Tatyana Finkelstein's Review of 5 Websites.

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June 16, 2006

Tatyana Finkelstein's
Review of 6 Websites.

Foreword

Site #1 Cut The Knot: http://cut-the-knot.org/

Site #2 SET Game: http://SETGame.com/

Site #3 EDC: Mathematics Projects http://edc.org/makingmath/

Site #4 Mathematical Interactivities http://mathematics.hellam.net/

Site #5 Jim Loy's Mathematics Page http://jimloy.com/math/math.htm

Site #6 The Online Test Page http://saab.org/

Conclusion

Foreword. I decided to only include free sites in this review. There may be a number of other sites which are quite likeable but are not fully accessible from the Internet. I am not looking at those because they usually require a school district's subscription (just like textbooks), and therefore not all viewers will be able to try them. I am looking for meaningful content that is useful to both students and teachers of mathematics, primarily in grades 5 through 9. As you can probably tell from my selection, I am generally focusing on the material which is more challenging than what is normally included in a typical school curriculum. But this happens only because usually, the harder the problem, the easier it is to invent some "interesting" computer presentations for it. My favorite website items would be in the form of a presentation of a particular mathematical problem which utilizes the multi-media capabilities of the computer in a significant, meaningful way. This could be either an interactive applet, or just a video. When evaluating a website, my most important criteria is whether a particular applet allows one to pose a problem which would otherwise be difficult or even impossible to describe, and whether it is "interesting" enough to make the student curious and keep her attention, and is thus likely to tempt her to work on a problem which she would otherwise be reluctant to approach.

I am looking for the brightest, most innovative, original ideas on how the computer can be made useful in teaching mathematics. And this means - it must never help you solve anything (because, simply put, if it does that, than, in the final analysis, you will have learned less instead of more). Therefore, I prefer that the computer asks you questions rather than answers them. There are just a couple of examples of such videos and applets on my own webpage. This also means that I will generally ignore a lot of very useful sites and software applications, such as some sophisticated graphing tools, modeling programs, calculators, or spreadsheets. Because these are good at helping solve problems, but they do not create a new challenge for the student, and often do part of the thinking which the student is supposed to, hopefully, do on her own. Thus, as useful and sophisticated as they are, their function can be compared to a driver offering a ride to a jogger who happens to be out trying to prepare for a marathon.

On the other hand, there are many interactive "fun" games and puzzles which can be very useful in teaching some of the most important mathematical concepts. For example, the computer can make an excellent opponent for a chess player. In my list, I am including Site #2 which features one such puzzle game, called "SET".

Also, I am not favoring websites for having a consistent theme, for being well designed, beautiful looking, or highly organized, or for the quantity of content, or for being popular. What I am attracted to are the best specific learning ideas, such as the best individual interactive applets, wherever they are located -- as opposed to looking for best "websites" as whole entities.

Please note that Site #1 is my favorite on this list; and at the bottom of the list is Site #4, a smaller site which is very promising, but is currently still in its very early stage of development.

[Rank: 46,703; Links 5,529] *

Site # Name: Author:

URL: This site is designed for Teacher and/or Student

Subject: Grade Level: from through

Copyright:

Date of Last Revision:

Restrictions on Use/Permissions Required:

This site is aligned to curriculum standards:

State Subject(s) Grade: from through

National Subject(s) Grade: from through

Describe how easy or difficult the site is to use:

How is the site being used? Provide specific examples. (Ex. (1) Student uses site to practice solving quadratic equations; “section C” allows student to change values and see effect on curve. (2) Provides teacher with curriculum unit on trigonometry using “section B” to show students how changes in variables effect quadratic equations.)

This site is for teachers, parents and students who seek engaging mathematics. Many of the topics are accompanied by Java illustrations. There are more than 700 Java applets.

Describe site content. Give examples:

This site is #1 on my list, and it stands out in its subject area as the hands-down winner, as no one else even comes close to it in terms of quality and amount of content. This is a huge, highly successful, and extremely popular website. Its only drawback is that as their volume of material increases, it becomes harder to pinpoint the best items. I suppose this is inevitable, since you cannot expect an author to create a feature called "the best choices on my website". He would be naturally reluctant to admit that only a fraction of the content is really excellent, while the rest may be average, or so-so, and is there just to create an impression of having a very large volume of material. Dr. Bogomolny appears to share my own longing for "fun and games" applications of technology (and when we say "technology" we actually use it as an exact synonym of the word "computer". These days, it just so happened that the long awaited convergence of various media to become just one universal machine is pretty much complete, as the computer is finally fast enough, and has enough storage to handle practically any type of interactive media. Just the enormity of the amount of material here would likely render any attempt on my part to point out some specific examples more or less futile. This is one of the few sites in existence today which can be simply given to a student to explore, and can keep her curios and engaged in a meaningful way for a long time, almost none of which would be wasted. One might ask, what is the useful function, if any, of all those interactive applets, when none of them helps you solve a problem? After all, most (probably, all) of these respective problems can be done on paper just as well. I suppose, the important point here is that creating a modeling interactive applet (and, in some cases, just a movie) to represent a problem does not take too much away from the process of "imagining" such a model, or making drawings on paper. What it adds is a positive emotional aspect created by the complex nature of the interactive process itself. This, in turn, keeps the student's attention, heightens her curiosity, keeps her thinking about the problem in question for a longer period of time, and ultimately makes the act of getting it solved more rewarding. All it does is it simply attempts to trick the student into being interested and doing something which she would otherwise be, at best, indifferent to, and may never actually try. <<“Most … mathematical concepts … can best be taught through problem solving.” (Van de Walle, 2001, p. 30)(1) Cut the knot brings us back to classical problem solving. This site has the mathematical games and puzzles that I was raised on, including Nim, Tower of Hanoi, and probabilities puzzles and different sized jugs of water puzzles. However, this site goes all the way to complex topics, such as theorems and paradoxes. This site might be a good resource for enrichment activities for the most advanced students, though the mathematically-challenged elementary teacher might find the site difficult to understand. . The site is for teachers and advanced (high school +) students. It seems to be affiliated with Amazon.com and has banners.>>(2)

Site is most often used for: (i.e. fact finding, practicing skills, understanding concepts)

Finding both (1) new material (problems, concepts), as well as (2) new interactive presentations of existing, or known material, and (3) to a lesser extent, link to a small number of other highly recommended websites.

Example: The 3 Jugs Problem

[Rank: 186,369; Links 1,479] *

Site # Name: Author:

URL: This site is designed for Teacher and/or Student

Subject: Grade Level: from through

Copyright:

Date of Last Revision:

Restrictions on Use/Permissions Required:

This site is aligned to curriculum standards:

State Subject(s) Grade: from through

National Subject(s) Grade: from through

Describe how easy or difficult the site is to use:

The "Set" part of this website provides both teachers and students with a series of tools for extensive exploration of the most important concepts of the Set Theory. Exploring the 81-card deck of the game "SET" offers a unique and fun approach to introduction and understanding of the definitions of cartesian product and function. SET became very popular with math teachers around the World. To be used effectively, it requires an investment of about $10 per student to purchase actual SET card decks for the class, because this manipulative must be readily available for every student in order to turn its exploration into a rewarding and successful experience.

Describe site content. Give examples:

The site offers a series of games, most of which require a specialized deck of playing cards. One of them, the game called SET is of particular interest to us. It was invented by Marsha Jean Falco in 1974, but only recently became very popular. It shares certain similarity with the Rubic's Cube in that both are closely related to serious mathematics. Both SET and the Rubic's Cube can be used very productively in the classroom. While the Rubic's cube offers an introduction to algebra and allows to create an excellent real-life prospective on groups and binary operations, thus providing a very natural way of generalizing from familiar sets, like rational numbers, to other algebraic structures. This makes them seem more intuitive, less abstract, more friendly. Likewise, "the game of SET" (not to be confused with the mathematical concept of "set") allows for a novel approach to the introduction of the mathematical Set Theory itself, but I believe SET to be much more valuable in that capacity compared to Rubic's Cube and many other approaches. Asked, "What does mathematics study?" (for example, "chemistry studies molecules and how they interact"), most of both teachers and students would offer things like "logical thinking" and "shapes and numbers". But the best answer, of course, is "sets". Because, in fact, every object in mathematics is a set! Therefore, it is crucial to have an excellent understanding of the concept of "set" (preferably before anything else can be done in mathematics). Not doing so would be like learning chemistry without ever explaining what a molecule is; or physics - without explaining "force"; or biology - without defining "a living thing". The game SET is unique in that it allows to introduce sets in the "fun and games" setting, while at the same time one can define Cartesian products (of the 4 sets called "properties", each consisting of 3 elements, yielding a total of 81 elements, or "points"). Each card now becomes an element of said Cartesian product. The concept of function can also be discussed. For example, removing 1 property (a "projection") results in fewer cards (27). This ability to materially enhance the comprehension of the concept of "set" actually makes the game SET one of the most valuable existing resources in teaching the fundamentals of mathematics. A very intriguing aspect of the game SET is its similarity to a game often used to train and test parrots' intelligence. In its latter form, a set of properties is defined (for example: color, shape, material, and size), just like we do in the game SET, except that the properties are made so that they are just a bit more "parrot-friendly". There may actually be more properties, and they may have more values then in the game SET. Example: 5 colors, 5 materials (wood, plastic, cardboard, metal, cloth), 5 sizes, and 5 shapes (circle, triangle, square, X, figure 8). Now we can make as many as 625 different objects. It turns out that a well trained parrot can distinguish them all from one another. Given a card, he can name all of its 4 properties. Given 2 cards, he can answer questions like "which property is the same?" and "which property is different?"(3) It seems imperative that our students should be given an opportunity to learn to do at least as well as this (albeit it a very well trained) parrot.

Site is most often used for: (i.e. fact finding, practicing skills, understanding concepts)

(1) Practicing the actual skill of playing the game of SET. This is by far the most common reason why people go to this site, however, actually playing the game itself is the least mathematically educational aspect of this story. It is the mathematics behind the game that matters. (2) Buying decks of SET for use in one's classroom. (3) Finding new mathematical topics and problems related to SET, including links to other related resources.

Example: the The daily SET competition

[Rank: 77,793; Links 115,280] *

Site # Name:

Author:

URL: This site is designed for Teacher and/or Student

Subject: Grade Level: from through

Copyright:

Date of Last Revision:

Restrictions on Use/Permissions Required:

This site is aligned to curriculum standards:

State Subject(s) Grade: from through

National Subject(s) Grade: from through

Describe how easy or difficult the site is to use:

Making Mathematics matched students and teachers in grades seven through twelve with professional mathematicians who mentored their work on open-ended mathematics research projects. The site is very useful for “enrichment”, i.e. if some students in a class are ahead of the rest of the class, they can work on one of the suggested projects. The site has all the necessary recourses including the statement of a problem, examples, warm-up problems, and extensions.

Describe site content. Give examples:

The site suggests 12 projects of various levels of difficulty. Some of my students have worked on the problems and did presentations of them for the rest of the class. Most of the students cited that experience as the most exciting and rewarding part of their school year. For example, this year two separate groups of four 7th graders worked for two weeks on the Amida-Kuji project. ("Amida-kuji" is Japanese for "network lottery"). According to Bob Stanton, it is used in Japan to choose turns in games and assign prizes. You start with as many vertical lines as there are people. Here is what we would have with six people numbered 1 to 6. (Pic. 2 below). The prizes are placed at the other end of the lines in any order. Kids were able to come up with a general algorithm to create any given rearrangement; they made simple and elegant conjectures and proved most of them. Such as: each horizontal line gets traveled exactly twice, each vertical line is traveled all the way through, etc. They attempted to find the way to predict how many horizontal lines each particular rearrangement will require; they created a couple of functions which allowed them to classify a given rearrangement. Throughout this experience, kids felt that they owned the exploration, they learned how to express their ideas so that the rest of the group can understand them, they struggled with the concept of “proof”, and they were looking for the next good question. Such experience was invaluable. One of the great assets of this site is the Teacher Handbook (http://www2.edc.org/makingmath/mathproj.asp#rsskil) which helps a teacher to master the art of posing a question, explore a problem, express her thoughts and conjectures in a mathematically sound way, and write a proof. I used the Teacher Handbook extensively and productively in my work. Especially, the “getting stuck, getting unstuck” part. Mathematical tools allow students to catch up on some important concepts and algorithms, mostly from the number theory, which are not part of the typical curriculum. Some of these concepts are essential and very engaging. There are links to the Cut_The_Knot site (#1 on our review list), where kids can play with interactive software. For example, the Euclid’s Algorithm. Finally, there are examples of completed student work, designed to help set the level of expectations for the new students.

Site is most often used for: (i.e. fact finding, practicing skills, understanding concepts)

In my experience, this site is mostly used by middle school and high school teachers to supplement their curriculum, especially for the gifted students.

Examples: Marion Walter’s Theorem Raw Recruits

Amida-kuji Pic. 1 Pic. 2 Pic. 3

[Rank: 589,085; Links 192] *

Site # Name: Author:

URL: This site is designed for Teacher and/or Student

Subject: Grade Level: from through

Copyright:

Date of Last Revision:

Restrictions on Use/Permissions Required:

This site is aligned to curriculum standards:

State Subject(s) Grade: from through

National Subject(s) Grade: from through

Describe how easy or difficult the site is to use:

"Mathematical Interactivities" is a collection of about 60 educational games and puzzles covering a wide variety of mathematical topics. Over 60 interactive multimedia resources - mostly online games written in Flash - to help in teaching and learning mathematics. Games, puzzles, and other interactive multimedia resources online for teachers and kids.

Describe site content. Give examples:

While much smaller in volume, and lower in ranking than other sites in our selection, I feel it is important to distinguish this site for including just the type of interactive applet which I consider to be one of the most promising in helping the math teacher. Consider the Frog Puzzle (below), and note that it is not written for the purpose of having students play this as an interactive video game. Rather, the purpose is to try and represent an essential math problem on invariants in the shape of a "fun" interactive, and thus coax a student into spending time thinking about the problem, then developing an answer and a proof. At the same time, since this applet is not offering any hints on how to solve the problem, there is no downside to it. Still in their early developmental stage, applets like this one are few and far between; it would take a lot of time, work, new ideas, and creativity, to bring them to a level where it would be possible to adequately assess their real-life usefulness.

Site is most often used for: (i.e. fact finding, practicing skills, understanding concepts)

(1) Just a small number of interactive models to present some specific math problems. (2) Also, explore a number of more general useful tools for educators by the author, David Hellam; all accessible from his domain's home page at http://hellam.net/ where he includes a couple of new educational websites; note http://educator.pro/ - however, at this time, it's still very much "a work in progress".

Example: The Frog Puzzle

Jim Loy's Mathematics Page [Rank: 56,678; Links 1,374] *

Site # Name: Author:

URL: This site is designed for Teacher and/or Student

Subject: Grade Level: from through

Copyright:

Date of Last Revision:

Restrictions on Use/Permissions Required:

This site is aligned to curriculum standards:

State Subject(s) Grade: from through

National Subject(s) Grade: from through

Describe how easy or difficult the site is to use:

Both Math and Puzzles areas provide large numbers of challenging and fun problems, suitable for a wide range of grades and skill levels. This is an informal collection of articles on many different mathematics problems. Topics include Algebra, Geometry, Calendars, Number Theory, and reviews of mathematics-related books.(5)

Describe site content. Give examples:

Probably, the best annotation of the Jim Loy's website would consist of actually visiting the site itself. Not limited to mathematics, it has an overwhelming array of diverse stuff, which is quoted, re-told, completely original, as well as linked-to. So, it appears inevitable that the author's favorite personality happens to be Isaac Asimov, and the author, in my view, appropriately, shares many of Dr. Asimov's interests and talents. Our main interest is, naturally, in the 2 chapters of his site: Math http://jimloy.com/math/math.htm ; and Puzzles http://jimloy.com/puzz/puzz.htm. I understand that Jim Loy, the person, is as well liked and popular, as is Jim's website.

Site is most often used for: (i.e. fact finding, practicing skills, understanding concepts)

This collection of puzzles provides a great way to entertain and educate young and old! From word arithmetic to geometry and beyond, these puzzles will not only be the favorite of substitute teachers (as they can be easily printed), but that of students and their parents, too. How You Can Use This Source: * Solve cryptograms and learn about decrypting. * Challenge your logic to solve the featured logic puzzles. * Learn about Geometry while solving the maze puzzle.(4) Jim Loy's collection is indispensable for its large volume of links, quotes, and original writing in so many areas, but also just for fun and enjoyment.

Example: Tower of Hanoi (the actual animation):

[Rank: 184,368; Links 1,080] *

Site # Name: Author:

URL: This site is designed for Teacher and/or Student

Subject: Grade Level: from through

Copyright:

Date of Last Revision:

Restrictions on Use/Permissions Required:

This site is aligned to curriculum standards:

State Subject(s) Grade: from through

National Subject(s) Grade: from through

Describe how easy or difficult the site is to use:

Practice in diverse areas of math; primarily with a view to taking specific aptitude tests; also included are some very useful algebraic graphing modelers, and other interactives.

Describe site content. Give examples:

One of the best organized math sites with excellent links. Mahematics Tests and Drills (http://www.saab.org/mathdrills/) allow a student to either practice specific technical skills (for example, factoring polynomials, solving linear equations, etc.), or take a test on specific subject (geometry, algebra, probability), or take a mixed test with a specific number of problems and have the answers checked instantly upon the completion of the task. In addition, the site has excellent links which are very easy to navigate. Click Useful Links, choose Cool Math Sites and you are in for a real treat. The Function Flyer (http://www.shodor.org/interactivate/activities/flyall/index.html) allows you to explore the impact of change in coefficients on the graph of a function.

Site is most often used for: (i.e. fact finding, practicing skills, understanding concepts)

A large variety of test models make it one of the most useful resources for preparation for all major math tests, but practicing using these interactive models is useful even regardless of any specific test's limitations. Extensive inter-linking with other domains makes navigation a bit confusing; one rarely knows for sure which domain she is on, or how she got there. But this confusion is a small price to pay for being able to easily access all these vast resources from a single starting point.

For additional perspective on Dr. Saab's work, see his home page (University of Missouri, Dept. of Mathematics), and here is some of his work in mathematics.

Example: Math Drill The 4-D cube

The Function Flyer

In conclusion, it should be noted that a prospective imaginary compilation of "the very best of the Net" (according to my selection criteria, as I described them earlier) would likely include a diverse collection of individual pages, or even smaller items (elements), rather than just top-level website references. These would simply be selected to represent the type of innovative "interactives" which are the best candidates for actually adding some meaningful, qualitatively superior new level to the learning process, instead of simply facilitating the act of solving a problem in question, or even creating a distraction. This brings me to an idea (a daydream, really) about the way the Internet may be re-organized to benefit our (very real) math student's needs. In this dream, the entire Internet would be re-arranged, or re-linked, in order to cater to a particular individual's requirements, as opposed to its current presentation as a "page-based" medium. The nice thing about this is that (at least in theory) it should not be necessary to change anything that's already out there. Rather, a new rendering technique may be proposed which would enable one to see and communicate with just the selection of desired material (from numerous sources, together), and nothing else. In our case, that would be a collection of the very best interactive items which are the most beneficial to a student in developing his skills of mathematical argument. The quality of a user's online experience would be determined by the functionality of an item, or a collection of items (which may exist individually, or be inter-related), irrespective of these items' physical location within the Cyber-Universe. This idea is, of course, very old, and is quite similar to something that is already widely available and is in fact quite ubiquitous: the results produced by a search engine, and those of a query in a relational database. But currently, a user's options, both in terms of the versatility of the search mechanisms itself, and in terms of the sophistication and capabilities of the front-end interface, are still quote limited. Perhaps, further development of these technologies would bring this dream closer to reality.

There is more on these and other ideas in this article, which was, incidentally, written by the author of our reviewed Site # 1: http://www.maa.org/editorial/knot/reform.html

*Site rankings are quoted according to Ranking.com; a lower ranking means a more popular site (referring to its top domain); link counts include links to all pages in the domain, even though our own focus may be on its subdomain, or a particular set of pages. For example, querying "doe.mass.edu" returns [Rank: 10,004; Links 217,194], which, however, applies to the entire "mass.edu", there is no separate ranking for the "doe" subdomain.

⁽¹⁾Van de Walle, J. A. (2001). Elementary and Middle School Mathematics, Teaching Developmentally, Fourth Edition. New York: Addison Wesley Longman, Inc.

⁽²⁾From http://members.cox.net/omegacatpress/543/543Web3.htm

⁽³⁾ This was published in a popular science magazine and seen on PBS, I am still looking for the exact reference, and am going to add it here.

⁽⁴⁾ From http://www.learn.motion.com/topten/matlinks/JimLoy.htm .

⁽⁵⁾ From http://mathforum.org/library/view/10675.html .

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