing, and math.

One of the specific goals of the IAE-pedia is to aid in the creation, collection, and dissemination of a Math Education Digital Filing Cabinet of materials designed to serve the needs of teachers and their students. The underlying model for this Digital Filing Cabinet is a collection of free, open source materials that can made available to teachers and students throughout the world.

The IAE-pedia contains a number of articles designed to help improve informal and formal math education. Some of these articles are listed below. The list given below also contains some links to articles located elsewhere.

The notion of "concrete," from concrete manipulatives to pedagogical sequences such as "concrete to abstract," is embedded in educational theories, research, and practice, especially in mathematics education. In this article, I consider research on the use of manipulatives and offer a critique of common perspectives on the notions of concrete manipulatives and concrete ideas. I offer a reformulation of the definition of "concrete" as used in psychology and education and provide illustrations of how, accepting that reformulation, computer manipulatives may be pedagogically efficacious.
  • Computational Thinking. Cuts across all disciplines. Includes an emphasis on math modeling that makes use of human brain and computers.
  • David Moursund Editorials. A collection of all of the editorials that David Moursund wrote for the Oregon Computing Teacher, The Computing Teacher, and Learning and Leading with Technology.
  • Empowering Learners and Teachers. This document includes a specific discussion of empowering students through teaching of reading and math. It includes the calculator and the digital watch in its examples.
  • Folk Math. A seminal article by Eugene Maier that draws a parallel between Folk Music (music that the ordinary people learn and do or use), and Folk Math. Contrasts Folk Math with School Math. See also: Eugene Maier. Gene is a world class math educator. This collection of short articles is well suited for use in preservice and inservice math education, and by others interested in the quality of math education that children are currently receiving.
  • Free Math Software. There is a huge and growing amount of free math software, math education software, math-oriented games, and so on.
  • History and Pedagogy of Mathematics. Each academic discipline has its own history and pedagogy. Both are important areas from the point of view of being a good teacher in the discipline.
  • Johnson, Jerry (n.d.). Math NEXUS. Jerry Johnson's Math NEXUS Website is an excellent example of a math education digital filing cabinet. It is designed to meet the needs of students and faculty interested in math education, and it serves as an outlet for his own personal creativity.
  • Lockhart, Paul (2002). A Mathematician's Lament. Retrieved 4/24/08: http://www.maa.org/devlin/LockhartsLament.pdf. Argues that math should be considered an art, compares with music and painting, and "blasts" our current math education system.
  • Math Education. Discusses different answers to the questions, "What is mathematics." Emphasizes the need for students to gain increasing insight into possible answers as they progress in their math studies.
  • Math Education Digital Filing Cabinet. Introduces the idea of providing preservice and inservice teachers with a free electronic filing cabinet (that is, virtual filing cabinet) of materials useful to them, their students, and their teachers.
  • Math Education Free Videos. This page contains brief summaries and pointers to videos useful in precollege math education, teaching teachers, and teaching parents.
  • Math Education Wars. Discusses the conflict between the math education traditionalists and the math education reformers.
The Math Forum Is...the leading online resource for improving math learning, teaching, and communication since 1992.
We are teachers, mathematicians, researchers, students, and parents using the power of the Web to learn math and improve math education.
We offer a wealth of problems and puzzles; online mentoring; research; team problem solving; collaborations; and professional development. Students have fun and learn a lot. Educators share ideas and acquire new skills.
  • Math Maturity. An introduction to a general measure of student progress toward learning mathematics for long term use and understanding.
Computational Thinking and Math Maturity: Improving Math Education in K-8 Schools.
Introduction to Roles of Computers in Problem Solving.
Computers in Education for Talented and Gifted Students.
Introduction to Using Games in Education: A Guide for Teachers and Parents.
The Mind and the Computer: Problem Solving in the Information Age.
College Student's Guide to Computers in Education.
  • Moursund Editorial: High Tech—High Touch. Explores the need for education to provide an appropriate balance between high technology and strongly people-oriented low or no technology. From the November 1985 issue of The Computing Teacher.
International and domestic comparisons show that American students have not been succeeding in the mathematical part of their education at anything like a level expected of an international leader. Particularly disturbing is the consistency of findings that American students achieve in mathematics at a mediocre level by comparison to peers worldwide. On our own “National Report Card”—the National Assessment of Educational Progress (NAEP)—there are positive trends of scores at Grades 4 and 8, which have just reached historic highs. This is a sign of significant progress. Yet other results from NAEP are less positive: 32% of our students are at or above the “proficient” level in Grade 8, but only 23% are proficient at Grade 12. Consistent with these findings is the vast and growing demand for remedial mathematics education among arriving students in four-year colleges and community colleges across the nation.
Moreover, there are large, persistent disparities in mathematics achievement related to race and income—disparities that are not only devastating for individuals and families but also project poorly for the nation’s future, given the youthfulness and high growth rates of the largest minority populations.
  • Oregon_Mathematics-OCTM. Email messages facilitating an increased level of communication and discussion among members of the Oregon Council of Teachers of Mathematics.
  • Problem Solving. Problem solving lies at the core of each academic discipline. Many of the general ideas and strategies used in problem solving in one discipline can transfer to other disciplines. This is especially true of math problem solving, since math is an important component of many other disciplines.
  • Science & Technology Museum Math Exhibit. Explores possible answers to the question, "What is math?" Analyzed some components of a science and technology exhibit on math. Explores the idea of making an elementary school classroom and overall curriculum more mathematical.
  • Two Brains Are Better Than One. Explores educational implications of human brain and computer brain working together to solve problems in math and other areas.
  • What is Computer Science? Computer science is a discipline closely related to mathematics. In many cases, today's Computer and Information Science Departments were "spun off" from Math departments. In many other cases, Computer Science and Math are still together in one college or university department.
MathWorld is the web's most extensive mathematical resource, provided as a free service to the world's mathematics and internet communities as part of a commitment to education and educational outreach by Wolfram Research, makers of Mathematica.
MathWorld has been assembled over more than a decade by Eric W. Weisstein with assistance from thousands of contributors. Since its contents first appeared online in 1995, MathWorld has emerged as a nexus of mathematical information in both the mathematics and educational communities. It not only reaches millions of readers from all continents of the globe, but also serves as a clearinghouse for new mathematical discoveries that are routinely contributed by researchers. Its entries are extensively referenced in journals and books spanning all educational levels, including those read by researchers, elementary school students and teachers, engineers, and hobbyists.
  • Women and ICT. ICT and mathematics overlap. Many of the women who were pioneers in the computer field were mathematicians.

References

Curriki (n.d.). Free math course materials. Retrieved 6/18/08: http://www.curriki.org/xwiki/bin/view/Main/GWTSearch#XWiki.AssetClass.category%3D%26__sortBy%3D%26XWiki.AssetClass.educational_level2%3D%26__advShow%3D0%26__atItem%3D1%26__terms%3Dlearn%20without%20limit%26__go%3D1%26__space%3D%26__special%3D%26XWiki.AssetClass.fw_items%3DFW_masterFramework.WebHome%26XWiki.AssetClass.instructional_component2%3D.

Most of these materials are for calculus courses.

Math Forum (n.d.). The Math Forum @ Drexel: People learning math together. Retrieved 9/1/07: http://mathforum.org/. Quoting from their Website:

The Math Forum is a leading center for mathematics and mathematics education on the Internet. Operating under Drexel's School of Education, our mission is to provide resources, materials, activities, person-to-person interactions, and educational products and services that enrich and support teaching and learning in an increasingly technological world.
Our online community includes teachers, students, researchers, parents, educators, and citizens at all levels who have an interest in math and math education.
Making math-related web resources more accessible
Want to use or develop educational technology? Visit Math Tools, the Forum's community digital library supporting the use and development of software for mathematics education. When a generic Web directory falls short of your mathematics needs, visit the Forum Internet Mathematics Library, which covers math and math education Web sites in depth. In our collaboration with the Mathematical Association of America, Mathematical Sciences Digital Library (MathDL), we collect mathematics instructional material with authors' statements and reader reviews; and catalogs mathematics commercial products, complete with editorial reviews, reader ratings and discussion groups. The Problems Library offers a convenient interface for searching and browsing the collective archives of the six Problem of the Week services.
Providing high-quality math and math education content
There's a lot of material on the Web, but how good is it, and how does it take advantage of new technologies or implement new pedagogy? We have worked with teachers, students, and researchers to put the best of their materials on the Web. This collaborative work is available via the Forum's Teacher Exchange: Forum Web Units. Teachers are invited to use the Web interface to contribute their own lessons.

MathNEXUS (n.d.). Mathematics Portal: News and ideas for teachers and learners of mathematics. Retrieved 4/15/08: http://mathnexus.wwu.edu/.

This is an excellent and growing set of math materials for preservice and inservice math teachers at all grade levels. Some examples of the categories of material being made available include:
  • Problem of the Week.
  • Quote of the Week.
  • Statistic of the Week
  • Humor of the Week
  • Website of the Week
  • Resource of the Week

Also from the Math Forum, see their Library. This is a very large collection of materials and links to materials.

Math Resources from the Southern Oregon Education Service District. Retrieved 2/5/08: http://www.soesd.k12.or.us/Page.asp?NavID=741. A nice collection of computer-based resources of use to teachers and to teachers of teachers.

Ma, Liping (1999). Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the U.S. Mahwah, NJ: Erlbaum.

Mishra, P. and Koehler, M. TPCK (n.d.). Technological Pedagogical Content Knowledge (TPCK). retrieved 9/1/07: http://www.tpck.org/tpck/index.php?title=TPCK_-_Technological_Pedagogical_Content_Knowledge.

PlanetMath.org. Retrieved 12/27/08: http://planetmath.org/?op=browse&from=books. 159 free college level math books. Quoting from the site:

All pages on this site are copyrighted by their respective authors.
Permission is granted to copy, distribute and/or modify these documents under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, with no Front-Cover Texts, and with no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".

Shulman, Lee (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14.

Technological Pedagogical Content Knowledge (TPCK) attempts to capture some of the essential qualities of knowledge required by teachers for technology integration in their teaching, while addressing the complex, multifaceted and situated nature of teacher knowledge. At the heart of the TPCK framework, is the complex interplay of three primary forms of knowledge: Content (CK), Pedagogy (PK), and Technology (TK). … the TPCK framework builds on Shulman's idea of Pedagogical Content Knowledge.
Additional information is available in an article by Mishra and Koehler. Here is the abstract of that 2006 article:
Research in the area of educational technology has often been critiqued for a lack of theoretical grounding. In this article we propose a conceptual framework for educational technology by building on Shulman’s formulation of ‘‘pedagogical content knowledge’’ and extend it to the phenomenon of teachers integrating technology into their pedagogy. This framework is the result of 5 years of work on a program of research focused on teacher professional development and faculty development in higher education. It attempts to capture some of the essential qualities of teacher knowledge required for technology integration in teaching, while addressing the complex, multifaceted, and situated nature of this knowledge. We argue, briefly, that thoughtful pedagogical uses of technology require the development of a complex, situated form of knowledge that we call Technological Pedagogical Content Knowledge (TPCK). In doing so, we posit the complex roles of, and interplay among, three main components of learning environments: content, pedagogy, and technology. We argue that this model has much to offer to discussions of technology integration at multiple levels: theoretical, pedagogical, and methodological. In this article, we describe the theory behind our framework, provide examples of our teaching approach based upon the framework, and illustrate the methodological contributions that have resulted from this work.

NLVM (n.d.). National library of virtual manipulatives. Retrieved 8/31/07: http://nlvm.usu.edu/en/nav/index.html. Quoting from the Website:

Project Information
The National Library of Virtual Manipulatives (NLVM) is an NSF supported project that began in 1999 to develop a library of uniquely interactive, web-based virtual manipulatives or concept tutorials, mostly in the form of Java applets, for mathematics instruction (K-12 emphasis). The project includes dissemination and extensive internal and external evaluation.
Learning and understanding mathematics, at every level, requires student engagement. Mathematics is not, as has been said, a spectator sport. Too much of current instruction fails to actively involve students. One way to address the problem is through the use of manipulatives, physical objects that help students visualize relationships and applications. We can now use computers to create virtual learning environments to address the same goals.
There is a need for good computer-based mathematical manipulatives and interactive learning tools at elementary and middle school levels. Our Utah State University team is building Java-based mathematical tools and editors that allow us to create exciting new approaches to interactive mathematical instruction. The use of Java as a programming language provides platform independence and web-based accessibility.
The NLVM is a resource from which teachers may freely draw to enrich their mathematics classrooms. The materials are also of importance for the mathematical training of both in-service and pre-service teachers.

Purplemath (n.d.). Retrieved 8/31/07: http://www.purplemath.com/http://www.purplemath.com/. Quoting from the Website:

Need help with algebra? You've found the right place!
Lessons: "How do you really do this stuff?" -- Purplemath's algebra lessons are written with the student in mind. These lessons emphasize the practicalities rather than the technicalities, demonstrating dependable techniques, warning of likely "trick" questions, and pointing out common mistakes. The lessons are cross-referenced to help you find related material.
Site Reviews: -- Tired of looking through page after page of search-engine hits trying to find a site that might have something useful? These categorized Internet links have all been reviewed by Purplemath.
  • Free Online Tutoring and Lessons
  • Quizzes and Worksheets
  • Other Useful Sites and Services
Only those sites with something immediately useful (and free) for algebra students are listed. You won't find math jokes, biographies, or recreational math sites here. Instead, check these review for sites containing lessons, tutoring forums, worksheets, articles on "how math is used in real life", and more.
Homework Guidelines: "How to suck up to your teacher." -- English teachers tell students explicitly how to format their papers. Math teachers, on the other hand, frequently just complain about how messy their students' work is. Neat homework can aid your comprehension and maybe make your teacher like you better. These Homework Guidelines for Mathematics will give you a leg up, explaining in clear terms what your math teacher is looking for.
Study Skills Self-Survey: "Do I have what it takes?" -- Much of your success or failure in algebra can be laid at the feet of your study habits. Do you have good math study habits? Take this survey and find out.

Roher, Doug and Pashler, Harold (2007). Increasing retention without increasing study time. Retrieved 9/1/07: http://repositories.cdlib.org/cgi/viewcontent.cgi?article=5990&context=postprints.

This five page research article includes a discussion of math learning. It begins by noting that most of what one learns in a course is not retained very long. It argues that a change in study habits can make a very large difference in long term retention.

Roughly speaking, the authors argue that the design of the seat work and homework in the typical math book is poor if one's goal is long term retention. In the two paragraphs that follow, Spacers divide their study time into two sessions with a space in between. Massers mass their study time into one concentrated session. Quoting from the article:

Because the experiments described thus far required subjects to learn concrete facts, it is natural to wonder whether the results of these studies will generalize to tasks requiring more abstract kinds of learning. To begin to explore this question, we have been assessing the effects of overlearning and spacing in mathematics learning. For example, in one experiment (Rohrer & Taylor, 2006), students were taught a permutation task and then assigned either three or nine practice problems. The additional six problems, which ensured heavy overlearning, had no detectable effect on test scores after one or four weeks. In another experiment with the same task (Rohrer & Taylor, in press), a group of Spacers divided four practice problems across two sessions separated by one week, whereas a group of Massers worked the same four problems in one session. When tested one week later, the Spacers outscored the Massers (74% vs. 49%). Furthermore, the Massers did not reliably outscore a group of so-called Light Massers who worked only half as many problems as the Massers (49% vs. 46%).
This apparent ineffectiveness of overlearning and massing is troubling because these two strategies are fostered by most mathematics textbooks. In these texts, each set of practice problems consists almost entirely of problems relating solely to the immediately preceding material. The concentration of all similar problems into the same practice set constitutes massing, and the sheer number of similar problems within each practice set guarantees overlearning. Alternatively, mathematics textbooks could easily adopt a format that engenders spacing. With this shuffled format, practice problems relating to a given lesson would be distributed throughout the remainder of the textbook. For example, a lesson on parabolas would be followed by a practice set with the usual number of problems, but only a few of these problems would relate to parabolas. Other parabola problems would be distributed throughout the remaining practice sets.

SAGE (n.d.). SAGE: Open Source Mathematics Software. Retrieved 6/22/08: http://www.sagemath.org/.

SAGE is a free open source alternative to Magma, Maple, Mathematica, and Mathlab. It is available for Windows, Mac OSX, and Linux. Quoting from the Website:
Use SAGE for studying a huge range of mathematics, including algebra, calculus, elementary to very advanced number theory, cryptography, numerical computation, commutative algebra, group theory, combinatorics, graph theory, and exact linear algebra.
SAGE makes it easy for you to use most mathematics software together. SAGE includes interfaces to Magma, Maple, Mathematica, MATLAB, and MuPAD, and the free programs Axiom, GAP, GP/PARI, Macaulay2, Maxima, Octave, and Singular.

Shodor (n.d.). The Shodor Foundation. Retrieved 8/31/07: http://www.shodor.org/. Quoting from the Website:

The Shodor Foundation is a non-profit research and education organization dedicated to the advancement of science and math education, specifically through the use of modeling and simulation technologies.
Welcome to the Shodor's Curriculum Materials portal. There are several ways for you to browse our resources:
Search by grade level
Find the materials that are specifically geared towards a particular educational level.
Search by subject matter
Locate all of the available Shodor resources for a subject or field of study.
Browse all projects
View a list of all of Shodor's Curriculum Materials projects. Use this tool to find a specific project that you have used before.
Interactive Teaching Environments
The Shodor Foundation staff and associates are developing interactive tools and simulations that enable and encourage exploration and discovery through observation, conjecture, and modeling activities. These Modeling and Simulation Technology for Education Reform (MASTER) tools are part of on-going collaborations with the National Center for Supercomputing Applications (NCSA) and other education organizations. Simulations and supporting materials developed by Foundation staff form the basis of international science collaborations presently demonstrating network technologies involving middle and high schools of the Department of Defense Education Activity (DoDEA).
A growing portfolio of MASTER tools are being fully integrated with new collaboration tools and on-line research facilities to create authentic scientific experiences. All tools, simulations, and supporting curriculum materials are designed in accordance with the National Science Education Standards and the National Math Education Standards.

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This component of the IAE-pedia documents is a work in progress. If there are few entries in the next four subsections, that is because the links have not yet been added.

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