Interactive Mathematics on the Internet

 

Barry Kissane

 

In recent years, an increasing number of web sites have offered significant opportunities for student to learn online, often through the use of Java applets. You can here explore some good examples of this genre of material that are publicly available, to consider their potential for mathematics learning. Note that your browser will need to be Java-enabled for many activities, while many others will use Flash; free downloads for these are available. Websites will generally offer advice on such technical matters.

 

A thorough examination of the examples below will take many hours, and so will not be possible in a workshop of an hour or two. You will need to be very selective here and return later for a closer look.

 

If you are in a workshop in a computer laboratory, do not download anything onto the computers unless you are sure that this is acceptable to the owners.

 

Workshop Instructions

 

1. Work with a partner, either sharing a computer or working side by side on separate computers. Discuss what you see.

 

2. Using the brief description as a guide, choose a site below that is (relatively) unfamiliar to you and matches one of your interests. The year levels suggested are approximate.

 

3. Read the suggestions before accessing the site, so that you know what to look for and what to do.

 

4. Once on the site, explore a couple of examples, to get a sense of what might be offered to students that is not usually accessible to them:

 

How could these applets be used by your students: at school? At home?

 

How could the applets be used by a teacher: for demonstration? on an interactive whiteboard? for class discussion? for ideas?

 

5. Repeats steps 2 to 4 to choose a range of sites, in order to see a range of possibilities and educational styles.

 

 

Level

Website

Suggestions

3-12

National Library of Virtual Manipulatives

A large and very impressive collection of applets, with substantial support provided for teachers and parents. Linked to the NCTM Standards. Can be purchased separately as a CD-ROM, avoiding the need for Internet access.

 

Click on Virtual Library to access a grid of content by level. Choose an element of the grid and then choose an applet to explore. (You can choose an age level or a content band instead if you prefer. Be careful: the grade levels provided are not very reliable.) Icons at the top of each applet offer suitable activities and instructions. Use the Back icon to return. For example, try Platonic Solids in geometry 9-12 to see a 3-D version.

 

5-8

Project Interactivate

A large collection of well-designed applets, also keyed to the NCTM Standards, mostly aimed at middle school level, but many of which are also potentially valuable for older students too. Excellent support materials for teachers are included.

 

Click on the Activities hexagon to access a complete list. Each Activity has a What? How? And Why? Link to help, if needed. [A great deal of help is provided, but this workshop doesn’t suit a close scrutiny of this today.] Choose a range to suit your interests. Some of my favourites are Histogram, Transmographer, Area Explorer and Fire!!

 

5-12

NRich

This large UK site based at Cambridge University contains many different kinds of resources for both students and teachers, including interactive resources. You will need to have a Flash player or Java in order to use most of these. Instructions for this are given on the site itself via the Help link on the left. A good overview of the site overall is available at the tour, which deals with much more than just interactive elements.

Interactive resources appear in many places on this site, and can also be found by using a search command. The site uses the term ‘interactivity’ to refer to interactive resources. A nice example to try is the virtual geoboard, which can be configured in various ways and elastic bands easily used. There are several probability examples, including some using the wonderful DIME recording sheet. A substantial set of resources is available here as a result of a search using interactivity. Choose some of these in your areas of interest.

5-10

BBC Maths File Game Show

This small collection of games from BBC TV’s Maths File has a variety focussing on various aspects of Number, Algebra, Data and Measurement, focussed on the UK curriculum. Your browser needs to have Shockwave installed. (There is a download link if necessary). The games are nicely produced, with an undertone of British humour (eg when a ‘prize’ is won.) Advice is offered for teachers, and some activity sheets for students are provided too.

Many of these games would work quite well on an interactive whiteboard, although they could also be used by individual students. Click Hypatia or Pythagoras to spin the wheel. To get a feel for the style, try Builder Ted, which is concerned with ordering decimals, or Bathroom Tiles, which involves transformations in the plane, or Train Race, which is concerned with using and interpreting means and medians.

8-12

Fendt Java applets

Walter Fendt has compiled a number of Java applets related to secondary school mathematics, in several languages as well as his native German. The English language applets offer interesting ways of interacting with mathematical ideas across a range of topics, especially geometry topics, and are designed for direct use by students.

To explore this site, choose some applets in an area of your interest. While the applets are a little old now, they still offer useful opportunities for students and demonstration opportunities for teachers. Note that the suite of applets can be downloaded for use in your own classroom, so that they can be used when Internet access is problematic. (Don’t do this if you are in a computer laboratory, unless it is your own.)

9-12

Euclid’s Elements

David Joyce has created this interactive online version of the most successful mathematics text in history, including all 13 books. Worth including here, even if Euclid has faded from many of our curricula!

 

There’s too much here to deal with in detail, of course. I suggest going to the Table of Contents and looking at some of the Quick Trip, such as Book I Proposition 47. Dragging coloured points allows you to see Euclid’s work come alive.

 

7-13

Waldomaths

This British site contains a large number of well-designed Java applets across a range of areas of mathematics. These are arranged in two groups: for ages 11-16 and ages 16-19. Applets are designed for students to be able to use by themselves, so that onscreen icons are useful.

There are many small, single-purpose applets, which would be useful to use on an interactive whiteboard for whole class teaching, while some are best suited to individual use. Try Reflections, Algebraic long division or Positive and negative numbers to see a range of interesting examples. Instructions are helpful and easy to find through the icons, usually starting with the i icon at the top.

7-12

Platonic Solids

An entry into the encyclopaedic MathWorld site, powered by Mathematica. Although much of this site assumes very significant mathematical background, it is still worth a look. The Platonic solids are nicely represented, and can be rotated various ways using the mouse. The Wolfram Demonstrations Project (below) also uses Mathematica.

 

After using the mouse to have a good look at the Platonic solids, including some of the links to individual solids down the page a bit, select Live 3D Graphics from the left column to access the extraordinary collection of mathematical objects, or use either the search engine or links at the left to explore this entire site. Don’t spend too long here for now … the idea is just to see what is now possible and available on the web. Much of what is on this site is not interactive.

 

6-12

WisWeb

This is a website of the Freudenthal Institute for secondary education (students of 12 to 18 years old) in the Netherlands. The main focus of the site is a collection of Java applets, developed and studied by FI staff. Some of these applets are in English; many are very interesting and offer environments for pupils to explore ideas.

Select the applets link on the opening page. Then show the applets after selecting Abstract to get some instructions Most of these applets are small and come with minimal instructions, yet some are likely to be quite engaging by students or by a class, especially the 3-D applets. For example, try Building houses, Impossible object or Cut-Outs, Nets. Each applet has an About link at the top, which gives (minimal) information.

7-12

Cut The Knot

This website contains a very large collection of interactive activities created by the idiosyncratic Alexander Bogomolny to highlight elegant and powerful features of mathematics.  Many aspects of secondary school mathematics are involved as well as many mathematical elements that are not part of the school curriculum.

 

As CTK is a very extensive collection of several hundred applets, no choices will be ‘typical’. Equations of the straight line, Riemann Sums and Polygonal Numbers illustrate three different possibilities, but you will find many others with other selections. To see the philosophy underpinning the collection, read the author’s manifesto, which is also accessible on each applet page.

 

9-12

Maths Online

This excellent Austrian website offers many Java applets that have been thoughtfully constructed to offer sound educational experiences for students learning mathematics, mostly at the senior end of secondary school. The site contains a good collection, together with other useful advice on using them and information about their development.

 

Start with the Gallery, which houses the collection. Try Analytical geometry 1, for example, which contains three different applets. For each applet, make sure that you read the Exercises and the Didactical Background to see the thinking employed in constructing them. You may then like to explore some other examples or to try an interactive test, some of which (like this one on derivatives) are in the form of puzzles.

 

9-12

xFunctions xPresso

David Eck’s pioneering work was freely available for many years, and has now been rewritten as a Java applet, available for free download. Various aspects of graphing functions are involved, including differential and integral calculus. (See also the JCM link below to see how these ideas here were extended.)

 

Go to the Launch xFunctions button a little way down the page to get a sizable version. xFunctions is a suite of programs, not just a graphing package. Instead of the Main Screen, try the Animation Utility, Derivatives Utility and Riemann Sums Utility to see the opportunities provided to students to explore the mathematical ideas involved. Lots of detailed help with the applet is provided on the page, but there won’t be time to explore this in depth in the workshop.

 

9-12

Loci

Loci was previously The Journal of Online Mathematics and its Applications, a major publication of the MAA over the past few years and a fertile ground for thinking about and discussing the role of interactive multimedia in learning mathematics. While a lot of the material is focussed on undergraduate mathematics, there are still interesting ideas here relevant to senior secondary school. Reviews or work using multimedia and interesting conversations about work in progress are located on the site.

 

The MAA Mathematical Sciences Digital Library (MathDL) offers a new environment for teachers and authors to consider together the educational prospects for multimedia on the web. The articles refer to educational use of the multimedia applets (mathlets) and frequently contain good practical advice about using them, from teachers and developers. A quick browse of the Resources will give you a sense of the material. Free materials are available (as for JCM below), often requiring very little (but some) HTML expertise to install. An example of this process is shown here, using a Dice applet referred to in a recent article.

9-12

Java Components for Mathematics

This website reports the work of the JCM project, an outgrowth of the work of David Eck, author of xFunctions. The project has produced Java components that can be freely adapted for teaching purposes, and this website offers a good deal of help in showing users how to do the adaptation. With only a minimal knowledge of HTML, mathematics teachers can use these components for their own purposes.

 

This site provides a good deal of practical help for beginners, although a little knowledge of HTML will be needed in order to take advantage of it. In order to use the components provided, you will also need to download the JAR file and store it on your website or computer to be accessed by the applets created. (A big advantage of this is that you can use the applets on a computer without web access.) There is an example here of an elementary use of the EpsilonDelta applet, essentially comprising the text copied from the JCM website and pasted into a text file constructed using a basic text editor. There is a small amount of extra text (all of which can be seen by viewing the Source of the page.)

 

9-12

Dynamic Geometry

Both Cabri Geometry and Geometer’s Sketchpad provide a mechanism for users to easily create their own dynamic geometry Java applets. The outstanding Geogebra software is available online and can also be downloaded to your computer for free.

 

There are some rudimentary examples of applets from Cabri and Geometer’s Sketchpad in use here, together with links to the software manufacturers’ pages, explaining the process in some more detail. Try perpendicular bisectors, for example. The Geogebra site will need a lot of time to explore adequately, but is well worth while.

 

6-10

The Learning Federation

The TLF is an agency of various Australian governments, sponsoring the development of a large number of multimedia and online earning objects for Australian schools. Many of the objects are for younger pupils, but the brief extends from K-10, so there are examples for older pupils too.

 

Explore the site to find out what the TLF is doing in your state, as well as other information about how it works. Use the Access Information link at left to find out more local details: select Schools to do this. There are some sample objects at the link provided here, but you can’t actually use the entire set of objects from this site. The entire mathematics and numeracy catalogue is accessible as a (large) download from the web.

 

K-12

MathTools

MathTools is a Digital Library of maths resources compiled by the Math Forum to help teachers teach and students learn in K-12 mathematics. The site has two major features: a catalog of software and a discussion forum. Above all else, MathTools is a community of users that includes students, teachers, mathematicians, researchers, and software developers.

 

This site (for teachers, not students) contains links to many other sites on this list, and also contains discussions among users of the various kinds of software. There is a good searchable index on the opening page, which allows you to find resources that fit a particular teaching or learning need. It’s a good idea to register in order to take advantage of the special features only available to registered users. Registration is free.

6-12+

Wolfram Demonstrations Project

This site has been constructed using Steven Wolfram’s extraordinary Mathematica software. It contains a very large (more than 2000) and growing number of interactive demonstrations that use dynamic computation to illuminate concepts in science, technology, mathematics, art, finance, and a remarkable range of other fields. While most are very sophisticated, many are useful at school levels. Free software Mathematica Player needs to be downloaded to use independently of the web.

The versions shown on the web are just quick Flash animations; to explore their use, first download the software (only onto your own computer and then download particular demonstrations. The site has a good About description, good FAQ file, search facilities and a useful Topics list. The related menus on the right of the screen offer many links. Start exploring with Nets of Polyhedra or Two Dice with Histogram or Elementary School Mathematics to see some of the range. Select a demonstration to preview on the web and then use the related menus to follow your interests. In most cases, the (small) downloaded versions provide many more opportunities than the preview shows.

9-13

Interactive Mathematics

This website contains a lot of descriptive text, which is mostly well-written in  a large font, and a number of interactive elements constructed by the author, Murray Bourne from Singapore. The interactive elements use LiveMath, Flash or Scientific Notebook. The first two of these need a free plug-in to be downloaded and installed, while the third requires the actual software. While the main purpose seems to be to provide good explanations and descriptions for secondary students, which it achieves well, the website has many nice uses of interactive applets.

The Flash highlights link at the top of the page provide some nice examples of applets and will give you a quick check to see if you already have the Macromedia Flash plug-in installed. The Sitemap will give you a good sense of the scope of the material, which may provide useful revision for students, regardless of animations. Instructions for installing plug-ins are useful (but only install them on your own computer!). There are many LiveMath applets on the site, with a range of quality. Some of these would be useful for demonstrations and class discussions.

10-12

Moving Man Applet

This nice Java applet has been constructed by Lisa Murphy at UIUC and can be used in the study of functions, line graphs, slope, the derivative, and basic motion concepts. Using a mouse, the student drags a stick figure back and forth across the top of the screen between its home and its school. Below, graphs of the motion appear as the motion progresses.

 

This is a nice applet for students learning about distance-time graphs and the early steps in understanding derivatives, and a good alternative to using a motion detector. The instructions on the home page are useful, but you can go directly to the applet here. While individual students might use this, it seems also that it would be a nice class discussion tool. Some teaching materials are also available on the site, which contains some physics applets too, concerned with water flow and electricity.

11-13

Mathematical Visualization Toolkit

This is a large Java-based package for plotting, both 2-D and 3-D as well as solving equations, developed at the Department of Applied Mathematics at the University of Colorado. The entire package can be downloaded as well as run on the Internet. The interface is easy to use and the suite of capabilities very powerful.

Explore the tools for graphing, drawing vector slope fields, numerical integration and linear algebra, as well as the various stand-alone applets under the applications menu. While much of this suite is more suited to early undergraduate work, especially in calculus and engineering, much of it is also very useful for the senior secondary years as well.

11-13

Calculus Applets

This is a large collection of Java applets for the Advanced Placement calculus AB and BC courses in the US, produced by Tom Downey using the excellent applets at Webcompmath. The applets cover the entire range of the courses and hence of introductory calculus in general. Each applet comes in a version shown on a web page, suitable for viewing in a browser window, and also as a separate resizable window with larger fonts and line widths, suitable for projection on a screen. The entire set of individual applets can be downloaded for classroom use, with appropriate attribution. A very rich resource for students or teachers of calculus.

The various applets are well-designed and reminiscent of David Eck’s superb applets of many years ago (see the xFunctions link above). The applets come with ready-made examples for students or teachers to explore, as well as helpful suggestions for exploring them. In addition, the various applets can be used independently (eg using your own functions rather than pre-set functions). Many of the applets rely on a graphing window. Try Derivative Function to see the general idea and then explore topics of particular interest.

11-13

Online Statistics Textbook

This online textbook, Online Statistics: An Interactive Multimedia Course of Study, is most suited to an introductory level tertiary statistics unit, although there is also much of interest here for senior secondary school. The text has a number of interactive elements, especially simulations and an analysis laboratory. There are various case studies provided for the user.

Check with the instructions regarding your computer settings and requirements. Essentially, you need a browser, Java and Quicktime (for the multimedia, which is optional). Notice that different Mode settings are possible. After checking the Instructions, go to the Contents page and choose a topic of your interest, or go direct to the List of Simulations and Demonstrations. There are some good simulations (which generally get you to answer some questions in advance). Make sure you try the Analysis Lab too.

10-12

GapMinder

This website is based around the remarkable Trendalyser software, which allows for graphs Gapminder is a non-profit venture promoting sustainable global development and achievement of the United Nations Millennium Development Goals by increased use and understanding of statistics and other information about social, economic and environmental development at local, national and global levels.

This wonderful website allows a user to compare countries on two different social dimensions simultaneously, as in a scatterplot, but also shows how the relationship changes over time as well as showing population growth over time. Easier to observe than to describe. Choose GapMinder World 2006 to interact with these ideas, linking statistics and global public health. (There is a Help key if necessary.)  The website also has some nice presentations of Hans Rosling using the software, which provide a powerful new way to see statistics of these kinds.

9-12

CTTE Interactive Objects

The Centre for Technology and Teacher Education has long been a pioneer of nice uses of technology, and this collection of Flash applets is no exception. You will need a Flash [player to use these, most of which are for older students (although some are for primary students as well).

These applets can be downloaded for offline use, which is very handy (and generous of the developers). Some of them also come complete with some suggested activities, although it is fairly clear in most cases how they might be used. The Quincunx and Sampling Distribution applets provide powerful visual demonstrations, while the Integer Cars may help students make sense of operations with negative numbers. Most of these would be useful with an Interactive Whiteboard for whole class use.

10-13

Classroom Demonstrations

These Mathematics Web Resources from Lawrenceville cover a wide range of areas, and would be very useful for an Interactive Whiteboard for whole class use. The resources have been made using GeoGebra and EquationPlotter and take the form of Java applets.

The applets are mostly concerned with mathematics of the senior years or the first undergraduate year, especially in algebra, calculus and trigonometry. Each offers a flexible interface to explore key features. Try Unit Circle or A Function and its Derivative to get  a taste of the materials, which would provide good classroom demonstration tools.

1-8

Interactive Teaching Programs

This suite of 32 Flash applets was produced as part of the National Numeracy Strategy of the DFES in the UK, and are designed to be used on an interactive whiteboard. They deal with a range of mathematical areas, and will help turn an IWB into a teaching tool with mathematical capabilities.

While there is a range among the set, teachers in primary school will certainly find some of these to be of value. Importantly, the applets can be downloaded to a computer, so do not require you to be online to use them after they have been downloaded. Different versions are available online Try Decimal Number Line and Fractions to get a feel for the applets. Each has some (minimal) on-screen instructions, and slightly more extensive teacher notes can be downloaded.

7-11

Math Open Reference

Described as a prototype, this free web-based secondary school geometry textbook developed by John Page incorporates student learning with interactive tools and Java and Flash animations. Many of the tools and animations refer to specific concepts or single-purpose mathematics tasks, such as constructions, and could be used for a range of purposes, including demonstrations in class or on an interactive whiteboard.

Most of the materials refer to aspects of geometry, although this includes some measurement materials as well. To see some examples, try Circumcircle, Coordinate Plane, Volume of Cube or Graphical Quadratic Explorer. The material is well-suited as a reference for students (hence the title and the ‘textbook’ notion). Recently, the materials have been expanded to include coordinate geometry and some graphing applets. Use the index to find specific topics.

8-12

CPMP Tools

 

The Core-Plus Mathematics Program is a curriculum development program based at the University of Western Michigan. CPMP-Tools is a suite of Java-based mathematical software, specifically designed to support student learning and problem solving in each strand of Core-Plus Mathematics. This tools component includes a number of Java tools organised into four curriculum areas: algebra, geometry, statistics and discrete mathematics.

You will need to download a java webstart link in order to access the tools, which are very powerful. The software includes four families of programs, depending on which level of course (of 4) is chosen. Each content-area tool includes specific custom tools and built-in copies of many data sets used in investigations and homework. For example, there is a CAS engine, a spreadsheet, data analysis, simulation and dynamic geometry software, among others. The Help files will provide detailed information. Although the software supports a particular curriculum (CPMP), it is widely applicable.

8-12

Eyeballing game

 

Mathias Wandel’s website is mostly concerned with woodworking and other practical pursuits. This link however is intended to test your skill at gauging straightness and squareness, important woodworking attributes, but may also provide opportunities to practice and develop estimation skills.

 

The connection to mathematics here is not very strong, as the tasks are mostly concerned with visual estimation. Nonetheless, mathematics is involved in finding centres of circles, line segments and triangles (especially!); in finding parallelograms and convergence points of lines and in bisecting angles. Move the cursor and release to enter a point; results are tallied (so that you can see your improvement – if any) and your success is located on a large histogram.

K-12

Maths is Fun

 

This is a large and well-organised website with many kinds of materials for students, many of which have an interactive element. The website provides separate links for different mathematics topics. (Of necessity, the site also contains advertising, so be careful with tapping links unless you are interested in what is being advertised.)

There are many explanatory and descriptive materials on the site, mostly written competently and easy for students to follow. Amongst the material, there are also many interactive activities. A good example of an interactive is the zoomable number line in the Number section. This allows the decimal number line to be explored in interesting ways.

 

Feedback or comments on any of this are of course welcome.

 

Last modified on 26 April 2015