Digital Filing Cabinet: Math Education

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Background for Digital Filing Cabinets

Every person on earth is both a lifelong learner and a lifelong teacher. Our senses are continually taking in information that our brains process and learn from. Your communications with others provide learning opportunities for them.

Some of us are called "teachers" because we focus a considerable amount of our cognitive and physical abilities to becoming better and better at helping others to learn. Teachers spend a great deal of their time, knowledge, and skills in helping others to learn. They strive to become better teachers.

There is a substantial [http://iae-pedia.org/Scholarship/Science_of_Teaching_and_Learning "science & scholarship" of teaching and learning. This discipline is often called SoTL, with the abbreviation standing for both the Science of Teaching and Learning and the Scholarship of teaching and learning.

Good teaching is both an art and a science. Learning, understanding, and making effective use of SoTL can help a teacher become a better teacher.

The Information Age Education Digital Filing Cabinet project is designed to help teachers build personal libraries of SoTL materials that they and others consider to be important to effective teaching in specific disciplines and at specific grade or cognitive levels.

The document you are currently reading is part of my personal Math Education SoTL library. I recommend that you browse it as an aid to beginning the development of your own personal Math SoTL library and/or as an aid to adding to your library.

Remember, your goal is to build a "Personal SoTL Library." You gain ownership (your library becomes personal to you) through a process of personally selecting materials that you deem important to you, and that you have carefully read, understood, and are working to make use of. You may take pleasure in sharing your materials with others.

Some General Education Ideas

A good teacher develops and makes use of a personal philosophy of general of teaching and learning. Personally, I like to draw on ideas from people and groups who that have thought very carefully about purposes and goals of education. Here is an example.

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.

I like to collect quotations that are consistent with and supportive of my insights into education. Click here to see my general collection of quotations.

For example, you have likely heard the statement "Think globally, act locally." This statement is often applied to environmental issues. But, think about what it means in terms of education. How might we do a better job of educating each individual student be a contributing member of their local community and a citizen of the world?

The world is making progress toward achieving free, universal elementary school education. This, of course, is merely a step toward providing free or very inexpensive 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. See, for example, progress in developing Supersized Online Courses (MOOCs).

The Information Age Education Digital Filing Cabinet projects contribute to providing universal free education to people of all ages throughout the world. They are also designed as a role model for preservice teachers, inservice teachers, and teachers of teachers.

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 document 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 that discipline.

My personal 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 country, 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.

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.

Just for the fun of it, in mid 2014 I spent some time browsing the web for data on the worldwide number of research mathematicians. The site http://mathoverflow.net/questions/5485/how-many-mathematicians-are-there provided some data. From that site, one might estimate the number to be 10,000 or so research mathematicians. The site http://en.wikipedia.org/wiki/Mathematical_Reviews indicated that, as of November, 2007, the Mathematical Reviews database contained information on more than 2.2 million articles.

The challenge we face is to examining this growing accumulation of math content knowledge in terms of how it affects elementary school, secondary school, and higher education math teachers—as well as the teachers of these 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 one of more Math Methods course. This person's "peak" math content is the material covered in Math for Elementary Teachers course or courses, which may or may not have College Algebra as a prerequisite. (Some colleges require this prerequisite, and others do not.)
  2. A typical secondary school math teacher has a math content preparation that lies in the range from 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 from a master's degree in math to a doctorate or post-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 members in a college or university will routinely be 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 they achieved while in school.

Math Pedagogical Knowledge

A relatively strong rule of thumb is that teachers teach the way they were taught. While teacher education programs and staff development for inservice teachers attempts to change this situation, teachers teaching the way they were taught is very strongly ingrained. The pedagogical knowledge gained by years and years of observing teachers (while being taught by teachers) can create a powerful mind set on how teaching is done.

Think about your own 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, a math class might begin with students handing in an assignment that they started working on during the previous math class. This is usually 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 or white board, and the teacher will likely use an overhead projector, computer projector, chalk board, 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 or mathematical task.

In college math courses, demonstration and lecture tend to dominate. The teacher begins a class period by responding to questions and explaining how to solve various problems that were in the homework assignment. The teacher lectures about and demonstrates the new material to be covered that day. 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 an answer from some volunteer in the class.

Similar "deep learning" of how to teach math occurs for preservice teachers in their childhood at home, in elementary school, and in secondary school. At all of these levels, their teachers tend to not be well versed in current research on the teaching and learning of math. It is hard for a teacher at any level to break the math education content, pedagogy, and assessment patterns they grew up with.

This creates an interesting and large challenge to the math education community as research suggests new and possibly 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?

Jo Boaler is an excellent example of a math education researcher who strives to change preservice and inservice teachers based on the research she and others are doing. See her 20-minute year 2014 video at https://www.youtube.com/watch?v=pOOW0hQgVPQ&list=TLfYG-pMQ9CxeXW_yWKOYCQFyyKOrpu75x.

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 type of computer use 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 the advantages or disadvantages of using computer-based manipulatives? This is still a challenging research question.

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 Chinese teachers have better PCK because their elementary and secondary school teachers had better PCK. Learn more about Liping Ma's recent work at http://www.nytimes.com/2013/12/18/opinion/q-a-with-liping-ma.html?pagewanted=all&_r=0.

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. A number of academic disciplines have now established undergraduate and/or graduate specializations in uses of computers to help represent and solve the problems in their disciplines. For example, we have Computational Biology, Computational Chemistry, Computational Mathematics, and Computational Physics. (Indeed, as a side note, I had a doctoral student in 1977 and another doctoral student in 1983 who did their dissertations on computers in art education.)
  2. 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. ICT brings some general aids to teaching and learning, such as computer-assisted learning, distance learning, and use of multimedia in classroom instruction. Each specific discipline has its own ways of doing this and its own discipline-specific materials that are relevant to such tasks.
  3. Massive Open Online Courses (MOOCs) are now widely used. See http://i-a-e.org/iae-blog/entry/harvard-is-investing-heavily-in-moocs.html. MOOCs can be used as stand-along courses or in a hybrid model of teaching in which the MOOC materials are combined with face-to-face classes.

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 them to become better teachers.

Physical and Digital Math Filing Cabinet (DFC)

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 the possible content of a 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 so on.
  • Math games and puzzles, playing cards, and information about suitable uses of the games and puzzles to improve the quality of math education that students using them are apt to gain.
  • Blank paper, ruled paper, and graph paper.
  • 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 rulers, compasses, protractors, 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.
  • Sets of laptop and/or tablet computers.
  • Etc. etc. etc.

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

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 use 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 to refer to at home and at school.

With physical materials, there also 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 at a different grade level or in a new school, who owns/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 Digital Filing Cabinet: 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 one 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 your students, other teachers, and so on.

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 computer where you keep a backup copy of your DFC. If your storage and backup are just on these two computers, you 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 give the person the physical medium, copy the material onto a thumb drive and watch as the person copies it onto his or her computer, and so on. In all of these cases you have considerable control over deciding who is to receive your material. Of course, you don't know who might later get copies from this person.

When storing your DFC material on a server, you may want to have it password protected, or you may want it to open to the general public. For example, the various Information Age Education documents are part of my Digital Filing Cabinet that I make available to anyone who chooses to access it. But on my personal computer and Google Drive I have other DFC materials that I do not share.

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 a password protection system.

Math DFC: Making It Yours

It does little good to collect lots of "stuff" into a personal 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 locate thousands (indeed millions) of articles on various topics that are relevant to teaching and learning math. Each time your search locates an article of personal use to you, think about adding it to your DFC. The author and title of the materials may be enough to jog your memory of the content.

However, if you suspect that will not suffice, than when you put an entry into the "references" section of your DFC, 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 differently 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 when the lesson plan is used again.

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 according to the categories you have created. The combination of indexing the materials using categories and also using a search engine is a powerful aid to finding what is in your DFC.

Moursund's Sample Collection of Relevant Materials

My personal Digital Filing Cabinet has two major components. The first is the two websites containing all the Information Age Education publications. One of my goals in starting IAE was to provide a vehicle for the collection, storage, and dissemination of such materials.

The second component is the collection of documents on my personal computers. As an example of this, when I am working on a writing project I will frequently make copies of various Web-based resources that seem like them may be useful. Typically I browse a document to make a quick decision on its possible value. If it looks like it might be useful, I copy excerpts or all of the document into a folder on my computer. In some sense, that file folder becomes a Digital Filing Cabinet just for the project I am working on.

My Math DFC given below contains a number of articles designed to help improve informal and formal math education. The list also contains some links to articles located elsewhere that I have found particularly valuable.

Notice that in most entries I include a short annotation to a document. This is an expansion of the author, title, and source information, helps me to remember the general content covered in the document. Sometimes the title is so clear that I don't need an annotation to jog my memory.

Documents Authored by Moursund

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.
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.
Explores some of the problems of helping teachers and their students learn to read, write, speak, and listen with understanding in the language of mathematics. Discusses the analogy of native born teachers o a language and "native" speakers of mathematics who are teaching math.
Cuts across all disciplines. Includes an emphasis on math modeling that makes use of human brain and computers.
Provides links to Web sites that contain a huge and growing amount of free math software, math education software, math-oriented games, and so on.
This IAE-pedia document discusses the features that help to make a math lesson plan especially good. I have written a free book on the same topic that is available online. See http://i-a-e.org/downloads/doc_download/230-good-math-lesson-plans.html for the PDF file and http://i-a-e.org/downloads/doc_download/229-good-math-lesson-plans.html for the Microsoft Word file.
I have written extensively about Project-based Learning. Do a search for this and other materials stored on the IAE sites using the search engine at http://pages.uoregon.edu/moursund/dave/.
Historically, most of the current Computer and Information Science departments in colleges and universities were formed by spin-offs from the Mathematics, Engineering, and Business departments. During my four years of teaching at Michigan State university we dod not yet have a Computer Science Department, but I had a joint appointment between Mathematics and Engineering. My initial appointment at the University of Oregon was a joint appointment between mathematics and the Computing Center.
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.
Explores a variety of ways to significantly improve our math education system. A major update to this file was completed in June 2014. One of the most popular IAE-pedia pages.
Discusses Data, Information, Knowledge, Wisdom, and Foresight and a quote from Arthur C. Clarke.
Discusses different answers to the question, "What is mathematics?" and emphasizes the need for students to gain increasing insight into possible answers as they progress in their math studies.
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.
This page contains brief summaries and pointers to videos useful in precollege math education, teaching teachers, and teaching parents.
An extensive personal collection of quotations (by people other than Dave Moursund) about math, math education, and education.
Discusses the conflict between the math education traditionalists and the math education reformers.
An introduction to a general measure of student progress toward learning mathematics for long term use and understanding.
This page is designed mainly for preservice teachers who are taking a math methods course. It presents some important math methods ideas from an Information adn Communication (ICT) point of view.
Provides some examples of how to make use of Project-based Learning in math instruction.
Note that Project-based Learning and Problem-based Learning are not the same thing.Both are abbreviated PBL, which is certainly confusing.
This IAE-pedia article discusses a wide variety of math placement tests and other math tests. Designed for secondary school math teachers and their students. Also oriented toward elementary school math teachers, teachers of math teachers, and parents of students of high school students. Includes (and discusses) a number of pre-algebra questions.
Here is a sample of some of the free books written by David Moursund are available in PDF and Microsoft Word formats. These and other IAE books are available at http://i-a-e.org/downloads/free-ebooks-by-dave-moursund.html.
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.
Explores the need for education to provide an appropriate balance between high technology and strongly people-oriented low or no technology.
  • Moursund, D. (May, 1995). Moursund Editorial: Computers and Mathematics Education. Retrieved 7/24/2014 from Moursund_Editorial:_Computers_and_Mathematics_Education.
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.
This collection grows over time. Many have been used in articles and books written by Moursund.
Explores educational implications of human brain and computer brain working together to solve problems in math and other areas.
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.
Over the years I have collected a number of articles that I found interesting. They are organized by year. Other "historical" collections of a related nature are also included.
ICT and mathematics overlap. Many of the women who were pioneers in the computer field were mathematicians.

Documents by Other Authors

  • Clements, D.H. (1999). 'Concrete' manipulatives, concrete ideas. Contemporary Issues in Early Childhood. 1(1), 45-60. Retrieved 7/25/2014 from http://www.gse.buffalo.edu/org/buildingblocks/NewsLetters/Concrete_Yelland.htm. The Web reference is for a slightly updated version of the original article. Notice mention of computer manipulatives in the abstract that is quoted from the article:
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.
Find free to use learning and teaching content from diverse content providers and educators from around the world.
What happens when classroom teachers from every country in the world take part in a global community of sharing curriculum and best practices? Teachers are empowered to create extraordinary learning experiences for their students. Barriers to equal access to education begin to lift—geography and politics become immaterial. And the economy benefits from a highly educated population. That’s why we founded Curriki, a nonprofit K-12 global community for teachers, students, and parents to create, share, and find free learning resources that enable true personalized learning.
We believe free and equal access to the best curriculum materials is possible and Curriki is leading the way.
  • Drexel University. (2014). Math Forum at Drexel. Retrieved 7/24/2014 from http://mathforum.org/ Math Forum @ Drexel. Quoting from the website:
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 to 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.
Hardy is usually known by those outside the field of mathematics for his 1940 essay, which is often considered one of the best insights into the mind of a working mathematician written for the layman. The book contains an extensive foreword by C.P. Snow.
Back issues of the free twice-a-month IAE Newsletter and several free books based on the newsletters are available.
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. For example, see http://mathnexus.wwu.edu/archive/resources/detail.asp?ID=176. This provides links to a number of valuable math education resources.
  • 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 math with music and painting, and "blasts" our current math educational system.
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. It contrasts Folk Math with School Math. See also: Eugene Maier. Gene is a close personal friend of mine and 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.
  • MathNEXUS (n.d.). Mathematics Portal: News and ideas for teachers and learners of mathematics. Retrieved 8/21/2016 from http://mathnexus.wwu.edu.
This is an excellent and growing set of math materials (from Jerry Johnson) 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
The MLC is a 501(c)(3) non-profit corporation with headquarters in Salem. Oregon. I (Dave Moursund) was one of three people who founded the organization in 1976, and I have served on the Board of Directors since then. The MLC publishes a K-5 math curriculum named Bridges. The MLC provides a number of free virtual math manipulatives and other free materials. See http://catalog.mathlearningcenter.org/free.
  • NLVM (n.d.). National Library of Virtual Manipulatives. Retrieved 6/27/2014 from http://nlvm.usu.edu. Quoting from the website:
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.
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.
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.
  • SAGE (n.d.). SAGE: Open Source Mathematics Software. Retrieved 6/27/2014 from 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.
Speaking loosely, all of these studies dealt with the same set of issues regarding effective and resourceful problem-solving behavior. Their results can be summed up as follows: It's not just what you know; it's how, when, and whether you use it. The focus here is on two sets of studies designed to he1p students deve1op self-regulatory skills during mathematical problem solving.
The studies were chosen for discussion because of (1) the explicit focus on self-regulation in both studies, (2) the amount of time each devoted to helping students develop such skills, and (3) the detailed reflections on success and failure in each.
  • Shodor (n.d.). The Shodor Foundation. Retrieved 6/27/2014: http://www.shodor.org/. Quoting from the website:
Our mission: to improve math and science education through the effective use of modeling and simulation technologies — “computational science.”
Shodor, a national resource for computational science education, is located in Durham, N.C., and serves students and educators nationwide. Our online education tools such as Interactivate and the Computational Science Education Reference Desk (CSERD), a Pathway Portal of the National Science Digital Library (NSDL), help transform learning through computational thinking.
In addition to developing and deploying interactive models, simulations, and educational tools, Shodor serves students and educators directly through workshops and other hands-on experiences. Shodor offers innovative workshops helping faculty and teachers incorporate computational science into their own curricula or programs. This work is done primarily through the National Computational Science Institute (NCSI) in partnership with TeraGrid, NCSA, and other NSF-funded initiatives.
We begin our inquiry into conceptions of teacher knowledge with the tests for teachers that were used in this country during the last century at state and county levels. Some people may believe that the idea of testing teacher competence in subject matter and pedagogical skill is a new idea, an innovation spawned in the excitement of this era of educational reform, and encouraged by such committed and motivated national leaders as Albert Shanker, President, American Federation of Teachers; Bill Honig, State Superintendent of Schools, California; and Bill Clinton, Governor of Arkansas. Like most good ideas, however, its roots are much older
  • TOMT (7/24/2014). The Oregon Mathematics Teacher. Retrieved 7/24/2014 from http://www.octm.org.
My impression is that TOMT is one of the best state-affiliate-of-NCTM publications.
The Wikipedia contains a list of free software. For example,see http://www.visualstats.org/. Quoting from the Website:
Visual Statistics brings the most complex and advanced statistical methods within reach of those with little statistical training by using animated graphics of the data. Using ViSta: The Visual Statistics System-developed by Forrest Young and Pedro Valero-Mora and available free of charge on the Internet-students can easily create fully interactive visualizations from relevant mathematical statistics, promoting perceptual and cognitive understanding of the data's story. An emphasis is placed on a paradigm for understanding data that is visual, intuitive, geometric, and active, rather than one that relies on convoluted logic, heavy mathematics, systems of algebraic equations, or passive acceptance of results.
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.

Author or Authors

The initial version of this page was created by David Moursund with help from Sonya Richardson.