Math Manipulatives

What is a virtual manipulative?

In "What are Virtual Manipulatives?," Patricia Moyer, Johnna Bolyard, and Mark Spikell (2002) define a virtual manipulative as "an interactive, Web-based visual representation of a dynamic object that presents opportunities for constructing mathematical knowledge" (p. 373).  Static and dynamic virtual models can be found on the Web, but static models are not true virtual manipulatives.  Static models look like physical concrete manipulatives that have traditionally been used in classrooms, but they are essentially pictures and learners cannot actually manipulate them.  "...[U]ser engagement distinguishes virtual manipulative sites from those sites where the act of pointing and clicking results in the computer's providing an answer in visual or symbolic form" (p. 373).  The key is for students to be able to construct meaning on their own by using the mouse to control physical actions of objects by sliding, flipping, turning, and rotating them. 

Currently, virtual manipulatives are modeled after concrete manipulatives such as base ten blocks, coins, pattern blocks, tangrams, spinners, rulers, fraction bars, algebra tiles, geoboards, and geometric plane and solid figures, and are usually in the form of Java or Flash applets.  The figure above was developed by Patricia Deubel of CT4ME to illustrate virtual manipulatives found on the Web, which are useful for mastery of basic skills and conceptual understanding of K-12 mathematics and calculus.

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What role can virtual manipulatives play in the classroom?

Principles and Standards for School Mathematics (NCTM, 2000) calls for study of both traditional basics, such as multiplication facts, and new basics, such as reasoning and problem solving.  Using manipulatives in the classroom assists with those goals and is in keeping with the progressive movement of discovery and inquiry-based learning.  Students learn best when teachers use multiple instructional strategies that combine "see-hear-do" activities.  All students learn in different ways. For some, mathematics is just too abstract.  Using tools, such as calculators, allows students to focus on strategies for problem solving, rather than the calculation itself.  Manipulatives also increase exploration possibilities to develop concepts and test hypotheses for students at all levels of ability.   Visit Teacher2Teacher for more on the role of manipulatives.  

Douglas H. Clements in " 'Concrete' Manipulatives, Concrete Ideas" proposes a reformulation of the definition of "concrete" manipulative and illustrates the pedagogical value of using computer manipulatives. He says, "Good manipulatives are those that are meaningful to the learner, provide control and flexibility to the learner, have characteristics that mirror, or are consistent with, cognitive and mathematics structures, and assist the learner in making connections between various pieces and types of knowledge—in a word, serving as a catalyst for the growth of integrated-Concrete knowledge. Computer manipulatives can serve that function" (Section: The Nature of "Concrete" Manipulatives and the Issue of Computer Manipulatives, par. 2).

Most students benefit from visual representations of concepts. According to Christopher Matawa (1998, p. 1), there are many Uses of Java Applets in Mathematics Education:

• Applets to generate examples. Instead of a single image with a picture that gives an example of the concept being taught an applet allows us to have very many examples without the need for a lot of space.

• Applets that give students simple exercises to make sure that they have understood a definition or concept.

• Applets that generate data. The students can then analyze the data and try to make reasonable conjectures based on the data.

• Applets that guide a student through a sequence of steps that the student performs while the applet is running.

• Applets that present ''picture proofs''. With animation it is possible to present picture proofs that one could not do without a computer.

• An applet can also be in the form of a mathematical puzzle. Students are then challenged to explain how the applet works and extract the mathematics from the puzzle. This also helps with developing problem solving skills.

• An applet can set a theme for a whole course. Different versions of an applet can appear at different stages of a course to illustrate aspects of the problem being studied.

While the research is scarce on mathematics achievement resulting from using virtual manipulatives in the classroom, Kelly Reimer's and Patricia Moyer's action research study (2005), Third-Graders Learn About Fractions Using Virtual Manipulatives: A Classroom Study, provides a look into the potential benefits of using these tools for learning.  Interviews with learners revealed that virtual manipulatives were helping them to learn about fractions, students liked the immediate feedback they received from the applets, the virtual manipulatives were easier and faster to use than paper-and-pencil, and they provided enjoyment for learning mathematics.  Their use enabled all students, from those with lesser ability to those of greatest ability, to remain engaged with the content, thus providing for differentiated instruction.  But did the manipulatives lead to achievement gains?  The authors do admit to a problem with generalizability of results because the study was conducted with only one classroom, took place only during a two-week unit, and there was bias going into the study.  However, results from their pretest/posttest design indicated a statistically significant improvement in students' posttest scores on a test of conceptual knowledge, and a significant relationship between students' scores on the posttests of conceptual knowledge and procedural knowledge.  Applets were selected from the National Library of Virtual Manipulatives. 

Boston Public Schools has a professional development initiative to provide teachers and students access to virtual manipulatives and technology equipment that directly support the district's math and technology curricula.   Partially funded by a NCLB state grant, SELECT Math contains alignments for Grades 6-8, Algebra I and II, and Geometry with a Scope and Sequence calendar describing which book or chapter is being used in math classes during each month of the school year.  Click on the individual book/chapter to see the related SELECT Math alignments, worksheets, and links to supporting virtual manipulatives. The project began in 2002 as a collaboration between the Boston Public Schools' Secondary Math and Instructional Technology departments, in conjunction with their partner, the Education Development Center, Inc.  CT4ME believes this initiative to be valuable for middle and high school math educators throughout the country.

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Using Virtual Manipulatives and Calculators and Overusing Manipulatives in General

In order to effectively use virtual manipulatives in the classroom, "teachers must have an understanding of how to use representations for mathematics instruction as well as an understanding of how to structure a mathematics lesson where students use technology...Teachers must also be comfortable with technology and be prepared for situations where computers may not be available or Internet connections are not working properly" (Reimer & Moyer, 2005, p. 7).  This author's own experience confirms that virtual manipulatives may take a while to download, and in some cases, the wait time might be frustrating.  Imagine the frustrations for a learner anxious to begin.  Plus, even when successfully downloaded, they might not work fast enough for learners who are accustomed to playing high speed, interactive video games.  In some cases, the footprint on the screen might be too small for learners with poor mousing skills or for those with limited dexterity to click on relevant icons or to perform the spins, rotations, flips and turns required. 

Teachers should be aware of problems that might arise from overusing both concrete and virtual manipulatives.  In The State of State Math Standards 2005, David Klein (2005) discussed nine problem areas in which state standards come up short.  Among those was concern for an overuse of calculators and manipulatives in that students might come to depend on them and focus on the manipulatives more than on the math.  "[M]any state standards recommend and even require the use of a dizzying array of manipulatives in counterproductive ways" (p. 11).  In this author's view such a reliance might have its roots in the quality of instruction, in part, and failure of the math educator to explicitly state and reinforce the link between the use of the manipulative, and development of concepts for understanding and properties of mathematics to be learned.  Such might be the case, for example when using algebra tiles for multiplying and factoring polynomials, if the educator failed to explicitly link the knowledge of the distributive property to that action.  

I have an interesting personal story to relate on the use of calculators.  One day our newspaper person, who was a middle school student at the time, knocked on our door to collect our monthly payment for the newspapers.  He dug out his calculator to multiply the weekly payment by four, which he should have been able to do mentally.  I asked him what he would do to figure out my bill, if his calculator no longer worked.  He said, "I'd go buy new batteries!" Klein (2005) stated that manipulatives are useful for introducing new concepts to elementary students, but, "In the higher grades, manipulatives can undermine important educational goals" (p. 11).  Among those are for students to develop skill fluency, conceptual understanding, and mathematical reasoning.  Many states' standards documents overemphasize calculator use, for example. 

This author would agree with Klein (2005) in that educators should not overly rely on calculator use at the expense of having students master basic skills and memorize basic facts, which are essential for higher order learning in mathematics.  In this sense drill and practice still have a role in teaching and learning mathematics.  According to E. D. Hirsch (1999), drill and practice may have a disparaging connotation as a pedagogical tool to teach skills and runs contrary to the progressive movement, but the method should not be slighted as low level. It is just as essential to complex intellectual performance as drill and practice are to the virtuoso violinist or the athlete on the playing field. 

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More on Calculators

This author would also emphasize that a graphing calculator is an essential tool for learners in mathematics courses such as algebra, trigonometry, advanced math, and calculus.  The visual display becomes a powerful tool for teaching and learning to show the link among conceptual, procedural, analytic, and investigative dimensions of learning mathematics. 

Atomic Learning has a series of tutorials on Texas Instruments calculators: how to use the TI-30XS and TI-84. Readers might be particularly interested in TI-Nspire, which combines graphing capabilities with computer features (e.g., save and review work).  You can see multiple representations of a problem on one screen, use "grab and "move" to observe patterns and relations, and much more. Texas Instruments teamed with Atomic Learning to provide online tutorials on the TI-Nspire handhelds.  Also see SimCalc MathWorlds software for TI-graphing calculators developed by the University of Massachusetts-Dartmouth's James J. Kaput Center for Research and Innovation in Mathematics Education.  Other tutorials and manuals for elementary, financial, graphing, and scientific calculators are available from manufacturers, such as Texas Instruments Online Tutorials, Casio Worldwide [manuals], HP, and Sharp.

With this being said, CT4ME has a number of virtual manipulatives that can serve you well in the classroom.  As one educator recently told me at one of my own conference presentations on this topic, "I don't have to worry about students flicking rubber bands at each other any more!"  She was using virtual geoboards. 

 

Here's a bit of history for you! From Slates, to Slide Rules and Software! Does anyone remember the slide rule?   "Throughout American history, teachers and parents have used objects--from colonial--era slates to modern electronic calculators--to help students master abstract mathematical concepts," according to The Smithsonian Institute, which has posted a highly informative display called Slates, Slide Rules, and Software: Teaching Math in America.  Read developments in math education and teaching with manipulatives from the Early Republic, to the Cold War, and Information Age.  Additional resources are provided.  http://americanhistory.si.edu/teachingmath/    Can you guess the origin of the square root sign from the picture?  Watch a Flash animation to find out.   The irrational number is the ratio of the circumference of a circle to its diameter.  It is often used in mathematics approximated as 3.14, but a computer has calculated its value to over 6 billion decimal places! Learn more about the history of pi.  Also visit the Joy of Pi.

Technology Requirements:

JAVA: You might need to download Java, a free plug-in from Sun Microsystems to interact with virtual manipulatives.

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Calculators and PDAs
Including Tutorials, Activities, and Software Enhancements

Basic Calculator:  Enter your basic problem followed by the equal sign, as in 6 * 2 =   Press "c" to clear the calculator for the next problem. The Javascript code for this calculator was provided free by http://javascript.internet.com/

Calculator City from 1728 Software Systems has online converters and calculators for topics studied in basic math, algebra, geometry, trig, calculus, and more.  Several include additional graphics and brief discussions of concepts associated with the calculator.

Calculators On-Line Center features over 19,000 calculators for mathematics, statistics, science, and engineering.  Calculators for mathematics range from those suitable for basic mathematics through calculus and higher level mathematics.  Don't miss this vast collection of specialized calculators by topic from J. Martindale.

Coolmath Online Graphing Calculator is free and has all the common operators and functions expected in scientific calculators and graphing calculators for graphing functions.  Great alternative for students who forget their own handheld graphing calculator, such as the TI-83.

Cyberchase Talking Calculator is a free basic online calculator (add, subtract, multiply, divide) from PBS Kids.

GCalc is a free online graphing calculator. "GCalc is designed to provide a basic, easy-to-use, well-balanced set of graphing functionality for algebra, pre-calculus, calculus and beyond."

Graphing Calculator Help from Prentice Hall for TI-82, TI-83, TI-85, TI-86, TI-89, TI-92, HP48G, CFX-9850, Sharp EL9600C, and Casio FX2. 

SimCalc MathWorlds software for TI-graphing calculators, computer, and TI-Navigator developed by the University of Massachusetts-Dartmouth's James J. Kaput Center for Research and Innovation in Mathematics Education.  Animations, real life examples, narrative stories, and more are used to explain math concepts.

Talking Calculator from Premier Assistive Technology is an onscreen full-function talking calculator that can be used with or without a screen reader.  "Every button and edit area talks. It is easy to use with large keys and contrasting colors. It has three display areas, so when the user adds a series of numbers, the total is always displayed, even as you are entering a new number, while always displaying any numbers in memory." Further, "it displays your entries and results as you work. It actually displays the equation so that you can easily see or hear your last process. When students are required to show their work, they can simply cut and paste the steps into a document." A download is available.

TI Math has activities for use with Texas Instrument graphing calculators (TI-Nspire, TI-Nspire CAS, TI-84 Plus Silver Edition and TI-89 Titanium) in various subjects, which include algebra 1, algebra II, geometry, precalculus, calculus, and statistics.

Tutorials from Texas Instruments.  TI provides a number of interactive tutorials for basic and graphing calculators.

Tutorials from Atomic Learning on Texas Instruments calculators: the TI-30XS, TI-84 and TI-Nspire handhelds.

WebGraphing.com provides online graphing (1D, 2D, & Interactive 3D) of functions, equations, systems of equations, inequalities in one and two variables, and piecewise functions, with tutorial analyses appropriate for students of algebra, precalculus, and calculus. There is also a forum for the different math levels.  What sets the function graphing calculators apart from other graphing calculators is the automatic display of asymptotes and discontinuities in standard mathematical notation, and the automatic determination of an optimal graphing window--one that includes all mathematical features of interest.  Dr. Barry Cherkas of the Mathematics and Statistics Department at Hunter College (CUNY) is President of WebGraphing.com.
 

PDA Handheld Computing Resources on the Web is brought to you from the Shiawassee Regional Education Service District in Michigan.  The collection includes links to ebooks, journals, peripherals, tutorial videos, hardware/software, assessment programs, math programs, and more to integrate the PDA into your instruction.

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Manipulatives on the Web

K-12 Algebra Applets--Arlen Strader of the Texas A & M University has two algebra tiles applet pages to help middle and high school students explore factoring of quadratic equations by arranging given tiles into a rectangle: Algebra Tiles1 (easy version-coefficients positive) and Algebra Tiles2 (hard version-coefficients positive and negative).  There are also applets for Cartesian coordinates, slope, parallel and perpendicular lines, and Pythagorean Theorem.

Ambleweb Numeracy Hour of Ambleside C.E. Primary School (UK) contains excellent interactive online Tools for Teachers to use with elementary students for demonstrating concepts and helping students to master basic skills.  There is a large buttoned online four function calculator for demonstrating how to use a calculator, an online resource for teaching about angles and how to use a protractor, a virtual tangram, counter squares of various sizes and purposes on which you can scribble using the left button on your mouse, applets for studying number sequences and patterns, a simple bar graph to introduce graphing data, teaching time on an analog clock, and interactive geometry.  There is also a section for interactive skill building at three levels of difficulty, puzzles, and games.  

Arcytech's Educational Java Programs are primarily for elementary and middle school students.  Award-winning applets help students learn to count money, tell time, use patterns to explore fractions and geometric shapes; use base-10 blocks to explore whole and decimal numbers, place value and algebra with blocks; use integer bars to explore basic math facts, fractions, and more; use fraction bars to explore equivalent fractions, ordering fractions, adding fractions, and more.  Additional applets help students explore the Pythagorean theorem and the value of Pi, and fractals.  Lesson plans are included.  Site developed by J. Bulaevsky.

Argyll Centre of Edmonton Public Schools in Canada hosts Jim Reed's Math and Science Resources.  His resources address elementary and secondary topics in numbers/operations, patterns and relations, shape and space, probability and statistics, algebra, geometry, and trigonometry. Students can engage with interactive learning activities, see movies, and manipulate excellent Java math applets to study concepts.  The site is also an excellent resource for the home-schooled student.

British and Metric Conversions from Annenberg Media's Learner.org. is an interactive Web site to learn how to convert measurements between the metric system and the English (sometimes called British) system, which is commonly used in the United States.  Learn the symbols, more about length, mass, temperature, volume, and then test your knowledge.

Collaborative Whiteboard from GE allows individuals or groups to draw anything freehand or type text.  You can display a  graphing grid, select and then draw geometric shapes, save the work and share it with others via email.

Erathothenes' Sieve--watch the primes up to 400 being generated with this Java display.

Euclid's Elements Dr. Dave Joyce of Clark University in Massachusetts brings all13 books of the Elements alive using a Java geometry applet to illustrate definitions and postulates.

ExploreLearning.com is a subscription based set of over 400 virtual manipulatives (simulations of math and science concepts), which the authors call "Gizmos."  The math series addresses the five NCTM strands for grades 6-12 and college developmental math, college algebra, and pre-calculus. Gizmos for elementary math for grades 3-5 are also being developed.  ExploreLearning Gizmos, with associated Exploration Guides, help students develop critical thinking skills and conceptual understanding.  The series is also correlated with national and state standards and major textbooks, and includes assessment and real time reporting.  A 30-day free trial is available for the teacher or home.  Sample Gizmo at the site can be immediately activated.  

Geometry 3-D Shapes is one of the Interactives from Annenberg Media's Learner.org.  Students can interactively learn about three-dimensional shapes, calculate surface area and volume, and discover some of the mathematical properties of shapes. It comes with tutorials and an interactive test.

Geometry Gallery from Saltire Software contains some excellent Java applets, which are easy to use and suitable for illustrating several theorems presented in a high school or post-secondary geometry course.  These relate to angles in parallel lines, circular configurations, Pythagorean triples and incircles. Students can drag points to explore concepts.  Four triangle calculators are presented.  By entering SSS, SAS, or ASA, students can see the unique triangle that is formed, but if SSA is entered the resulting triangle is not unique (two are possible) or no triangle is possibly formed with given information.  Some advanced geometrical configurations are explored.  There is a geometry atlas of algebraic formulas with associated visuals. Saltire's work has been assisted with a number of National Science Foundation grants.

Harcourtschool.com E-Lab: Extensive collection of interactive applets for grades 3-8.  Activities to accompany each are provided.  You can download an E-Lab activity and use it without being connected to the internet.  While working on internet, Microsoft Java VM is recommended.  Later versions of Java will be supported soon for online activity.

Hispanic Math, available in English and Spanish, is an interactive multimedia tutorial program for grades 4-6 that combines complex action script with flash animation to deliver developmentally appropriate instruction on the relationship between perimeter and area. "To complete each lesson , students must help the animated characters "Kiri" and "Karlo" through a variety of adventures that require the use of measurement devices both on and off the computer screen. Each bilingual lesson features diagnostic testing, multiple learning paths, and flash animations in which working adults model various occupations such as cartography, cattle ranching, and banking." This is a project of Arizona State University and Technology Based Learning and Research.  Excellent and engaging!

HKCEE Mathematics contains several Java applets for investigating trigonometry ratios and the geometry of circles, which high school students can manipulate.

Illuminations Activities from the National Council of Teachers of Mathematics (NCTM) contain Java applets identified by grade band to investigate various concepts within NCTM identified strands.

Interactive Mathematics promotes learning math while you play with it!  Primarily for upper secondary and college level students; basic math includes algebra, complex numbers, statistics, logarithms, trigonometry, plane analytic geometry, and graphing.  Calculus includes differentiation, integration, and higher topics in calculus.  Lectures and full examples are provided.  Site provides links to LiveMath plug-in, Scientific Notebook, and Flash.

Interactive Resources from the UK has four math packs, each with a wide range of excellent interactives for classroom use with preK-6 students.  Explore numerous concepts related to NCTM strands.  Some include whiteboard (drawing on the figures) capabilities.  There are rulers, protractors, geoboards, Venn diagrams, number lines, and so on. Don't miss these teaching tools.

Key Curriculum Press JavaSketchpad has several applets for use with secondary math students and above.  See the interactive manipulative on the Pythagorean Theorem and the JavaSketchpad DR4 Gallery.  In the Gallery you will find demonstrations from geometry (e.g., centroid), trigonometry and algebraic geometry (e.g., sine waver and conic sections), calculus (e.g., box volume and falling ladder), mechanics, and an applet on least squares.

The Learning Equation (TLE) is brought to you by R. Blond of Edmonton, Alberta, Canada.  TLE features JAVA applets that help students to interactively explore complex mathematics concepts typically studied in depth from grades 9 through post-secondary.  Among topics are triangle relations, slopes, circle geometry, conic sections, exponential and trig functions; translations, reflections, stretching; and more. 

Learning Objects from Brock University (Canada) Department of Mathematics are tools to help students explore mathematical concepts interactively.

• The Great Balancing Act: explore the mean and median
• Parabola games: explore parabola properties, transformations and their use
• Leap Frog: explore number patterns
• Fire, Fire!: explore trigonometric ratios
• Transformations: an introduction to transformational symmetries
• Markov chains: What happens after years of movement in cities and suburbs, an introduction to Markov chains

Manipula Math with JAVA by International Education Software contains a collection of over 280 applets suitable for teaching math concepts to middle school, high school, and college students.  Applets address middle school geometry and the Pythagorean Theorem, trigonometry, calculus, vectors, complex numbers, conics, and miscellaneous topics.

Math Doodles is by Daren Carstens, who is the developer of the award winning software Math Arena.  He offers several interactive puzzles and games for elementary learners to have fun.  But there is learning, too, as students practice addition, subtraction, money skills, and pattern matching (shape, number, color), for example.  Among current games are Connect Sums, Number Jump, Polyomino Shift,  Double Traits, Sum Stacker, Time Shuffle, and more.  Older learners also benefit, as students really need to think as they do these.

Math Playground: Animated characters created with Flash add to the appeal of this learning site for grades 1-6.   Among resources are some virtual math manipulatives, such as a protractor, fraction bars, equivalent fractions, percents, a spinner for probability, function machine, pattern blocks, and a geoboard.

Mathematics Open Reference is a free plane geometry textbook for high school learners.  Its focus is on using interactive materials and animations to develop concepts.

MathTools from the Math Forum contains a number of interactive tools for understanding concepts related to K-12 mathematics and calculus.  Help kindergarten students understand concepts.  Introduce topics with pattern blocks, number lines, fraction bars and more.  Algebra concepts include number systems, integers and integer operations, properties (commutative, associative, distributive), multiplying/dividing, using variables, equivalent equations, inverse operations, graphing, linear equations and systems, quadratic equations, factoring, and more.  Geometry interactives for plane and solid topics are extensive.  Trigonometry includes the Law of Sines and Law of Cosines.  There is a range of calculus tools for differentiation and integration topics.

Mathsnet.net is a highly interactive site for doing mathematics, featuring concepts in number, geometry, algebra, graphs, data handling.  JAVA, Javascript, or plug-ins may be needed.  The geometry is of particular interest.  There are curriculum, articles, books, and a section for download, including free software to explore topics interactively.  Don't miss this award winning site brought to you by B. Dye in England.

Math Warehouse is "dedicated to dynamic Math lessons, demonstrations, and interactive activities." There are topics in algebra, geometry, trigonometry.  Worksheets are available for download, too.

MegaMaths, brought to you by the BBC, is a School Radio series for elementary students aged 7-9.  Each 15 minute episode promotes a whole-class approach to mental maths activities through a series of lively and stimulating quizzes. 

National Library of Virtual Manipulatives for Interactive Mathematics contains manipulatives and associated online activities for each of the five strands identified in standards set forth by the NCTM.  This is an ongoing project of Utah State University and a must see site for grades preK-12.  Select a manipulative from the grade level you teach and an activity from one of the strands.

Number Line Applet: Introduce addition and subtraction of integers with this number line.  Students see the concept after clicking onto the integers  (-35 to 35) for the problem.  From MSTE at University of Illinois-Urbana_Champaign.

PBS Teacherline Interactives --explore with cubes, floor tiles, frog in a well (number patterns and relations), geoboard, histograms, making a rule, mixing orange juice, number cruncher, the race, real-life data, space blocks, surface area, and tesselate.

Principles and Standards for School Mathematics Electronic Examples--interactive applets for investigating concepts from NCTM.  Examples are divided into grade bands.

• preK-2: Investigate patterns, triangles and properties of polygons, geometry and measurement; develop spatial skills by using tanagrams, learn estimation strategies
• 3-5: Communicate about math using games, use  simulation software to investigate distance, rate, and time; investigate data
• 6-8: Visualize the concept of multiplication, investigate rate of change, length, perimeter, area, volume, congruence, similarity, and symmetry, and the Pythagorean Theorem
• 9-12: Explore vectors; use graphs, equations, and tables to investigate elimination of medicine from the body; explore inscribed figures, linear regression, and linear functions.    

Project Interactivate, which is funded, in part, by the Office of Dependent Education of the Department of Defense Education Activity (DoDEA), includes a set of JAVA-based tools to support courseware for grades 6-8, although some activities are appropriate for grades 3-5.  Activities that use the tools support NCTM standards and concepts in number and operation, geometry and measurement, algebra and functions, probability and data analysis.  Lessons supported by discussion of concepts are included.  The project is listed by The Shodor Education Foundation, Inc.

Seeing Math by the Concord Consortium has eight free secondary math interactives written in Java to assist algebra learners in the study of functions.  "Each interactive provides a real-time connection between representations of the mathematics (symbolic, graphical, etc.), so that changes in one representation instantly cause changes in the other."  Interactives include a qualitative grapher, piecewise linear grapher, linear transformer, quadratic transformer, function analyzer, system solver, plop it and proportioner.  A user's guide, warm up exercise, frequently asked questions, and sample activity are provided. 

Statistics: Are you teaching your students about statistics and survey research?  Read Statistics Every Writer Should Know by Robert Niles.  Learn about mean, median, percent, per capita, standard deviation, margin of error, data analysis, and more.  Link to sites for data sources and interactive help to select the right statistical test.

• Descriptive Statistics (introduction to Mode, Median, and Mean) by Jay Hill at the University of Illinois.
• Chi-Square by Amar Patel. The material in this lesson, which is correlated to NCTM standards for grades 9-12,  will help students to understand statistical terms such as fairness, expectation, significance, and chi-square.  Excel spreadsheet is needed.  Content could be used with grade 8 and above, however.  Cartoon characters add humor to the discussion. 
• Tools:  Generate random numbers, sample sizes, charts and graphs, and perform statistical calculations using these tools:

• Research Randomizer
• Random.org
• The Survey System Sample Size Calculator 
• Create a Graph, brought to you by the National Center for Education Statistics, is an interactive site that helps students understand how to create area, line, and bar graphs, and pie charts.  You might point out to students the 10 Common Mistakes Found on Graphs, noted by CanTeach in Canada.  
• If you are looking for data to use when creating graphs, see the work of Robert Niles on finding data on the Internet.
• If you have developed a bar graph using Microsoft Excel, show students how to Create a Pictograph from that data. 
• Web Pages that Perform Statistical Calculations

Help your students to understand the beauty of mathematics found in nature.     Explore fractals with this unit by Cynthia Lanius, which is appropriate for elementary and middle schools learners and even adults.   You will learn about the importance of fractals, properties of fractals, create a few, and get a series of links to other sites on the Web that address this topic. Amazing Seattle Fractals! will benefit high school learners and above.  The developer provides tutorials to learn more about fractals and how to create fractal art.  Users can download free fractal software programs and view some fractal art galleries.

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References:

Clements, D. H. (1999). Concrete' manipulatives, concrete ideas. Contemporary Issues in Early Childhood, 1(1), 45-60. [Update online]. Available: http://www.gse.buffalo.edu/org/buildingblocks/Newsletters/Concrete_Yelland.htm

Hirsch, E. D., Jr. (1999). The schools we need and why we don't have them. New York: Doubleday. ISBN: 0-385-49524-2.

Klein, D. (2005, January). The state of state math standards 2005. Washington, DC: Thomas B. Fordham Foundation.  http://www.edexcellence.net/foundation/publication/publication.cfm?id=338&pubsubid=1117

Matawa, C. (1998). Uses of Java applets in mathematics education.  Paper presented at Asian Technology Conference in Mathematics, 1998.  Available: http://www.atcminc.com/mPublications/EP/EPATCM98/ATCMP016/paper.pdf

Moyer, P. S., Bolyard, J. J., & Spikell, M. A. (2002). What are virtual manipulatives? [Online]. Teaching Children Mathematics, 8(6), 372-377.  Available: http://my.nctm.org/eresources/article_summary.asp?URI=TCM2002-02-372a&from=B 

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author. Available: http://standards.nctm.org/ 

Reimer, K., & Moyer, P. S. (2005). Third graders learn about fractions using virtual manipulatives: A classroom study. Journal of Computers in Mathematics and Science Teaching, 24(1), 5-25.

Additional reading on math manipulatives:

For a current review of the literature, see David Young's (2006, April) Virtual Manipulatives in Mathematics Education. Available: http://plaza.ufl.edu/youngdj/talks/vms_paper.doc

Selected references relating to the use of math manipulatives, a list of resources by Dr. Garry Taylor of Northern Arizona University.  Note that CT4ME is one of the Web resources. 

Durmus, S., & Karakirik, E. (2006, Jan). Virtual manipulatives in mathematics education: A theoretical framework. The Turkish Online Journal of Educational Technology, 5(1), article 12. Available: http://www.tojet.net/articles/5112.htm  [Note: CT4ME is cited in this article.]

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See other pages of the section on Math Web Resources and Standardized Test Preparation:  Math Resources and Standardized Test Preparation.

To cite this page, use the following format:

Deubel, P. (fill in year from last revised). Math manipulatives [Online]. Retrieved [fill in date] from Computing Technology for Math Excellence at http://www.ct4me.net/math_manipulatives.htm