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# Old and New Math Manipulatives: A Paradigm Shift

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Quoting from National Library of Virtual Manipulatives (n.d.):

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 [computer-based] learning environments to address the same goals.

Initial funding for the National Library of Virtual Manipulatives project was provided by a 1999 grant from the National Science Foundation. This site provides a free trial subscription for Windows and Mac computers to its collection of virtual manipulatives.

Long before the development of reading and writing, people used their fingers as math manipulatives. Marks on a stick used to keep track of the passage of time are a type of math manipulative. Stones in a bag—one stone for each sheep in a flock—are a type of math manipulative.

Counting boards and the abacus date back well over 2,000 years. An illiterate person can learn to use an abacus to do addition and subtraction. The abacus is such an excellent math manipulative that it is still used today both in education and in some market places. See http://www.ee.ryerson.ca/~elf/abacus/history.html.

We have words such as one, two, three, four, and five for counting. We have symbols such as 1, 2, 3, 4, 5 and I, II, III, IV and V to represent the words. I think of the written symbols as a type of math manipulative. Spend some time thinking about the similarities and differences between the written (concrete) words and symbols for numbers and the abstract concept that these represent.

In math, we have an abstract mathematical concept called numbers. We make use of concrete manipulatives we call blocks, rods, and flats to help make numbers more concrete to students. We make use of written symbols to represent numbers. We teach students to manipulate the written representations and we teach students to manipulate the concrete representations. In both cases, we want students to have useful, mental representations of the abstract concept of numbers that will help them to represent and solve a wide range of number-related problems.

Some History of Concrete and Virtual Manipulatives in Math Education

The National Council of Supervisors of Mathematics (in 1979) and the National Council of Teachers of Mathematics (in1980) both published position papers approving the use of calculators in elementary school mathematics. See http://i-a-e.org/newsletters/IAE-Newsletter-2012-91.html and the book Adding it up: Helping children learn mathematics available at http://www.nap.edu/openbook.php?record_id=9822&page=366. NCSM and NCTM both supported integration of this tool into the K-8 math curriculum as an aid to doing math and as an aid to learning math. This can be thought of as part of the history of introducing virtual manipulatives into the math curriculum.

In the United States, prior to the late 1980s, manipulatives and student collaboration were nonexistent in elementary math classes. After 1989, due to a decision by the National Council of Teachers of Mathematics (NTCM), more creativity began to emerge in these elementary schools.This creativity took the form of manipulatives that modeled the addition, subtraction, multiplication, and division students used to have to memorize from practice.

Doug Clements, in his 1999 article 'Concrete' manipulatives, concrete ideas, summarizes early research on math manipulatives. Quoting 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. While such widely accepted notions often have a good deal of truth behind them, they can also become immune from critical reflection. In this article, I will briefly consider research on the use of manipulatives and offer a critique of common perspectives on the notions of concrete manipulatives and concrete ideas. From a reformulation of these notions, I re-consider the role computer manipulatives may play in helping students learn mathematics, providing illustrations from our empirical research.

Students who use manipulatives in their mathematics classes usually outperform those who do not, although the benefits may be slight. This benefit holds across grade level, ability level, and topic, given that use of a manipulative "makes sense" for that topic. Manipulative use also increases scores on retention and problem solving tests. Attitudes toward mathematics are improved when students have instruction with concrete materials provided by teachers knowledgeable about their use. [Bold added for emphasis.]

Notice the 1999 date of Clement’s article and the 1999 award date for the National Science Foundation grant that funded the creation of the National Library of Virtual Manipulatives. Computer-based math manipulatives were already in use many years before that date.

A Broader View of Math Manipulatives

I believe that the math education community’s definition of math manipulative is far too restricted. The basic idea of a math manipulative is that it is a concrete or computerized aid to learning and doing math.

A mechanical or electrical calculator can be thought of as a type of math manipulative useful for both learning and doing arithmetic. During my childhood I made extensive use of a mechanical adding machine that printed its results on a roll of paper. It was a wonderful toy, aid to learning, and practical tool.

Progress in transistor technology has led to the development of very powerful and relatively inexpensive hand-held calculators. Quoting from http://en.wikipedia.org/wiki/Calculator:

The first solid state electronic calculator was created in the 1960s, building on the extensive history of tools such as the abacus, developed around 2000 BC; and the mechanical calculator, developed in the 17th century. It was developed in parallel with the analog computers of the day.

Pocket sized devices became available in the 1970s, especially after the invention of the microprocessor developed by Intel for the Japanese calculator company Busicom.

The steadily improving capabilities of calculators and computers facilitated the development of a type of software called a Computer Algebra System. The first such systems were developed during the 1960s. A computer algebra system (CAS) is a software program that facilitates symbolic mathematics. The core functionality of a CAS is manipulation of mathematical expressions in symbolic form. See http://en.wikipedia.org/wiki/Computer_algebra_system. Quoting from this website, a CAS may include capabilities such as:

• arbitrary-precision numeric operations
• change of form of expressions: expanding products and powers, partial and full factorization, transforming logic expressions
• drawing charts and diagrams
• exact integer arithmetic and number theory functionality
• matrix operations including products, inverses, etc.
• plotting graphs and parametric plots of functions in two and three dimensions, and animating them
• series operations such as expansion, summation and products
• solution of linear and some non-linear equations over various domains
• statistical computation

Such systems can solve or help in solving or exploring a very wide range of math problems. Many versions of CAS are available free on the Web (Moursund, n.d.).

I think of a CAS as a new type of math manipulative. This is, of course a considerable broadening of the original concepts of physical (concrete) math manipulatives and virtual math manipulatives that currently are an integral component of math education. It is a paradigm shift that I believe will eventually be commonly accepted throughout all levels of math education. This paradigm shift is being helped by the movement toward every student having a tablet or laptop computer.

Final Remarks

Spreadsheet software provides an excellent example of a math manipulative that has long been used in education. Such software typically includes provisions for graphing and plotting. See http://www.ericdigests.org/2003-1/math.htm. Calculators that graph functions and solve equations are now required in many secondary school math courses.

Computer-aided design software has transformed courses in mechanical drawing. See http://en.wikipedia.org/wiki/Computer-aided_design. Music composition software has been designed for use by students of all ages. Such systems facilitate students composing in an environment in which the computer can play the compositions that are created.

This list is easily expanded. My point is that the paradigm shift I have been discussing has already occurred in many different disciplines. I believe that math education has been a laggard.

What You Can Do

A large number of free software programs designed to help represent and solve math problems is now available. Such software can help to make abstract concepts in math more concrete and accessible to students. Such software is routinely used by people who use math to help solve problems in their disciplines of study and work.

Explore and experiment with software that may be appropriate for use by the students you teach. I suggest you place emphasis on how such software is used to help represent and solve math-related problems across the curriculum being taught in our schools.

References

Clements, D. H. (1999). 'Concrete' manipulatives, concrete ideas. Contemporary Issues in Early Childhood. 1(1), 45-60. Retrieved 4/18/2013 from http://www.gse.buffalo.edu/org/buildingblocks/NewsLetters/Concrete_Yelland.htm.

Moursund, D. (1/15/2013). A tablet computer and connectivity for every student. IAE Blog. Retrieved 4/19/2013 from http://i-a-e.org/iae-blog/a-tablet-computer-and-connectivity-for-every-student.html.

Moursund, D. (n.d.) Free math software. Retrieved 4/18/2013 from http://i-a-e.org/iae-blog/using-ict-to-improve-education-consider-three-questions-instead-of-two.html

National Library of Virtual Manipulatives (n.d.). A library of uniquely interactive, web-based virtual manipulatives. Retrieved 4/18/2013 from http://nlvm.usu.edu/en/nav/siteinfo.html.

Suggested Readings from IAE and Other Publications

You can use Google to search all of the IAE publications. Click here to begin. Then click in the IAE Search box that is provided, insert your search terms, and click on the Search button.

Here are some examples of publications that might interest you.

Free math manipulative apps from the Math Learning Center. See http://i-a-e.org/iae-blog/free-math-manipulative-apps-from-the-math-learning-center.html.

Using Information and Communication Technology (ICT) to improve education: Consider three questions instead of two. See http://i-a-e.org/iae-blog/using-ict-to-improve-education-consider-three-questions-instead-of-two.html.

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Moursund is an Emeritus Professor of Education at the University of Oregon. His professional career includes founding the International Society for Technology in Education (ISTE) in 1979, serving as ISTE’s executive officer for 19 years, and establishing ISTE’s flagship publication, Learning and Leading with Technology. He was the major professor or co-major professor for 82 doctoral students. He has presented hundreds of professional talks and workshops. He has authored or coauthored more than 60 academic books and hundreds of articles. Many of these books are available free online. See http://iae-pedia.org/David_Moursund_Books. In 2007, Moursund founded Information Age Education (IAE), a non-profit company dedicated to improving teaching and learning by people of all ages throughout the world. See http://iae-pedia.org/Main_Page#IAE_in_a_Nutshell.